Space-Fleet Academy

The Sarfatti Lectures in Star Ship Science

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Star Gates and Time Travel

A Review by Jack Sarfatti of Matt Visser's book,

Lorentzian Wormholes

(AIP, 1995)

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Feynman Lectures on Gravitation

-- but don't forget to come back here! :-)

Let's start at the end and work backwards to the beginning. (Did you ever wonder why Hebrew is read right to left instead of left to right? :-)) The final statement of the book is Research Problem 27 "Quantize gravity!".

Visser's "Coda" is that while speculative his book is conservative. Research in quantum gravity is "still rather murky and confusing". His own personal choices are "Topology change is bad. Traversable wormholes are good. Time travel is bad." (p. 374) By time travel he means time travel to the past. Certainly at the semi-classical level it looks like Hawking is correct, vacuum fluctuations disrupt the traversable wormhole before it can be used to violate causality and travel into the past. The only loop hole is in Peter Holland's book,''The Quantum Theory of Motion'' (Cambridge, 1995) in which an alternative to the WKB method of the semiclassical approximation is possible within Bohm's paradigm. Traversable wormholes will still be good for fast interstellar travel even if they can't be used to alter the past.

Let's go to Chapter 17. Distinguish two kinds of time travel paradoxes. "A logical inconsistency in an apparently plausible argument. An apparent inconsistency in a perfectly correct argument" (p. 203). We have "consistency paradoxes" (17.2.1 p.212) like the "grandfather paradox" in which you kill your grandfather before your father is conceived. There are several possible solutions to this class of time travel paradox. The second is the "bootstrap paradoxes" (17.2.2 p.213) as given, for example, in the movie "Star Trek IV - The Voyage Home" "Ask yourself: Who discovered 'transparent aluminium'?" (p.214).

Quantum computer theorist, David Deutsch, calls this the "principle of the philosophy of science". When it is violated, complex ideas appear spontaneously. I find a book, I send it back in time to myself and publish the book. Who wrote the book? No one and "while there are no logical inconsistencies ... the purported effects are certainly weird." (p.213). This is Bergson's "principle of creative evolution" and Terence Mc Kenna's "principle of novelty" in which the universe is a "novelty-creating engine" pulled by "the transcendental object at the end of time" which sounds a bit like Frank Tipler's "Omega Point". The idea is that the evolution of the universe is pulled to the future rather than pushed by the past -- a combination of both is in the Wheeler-Feynman action-at-a-distance classical electrodynamics, and in Aharonov's double state vector theory of quantum mechanics. Once can make the case that all intuition and creative thought by artists, craftsmen and scientists involves the subconscious receptions of ideas from the future which literally create themselves.

Next we leap to Chapter 19. There are four possible responses to time travel paradoxes. "1. The radical rewrite conjecture. 2. The Novikov consistency conjecture. 3. The Hawking chronology protection conjecture. 3. The boring physics conjecture." (p.249) The radical rewrite conjecture makes spacetime a non-Hausdorff manifold. (19.1.2) One can have "point doubling" and "line splitting".

"Thus one can, for instance, impose the Einstein field equations in the usual manner. Every now and then, however, a 'temporal anomaly' induces a 'non-Hausdorff wavefront' which duplicates the whole universe... This structure is now sufficiently rich to be able to offer a 'multiple history' resolution of the consistency paradoxes... The moment x that he/she/it steps out of the time machine into the past... a 'temporal anomaly' occurs. A new history (a new timeline) is initiated... the traveler can change history all he/she/it likes ..." (p.253).

An alternative is the Novikov consistency conjecture (independently discovered by me ) in which there is only one actual Hausdorffian universe, but all would-be causality-violating mechanisms (including our free will) are globally constrained to fail around closed time like curves or in communications on nonlocal quantum connections which violate standard quantum mechanics but not all possible extensions of it. This, so far, is all classical physics.

What about quantum effects? 19.2.2 lists three possibilities.

1. The "test particle limit" in which spacetime is fixed and Hausdorff and we look at the quantum mechanics of a particle on a closed timelike world line.

