# Wormholes & Time Machines - for *experts* in GR/maths

EDIT: 21 Jan - Response to the Lubos Expansion appended [in progress, not yet complete]

EDIT: 23 Jan - Visser's calculations appended

EDIT: 26 Jan - Peter Shor's thought experiments rebutted

Summary to Date (26 Jan)

The question is: are the Morris, Thorne, Yurtsever (MTY) and Visser mechanisms for converting a wormhole into a time machine valid? The objection to the former is that the "motion" of a wormhole mouth is treated in an inadmissable manner by the former, and that the valid mathematical treatment of the latter is subsequently misapplied to a case in which a sufficient (and probably necessary) condition does not apply (existence of a temporal discontinuity). It is maintained that extant thought experiments lead to incorrect conclusions because in the former case correct treatment introduces factors that break inertial equivalence between an unaccelerated rocket and co-moving wormhole mouth, and in the latter case especially do not respect the distinction between temporal coordinate values and spacetime separations.

Given that the detailed treatment of the Visser case is reproduced below, a valid argument in favour of a wormhole time machine must show how an interval $ds^2=0$ (the condition for a Closed Timelike Curve) obtains in the absence of a temporal discontinuity.

In considering the MTY (1988) paper, careful consideration should be given to whether the authors actually transport a wormhole mouth or just the coordinate frame that is convenient for describing a wormhole mouth if one happened to exist there.

Issues concerning quantum effects, energy conditions, and whether any created time-machine could persist etc. are off topic; the question is solely about the validity of the reasoning and maths concerning time-machine creation from a wormhole.

The original postings in chronological order are below.

Over at Cosmic Variance Sean Carroll recommended this place for the quality of the contributions so I thought I would try my unanswered question here; it is most definitely for experts.

The question is simple, and will be stated first, but I'll supplement the question with specific issues concerning standard explanations and why I am unable to reconcile them with what seem to be other important considerations. In other words, I'm not denying the conclusions I'm just saying "I don't get it," and would someone please set me straight by e.g. showing me where the apparent counter-arguments/reanalyses break down.

The question is: How can a wormhole be converted into a time machine?

Supplementary stuff.

I have no problem with time-travel per se, especially in the context of GR. Godel universes, van Stockum machines, the possibilities of self-consistent histories, etc. etc. are all perfectly acceptable. The question relates specifically to the application of SR and GR to wormholes and the creation of time-differences between the mouths - leading to time-machines, as put forward in (A) the seminal Morris, Thorne, Yurtsever paper (Wormholes, Time Machines, and the Weak Energy Condition, 1987) and explained at some length in (B) Matt Visser's Book (Lorentzian Wormholes From Einstein To Hawking, Springer-Verlag, 1996).

A -- Context. MTY explore the case of an idealised wormhole in which one mouth undertakes a round trip journey (i.e. undergoes accelerated motion, per the standard "Twins Paradox" SR example).

What is unclear to me is how MTY's conclusions are justified given that the moving wormhole mouth is treated as moving against a Minkowskian background: specifically, can someone explain how the wormhole motion is valid as a diffeomorphism, which my limited understanding suggests is the only permitted type of manifold transformation in relativity.

Elaborating... wormhole construction is generally described as taking an underlying manifold, excising two spherical regions and identifying the surfaces of those two regions. In the MTY case, if the background space is Minkowski space and remains undistorted, then at times t, and t' the wormhole mouth undergoing "motion" seems to identify different sets of points (i.e. different spheres have to be excised from the underlying manifold) and so there is no single manifold, no diffeomorphism. [Loose physical analogue: bend a piece of paper into a capital Omega shape and let the "heels" touch... while maintaining contact between them the paper can be slid and the point of contact "moves" but different sets of points are in contact]

I'm happy with everything else in the paper but this one point, which seems to me fundamental: moving one wormhole mouth around requires that the metric change so as to stretch/shrink the space between the ends of the wormhole, i.e. the inference of a time-machine is an artefact of the original approach in which the spacetime manifold is treated in two incompatible ways simultaneously.

Corollary: as a way of re-doing it consistently, consider placing the "moving" wormhole mouth in an Alcubierre style warp bubble (practicality irrelevant - it just provides a neat handle on metric changes), although in this case v is less than c (call it an un-warp bubble for subluminal transport, noting in passing that it is in fact mildly more practicable than a super-luminal transport system). As per the usual Alcubierre drive, there is no time dilation within the bubble, and the standard wormhole-mouth-in-a-rocket thought-experiment (Kip Thorne and many others) produces a null result.

