Hyperbolic Geometrodynamic Warp Drives

& Beyond

Draft Copy June 19, 2003

Paul Hoiland

Edward Halerewicz, Jr.

This e-book was originally intended as a reference source for the E Somino Ad Astra (ESAA) online Group. Actually the group name was somewhat of a misnomer as ESAA was really the motto of the group, with the range of discussions evolving within the group the more dignified name Advanced Theoretical Physics Group (ATPG) was christened or ESAA-ATPG. While the original ESAA-ATPG has disbanded and allowed for the founding of the Journal of Advanced theoretical Propulsion Methods (http://www.joatp.org ), the ESAA spirit continues on with the authors with the public availability of this document. It is the hopes of the authors that this document may go beyond a simple reference source and inspire new knowledge and frontiers in science. As such a single copy of this E-book may be saved and printed for private purposes but it is not meant to be commercially distributed and otherwise used without the explicit written permission of the authors © 2003.

About the Authors

Paul Hoiland is a theoretical physicist and engineer who has a background in General Relativity as well as string theory. He has published a number of popular scientific articles in the New York Post, as well as in scientific journals. He is a self proclaimed educator of science, he is also the founder of the Transtator Industries research group and the Journal of Advanced theoretical Propulsion Methods (http://www.joatp.org).

Edward Halerewicz, Jr. is an independent researcher, with an interest in exploring the limits of modern science as well as popularizing such concepts to the public. He was one of the founding members of the ESAA group, and has worked to bring the warp drive into acceptable limits of existing theory. http://da_theoretical1.tripod.com/index.htm

PREFACE

The initial intent of this E-book was to give a general survey over what has become to be known as the “Warp Drive” class of metrics associated with gravitational physics or more specifically General Relativity. The purpose of the warp drive metric survey and related fields given within this E-Book is meant to act as an aid for scientific researchers as well as to answer questions curious parties may initially have regarding the behavior of warp drive metrics. As such the organization of this E-Book is presented so that the reader first learns the basic underlying physics behind warp theory. The reader then progress to a general description of a warp drive metric, which acts as an initial review as well as to possibly catch up others who may be unfamiliar with warp theory. However it is not the intent of this book to cover all the background physics as there are number of excellent sources available the reader to accomplish this task. After the basics of Warp Drive theory have been introduced a great effort is placed on how it may be theoretically possible to generate a “Warp Field.” Therefore the layout of this E-Book can be seen as a source to aid progressive research into the subject matter. As such a general overview of warp drive theory is attempted while more specific information can be found by searching the references within, and hence the overview of warp drive metrics presented may not be as comprehensive as a standard text book as active research continues. And so we arrive at the original intent of producing this E-book for ESAA-ATPG, which we hope the reader will appreciate.

The broader scope of this E-Book is to also explore aspects of theoretical physics which go beyond epistemology purposes and to investigate there practical uses for the benefit of society as a whole. The specific practical use of General Relativity which this E-Book attempts to cover is how it may be possible to use warp drive metrics as a means of allowing interstellar communication and transportation on human time scales. This could have several benefits for the collective world society, such as the stimulation of economic growth, the potential for practical human colonizations of space, the ability to protect our planet and home from celestial hazards, and the possibility of unprecedented peaceful international cooperation amongst all nations. While these benefits may not be realizable at present given time they should begin to reveal themselves and this the authors feel should be the direction in which the physical sciences should follow. It is hoped that the practical uses of the theoretical possibilities known today may lead to this direction and it is hoped that this document serves as a good start for this vision of the future.

Contents

1 Underlying Physics

1.1 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . .

1.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1 A Metric Defined . . . . . . . . . . . . . . . . . . . .

1.2.2 Minkowski Metric Tensor . . . . . . . . . . . . . . . .

1.2.3 Wormholes . . . . . . . . . . . . . . . . . . . . . . . .

1.2.4 Inflationary Universe . . . . . . . . . . . . . . . . . . .

1.3 What is the Warp Drive? . . . . . . . . . . . . . . . . . . . . .

2 Warp Drive Fundamentals

2.1 Alcubierre’s Warp Drive . . . . . . . . . . . . . . . . . . . . .

2.1.1 Flaws of the Warp Drive . . . . . . . . . . . . . . . . .

2.2 The Energy Conditions . . . . . . . . . . . . . . . . . . . . . .

2.3 Wormholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 GR FTL Ambiguity . . . . . . . . . . . . . . . . . . . . . . . .

2.4.1 Accelerated Observers . . . . . . . . . . . . . . . . . .

2.4.2 tetrads . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 After Shocks

3.1 Van Den Broeck . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1 Quantum Fields . . . . . . . . . . . . . . . . . . . . . .

3.2 Warp Drives & Wormholes . . . . . . . . . . . . . . . . . . . .

3.3 Time Travel & Causality . . . . . . . . . . . . . . . . . . . . .

3.3.1 The Two-Dimensional ESAA Metric . . . . . . . . . .

3.4 Photon-ed to Death . . . . . . . . . . . . . . . . . . . . . . . .

4 Null Manifolds

4.1 The Causal Photon Problem . . . . . . . . . . . . . . . . . . .

4.1.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . .

4.2 The Shield Effect . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1 Cauchy Geodesics . . . . . . . . . . . . . . . . . . . .

5 Negative Energy Problems 45

5.1 Quantum Theory in Brief . . . . . . . . . . . . . . . . . . .

5.2 Virtual Particles . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 The Casimir Effect . . . . . . . . . . . . . . . . . . . .

5.2.2 Casimir Tensor . . . . . . . . . . . . . . . . . . . . . .

5.3 Quantum Inequalities . . . . . . . . . . . . . . . . . . . . . . .

5.4 Exotic Energies Nullified . . . . . . . . . . . . . . . . . . . . .

6 The Mysterious Vacuum

6.1 EM Stress-Energy . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Vacuum Squeezing . . . . . . . . . . . . . . . . . . . . . . . .

6.3 PV Warp Motion . . . . . . . . . . . . . . . . . . . . . . . . .

7 The Affects of Scalar Fields

7.1 The Anti-Gravity Non Sense . . . . . . . . . . . . . . . . . . .

7.2 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . .

8 EM ZPF Force

8.1 EM Field in gauge theory . . . . . . . . . . . . . . . . . . . .

8.2 Ghost Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Quantum Entanglement

9.1 Bell States . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . .

9.3 Boost Entanglement . . . . . . . . . . . . . . . . . . . . . . .

10 Vacuum Engineering

10.1 Non-gravitational Effects . . . . . . . . . . . . . . . . . . . . .

10.2 The GSL Issue . . . . . . . . . . . . . . . . . . . . . . . . . .

10.2.1 Dealing With Entropy Issues . . . . . . . . . . . . . .

10.3 Quintessence Scalars . . . . . . . . . . . . . . . . . . . . . . .

11 Beyond the Fourth Dimension

11.1 Superstring Theory in Brief . . . . . . . . . . . . . . . . . . .

11.2 Type I: Bosonic strings . . . . . . . . . . . . . . . . . . . . . .

11.3 Type IIA: Fermionic strings . . . . . . . . . . . . . . . . . . .

11.4 Heterotic strings . . . . . . . . . . . . . . . . . . . . . . . .

11.5 Type IIB: Compactified Topologies . . . . . . . . . . . . . .

11.6 Shortcuts in Hyperspace . . . . . . . . . . . . . . . . . . . . .

12 Navigation Issues

12.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . .

13 Ship Structure Requirements

13.1 Initial Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . .

13.2 Structural Requirements . . . . . . . . . . . . . . . . . . . . .

13.2.1 Shielding

13.2.2 Hull Materials

13.3 Oxygen Generation

13.4 Cooling / Heating / Humidity

13.5 Electrical Systems

13.6 Sensor Equipment

13.7 Computer Systems

Bibliography

Chapter 1 - Underlying Physics

“Basic Warp Design is requirement at the Academy. Actually the first Chapter is called Zephram ^Cochrane1 .” – Lavar Burton from the 1996 film Star Trek: First Contact

The Warp Drive is an incredibly difficult branch of gravitational physics which tests the theoretical limits of General Relativity. The goal of this chapter is to let the reader explore the underlying physics behind ‘warp drives,’ as well as to begin to grasp its great difficulty. First and foremost the Warp Drive is a topological (geometrical surface) model which may seem to allow for Faster Than Light (FTL) travel as seen by a distant non local observer located in ‘flat’ spacetime. What we mean by at spacetime is a preferred model of physics which is valid in accordance to the theory of Special Relativity (or non gravitational physics). It is actually from ‘flat’ Minkowski space which we inherent the misleading golden rule that no body can exceed the speed-of-light c. This is because the symmetries found in Minkowski space are based on the assumption that spacetime is at. So that all relative motion is based upon the seemingly universal constant c; which stems from the second postulate to the theory of Special Relativity (in fact, early relativity theory earns its name because of the special assertion that the dynamics of physics take place in Minkowski space).

Using a more general approach for a potential “FTL” interpretation of a moving body, one can easily see how the curvature associated with General Relativity could allow for “warp” propulsion. As one can say that if space-time is not completely at (i.e. having a non-trivial curvature) as seen by a preferred observer that observer may record different transit times for a single distant vector at different displaced locations in spacetime (this discrepancy of the local behavior of physics in spacetime is the very background from which warp drives originate). It is within this chapter where we will investigate the ‘strict’ rules of Special Relativity and later modify them in accordance to the bizarreness of General Relativity. This modification then allows us to discuss the possibility of apparent FTL travel, although it will become clear that this pursuit comes at a great cost.

This book is written for those who are familiar with both the special and general theories of relativity. The brief introduction which this chapter covers on the two theories are only meant to be a refresher to the reader. This is done so that you the reader will have a conceptual source of the physics behind the Warp Drive. Although we have also attempted to describe the warp drive to a general but knowledgeable reader who has a curiosity on the subject of spacetime shortcuts. As such we explain key concepts as they are needed and omit critical pieces of background material (unless needed) to the general reader. Whether we are successful at making the book accessible to both the technical and general reader or not, only you can judge. That said we will give very brief treatises on the background for Warp Drives, primarily for historical and conceptual reasons.

1.1 Special Relativity on Speed Limits

Let’s start by considering the well-known velocity-of-light speed limit, as viewed by Special Relativity (SR) and by General Relativity (GR). In the context of SR, the speed-of-light is the absolute speed limit within the Universe for any object having a real mass, there are two reasons for this. First there is the limit imposed by what we in physics term a Lorentz invariant frame. Simply explained, it's giving a fast material object additional kinetic energy and has the main affect of causing an apparent

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¹The father of warp drive in the ‘Star Trek Universe.’

increase in mass-energy of the system rather than speed, with mass-energy going infinite as the speed snuggles up to the velocity-of-light. This occurs mathematically because we can attribute kinetic energy to a system by the following formula

KE = ½ mv² (1.1)

Now the reason why a mass (in reality momentum) increase occurs is not presently understood (that is beyond mathematical computation), it is simply an observed phenomena. For example the acceleration rate of a bullet can be calculated with the aid of a mathematical function known as the beta function , but why this function is accurate gets into metaphysics.

A similar phenomenon was noted in a series of experiments conducted by Antoon Hendrik Lorentz, as the acceleration rates of electrons also seemed to be strangely governed by the beta function. Thus for all intents an purposes acting to increase the mass of an electron. However the mass-energy increase only exist relative to an external observer, the mass-energy increase is actually a consequence of the dynamics of the a system in relation to momentum. Which is given through the equality p = m₀v, thereby leading to an unknown coupling constant which we call gamma, so that the momentum of the system is governed by p = γ(m₀v) where . Therefore it is clearly seen that it is the momentum of the system which increases and not the rest mass m0, as a result the total kinetic energy of a system can be calculated from (1.2)

By this mechanism, relativistic mass increase limits bodies with mass to sub-light velocities, this result comes from the fact that as v → c the “accelerated mass” m₀ increases asymptotically. This then requires a massive body to obtain an infinite kinetic energy to maintain the equality v = c. In theory, with “light masses” (relative to the gamma function) you could get close to c, which is intentionally done with the world’s particle accelerators at will. But, you can never reach c as long as we have to deal with what is called inertia (the resistance of matter to accelerate) in the classical sense.

There is also a second FTL prohibition supplied by Special Relativity. Suppose a device like the “ansible” of LeGuin and Card were discovered that permitted a faster-than-light or instantaneous communication, then an absolute frame would form. However, since SR is based on the treatment of all reference frames (i.e., coordinate systems moving at some constant velocity) with perfect even-handedness and democracy [gravitation is often said to be the most democratic force in the Universe, we will explore this in later chapters]. Therefore, FTL communication is implicitly ruled out by Special Relativity because it could be used to perform “simultaneity tests” of the readings of separated clocks which would reveal a preferred or “true” (universal) reference frame for the Universe as a whole. The existence of such a preferred frame is in conflict with the building blocks of Special Relativity. However, in more recent times that second prohibition has begun to be looked at more closely. Recent articles with the Los Alamos National Laboratories (LANL) e-print server and other publication mediums have begun to ask further questions about the preferred reference frame issue. There are modifications to Einsteins original theory of relativity that do allow for such.

Secondly, experiments with quantum entanglement and quantum tunneling have also added to the reason this secondary prohibition has been brought into question. It is from these contradictions with other existing physical theories which allows us to seriously explore the question: “Is FTL travel really possible?”

1.2 General Relativity

General Relativity treats Special Relativity as a restricted sub-theory that applies locally to any region of space sufficiently small that its curvature can be neglected. General Relativity does not forbid apparent FTL travel (as seen by a preferred topological observer) or communication, but it does require that the local restrictions of Special Relativity must apply (that is Special Relativity applies to ones local geometry). Light speed is the local speed limit, but the broader considerations of General Relativity may provide an end-run way of circumventing this local statute. It is possible to change the metrics of a local region of space-time to one in which the local limit on c has been increased. It is also possible to alter them so that c is reduced. Neither of these situations would violate anything Special Relativity dictates (although it may require a recalculation for the value of c). They would simply be using a special frame of reference different from the ideal one (which we associate with the vacuum state of the Universe). For example within those modified regions Lorentz invariance would still apply. You could not out run light within that region. But, you could out run light within our normal space-time reference.

So you may ask why does General Relativity allow for apparent FTL travel, and not Special Relativity? In the earlier section we explored Lorentz invariance and found why c ≤ v travel can not be overcomed by massive bodies (or for that matter light-like propagations). But what is so special about General Relativity you ask? First General Relativity is a physical theory based upon a differential geometry along with the second postulate of Special Relativity; that c = constant. Now one has to ask how does this geometry affect the apparent value for c? The question is not really valid as Lorentz invariance still applies (see Section 1.1). However the affects of geometry on an observer modifies the way in which that observer records events. An altered geometry can affect the rate at which clocks tick for example. If the rate at which a clock ticks slows down locally an outside observer will measure, this time lapse. However for the occupant in the curved region there is no time displacement. In plain English this tells us that since time slows, vector speeds may appear to decelerate in certain regions as well. This is a consequence of the time dilation affects of Special Relativity, mathematically represented as: (1.3)

Meaning that outside these regions time will appear to move faster to an observer with a slow clock. V’ola this leaves us with “shortcuts” in spacetime, thus the speed-of-light locally depends on the rate at which your clock ticks (often refereed to as coordinate time). However, this can not do justice to the grandiose of that remarkable theory. We will be gradually pulling out tricks from GR as well as other theories to explain how the warp drive can work throughout this book.

Before closing on this section let us review the foundations of GR as they will be discussed throughout this book. One should keep in mind that within the frame work of General Relativity geometry and energy are connected in a most fundamental way. The exact relation is given through the seemingly simple formula

G_ij = T_ij (1.4)

G is known as the Einstein tensor which represents the geometry of space and time. T is known as the stress-energy tensor and acts to measurement the energy within spacetime. The subscripts i,j represent a chosen coordinate system, or measurement system (this is an imaginary metric to keep a chosen tensor positive for conceptual reasons, although physically this is not the case). If you noticed we called Eq. 1.4 seemingly simple, the reason for this is because G actually represents a complicated set of differential geometry known as Riemannian geometry. The differential geometry is affected by classical physics through partial derivatives known as Christoffel symbols. Although Riemannian geometry is very complex we can generally write a spacetime geometry in the form of several tensors such as R^a_bjk . Although solving the equations is a bit overwhelming to new comers as the geometry reduces (or contracts) under special operations by something we call a Ricci tensor R^a_b . This means that for computations involving GR you will often see Eq. 1.4 expressed in some variant of

(1.5)

The new term g_ab is called a metric tensor because it acts as a tool to measure space. It is from the concept of a metric tensor where we can take measurements of spacetime as mentioned earlier in the section to make judgments about the curvature of spacetime. This very abstract geometry G intern affects T and T can later affect G (also note that this does not represent Newton’s Gravitational constant, but a tensor space). However if we generically set T = 0, we notice that we can build a geometrical model for the law of conservation of energy, if we choose. This is a very important concept which will tell us the forms of geometry allowed and not allowed by GR. We shall come to these terms later in the book. Geometry can affect energy and vise vesra, this pull and tug allows physicists to judge what are and what are not valid conditions for GR based on known empirical data. Any drastic change in one of the tensors can greatly affect the other and is why there is such a great skepticism against the concept of a Warp Drive.

1.2.1 A Metric Defined

A metric is a distance on some surface. For example, in at Cartesian space the metric is simply the straight line distance between two points. On a sphere it would be the distance between two points (given by Gallian coordinates), as confined to lie along the surface of the sphere. In the space-time of Special Relativity the metric is defined as

ds² = c² dt² − dx² − dy² − dz² (1.6)

You can think of the metric as the distance between points in space-time. Related to the metric is the very important idea of a metric tensor, which is usually represented by the symbol g which is called a determinant, sometimes with the indices explicitly shown (e.g., g_ab). The metric tensor is one of the most important concepts in relativity, since it is the metric which determines the distance between nearby points, and hence it specifies the geometry of a space-time situation.

One can represent the metric tensor in terms of a matrix as we show below. In four-dimensional spacetime (whether at or curved by the presence of masses) the metric tensor will have 16 elements. However, like all other tensors which you will experience in this book, the metric tensor is symmetric, so for example g₁₂ = g₂₁. This means that there are 10 independent terms in the metric tensor, these are also known as scalar components.

(1.7)

Misner, Thorne and Wheeler have a unique way of picturing the metric tensor g as a two slot machine g(_,_). The precise numerical operation of the g machine depends on where you are on a surface (i.e., what the local curvature is). When one puts in the same vector, say u, into both slots of the g machine, one gets the square of the length of the vector u as an output. If one puts two different vectors u, v then the g machine gives the scalar (or dot) product of the two vectors u and v.

You might say so what is the big deal – it’s just something to calculate the dot product of two vectors. The subtle difference is that in curved space-time the “angle” between vectors depends on where you are in space-time, and therefore g is a machine which operates differently at different space-time points. Viewed backwards, if you know how to calculate the dot products of vectors everywhere, you must understand the curvature of the space (and therefore the g machine specifies the geometry). For this reason, the metric tensor enters into virtually every operation in General Relativity (and some in Special Relativity).

1.2.2 Minkowski Metric Tensor

In the at spacetime of Special Relativity, g is represented by the Minkowski Metric. The Minkowski metric tensor is that of at space-time (Special Relativity) is written as follows:

(1.8)

It is more conventional to express tensor relationships in terms of summations, rather than as matrices. We can express the tensor relationship for the space-time interval in the following way:

(1.9)

which yields, if one uses the terms dx⁰ = cdt,dx¹ = dx,dx² = dy,dx³ = dz

one receives ds² = cdt² − dx² − dy² − dz² (1.10)

and writing Eq. 1.10 in matrix form gives us

(1.11)

Note that the Minkowski metric tensor has only diagonal terms. When we go to the curved space of General Relativity in the presence of masses we will employ tensors which have off-axis components. This is yet another example of why Eq. 1.4 in Section 1.2 is deceptively simple. Since our topic again is not so much GR, but Warp Drives, we bring up these important background issues as needed.

1.2.3 Wormholes

To explore what we mean by a topological (or geometrical) shortcut in GR, we start with the very popularized concept of a wormhole. A wormhole is a gravitational/topological shortcut capable of connecting two widely separated locations in space, say 26 light-years apart. A body might take a few minutes to move through this shortcut trough space at velocities v << c in a region known as the neck of a wormhole. However, by transiting through the wormhole the body has traveled 26 light-years in global (larger) spacetime in just a few minutes, producing an effective speed of a million times the conventional velocity-of-light to an outward observer. The body simply moved through a region that has a shorter distance to its destination. By taking this short cut an external observer will record a faster than light transient time. Yet, in reality it never exceeded c in the local metric of the global manifold, its simply a topological trick!

However these wormholes come at a great cost, although they are mathematically viable they are not necessarily physically viable. To elaborate, wormholes are generally considered as nothing more than a mathematical curiosity (however there is a minority in the physics community that believe they may be physically possible). The reason for the debate over their physical nature is that they were accidentally discovered by a thought experiment of Einstein and Rosen, initiated through a symmetry operation in the mathematics of GR (as a result a wormhole is sometimes refereed to as an “Einstein-Rosen Bridge”). As a black hole forms the null geodesics (the path of light) of the dying star converge to a singularity. At the singularity the known number system completely fails and all meaningful information about the dead star is lost. However because GR is based upon symmetries, mathematically there should exist an imaginary copy of the collapsing black hole. If the imaginary (mirror) copy somehow converged, the singularity is adverted and a transversable hole (wormhole) is formed in space. There have been a number of topological studies performed such that one does not need an imaginary black hole to collide with a real one for a wormhole to form. However the gravitational forces acting on the throat of the wormhole are enormous and strongly relativistic in nature, which causes the “neck” to collapse. Thus the only way left to hold open the wormhole’s neck is the mysterious acceptance of a form of energy called negative energy or exotic energy, which just happens to violate the laws of thermodynamics (such that the T = 0 term in Section 1.2 becomes T < 0).

1.2.4 Inflationary Universe

Another example of FTL behavior in General Relativity is the expansion of the Universe itself. As the Universe expands, new space is seemingly being created between any two separated objects. The objects may be at rest with respect to their local environment and with respect to the Cosmic Microwave Background (CMB) radiation, but the distance between them may grow at a rate greater than the velocity-of-light. According to the standard model of cosmology, parts of the Universe are receding from us at FTL speeds, and therefore are completely isolated from us (resulting from our corner of the Universe forming a local cosmological horizon). The recently observed accelerated expansion of the Universe by S. Perlmutter [1] and others even shows the presence of this affect more than the original standard model. Adding in variances in the velocity-of-light over time we find that at one time the Universe had a higher velocity-of-light than it now does. So, the idea of altering the local velocity-of-light, or of traveling FTL in a normal space-time region is not new to physics. It is simply determined by finding methods based upon General Relativity to do an end-run around the local restrictions of Special Relativity.

The present “accelerating” (in quotations as this is a non Lorentzian expansion) Universe gives clear evidenced that seemingly apparent FTL travel is possible and is presently at work within our Universe. The concept of an accelerating Universe was first conceived by ad hoc adaptation Einstein made to his famous field equation (see Eq. 1.5). He added a term to the right hand side of his equation called the Cosmological Constant, represented by the Greek symbol Λ. Einstein invented this concept to maintain the idea of a static Universe, until Edwin Hubble showed that the Universe is in fact expanding. Einstein later withdrew this term calling it “the biggest blunder of my life”. What is even more strange is that if you put the term to the left side of the Einstein Field Equation (Eq. 1.5), it acts to remove energy, thus the Cosmogonical Constant defies the law of conservation of energy as T mentioned in Section 1.2 can takes values which are T ≤ 0. It is rather disconcerting that something in the large-scale of the Universe is violating the very nature of GR, nothing is sacred anymore. However, this does not necessarily mean that GR is wrong, and that the energy conditions as far as GR is concerned are not violated. GR for example cannot take quantum action into affect because quantum theory has non locality as its basic building blocks, which violate the principles upon which relativity is built. There is also the possibility that we are missing something and the Universe’s apparent acceleration may exist because we are unaware of something else that may exist. So that said even the “accelerating Universe,” does not provide evidence that objects such as wormholes do in fact exist.

1.3 What is the Warp Drive?

Now to the heart of the matter, what exactly is a Warp Drive, how does it relate to relativity, and what exactly does it do? The Warp Drive was brilliantly conceived by Dr. Alcubierre in 1994 in a short letter to the editor of the journal of Classical and Quantum Gravity [2]. Alcubierre’s vision of a way to allow for apparent FTL travel as seen by an outside observer in GR was the Warp Drive. His method of travel is somewhat connected to the expansion of the Universe, but on a more local scale. He has developed a “metric” for General Relativity, a mathematical representation of the curvature of space-time, that describes a region of at space surrounded by a warp or distortion that propels it forward at any arbitrary velocity, including FTL speeds. Alcubierre’s warp is constructed from hyperbolic tangent

Figure 1.1: The Alcubierre Warp Drive

functions which create a very peculiar distortion of space at the edges of the at-space volume. The hyperbolic structure was chosen so that such a spacetime would not produce a Causal Time-like Curve (CTC), or what the general public calls a time machine. In affect, new space is rapidly being created (like an expanding Universe) at the backside of the moving volume, and existing space is being annihilated in front of the metric. The effect is what we term a dipolar region of space. It causes space-time to flow, think of a snow plow, it eats the snow and then spits it out. This is in affect how Alcuiberre’s Warp Drive functions. Although this spacetime region creates a violation of the law of conservation of General Relativity represented by . This finding is not really to surprising in the present, for example Olum has shown that FTL travel requires negative energy [3]. So the relevant question now is that since Warp Drives violate the law of conservation of energy can they exist in reality. Unfortunately there are no easy answers to this question, but throughout this book we try to examine this possibility. Welcome to the wonderful world of differential travel schemes using c in GR, you now have the background to tackle the bigger questions.

Chapter 2 Fundamental Warp Drive Metrics

"Captain’ I cannota’ change the laws of physics". –Chief Engineer “Scotty” from the Star Trek series

We now wish to introduce the reader to the heart of the matter, what is a Warp Drive? The Warp Drive is anything but clear, it is often contradictory, and is generally counter intuitive. From here we hope the reader grasps how a basic Warp Drive metric behaves, and why even the most simplistic models of Warp Drives are littered with countless problems. The intent however is not to show the general affects of a warp metric, but to show the nature of the beast. In more general terms we are modifying GR to intentionally allow for an arbitrary shortcuts in spacetime. In the previous chapter we encountered how these shortcuts have sever limitations and that they are geometrical in nature. But generally we have not been clear about the geometrical nature of these shortcuts and why they violate some of our most cherished laws of physics.