2. The "test field limit" same as above but use quantum field theory rather than particle mechanics.

3. "Permit spacetime itself to participate in the quantum fluctuations." This introduces non-Hausdorff topologies. David Deutsch [1] identifies non-Hausdorffian splitting spacetimes with the Everett-DeWitt "many-worlds" interpretation of quantum mechanics. How does Bohm's theory fare in non-Hausdorff topology? Visser also asks how Einstein-Podolsky-Rosen (EPR) quantum connections fare. "... since Einstein-Podolsky-Rosen correlations are nonlocal the picture of universes splitting at a point, followed by the split propagating at light speed, has too much a flavor of 'local realism' to be compatible with quantum mechanics. ... In the presence of closed timelike curves it is far from clear what happens to EPR correlations." (p.257)

".. the universe might not even have a unique past ... if one takes Feynman's 'sum over paths" notion of quantum mechanics seriously, then all possible past histories of the universe should contribute to the present state of the universe. " (p. 258)

Note that Einstein, Tolman and Podolsky wrote a paper "Knowledge of Past and Future in Quantum Mechanics (p.135 of Wheeler and Zurek's "Quantum Measurement" (Princeton, 1983) that asserts "the principles of quantum mechanics actually involve an uncertainty in the description of past eventsmwhich is analogous to the uncertainty in the prediction of future events." Remember Orwell's "1984"? -- it may be closer to reality than we imagine. :-)

19.2.3 is on interpretations of quantum mechanics. Visser lists six.

1. Copenhagen -- ''the official party line''

2 Everett's "relative state" popularized as "many worlds" -- the favorite of quantum cosmologists.

3.Bohm's nonlocal hidden variable "Works well for non-relativistic systems but does not seem to have a clean relativistic extension. Highly nonlocal. Not really popular with anybody" (p.258).

4. Cramer's transactional interpretation This is a reworking of the Wheeler-Feynman action-at-a-distance classical electrodynamics for the Schrodinger equation. Hoyle, Narlikar and Bennett have shown that quantum spontaneous emission induced by virtual photons in the vacuum fluctuations here-now are actually advanced effects from the future that also explain classical radiation reaction. Furthermore, the standard post-inflationary open universe Big Bang cosmology does not obey the Feynman-Wheeler total absorption boundary condition which means that there are unbalanced advanced effects of the future on the past if that cosmology is the right one.

5. Decoherence (Gell-Mann, Hartle, Zurek, Zeh). "Consistent histories"

6. "Shut-up-and-calculate non-interpretation: Extremely popular when teaching"

19.2.4 is good it is called "Parametrized post-classical formalism?" (p.259) Visser points out that (Post-Newtonian) PPN tests of general relativity have been useful. There are other weak-field alternatives to Einstein's general relativity like Dicke-Brans and Einstein's always wins experimentally. Weinberg's nonlinear quantum mechanics is a PPC as is Bohm's idea of making the particle a source of its own wavefunction. The latter is a nonunitary theory that will, I suspect, give the "bootstrap" pseudoparadox discussed above.

19.2.5 has more PPC technical details for quantum field theory. There is a consensus that time travel to the past demands "nonunitarity" in the corresponding quantum theory. (p.260) The general argument given by Visser is:

1. "Perturbative unitarity of a quantum field theory is equivalent to the existence of Cutkosky rules for the Feynman diagrams." p.260

2. Cutkosky rules are equivalent to Feynman iepsilon rule for the boundary condition of the propagator (i.e. choice of contour in complex energy plane around the mass shell poles of the propagators).

3. This Feynman rule is equivalent to the use of time-ordered products of fields in the propagator.

4. There is no such time order in presence of closed timelike world lines.

David Deutsch offers an experimental test of the many-worlds interpretation of quantum mechanics. One must acquire a time machine. Use non-Hausdorffian "many time lines" and identify them with the many worlds. Accept the loss of unitarity (i.e., total probability is not conserved) in presence of closed timelike curves. This loss of unitarity is rigorous for a single-timeline Hausdorrf universe subject to Novikov global self-consistency. [2] Deutsch goes beyond that to the non-Hausdorff case in which "the quantum physics of the entire class of possible timelines is assumed to be described by a single quantum density matrix that is subject to Novikov style consistency constraints." (p.261).