B -- Context. (s18.3 p239 onwards) Visser develops a calculation that begins with two separate universes in which time runs at different rates. These are bridged and joined at infinity to make a single wormhole universe with a temporal discontinuity. The assumption of such a discontinuity does indeed lead to the emergence of a time machine, but when a time-machine is to be manufactured within a single universe GR time dilation is invoked (one wormhole mouth is placed in a gravitational potential) to cause time to flow at different rates at the two ends of the wormhole (to recreate the effect described in the two universe case in which time just naturally flowed at different rates). However, in the case of a simple intra-universe wormhole there is no temporal discontinuity (nor can I see how one might be induced) and the application of the previously derived equations produces no net effect.

Thus, as I read the explanations, neither SR nor GR effects create a time-machine out of a (intra-universe) wormhole.

Where have I gone wrong?

Many thanks,

Julian Moore

Lubos' initial answer is informative, but - like several other comments (such as Lawrence B. Crowell's) - the answerer's focus is on the impossibility of an appropriate wormhole per se, and not the reasoning & maths used by Morris, Thorne, Yurtsever & Visser.

I agree (to the extent that I understand them) with the QM issues, however the answer sought should assume that a wormhole can exist and then eliminate the difficulties I have noted with the creation of time differences between the wormhole mouths. Peter Shor's answer assumes that SR effects will apply and merely offers a way to put a wormhole mouth in motion; the question is really, regardless of how motion might be created, does & how does SR (in this case) lead to the claimed effect?

I think the GR case is simplest because the maths in Visser's book is straightforward, and if there is no temporal discontinuity, the equations (which I have no problem with) say no time machine is created by putting a wormhole mouth into a strong gravitational potential. The appropriate answer to the GR part of the question is therefore to show how a time machine arises in the absence of a temporal discontinuity or to show how a temporal discontinuity could be created (break any energy condition you like, as far as I can see the absolute discontinuity required can't be obtained... invoking QM wouldn't help as the boundary would be smeared by QM effects.

Lubos said that "an asymmetry could gradually increase the time delay between the two spacetime points that are connected by the wormhole", I'm saying, "My application of the expert's (Visser's) math says it doesn't - how is my application in error?"

The SR case is much trickier conceptually. I am asserting that the wormhole motion of MTY is in principle impossible because, not to put too fine a point on it, the wormhole mouth doesn't "move". Consider a succession of snapshots showing the "moving" wormhole mouth at different times and then view them quickly; like a film one has an appearance of movement from still images, but in this case the problem is that the wormhole mouth in each frame is a different mouth. If the background is fixed Minkowski space (ie remains undistorted) at times t and t' different regions of the underlying manifold have to be excised to create the wormhole at those times... so the wormhole manifolds are different manifolds. If the background is not fixed Minkowski space, then it can be distorted and a mouth can "move" but this is a global rather than a local effect, and just like the spacetime in an Alcubierre warp bubble nothing is happening locally.

Consider two points A & B and first stretch and then shrink the space between them by suitable metric engineering: is there are time difference between them afterwards? A simple symmetry argument says there can't be, so if the wormhole mouths are treated as features of the manifold rather than objects in the manifold (as it seems MTY treat them) then the only way to change their separation is by metric changes between them and no time machine can arise.

Of course, if a time-machine could be created either way, it would indeed almost certainly destroy itself through feedback... but, to repeat, this is not the issue.

Thanks to Robert Smith for putting the bounty on this question on my behalf, and thanks to all contributors so far.

Edit 2: Re The Lubos Expansion

Lubos gives an example of a wormhole spacetime that appears to be a time-machine, and then offers four ways of getting rid of the prospective or resulting time machine for those who object in principle. Whilst I appreciate the difficulties with time-machines I am neither for nor against them per sunt, so I will concentrate on the creation issue. I have illustrated my interpretation of Lubos' description below.

As I understand it, there is nothing in GR that in principle prevents one from having a manifold in which two otherwise spacelike surfaces are connected in such a way as to permit some sort of time travel. This is the situation shown in the upper part of the illustration. The question is how can the situation in the upper part of the illustration be obtained from the situation in the lower part?