In this chapter we explore in relative detail the geometrical requirements of a warp drive spacetime. From here it is learned how Warp Drives use geometry to satisfy a remote observers local definition of FTL travel. When we explore the geometry in detail we also become more aware of the violations of GR explored in Chapter 1. Generally from this chapter we learn of the specific nature of warp drives, and why there Nature defies General Relativity.

2.1 Dr. Alcubierres Original Metric

To explore how The Alcubierre Spacetime works we will explore the mathematics of his paper [2] throughout this section. To do so let us imagine that you are in space ship who’s position is given in Cartesian coordinates by an arbitrary function x_s(0, 0). Now imagine that your space ship is coasting so that its future position is determined by an arbitrary function of time x_s(t), this function yields the velocity of your space ship by the standard relation v_s(t) = dx_s(t)/dt. Therefore the Alcubierre Warp Drive can be imagined as special case geometry which brings (or ”annihilates”) a point in space to in front of an observer as well as pushing (generating space behind) an observer from their point of origin. In order to understand the metric further we will need to construct an arbitrary function f as to produce the contraction/ expansion of volume elements as proposed by the Alcubierre Spacetime.

One would then have to form a bubble around oneself, lets say that three spatial coordinates are given with x = y = z = 0. With the ship’s position x_s = 0, the invariant lengths of this space are then given by:

(2.1)

One can now write the arbitrary function of the warp drive spacetime as a function of the ship’s positions r_s.

(2.2)

When the velocity of the warp drive is set to v = 1 then the Warp Drive Spacetime can be described by the metric:

ds² = − dt² + (dx − v_s f(r_s)dt)² + dy² + dz² . (2.3)

2.1.1 Problems with the Alcubierre metric

The initial problem of the warp drive was recognized by Alcubierre himself in his breakthrough paper. It was even more focused upon by Dr.s Ford and Pfenning [4] in their works. The problem being a weak conservation principle in General Relativity (GR) is broken, simply known as the Weak Energy Condition (WEC). The formula for a non-trivial space is given by T_abV_a V_b ≥0 for all time like vectors V_a . The problem which arises is that it causes the flat spatial metric within the zero Christoffel space to obtain a positive curvature. Thus violating relativity by a local expansion of space-time (that is the creation of an exotic energy region). Further, there is a restraint placed upon any negative energy condition that an equal or greater rebound pulse of positive energy must follow the initial negative energy. We know from Quantum Mechanics that such violations do occur at the sub-atomic level. The fundamental question is can they exist at the Macroscopic level. Recent work with what has become known as the Casimir Effect, which demonstrates that they can. This has now brought up a new energy condition referred to as the Average Weak Energy Condition (AWEC). Basically, under it negative energy can exist in a local region of space-time as long as globally the composite energy condition remains positive.

The idea of space-time expanding and contracting is not at all new, it should be noted that Einstein predicted such an affect. From the local metric g_ab acting on a Riemannian surface geometry, the curvature connection of space-time is derived directly from a metric with a signature of rank 2 (referring to the indices of the metric). Thus the curvature connection would be given in a linear form do to the presence of the metric, hence space would be required to expand and contract from the metrics signature. This expansion and contraction of spacetime is called gravitational radiation, which is assumed to propagate at the speed-of-light c (although there is some recent in direct evidence that points to this). The key between this idea and Alcubierre’s proposal is that his paper requires two separate sets of gravitational waves. However, do to the spatial metric proposed by Alcubierre one set of the gravitational waves would be required to defy the WEC since his original metric required a large negative energy it would also violate the AWEC. Both of these issues alone make this metric a good exercise in math, but not valid as far as an honest method to exceed c.

2.2 Introduction to the Energy Conditions

Most of the so-called energy conditions fail at some level whether it be from special case topologies to the inclusion of quantum field theories. To account for these bugs physicists have collectively decided to come up with an “averaging condition”. The same trick that one uses to find the average velocity of a trip under stop and go conditions. An example of the “averaging conditions” is the Average Weak Energy Condition (AWEC).

The Strong Energy Condition (SEC) holds that all time like vectors l are always non negative (even in the presence of strong curvature) represented by the formula

(2.4)

In GeneralRrelativity a Ricci curvature (an important rank 2 tensor for metric curvatures), which can take positive and negative values for R_ab depending on the local curvature of the geometry, one can add a time-like vector to R by the operation R_abl_a l_b , so generally the SEC implies that even when there is a non trivial gravitational curvature present the null geodesics remain unchanged. It is however widely known that this is not the case in GR for all observers.

To account for this the SEC is then taken as an averaging means, which is known as the Average Energy Condition (ANEC). As such on a physically large metric the vector a may violate locally its value for ds² = 0 (i.e. appear to move faster or slower than it should) but over all the time like vector always remains non negative (even when being affected by curvature). T represents a stress-energy tensor (measure of spatial energy density) in short it basically states “that there should be no such thing as negative energy, even if spacetime is distorted” (its a strong argument because it intuitively wouldn’t always work hence its name).

The Dominant Energy Condition (DEC) in short states that “the flow of energy within matter (or matter-energy) is always less than the speed-of-light.” This acts to conserve energy by assuming the momentum of body can be given with the operation −T_ab l_b so if a body moves forward with a vectorl a then conservation is given with the formula

(2.5)

which we explored conceptually in Chapter 1. This also would imply that energy is always non negative which is what the Weak Energy Condition (WEC) states:

T_ab l_a l_b ≥0 (2.6)

or that “all time like vectors are non negative”. In short all these energy conditions try to say there is always positive energy in a gravitational field, but on occasion negative energy can appear to form, but can be removed with averaging techniques.

There are several run off energy conditions which stem from these ideas. Take the Generic Energy Condition (GEC), for example. One may even conceive their own special energy condition, if need. The GEC simply states that the WEC always holds, or more correctly that every time like (or null) geodesic contains a point where

. (2.7)

It should be obvious that all these energy conditions are interconnected. This is to be expected as the energy conditions state how the dynamics of spacetime affect T and vise versa, as this is the very fabric of GR.

2.3 Wormholes

Topological studies have also revealed other potential shortcuts in spacetime as opposed to Warp Drives alone. An excellent example of such an object is a wormhole. However, they are not something you cannot just immanently "dive" right into. Wormholes are based initially off the Schwarzchild geometry whose solution is given by

(2.8)

This space describes a spherical curvature in spacetime, an extreme curvature resulting in a cusp or singularity is what is refereed to as black hole. As geodesics caught within the Schwarzchild geometry are no longer able to escape to the outside Universe unless the geometry is modified. The Schwarzchild geometry also has an imaginary symmetric counter part which would be sufficient to act as a topological tunnel in spacetime. For a Lorentizian wormhole to remain open would require exotic energy as the gravitational forces created by the now partial horizon collapse the tunnel. In order to consider the geometry of wormhole, take some Lorentizian static geometry --say …

. (2.9)

This structure insures that this surface has an acceleration equivalent to c its invariance is again Lorentzian in nature an hence the name Lorentzian wormhole. However, this only insures a spacetime with Lorentzian signature. We want to limit the minimum radius to some arbitrary function --say b(r) -- so that (relativistic) gravitational forces won't completely overwhelm our chosen geometry

(2.10)

where Ω² = dθ² sin²θdφ² and Ф(r) is known as the red-shift function as geodesics falling into the partial horizon there frequencies become red-shifted. Any body attempting to leave the wormhole, thus necessarily becomes blue-shifted, since a gravitational wormhole is Lorentzian in nature. This again

Figure 2.1: Lorentzian Wormhole

requires a form of exotic energy, and it is the exotic energy which maintains the critical radius b(r). If exotic energy is no longer present the bridge collapses and a complete horizon is formed. Hawking radiation can be argued to posses negative or exotic energy explaining the thermal radiation which a black hole is capable of radiating. So again we see the relation that FTL travel requires geometry, and FTL geometry requires negative energy.

2.4 Ambiguity on the speed-of-light issue in GR

In the previous pages we have discussed some of the rather popular known apparent FTL shortcuts in spacetime. However, these shortcuts are not as clear as they appear to be, there is some ambiguity on what FTL means in GR. Just because one shortcut will allow for apparent FTL travel for observer does not necessarily mean that it will appear to be FTL for all observers. In the following subsections we will investigate why such shortcuts in spacetime far from declare that FTL travel is possible for GR. In some since it can be said the apparent FTL travel is possible locally, but not globally in spacetime, in fact there are number of papers written on this subject alone. We investigate these ambiguous issues so that we can have a clear understanding of warp drives, so that we can avoid easy mistakes.

2.4.1 Accelerated Observers

There is heavy criticism for FTL travel an seen from an external observer in GR, because this often requires the acceleration of some frame for some arbitrary time. While this allows for the existence of FTL within General Relativity, this form of FTL travel is completely ambiguous and is only locally viable. A moving observer in spacetime affects the rate of clock ticks (sometimes referred to a lapse function α) in his time frame according to his vector displacement ξ.

δτ = α = [ − g(ξ,ξ)]^1/2 (2.11)

An acceleration occurs if there is a second frame with its own vector displacement this space is called a momentarily co-moving frame. Such an acceleration is limited by ones spatial displacement l, such that l << 1/g, it’s actually limited generically to gl << 1 where v = c = 1 which complicates matters greatly. The acceleration is relative to a four velocity (a vector with 4 components) corresponding to a = du/dτ , which is orthogonal (at right angles) to four velocity

(2.12)

Eventually for the acceleration occurring in space and time one obtains the following values

t = g⁻¹ sinh gτ, x = g⁻¹ cosh gτ

expressed in a different form such as x² − t² = g⁻² one can see how the hyperbolic functions form in the first place (and perhaps why Alcubierre chose this method for his Warp Drive model). The striking thing about an accelerated observer is the fact that if a body was in a geodesic (free fall) with an equivalent acceleration of 1 g or 9.8 m/s² for a period of one year such a body would obtain relativistic velocities where gl << 1 is seen.

If one were 1 light-year a head of a photon the slow acceleration rate of 1g would be enough to out run a photon, of course this is an ideal case, which doesn't take Lorentz invariance into consideration. That is the hyperbolic acceleration is tangential and parallel to the boundary of a light cone.

2.4.2 tetrads

A way of attacking the discrepancy between the observations made by a rest observer an accelerated observer in a momentarily co-moving frame is to consider an infinitesimal acceleration. Using an infinitesimal acceleration the discrepancies can become trivial and one can compare the acceleratory frame with the reset one, this is done with a base vector called a tetrad. The base vectors (meaning they are not vectors of their own right) e'₀ , e'₁ , e'₂ , e'₃ make up the bases of a tetrad. Since there is zero rotation in a Lorentz frame for e'₂ and e'₃ , the reaming values for the frame in acceleration become

e'₀ = u (2.13)

e'₁ = g⁻¹ a (2.14)

aside for the acceleration in the x,t frame the metric is Minkowski in nature such that

e_µ_' e_v_' = η_µ'v' (2.15)

Although this tactic minimizes discrepancies, it does not tell us how to compare the frame with one another. In order to do so one must parallel transports the tetrads along the geodesic of the reset observer so that they become geometrically compatible.

To do so, a technique called Fermi-Walker transport must be used. One must set up a tetrad have (hyperbolic acceleration) zero Lorentz transformation in the 1,2 plane one then must align this with the world line of a spacetime event t. This particular alignment is necessary so that there exist a null acceleration in the rest frame. So in the end the classical conception of gravitation suggest that FTL travel is possible locally and only locally, (and this dependence lies heavily on the observers' location in spacetime). There exist no global frame for which FTL travel is possible within classical frame work of General Relativity. This is however meant to discuss subliminal accelerations in a causal Universe, the acceleration of space itself is not taken into context here. However it is this is what need to explain the vastness of the Universe if a Big Bang event did take place some 14 billion years ago. At some point in history the Universe must have accelerated beyond c in order for the Universe to have arrived at its present state, in this context we can explain FTL within GR.

2.5 Cosmology

The evolution of the Universe can be considered simply with parametric function that relates its size and flow of time a(t), this is called a cosmologic horizon (which is a region where a light cone is SR can not be expanded). This can in general be taken as the curvature of the Universe as a whole and is governed by

ds² = − dt + a²(t)[dχ² + sinχ² [dθ² + sin²χ(dθ² + sin²χ dΦ² )] (2.16)

The famous Hubble red-shift can be incorporated into a line element of the form

ds² = − dt² + a²(t)d dχ²(2.17)

Now the Big Bang theory has a huge problem in this area, it must account for how portions of the Universe could have expanded past a cosmological horizon. This is accomplished by adding a “cosmological constant’ to the Einstein Field Equation, usually represented with Λ. Such that the Eq. 1.5 becomes

G_ab = R_ab − ½ g_abR + Λ g_ab (2.18)

Since the Einstein tensor can be written in the form G_ab = T_ab, the addition of the cosmologic term affects the properties of the stress energy tensor. T_ab so that it affectively becomes T_ab = − λgT, that is to say a region of negative energy is created, meaning that Eq. 2.5 is violated.

What we have learned thus far is that somehow a cosmological horizon a(t) is related to the Cosmological Constant Λ. And that this connection is somehow related to the expansive nature of the Universe which can exceed our local value for c. And that this must somehow be powered by some form of exotic energy. This frame is very peculiar in comparison to classical General Relativity as it allows for apparent FTL travel globally, without the pesky side affects of accelerated observers. It would seem that the large-scale Universe is somehow involved in a conspiracy against GR, or just perhaps GR is not a complete theory. Quantum Theory even gives us another candidate, under certain circumstances a black hole can radiate energy known as Hawking radiation. Over time a black hole can radiate enough energy to shrink to microscopic size. Thus forming a quantum version of a(t), known as a baby Universe. As the black hole fades away it can momentarily exceed our value of c and radiate in an explosion of energy. There is a certain interconnectedness going on which tells us there is some underlying physics here. However all these signs are completely ambiguous as no theory can currently resolve all these issues.

Chapter 3 After Shocks

“Warp Drives can not be ruled out.” -- Stephen W. Hawking

So now we have a strong theoretical bases to travel distant places within a coordinate system which may be shorter than other potential paths. Thus making travel to regions beyond a cosmologic horizon possible in finite time without requiring new laws of physics. But what of Alcubierre’s thesis, what would be the next logical steps to develop these shortcuts? We now take the reader away from demonstrating that shortcuts exist in spacetime. Our intent is to now show that these shortcuts can be modified to fit some desired criteria. Although the energy condition violations are very troublesome we have shown cases where they are explicitly violated in the real Universe. They however fall far short of proving that these kinds of topologies can exist in spacetime.

Since we have the background information out of the way we can now explore more deeply into the warp drive. Are there other types of warp metrics, is it possible to make Warp Drives which do not violate the standard energy conditions in GR? These are the topics that we’ll be exploring in this chapter, this is the stuff that warp drives are made of.

3.1 Van Den Broeck

From Alcubierre's paper one can calculate the energy needed for create a warp field. The problem is that the required energy for a space ship with a warp bubble radius of 100 meters will have a mass-energy content 10⁹ times greater than that of the observable Universe. In 1999 Chris Van Den Brock proposed a metric that would reduce the energy requirements for a warp drive system [5]. This metric has a space-time distortion for which the total negative mass needed is of the order of a few solar masses, accompanied by a comparable amount of positive energy. His space-time geometry is represented by the following equation

ds²= dct²− B² [(dx − βfdct)² + dy² + dz² ] (3.1)

where B can be any function that is large near a space ship and one far from it (that is the range value of the continuous piecewise function) though he used a specific top hat function for his example calculations.

(3.2)

Figure 3.1: Van Den Broeck Bubble

That not only brought the negative energy requirements down to a one-day reachable goal but it also solved one of the quantum mechanics' energy condition violations. A more general case of an Alcubierre–Van Den Broeck type Warp Drive spacetime can be written [33]:

ds² = (A(x^µ)² + g₁₁ β² f²)dct²− 2g₁₁ β f dct dx¹ + g_ijdxⁱ dx^j (3.3)

where the metric tensor reduces to that of Special Relativity far from the ship, and dx¹ represents a coordinate distance displacement. The problem was this metric is very long and complicated, due to its Kerr type metric instead of a regular Schwarzchild format.

However, some of the original members of the online group, ESAA, picked up on this original idea, changed it into a regular metric [31]. This new metric allowed an arbitrary lowering of the energy requirements by modifying the lapse function A, so that the Warp Drive could be argued to be realistic some point in the distant future.

3.1.1 Quantum Fields

Then Pffening and Ford as well as others showed that quantum fields can affectively change the behavior of warp bubble, specifically limiting its size [30]. Take the width of a warp bubble by considering the following piecewise function

(3.4)

where ∆ = [1 + tanh²(σR)]² /2σ tanh(σR) . Affectively the bubble radius would be limited to ∆. To calculate the size of the bubble is relatively straight forward one inputs this function into the Alcubierre Warp Drive. Then one performs a total energy calculation of the system. But aghast! you say that can not limit bubble size. That is a correct assumption but if you calculate the negative energy states allowed by Quantum Field theory ∆E ≤ t_planck ħGc, then only an arbitrary amount of negative energy is allowed to exist and this limits the bubble size to a few orders of the Planck length. The issue of negative energies have also been addressed, quantum fields do allow for large-scale negative energy densities. However they are always over compensated with an amount of positive energy afterward, this is known as the Average Weak Energy Condition (AWEC) -- see Section 2.2 -- meaning that at the surface quantum field theory wont allow for Warp Drives.

3.2 Warp Drives & Wormholes

A peculiarity of Warp Drives is that they somewhat resemble wormholes from a topological stand point. For example, let us examine the Kransnikov two-dimensional (wormhole-like) spacetime [7]:

ds² = (dt − dx) (dt +k(t,x) dx) = − dt² + [1 − k(x,t)] dx dt +k(x,t)dx² (3.5)

with

k(x,t) ≡ 1 − (2 − δθ_ε (t − x) [θ_ε(x) − θ_ε (x + ε − D)] (3.6)

where

(3.7)

Figure 3.2: The Krasnikov Superluminal Subway

In this context FTL travel is possible for a traveler’s forward light cone but not an observer’s past light cone. As an observer travels through this space only standard relativistic effects are noticed, however on the return trip, it produces effects characteristic of warp drive. The neat little trick using this models is that as you travel to a nearby star you travel nearly at the speed-of-light, so that it would only appear to take a few days (i.e., standard relativistic effects). While making your trip you would simply drag space behind you, and to return you would just ride back on “tube” which you have been “towing” with you. Thereby on the return trip you end up transversing less space by shorting the geodesic on the return trip. The downside is that this equation is only two-dimensional, and as in the case of the Alcubierre metric a tremendous amount of negative energy is required to maintain the affect.

The metric shortens the return path by the affective action

. (3.8)

Consequently it has an affect on the spatial coordinate such that

. (3.9)

Finally one can calculate the round trip travel time of this special geodesic

. (3.10)

Superluminality in only one direction and only in a tube region. The tubes' thickness is also restored to a volume of a few Planck lengths. Further investigation reveals that two warp bubbles somehow coupled at spatial distance can be constructed into a time machine. This does somewhat seem like a time machine in which no flagrant time paradoxes occur. It has generally come to be accepted that if time machines exist, they cannot go further back than there moment of Creation (this is in the context of spacetime topologies). This thus naturally leads us to consider the possible affects of a warp drive as a time machine.

3.3 Time Travel & Causality

As with the case of wormholes warp drives can be engineered to form there own form of ”time machines”. Specifically if two warp drives spacetime from distant corners of galaxy were to collide there topology may produce a time machine. This conclusion should not come as to surprising if one is familiar with the dynamics of wormholes (see Section 2.3). I f two distant wormhole interact there combined rotation can curve spacetime into a CTC, which is the GR equivalent of a time machine. The motion of a warp bubble through spacetime is almost identical to that of a wormhole if it is written in cylindrical coordinates. The concept of a warp drive time machine arose from the work of Allen Everett [8] and later followed by Pedro González-Díaz [10].

Everett considered a very simple modulation to the Alcubierre Warp Drive to consider the possibility of creating CTCs [8]. Consider the acceleration of a warp bubble to some distance a ≡ 0 < x < D/2 ˙ and − a = D/2 < x < D ˙ , so the spacetime evolution can be given with x = D and . If we consider a > 4/D then we have D²− T² = D² [1 − 4/(aD)] > 0 which implies that such an acceleration of a warp bubble would beat a beam of light traveling outside the warped region. In affect the warp drive becomes a generalization of a wormhole, however these coordinates are not well enough defined to find the underlying physics. This can be accomplished by setting the displacement coordinates

(3.11)

In this context physical clocks outside a warp metric can take the values dτ = dt'. From this it would appear that on the x-axis in external spacetime that two ships along the transversed distance move towards one another at infinite speed. When the negative axis ship reaches x = 0, it will then appear to vanish. This is very much similar to the behavior of tachyons. However, tachyons are spacelike while the Alcubierre Warp Drive is time like. From this the only time causal loops can occur is if there is a CTC present in the metric. A CTC or “time machine” can be created if the coordinate t' was running backwards through time. To an outside observer it would appear that two warp bubbles have collided. If one considers h_µυ as the Alcubierre metric and k_µυ as the time reversed coordinate then in terms of the Einstein tensor, this time machine can be written as

G_µυ = η_µυ + h_µυ + k_µυ (3.12)

From this mechanism it is similar to a CTC that is formed by two parallel intersecting gravitational sources for a Gott-like time machine. Although there are differences between the coordinates, it can still be seen that there is a causal loop connection.

Warp Drives like black holes can produce horizons, an apparent horizon and an event horizon. Hiscock revealed this analogy by modify the Alcubierre Warp Drive [9] so that its coordinates would be obtained more easily that it was for Everett. The line element for the Alcubierre warp drive is now written in the form

(3.13)

with

A(r) = 1 − v²₀ (1 − f(r))² (3.14)

And after considering a co-moving frame his new coordinates reveal

(3.15)

The event horizon and coordinate singularity form at the location A(r₀) = 0 or f(r₀) = 1 − 1/v₀. The problem with this coordinate system is that it reveals that the topology becomes unstable when v_b > c. The problem that had been posed concerned the fact that there was no causal connection between the craft and the warp field after the field was engaged.

Finally González-Díaz considered the time travel applications of Warp Drives in two dimensions. In short, González-Díaz showed that a Warp Drive in a multiply connected nonchronal spacelike region. Which has horizons that form near the Planck boundaries satisfying the Ford and Roman energy conditions [30] for apparent timelike regions. Lastly González-Díaz showed a relation between gravitational kinks (the generator for CTCs using cosmic strings) and the two-dimensional Warp Drive [10]. González-Díaz showed that Eq. 3.13 can be regarded as a gravitational kink of the topology of Eq. 3.15. If we now write dt' = A^1/4dt and dr' = A^−1/4dr so that

(3.16)

Horizons are formed by gravitational kinks or cusp which occur at a = 3π/4 such that a complete Warp Drive (in four dimensions) is described with line element

ds² = − [A(r) ± 2kv₀] dt² + 2k dt dx + dy² + dz² . (3.17)

Using this set-up there can remain stability of quantum fields in contrast to Hiscock’s work [9]. Another important aspect of González-Díaz is that he showed that a special kind of time machine can be created with these gravitational kinks. He showed that its possible to generate a CTC locally within a warp bubble, so that one can set up the initial state for a warp drive or to obtain causal control over it at luminal speeds. A three hyerbolid in Misner space is given by the line element ds² = − dv² + dw² + dx² . Now including the two-dimensional warp drive by the coordinates

(3.18)

x = F(r)

where

(3.19)

with ρ ≡ ρ = − v₀ [1 − f(r)]² ˙ and A = 1 + ρ(r). The spacetime of the warp bubble then becomes multiply connected (non chronal) by the relation

(3.20)

This relation takes place by embedding the two dimensional warp drive into a three-dimensional Euclidean space by means of a Misner geometry. This interconnectedness within the bubble is quite different from the Everett case and can allow for causal control

3.3.1 The Two-Dimensional ESAA Metric

Taking the generic ESAA metric is written as [31]

ds² = − A² dt² + [dx − v_s f(rs) dt]² (3.21)

or in full form

ds² = − A² dt² + v_s g(rs)² dt² + 2v_s g(r_s) dx₀ dt + dx₀² (3.22)

gives us the ESAA ship frame metric in two dimensions. The horizons for this metric can be written as:

(3.23)

with

(3.24)

thus

H(r_s) = A² − v_s g(r_s)² (3.25)

The ESAA group next redefined the Pfenning Piecewise function into the function of A. Defining now the value of A in the Pfenning integration limits R−(δ/2) to R+(δ/2) being δ = 2/σ. The Pfenning limits would be the following: R−1 and R+1. For r_s < R−1 the lapse function A = 1, for R−1 ≤ r_s ≤ R+1 A = large, for r_s > R+1 A = 1. It must now be remembered that A cannot be tied to velocity so that it changes as velocity changes. And now the values of the Pfenning Piecewise function

(3.26)

and the ESAA ship frame Piecewise function

(3.27)

the Pfenning Piecewise and ESAA Piecewise functions are defined in function of the lapse function A [31].

3.4 Photon-ed to Death

There’s been a lot of discussion on how Warp Drives affect the null geodesics of photons. Generally the paths of null geodesics are expanded in front of a ship and contracted behind if there is a warp drive geometry present. From this we can come to the general conclusion that as light approaches a warp metric it becomes blue-shifted. And conversely any photons as seen from the rear of the warp metric become red-shifted. The more precise definition of what goes on was given by Clark, et al. [11]. To do so calculate the null momentum of a photon, such that it can be conserved:

p_α p_β,α= 0 (3.28)

such that we can now write

. (3.29)

The geodesic path of a photon can now be calculated from

. (3.30)

As a photon enters a “warp field” it become deflected means of

p^t = 1p^x = cos(θ₀) + v p^y = sin θ₀) . (3.31)

The propagation of those photons as seen at infinity is

. (3.32)

These values then give us a relatively easy way to calculate the approximate energy of these photons:

E₀ = − p^α u_α|_r = 0 = p^t|_r = 0 (3.33)

We can also probe the effects of this apparent energy increase in refreeze to SR. Take the concept of boots and transforms. More generally, the coordinates (t, x, y, z) are the interval between any two events in 4-dimensional space-time (a 4-vector) change when the coordinate system is boosted or rotated, but the space-time separation s of the two events remains constant

s² = − t² + x² + y² + z² = constant (3.34)

However, with an inflation or moving frames situation the coordinate system is not boosted. It is the space-time separation of the two events that is altered to produce a change in the coordinate system. In the first, when there is a boost there is a Lorentz transform involved. Thus, we get a change in inertia mass and time. But in the second since no boost has occurred the actual Lorentz transform present is that it would have in a normal space-time condition even though we measure an apparent velocity increase.