OK, now to Chapter 2 which is a quick review of classical general relativity. Start with the differential square ds^2 of the invariant spacetime interval between two nearby spacetime events in terms of the second rank metric tensor g. Introduce the affine connection which "governs the 'acceleration' of a freely-falling particle in a gravitational field" (p.10) This is a timelike geodesic for an ordinary massive subluminal small "test particle". The geometrical distortion of the stress-energy of the test particle can be safely ignored. Geodesics are the "straight lines" of a warped spacetime. They are the extremums of a global "action" which gives a local geodesic equation. The slower-than-light timelike geodesic for a freely falling test particle is a maximum not a minimum as you might first expect. This corresponds to the fact that the twin who accelerates in starship returns younger than the twin left behind on a geodesic. The earth's motion is approximated by a geodesic in this example. The full fourth rank curvature tensor is built from the connection and it is contracted with the metric tensor g into a second rank Ricci tensor which in turn is contracted again into a Ricci scalar. The second rank Einstein tensor describing the warped spacetime geometry G is the Ricci tensor minus (1/2) Ricci scalar multiplied by the metric tensor. Einstein's field equations are

Einstein Geometrical tensor = ( 8pi Newton's constant/square of speed of light) Stress-Energy Tensor of Matter and Radiation

The stress energy tensor T has dimensions of [energy/spatial volume = force/area] its time-time component is the local energy density at a point event. Its mixed space-time components is the flux of energy (i.e., a generalized Poynting vector) and its space-space components are the stresses which are generalized pressures. For example, Txy is the y-flow of x-momentum. Ttx is the x-flow of energy. A good reference for this is the Feynman Lectures in Physics Vol II. These local field equations can be derived from a global action that breaks into three terms. The first is the integral over all of spacetime from the Big Bang to the End Time (if there is one) of the Einstein-Hilbert Lagrangian which is the Ricci scalar multiplied by the square root of the determinant of the metric tensor. The second term is the Gibbons-Hawking surface term which is an integral over a three-dimensional boundary of the trace K of the second fundamental form multiplied by the square root of the induced 3D-metric. It cannot not always be safely ignored. (p. 12) The final term comes from the Lagrangian of all the quantum fields of matter and radiation.

Weak fields (2.2) correspond to a linearization of the highly nonlinear Einstein field equation. There is an anology to Maxwell's electromagnetic field equations here which significant differences such as the nature of the polarization of the waves. One gets Newtonian gravity in this weak field limit when the speeds of particles are small compared to the speed of light.

Strong fields (2.3) has the ADM formalism (2.3.1) which is a 3+1 split in a coordinate patch. The metric tensor has a time-time component of the form minus square of the lapse function plus contraction of the 3D metric tensor with the shift function. The mixed time-space components are the shift function ans the space-space components are the afore-mentioned 3D metric tensor components. The lapse and shift functions describe how the 3D spacelike slices are put together (better than Humpty Dumpty) into a 4D spacetime.

Black holes and horizons are discussed in (2.3.2). Horizons are 2D one-way membranes permitting passage of particles on timelike and lightlike worldlines in one space direction only. If the 2D surface is closed, these particles cannot escape from the inside the horizon to the outside. Furthermore, relative to an observer on the outside, the horizon is an infinite redshift surface in which time literally stops.

Visser lists several types of horizon.

(2.3.3) First is the "event" or "absolute" horizon in an asymptotically flat spacetime. The surface of a black hole is such an event horizon. Note, you can freely fall on a timelike radial geodesic through the event horizon of a large black hole and not feel any tidal force there until you get close to the singularity which radially stretches and tangentially crushes you out of existence. The outside observer will never see you fall through the black hole. Your image will be red-shifted out of detector range before you reach the surface, but the proper time you experience will be too short for you unless you are suicidal. In contrast if you are firing your rocket engines and hovering at a fixed distance above the black hole, you will feel tidal forces that get infinitely strong as you unsuccessfully try to approach to zero distance above the surface of the hole. Thus, the gravitational forces you feel depend very strongly upon the nature of your worldline. Hovering is a non-geodesic worldline near the black hole requiring firing of the star ship engines radially away from the horizon.

Note from Bruce Bowen (bbowen@megatest.com) clarifying this last point.

Two references are:

Gravitation , by Misner, Thorne and Wheeler; aka MTW

Essential Relativity, Wolfgang Rindler; aka Rindler

Fact #1. The radial proper distance to the event horizon (r=2M) from any Schwarzschild "r" is finite. See MTW, eq 31.9, pg 824.

Fact #2. The local static acceleration or "weight" of a stationary observer diverges or goes to infinity as "r" approaches 2M.