Now consider the illustration below of a spacelike surface with a simple wormhole (which I think is a valid foliation of e.g. a toroidal universe). As time passes, the two mouths move apart thanks to expansion of space between them, and then close up again by the inverse process (as indicated by the changing separation of the dotted lines, which remain stationary)

Edit 3: Visser's calculations reproduced for inspection

Consider the result in the case where there is no temporal discontuity using the equations derived for the case where there is a discontinuity, given below

Visser, section 18.3, p239 The general metric for a spherically symmetric static wormhole

$$ds^2~=~-e^{2\phi(l)}~dt^2~+~dl^2~+~r^2[d\theta^2~+~sin^2\theta~d\psi^2]~~~~~(18.35)$$

Note that "there is no particular reason to demand that time run at the same rate on either side of the wormhole. More precisely, it is perfectly acceptable to have $ф(l=+\infty)~\neq~ф(l=-\infty)$"

Reduce to (1+1) dimensions for simplicity and consider $$ds^2~=~-e^{2\phi(l)}~dt^2~+~dl^2~~~~~~~~~~(18.36)$$

The range of l is (-L/2,+L/2) and l=-L/2 is to be identified with l=+L/2. Define

$$\phi_\pm\equiv\phi(l=\pm~L/2); ~\Delta\phi\equiv\phi_+~-~\phi_-~~~~~~~(18.37)$$

at the junction $l=\pm~L/2~~$ the metric has to be smooth, ie. $ds = \sqrt{g_{\mu\nu}{dx^\mu}{dx^\nu}}$ is smooth, implying $$d\tau=e^{\phi_-}dt_-=e^{\phi_+}dt_+~~~~~~~~~(18.38)$$ Define the time coordinate origin by identifying the points $$(0,-L/2)\equiv(0,+L/2)~~~~~~(18.39)$$ then the temporal discontinuity is $$t_+=t_-e^{(\phi_-~-~\phi_+)}~=~t_-e^{(-\Delta\phi)}~~~~~~(18.40)$$ leading to the identification $$(t_-,-L/2)\equiv(t_-e^{-\Delta\phi},+L/2)~~~~~~~~~(18.41)$$ which makes the metric smooth across the junction. Now consider a null geodesic, i.e. ds=0, which is $${dl\over{dt}}=\pm{e^{+\phi(l)}}~~~~~(18.42)$$ where the different signs correspond to right/left moving rays. Integrate to evaluate for a right moving ray, with conventions $t_f$ is the final time and $t_i$ is the initial time $$[t_f]_+=[t_i]_-+\int_{-L/2}^{+L/2}e^{-\phi(l)}dl~~~~~(18.43)$$ then apply the coordinate discontinuity matching condition to determine that the ray returns to the starting point at coordinate time $$[t_f]_-=[t_f]_+e^{\Delta\phi} = [[t_i]_-+\oint{e^{-\phi(l)}}dl]e^{\Delta\phi}~~~~~(18.44)$$ A closed right moving null curve exists if $t[_f]_-=[t_i]_-$, i.e. $$[t_i]^R_-={{\oint{e^{-\phi(l)}dl}}\over{e^{\Delta\phi}-1}}~~~~~(18.45)$$

Edit 4: Peter Shor's thought experiments re-viewed

Peter Shor has acknowledged (at the level of "I think I see what you mean...") both the arguments against wormhole time machine creation (the absence of the required temporal discontinuity in spacetime if GR effects are to be used, and that wormhole mouth motion requires metric evolution inconsistent with the Minkowksi space argument of MTY) but still believes that such a wormhole time-machine can be created by either of the standard methods offered a thought experiment. This is a counter to those thought experiments and whilst it does not constitute proof of the contrary (I don't think such thought experiments are rich enough to provide proof either way), I believe it casts serious doubt on their interpretation, thereby undermining the objections.

The counter arguments rely on the key distinction between the values of the time coordinate and the separation of events ($ds^2$). Paragraphs are numbered for ease of reference.

(1) Consider the classic Twins Paradox situation and the associated Minkowksi diagram. When the travelling twin returns she has the same t coordinate ($T_{return}$) as her stay-at-home brother (who said they had to be homozygous? :) ) As we all know, despite appearances to the contrary on such a diagram, the sister's journey is in fact shorter (thanks to the mixed signs in the metric), thus it has taken her "less time" to reach $T_{return}$ than it took her brother. Less time has elapsed, but she is not "in the past".

(2) Now consider the gravity dunking equivalent Twins scenario. This time he sits in a potential well for a while and then returns to his sister who has stayed in flat space. Again their t coordinate is the same, but again there is a difference in separations; this time his is shorter.

(3) Now for the travelling/dunking Twin substitute a wormhole mouth; then the wormhole mouths are brought together they do so at the same value of t. The moving mouths may have "travelled" shorter spacetime distances but they are not "in the past"

(4) Suppose now we up the ante and give the travelling/dunking Twin a wormhole mouth to keep with them...