A craft in a warp field is held in an at rest frame. It undergoes no change whatever, except an apparent one of a coordinate change induced by the fields compaction and expansion region. Particles on the outside have undergone no real change except an apparent one: since it was only the separation that has changed. Their spectrum will have changed, but this spectrum change is simply a false picture produced by a change in the space-time separation, and not any real boost. So from this perspective Eq. 3.33 is really a pseudo gain of energy caused by interpreting the Doppler expansion of the volume elements θ as local to the real metric.

Chapter 4 Null Manifold Inconsistencies

“The shortest distance between two points is zero.” – Sam Neill from the 1996 film 'Event Horizon'.

In this section we explore the manifolds in which General Relativity finds itself embedded in. Manifolds in spacetime may have a global structure however there are some local variances in a field which alter its local properties. It is often difficult to differentiate what is a real and artificial topology as a local observer only has partial information. In this chapter we will focus on the apparent behavior of null geodesics. This affects a great deal number of things, the emission of radiation, the causality of events, differentia between apparent an real SR affects in a local geodesic. In short in this chapter we discuss some of the wishy-washy areas within the manifolds of Warp Drives.

4.1 The Causal Photon Problem

In March of 2002, Dr. Natario a researcher in Portugal, published an article in Classical and Quantum Gravity that raised a new problem with warp drive [12]. This problem has to do with highly blue shifted photons entering the field and impacting the region of the craft (these effects were brie y outlined in Section 3.4). It was at this point that the group known as ESAA began seeking what answers they could find for this problem. One of the first answers to this problem was raised by one of the authors in looking over the issue. That answer deals with the improper way that Dr. Natario dwelt with the differing Lorentz Invariant frames within the metric itself. It has also been raised rightly that Dr. Natario only examined the original metric proposed by Dr. Alcubierre which was never engineered to be a working metric.

4.1.1 The Cauchy Problem & Dr. Natario’s Article

A Cauchy surface is an equal-time surface. As a result there is a confusion between the concept of “Cauchy surface” and “time”. In solving the Cauchy problem, what you’re actually doing is providing a relation between the data q(S) which resides on one Cauchy surface S to the data q(T) which resides on another T. So, when one talks about evolution, one is talking about “evolving” quantities which are actually functionals q(S) of Cauchy surfaces S, not functions q(t) of time t. But in Newtonian Physics, you can get away with this confusion because the two concepts coincide. Basically, you have the equivalence:

Historical Time = Coordinate Time

where “Historical Time” is the arena in which evolution takes place (S) “Co-ordinate Time” is (t). In Relativity this is not the case. So the issue invariably arises that when one generalizes a theory from its Newtonian form to its Relativistic form: which concept of time applies, “coordinate” or “historical?” Here, with the Warp Metric we actually have a static inner region around the craft that is Newtonian, and an outer bi-polar region that is relativistic; it has an expanded Lorentz Invariant Frame. So, the concept of a shielded area is pre-built into Warp Mechanics in the first place. However, this complicates the inherent Cauchy Problem. In more technical terms a Cauchy surface corresponds to

(4.1)

where p is a point on some manifold M and I is causal past light cone. S is the subset of a spatial path J⁺ such that

. (4.2)

A Cauchy surface is then further defined as the orthogonal set Σ to the sum of the surfaces or simply D(Σ) = M . From here a relation can be made to a metric tensor such that (M , g_ab) ≈ Σ.

Natario's basic mistake was that he didn't take into account the effects of the expanded Lorentz Invariant frame have upon the incoming photons' blue shift. Yes, they are blue-shifted. But, they are not as hot as he portrayed them in his article. Secondly, he didn't apply a proper Cauchy Evolution to the issue. He failed to consider even the simply shielding that Newtonian frame could provide around the craft. Coming from a region experiencing blue-shifting and crossing into a free-fall region marks the crossing of a boundary. The evolution of those particles has now been changed. Going from a hyper accelerated region into a region in total free fall is close to having a speeding car slam into a brick wall. The photons are bound to bleed-off energy fast. In Relativity, the state W(S) (classical or quantum) is actually a functional of S. The state W(S) is really nothing more than a summary of the Cauchy data for S. When solving the evolution problem, all you’re doing is providing a map:

E(T,S) : W(S)| '→W(T)

That relates the state on T to the state on S. In a more technical format the proper map is Eq. 4.2.

In Quantum Theory, for instance, this map would be the (generalized) evolution operator U(T, S). Though it’s not widely recognized, this operator is well-defined even where S and T are overlapping or intersecting Cauchy surfaces. Now when making the transition from the Lagrangian formulation to the Hamiltonian formulation in Newtonian space, one takes a space-time region V whose boundary dV has the shape:

dV = V_f − V_i + V₀V_i

where the initial hypersurface V_f is equal to the final hypersurface. With the “same” 3-volume at a later time V₀ which is the 2+1 dimensional “side” boundary and the only way to remove the “boundary” terms is to push the side boundary V₀ off to spatial infinity. In a Relativistic space, one has the rather unusual phenomenon whereby finite (compact) space-time regions V exist which are bounded solely by space-like surfaces: dV = V_f − V_iV . Where the compact N-region V_i is the “initial” N−1 space-like boundary V_f with “initial” N−1 space-like boundary. The space-like hypersurfaces intersect on a common N−2 boundary:

d(V_i) = d(V_f ) = V A = N − 2_Anchor

Here again a point Natario seems to have neglected. Consider the case where you have a Minkowski space and you have two Cauchy surfaces S,T which coincide up to the Anchor; but with surface S following V_i, and surface T following V_f :

S = c + V_i T = c + V_f dc = V A = d(V_i) = d(V_f )

where c goes out to spatial infinity.

The basic principles underlying a purely local formulation of the Cauchy problem are:

Evolution: The evolution E(T,S) : W(S) '→ W(T) is a function only of the physics within region V .

Factors: The state W factors such that W(S) = W1(V_i) × W2(c)W(T) = W1(V_f ) × W2(c) with E(V_f , V_i) : W1(V_i) '→ W1(V_f )

Transition: The transition E(V_f ,V_i) is independent of the Lagrangian out-side the region V .

and most importantly:

Global Transition: The transition E(V_f, V_i) is independent of the global structure of space-time outside of V .

When one formulates the Lagrangian principle of least action over the region V , the result is a local generalization of the Hamiltonian formulation which applies solely to the region V . It does not apply once the Boundary is crossed. And yet, Natario in his work assumes it does. He never once treats the Warp Region in its basic parts formed by the changes in the Lorentz frame. Nor, did he properly treat the crossing of the internal shield boundary around the ship itself. Continuing on, we find: The surfaces V_i and V_f are connected by a homotopy of surfaces all sharing the common anchor V

V_i = V(0) V_f = V(1) d(V(t)) = V A

for t between 0 and 1 V U V (t) as t ranges from 0 to 1. The homotopy parameter t is what plays the role of “historical” time and is the evolution parameter in the generalized Hamiltonian formulation. Here again, the common anchor issue was ignored in Natario's work.

An Incoming Photon will enter a region in the forward area where the local limit on c is less than one observes normally in space-time. While the photon will be entering this region at the velocity of the overall field, it will lose energy as it enters this forward region. An example being that if the overall field velocity is say 2c yet this compression region has an overall maximum of 1c in that region then the photons will shed their excess energy as they pass through this region. Again when they cross into the near region of the craft which is in free fall they will pass through a horizon. But since the local velocity-of-light here is the same as that of the frontal region in this case they will not shed energy again. However, once the photons pass through this region they will again gain energy since the rear region of the field has a higher value of c. Dr. Natarios article never took this into account. He treats the whole field and its differing regions as if the rear regions velocity was all that one needed to account for.

Secondly, Dr. Natario did not take into consideration what effects an expanded Lorentz frame has upon the overall energy of those photons in question. When Einstein wrote down his postulates for Special Relativity, he did not include the statement that you can not travel faster-than-light. There is a misconception that it is possible to derive it as a consequence of the postulates he did give. Incidentally, it was Henri Poincare who said “Perhaps we must construct a new mechanics, . . . in which the speed-of-light would become an impassable limit.” That was in an address to the International Congress of Arts and Science in 1904 before Einstein announced Special Relativity in 1905. It is a consequence of relativity that the energy of a particle of rest mass m moving with speed v is given by Eq. 1.2. As the speed approaches the speed-of-light, the energy approaches infinity. Hence it should be impossible to accelerate an object with rest mass up to the speed-of-light and particles with zero rest mass must always travel at exactly the speed-of-light otherwise they would have no energy. This is sometimes called the “light speed barrier” but it is very different from the “sound speed barrier”.

As an aircraft approaches the speed-of-sound, it starts to feel pressure waves which indicate that it is getting close. With some more thrust it can pass through. As the light speed barrier is approached (in a perfect vacuum) there is no such effect according to relativity. However, there is a logic flaw in this that is commonly missed. If you expand the c to a higher value then what we now call light speed, 3×10⁵ km/s, is no longer the barrier. The new value of c becomes the barrier now. In this altered region one still cannot travel faster than light. But one can travel faster than light does now. So, when you view warp drive as FTL, it is really a drive with an altered space-time where the upper limit on c has been changed so that we can travel faster than light in normal space-time, yet slower than light in the new space-time region. One has simply changed the barrier, not violated relativity. So in essence, even Warp Drive mechanics proves relativity is correct. In 1842, the Austrian physicist Christian Johann Doppler noted that the wavelength of light, sound, or any other kind of propagating energy measured by a moving observer will be shifted by a factor of: f' = f(v +v₀)/v where v is the velocity at which the observer is approaching or receding from the source and c is the speed at which the wave propagates. But since c in a warp region has been shifted higher then v/c which means it has a different effect in a warped region. Examine the following: If c = 2c and v = 1c, then v = 1/2 = 0.5 . Now the Doppler shift of plane light waves in vacuum which arrive with an angle φ with respect to the direction of travel is:

However, in a hyperspace region the c unit has a different value. The value of a 2c warp region would fall nearly on the classical line of a graph if one compared a classical one to one done using SR and one using the modified Lorentz frame. In fact, as the lightcone of the region is expanded the classical is more closer followed as long as our ships velocity for each region remains at 0.5 of the regions total velocity range. So in reality those incoming photons would not hit the ship with any more energy than a craft would experience if it was traveling at 0.5c in normal space-time. So, Dr. Natario's examination of Warp Drive failed simply because he didn't take into account changes in the Lorentz Invariant frame. This does not imply that a blue-shift of photons entering the field does not take place. Indeed, the photons at the rear of such a bi-polar region are highly blue shifted. But, in the front region coming at the craft they are not shifted beyond a certain point. Recent, none other than Dr. Pfenning, who wrote against the original Warp Metric from Alcubierre stated in his own published article that Natario was incorrect in assuming the problem he posed rules out warp drive. If this is not enough of a rebuttal against Dr. Natarios article, the next step in the metric development is.

Even at a region slope that mnemics some fraction of a normal c condition as far as incoming photons are other particles there is still a vast danger to our craft and its occupants. For “starship” travel to ever become a viable reality, shielding against radiation must be developed beyond the bulky type we presently use in reactors. The answer is found in that Cauchy problem's solution and its primate shielding. The key is to enhance this effect and to add other layers of shielding. The focus within the following is to shield against incoming photons. However, this same effect could be applied to massive particles and even large macro objects. At one point the dream of what was called the Star Wars Program was to develop a shield against an incoming ICBM attack. This enhancement of a natural shielding effect has the potential of making such an attack ineffective once fully developed.

4.2 The Shield Effect

A warp driven body traveling with a speed of magnitude v ≥ c may encounter hazardous radiation and debris. Under such a scenario, how can one limit these potential hazards while at warp? The ESAA group proposed a variation of the Van Den Broeck metric to generate a ‘gravitational shield’ to protect a body from such hazards.

The new metric requires a revision of the Van Den Broeck function Region B, such that:

(4.3)

where D is the radius of the Van Den Broeck warp region and R is the radius of the warp bubble. The general behavior of B is 1 at the ship and far from it, while it is large in the vicinity of D. The exponent P is a free parameter.

The resulting line element of the new metric is

ds² = [1 − B²[v_s S (r_s)²] ]dt² − 2v_s S (r_s)B² dx₀dt − B² dx₀² (4.4)

where S = 1 − f(r_s). The resulting spacetime curvature affects the behavior of photons in the ship frame. After solving the null geodesics of this space it is found the apparent velocity of photons become

v = − v_s S (r_s) ±1/B (4.5)

This is actually the energy of an incoming photon as seen from the local metric. Thus this represents a shift in Doppler Frequency f' = f(v+v_o)/v, thus there is an energy increase as seen by v = c = λν = 1. The qualitative analysis of the photons speed can be broken up into 5 regions: the Ship; B; between the B and f; f; and finally outside of the f_region. At the location of the ship, S = 0 and B = 1. At a distance D from the ship in the B warped region, g = 0 and B is large. In between the B and f warped regions, the space is effectively flat, so S = 0 and B = 1.

Note again that the photons entering the f region will be Doppler Shifted due to v_s. The B warped region is designed to slow them down. Observe that the photon going backward never changed sign. A horizon occurs when the photon changes the direction of the speed. This important topic will be discussed later.

The exact description of what happens to matter entering the space-time is quite complex. Since a time-like interval is appropriate to matter, the equations earlier in section would have the equalities replaced by inequalities for equations with based on 4.5. The descriptions of such behavior however remain unchanged qualitatively. Here, we offer a qualitative explanation with more details to follow in a future work. Pieces of matter too small to be disrupted by the tidal forces will be slowed down in the B warped region, potentially to impact the ship at relatively slow speeds. For larger pieces of matter, they will become tidally disrupted in the B regions. The trajectories of the pieces are complicated, but the pieces are not expected to impact the ship at high speeds.

Refine B = B₁▪B₂ where:

B₁ = ((1/2)1 + [tanh(σ(r_s − D)] ² ) ^−P

B₂ = ((1/2)1 + [tanh(σ(r_s − S)] ² ) ^−P

where S is a distance from the ship such that S > R. B₁ is 1 at the ship and far from it, while it is large at a distance D from the ship. B₂ is 1 at the ship and far from it while large at a distance S from the ship. This produces two regions where the matter is slowed down, one at D and the other at S. S is outside the f region but involving the f region as a protection layer which will deflect incoming tidal shifted matter that otherwise would enter f region at high speeds and disrupt possible the Warp Field. For a rigid body, the gravitational gradients in B regions will tidally disrupt hazardous objects in the ships neighborhood. This is a property of B space-time, any natural object will be disrupted and deflected (if not on a head-on collision).

The qualitative description seen from a remote frame for an incoming particle or photon approaching the ship from the front. outside f region at a distance S from the ship S > R bubble radius f(r_s) = 0 then Eq. 4.5 at a distance S, then meteors and debris will be slowed down before entering f warped region. Outside the region S and B is again 1 and the speed of the incoming photon seen by a remote frame is again v = − 1.

A photon sent by ship forward but seen by a remote frame outside f region in the vicinities of S v = 1/B the photon seen by the remote frame never changes sign so the photon sent by ship but seen by the remote frame will cross f region and the outermost B region designated to protect f region from impacts and arrive at normal space.

4.2.1 Geodesics and Expansion, Cauchy?

Lets now take a closer look at Clark, et al's work (see Section 3.4). Let us examine the null geodesics and focus on the Christoffel symbol. We will find that a “warp shell” deflection of radiation discussed in Section 4.2 on the shield approximated from a two-dimensional Schwarzchild geometry with coordinates (r,θ). However the Christoffel symbols give us a more telling description of the deflection. In effect a shield deflection corresponds to an expansion of volume elements, and in the two-dimensional case there would be no deflection (revealing the partial case where Natario's work [12] holds).

Take the simplest geodesic in the work of Clark, et al. [11]:

(4.6)

Γ ^y_tt basically modifies g_tt by some vector v^y and its derivative. The Clark, et al. work shows that the Christoffel mechanism is only working on the warped region -- i.e., r = 90^o. It is not, however, present within the bubble. In this case the Christoffel symbol is replaced with vfv(p^t)² , for the 90^o case it might be more reasonable to introduce some boundary say | _r<100. What is interesting is that p^x = cos(θ₀) + v ˙ and p^y = sin(θ₀)˙ .

In the case of the warp shields we have a photon with an apparent velocity given by Eq. 4.5, so according to p^x this value should then be:

cos(θ₀) v_s S(r_s) + A/√B (4.7)

which measures the vertical angles. This is the result from interpreting Eq. 4.4 with the following Christoffel symbol

Γ ^c_ab = ½ g^cd [∂_ag_bd + ∂_bg_ad − ∂_dg_ab] . (4.8)

However an incoming photon does not see v_s S(r_s) until it encounters the functions A and B, thus the lapse function(s) produce an independent hypersurface. This surface is need for photons which exist in three-dimensional space. Such that it is seen that an incoming photon is deflected by the function S(r_s) only when as it encounters the lapse functions of A and B.

This can be thought up as a warp drive with positive expansion now take p^y , there is no expansion by A = large and B = large, light coming into the metric is purely based on the function S(r_s) and not A and B. Since in the two-dimensional case A = 1 and B = 1 are embedded in S(r_s) because of lack of expansion. Of course θ₀ is meant to be the angle of the emitted photons, so some re-labeling is needed so that it is not confused for a volume element. That is, the volume expansion occurs via

. (4.9)

The reason is that from this point, p^y is within the S(r_s) and not part of it! So a deflection of radiation occurs if there is a positive expansion, if there is no expansion, then there is no deflection of light. If the light is not deflected its energy remains unchanged, replicating a partial Natario Effect (this is because in the two dimensional case the photons energy have no where to run by ρ = y² +z² = 0, and remains in the system). In other words, what we have here is a “deflector shield” for incoming radiation.

This also reveals a connection to the Cauchy problem, being q(t) p^x and S(t) p^y . So that it can be seen that Natario’s error resulted from the classical case s(t) where he assumed s(t) = q(t) and thus neglected s(t) but not q(t).

Chapter 5 Negative Energy Problems

“If you’re not confused by quantum physics, you don’t understand it”. – Niels Bohr

While the idea of a warp drive is very much topologically sound it is far from being physically sound. For example, it's mathematically possible to place Asian and Australia 180^o apart. Although the question to next arise is: is this physically possible? The answer is generally 'no' unless say you have some kind of doomsday device on hand. Now we are faced with the question is there a doomsday device for General Relativity to allow for the energy conditions associated with warp drives? Viewing the classical energy conditions in GR (see Section 2.2), the answer is most certainly 'no'.

But this is what the GR cop imposes on us. We can, howeve,r explore the properties of negative energy to examine its rules. We now investigate the nature of exotic energy as seen through quantum fields. There the nature of non-locality violates the GR energy conditions as a rule. The exploration is done in order to see if such findings can be applied to a gravitational field. But for this chapter we shall put GR on the "back burner" and dive into the mysterious of the quantum theory.

5.1 Quantum Theory in Brief

In quantum theories, energy and momentum have a definite relationship to wavelength. All particles have properties that are wave-like (such as interference) and other properties that are particle-like (such as localization). Whether the properties are primarily those of particles or those of waves, depends on how you observe them. For example, photons are the quantum particles associated with electromagnetic waves. For any frequency f, the photons each carry a definite amount of energy (E = hf). Only by assuming a particle nature for light with this relationship between frequency and particle energy could Einstein explain the photoelectric effect (this is the very origin of quantum theory).

In classical physics, quantities such as energy and angular momentum are assumed to be able to have any value. In quantum physics there is a certain discrete (particle-like) nature to these quantities. For example, the angular momentum of any system can only come in integer multiples of h/2, where h is Planck’s Constant. Values such as (n+1/4)h are simply not allowed (such particles are known as fermions). Likewise, if we ask how much energy a beam of light of a certain frequency f deposits on an absorbing surface during any time interval, we find the answer can only be nhf where n is some integer. Values such as (n+1/2)hf are not allowed (these particles are known as bosons). To get some idea of how counter-intuitive this idea of discrete values is, imagine if someone told you that water could have only integer temperatures as you boiled it. For example, the water could have temperatures of 50^oC, 53^oC or 55^oC, but not 55.7^oC or 86.5^oC. It would be a pretty strange world you were living in if that were true.

We can also talk about quantum states for freely moving particles. These are states with definite momentum, and energy. The associated wave has frequency given by f = pc/h where c is the speed-of-light. A state is described by a quantity that is called a wave-function or probability amplitude. It is a complex-number-valued function of position, that is a quantity whose value is a definite complex number at any point in space. The probability of finding the particle described by the wave function (e.g., an electron in an atom) at that point is proportional to square of the absolute value of the probability amplitude. The rule for probability in quantum mechanics is that probability is the square of the absolute value of the relevant probability amplitude:

P(A,B) = |< A | B >|² . (5.1)

You have here, then, the basis of coherence and decoherence.

It is formulated in a well-defined mathematical language. It makes predictions for the relative probabilities of the various possible outcomes, but not for which outcome will occur in any given case. Interpretation of the calculations -- in words and images -- often leads to statements that seem to defy common sense – because our common sense is based on experience at scales insensitive to these types of quantum peculiarities. Because we do not directly experience objects on this scale, many aspects of quantum behavior seem strange and even paradoxical to us. Physicists worked hard to find alternative theories that could remove these peculiarities, but to no avail. The word "quantum" means a definite but small amount. The basic quantum constant h, known as Planck’s constant, is h = 6.626075 × 10⁻³⁴ j/s. Because the particles in high-energy experiments travel at close to the speed-of-light, particle theories are relativistic quantum field theories.

Subatomic particles known as virtual particles can come about due to a relation known as the Heisenberg Uncertainty Principle. In 1927, Heisenberg formulated a fundamental property of quantum mechanics which said that it is impossible to measure both a particle’s position and its momentum simultaneously. The more precisely we determine one, the less we know about the other. This is called the Heisenberg Uncertainty Principle. The mathematical relation is:

∆x ∆p ≥ ħ/2 (5.2)

where ħ = h/2π. This means that the uncertainty in the position (x) times the uncertainty in the momentum (p) is greater than or equal to a constant (h-bar divided by two). This principle can also be written in terms of energy and time:

∆E ∆t ≥ ħ /2. (5.3)

This means that the uncertainty in the energy of a particle multiplied by the uncertainty of time is greater than or equal to a constant (ħ/2). So for a very short time the uncertainty in the energy can be large.

5.2 Virtual Particles

Particles that can be observed either directly or indirectly in experiments are real particles. Any isolated real particle satisfies the generalized Einstein relativistic relationship between its energy E, its momentum p, and its mass m (see Eq. 1.2). Notice that for a particle at rest p = 0, this becomes E = mc² . This is the minimum possible energy for an isolated real particle.

A virtual particle expresses the quantum exchanges of energy often in terms of objects called Feynman diagrams. These diagrams are a shorthand hand representation for extensive calculations which gives the probability of a particular process. Richard Feynman was the physicist who developed that tinker-toy method for particle physics still used today to calculate rates for electromagnetic and weak interaction particle processes. The diagrams he introduced provides a convenient shorthand for the calculations for rather elaborate processes. They are a "code" that physicists use to talk to one another about their calculations of particle/wave behavior in quantum theory.

Figure 5.1: A Feynman Diagram of electron-positron pair.

A Feynman diagram works in the following way. An electron emits a photon A. An electron absorbs a photon A. There is also a symmetry where a positron emits a photon A and a positron absorbs a photon A photon produces an electron and a positron (an electron-positron pair). An electron and a positron can meet and annihilate (disappear) producing a photon (notice that all six of these processes are just different orientations of the same three elements.) Because Feynman diagrams represent terms in a quantum calculation, the intermediate stages in any diagram cannot be observed. Physicists call the particles that appear in intermediate/unobservable stages of a process “virtual particles”. Only the initial and final particles in the diagram represent observable objects, and these are called “real particles”. These come about, we believe, due to quantum probability which ultimately is singled out by the presence of an observer.

From the Heisenberg Uncertainty Principle, we have a high probability that virtual particles -- demanded by the diagrams and equations -- can exist by "borrowing" energy under the Uncertainty Principle. In many decays and annihilations, a particle decays into a very high-energy force-carrier particle which almost immediately decays into a low-energy particle. These high-energy, short-lived particles are virtual particles. The conservation of energy seems to be violated by the apparent existence of these very energetic particles for a very short time. However, according to the above principle, if the time of a process is exceedingly short, then the uncertainty in energy can be very large. Thus, due to the Heisenberg Uncertainty Principle, these high-energy force-carrier particles may exist if they are short lived. In a sense, they escape reality’s notice. The bottom line is that energy is conserved. The energy of the initial decaying particle and the final decay products is equal. The virtual particles exist for such a short time that they can never be observed. But their effects can be observed, especially when considered over a long scale of time.

5.2.1 The Casimir Effect

Even though virtual particles are short-lived, one can observe their effects indirectly. One way of doing this is through an experiment called the Casimir Effect. One has two parallel metal plates a short distance apart. The plates act like mirrors for the virtual particles and anti-particles. This means that the region between the plates is a bit like an organ pipe and will only admit light waves of certain resonant frequencies. The result is that there are slightly fewer vacuum fluctuations -- or virtual particles -- between the plates than outside them where vacuum fluctuations can have any wavelength. The reduction in the number of virtual particles between the plates means that they don’t hit the plates so often, and thus don’t exert as much pressure on the plates, as the virtual particles outside. There is thus a slight force pushing the plates together. This force has been measured experimentally.

Figure 5.2: The Casmir Effect, virtual pairs contribute positive pressure to metallic plates.