The per unit weight of an object (ie, local proper acceleration in conventional units is:

(GM/r^2) (1 - 2GM/(c^2 r))^(-1/2)

See Rindler, eq 8.71, pg 148.

[note: An easy way to remember the above is Newton's formula divided by the redshift factor.]

By now the cogs in the brain of the astute physicist or mathematician should be turning and he or she should begin to realize that the weight diverges over a finite distance, and this diverging is occurring outside the event horizon. So the static tidal forces, defined to be: d(weight)/d(proper distance), or the weight gradient, also diverges as one approaches the event horizon.

So now the budding general relativist is thinking to himself, "I'm so confused. How can this be? I thought there are finite tidal forces at the event horizon?"

Well let's look at this situation from a free-falling observer. Since the event horizon represents a stationary light-like surface, by definition any infaller falls thru the horizon locally at the speed of light. So he is locally Lorentz-contracted to zero thickness with respect to a stationary observer at the horizon (for the purists, use limits and pretend you are a small distance epsilon outside the horizon, and then let epsilon -> 0.) Also at the event horizon the angle of escape for light (and thereby anything else as well) is zero degrees with respect to the vertical, all freefalling trajectories are vertical at the event horizon. Since the infaller is of zero thickness in the radial direction (ie, flat ) his d(proper distance) in the radial direction is zero To get the tidal force on the faller we multiply the static tidal force times the faller's radial extent. We get: infinity * zero which is an indeterminate form. If you work this out in detail you will see that the above product remains finite for a free falling object. So the tidal forces on a freefaller remain finite, although they vary depending on his 4-velocity. Since the faller is in freefall, his static weight is of course zero.

(2.3.4) defines a "trapped surface". Any point on a trapped surface can be a Huygens source both inward and outward propagating wave fronts of light. If the area of both these wave fronts decrease as a function of time from past to future then the surface is trapped. One of these wavefronts is inside an "apparent horizon". Visser is not too clear here.

(2.3.5) defines the "Cauchy horizons" which are "associated with the onset of unpredictability" (p.17) There are several types of initial data on a given spacelike surface in this classical theory. All of them violate the quantum Heisenberg uncertainty relations in the Smoky Dragon Copenhagen interpretation. The situation is more subtle in the Bohm hidden variable interpretation.

1. Source particle positions and initial velocities.

2. Gauge force field configurations and their time rates of change.

3. 3D metric configuration and its time rate of change in the ADM formalism.

Assuming the principle of retarded causality, the future Cauchy horizon of the given spacelike surface is the boundary of the region from which all past-directed causal worldlines (i.e. timelike or null lightlike) intersect the spacelike surface.

(2.3.6) "A particle horizon occurs whenever a particular observer never gets to see or be influenced by the whole spacetime." This is a purely classical notion that ignores the spacelike quantum influences of standard quantum mechanics which are in violation of Bell's theorem and which have been detected experimentally even though they cannot be used to send intelligible decodable messages faster than light.

(2.3.8) introduces the "Killing vector field", but you need to read a standard text to understand what it is.

In any stationary spacetime the event horizon and the outermost apparent horizon coincide. Loosely speaking, stationary <--> time independent. More precisely, stationary <--> the metric posesses a (globally defined) timelike Killing vector ... Because the spacetime is stationary it has a unique time function such that the components of the metric are time independent ... p.18

(2.3.9) is the spherically symmetric Schwarschild metric tensor with vanishing Ricci and stress-energy tensors. The "coordinate singularity" is at the Schwarzschild radius is 2GM/c^2 , where M is the mass seen by a far away observer where the metric is flat in four dimensions. If the actual matter is inside the Schwarzschild radius we have a black hole. The "physical singularity" is at zero radius. Note that, for the black hole case, a geodesic observer falls through the coordinate singularity with no ill effect if the Schwarzschild radius is large enough, but, as remarked above, a hovering non-geodesic observer outside but very close to the event horizon at the Schwarzschild radius, will feel infinite tidal force at this coordinate singularity horizon (as explained by Bowen above). Therefore, it is misleading to say that the Schwarzschild singularity is not "physical". This is explained most dramatically in the "Prologue" to Kip Thorne's Black Holes and Time Warps (W.W. Norton, 1994). Thus, I take issue with Visser's remark (also found in many other books -- see also 2.3.3 above)

In pre-1960s books and papers this is often referred to as the "Schwarzschild singularity". This is is a complete misnomer as there is no singularity at r = 2GM/c^2; all that happens is that the coordinate system is breaking down there. The coordinate singularity is no more physical than the singularity occuring at the north and south poles of the Earth when using latitude and longitude coordinates." p.20

Compare this with Kip Thorne's remark for a 10 solar mass black hole with a Schwarzschild radius of 185 km.