(5) According to the usual stories, Mr A can watch Ms A receding in her rocket - thereby observing her clock slow down - or he can communicate with her through the wormhole, through which he does not see her clock slow down because there is no relative motion between the wormhole mouth and Ms A. Since this seems a perfectly coherent picture we are inevitably led to the conclusion that a time machine comes into existence in due course.

(6) My objection to this is that there are reasons to doubt what is claimed to be seen through the wormhole, and if the absence of time dilation is not observed through the wormhole we will not be led to the creation of a time machine. So, what would one see through the wormhole, and why?

(7) I return to the question of the allowable transformations of the spacetime manifold. If a wormhole mouth is rushing "through" space, the space around it must be subject to distortion. Now, whilst there are reasons to doubt that one can arrange matter in such a way as to create the required distortion (the various energy condition objections to the original Alcurbierre proposal, for instance), we are less concerned about the how and more concerned with the what if (particularly since, if spacetime cannot "move" to permit the wormhole mouth to "move", the whole question becomes redundant). The very fact that spacetime around the "moving" wormhole mouth is going to be distorted suggests at least the possibility that what is observed through the wormhole is consistent with what is observed the other way, or that effects beyond the scope of the equivalence principle demonstrate that observation through the wormhole is not equivalent to observing from an inertial frame. Unfortunately I don't have the math to peform the required calculations, but insofar as there is a principled objection to the creation of a time machine as commonly described, I would hope that someone would check it out.

(8) What then for the dunking Twin? In this case there is no "motion" of the wormhole mouth, so no compensating effects can be sought from motion. However, I believe that one can apply to the metric for help. Suppose that the wormhole mouth in the gravitational well is actually embedded in a little bit of flat space, then (assuming the wormhole itself is essentially flat) the curvature transition happens outside the mouth and looking through the wormhole should be like looking around it: Mr A seems very slowed down. If, instead, the wormhole mouth is fully embedded in the strongly curved space that Mr A also occupies, then the wormhole cannot be uniformly flat and again looking through the wormhole we see exactly what we see around it (at least as far as the tick of Mr A's watch is concerned.) but the transition from flat to curved space (and hence the change in clock rates) occurs over the interior region of the wormhole.

(9) Taken with the "usual" such thought experiments, we now have contradictory but equally plausible views of the same situations, and they can't both be right. I feel however that no qualitative refinements will resolve the issue, thus I prefer the math, which seems to make it quite plain that the usually supposed effects do not in fact occur. Similarly, if you disagree that the alternative view is plausible, since the "usual" result does not seem plausible to me,maths again provides the only common ground where the disagreement can be resolved. I urge others to calculate the round-trip separations using the equations provided from Visser's work.

(10) I say the MTY paper is in error because it treats spacetime as flat, rigid Minkowskian and then treats the "motion" of a wormhole mouth in a way that is fundamentall incompatible with a flat rigid background.

(11) I say Visser is in error in applying his (correct) inter-universe wormhole result to an intra-universe wormhole where the absence of the temporal discontinuity in the latter nullifies the result.

(12) These objections have been acknowledged but but no equally substantive arguments to undermine them (i.e. to support the extant results) has been forthcoming; they have not been tackled head on.

(14) I am not comfortable with any of the qualitative arguments either for against wormhole-time machines; an unending series of thought experiments is conceivable, each more intricate and ultimately less convincing than the last. I don't want to go there; look at the maths and object rigorously if possible.

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"The question is: How can a wormhole be converted into a time machine?" -- that must be also a title for your question. –  Kostya Jan 14 '11 at 13:33
What? A question about me? Time Machines are awesome. :) –  rightfold Jan 15 '11 at 1:10
You say "Consider two points A & B and first stretch and then shrink the space between them by suitable metric engineering: is there are time difference between them afterwards? A simple symmetry argument says there can't be" But this is exactly the twin paradox in special relativity. How does the fact that the objects are wormhole mouths make any difference? I guess I don't understand your question. –  Peter Shor Jan 20 '11 at 18:54
@Julian: If you move the wormhole mouths with the Alcubierre warp-bubble, then they presumably have no time dilation. But if you move them any other way, they should have the same time dilation you get from GR. Why should they behave any differently? –  Peter Shor Jan 21 '11 at 10:58
@Julian: there is a difference between gravitational potential and curvature. You should learn general relativity better. At the event horizon of a large black hole, there is low curvature but very high gravitational potential. Look in Wikipedia (or at the equations) if you don't believe me. –  Peter Shor Jan 28 '11 at 17:31

If you have a charged black hole, and a strong enough magnet, you should be able to move the charged black hole without any theoretical difficulty. A wormhole mouth shouldn't be any different.