So virtual particles actually exist and produce real effects. The Casimir Force can be calculated by taking the integral of an electromagnetic stress-energy tensor:

(5.4)

where T_zz = ½ (H²_┴ − H²_x + D²_┴ − E²_z ) (5.5)

such that with the aid of Green’s functions the force without plate presence is given from

. (5.6)

This is the force of the volume stresses, having and energy

f = − 8.11 MeV fma⁻⁴ = − 1.30 × 10⁻²⁷ Nm² a⁻⁴ (5.7)

require electron density n ≤ 1/a² . The transverse area of the EM field for two separated plates is

. (5.8)

Because there are fewer virtual particles or vacuum fluctuations between the plates, they have a lower energy density, than in the region outside. But the energy density of empty space far away from the plates must be zero. Otherwise it would warp space-time, and the Universe wouldn’t be nearly flat. So the energy density in the region between the plates must be negative. We thus have experimental evidence from the bending of light that space-time is curved, and confirmation from the Casimir Effect that we can warp it in the negative direction. This shows we can engineer the vacuum, and thishas been duplicated many times over. The fundamental question isn't so much can we generate a warp metric field. We could do so in simulated fashion with a specifically designed Casimir Effect setup. The question is can we enlarge one to surround a craft, and can we design a working usable metric that eliminates the many proposed problems.

5.2.2 Casimir Tensor

To apply the Casimir Effect to GR, we must modify to scalar force into a tensor format. In order to do so, write the Casimir energy in the following form:

t_ab = (ħcπ² /720 l⁴ ) diag[−1, 1, 1, −3] (5.9)

Such that the Casimir coupling constant becomes (G/c⁴)ħcπ² /720 l⁴ . In terms of Eq. 1.5, we can write

(5.10)

This results in a quantized gravitational constant, which is in itself contradictory. We can now write Eq. 5.10 in terms of energy such that

(5.11)

This indeed looks very similar to the Planck length l_p = (Għ/c³)^1/2 although the factor for Eq. 5.11 would be much larger. In this regard the Casimir Effect can be seen as a Macroscopic quantum gravity effect. Which means these virtual particles undetected by GR can have adverse effects on geometry

G_ab = − kh_ab or G_ab = − kT_ab +h_ab. This is reminiscent of the addition of the Cosmological Constant to the Einstein Field Equation. This method gives a possible explanation for the arrival of the Cosmological parameter. More importantly this tells us that h_ab somehow couples to a gravitational field by means of a scalar field. This is of importance as it may allow us to modify the geometry of spacetime by this coupling.

5.3 Quantum Inequalities

There are always quantum energy measuring problems dealing with negative energy. Richard Feynman had a unique way of describing energy calculations in quantum theory: “There are three different ways of specifying energy: by the frequency of an amplitude, by the classical sense E₀, or by inertia.”

Now take the total energy of semi-classical General Relativity

(5.12)

where b(t) = t₀/ [π(t² + t₀²)] is a sampling function, at unity E ≤ kħc(ct₀⁾⁻⁴ and k ≤ 3/(32π²), resulting in

∆E ≤ E(4π/3) (ct₀)³ ≤ (4πk/3) ht₀^-1 ≤ (8πt₀) (5.13)

carrying signals have limiting velocity limiting energies to ħ/t₀, although along with Eq. 5.3 sampling times become nearly impossible to predict. Noting Feynman’s words, we see another problem with the calculation it blends the classical with the amplitude approach:

(5.14)

This amplitude can play nice with General Relativity if it's a variation of Hilbert space. If ψ(s) is a function of real space, we can write

(5.15)

(5.16)

This then allows us to write an imaginary metric such that

(5.17)

although the sampling time problem still remains (in fact if one continues this operation on finds a Schwarz inequality, so this approximation isn’t really valid either). From this we get that if large amounts of negative energy can exist for small sampling times. They are always compensated by large pulses of positive energy by means of Eq. 5.2. Ford and Pfenning have expressed a unique massless scalar field inequality for the stress energy tensor:

(5.18)

This unique sampling energy requires the input of the warp function (df(r_s)/dr_s)² so that its specific energy can be evaluated. The sampling time is related to the metric signature of its function (roughly its inverse). Meaning it adopts from GR the “space-time to length” convection, this requires that the Heisenberg energy-time uncertainty be limited to a small spatial region, for negative energy this limit is roughly near the Planck length. The side affect of this is that it modifies the limits in which energy calculations are made in GR, such that it requires an absurd energy requirement that happens to exceed the visible mass of the Universe. However if you modify the spatial geometry you can also affect the sampling time function, which reduces the energy requirements. This was explicitly proven by the ESAA group by the inclusion of a large lapse function into the energy tensor [31].

5.4 Exotic Energies Nullified

Ford and Pfenning have stated that Quantum Inequalities can restrict negative energy requirements within General Relativity, a somewhat haphazard conclusion with the known discrepancies between General Relativity and quantum theory. However, Kransnikov has brought this reasoning into question in the case of topological shortcuts in spacetime [36]. This is first done by considering a cylindrical spacetime which can be generalized to the Krasnikov tube and Alcubierre geometry respectively. If one distorts the cylinder, it transports null vectors along the traversed trajectory. Thus a suitable global means of topological travel becomes possible. This is further supported by a paper from Olum and Graham where to interacting scalar fields can violate QI’s near a topological domain (similar to the cylindrical tube generalization). From this, Krasnikov then goes on to argue if an electron with suitably small radius can have unphysically values does not mean this is the case. For example, if there is a different field present as there is in quantum field theory, the energy can be removed. Thus the unphysically large mass is a product of a trivial assumption and not a physically reliable result. Take the Big Bang era, say a non-trivial wormhole forms with signature

(5.19)

if enough negative energy is present by the new polarization the wormhole will hold open. However, if

Figure 5.3: Replacing a spherical geometry (white region) with hyperbolic scalar (yellow region) dramatically lowers negative energy requirements within spacetime.

there is not enough energy the throat region continues to collapse to some minimum radius until the proper polarization is reached. If the throat region is spherical (not taking the Lorentzian walls into affect) this creates several shells with of order thickness π(ξ)₀ << 1. What would happen if the throat was no spherical so the wormhole would have the following structure

ds²= dt2 − 4[(ε + r² )dr² + r² dψ² ] − (r²₀ − r² cos 2ψ)² dψ² (5.20)

that is the outer mouth of a wormhole is governed by the standard red-shift function while the inner tunnel is at as seen from Euclidean space. The Quantum Inequality restrictions only hold for the wormholes boundary and are not present in the inner region. This makes it possible to expand the inner radius of topological short cut to counter react semi-classical violations of General Relativity. What would once take more energy in the observable Universe only requires the energy of a typical supernova explosion. The energy violations can even be further reduced by applying these methods to the Van Den Broeck bubble to reduce the exotic energy requirement down to a few kilograms.

The reason why so little energy is required in this case is because a special modification was made to the spatial geometry. As we saw in Section 5.3, modifying spacetime geometry can modify so called negative energy requirements. We can now discuss this affect in terms of the Casimir Effect thanks to a work of Masafumi Seriu [13]. In this model we can calculate the amount of Casimir energy by measuring the amount of Dirac energy in spacetime which is coupled to matter by means of a scalar field:

(5.21)

by mapping this field onto a tours spacetime of order

ds² = − dt² + V dh_ab dξ^a dξ^b (5.22)

Seriu was able to find that Casimir Force exerted by the local matter was enough to modify the Lagrangians of a gravitational field. This puny amount of energy was even found to be strong enough to shrink the over all volume of the torus in question. Further still it was found that the generic scalar field could be generalized as the Cosmological Constant. Such that if this tiny amount of energy was applied to spacetime, it could modify the inner boundaries to be super at, such that the energy requirements lessen. Necessarily the inner region has a lower energy then the outer boundary held by infinity.

Chapter 6 The Mysterious Vacuum

“Empty Space is not so empty.” –Stephen Weinberg

In this chapter we further explore the strange world of Quantum Mechanics. It is often stated that “nothing” exists in a vacuum. This obviously outdated assertion is maintained by classical mechanics. There is always “something” in a vacuum in the voids of space -- e.g., there is always electromagnetic radiation present. This radiation can intern interact with matter and in tern matter can interact with spacetime. This is of interest because GR does have in its nature vacuum equations for the behavior of spacetime. This means that the vacuum can have unique properties which derive the properties of G_ab. We now explore how this vacuum (massless) energy can interact with spacetime and what new dynamics they reveal.

6.1 Electromagnetic Vacuum Stress-Energy Tensor

Maxwell’s Equations (see Section 7.3) can be written in curved spacetime in the form of tensors as

(6.1)

. (6.2)

From this tensor notation we can calculate the momentum of the energy j and represent an electromagnetic stress-energy tensor:

(6.3)

Thus by means of Eq. 1.4 radiation can contribute to the energy of a gravitational field, leaving it non zero. This means a gravitational field can exist even in the abyss of a vacuum. Further radiation interacts with gravitational fields differently than matter source. However they can contribute to matter sources by a Klein-Gordon field:

(6.4)

which can be represented in tensorial notation by:

. (6.5)

This is actually somewhat contradictory to the vacuum solution to the Einstein Field Equation, namely:

. (6.6)

This is only an apparent contradiction and not a real one -- as one must be careful as to what they mean by a vacuum solution -- for Eq. 6.6 a vacuum solution implies zero curvature. However, for T_ab it implies a massless source.

In short Eq. 6.6 implies that T_ab = 0 for a “vacuum solution”. While coherent in the sense of relativistic mechanics, things get rather nasty when we consider the affects of quantum field theory. So that vacuum solution for Eq. 6.6 can have values so that T_ab ≠ 0 and T_ab ≤ 0. This brings non-locality into the range of relativistic mechanics so that our assumptions of spacetime fail (although they can be recovered by Averaging Techniques -- see Section 2.2).

6.2 Vacuum Squeezing

Vacuum Squeezing discoveries by Morris-Thorne-Yurtsever (MTY) [34] suggest that a Lorentzian wormhole can be stabilized by creating a zone of negative energy in the wormhole throat, the place where it has maximum space-time curvature. They suggested creating the needed negative energy region by using a “parallel plate capacitor” made with a pair of superconducting spheres carrying huge electrical charges and separated by a very small gap, employing the Casimir Effect (see Section 5.2.1) to make the zone of negative energy by suppressing electromagnetic fluctuations of the vacuum and reducing the vacuum energy to a value less than zero. This same method could also be employed to generate a warp metric if that negative energy created by such a method was focused outward from the ship towards the rear region, while in the forward region a similar high energy EM field was focused. The high energy field in the forward region would simply add to the local ZPF energy, while the negative energy focused rearward would subtract from the energy. This could be formatted into stages to give us the desired effect of our warp metric which in all cases varies in strength backwards.

Figure 6.1: (Top) Waves of light ordinarily have a positive or zero energy density at different points in space. (Bottom) But in a so-called squeezed state, the energy density at a particular instant in time can become negative at some locations

To compensate, the peak positive density must increase. As an example, researchers in quantum optics have created special states of fields in which destructive quantum interference suppresses the vacuum fluctuations. These so-called squeezed vacuum states involve negative energy. More precisely, they are associated with regions of alternating positive and negative energy. The total energy averaged over all space remains positive; squeezing the vacuum creates negative energy in one place at the price of extra positive energy elsewhere. To compress a wavelength one needs some state function and the compression r of some wave function such that:

(6.7)

The nature of such a compression can then be governed by:

. (6.8)

Another proposed method involves something from nature being duplicated. The theory of quantum electrodynamics tells us that the vacuum, when examined on very small distance scales is not empty at all; it seethes witha kind of fireworks called vacuum fluctuations. Pairs of virtual (energy non-conserving) particles of many kinds continually wink into existence, live briefly on the energy credit extended by Heisenberg’s uncertainty principle, and then annihilate and vanish when the bill for their energy debts falls due a few picoseconds or femtoseconds later. These vacuum fluctuations can be squeezed in the same way that light beams or systems of atoms can be squeezed, and the result is a vacuum that has an energy less than zero. In other words, a region of negative energy of just the kind needed to produce a warp field.

In complex quantum systems containing many semi-independent quantum oscillators, the zero-point motion represents a quantum noise that is superimposed on any measurement of the system. If an attempt is made to cool the system by removing the heat energy, the zero-point motion represents a rock bottom, a limit below which straightforward cooling cannot go. There is a technique, however, for making the system colder. In Quantum Mechanics the energy and the frequency of a quantum oscillator system are interchangeable, differing only by a constant multiplier. Further, in the context of Heisenberg’s uncertainty principle, the conjugate variable to the frequency is the phase or -- in other words -- the starting angle for an individual quantum oscillation. Phase is difficult to measure and is usually ignored in characterizing complex quantum systems. However, it has its uses. Recently it has been realized that in many quantum systems the limits to measurement precision imposed by zero-point motion can be breached by converting frequency noise into phase noise, keeping the product within the limits dictated by the uncertainty principle while reducing the variations in frequency (and therefore energy). If this technique is applied to a light beam, the result is called squeezed light. Recent work in quantum optics using squeezed light has demonstrated that old measurement noise limits considered unbreachable can now be surpassed with ease.

Hochberg and Kephart [35] used a technique of General Relativity called a Rindler transformation to show that over a period of time the vacuum in the presence of a gravitational field is squeezed. They found that near com-pact gravitational objects like black holes, substantial squeezing of vacuum fluctuations occurs at all wavelengths greater than about the Schwarzschild radius of the object.

There are two important consequences of the results of Hochberg and Kephart . First, in the MTY work on Lorentzian wormholes it was found necessary to violate the Weak Energy Condition. It does not have the status of a physical law, but it is a condition that holds in normal situations. There were speculations that its violation might be in conflict with quantum gravity, making stable Lorentzian wormholes impossible. This is apparently incorrect. Hochberg and Kephart have demonstrated that the natural and inevitable squeezing of the vacuum as it evolves in a strong gravitational field is in violation of the Weak Energy Condition. This places not only warp metric work on a more stable ground, it also supports some of the findings that the only energy condition that seems to never be broken is the AWEC. It also raises the possibility that we could generate warp fields needed negative energy via quantum black holes and then amplify this effect. We have some in the field of physics predicting we may generate such objects soon in our accelerator experiments, which is not beyond the scope of the next 50 years. With the proper funding for research on all of these methods and more, it is possible this approximation itself could be cut in half.

6.3 Polarized Vacuum Approach to Photon Motion Through A Warp Field

In the weak field approximation the general relativistic curvature of spacetime can be mimicked by a vacuum having variable dielectric properties in the presence of matter (as conjectured by Wilson, Dicke, Puthoff and others) [25]. This Polarized Vacuum (PV) approach can be utilized to discuss the motion of photons through a Warp Field. The reason for this concerns the Stress Energy Tensor from original General Relativity, T_ab. That Tensor involves amongst other aspects the local dielectric constant of the vacuum or ZPF. The Dielectric Constant also co-controls the propagation velocity of an electromagnetic wave like light through it. The reason c = 3 × 10⁵ km/s in a vacuum is because that vacuum has a certain refractive index partly controlled by the dielectric constants value.

In other words, change the dielectric constant and you not only change the velocity of propagation for photons you also change the local Stress-energy-tensor. The following sets of arguments explains how this works. The equation that determines c in a vacuum or any medium is: . To derive the velocity factor of our medium where K ≠I is the dielectric constant of the medium and then

V_wave = 300×10⁶·v_f

to derive the velocity of photons in this medium. Since the velocity changes with the dielectric changes, the wavelength does also. The only thing remaining constant is the frequency. The formula for wavelength is:

Thus, we can now derive two issues, (1) that of a specific photons motion through our field and (2) that of the velocity of our craft.

The polarizable vacuum is based of the dielectric changes of the primitivity and permeability of a vacuum related to the speed-of-light one has K(c). The general structure of spacetime then becomes

(6.9)

with dx = dx₀/ √K. In the case of a spherically symmetric space, such as the Schwarzchild geometry

. As such K can be seen to modify the vacuum to produce spacetime curvature (and vise versa). If K = 2 in the forward region and K = 0.5 in the aft region, then overall v_f = 0.707c. However, to our outside observer a photon or a light beam passing through this warp field would display a motion that dropped from c to 0.707c in the forward area of the field and yet, in the rear region it would appear to travel at 1.414c. Yet when actual travel time was measured, the photon would have crossed the region at an average of 0.707c.

Now let's change velocity again with K = 2 in the forward region and K = 0.25 in the rear region. This yields a velocity of 0.707c in forward region and 2c in rear region with an overall velocity of 1.293c. For our outside observer, our photon would show a travel time faster-than-light. If we changed our photon to, say, an electron whose original velocity was 0.5c, it would show a travel time of 0.643c.

Thus, in one region of our warp field particles will travel less than c, while in the rear region they will travel faster than c. At some region along their path through the field, they will travel at c. This establishes another horizon to our warp metric. This Horizon concerns visual effects for an outside observer at rest and also for that of a person on our craft. Also -- as mentioned earlier -- you have the Hiscock Horizon [9] along with the outer horizon of the field and the inner horizon at the ship which form a Casimir Region with the warp field region in between these two [21].

Now as concerns wavelength for our photons they will undergo a red-shift in the forward region and a blue-shift in the rear region. At this point let's examine the cutoff issue which shall control this from runaway effects. G. Shore found that at low frequencies, the phase velocity for photons v_phase begins less than 1 showing birefringence instead of conventional subliminal motion. In the high frequency case, v_phase approaches c with a W−4/3 behavior. The focal point is the comparison of the calculation V_phase(W) via the effective action. He found evidence for the lightcone being driven back to K₂ = 0. You have actually a dual lightcone for the wave normal co-vectors where one set remains normal and the other is expanded. Thus a dual image forms with two timelines. The frequency range allowed by this is: Λ/L >> Λ/Λc >> Λc² /L². So superluminal photon action is possible between these ranges.

But an outside observer would view an odd situation where a mixed set of lightcones existed for the region. He would see the ship frozen at c in one set and exceeding c in a smeared out format in another. For the low range photons, the range would run from v ≤ c in a similar effect. Basically for a craft going to Warp, an outside observer would view two actions at once. He would see one image of the craft reaching c and another of the craft exceeding c in a blue-shifted smeared effect. This is because of the birefringence issue where two lightcones are generated in the wave normal co-vectors [22].

The question then becomes how does this effective cutoff effect our warp field. In the before-mentioned PV examination with each velocity change upward, the lightcone is expanded further in the rear region. At some point the expansion of the lightcone will exceed the frequency range allowed by Shore’s findings. At this point the field itself will be forced to a K₂ = 0 condition by a forced shift in the dielectric constant. Thus, even Warp Drive under proposed metric methods has an upper bound on velocity ranges. That range will fall within the confines where the AWEC is maintained as far as Casimir Effects go for generation of negative energy ZPF states. Also, in between this range as velocity increases, stresses due to gravametric distortion effects will increase and they would have to be engineered around for every increase in velocity.

Now going back to , if K = 0 then v_f = 1 which would violate the AWEC. So any value of 0 < k < 1 that meets the requirements of the Λ condition established by Shore can be utilized within the rear area of a Warp Metric. Basically, this also implies that no energy condition can exist in space-time where the energy condition of the coherent ZPF is less than zero with its normal coherent state seen as an above zero average. If we renormalize as most do and consider the baseline ZPF as zero, then no energy state for the coherent ZPF can ever reach its negative infinite point. This would also put some restraints upon Quantum Field Theory for probability states and show that renormalization has to take place in Nature by an inherent requirement of space-time. Thus, the AWEC is supported by this research findings and is an inherent quality of space-time itself. As such, the AWEC is the only energy condition that cannot be violated under special cases.

A close examination of Warp metrics shows that many different Horizons exist within them. We have the Outer Event Horizon at the edge of the field beyond which space normal lightcone conditions exist. Within this boundary and extending to the ship’s own frame, we have the Casimir Formed altered ZPF region in which Photons exist in either an expanded or collapsed lightcone condition. This forms the Propagation Horizon area. And in the Ship’s frame we have a restoration Horizon within which the lightcone is the same as that beyond the outer event horizon. This Horizon forms the inner Casimir junction. Following the Kerr Metric approach the outer and inner Horizons correspond to those of the Outer and Inner Event Horizons. The region in between is the same as that region found in Kerr Black holes where photon motion is dragged along by the rotation of the field altering their individual lightcones and their frequency state.

Indeed, we now know that for propagation in vacuum space-times (solutions of Einstein’s field equations in regions with no matter present, such as the neighborhood of the event horizon of black holes), there is a general theorem showing that if one photon polarization has a conventional subliminal velocity then the other polarization is necessarily superluminal. In fact, gravity affects the photon velocity in two distinct ways: first, through the energy momentum of the gravitating matter; and second, through the component of the curvature of space-time that is not determined locally by matter, the so-called Weyl curvature. It is this that produces birefringence. Such motion is genuinely backwards in time as viewed by an outside observer.

In Special Relativity, a causal paradox requires both outward and return signals to be backwards in time in a global inertial frame. In General Relativity, however, global Lorentz invariance is lost and the existence of a sufficiently superluminal return signal is not guaranteed. The quantum violation of the SEP certainly permits superluminal motion but with photon velocities predetermined by the local curvature. And that curvature also determines the local dielectric constant under PV methods. Thus, expanded lightcone regions showing birefringence are possible at not only the sub-atomic level but also in the macro world. We observe a similar v ≥ c condition in the refraction of light through differing mediums like say air and water. The fundamental difference is the above case would display v ≥ c birefringence motion with two separate lightcone conditions.

Investigations of superluminal photons in the FRW space-time show that photon velocity increases rapidly at early times, independently of polarization. Recent work on the rather different subject of cosmologies in which the fundamental constant c varies over cosmological time has shown that an increase in the speed-of-light in the early Universe can resolve the so-called ”horizon problem”, which motivates the popular Inflationary model of cosmology. The key to this whole subject being able to maintain causality is the issue of an expanded lightcone. With the lightcone expanded -- even though an outside observer views two time elements -- both sets remain causally connected. The at c lightcone is the cause of the superluminal light-cone, even though the superluminal one appears before the other. It's just that time is shifted for one lightcone ahead. An extreme case of such is actually encountered in quantum entanglement where an action at one point in space produces the same action in another non-causally (by c) connected region. The two events happen at the same time. But one is the cause of the other. This involves what is called Backwards Causation or -- in the case of entanglement -- instant causation. In both lightcones time remains causal. It is the outside observer who sees a time causality problem due to conflicting light cones. But the idea that one event causes another event remains in force. Thus, the problem concerns observation and not the actual flow of time backwards.

Before we go further into the issue of how to produce such an enlarged state, we wish to focus on an issue that will arise when we discuss the subject of a pulsed field or a continuous one. That being the entropy of a warp field.

Now the internal state of the warp field between these two horizons is associated with an entropy of S. Slightly simplified, S = ln(N) where for our region S = 1/4A. Now assuming there are n degrees-of-freedom within this system, the total number of spin-2 states for our warp field would be N=2n implying n = ln(N) / ln(2). If we allow -- for the sake of simplification -- that the internal entropy is Boolean, then we get the number N = A/4 ln(2). This would yield that the degrees-of-freedom per unit Plank area is of the same order as one Boolean d.o.f. The bound on this system then becomes S <= 1/4A.

However, in a generally flat FRW or under PV, there will be a size over which that last condition will be violated. Given that evidence exists for a boundary condition to be maintained, then an upper bound on fields of this order must be that in no case can S ≥ A with the general stable case being S <= 1/4A. In the modified warp metric of the ESAA-Hiscock-Holodoc version, the large internal surface area inside of the horizon and the lowering of the negative energy requirement assures a near stable case. This is because as A increases, then S can have a higher value within the two bounds with large values of A allowing S to fall within the stable range. Taking Mukhanovs theory [15] on nlack hole entropy and applying it to our Warp field metric, we find that its proportionality to horizon area is conditioned upon the degeneracy degree gn, the number of states |njmqs > with a common area eigenvalue an being given by gn=kn when k is some integer greater-than-1 or less-than-1 depending upon the specific area of the field involved. Following the quantization law [16], the horizon may be considered as parceled into n patches of area a₀. If each can exist in k different states, then the law g = kn is immediate [17]. The reason is self evident. At each patch there is not only the component of the individual field states stress energy density, but also the individual particles present within that vacuum patch add to the total state while an average could be assumed from a normalized vacuum state. The actual value will vary non-linearly.

To get a deeper understanding, let's examine some aspects of Photon Propagation in a Warp Field. For this we will establish some information.

Notice that in the first three terms k plays a role, which from our earlier work we know is partly controlled by the dielectric value. Thus, three aspects of a Photons motion are directly involved with changes in the dielectric constant. These are the phase, front, and group velocity. Now given that the Front velocity is greater than or equal to the Signal velocity which is greater than or equal to the group velocity and that under conditions of less-than-1 dielectric value, all would be greater than c in normal space-time. We must assume that all of these have values c >= c_norm. As such, since information carriers in part of a warp field faster than normal and since two lightcones are generated by our field, we have two velocities of energy and information flows. Yet, externally we have only one signal velocity regardless of the set. This is because the speeded up photon set -- while ahead in time -- drops back to space-time normal limits once outside of the field. The only tell tale of its passage through a different space-time will be its increased energy and its mirror copy behind in time. It is that timing space and its energy that encode the information about the space-time it traveled through. However, any photons that have passed the cutoff will appear as extreme high energy photons traveling at c without the mirror copy in time. In this case there is an information compaction due to the cutoff effects on that information.

Outside of abnormal energy levels, no solid encoded information on the space-time region it crossed is maintained. This leads to a short discussion of what would happen if a starship breached that cutoff barrier. Since the cutoff is the Modified Lorentz equivalent to the Light Barrier in normal spacetime, in theory no ship possessing mass could ever reach this barrier. But if it were possible, the craft would be forced backwards in time to an "at-c" condition. It would literally assume tachyon motion in its jump out of Warp. Since information loss would occur, it must be assumed information loss on material structure of the craft would occur. But since this new barrier is like the "before-c" barrier, from relativity the craft would appear to both space-time regions to be frozen in time.

We also have the other issues on entropy from our other discussions on photon motion. These can form + and - components to the above. So in essence the trackable signature of high-energy photons of a craft in warp would at best be an on-again/off-again occurrence. The strongest signature would be in high density areas such as gas clouds due to the added particle density of the local vacuum.

One noteworthy issue when it comes to the passage of photons from a warp metric region is the similarity theorem [19] of Fourier transform which states that compression of support of a function corresponds to expansion of the frequency scale so that lim_k→0E(ω≤c) / Ea(0) → 0. Above E(ω, =Ωc) with frequency is the electromagnetic energy of modes smaller than a fixed frequency, w ≤ Ωc which shows that high-frequency photons are less hindered by backscatter from the horizon than low-frequency ones. So the higher into Warp a craft goes the more it will display its high-frequency photon signature up to the cutoff of the warp region it is in. Thus some radiation from our field is trapped within the horizon while other high-energy portions are not. Basically, from an outside observer's perspective, there is a loss in entropy information due to the embedding of our hyper space-time region. But this is an observer issue just like the causality problem and not a "real" loss. This trapped radiation will appear again when our craft drops its warp field and re-enters normal space-time as a burst of mixed frequency photons.