Bidding farewell to your crew, you climb into a space capsule and drop out of the belly of the starship and into a circular orbit along side it. You then blast your rocket engines ever so gently to slow your orbital motion a bit. .. Your goal, by this gentle, safe inward spiral, is to reach a circular orbit just above the horizon, an orbit with circumference just 1.0001 times larger than that of the horizon itself ... as you float in your capsule... with your feet toward the hole and your head toward the stars ... at a circumference of 100,000 km, the head to foot difference is 1/8 g ... At a circumference of 30,000 km it is 4 g ... you continue to 20,000 km and a 15 g stretching force ... If your capsule spirals in much farther, your body will give way, you will be torn apart! There is no hope of reaching the horizon's vicinity. pp 34-35

In fact, then, the Schwarzschild coordinate singularity is also physical if you are on the wrong world line. Therefore, the pre-1960's book Visser mentions had a valid point and his charge of "misnomer" is really not justified. Visser's view is common in modern books. Only Kip Thorne's book appears to portray the situation accurately. Even though the curvature tensor is not infinite at the coordinate singularity, Thorne claims infinite tidal force there for an observer on a timelike non-geodesic circular orbit.

Fig. 2.2 p. 21 is the Kruskal diagram in "isotropic coordinates" for the maximally extended Schwarzshild spacetime. Solid crossed lines intersecting at right angles denote the event horizons at the "coordinate singularity" while two hyperboloids are the actual "physical singularities" (i.e. r = 0) where the fourth rank curvature tensor is infinite.

Fig. 2.3 p. 22 is the Penrose diagram for this same solution. The two inside dashed diagonal lines are the "coordinate singularity", the two horizontal solid lines are the "physical singularity". The maximal analytic extension has two asymptotically flat regions each with a coordinate patch. The original Schwarzschild solution was only a single coordinate patch. We will show later how this is the basis for the Einstein-Rosen "bridge". Visser also discusses the charged black hole and the possibility of a "naked singularity" when the charge is bigger than the mass (in suitable dimensions).

Chapter 3 is a brief overview of special relativistic quantum field theory in flat spacetime. (3.1) describes canonical quantization in the Schrodinger picture where the field operators are time-independent and the state vector in Hilbert space is time-dependent. Interacting fields require the interaction picture where a unitary transformation generated by the free parts of the fields is made. The math here is that of functional derivatives. The state vector in Hilbert space is a functional of the field configuration over an entire 3-geometry or spacelike surface. One has to use the formal square root of the negative Laplacian. The infinite ground state energy is the trace of this pseudodifferential operator (eq 3.14, p.33). The two-point vacuum field correlation function (eq. 3.15, p.33) is of particular interest. It gives a quantum field fluctuation of order square root of Planck's constant h divided by the length scale L of the fluctuation (eq. 3.22, p.34).

(3.2) is on Feynman functional integrals in the Heisenberg picture in which the field operators are time-dependent and the state vector is time-independent. The functional integral that defines the statistical ensemble expectation value of any quantum property involves an "integration measure" on the set of all possible paths in the classical configuration space of the generally many-particle or field system. The integrandis the property multiplying a unimodular phase factor in which the phase is the classical action of the path divided by hbar. Feynman's approach gives the same vacuum fluctuation as does the canonical method. (p.37)

Chapter 5 is on the Einstein-Rosen bridge from the Schwarzschild solution of Chapter 2. It was first introduced as a model for an elementary particle with the particle as the bridge connecting two flat spacetime sheets. It is only a formal device and has no immediate physical application for fast starship travel application. A faulty detection of a hidden black hole might look like such a bridge. Its importance is historical as a precursor to traversable wormholes.

Coordinate artifacts will not protect one from an untimely demise should one be so foolish as to attempt crossing an Einstein-Rosen bridge. p 51

Chapter 6 is on the spacetime foam of quantum gravity. (6.1) is on the tiny Wheeler wormholes with trapped electric flux derived from the Einstein-Rosen bridge solution. Such wormholes on the Planck scale of 10^-33 cm are virtually excited in the quantum vacuum fluctuations of the metric geometry itself - or so it would seem. The size of this metric fluctuation is Planck scale/L.

