More on the Work

Numerical evaluations

Related links

Closing

1 Introduction

A free discussion forum was established in late December 2000 by Fernando Loup in honor of Miguel Alcubierre's work  on the warp drive spacetime.  Loup had a dream of showing both the scientific community and the world at large the reality of the warp drive spacetime. The discussion forum was open to anybody and everybody willing to entertain the idea of a working warp drive spacetime.  The discussion forum the Alcubierre Warp Drive hosted by  yahoo (currently available at the following URL: http://clubs.yahoo.com/clubs/alcubierrewarpdrive), is what brought the authors of this work together to envision a "modified" warp drive.

1.1 The Spacetime

The main concept behind our proposed warp drive came directly from independent work developed by David Waite (the early work, and subsequent others are available thorough Modern Relativitity).  Previous research work into the energy requirements of the warp drive focused on [global] remote frames, so Waite began to investigate the local energy requirements of the warp drive spacetime within the [local] ship's frame, this is where the story of our warp drive begins.  By adding a time dilatation coefficient of order A(ct,r) to the ship's frame one can reduce the stress energy energy terms T_ab and T^ab, simply by making A arbitrarily large in the warped region!

1.2 The Discussions

The online discussion forum the Alcubierre Warp Drive, began as a novel experiment, where members would answer questions that interested parties had for the new science.  The early discussions within the Alcubierre Warp Drive began with philosophical debates, random brain storming, and slowly evolved to include mathematical discussions.  It was during this new phase where Waite joined the club discussion, and we carefully began discussing the ramifications of his proposed warp drive.  Unfortunately as soon as the mathematics became the bulk of the discussion, most of the philosophical debates were lost, but at this point Fernando Loup and Edward Halerewicz began to describe the possible consequences of Waite's spacetime.  During the discussion of Waite's idea the club was very grateful to receive advice and guidance from the very busy but gracious Dr. Alcubierre.  And it was at this point that simply entertaining the idea of a new warp drive became a much more formal process, and a new theory of its own right began to form.

2 The Work

It can be shown that negative energy requirements within the Alcubierre spacetime can be greatly reduced when one considers the Einstein tensor in the ship's frame.  Thereby reducing the negative energy requirements of the warp drive spacetime arbitrarily as a function of A(ct,r).  With this function new quantum inequality restrictions are investigated in a general form.  Finally a potential method for generating a warp velocity greater than that of light is presented.

So what are the results of our applied efforts you ask?  Waite started with a new approach on how the energy requirements within the warp drive can be greatly lowered, by simply looking at the energy requirements of the ship's frame.  Although this approach hadn't been used in the previous literature, this alone could not lower the extreme magnitudes negative energy required by the existing literature.  However by introducing  an arbitrary time dilatation coefficient A(ct,r), as similarly done by Hiscock, the negative energy densities from equation 19 of Alcubierre's work can be reduced greatly.  The coefficient A(ct,r) has the following properties a=A=1 in the ship's frame and far from it, and A has large values within the bubble walls R which act to lower the negative energy densities.

Figure 1:  The ESAA spacetime represented one dimensionally in Cartesian coordinates, with a=1 this spacetime appears hyper-dimensional in comparison to the Alubierre spacetime f(rs)=0.  This figure represents the varying Warp Drive parameter N the minimum value N=B(rs)=1 gives the Alcubierre top hat f(rs)=1 (black).  The median (gray) and maximum (white) represents an energy reduction due to a decreased volume as a factor of the warped region with an arbitrary boundary a=1+(R+(D/2))^N, when R=100 m the "maximum warp" factor is N=9.3.

2.1 Possible solutions to the Einstein Tensor

From GR, the meaningful solutions (i.e. solutions which corresponds to classical physics) for the stress energy calculations result from the contrivaratint solutions T^ab.  We had entertained the idea of a covariant solution (simply because this was not done in the previous literature), but quickly withdrew it because it greatly complicated the warp drive we chose (see the links within Section 2.3 for reference).  But in studying the relationships between covariant and contrivariant solutions, we have found an explicit case where a covariant solution can act to subtracted energy from the proper time coordinates.  We achieved this result by causing the warp drive to behave similarly to the Alcubierre top-hat function.  This result is however erroneous in reference to our work because we modified our result to coincide with Pfenning's piecewise function, thus we arrive at a completed total energy calculation of order (by implementing the boundaries associated with figure 1):

E_PI=-(b² c⁴/12G)*(1/D²)*(1/(R+{D/2})^4N)*((R+{D/2})³-(R-{D/2})³/3)

thus arriving at an energy of approximately equal to one galactic mass.  When this is however extended to a five-dimensional Anti de-Sitter space, the integral need not be confined to the the boundary R+(D/2), thus resulting in an energy approximately equal to100,000 solar masses.  These results however are purely arbitrary by defining N=1, R=100, and D=16 for A(ct,r), for the above equation the maximum warp factor is given by N=19.8 (see figure 1 above), thus resulting in a minimum energy requirement of -0.024 grams, for one possible boundary of the warp drive spacetime.

2.2 Numerical evaluation of A(ct,r)

To obtain a better understanding of the chosen time dilation coefficient click here.

2.3 Exact numerical evaluation of the ESAA spacetime

The exact contrivariant soultions of the Einstein Tensor, in regards to the Waite spacetime, provided by GRTensorII package through Maple VI, are provided here.

3 Related Links

Although there are great number of useful Web Links pertaining to the warp drive, we have chosen to only to identify links affiliated to the authors of this work.

Alcubierre Warp Drive (Open warp drive discussion forum available through yahoo)

Modern Relativity (affiliated page of David Waite)

Loup Warp (homepage of Fernando Loup)

Warp Drive Today (popular description of the warp drive by Edward Halerewicz)

4 Closing Remarks

In conclusion it can be shown that the warp drive spacetime can not be ruled out because of `unphysical energy conditions', by simply choosing an arbitrary function the energy reductions become rather dramatic.  It is also however noted that an arbitrary function is indeed a mathematical trick, within a four-dimensional spacetime there is no mechanism to make N>1 as far as we know.  However, it is somewhat easy to consider N=2 for a "hyper drive" spacetime, in our discussion this would describe a five-dimensional Anti de-Sitter spacetime, and without experimental results this may also be considered a mathematical trick (in which the "warp bubble" can be interpreted to have velocity that is 100 times that of light).  It is therefore noted that the warp drive may have several limitations within semi-classical general relativity, such as the violation of the Null Energy Condition (NEC), thus post relativistic corrections would make it appear that the warp drive spacetime still has life in it!

Copyrighted © 2001: Edward Halerewicz, Jr.

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