This brings to mind a double energy loss issue. These extra trapped photons will tend to excite the local vacuum state and will require extra compensation in the field setup, as does the energy loss through cutoff effects as the faster one goes. Basically, we have a decay issue that obeys entropy. It is this effect we can either work with or work around.

Chapter 7 The Affects of Scalar Fields

“Elementary, my dear Watson.” – Sherlock Holmes

In this chapter we consider the affects scalar fields have when coupled to gravitation. As a rule any suitable gravitational topology must have some conserved energy condition which is not violated in order to be considered physical. We are aware of specific cases where this is violated in quantum theory, although as a rule it appears the AWEC always holds. However, these interpretations can be viewed as the generalist approach towards gravitation. There are also specific conditions one should consider.

First, allow us to generalize the role of a scalar field. Take a metal box and insert a metallic pendulum into the box. Next, create an external sound wave and have it interact with the box. This will create an acoustic vibration. Now begin swinging the pendulum such that it makes contact with the metallic box. When the pendulum hits the same spot, the sound wave does it forms a scalar field within the acoustic vibration. This then affects the way in which the sound wave behaves. It is really almost too elementary to think about, although when coupled to gravitation and other forces it appears to take on a “magic” role. (In the case of a sound wave, it may do nothing more than change the pitch of a note). And some believe scalar solutions are indeed magic solutions. But they are not; in fact, they are rather elementary solutions.

However, the existence of the cosmological constant reveals some of the bizarreness of scalar fields. Having an in-depth look at these scalar fields will allow us to consider new kinds of geometries and energy conditions not available through a generalist approach to GR. We now begin our look into how scalar fields interact with ordinary field theories.

7.1 The Anti-Gravity Nonsense

You will often times hear lone inventors claiming they have discovered “anti-gravity”, a supposed force which reacts negatively toward gravity. The closet thing to this is the cosmologic term Λ. However, this is not really anti-gravity as it is a scalar field coupled to gravity (something which reacts with gravity that modifies its assumed behavior). What these lone inventors are actually claiming is that objects appear to levitate through the electromagnetic force.

This is not too hard to do as the electromagnetic force is 10⁴³ times stronger than the gravitational force. So anti-gravity is not really a gravitational force. It's the absence of one! The whole concept of anti-gravity has a lot of errors; its descriptions show it's a flawed concept. For example, an EM field counter-reacting gravitational forces was once believed to be “presumably less at high altitudes and zero in vacuum”. The prior hypothesis was later disproved by Brown during experiments in France where it was shown that in fact the efficiency of his model "flying saucers" actually increased when own in high vacuum.

Now this makes some sense considering you have an altered dielectric condition in a vacuum state. Given that the inertia effect of the c limit climbs the higher the dielectric of the medium, there would be more inertia on an object in normal air than in a vacuum following normal theory. But the efficiency issue would still stand. The Biefield-Brown effect is simply an issue of EM as a force being stronger than gravity. It is not "anti-gravity" as many think. However, the scalar coupling issue transcends that issue altogether. We are not after a remake of the Biefield-Brown effect but simply using quantum effects to our advantage.

7.2 Scalar Fields

The massless scalar in KK and the Brans-Dicke scalar are not as far removed. If you have only gravity and a scalar field, there’s a lot of flexibility. That’s because you can do a field redefinition,

g_ab → f(φ) g_ab that moves the scalar coupling around in the action.

In Brans-Dicke theory for instance, there’s a term in the Lagrangian of the form φR but by redefining the metric to φ⁻¹ gab you can absorb this into R, at the expense of changing the “kinetic energy” term for φ. In string theory, such a choice of scaling is sometimes called a (conformal) frame. The scaling in which the purely gravitational part of the Lagrangian is just R -- with no scalar coupling -- is the “Einstein frame”. If the scalar couples to other forms of matter, the metric appears in the matter couplings as well and a field redefinition involving g_ab changes the couplings of matter to gravity. This can lead to a breakdown of the principle of equivalence. Different types of matter will couple to different effective metrics because of the different dependence on φ in each term. This leads to some important experimental constraints which essentially require that any Brans-Dicke-like scalar field be very nearly constant.

However, it does not eliminate such a field being engineered to be non-constant. If your matter fields happen to have a “conformal” coupling --that is, a coupling that’s invariant under local rescalings of the metric -- then you can do your field redefinitions without violating the principle of equivalence. In particular, electromagnetism in four dimensions is invariant under such rescalings -- the metric appears only in the form √g g^ab g^cd and it’s easy to check its invariance. This is why we feel the EM coupled Scalar is the key to generating a warp metric field.

Now, there is a well-defined set of equations called the Einstein-Maxwell equations that already unifies electricity and magnetism with gravity. They are discussed in standard books on General Relativity. Some of this is brought up in that slight modification to GR article. The stress-energy tensor for electromagnetism fits nicely into the right-hand side of Einstein’s equation. Maxwell’s equations generalize easily to curved space-time. But this partial unification doesn't solve Quantum Gravity issues, and it is only the scalar that transfers over in most viewpoints. The reason for this is the lack of noted bi-polar gravitic effects in Nature. However, this does not mean they do not exist. QM and QED are prime examples of some noted bi-polar effects as well as the Einstein-Bosen issue. The reason they don't appear globally in Nature -- except perhaps with Inflation and the Dark Matter issue -- has to do with the AWEC. Locally it can be violated and still hold globally. Nature has placed that constraint issue upon us. However, the key is the local issue which is what we are after here. We only want a Casimir-like local violation of energy conditions. Then it can still obey the AWEC. If we take

, (7.1)

when λ(x) > 0 then the above is T < 0 and the stress-energy density becomes negative with noted changes in the PV state of the vacuum. This is the Quintessence solution from theory. Altering λ gives us normal dark matter with positive gravity and the dark energy field that powers the accelerated inflation. It requires a non-classical phase of the vacuum state which - while scalar -- doesn't radiate as normal EM waves do. As such, it is non-detectable. Now a transverse scalar field component from EM would fulfill such an issue neing that it is transverse or TEM in nature. As mentioned before, unless its coupling was outside of the constrained range we would never notice its effects. But the Quintessence field is just such an abnormal coupling that would display itself through an altered vacuum state. Here is where observation evidence from nature may be our key to developing a warp field generator.

Now we wish to back tract a bit and explain a commonly assumed error concerning the direction of an EM fields propagation. Consider a magnetic dipole lying in a plane -- say, the x-y plane -- and a charged particle moving in the same plane. Because of the dipolar nature of the magnet, it’s a little hard to attribute a single “direction of propagation” from the magnet to the particle. Logic says every straight line between the magnet and the particle lies in the x-y plane. So presumably if the direction of propagation and the direction of acceleration are identical, the net acceleration of the particle should also be in the x-y plane. But this isn't the real case. The magnetic force on the particle in this set-up points entirely in the z direction -- perpendicular to the plane -- with a sign that’s determined by the charge and the direction of the particle. We submit that it follows the scalar's transverse field. We don't detect it normally because all EM detection methods are set to determine the amplitude changes in the x-y plane.

Then if the scalar is the carrier, it follows that this carrier is the underlining fundamental element. Since the scalar is minus the x-y plane, it must be of a 2D format designated by a z-t plane. Thus, we have come full circle on this back to M-Theory where the fundamental unit of Nature from which all particles -- some of which are the carriers of force -- are comprised of vibrating 2D manifolds. Yet to achieve their vibration, one has to have the x-y aspect to account for amplitude and the z-t aspect to account for frequency. To account for all the symmetry changes needed, one has to have a multidimensional frame. But more on this later.

Now, there may be a relation between the identity here with the scalar and the long sought Higgs field. This would then provide an answer to why we never see the Higgs particle, just its effects. One cannot detect it by normal means unless it displays some odd effect in the structure of space-time like the arrow of spin, or in extremal cases with the dark energy issue. It is otherwise embedded and hidden from detection.

Now inflation as introduced by Linde and Guth is “only” a scalar field minimally coupled to gravity. The evolution of this field (slow roll) in a space-time background -- which is often a FRW metric --provides an equation for the expansion factor in the metric a(t). The growth of the Universe during the inflationary era is of the order: N = ln[a(t)/a(0)] > 60 and it solves a lot of problems. In conclusion, inflation is not a space effect; it is a space-time effect even if in the metric only changes the factor in front of the spatial's terms of FRW metric. But the inclusion of a non-minimally coupled scalar to gravity would change the effect. Instead of a slow roll in a space-time background, you get a rapid expansion over a short period. If we can balance this fast roll with a fast decay due to Hawking bleed off, we get a continuous warp field. If not, then a pulsed mode will have to be utilized.

7.3 Maxwell's Equations under Vacuum Conditions

Imagine any closed surface in 3D physical space. The net static electric flux through that surface is the two-dimensional integral of the scalar product E▪dA of the electric vector E at the area element dA with the area element.

The direction of the vector area element is normal to the actual infinitesimal element of area of the surface. The total charge inside the surface is the three-dimensional integral of the charge density multiplied by volume element dV.

(7.2)

Gauss’s law in integral form says that the total electric flux through the closed surface is proportional to the enclosed electric charge. In local or differential form, Gauss’s law says that the divergence of the electric field at a point in space is proportional to the charge density at that point.

The original static law of Ampere (in integral form) was that the ”circulation” (i.e., 1-D closed loop line integral of the scalar product of the magnetic field vector B with the infinitesimal tangential line element dl) is proportional to the electric current passing through any interior 2-D surface which has the closed loop as its boundary.

(7.3)

Note that this law is essentially topological because one can distort the shape of the surface without tearing it without changing the result. In fact, as shown by John Wheeler, all of Maxwell’s electromagnetic field equations have the same profound topological foundation that ”the boundary of a boundary is zero”.

Maxwell’s stroke of genius was to add the displacement current term which is the rate of change of the electric flux through the same surface. This implies there can be an effective electric current even in empty space where there are no actual charges. It is this displacement current which explains how transverse electromagnetic waves of radiation can propagate to infinity. Here again my key focus on the scalar issue appears. In local differential form, this Maxwell equation says that the vector curl of the magnetic field at a point in space at a given time equals the material vector electric current density plus the partial time derivative of the electric vector at that same point in space and time.

(7.4)

Faraday’s Law is what makes generators possible. In integral form, the circulation of the electric field around a closed loop is proportional to the rate of change of the magnetic flux through any surface whose boundary is the loop.

In local differential form, the curl of the electric field at a given place and time is proportional to the partial time derivative of the magnetic field vector at that same place and time. To make an electric generator, simply rotate a metal wire loop in a static magnetic field such that the surface with the loop as boundary is ant different angles to the direction of the magnetic field. Therefore, the magnetic flux through that area is changing in time. This causes an electric field in the wire which pushes on mobile electrons in the metal causing a periodic electric current.

One can also make an electric motor by hooking the loop to a battery. Now we need some additional ideas. First the Lorentz-force Law which says that a magnetic field exerts a force on a moving charge that points at right angles to both the direction of motion of the charge and to the direction of the magnetic field. In fact, this Lorentz force is proportional to the vector product of the magnetic field vector with the charge velocity vector all multiplied again by the size of the charge divided by the speed-of-light in vacuum. One can then see that a force couple or torque is made on the current-carrying loop in the magnetic field causing the loop to rotate.

(7.5)

Electric charges are electric monopoles. No analogous magnetic monopoles have ever been observed -- only magnetic dipoles in which the positive magnetic charge is separated from the negative magnetic charge. One of Maxwell’s equations is that the total magnetic flux through any closed surface is zero. This is the magnetic analog to the electric Gauss’s Law, but now it says that there is always zero magnetic charge. Some Grand Unified Theories predict magnetic monopoles, but so far no one has seen them. String Theory has the added beauty of being nearly magnetic monopole free.

The equations then read as Maxwell’s four equations:

(7.6)

(7.7)

(7.8)

(7.9)

where H is the magnetic field (A/m), E is the electric field (V/m), j is the vector current density (A/m²), ε₀ = 8.8542x10−¹² F/m is the permittivity of free space, µ₀ = 4ˇ × 10⁻⁷ H/m is the permeability of free space, and ρ is the scalar charge density (C/m³ ). The ρ value is our focus here. By a theorem of vector calculus, if and only if there exists a vector field A such that . A Call A the vector magnetic potential.

We will find a characteristic solution E_c and a particular solution E_p. The total electric field will then be the sum E_c + E_p.

Characteristic Solution: . By another theorem of vector calculus, this expression can be true if and only if there exists a scalar field φ such that . Call φ the scalar electric potential.

Particular Solution: , and allows us to write . We may obtain a particular solution E_p by simply choosing E_p = − µ₀δA/δt.

Total Electric Field: .

Let us now substitute the expressions derived above (7.10)

and from Eq. 7.6, we obtain . We have the identity , so that the equation we derived above becomes

(7.11)

or after some simplification,

(7.12)

since may be manipulated by taking the curl of both sides. The terms involving the gradient then vanish leaving us with the identity

(7.13)

from which we may infer -- without loss of generality -- that or . We recognize that ε₀µ₀ = 1/c² where c is the speed-of-light. We now introduce a new operator

□² = (7.14)

so that □² A = j , from which from which we may infer -- again without loss of generality -- that

(7.15)

by distributing the operator . The relation becomes

(7.16)

Substituting for simplifies to

(7.17)

0r recalling that ε₀µ₀ = 1/c² and using the operator □²ψ = − ρ/ε₀ is derived. There is then seen a very close tie to the that 4ε₀µ₀ = 1/c² issue. If you affect this value, you change the whole vacuum solution. There are two known possible methods by which to accomplish this. One is to let the scalar affect it. The other is to change the ZPF's energy level.

Now, if φ is raised while ε₀µ₀ = 1/c² is constrained the value of □²ψ will increase. But if ψ is lowered then □²ψ will decrease. If we instead constrain P and lower ε₀µ₀ = 1/c²□²ψ to increase and an increase in ε₀µ₀ = 1/c²would decrease □²ψ. So the effects -- while similar --are reversed under a Polarized Vacuum approach.

If we focus on the one value of ε₀µ₀ = 1/c² that can be varied -- its dielectric -- then we can alter the vacuum solution at will. The reason goes back to what provides the vacuum with an energy density in the first place -- its virtual particle components. If you want inflation, you must cause the field to expand in opposite of the effect gravity has. This is the key to anti-gravity effects. If you want an increase in gravity, you must increase the energy density present. The two together give us a bi-polar space-time field from which we can generate a warp metric.

Chapter 8 EM ZPF Force

“The Casimir Effect is a real observable quantum force viewable in laboratory.” –Michio Kaku

Whether in the form of radio waves or visible light or virtual particles or real particles, all electromagnetic energy is inherently similar. The behavior of electromagnetic energy is governed by basic wave theory, which is described by Maxwell’s equations in a background described by General Relativity. These equations describe electromagnetic radiation as traveling at a velocity c equal to 3×10⁸ m/s in a sinusoidal, harmonic fashion. The electromagnetic wave is propagated in a direction perpendicular to the electric and magnetic fields. According to this, electromagnetic waves are characterized by amplitude, wavelength, period, frequency, and velocity.

Remember that wavelength of an electromagnetic wave is the distance between crests and the period of an electromagnetic wave is the time between crests. The velocity of an electromagnetic wave is the wavelength (distance) divided by the period (time). This is not an especially useful description of velocity because many applications of remote sensing make reference to wavelength and frequency, not wavelength and period. However, since the definitions above tell us frequency is the inverse of the period, it is not too difficult to express velocity in terms of frequency and wavelength v = λ/T = λf: A very important point to make at this time is that this relationship holds for all waves. Velocity always equals frequency times wavelength. Beyond this, electromagnetic waves are a special case because the velocity of electromagnetic waves is essentially constant. Electromagnetic energy travels at the speed-of-light governed by the formulas mentioned above.

8.1 EM Field in gauge theory

Now in EM theory and in general wave model theory there is something called coherence and decoherence. But before we explain this, we will introduce the relations between Ghost Scalars and the EM connection. In the Standard Model, the electroweak interaction is described in terms of the SU(2) × U(1)SU(2) symmetry which has n²−1 = 3, T1, T2, and T3. The generators follow the anti-commutation relation [T_i, T_j] − i ε_ijkT_k. In the fundamental representation, the generators are given by the Pauli spin matrices:

(8.1)

Furthermore, one defines T_± = T₁ ± iT₂/ √2 leading to

[T₊, T₋] = T₃, [T₃, T_± ] = ± T_± , T₌T₊⁺ (8.2)

The covariant derivative in the electroweak interaction is

D^µ = δ^µ − ig(W^µ₁ T₁ + W^µ₂ T₂ + W^µ3 T₃) − ig'B^µ Y. (8.3)

Here, g is the SU(2) coupling constant and g' is the U(1) coupling constant. The U(1) charge Y is called the weak hypercharge. B^µ is the U(1) field, and the three W^µ fields are the SU(2) fields W^± = − ( ± W₂ − W₁) / √2.

Instead of g and g' one usually applies the electromagnetic coupling constant e and the angle θ_w, defined through

(8.4)

SU(2) is a non-Abelian group. It contains infinite contributions to the propagator arising from certain diagrams. ’t Hooft has proven that SU(2) can be renormalized. In order to do so, one needs to choose a specific gauge. The gauge-fixing part in the Lagrangian and the associated ghost fields are discussed next because it is these Ghost fields that we want to focus on. The quadratic parts in the Z and χ fields in the Lagrangian reads

(8.5)

One now fixes the gauge by using the term in the Lagrangian

(8.6)

Here, the real positive parameter η_Z relates different gauges. In the Landau gauge it is chosen 0; in the Feynman gauge it is 1. Measurable quantities should be independent on the choice of gauge. Adding this gauge-fixing term to the Lagrangian, we see that the terms for χ yield the usual propagator for a scalar boson with mass mixed terms with both χ and Z are removed by integration in parts. The remaining terms yield the propagator for the Z-boson. T he second part contains a pole on the unphysical squared mass η_Z m_Z² . Its effects must cancel out with the effects of the propagator of χ and of the ghost fields. For the W-sector, the same procedure yields also the usual charged scalar propagator for the ф^± particles with a mass-squared of η_wm_W² and a W propagator with an unphysical part with mass η_wm_W² which has to cancel in all observeables. For the photon, one has the gauge-fixing term

(8.7)

The gauges for the Z, W, and A fields may be chosen independently (the η-factors are three -- possibly different -- real non-negative numbers).

Figure 8.1: Propagators of the gauge bosons

The second step in renormalizing the theory leads to the introduction of ghost fields. These are mathematical constructs obeying a Grassmann algebra. The procedure of fixing the gauge and introducing ghost fields is necessary to remove contributions to the action of

(8.8)

that are only connected via gauge transformations. The action is given by integrating the Lagrangian density for all possible paths and all possible field configurations.

Figure 8.2: Scalar Propagators

In the formula above, ∫DA, ∫DW, ∫DZ stand for the functional integrals over the gauge fields. Since a gauge transformation leaves the Lagrangian invariant, one wants to remove these infinite contributions. The ghost fields are non-physical fields since they follow Fermi-Dirac statistics but have boson propagators; they can only enter inside loops in Feynman diagrams. In this since they share a similar nature with the virtual particles. Yet, we now know they actually exist. So perhaps these scalars do, too.

Figure 8.3: Propagators of Ghost Scalars

8.2 Ghost Scalars

Consider a Ghost Scalar field denoted by

(8.9)

∂_µ implies differentiation with respect to x and ∂_ρ implies differentiation with respect to x'.

The operation of Hermitian conjugation transforms a Ghost Scalar field to itself whereas an anti-ghost is transformed to itself with the opposite sign. Thus the rules of Hermitian conjugation ghosts are

(G.c)⁺ = G.c

(G.C)⁺ = − G.C (8.10)

(W + .C)⁺ = W − .c

(W + .C)⁺ = − W − .C

(anti)ghosts are scalar, anti-commutative fields [23]. The nonzero propagators for these fields are:

< 0 |T[A+.c(p₁), A.C(p₂)] | 0 >=< 0 |T[A+.C(p₁), A.c(p₂)] | 0 >= ScPr(p₁, p₂, M) (8.11)

where M is a mass of the prototype particle. This equality of masses is a consequence of the choice of the t’Hooft-Feynman gauge.

The appearance of the Ghost Scalars in gauge theories can be explained in the following way. In the case of the t’Hooft-Feynman gauge, the quantum field of massless gauge particle has two unphysical components because the four-dimensional vector field describes a particle with two polarization states. The contributions of Faddeev-Popov ghost and anti-ghost compensate the contribution of these unphysical polarizations.

Now before we discuss the usage of Ghost scalars in creation of warp metric fields, the scalar components of regular EM fields must first be discussed. Whittaker in 1904 [24] showed that all EM fields and waves can be decomposed into two scalar potential functions. For the T.E. degrees of freedom, the components of the charge-current four-vector are

. (8.12)

The components of the T.E. vector potential are

(8.13)

and those of the e.m. field are

(8.14)

These components are guaranteed to satisfy all the Maxwell field equations with T.E whenever satisfies the inhomogeneous scalar wave equation

(8.15)

The charge-current four-vector S_µdx^µ = S_t +S_zdz +S_xdx +S_ydy for each set determines and is determined by appropriate derivatives of the scalar source S(t, z, x, y). Similarly, the vector potential A_µdx^µ = A_tdt + A_zdz + A_xdx + A_ydy as well as the electromagnetic field

≡ Ê_long dz Λ dt + Ê_y dy Λ dt + Ê_x dx Λ dt + (8.16)

B^{^}_x dy Λ dz + B^{^}_y dx Λ dz + B^{^}_long dx Λ dy

determine and are determined by the scalar wave function ψ(t, z, x, y) with the result that the Maxwell field equations are satisfied whenever this is valid. For the T.M. degrees of freedom, the source and the electromagnetic field are also derived from a solution to the same inhomogeneous scalar wave. However, the difference from the T.E. case is that the four-vector components of the source and the vector potential lie in the Lorentz (t,z)-plane. All the corresponding T.M. field components are derived from the scalar ψ(t, z, x, y). These components are guaranteed to satisfy all the Maxwell field equations with the T.M. source whenever ψ satisfies the inhomogeneous scalar wave equation. Thus, the two scalar potentials Whittaker introduced. There are also the T.E.M. degrees of freedom. For them the Maxwell four-vector source

(8.17)

is derived from two functions I(t, z, x, y) and J(t, z, x, y) scalars on the 2-D Lorentz plane and the 2-D-Euclidean plane respectively. They are, however, not independent. Charge conservation demands the relation

(8.18)

has the same form, but only the difference Φ − ψ is determined by the field equations. Indeed, the T.E.M. field components are derived from this difference for E_long : F_zt = 0:

(8.19)

and for B_long : F_xy = 0

(8.20)

This e.m. field satisfies the Maxwell field equations if any two of the following three scalar equations,

(8.21)

are satisfied. The last equation is, of course, simply the conservation of charge equation. Furthermore, it is evident that the T.E.M. field propagates strictly along the -axis, the direction of the Pointing vector. Now compare to the Ghost Scalar which has propagation in the X-axis. However, this axis was a matter of choice. If we switch it to the z-axis, then the two fields have similar properties.

Now, Hisa-Aki Shinaki and Sean Hayward in a recent article considered the effects of EM scalar fields and Ghost fields upon the Morris-Thorne Solution to Wormholes [24]. Basically, following their line one can consider both wormholes and warp metrics as black holes under negative energy densities. The only question has always been what defines this negative energy and how could such be generated. Related to this is an issue -- first brought forward by such men as Pfenning [25] -- of concerns over what is called the Weak Energy Condition (WEC) and the Average Weak Energy Condition (AWEC). Work on Warp Metrics has answered many of those concerns [26], and indeed the only real energy condition that appears to never be violated is the AWEC [27]. In this article they treated this solution using a 4D regular GR approach with numerical simulation. What they found was that the presence of a mass -- possessing normal EM-like qualities -- can collapse the actual field. But the introduction of an EM field with a negative scalar can not only keep the field stable, but it also can also expand such a field.

Given the current model of virtual components pictures them as wormholes where the virtual particles appear and disappear [28], it also follows that an EM field with a negative scalar focused upon such could expand that wormhole around a craft via inflation methods. This, when viewed from their perspective of effects of both positive and negative scalars lends itself to an engineered approach to Warp Drive. This is exactly the transtator effect we first proposed in http://groups.yahoo.com/group/alcubierrewarpdrive2/files/On. Only here the ZPF around a craft is altered into the proper metric in a pulsed mode that allows for the required collapsing pulse from violations of the WEC. Instead of enlarging just one virtual particle into a stable wormhole, we now alter the space-time region around a craft into a similar stable solution. We keep the region in a freely altering collapse/expansion state at a fast enough rate that -- even though the craft is moving forward along the axis of motion in a quantum step mode --the craft itself experiences this motion as if it were a continuous motion.

What is involved here is Gaussian Pulses can influence a Klein-Gordon field. The exact breakdown of an individual virtual particle is as a miniature Klein-Gordon Field following the Compton Vortex model [29] and our own String Theory-derived version [30]. Now, if you treat the Klein-Gordon Field as a simple white Gaussian noise field of overlapping short duration pulses and we focus on the virtual electrons, we then find these to have an average de Broglie wavelength and frequency. Now to add in an equal Gaussian field whose phase is sin(2πfct−180) instead of sin(2πfct), one finds that the two fields will cancel out leaving a negative energy condition as compared to a normal coherent vacuum state. If the -180 is exchanged for a field whose amplitude and phase operates on a constructive basis, then the overall energy of the vacuum will be raised locally within a sphere around the craft whose radius is denoted by the range at which the Gaussian signal drops below the normal ZPF's level following the 1/r² rule.

The Uncertainty Principle permits these virtual particle events to occur as long as they are completed within an extraordinarily brief period of time, which is of the order of 10⁻²³ seconds. This is what controls our Transtator field's total sweep time. The frequency range is determined by the upper and lower bounds on the energy average for virtual electron positron pairs when figured back into de Broglie wavelength for them. The phase reversal is simply assumed to be 180 degrees reversed from the plane in all directions from the craft itself at that point of reference given a uniform spherical field radius with a boundary formed when the field strength drops below the cutoff to be effective by the 1/r² rule.