(6.4) deals with the unsolved problem of topology changes in 3D spacelike slices of 4D spacetime. Does the Feynman path integral include metrics with different numbers of Wheeler wormholes in the spacelike slices? A very important clue is that topology change requires violation of causality in classical general relativity . (p.63) This idea is further developed in (6.5) at the classical level. If you believe in causality at all costs then you must reject topology changes at least in the classical limit. See "Topology change in classical general relativity" gr-qc/9406053 by A. Borde. The essential conflict between causality and topology change reaches down from the classical to the quantum level as we see below. The emphasis on this conflict is, perhaps, the most valuable contribution of Visser's book to the ongoing discussion.

(6.6) deals with topology change at the quantum level. (6.6.1)'s "Theorem 12" on p.68 is

Canonical quantization of the gravitational field is incompatible with topology change.

(6.6.2) considers topology change in the Feyman path formalism. The particular "i epsilon" rule of Feynman for the contour in the complex energy plane around the mass shell poles of the propagator use both retarded waves of positive frequency and advanced waves from the future of negative frequency. One can "Wick rotate"in the complex energy plane from Lorentz topological signature -+++ to Riemannian signature ++++ without having to cross a mass shell pole. As Hawking says there are no "obstructions" in relativistic quantum field theory in flat spacetime. The situation is different in curved spacetime. One cannot make an unambiguous separation into positive and negative frequencies. The Wick rotation is obstructed. Unlike canonical ADM quantization, the Feynman "inegration measure" is broad enough to include all possible views.

... one is led to at least ten different models for what it means to quantize gravity via functional integrals. p.69

(6.6.3) Shows that topology change by quantum tunneling is possible

if and only if changes in the topology of (3D) space are already kinematically allowed at the classical level. p.71

That means causality-violation as with closed-timelike curves (CTC's), for example. Topology change can cause violations of the strong equivalence principle if one wants to hang on to causality at all costs -- the way I read Theorem 11 on p. 67. I could be wrong here. Pundits please correct! :-) One can keep the equivalence principle and have topology change if one violates causality -- which I want to do any way.

Chapter 8 is on the cosmological constant. The Einstein field equation with the cosmological constant is

Einstein = StressEnergy - cosmologicalconstant metric

The effective zero-point vacuum fluctuation energy density for the given cosmological constant is

vacuum energy density = cosmologicalconstant/8pi Newton's constant

Ab initio estimates of the cosmological constant seem to imply that the universe should be about one Planck length in diameter. This is in violent disagreement with the observational evidence. p. 81

(8.2) is on the observational limit on the cosmological constant. Start with the standard FRW metric. The Einstein equations reduce to two equations (8.19) and (8.20) (p.85). We need measurements of the Hubble constant and the scale of the universe. If we assume that the universe is both spatially flat and that the cosmological constant is zero one gets a critical matter density for the universe. (eq. 8.23) p.85 This is compared to the observed density which is much less than the critical density giving rise to the missing mass problem. Estimates on the size of the cosmological constant based on this argument make it extraordinarily small which conflicts with the size of the zero point energy with a Planck scale cutoff. There is some very important physics missing here. This dilemma is a crisis comparable to the one Max Planck faced with black body radiation at the end of the 19th century.

(9.2) Euclidean wormholes are in Hawking's imaginary time. Is imaginary time physical or simply a formal trick? Is most of the universe in imaginary time where the topological signature of the metric is ++++ rather than the Lorentzian -+++?

Euclidean wormholes are ... "instantons" in the gravitational field. ... They underlie Coleman's attempt [3] at explaining the cosmological constant problem. p.93

However, there are objections to Sidney Coleman's use of "baby universes":

... the presence of Euclidean wormhole solutions ... completely destroys the Euclidean quantum theory by producing highly nonlocal effective Euclidean action which is violently unbounded from below. W. D. Unruh, "Quantum coherence, wormholes and the cosmological constant", Phys Rev D. 40:1053-1063, 1989.

Chapter 11 is on traversable wormholes without event horizons sustained by negative energy exotic matter and without infinite tidal forces to rip would-be travellers apart. The key paper here is by Kip Thorne and his student M. S. Morris "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativty." Am. J. Phys. 56:395-412, 1988. They simply assumed a traversable spacetime geometry and then computed what the stress-energy tensor would have to look like in order to sustain the hole as a star gate for interstellar travel.