Since the vacuum permeability and permittivity are also energy-related quantities, they are directly proportional to the energy per unit volume (the energy density) of the ZPF. It follows that if the energy density of the ZPE ever increased, then there would be a proportional increase in the value of both the permeability and permittivity. The reverse holds for lowering it. It is established physics that the velocity of a wave motion squared is proportional to the ratio of the elasticity over the inertia of the medium in which it is traveling. In the case of the vacuum and the speed-of-light c, this standard equation becomes c² = 1/(UQ). As noted above, both U and Q are directly proportional to the energy density of the ZPE. It therefore follows that any increase in the energy density of the ZPF will not only result in a proportional increase in U and Q, but will also cause a decrease in the speed-of-light c. Here again, the reverse also holds.

Now, since we can only effect those virtual particles in our plane we established, we can at best only lower the total energy by a certain amount. Once more we encounter a engineering problem. However, circular polarization methods can increase our effectiveness to make up some difference here. But even taking this 1/4 energy value gives us some ability to engineer the vacuum. The reason is that the only aspect under PV models that effects acceleration is that in the plane. This is seen in Casimir effects when the measurement of the negative energy state within is based upon the plane of the plates. So our 1/4 is actually a high drop in the total available energy of the ZPF given our field is spherical and not flat.

Returning now to our figure of 10⁵² ergs/cc and taking 1/4 of this, we see we can effectively in conservative terms alter the ZPF within a range of 2.5x10⁵² ergs/cc. I f we even follow the renomalized approach where the ZPF cancels to 10⁻¹²⁰,we still have a wide range we can utilize even with only the virtual electron positron pairs canceled.

Turning now to a Morris-Thorn Solution to Wormholes, taken from http://home.aol.com/zcphysicsms/wormholes.htm, The Morris-Thorne Wormhole can be expressed by the following interval:

(8.22)

w is allowed to run from −∞ to +∞ . r is a function of w that reaches some minimal value above zero at some finite value of w. In the limit that w → ±∞ , r = |w| . Φ(w) is just a gravitational potential allowed by the spacetime geometry. Traveling along the -w direction, one falls ”inward” toward the hole with decreasing r until reaching the minimal value of r. This minimal value is the radius of the wormhole throat. Continuing along the -w direction, one finds himself falling “outward” again away from the hole again with increasing r. This type of wormhole has great advantages over the space-like wormholes associated with black holes previously discussed. For instance, there is no event horizon associated with this wormhole. The coordinate speed-of-light is nowhere zero. Because of this, remote observers can observe travelers cross from one external region to the other. Also, with this kind of wormhole timelike geodesics connect the different external regions. So one needn’t travel at locally faster than c speeds to transverse the wormhole [31].

This type of wormhole solution lends itself to a PV method very well. If we consider the metric denoted changes in space-time structure as changes in the local dielectric of the coherent vacuum state, then creation of this space-time region could be accomplished along EM methods. Turning to the Stress Energy Tensor of this region, we find that

(8.23)

(8.24)

(8.25)

(8.26)

All other T^µν = 0. This solution -- though based upon a Kerr Metric -- is basically the same as a regular Warp Metric. In one region you have the compression of space-time, accomplished via an increase in the local dielectric constant. In the other region you have an expansion of space-time, accomplished via a decrease in the local dielectric constant. If a craft was placed in the center of the minimal area of r and its field generators denote an added boundary along Casimir lines within which the space normal vacuum solution is re-stored, then this solution could be modified into a Warp Drive. So in essence there does exist a decent Kerr type metric solution to Warp Drive. In fact the central r region can be opened up under this format to the same as found under regular warp drive.

Now back to the concept of coherence and decoherence. The phase of each component must be chosen such that the waves interfere destructively to give zero amplitude within the range of the Transtating Warp Field Generator. They must also be forced to quantum tunnel at a rate that places them ahead in time of the field flow in any given region. This is to ensure our transtating signal moves faster than the region flows so the effect is created ahead of the flow it actually generates.

It has long been known that a quantum tunneled EM wave can during its tunneling process exceed the velocity-of-light across the same given region. We are not after a constant faster than light signal. We are after a signal that gets enough of a boost ahead in time during its tunneling process so that when it emerges it is ahead of the same signal in regular motion and ahead of the movement of space-time through the region in question. If the phase of the transtating signal is proper, there will occur a canceling of the ZPF virtual particles of the same frequency within the range of the signal. That canceling will in theory produce the same effect a Casimir plate would have on the local field. The energy density will lower for a brief instant in time to whatever level we have chosen to create our desired warp metric region. We will then have bi-polar flow of space-time as long as that effect holds. But since two basically EM waves can pass through each other -- only canceling for a brief instant in time -- we only get a pulse effect bi-polar flow. This is one reason this type of field must be pulsed.

The other concerns the return pulse issue from a WEC violation. This collapse stage is used to restore normal space-time conditions to release the trapped photons from the rear region before they can bleed off as regular Hawking radiation and collapse the field normally. So the craft will then move forward in quantum steps whose rate can be high enough to give the effect of continuous motion.

It must also be remembered that what will appear to the craft as a large internal region is simply a thin shell of altered space-time around the craft. Everything that we must create in this effect will be within that thin quantum flux shell region. This is because as the signal propagates outward its signal passing through the virtual particles will only produce the desired effect in that local region (remember the effect only exists as the fields move through each other). Thus, the thin shell issue.

However, in that instant we will be creating an altered space-time region which becomes embedded due to motion of space-time itself. The ship -- though in free fall -- will move forward at the rate of local space-time determined by the dynamics of the metric itself as if it was inside an enlarged quantum state. Indeed, that is exactly what it will be inside: an envelope of space where Planck's constant has an enlarged local value. This is why we discussed the Ghost Scalar issue and its ability to enlarge a quantum state. By mimicking that type of field we could, in theory create just such an enlarged region and take advantage of its effects.

Now, returning to our original discussion of Scalar Fields, we shall examine what effect an engineered scalar EM wave would have upon space-time. Consider one difference is the x-plane versus the z-plane. If we were to alter the z-plane to that of the x-plane, we would have some effect of that EM Scalar upon the Ghost Scalar. But these two cannot be turned into each other by any simple operation. They are fundamentally different scalars. So a Ghost Scalar field cannot be generated with an EM signal. The best you could hope for is some desired effect of the combination of the two fields. However -- as mentioned early -- the sole function of a Ghost Scalar field is to cancel infinites. Applied to a spin two field coupled to a Ghost Scalar, you might lower gravity within a region with its additional energy but it would depend upon the coupling involved. The normal non-minimal coupled scalar to gravity takes the form

(8.27)

which isn't far removed from Einstein's field equation for gravity with the cosmological term added. If −V(ψ) was added to with the Ghost Scalar combined with an EM Scalar, you would only complicate the matters by an additional two scalar fields. However, if −V(ψ was the additional term in cases where the scalar was positive, you would have a subtraction of energy from the field; and when the scalar was negative, you would have an increase in the scalar effect. Since we found before that the scalar can alter the local Stress Energy, this would give us a means to alter gravity. However, if the actual metric for gravity coupled to a scalar was

, (8.28)

you would have the cosmological term tied to the EM energy of the ZPF which is the only field possessing charge that doesn't zero globally. If you remember the range allowed by a scalars variance earlier you will find that in most conditions, except perhaps in superconductors or Einstein-Bosen states the EM scalar never transcends this range.

Chapter 9 Quantum Entanglement

“The Lord is subtle but not malicious … … I take that back -- maybe He is malicious.” –A. Einstein

Many of the technological developments of the past century are based on the principles of a strange theory called Quantum Mechanics. This is true for lasers, nuclear reactors, semiconductors and many other systems that have changed our lives. Apart from these technologies, the advent of Quantum Mechanics introduced a new way of looking at Nature, attracting the attention of philosophers as well as scientists. Quantum Mechanics predicts the existence of entangled states that have counterintuitive properties. These states can involve two-or-more objects. And their unusual properties mean that when we perform a measurement on one object, the others become instantaneously modified wherever they are! Entangled states are needed for teleportation experiments -- in which one can transfer the (quantum) properties of one object to another in a different location -- and in quantum communication and computation. To date, entangled states of a few atoms and photons have been prepared in a number of experiments. But some applications need much bigger entangled objects, formed by a large number of atoms.

Particles small enough for their behavior to be governed by the laws of Quantum Mechanics can exhibit a remarkable property known as entanglement. A pair of quantum particles can exist in entangled "superposition", a mixture of states that resolves only when some physical property such as spin or polarization is measured. Quantum entanglement is a fundamental requirement for quantum computing. But until now it has been possible only to generate entanglement between microscopic particles. Using a new method of generating entanglement, an entangled state involving two Macroscopic objects -- each consisting of a Cesium gas sample containing about 10³ atoms -- has now been created. The entangled spin state can survive for 0.5 milliseconds -- a long time in this context, --bringing practical applications of quantum memory and quantum teleportation a little closer [34].

9.1 Bell States

As shown, when an off-resonant pulse is transmitted through two atomic samples with opposite mean spins J_x1 = − J_x2 = J_x, the light and atomic variables evolve as

Ŝ^out_y = Ŝ_y +α Ĵ_x12,Ŝ^out_{z =}Ŝⁱⁿ_z(9.1)

Ĵ^out_y1 = Ĵⁱⁿ_y1 + β Ŝⁱⁿ_z, Ĵ^out_y2 = Ĵⁱⁿ_y2 + β Ŝⁱⁿ_z, Ĵ^out_z1 = Ĵⁱⁿ_z1 , Ĵ^out_z2 = Ĵⁱⁿ_z2 (9.2)

where S and J are constants. Eq (9.1) describes the Faraday effect (polarization rotation of the probe). Eq (9.2) shows the back action of light on atoms -- that is, spin rotation due to the angular momentum of light. As a result, the sum of the z components and the sum of the y components of the spins of the two samples are known exactly in the ideal case, and therefore the two samples are entangled according to the condition specified by

δ J²₁₂≡ δ J²_z12 + δ J²_y12 < 2 J_x (9.3)

Now in physics there is something commonly referred to as Bell's Inequality. When Bell published his inequality, he argued that it showed that in order to be consistent with experimental data, any theory of quantum physics must abandon at least one of the following: (1) the objective reality of the properties of subatomic particles or (2) Einsteinian locality, which is to say the requirement that no influence can travel faster than light. However, for any conclusion that someone has reached on this topic, another has come along and contradicted them;. Fr example, in 1982 Arthur Fine argued that abolishing objective reality wasn’t enough to escape from the inequality. Willem de Muynck has been arguing since 1986 that abandoning locality doesn’t help. And Thomas Brody argued from about 1985 onwards that Bell’s inequality doesn’t require us to abandon either of these things.

First, let us review the role of time in Relativity theory because it is this that underlies the strangeness of abandoning Einsteinian locality. Einstein suggested that time should be viewed as another dimension, with much in common with the three spatial ones we are all familiar with. However, the relations between time and the spatial dimensions are not quite the same as the relations amongst the spatial dimensions themselves.

Rotations in space will always conserve any length r, which can be found from Pythagoras theorem: r² = x² + y² + z² where x, y and z are distances in the three spatial directions respectively. What they will not conserve is distance in a particular direction. In order for these facts to be true, it can be shown that the familiar trigonometric relations must apply. That is, if we rotate an object a degrees about the z-axis, we can write x = r sin a and y = r cos a.

Now, in Relativity theory we find relations which are closely related to these but have rather different consequences. The crucial difference is that "rotations" in spacetime conserve -- not distances -- but space-time intervals which are found from a slightly modified version of Pythagoras’ Theorem:

R² = x² + y² + z² − t² . As you can see, the time difference between two events subtracts from the space-time interval. Just as it can be shown that the trigonometric relations must follow to conserve distances, it can be shown that in order to preserve a spacetime interval when something accelerates (a "rotation" in spacetime), their close cousins -- the hyperbolic trigonometric functions -- join in. In many senses, the hyperbolic and standard trigonometric functions are very similar; the mathematical analogies between the two groups are very powerful (all the trigonometric identities -- as well as the series expansions -- are the same in hyperbolic geometry apart from sign changes).

However, in practice they have quite different effects. If one keeps increasing an angle, for instance, one quickly gets back to the starting position (one full turn). The hyperbolic functions are not periodic like this. Instead, as the hyperbolic angle increases-and-increases -- which means that an object is getting faster-and-faster -- what we get are ever-shrinking spatial separations and ever-increasing time separations. These are the Lorentz-Fitzgerald contractions -- time dilation and length contraction. The upshot of all this is that observers traveling at different speeds will always agree on the size of the space-time interval between two events. But they will disagree on both the distance between them and the times at which they occur.

Now we know from our earlier discussions that Lorentz frames can be expanded or opened up to yield a higher limit (see Chapter 1). This can yield a unique situation where you have dual lightcones expressed and this causality-based issue can seem to be violated even if it isn't violated in actuality. Events outside the light cones are spacelike-separated from 'A'; the space-time interval between them is negative, and outside observers will disagree about which comes first. The idea that no influence can travel faster than the speed-of-light -- thereby connecting two spacelike-separated events -- is known as locality. It is widely seen as a natural consequence of Relativity because any violation of the locality principle entails a violation of the normal assumption that a 'cause' must precede its effects.

The mathematical descriptions provided by Quantum Mechanics do not satisfy the locality principle. When something has an effect on one of two particles, the wave function of the other changes simultaneously and they are considered entangled. However, if those so-called "hidden variables" are not violating Relativity but simply moving through a higher dimensional region or an embedded region where the limit on velocity is higher than in normal space-time -- yet still Lorentz invariant -- then causality is persevered. Yet, the effects of what was once theory have now been proved with experiments. It is these experiments that hold some promise in the negative energy generation question.

Now, the reason we bring up this issue is that we know negative energy states exist in quantum mechanics. What to-date has not been brought up is that one could -- in theory -- entangle a larger such group of states and project it outward from the craft to alter the local space-time energy around a craft into a viable warp metric region. That 0.5 milliseconds time is more than enough to create a viable pulsed warp metric region if we utilize quantum tunneling to project our entangled region outward.

The best known negative energy state that can be generated on a decent scale is the before referenced Casimir Effect. The energy present in such a situation is expressed by P = − (π² / 720d⁴). If we could utilize the charged plate effect to increase this effect's strength and then entangle such a case to the local near field space-time of the craft, one would have a viable method of generating a warp field. We say "viable" because both are based upon solid theory and solid experimental evidence. Combined together, you have what one of the authors (Paul) calls a Quantum Entanglement Enhanced Vacuum Transtating Method.

9.2 Quantum Teleportation

Quantum Teleporation deals with exchanging bits of information from one point to another by means of EPR-B (Einstein, Podelksy, Rosen-Bell) entanglement through superposition. The simplest way to consider the effects of teleportation is to take 3 spin 1/2 particles such as an electron. Have two of the wave functions strongly EPR-B coupled (call these wave functions 'Alice' and 'Bob', or α,β) and then one can transfer the information of the unknown superposition state:

(9.4)

To do so one must have a Bell state:

(9.5)

Figure 9.1: Quantum Teleportation of photons by means of EPR states in action

such that φ₁ becomes:

. (9.6)

In order to exchange such information, a group of unity operators are needed to exchange information.

(9.7)

By this transformation we have disentangled particle 3 from particle 2 and produced the state (α| ↓>3 + β ↓>3) on particle 3. Thus the information represented by [a,b] has been perfectly "teleported" from particle 1 to particle 3 without our having measured a or b directly. This is what is meant by 'quantum teleportation'. Howeve,r as the functions Ψ are considered to be nothing more than potentials, it becomes ambiguous to ask when an how such a teleportation takes place.

The Bohm interpretation is that there exists some quantum function Q to the potential Ψ. This is done by decomposing the Schrödinger wave equation radially so that

(9.8)

as such the quantum function which determines the states is given through . If we now include several particles through this relation, one receives a "spooky" non-local effect that is:

(9.9)

9.3 Boost Entanglement

A recent phenomena has been observed with the entangled of particles and boosted (accelerated) frames.

Chapter 10 Vacuum Engineering

“... in attempting to turn fiction into reality for the future, progress would result today.”

–Majel Barrett-Roddenberry

We have shortly covered generalizations of modern physical theories which relate to Warp Drive physics. We are quite aware of the ambiguity issues, the "maybe it is, maybe isn’t" possible frames of thought. There are certainly enough mathematical issues open so that one could get a good exercise in both physics and mathematics. However, if Warp Drives are actually physical entities, then how would one go about creating a warp field? That is the question this book attempts to address -- how do we get from theory to a working spacecraft? In this chapter we will consider several theoretical points of views which may allow us to control the vacuum to engineer a Warp Drive Spacetime.

10.1 Non-gravitational Effects

In GR, one can consider the gravitational effects of non-gravitational fields such as a Klein-Gordon field or an EM field. (The classical approach to doing this was to use the principles of equivalence and minimal gravitational coupling plus intuition where these principles prove ambiguous; see the textbook by Wald for a modern approach). The field energy of the non-gravitational field will in general certainly cause light to bend. A simple example in which to study this is the Melvin magnetic Universe

(10.1)

with r ≥ 0. The vector potential is q/2(1 + (qr/2)²)d/du and the magnetic field is q/(1 + (qr/2)²)e_z = q/(1 + (qr/2)² )³ d/dz.

Using the PPN formalism, one can compare the predictions of "alternative" theories of gravitation to GR such as the Brans-Dicke scalar-tensor theory (confusingly, this is nothing other than a Klein-Gordan field plus GR; but most "alternative" gravitation theories are not equivalent to GR plus some field!) (Conformally flat: the only metrics permitted are scalar multiples of Minkowski metric). The simple reason: Maxwell’s equations are conformally invariant but predict no bending of light! However, there is a scalar theory which comes pretty close to reproducing GR, and in particular gives the same prediction for light bending and most other ”standard tests” as does GR. This is the theory of Watt and Misner (http://xxx.lanl.gov/abs/gr-qc/9910032 [6] ) which permits only metrics of form

ds² = − e^2f dt² + e^−2f (dx² + dy² + dz² ) (10.2)

where f statisfies a nonlinear Klein-Gordon equation. This theory cannot reproduce the gravitomagnetism predicted (correctly) by GR because no off-diagonal terms in the metric are allowed. Another example is the Polarized Vacuum version. But here the metrics remain the same as GR with the exception of the K value changes. However, it has the place as one conformally at version which does predict the bending of light.

However, a non-minimal coupled EM scalar forced to such a condition could violate the NEC on a local scale and thus bend light or space-time in totally the opposite fashion [7]. This is exactly what we are after when it comes to the Warp Field Metric Generation.

10.2 The GSL Issue

Assign an entropy S_bh to the warp field, given by a numerical factor of order unity times the area A of the black hole in Planck units. Define the generalized entropy S₀ to be the sum of the ordinary entropy S of matter outside of a black hole plus the black hole entropy

S' ≡ S + S_bh (10.3)

Finally, replace the ordinary second law of thermodynamics by the generalized second law (GSL): The total generalized entropy of the warp region never decreases with time,

∆S' ≥ 0 (10.4)

Although the ordinary second law will fail when matter is dropped into a black hole, such a process will tend to increase the area of the black hole. So there is a possibility that the GSL will hold.

“Conservation of energy requires that an isolated black hole must lose mass in order to compensate for the energy radiated to infinity by the Hawking process. Indeed, if one equates the rate of mass loss of the black hole to the energy flux at infinity due to particle creation, one arrives at the startling conclusion that an isolated black hole will radiate away all of its mass within a finite time. During this process of black hole “evaporation”, A will decrease. Such an area decrease can occur because the expected stress-energy tensor of quantum matter does not satisfy the null energy condition -- even for matter for which this condition holds classically -- in violation of a key hypothesis of the area theorem.

10.2.1 Dealing With Entropy Issues

If we utilize a pulsed field then we simply engineer around this problem. It is also worth noting that the GSL issue is violated in Nature with black holes once the effect of Hawking Radiation is taken into account unless you adjust for it. In this case, then, the definition of S_bh will correspond to the identification of k/2π as temperature in the first law of black hole mechanics. We then have S_bh = A/4 in Planck units. Although A decreases, there is at least as much ordinary entropy generated outside the black hole by the Hawking process. This is the bleed-off of energy brought up before and the return pulse that compensates for the fields negative energy.

Taking Mukhanov's theory [8] on black hole entropy and applying it to our warp field metric, we find that its proportionality to horizon area is conditioned upon the degeneracy degree g_n, the number of states !njmqs¿ with a common area eigenvalue a_n, being given by g_n = kⁿ , when k is some integer greater than or less than 1 depending upon the specific area of the field involved. Following the quantization law [9], the horizon may be considered as parceled into n patches of area a₀. If each can exist in k different states then the law g = kⁿ is immediate [10]. The reason is self-evident. At each patch there is not only the component of the individual field states stress energy density but also the individual particles -- present within that vacuum patch -- add to the total state. While an average could be assumed from a normalized vacuum state, the actual value will vary nonlinearly.

We also have the other issues on entropy from our other discussions on photon motion. These can form + and - components to the above. So in essence the trackable signature of high-energy photons of a craft in warp would at best be an on-again/off-again occurrence. The strongest signature would be in high-density areas such as gas clouds due to the added particle density of the local vacuum.

Now, this issue is closely related to the issue of energy condition violation. A particular case that energy violation is expected to arise is in the inflationary Universe scenario of the GUTs, where a supercooled metastable state of unbroken symmetry (a "false vacuum") gives rise to an effective cosmological constant Λ dominating other fields present. This, in turn, gives rise to an effective energy density µ' = A/k and effective pressure p' = − A/k, so (p + u)' = 0 which provides a limiting case. This allows energy increase while conserving energy-momentum. The matter expands while its density stays constant because of negative pressure provided by the negative energy density present. Now (µ + 3p)' = − 2A.k, so if A > 0 then a violation of the energy condition must take place under certain circumstances. This expansion solves the causal problems raised by horizons in the Universe because causally connected regions are now much larger than a Hubble radius. Turning to our warp field, since the field is pulsed, our warp field forms an expanding bubble out to the region effected by our warp field generator itself.

At the end of this expansion cycle the inflation-like field collapses again due to the required return pulse from WEC violation. Since the effect is a complete cycle, no actual GSL violations take place. In fact -- due to cut-off effects and other sources of Hawking Radiation from our field -- its overall energy-to-energy loss is far below unity. Like any system know it displays a perfect example of normal entropy and as such, it obeys the 2^nd law of theromodynamics.

10.3 Using Non-Miniumal Copuled Scalars & the Quintessence Connection

But, we suspect there may be a possible way to engineer around this issue through the usage of a scalar non-minimally coupled to gravity. Einstein's Theory of Gravity with a non-minimal coupled scalar field is stable for all sectors when ξ ≤ 0 and ξ ≥ 1/6. Outside of this range you can encounter negative energy state modes [11]. The question is could we use an Engineered EM scalar-coupled non-minimally-to-gravity to produce an effect outside of the allowed stable range. Since the implication is that such a field wants to hyper inflate, we again would have to run the field pulsed to control its inflation or balance the field in such a way that the Hawking radiation decay of our embedded hyperspace region counters its tendency to hyper inflate. We would simply then be utilizing thermodynamics to our advantage. We can then start with a normal GR field equation with the scalar added:

(10.5)

where in normal convention 8πG = 1 and the metric has a timelike signature. V(ψ) is the coupling scalar potential which can be engineered to assume any value. Our specific Warp Metric Region would be the determinate of the value here. The Conformal factor,F(Psi) ≡ 1 − ξ ψ² in the Langrangian multiplies the scalars own curvature those the degrees-of-freedom of the gravity field and entangled to the scalar fields degrees-of-freedom. If we do a conformal transform of the metric so that

G_µν = Ω² g_µν(10.6)

This transforms the Ricci scalar as

R₁ ≡ R[g] = Ω⁻² [R − 6g^µνD_µD_ν (log Ω) − 6g^µν D_µ (log Ω)D_ν (log Ω)] (10.7)

where D_µ[g] are covariant derivatives. We use the scalar coupled form of GR again to derive an Einstein-Hilbert field with a coupling scalar

L ≥ √g − ((F(ψ)/2Ω²)R₁ + 1/2g_µν (ψ_µψ_nu/Ω² + 6((Ω_µ/Ω) (Ω_ν/Ω)) − V(ψ)/Ω⁴(10.8)

Then we get

(10.9)

You will now notice that the variable is F(ψ) and that we can control the energy tensor from GR through the coupling scalar field.

Now, the stationary vacuum Einstein equations describe a two-dimensional -model which is effectively coupled to three-dimensional gravity. The target manifold is the pseudo-sphere SO(2,1)/SO(2) ~ SU(1,1)/U(1) which is parametrized in terms of the norm and the twist potential of the Killing field. The symmetric structure of the target space persists for the stationary EM system where the four-dimensional coset SU(2,1)/S(U(1,1) × U(1)) is represented by a Hermitian matrix Ф comprising the two electro-magnetic scalars -- the norm of the Killing field and the generalized twist potential. The coset structure of the stationary field equations is shared by various self-gravitating matter models with massless scalars (moduli) and Abelian vector fields.

For scalar mappings into a symmetric target space Ĝ/Ĥ, Breitenlohner et. al. [12] have classified the models admitting a symmetry group which is sufficiently large to comprise all scalar fields arising on the effective level within one coset space G/H. A prominent example of this kind is the EM-dilaton-axion system, which is relevant to N = 4 supergravity and to the bosonic sector of four-dimensional heterotic string theory: The pure dilaton-axion system has an SL(2,IR) symmetry which persists in dilaton-axion gravity with an Abelian gauge field [13]. Like the EM system, the model also possesses an SO(1,2) symmetry arising from the dimensional reduction with respect to the Abelian isometry group generated by the Killing field. So there is a tie between the two as far as geometry goes.

Now a Einstein-Maxwell dilation field is given by S = 1/16π function of

, (10.10)

where R is the Ricci scalar, ф is the dilation field, and F_ij is the EM field tensor. β here is what regulates the coupling. β = 1 is the low energy coupling from String Theory Varying with this we get G_ij= − 8πT_ij , (e^−2βф F^ij); i = 0 so the coupling can be varied even in dilation field work to alter the metric.

In a quantum-field representation of an Einstein-Bose condensate described by a field with non-vanishing vacuum expectation value ν, possibly depending on the spacetime coordinate x. It is interesting to insert the action of this field -- suitably "covariantized" -- into the functional integral of gravity, expand the metric tensor g_mn in weak field approximation, and check if the only effect of ν (x) is to produce gravity/condensate interaction vertices proportional to powers of . One finds that, in general, this is not the case. The quadratic part of the gravitational action is modified as well by receiving a negative definite contribution. It can thus be expected that the condensate induces localized instabilities of the gravitational field.