Does the deduced distribution of stress-energy violate any deeply held physical principle? The answer is that the stress-energy distribution near the throat of a wormhole is certainly peculiar, but that it does not seem to be incompatible with known physics. p.100, Visser

Traversable wormholes must violate one or more classical energy conditions near their entrance and exit "throats". The traversable wormhole metric tensor model is time independent, non-rotating with a spherically symmetric bridge connecting two flat spacetime regions that could be in the same or in different branches of the universe in the non-Hausdorff case discussed above. The metric has the effective Schwarzschild form (eq.11.5, p.102)

ds^2 = -e^2redshift(r) dt^2 +dr^2 / [1 - shape(r)/r] + r^2 [dtheta^2 + sin^2theta dphi^2 ]

Each branch can have its own redshift and shape functions. The redshift function cannot go infinite in a traversable wormhole, otherwise, as we saw above, the star ship is in danger of being ripped apart by infinite tidal forces. The proper radial distance is l(r). It is a definite integral over r (eq.11.6, p.103) of the square root of the shape factor 1/[1 - shape(r)/r]. Note in the Schwarzschild solution the shape function is 2GM/c^2. The throat is the minimum of r as a function of l. But

dr/dl = +- square root[1 - shape(r)/r] (eq.11.0, p.104)

Therefore shape(r*) = r* at the throat. Vissser shows that the slope of the shape function is less than 1 at the throat. (eq. 11.17, p.105). One can then get formulas for the Einstein tensor and the stress-energy tensor (i.e. 11.3, 11.4). The stress-energy tensor has three non-zero functions. First is rho which is the energy density. Second is tau which is the radial tension minus the radial pressure. Third is p the transverse pressure. All three are definite functions (eqs. 11.36 - 38, p. 108) of the redshift and shape functions of this simplest star gate model.

The key result is the inequality (eq. 11.56, p. 109) that at the throat r* of the traversable wormhole the energy density rho must be smaller than tau.

Note when the wormhole connects two parts of the same universe and if the redshift functions are different in each part, then the gravitational field is not conservative (p. 110 and p. 239. There is also the issue of non-orientability which seems to violate quantum mechanics.

[11.6] deals with transport of mass and charge through the traversable wormhole.

For a sufficiently large net transfer of mass the final mass of the "emitting" wormhole mouth becomes negative. ... If the mass of the "emitting" mouth of the wormhole does become negative one has the possibility of a runaway reaction: its mass now being negative the "emitting" mouth will now gravitationally repel the ambient medium. The "absorbing" mouth will continue to accrete matter, becoming ever more massive. p.113

See J. G. Cramer, R. L. Forward, M.S. Morris, M.Visser, G. Benford, and G. A. Landis "Natural wormholes as gravitational lenses", Phys. Rev D, 51:3117-3120, 1995 for more details on this negative mass possibility.

Chapter 12 is an exposition of the classical energy conditions. 12.3.2 discusses the quantum zero point energy Casimir effect which violates the null, weak, strong and dominant conditions. However, this effect appears to be much too weak to be of practical use for star ship engineering of artificial traversable wormholes.

[12.3.4] discusses the important idea of squeezed vacuum states to generate exotic matter. See

D. F. Wallis, "Squeezed states of light", Nature, 306:141-146, 1983

D. Hochberg and T.W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Phys. Lett B, 268:377-383, 1991

Squeezing makes a small fluctuation in one observable paid for by a large fluctuation in a noncommuting observable in accord with the Heisenberg uncertainty principle in the Copenhagen statistical ensemble sense. Visser remarks:

This trick can be used to excavate energy from one place in the ordinary vacuum state at the cost of piling up energy elsewhere. The regions from which energy has been excavated are lower in energy than an equal volume of ordinary vacuum and so by definition have a locally negative energy density. p.127

Chapter 13 deals with more practical "engineering considerations" for keeping the tidal forces small. . Let V be the observer's four vector velocity. I suppress the indices. The motion need not be geodesic. Consider two points on the observer's body separated by four vector displacement e. Starting in the observer's rest frame, it is easy to see that the frame-invariant inner product eV = 0. The difference in acceleration between the two points is the four-vector da

da = -RVeV (13.1) p.137

where R is the full fourth rank Riemann curvature tensor so there is a contraction over three of the four indices. This is not the equation for geodesic deviation which it superficially resembles. One can show that daV = 0. Eqs. 13.7 and 13.8 on p. 138 then give both the radial and transverse tidal acceleration differences on the observer in terms of the redshift and shape functions of the traversable wormhole star gate. Only the transverse part of the tidal force is velocity-dependent.