Let us consider the action of the gravitational field g_mn(x) in its usual form:

10.11)

where k²= 8πG, − R(x)/k² is the Einstein term and Λ/k² is the cosmological term which generally can be present. ∫* represents the integral in d⁴x√g(x).

It is known that the coupling of the gravitational field with another field is formally obtained by "covariantizing" the action of the latter. This means that the contractions of the Lorentz indices of the field must be effected through the metric g_mn(x) or its inverse g^mn(x) and that the ordinary derivatives are transformed into covariant derivatives by inserting the connection field.

Moreover, the Minkowskian volume element d⁴x is replaced by [d⁴x √g(x)], where g(x) is the determinant of the metric. The insertion of the factor √g(x) into the volume element has the effect that any additive constant in the Lagrangian contributes to the cosmological term Λ/k² . For instance, let us consider a Bose Condensate described by a scalar field F(x) = v + f(x), where v is the vacuum expectation value and [m|v|²] represents the particles density of the ground state in the non-relativistic limit. The action of this field in the presence of gravity is ,

(10.12)

Considering the total action [S_g + S_F], it is easy to check that the ”intrinsic” gravitational cosmological constant Λ receives the contribution (1/2) m²|v|² k² . Astronomical observations impose a very low limit on the total cosmological term present in the action of the gravitational field. The presently accepted limit is on the order of ΛG < 10⁻¹²⁰ , which means approximately for Λ itself, that |L| < 10⁻⁵⁴ cm⁻² .

This absence of curvature in the large-scale Universe rises a paradox called ”the cosmological constant problem". In fact, the Higgs fields of the standard model as well as the zero-point fluctuations of any quantum field -- including the gravitational field itself -- generate huge contributions to the cosmological term which somehow appear to be ”relaxed” to zero at Macroscopic distances. In order to explain how this can occur, several models have been proposed. A model in which the large-scale vanishing of the effective cosmological constant has been reproduced through numerical simulations is the Euclidean quantum gravity on the Regge lattice. From this model emerges a property that could turn out to be more general than the model itself. Iif we keep the fundamental length L_Planck in the theory, the vanishing of the effective cosmological constant L -- depending on the energy scale p -- follows a law of the form |L| (p) G⁻¹ (L_Planckp)^γ where γ is a critical exponent. We find no reason to exclude that this behavior of the effective cosmological constant may be observed in certain circumstances. Furthermore, the model predicts that in the large distance limit, Λ goes to zero while keeping negative sign. Also, this property probably has a more general character, since the weak field approximation for the gravitational field is ”stable” in the presence of an infinitesimal cosmological term with negative sign; conversely, it becomes unstable in the presence of a positive cosmological term.

There appears to exist -- independent of any model -- a dynamical mechanism that "relaxes to zero" any contribution to the cosmological term. This makes the gravitational field insensitive to any constant term in the action of other fields coupled to it. Nevertheless, let us go back to the previously mentioned example of a Bose Condensate described by a scalar field F(x) = v + f(x). If the vacuum expectation value v is not constant but depends on the spacetime coordinate x, then a positive ”local” cosmological term appears in the gravitational action S_g. This can have important consequences. Let us suppose that v(x) is fixed by external factors. Now, decompose the gravitational field g_mn(x) as usual in the weak field approximation. That is, g_mn(x) = δ_mn + kh_mn(x). The total action of the system takes the form

(10.13)

where µ²(x) = d_mv * (x)d^m v(x) +m |(x)2 ; S_hv = ∫ *d^m v * (x)dⁿv(x)kh_mn(x) ; S_f= ∫ *m²|f(x)|² +m² [v(x)f *(x)+c.c.]+[d_mf *(x)d_nf(x)+d_mf ?(x)d_nv(x)+c.c] g^mn(x).

In the action S, the terms S_hv and S_f represent effects of secondary importance. The term S_hv describes a process in which gravitons are produced by the "source" v(x). The term S_f contains the free action of the field f(x) describing the excitations of the condensate and several vertices in which the graviton field h_mn(x) and f(x) interact between themselves and possibly with the source. None of these interactions is of special interest here. They are generally very weak due to the smallness of the coupling k. The relevant point is that the purely gravitational cosmological term Λ/k² receives a local positive contribution [(1/2)µ²(x)] that depends on the external source v(x). We shall use "critical regions" to designate those regions of space-time where the following condition is satisfied:

Λ/k² + (1/2)µ²(x)] > 0 . (10.14)

In these regions the gravitational Lagrangian is unstable and the field tends to be "pinned" at some fixed values. A numerical estimate of the magnitude order is µ²(x) in the case of a superconductor. To this end we recall that the Hamiltonian of a scalar field F of mass m is given by

(10.15)

In our case F describes a system with a condensate and its vacuum expectation value is < 0 |F(x)| 0 >= v(x). Then in the Euclidean theory µ² is positive definite and we have < 0 |H| 0 >= (1/2) ∫d³xµ²(x). In the non-relativistic limit -- appropriate in our case -- the energy of the ground state is essentially given by [NVm_c] where m_c is the mass of a Cooper pair (of the order of the electron mass; in natural units m_c10¹⁰cm⁻¹), V is a normalization volume and N is the number of Cooper pairs for unit volume. Assuming N 10²⁰ cm⁻³ at least, we obtain µ²(x) Nm_c > 10³⁰ cm⁻⁴ . We also find in this limit v Nm_c^−1/2 .

This small value -- compared with the above estimate for µ²(x) --supports our assumption that the total cosmological term can assume positive values in certain states. In fact the positive contribution in a condensate is so large that one could expect the formation of gravitational instabilities in any superconductor or superfluid subjected to external e.m. fields or not. A conclusion in contrast with the observations at present. We may then hypothesize that the value of Λ/k² at small-scale is larger than that observed at astronomical scale and negative in sign such that it would represent a ”threshold” of the order of 10³⁰ cm⁻⁴ for anomalous gravitational couplings.

The Poynting vector is the very component in the S = E × H triad that has to be longitudinal by the laws of vector multiplication. This longitudinally-oriented ”pulsation” shows the ZPF exists and influences matter. The Aharonov-Bohm effect showed that potentials can have observable effects on charges even when no fields are presen, the potentials are the cause and the fields are the effects. When the E and H fields themselves have been phase-cancelled, the Poynting vector should still exist. We should have a scalar product still present that propagates longitudinally just like particle-waves. The real question that nobody has ever answered is how do you account for polarization effects. We believe the answer comes from the polarized vacuum approach. We know that the Higgs field provides the background against which the spin arrow is aligned in the standard model. If that background is itself an embedded region, then the Poynting vector can be longitudinal and still be polarized since it would carry the imprint of polarization from its original canceled effects. So much for Beardon's abandonment of longitudinal explanations for scalar effects. If you utilize PV and the ZPF, there is no reason to resort to ether based theory to explain certain effects.

There is a simple relation between Killing vectors and the desired EM vector potentials which was found by Papapetrou [14] (1951). Spinning Extended bodies deviate from their geodesic:

This is similar to Frenkel Equation (1926). Where in at space, Papapetrou's definitions of momentum and spin the same equations for a spinning particle, being:

That is admittedly a force-free helical path (i.e., the classical Zitterbewegung). A connection to a Clifford Manifold can also be shown. A space is said to be "polygeometric" when physical quantities are uniquely associated with different geometric elements. A polyvector can add unlike geometries:

The Invariant Modulus are given with

or e² − p'² = n²₀ + (S/λc)² where we can interpret that spin contributes to mass. A variation in the local Largrangian can affect topologcial structures as well through Non-Holonomic Mechanics. Indeed, one can easily use Killing vectors to find -- exact -- electrovacuum solutions via the Ansatz method if one uses ONBs via Vierbeins or real tetrads and frame fields instead of coordinate bases. We seek simultaneous solutions to the source-free Maxwell equations and the EFE with stress-momentum-energy tensor arising entirely from the energy-momentum of the EM field's scalar.

The concept of scalar field coupling to gravity is not new at all. But recent M-Theory work has brought it back into popularity concerning Killing vectors and certain Dark Energy solutions to the accelerated expansion issue. But there is a certain connection here that we believe worth looking into on the issue of generating an actual warp metric.

We mentioned earlier the Biefield-Brown effect. Here is where we show that idea for what it actually is. Military files on Bifield-Brown effect: the efficiency of propulsion by the electric wind, as exemplified by the model flying saucers is of the order of 1.4 percent. At speeds higher than 2.3 ft/sec attained by the model flying saucer, the efficiency might be substantially improved. It is presumable less at high altitudes and zero in a vacuum.* If the efficiency of conversion of fuel energy into electrical energy is 21 percent, the overall efficiency of propulsion of the model flying saucers by the electric wind is 0.3 percent. This compares with about 25 percent for a propeller-driven airplane and about 15 percent for a jet airplane.

Chapter 11 Beyond the Fourth Dimension

“String theory can explain not only the nature of particles but that of space-time as well.”

–Michio Kaku

Thus far, hyper-fast travel techniques have been discussed within the paradigms of General Relativity and of Quantum Mechanics. We have thus laid out the tantalizing possibilities for apparent superluminal travel within the confines of established scientific fact. However, there still remains the open possibility of uniting the properties of the microscopic world with the Macroscopic one. Currently the laws of physics that describe the properties of matter and that of spacetime are attempting to be bridged by mathematical symmetries known as superstring theory.

11.1 Superstring Theory in Brief

Around the mid 1950s most of what was known in the physics world was explained including electrodynamics, mechanics, and thermodynamics. It was hoped that just as the electric and magnetic forces were unified into electromagnetic theory that the gravitational and electromagnetic forces could be unified. Many attempts to unite the two forces have failed. While the two share the same "power law", they fail to interact through similar methods.

One way around this problem was through the use of group symmetry which was found useful in describing quantum theory. This attempt was applied to the gravitational and electromagnetic fields by embedding them into a 5-dimensional continuum by Theodor Kaulza. This group approach was submitted to Albert Einstein who let the idea sit a full year before submitting for publication. The idea had some mild success but there were a lot of terms left over in the field equations which resulted in the symmetry breaking of the 5-dimensional continuum.

The idea was later to be resurrected by Oscar Klein. The reason we do not see the symmetry breaking affects in spacetime is that the extra dimension is compactified to microscopic size. It was explored as a mathematical curiosity for a while, but it was far removed from an accurate representation of the physical and to an extent was ignored. Most of the research in physics after this time was devoted to exploring the paradoxical affects for the quantum theory of matter which would breakdown into further subcategories which lead to the development of modern Quantum Field Theory.

Superstring theory evolved early on as a string model for particle physics to describe the energy conditions involved in elastic collisions. This model assumed that energy was stored in a string-like structure held by two separated anchors. The energy of each particle depended on the length separation of the anchors, and hence each particle would vibrate at a certain frequency. While this model could explain the energy released in elastic collisions, it failed to describe the nature of known particles -- for example, requiring them to have unrealistic radii. One way to resolve this problem was to use group symmetry. This could relate certain particle classes and make the string theory of energy useful in approximation.

As mathematical rules for the quantum theory emerged, its seemingly mysterious properties were at last being revealed to the extent that science could explain them. The development of Quantum Electrodynamics and Quantum Chromodynamics made the quantum theory of matter less mysterious, leaving gravitation once more the "odd ball" out in the physical picture of the Universe. The symmetries of string theory were applied to gravitation forming a so-called "unified field theory" between matter-energy and space-time. This would be useful theory in physics if it could be verified as it could explain the complex interactions which build up the Universe in which we live.

However, many physicists have gone off the building blocks of physics and attempt the construct a 'Theory of Everything'. This naive concept that everything known to modern physics can be explained purely mathematically is presently called M-theory, which is a unified version of superstring theory. To discuss how M-theory can be considered a unified field theory, one most go through the various aspects of string theory. There is not one single mathematical set of equations which explains the complex interactions between energy and spacetime, but at least five according to present string theory.

(1) Type I :The bosonic string

(2) Type II : Fermonic

(3) Type II : Heterotic

(3) Type II without "left-hand"

(4) Type IIB : Compatified topology

11.2 Type I: Bosonic strings

The bosonic string exists on a worldsheet X^µ (τ,σ) where τ,σ represent space and time respectively. The motion of a bosonic string along a worldsheet is given through the Nambu-Goto action:

(11.1)

There is a duality given in space and time by the prerequisite definition τ = σ⁰,σ = sigma¹ . α⁰ in an inverse string tension with mass dimension −2, to extend a metric of n-dimensions into string theory requires that:

g_ab = G_µυ(X) ∂_αX^µ ∂_βX^υ . (11.2)

Thus an independent action for motion on a worldsheet is given through

. (11.3)

Consequently an energy momentum tensor can take the form of

(11.4)

With G_µυ = η_µυ one obtains Weyl invaranices γ_ab → e^φ(τ,σ)γ_ab. Thus under the rotation of γ_ab = η_ab, which give rise to the light cone coordinates of the world action as (τ, σ): σ⁻ = τ − σ , σ⁺ = γ+ σ. A gauge fix to a Minkowski metric leads to

(11.5)

where

(11.6)

which gives X i (i = 2, . . . , d − 1) two-dimensional super symmetry:

(11.7)

where ψ_± are two dimensional Majoranano-Weyl spinors:

(11.8)

with where

(11.9)

This is verified through a Clifford Algebra {p^a,p^b} = − 2n^ab . This action gives rise to the inclusion of fermions into the picture of string theory.

11.3 Type IIA: Fermionic strings

Type IIA theory can be obtained from the dimensional reduction of the 11-dimensional supergravity theory and given by the Cremmer-Julia-Scherk Lagrangian

(11.10)

where Y is the 11-dimensional manifold. If we assume that the 11-dimensional spacetime factorizes as Y = (X × S¹) R, where the compact dimension has radius R, the usual Kaluza-Klein dimensional reduction leads to get the 10-dimensional metric: a scalar field and a vector field. A₍₃₎ from the 11-dimensional theory leads to A₍₃₎and A_P(2) in the 10-dimensional theory. The scalar field turn out too be proportional to the dilaton field of the NS-NS sector of the Type IIA theory. The vector field from the KK compactification can be identified with the A^IIA field of the R-R sector.

From the three-form in 11 dimensions are obtained the RR field A₍₃₎ of the Type IIA theory. Finally, the A₍₂₎ field is identified with the NS-NS B-field of field strength H₍₃₎ = dB₍₂₎. Thus the 11-dimensional Lagrangian leads to the Type IIA supergravity in the weak coupling limit (Ф → 0 or R → 0). The 10-dimensional IIA supergravity describing the bosonic part of the low energy limit of the Type IIA superstring theory is

(11.11)

where H₍₂₎ = dB₍₂₎, F₍₂₎ = dA₍₁₎ and F^-₍₄₎dA₍₃₎− A₍₁₎ Λ H₍₃₎.

It is conjectured that there exist an eleven dimensional fundamental theory whose low-energy limit is the 11-dimensional supergravity theory and that it is the strong coupling limit of the Type IIA superstring theory. This is M-Theory. But it must incorporate the above in it. Just as the M-theory compactification on S¹_R leads to the Type IIA theory, Horava and Witten realized that orbifold compactifications leads to the E₈ × E₈ heterotic theory in 10-dimensions HE M / (S¹/Z²) ↔ HE where S¹/Z²is homeomorphic to the finite interval I and the M-theory is thus defined on Y = X × I . From the 10-dimensional point-of-view, this configuration is seen as two parallel planes placed at the two boundaries ∂I of I. Dimensional reduction and anomalies cancellation conditions imply that the gauge degrees of freedom should be trapped on the 10-dimensional planes X with the gauge group being E8 in each plane. While that gravity is propagating in the bulk and thus both copies of X’s are only connected gravitationally.

11.4 Heterotic strings

11.5 Type IIB: Compactified Topologies

11.6 Shortcuts in Hyperspace

Chapter 12 Navigation Issues

“Second star to the right and straight on to morning.” –William Shatner from Star Trek VI

Assume that one has a created a working Warp Drive metric. Then how does one get from Earth to some interstellar distance? This is no easy task. As the cosmos are dynamic, we see a historical view of our Universe. If we want to travel to some star system outside our solar system, we’re going to run into problems because our maps of the Universe are outdated. We now begin to explore how one may navigate a Warp Drive spaceship with these problems in mind.

12.1 Measurement

Consider a set of 10 measurements of leaf-size: (x₁, x₂, . . . , x₁₀). where x₁ is the size of the first leaf, etc. According to some expert, leaf sizes are supposed to be “normally” distributed with mean and standard deviation. Knowing all these numbers you could now calculate the quantity known as chi-square χ² .

(12.1)

where in this case there are 10 x values, so k = 10. (This formula says find how each x deviates from the mean: square each difference; add up all the squared-differences; and divide by the standard deviation squared.) More general versions of this formula would allow different means and standard deviations for each measurement.

Roughly speaking, we expect the measurements to deviate from the mean by the standard deviation, so |(ξ − µ)| is about the same thing as σ. Thus in calculating chi-square, we’d end up adding up 10 numbers that would be near 1. More generally we expect χ² to approximately equal k - the number of data points. If chi-square is ”a lot” bigger than expected, then something is wrong. Thus one purpose of chi-square is to compare observed results with expected results and see if the result is likely.

In biology the most common application for chi-squared is in comparing observed counts of particular cases to the expected counts. For example, the willow tree (Salix) is dioecious -- that is -- like people. And unlike most plants, a willow tree will have just male or female sex organs. One might expect that half of the willows are male and half female. If you examine N willow trees and count that x₁ of them are male and x₂ of them are female, you will probably not find that exactly x₁=N and x₂=N. Is the difference significant enough to rule out the 50/50 hypothesis? We could almost calculate the chi-squared but we don’t know the standard deviation for each count.

Never fear: most counts are distributed according to the Poisson distribution, and as such the standard deviation equals the square root of the expected count. Thus we can calculate x₂: In our simple willow example, there are just two cases. So k=2 and the expected results are: E₁=N and E₂=N. Note that the E_i are generally not whole numbers even though the counts x_i must be whole numbers. If there were more cases (say k cases), we would need to know the probability p_i for each case and then we could calculate each E_i=p_iN where N is determined by finding the total of the counts:

(12.2)

Finally it should be noted that the technical differences between a Poisson distribution and a normal distribution cause problems for small E_i. As a rule of thumb, avoid using x₂ if any E_i is less than 5. If k is large, this technical difficulty is mitigated [35].

This could also be utilized to give an approximate particle distribution in the path of our warp drive craft -- given the data we currently have from observation and theory -- to determine a standard field strength a warp drive ship should run on for any given velocity range. This standard could them be utilized as part of the regular main engineering program for the StarDrive system. If larger objects were detected, then we simply have the program up the shield strength to counter them.

This brings up another area that needs research. How do we scan ahead of a craft in warp? The answer at present is hard to find. We have the c limit in normal space-time to deal with. Even gravity waves seem to be limited to c. Yet, a Starship will need a sensor system that can work while in warp. The answer may be that we will simply have to scan for objects moving less than c, which seems to be the normal anyway. Their gravity waves would move at c and thus well in front of them. Those waves will enter the field; and if we can develop a decent gravity wave detection system in the near future, we can use that as a sensor once it is zeroed to the normal fields effects. The only problem then would be an energy wave front hitting the craft and its field moving at c. We would not have the time to respond before it hit. Shields being up and at whatever level they were at would provide some protection. But the danger of damage would still remain in that extreme case. Space exploration has a certain risk to it. We will never be able to totally eliminate risk. But we can think ahead and eliminate some risks.

What type of sensors are needed besides this? Besides the normal radar type for out of warp conditions, such a craft would need the best of those we now employ to monitor weather and other conditions on Earth. It would need these to explore planets for possibly life signs. We'd need to know what type of atmosphere a planet had, if it had any forms of plant life, etc. We need to know if a civilization exists on such. We need data on biological issues that could effect humans.

The fundamental purposes of star travel is that we will eventually need new places to survive on. We have a desire to learn and increase our knowledge. Contact with other civilizations. Finding new resources to utilize to supplement those we presently use.

All of these issues require advanced sensor data. A starship will need the best there is to offer in this field and some not even developed at this point.

The effect of a passing gravitational wave would be to cause a change in the warp field we generate. Such a change could be measured as changes in the field. The principle would be similar to those proposed to use pulsars as gravity wave detectors.

In the ideal case, the change in the observed frequency caused by the GWB should be detectable in the set of timing residuals after the application of an appropriate model for the rotational, astrometric and -- where necessary -- binary parameters of the pulsar. All other effects being negligible, the rms scatter of these residuals would be due to the measurement uncertainties and intrinsic timing noise from the neutron star. Detweiler [1] has shown that a GWB with a at energy spectrum in the frequency band ∫ ± ∫/2 would result in an additional contribution to the timing residuals σ_g. The corresponding wave energy density ρ_g (for ∫T >> 1) is

(12.3)

An upper limit to ρ_g can be obtained from a set of timing residuals by assuming the rms scatter is entirely due to this effect (σ = σ_g). These limits are commonly expressed as a fraction of ρ_c the energy density required to close the Universe:

(12.4)

where the Hubble constant H₀ = 100 h km/s⁻¹ . Mpc. Romani and Taylor applied this technique to a set of TOAs for PSR B1237+12 obtained from regular observations over a period of 11 years as part of the JPL pulsar timing program. This pulsar was chosen on the basis of its relatively low-level of timing activity by comparison with the youngest pulsars, whose residuals are ultimately plagued by timing noise (4.4). By ascribing the rms scatter in the residuals (σ = 240 ms) to the GWB, Romani and Taylor placed a limit of ρ_g/ρ_c < 4×10⁻³ h⁻² for a center frequency f = 7×10⁻⁹ Hz.

This limit -- already well below the energy density required to close the Universe -- was further reduced following the long-term timing measurements of millisecond pulsars at Arecibo by Taylor and collaborators. So if we can detect long and large-scale effects with this method, we could apply a similar model on the field we generate to detect outside objects in near-field and correct our shields and navigation data from this.

The Field would act as the reference clock at one end of the arm sending out regular signals which are monitored by equipment on the craft over some time-scale T. The effect of a passing gravitational wave would be to cause a change in the observed field frequency by an amount proportional to the amplitude of the wave. For regular monitoring observations of a pulsar with typical TOA uncertainties, this “detector” would be sensitive to waves with dimensionless amplitudes > TOA/T and frequencies as low as 1/T. For warp drive, we would need far greater sensitivity. However, our near-field location to the field we monitor and the increase on amplitude our field causes would make this possible.

Searching for passing gravity waves is a delicate art since it involves sensing deformations much smaller than the size of an atomic nucleus in huge detectors meters or kilometers in size. In the resonant detector approach, this means watching for longitudinal vibrations in chilled automobile-sized metal cylinders. In the LIGO approach, the deformation is the change in the separation of distant mirrors attached to test masses. Gravity waves strong enough to be detected will most likely come from events such as the coalescence of black holes or neutron stars, and these are rare. But if the waves are allowed to naturally effect our field and the effects are amplified by the field itself, then our lower limit can be increased to a range capable of detecting smaller objects -- of, say, planet-size or smaller -- just from perturbations of the warp field. We'd have eyes working ahead of the object in question because the gravity waves move faster than the object.

GEC is not only striving to have the sensitivity to record gravity waves from events out to distances of 100 million light-years but is also hoping to be able to locate the source of the waves in the sky. This would be accomplished partly by the use of more sensitive individual detectors and by improving the comparative time resolution of signals so that the network could be operated more as a true interferometer, much like networks of radio telescopes. A similar method could be used onboard a craft. This would give us near pinpoint view while in warp when combined with a computer system that alters the incoming photon images into something that resembles real-time events. With these two combined with other sensor data, you have warp capable sensors using present method [36].

The forefront of research on this is superconductors. Some recent articles in LANL have noted they have the potential of not only working as good gravity wave detectors, but they also might be able to generate simulated gravity waves. Again, a fundamental area that needs further research as man's quest for the stars moves on.

Three different statistics for stochastic background searches are currently under investigation: (a) the standard cross-correlation statistic, (b) the maximum-likelihood statistic, and (c) a robust version of the cross-correlation statistic for non-Gaussian detector noise. Monte Carlo simulations have shown that for simple models the robust statistic performs better than the standard cross-correlation statistic; while the maximum-likelihood statistic performs either better-or-worse than the standard cross-correlation statistic, depending on how well one can estimate the auto-correlated detector noise in advance.

There are a number of important unresolved problems facing stochastic background searches. These include: (1) how to best estimate the auto-correlated

detector noise, (2) how to set a threshold for signal detection, (3) how to deal with non-Gaussian and/or non-stationary detector noise, (4) how to distinguish cross-correlated environmental noise from a cross-correlated stochastic gravity-wave signal, and (5) whether-or-not one can claim detection (or only set an upper limit) given a two-detector cross-correlation.

We have already touched on the issue before of eliminating noise from a system. Our field itself will generate noise that must be zero-ed out. With a working field, we can simply take the average of the field and cancel for such. So some of these problems not only have solutions but also they simply need further research and development. For example, the resonant detectors presently in operation show non-stationary noise and also future detectors are expected to have non stationary noise. Another important point to be solved is the possibility of correlated noise in the outputs of the detectors. Joe Romano -- in collaboration with B. Allen, J. Creighton, L. S. Finn, E. Flanagan and others -- is studying the different statistics for this search and work is in progress in trying to solve the problems.

A stochastic background of g.w. may also be of astrophysical origin. In fact the emission of g.w. from a large number of unresolved astrophysical sources can create a stochastic background of g.w. The work done by the Rome group (Prof. V. Ferrari and collaborators) on the study of the stochastic background of gravitational radiation is focused on the contributions given by cosmological populations of astrophysical sources -- i.e., sources distributed throughout the Universe from the onset of the galaxy formation up to the present time. In collaboration with Raffaella Schneider and Sabino Matarrese, they have elaborated a general method to evaluate the spectral properties of the stochastic background generated by astrophysical sources, whose rate of formation can be deduced from the rate of formation of the corresponding stellar progenitor. The star formation rate they use is based on observations collected by the Hubble space Telescope, Keck and other large telescopes on the UV emission of stars forming galaxies up to redshifts of about z 5. The current limits on this and some of that background are shown in the following graph.

Chapter 13 Thoughts on Warp Vehicle Structure Requirements

“The Flux Capacitor is what makes the whole thing work.” –Dr. E. Brown from Back to the Future

How to build an interstellar spaceship -- assuming it is possible to generate a Warp Drive spacetime --how can you construct a spaceship? This is going to require revolutionary new technologies combining abstract theories with our bulky classical machines. We now consider how it may be possible to construct a Warp Drive spaceship with known technologies and the theories explored in this book.