If the star gate has a zero redshift function, there is zero radial tidal force (eq. 13.71, p.146) and the velocity-dependent transverse force is given in eq. 13.72 and is determined by the shape function and is proportional to the square of the diverging gamma factor relative to the rest frame of the throat. So if you go too fast you will crush the ship transversely. Traversable wormholes can be used to probe the interior of blackholes. (p.152)

... "infinitesimal" perturbations of the Schwarzschild metric can easily lead to wormholes that are "traversable in principle". p. 148

Visser claims that so far, no one has done a self-consistent semi-classical computation in which we have a classical spacetime geometry whose statistically averaged quantum field stress energy tensor obeys the Einstein field equations with precisely that geometry as the solution. (p.149)

The idea for advanced Dyson engineers is to artifically design and construct the redshift and shape functions of the stargate by judicious placement of exotic negative energy density matter for maximum comfort of the travelers.

... any and all travelers passing through this class of wormhole must pass through a region containing exotic matter. It is far from clear what the direct, nongravitational effect of exotic matter would be on any potential visitor to the exotic region. p. 143

The tension of the exotic matter at the wormhole throat is

tau(r*) = 5 10^+42 (1meter/r*)Joules/meter^3 13.60 p. 143

The corresponding tension for ordinary matter is 1.6 10^+11 Joules/meter^3. This means the wormhole throat must be of the order of a light year across if the tension in the exotic matter keeping it open is to be of the same order of magnitude as that of ordinary matter.

(15.1.2) uses the thin shell formalism of Ch 14 to describe "cubic wormholes" whose 3D throats consist of 6 flat planes, 12 quarter cylinder edges, eight octants of a sphere as corners. (p.168). (15.1.4) Fig. 15.4 is the "dihedral wormhole" whose throats are 2D rectangles (p.173). Eq. 15.28 on p.174 is an exotic mass estimate to sustain a star gate. This is -10^-3 solar mass (scale of throat in meters).

So building a one metre wide thin-throat traversable wormhole requires about one Jupiter mass of exotic negative energy matter ... the total mass of the wormhole as seen from infinity ... is in fact zero. The non-linearity of the Einstein field equations leads to this effect. p.174

We do not know how exotic matter couples to ordinary matter, or what the relation of exotic matter is to "missing mass", "dark" and "shadow" matter which may be 90% of the matter in the universe.

Can a sufficiently advanced civilization manufacture star gates? (13.3) Such a feat requires topology change and that requires eating the Sacred Cow of microcausality in flat spacetime quantum field theory. That is a profound change. The Green's function method seems large enough to accommodate it however. There is evidence quoted by Charles Bennett in papers in Physical Review A in mid 1980's on "Pre-causal quantum mechanics" that cites data indicating large violations of causal dispersion relations. This series of refereed papers in a major journal seems to have been ignored.

One's best bet might be to find a wormhole that was built into the fabric of spacetime back in the big bang, and by suitable means to modify it and make it suitable for traversal. p. 144

One advantage of the primordial wormhole is that we may be able to use it to travel back in time to near the big bang if we can overcome the disruption of the hole by vacuum fluctuations as we try to travel along a closed timelike worldline. Visser is pessimistic on this possibility. However, some UFO reports are consistent with the idea that time travellers from our future are making contact. The evaluation of this evidence is beyond pure physics. Most physicists are not rational about it.

[1] "Quantum mechanics near closed timelike lines." Phys. Rev. D. 44:3197-3127 (1991) and Scientific American March 1994 pp 68-74.

[2] Friedman, Papastamatiou, Simon "Failure of unitarity for interacting fields on spacetimes with closed timelike curves", Phys. Rev. D, 46:4456 (1992) - this is for single timeline Hausdorff spacetime with Novikov self-consistency.

[3] "Why there is nothing rather than something: A theory of the cosmological constant." Nuclear Physics B, 310:643-668, 1988