13.1 Initial Thoughts

Getting to the Stars will make a trip to the Moon seem like a piece of cake. Along with the daunting engineering problems that will have to be solved before humans can travel to the stars, a number of biomedical concerns will also need to be addressed. Particular attention will have to be paid to protecting crews from space radiation and providing them with oxygen, water and food during the long journey. While shielding crews from radiation is a primary concern of engineers, providing people with life supplies inflight is an area for biomedical specialists. In Russia, specialists of the Institute of Biomedical problems (IBMP) have developed a concept for a closed-loop life-support system (LSS) for interplanetary voyages. Their idea is based upon 3 key proposals.

The first is to put an oxygen tank and a CO₂ absorber inside the space-craft. The second is to disintegrate water through hydrolysis in order to get Oxygen – Carbon Dioxide will still have to be removed from the spacecraft by a physical-chemical CO₂ absorber. The third way is to grow plants which will generate O₂ and absorb CO₂. The latter -- basically taking a piece of biosphere into space -- seems to be the most promising for long interstellar flights.

Mir showed long-term flight works. During 15 years in orbit, Russia’s Mir space station survived fires, collisions and equipment failure. It even out-lasted the government that launched it. Its name meant "Peace", but Mir’s great accomplishment was endurance. Launched on Feb. 20, 1986, it was occupied almost continuously by more than 100 visitors from 12 countries. One cosmonaut -- Valeri Polyakov -- stayed aboard for 438 consecutive days. Another -- Sergei Avdeyev -- spent a total of more than 2 years in space over 3 flights.

As a result, said Rick Tumlinson of the Space Frontier Foundation, “the Russians know more about long-duration space flight than we have ever come close to.” But in the end, the sheer financial strain of maintaining a troublesome "old clunker" doomed the 143-ton spacecraft.

But that was an orbiting station that could be resupplied. It was not a venture to the stars that had no hope of resupply unless it gets back to Earth or found an Earth-like planet. Starship missions cannot count on anything but themselves. They have to be self sustained missions with a long-term survival capability given the distances and travel times involved even with warp drive for any chance of success.

Mir showed us some of the challenges that await starship travel. But it only touched the surface. Most of us in the older generation remember Apollo 13. They barely made it back from a simply trip to the Moon. They had the advantage of many engineers on Earth helping them. Those on a starship will not have this advantage. They will be forced to fight on their own to survive. This ending presentation on starship design is intended to address some of these issues on getting to the stars and what is needed to get there.

For this type of spaceflight to be valid, it has to be safe within reason. We have to not only design the type of ship to achieve such a mission (which has never been done) but we also have to think about the biological issues of spaceflight to the stars. We also have to think ahead on the What-Ifs. The Universe around us offers much promise. But it also holds many dangers. We may seek to come in peace for all mankind. But we also must be ready to fight to survive. Those who take that first voyage to the stars will have to be ready to do what it takes to survive. They will also need a craft that can survive.

This, then, shall be the last section of this course on Warp Drive. We will look at some of that right stuff it will take for such a craft to become a reality. We shall look at both the engineering requirements known to date and at those biological issues. This is not meant to be exact or the "last word". It is simply added to show what faces mankind if they choice to embark on such a mission. With all the expense that would be involved in such a mission, we do not want a repeat of the Challenger event. Perhaps on that word it is best to dedicate this section to those men and women who died on Challenger that day and whose spirit shall live on till the 23^rd century and beyond.

13.2 Structural Requirements

The safety of spacecrafts at a long-term spaceflight requires the creation of special protection from damage by meteoroids and orbital debris. As of now, this danger is the most important problem of spacecraft survivability. The known principles of designing of the protection demand the very massive shields. With an overall outer surface of a pressurized cabin more than 100 m², the shielding mass becomes critical importance characteristic in spacecraft design. For example, the specific mass of an ISS Module shielding is equal about 10 kg/m². Its delivering cost to the orbit is more than $10,000(US) for each kilogram. Evidently even the saving 1 kg/m² in the shielding weight leads to a considerable economy of the finance expenditure for space missions. When we begin to consider long-term mission such as star travel with durations -- even given the development of warp drive -- of a year-or-more, the problems that have confronted mankind so far become even more critical.

Starships by the very long-term usage and by the stresses involved in star flight will demand a new level of engineering and development of new materials to accomplish their missions. The first and foremost goal must be to develop materials that are strong, lightweight, have high stress capabilities, and provide some basic shielding against radiation and impact on their own. Along these lines composites would seem to hold the key.

13.2.1 Shielding

To get a perspective of why not only field-generated shields are needed but also why the hulls must provide shielding in case of shield failure, the Space Physics Textbook by the Space Physics Group of Oulu lists what high-energy particle radiation can do to equipment and crews. This is based upon their study of the Van Allen Belt. It degrades satellite components, particularly semiconductor and optical devices -it induces background noise in detectors. Iit induces errors in digital circuits. It induces electrostatic charge-up in insulators. It is also a threat to the astronauts.

Starships will have to deal with radiation levels at times vastly higher than the Van Allen Belt produces. On Earth, we are protected from these particles by the atmosphere which absorbs all but the most energetic cosmic ray particles. During space missions, astronauts performing extra-vehicular activities are relatively unprotected. The fluxes of energetic particles can increase hundreds of times -- following an intense solar are or during a large geomagnetic storm -- to dangerous levels. In the deep voids of interstellar space, we have little current knowledge as to what may confront us. At present we have only sent probes to the outer edges of our solar system. Mankind's first trip to the stars will be a journey truly where no man has gone before.

The safe dose limits on radiation can be found in Radiation Safety by the International Atomic Energy Agency, Division of Radiation and Waste Safety. The maximum safe limit is 25 rems per day. The dose limits for practices are intended to ensure that no individual is committed to unacceptable risk due to radiation exposure. If we adopt the 15-millirem standard the government now uses, the question that confronts us here is to provide shielding that meets-or-exceeds the safe daily limit. To give one an ideas of how strict this 15-millirem standard is, the EPA stated that the standard means that a person living 11 miles from the Yucca Mountain site -- the distance to which the standard applies -- will absorb less radiation annually than a person receives from 2 round-trip transcontinental flights in the United States.

The EPA also stated that background radiation exposes the average American to 360-millrem of radiation annually, while 3 chest x-rays total about 18-millirem. The reason the authors favor this is that there will most likely be radiation issues onboard the ship for power generation equipment. So limiting outside radiation will be of prime importance.

13.2.2 Hull Materials

The 3 main materials used for the hull of a spacecraft are (for the hull) a mix of mainly Aluminum and Magnesium and (for the supports that hold the hull together) Titanium. These materials are very strong and very light. Hopefully even better materials will later be discovered that are even more efficient. However, future spacecraft of the starship type will require far different materials in part of their hulls.

One aspect that was brought up earlier is the need to produce a variable barrier for quantum tunneling. This will require some form of "smart material" that can vary its properties via electrical control. This could be a surface coating over a regular hull skin or take the actual form of part of the hull itself in the area of the field generator. Secondly, if the EM transtator method is utilized, an embedded system of waveguides will have to be incorporated into the hull structure to emit the transtating field itself.

"Smart materials" will most likely hold the key to such a craft's production. Materials that can alter their strength and tensile properties to compensate for stresses involved during flight will have to be developed. Even given the external radiation shield ability, better methods on internal radiation shielding without excess weight will have to be developed along-the-way especially if atomic power is utilized for power generation. As to those present materials -- since a starship has less weight requirements -- we would suggest Titanium-based alloys are the way to go here.

Here on Earth, increasing the percentage of Oxygen to slightly above 21% dramatically increases the probability of fires. According to The Anthropic Cosmological Principle (p. 567) by Barrow and Tipler, “ the probability of a forest fire being started by a lightning-bolt increases 70% for every 1% rise in Oxygen concentration above the present 21%. Above 25% very little of the vegetation on land would survive the fires ... At the present fraction of 21%, fires will not start at more than 15% moisture content. Were the Oxygen content to reach 25%, even damp twigs and the grass of a rain forest would ignite.”

So we also must stay below the 21% range to lower the danger of fire. On Earth the atmosphere is a mixture of gases that becomes thinner until it gradually reaches space. It is composed of Nitrogen (78%), Oxygen (21%), and other gases (1%). Of the 1%, Carbon Dioxide is a major component.

Now given that starships with their unique warp drive may not be as weight subject as craft we have at present, they could be built in orbit above the Earth. As such, if the size was kept large enough we could utilize something akin to the Biosphere in Arizona. Since plants utilize Carbon Dioxide, a waste product of our own breathing and produce oxygen we might be able to extend such a craft's operating time through such methods with backup Oxygen supplies for an emergency situation. There are also Carbon Dioxide scrubbers and rebreathing systems currently in use that can add to the extended operating times required.

13.4 Cooling / Heating / Humidity

In orbit, our present craft encounter extreme temperature ranges. The Sun-ward side of a craft may a 200-or-more degree temperature. Yet the opposite side of the craft can be well below freezing. This raises issues on hull materials that can perform well across a wide range. In deep space away from our Sun, the cold side of our present craft would be "nice" in comparison. To have the crew survive properly for long terms, we need to keep the inside temperature within a safe range. Most average people can feel comfortable in the range of 68 to 72. We can survive in ranges higher and lower than this. But for long-term missions, comfort will become an issue.

We also have a humidity issue. Moisture and electrical equipment do not mix. Since much of the equipment on a starship will be electrical in nature, we will have to provide means of controlling this. Here again comfort will enter in. Too dry and too wet can both cause trouble. The temperature of the air determines its moisture-carrying capacity. Warm air absorbs more moisture than does cooler air. If moisture-laden air comes in contact with a cold surface, the air is cooled and its moisture-carrying capacity is reduced. Therefore, some of the moisture may be condensed out of the air and condensation forms on the cold surface. This phenomenon is illustrated by the condensation on an ice-filled glass in hot weather or condensation on windows in cold weather.

Basic construction techniques and procedures can prevent or minimize a lot of these problems. Generally, people are comfortable when the temperature and relative humidity are maintained 68 to 72 ^oF and 25 to 50 percent relative humidity. Normally mildew is a problem at a maintained relative humidity level above 60 percent, and static electricity is noticeable in cold weather at levels of 20 percent or lower. Both of these as well as inside moisture condensations are to be avoided on spacecraft.

Air conditioners do a good job of removing moisture. However, we also have a heating problem when in deep space. Dehumidifiers would seem to pose the proper answer here. The water from dehumidifiers can be piped directly to recycling systems onboard the craft and utilized for drinking, cooking, and for plants for Oxygen generation once filtered. This could be supplemented by some water collected from AC systems onboard during their limited usage. Active heating has already been utilized on present spacecraft. However, if we can utilize fusion in the next 20 years some heat recirculation could be accomplished with that.

13.5 Electrical Systems

Before we started building the ISS, electric power systems on U.S. space vehicles operated at the nominal 28 volts DC inherited from the aircraft industry. At such low voltages, plasma interactions were negligible and were not a consideration in spacecraft design. High-power systems such as those on the International Space Station operate at higher voltages to reduce power loss and system weight. But they suffer the drawback of interacting with the Ionospheric plasma in several different ways:

Internally, composites are the way to go. In an advanced society like ours, we all depend on composite materials in some aspect of our lives. Fiberglass -- developed in the late 1940s -- was the first modern composite and is still the most common. It makes up about 65 per cent of all the composites produced today and is used for boat hulls, surfboards, sporting goods, swimming pool liners, building panels and car bodies. Most composites are made up of just 2 materials. One material (the matrix or binder) surrounds and binds together a cluster of fibers or fragments of a much stronger material (the reinforcement). One modern advance has been Carbon-based composites which are both strong and lightweight. Certain spacecraft parts could be fashioned from this material.

Ceramics, Carbon, and metals are used as the matrix for some highly specialized purposes. For example, ceramics are used when the material is going to be exposed to high temperatures (e.g., heat exchangers) and Carbon is used for products that are exposed to friction and wear (e.g., bearings and gears). Here again, these have applications in starship design.

Kevlar is a polymer fiber that is immensely strong and adds toughness to a composite. It is used as the reinforcement in composite products that require lightweight and reliable construction (e.g., structural body parts of an aircraft). Composite materials were not the original use for Kevlar; it was developed to replace steel in radial tires and is now used in bulletproof vests and helmets. Its present usage in aircraft could be adapted to spacecraft usage.

In thinking about starship designs, it is worth remembering that composites are less likely than metals (such as Aluminum) to break up completely under stress. A small crack in a piece of metal can spread very rapidly with very serious consequences (especially in the case of aircraft). The fibers in a composite act to block the widening of any small crack and to share the stress around. They also have the advantage of being able to easily be sealed which for spacecraft is a plus. It is possible with near-term future developments in this area for the whole structure of a spacecraft to be built from composites.

Even the circuits of the electronic equipment could be "grown" and etched into these composite materials, just as we now use composite based boards for our integrated circuit systems. This would give use a smoother layout for our craft.

13.3 Oxygen Generation

To survive, we must breath. To breath, we must have Oxygen. As a result of the Apollo 1 spacecraft fire, the use of a pure Oxygen atmosphere during launch and ascent of Saturn V was abandoned by the U. S. space program. This is currently still the practice of NASA and one can assume it will remain the same into the future. However, we live in an atmosphere at average 15 psi on Earth. According to the Apollo Lunar Surface Journal, the values given for the Apollo 12 cabin pressure were 4.8 psi.

At Earth’s atmospheric pressure of 14.7 psi, this correlates to 55% to 60% oxygen content, which gives an oxygen partial pressure of 8.1 to 8.7 psi. NASA's decision to use a two-gas atmosphere (60% Oxygen, 40% Nitrogen) during manned Apollo on-the-pad preparations and in pre-orbital flight reflects a basic inability to make the spacecraft flameproof after 14 months of redesign that cost more than $100 million and added about 2,000 lb. to the system. Given the necessary electrical equipment on a starship and the need to cut weight, we doubt that in 20 years time such will have changed. So, the two-gas approach -- or even a more-Earth atmosphere combination -- makes total sense.

1. Conducting surfaces whose electrical potential is highly negative with respect to the plasma undergo arcing. Such arcing not only damages the material but also results in current disruptions, significant electro-magnetic interference, and large discontinuous changes in the array potential. Further, inbound ions -- accelerated by the high fields -- will cause sputtering from surfaces with which they impact.

2. Solar arrays or other surfaces whose charge is biased positively with respect to the plasma collect electrons from the plasma, resulting in a parasitic loss to the power system. Since the mass of an electron is much less than an ion, the magnitude of current density is much greater for surfaces with positive bias. At bias potentials in the 200-volt range, sheath formation and secondary electron emission from the surface causes the entire surrounding surface -- normally an insulator -- to behave as if it were a conductor. This effect -- called ”snapover” -- results in large current collection from even a very small exposed area.

3. Currents collected by biased surfaces also significantly affect the potentials at which different parts of the spacecraft will ”float”. (See the "Floating Potential" section below.) Because of their large mass and low mobility, ions collected by negatively biased surfaces result in a relatively small plasma current density. Lighter-weight electrons, on the other hand, are readily collected by positively biased surfaces. RAM and wake effects around the moving spacecraft further complicate the picture.

The worst situations occur when the spacecraft power system uses a negative ground, so that large surfaces are negative and collect slow moving ions to balance the current from electron collection. In this arrangement, parts of the spacecraft will be biased with respect to the Ionosphere to a level very near the maximum voltage used on the solar arrays. This charging of spacecraft as they float in the plasma of space --if not controlled -- can lead in some cases to mission degradation or worse. Loss of contact with some geosynchronous (GEO) satellites has been attributed to arcing caused by differential charging of the vehicle. There is also the problem of floating potential in low-Earth orbit (LEO), as with the space station. This problem for starships with their field generators will be even more pronounced.

In the GEO space environment, the major interaction of concern is differential charging of various parts of a spacecraft, leading to high electric fields and arcing between spacecraft components. In GEO, the ambient plasma thermal current is insufficient to discharge spacecraft surfaces rapidly. You can have a buildup of current to the actual potential involved very rapidly. Electrons can build up in space and can lead to increasing electric fields to adjacent conductors beyond a breakdown level of whatever insulators we employed.

Spacecraft surfaces charge up to (or "float" at) potentials that result in no net current collection from the plasma. If a spacecraft has no exposed conductive surfaces, it will float within a few volts of the surrounding plasma potential ("plasma ground"). Typically, a spacecraft will charge negatively in order to collect enough of the less mobile heavy ions. If conductors of similar area but at different voltages are exposed to the plasma, a rule-of-thumb is that the most negative surfaces will oat negative of the plasma about 90 percent of the total voltage difference between the surfaces.

In addition to the surface voltage, a spacecraft’s floating potential also depends on how it is grounded to the power system. For example, a spacecraft grounded to the negative end of a solar array (160 volts DC end to end) will float at about 140 volts negative from plasma ground, which is where the negative end of the array will float. If grounded to the positive end of the solar array, the spacecraft will oat close to plasma ground. Given the high voltage potentials involved in starship field generator designs, this voltage potential issue is even more dramatic. As the floating potential increases more negatively (greater than 30 volts negative of plasma ground), the severity of these effects also increases.

For the ISS, the grounding of the space station’s photovoltaic arrays (the negative end of the arrays are grounded to the structure) place the ISS at large negative electrical potentials (-160 volts DC) relative to the ambient space plasma when the arrays are producing power. In order to avoid sputtering and to lessen other undesired results of this grounding scheme, plasma contactor devices in the hull emit a steady stream of plasma from the station to raise its potential to within a few volts of the ambient plasma. This method could well be adapted to starship designs whose external plasma field potential will very according to warp field effects.

The plasma contactor acts as an electrical ground rod to connect the space station structure to the local environment and harmlessly dissipate the structure charges. A lot of the specifics of this issue alone will be determined by whatever power generation system the first starships employ. At present -- disregarding any future breakthrough in fusion power -- the best candidate is a fission reactor similar to those utilized on US submarines. The problem is cooling and the requirement of it being a closed-system to avoid excess use of water for the crew which will be a minimum. Here again, shielding issues enter in for long-term exposure protection. Solar power is not reasonable due to the long distances away from a decent solar source. A possible method of some version of MagnetoHydroDynamic (MHD) backup could exist to supplement energy onboard the craft as well as regular fuelcell power sources for battery-type backups. If we could develop the proper superconductors to utilize the older proposed Bussard collector system and proper fusion systems could be developed, then this would also hold promise for power generation coupled with MHD type systems.

While anti-matter holds possible future promise, a lot of what is needed for this to be developed will involve coming up with working fusion-based systems. At present this is still only a dream in the experimental stage. It might be possible to achieve a pulsed fusion system instead of a constant system in the near future. Again, development of a Bussard collector system could provide the needed fuel for such a power generator.

NASA currently has over 30 spacecraft in Earth-orbit and in deep space on missions of scientific exploration. Most of these missions rely primarily on solar power -- generated using large wing-like solar arrays -- along with battery back up for electrical power to operate the on-board scientific equipment, cameras, radio communication systems and computers. Missions which operate in regions of space where solar arrays cannot provide adequate power rely on radioisotope power systems (e.g., Radioisotope Thermoelectric Generators [RTGs]) which generate electrical power from the heat of decaying Plutonium in specially designed rugged containers. Starships fall within this area. The problem is they only deliver small power at the present and would require far larger more long-term system for a starship.

The RTG design used on the Cassini mission generates electrical power by converting the heat from the natural decay of Plutonium through the use of solid-state thermoelectric converters. It is about 113 cm (about 44 in) long and about 43 cm (about 17 in) in diameter, and contains about 10.8 kg (about 24 lb) of Plutonium Dioxide. RTGs have been designed to contain their Plutonium Dioxide in the event of launch or reentry from Earth orbit accident. NASA is working with the Department of Energy to identify power requirements of future spacecraft and to design smaller and more efficient power systems. These power systems may only need to carry about 2-3 kg (about 4-7 lb) of nuclear material for power generation.

13.6 Sensor Equipment

When man first ventures to the stars, sensors for such a craft will have to be developed. We will not only need the normal sensors like radar, IR, EM,etc., but also sensitive gravity wave detectors, sensors designed to detect signs of biological activity, and telescopic systems designed to detect planets at a vast distance will be required. The list goes on. We will also have to have these tied into a dedicated computer system to sort through the information.

Navigation alone will require its own sensor systems and a means of properly plotting a source in areas no one has traveled before. Information gathered along-the-way will help update our navigational data as we explore outward. We would suggest that pulsars could be used as reference points to determine our position as we travel. But an initial map -- based upon data that we have at present -- would have to be built into the computer systems data.

Also an extensive database of information will be needed by the craft in the event of trouble, since at present without some means of FTL communication a starship would not have the luxury of "calling home" for advice and help.

13.7 Computer Systems

Computers will have to be designed to "network" on a starship. The computer behind Navigation has to interface with not only the Engineering Propulsion computer, but also with the Science Monitoring computer tied to the sensors and with Life Support computer control. Fiber-optics hold promise in the network of this system as well as advanced multiprocessors. Another area is quantum computers. If developed, they could vastly increase the processing speed and memory of a starship computer system.

In the area of networking we are looking at something better than a T3 connection between each subsystem in the net. Speed is vital on a craft traveling in excess of the speed-of-light. If another car pulls out in front of you while doing in excess of 100 mph on the Autobahn, you have to break a mile in advance to slow down. Likewise, in space at the speeds we've been discussing you have to be able to drop from warp or change your course faster than man has ever achieved before. At present not only are our sensor systems too slow, but even our computer systems are also too slow compared to what is needed. At c-- traveling at 186n300 mps --even 0.000001 second equals 0.1863 miles. Take that example and figure it for say, 100 c , and you have 18.63 miles in 1 millisecond. At 1000 c you have 186.3 miles in the same time. Even though our present computers operate faster than a millisecond, they are still somewhat slow for the task given the main bus speed for interface to external devices. This is one area that will require further technological advances.

Bibliography

[1] S. Perlmutter, et al. Measurements of Ω and Ω From 42 High-Redshift Supernovae, Astrophys. J. 517 1999 565-86 astro-ph/9812133

[2] Miguel Alcubierre Class. Quantum Grav. 11 (1994), L73-L77

[3] K. Olum, Superluminal Travel RequiresNnegative Energies, Phys. Rev. Lett., 81, 1998 3567-70 gr-qc/9805003

[4] Michael J. Pfenning, L.H. Ford, Class.Quant.Grav. 14 (1997) 1743-1751

[5] Chris Van Den Broeck, Class.Quant.Grav. 16 (1999) 3973-3979

[6] F. Loup, et al. A Causally Connected Superluminal Warp Drive Space-Time, gr-qc/0202021

[7] S. V. Krasnikov, Hyperfast Interstellar Travel in General Relativity, Phys. Rev. D, 57, 1998 4760 gr-qc/9511068

[8] Everett A. Phys. Rev D 53 1996 7365

[9] W. A. Hiscock, Quantum Effects in the Alcubierre Warp Drive Spacetime, Class. Quant. Grav., 14 1997 L183-88, gr-qc/9707024

[10] P. F. Gonz´alez-D´ıaz, On the Warp Drive Space-Time, Phys. Rev. D, 62 2000 gr-qc/9907026

[11] C. Clark, W. A. Hiscock, and S. L. Larson, Null Geodesics in the Alcubierre Warp Drive Spacetime: the view from the bridge, Class. Quant. Grav. 16 1999, 3965-72 gr-qc/9907019

[12] J. Natario, Warp Drive With Zero Expansion, Class. Quant. Grav., 19 2002 1157-66 gr-qc/0110086

[13] Seriu M. Back-Recation on the Topological Degrees of Freedom in (2+1)-Dimensional Spacetime, Phy. Rev. D 1995 gr-qc/9511046

[14] From http://www.geocities.com/lifsspeed/

[15] Watt and Misner gr-qc/9910032

[16] P. Di Vecchia, A. Lerda, P. Merlatti, N=1 and N=2 Super Yang-Mills Theories from Wrapped Branes, hep-th/0205204

[17] V. F. Mukhanov, JETP lett. 44,63 (1986)

[18] A. Ashtekar and K. Krasnov, in Black Holes, Gravitational Radiation and the Universe

[19] J. D. Berkenstein, in Proceedings of the VIII Marcel Grossmann Meeting on General Relativity gr-qc/0205022

[20] Breitenlohner and Maison. Supersymmetric Regularization of Gauge Theories (1977)

[21] Aharonov-Bohm effect http://psroc.phys.ntu.edu.tw/cjp/v40/130.pdf

[22] The Papapetrou Equation Derived as a Geodesic in a Non-Holonomic Clifford Manifold, http://www.clifford.org/wpezzag/talk/98oregon/

[23] A. Ashtekar and K. Krasnov in Black Holes, Gravitational Radiation and the Universe

[24] R. Bracewell. The Fourier Transforms and its Applications Mc-Graw-Hill 1965

[25] H. E. Puthoff, Polarizable-Vacuum Approach to General Relativity, gr-qc/9909037 (1999)

[26] G. M. Shore gr-qc/0203034

[27] O.J.F. Martin and N.B. Piller. Electromagnetic Scattering in Polarizable Backgrounds. Phys. Rev. E, 58:3909-3915, 1998

[28] Yuri Obukhov and Guillermo Rubilar. Fresnal Analysis of Wave Propagation in Nonlinear Electrodynamics gr-qc/0204028

[29] E. T. Wittaker, On an Expression of an Electromagnetic Field due to Electrons by means of Two Scalar Potentials, 1904

[30] Ford and Pfenning gr-qc/9905084

[31] F. Loup, D. Waite, E. Halerewicz, Jr. Total Reduced Energy Requirements for a modified Alcubierre Warp Drive Spacetime gr-qc/009013

[32] P. Hoiland. A Modification to M-Theory http://www.theory.mybravenet.com &

http://www.superstringtheory.fanspace.com

[33] Modern Relativity. http://home.aol.com/zcphysicsms/

[34] M Morris, K Thorne, U Yurtsever. Wormholes, Time Machines, and the Weak Energy Condition. Physical Review Letters, Vol. 61, No. 13, pp1446-1449, 26 Sep 1988

[35] D. Hochberg and T. Kephart. Vacuum Squeezing. Phys Letters B, No. 268, pp377, 1991

[36] The Quantum Inequalities Do Not Forbid Spacetime Short-Cuts. gr-qc/0207057

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