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Right, but this still doesn't directly answer the question whether the wormhole may become a time machine, does it? –  Luboš Motl Jan 20 '11 at 8:42
@Lubos & Peter. I've added some extensive clarification to the original question to help focus on the points at issue. –  Julian Moore Jan 20 '11 at 9:34

this answer has been expanded at the end.

I am convinced that macroscopic wormholes are impossible because they would violate the energy conditions etc. so it is not a top priority to improve the consistency of semi-consistent stories. At the same moment, I also think that any form of time travel is impossible as well, so it's not surprising that one may encounter some puzzles when two probably impossible concepts are combined.

However, it is a genuinely confusing topic. You may pick Leonard Susskind's 2005 papers about wormholes and time travel:

http://arxiv.org/abs/gr-qc/0503097

http://arxiv.org/abs/gr-qc/0504039

Amusingly enough, for a top living theoretical physicist, the first paper has 3 citations now and the second one has 0 citations. The abstract of the second paper by Susskind says the following about the first paper by Susskind:

"In a recent paper on wormholes (gr-qc/0503097), the author of that paper demonstrated that he didn't know what he was talking about. In this paper I correct the author's naive erroneous misconceptions."

Very funny. The first paper, later debunked, claims that the local energy conservation and uncertainty principle for time and energy are violated by time travel via wormholes. The second paper circumvents the contradictions from the first one by some initial states etc. The discussion about the violation of the local energy conservation law in Susskind's paper is relevant for your question.

I think that if you allowed any configurations of the stress-energy tensor - or Einstein's tensor, to express any curvature - it would also be possible for one throat of an initial wormhole to be time-dilated - a gravity field that is only on one side - and such an asymmetry could gradually increase the time delay between the two spacetime points that are connected by the wormhole. For example, you may also move one endpoint of the wormhole along a circle almost by the speed of light. The wormhole itself will probably measure proper time on both sides, but the proper time on the circulating endpoint side is shortened by time dilation, which will allow you to modify the time delay between the two endpoints.

Whatever you try to do, if you get a spacetime that can't be foliated, it de facto proves that the procedure is physically impossible, anyway. Sorry that I don't have a full answer - but that's because I fundamentally believe that the only correct answer is that one can't allow wormholes that would depend on negative energy density, and once one allows them, then he pretty much allows anything and there are many semi-consistent ways to escape from the contradictions.

Expansion

Dear Julian,

I am afraid that you are trying to answer more detailed questions by classical general relativity than what it can answer. It is clearly possible to construct smooth spacetime manifolds such that a wormhole is connecting places X, Y whose time delay is small at the beginning but very large - and possibly, larger than the separation over $c$ - at the end. Just think about it.

You may cut two time-like-oriented solid cylinders from the Minkowski spacetime. Their disk-shaped bases in the past both occur at $t=0$ but their disked-shaped bases in the future appear at $t_1$ and $t_2$, respectively. I can easily take $c|t_1-t_2| > R$ where $R$ is the separation between the cylinders. Now, join the cylinders by a wormholes - a tube that goes in between them. In fact, I can make the wormhole's proper length decreasing as we go into the future. It seems pretty manifest that one may join these cylinders bya tube in such a way that the geometry will be locally smooth and Minkowski.

These manifolds are locally smooth and Minkowski, when it comes to their signature. You can calculate their Einstein's tensor - it will be a function of the manifold. If you allow any negative energy density etc. - and the very existence of wormholes more or less forces you to allow negative energy density - then you may simply postulate that there was an energy density and a stress-energy tensor that, when inserted to Einstein's equations, produced the particular geometry. So you can't possibly avoid the existence of spacetime geometries in which a wormhole produces a time machine sometime in the future just in classical general relativity without any constraints.

The only ways to avoid these - almost certainly pathological - configurations is to

1. postulate that the spacetime may be sliced in such a way that all separations on the slice are spacelike (or light-like at most) - this clearly rules "time traveling" configurations pretty much from the start

2. appreciate some kind of energy conditions that prohibits or the negative energy densities

3. impose other restrictions on the stress-energy tensor, e.g. that it comes from some matter that satisfies some equations of motion with extra properties

4. take some quantum mechanics - like Susskind - into account

If you don't do either, then wormholes will clearly be able to reconnect spacetime in any way they want. This statement boils down to the fact that the geometry where time-like links don't exist at the beginning but they do exist at the end may be constructed.

All the best Lubos

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Hi Lubos (quality input already!) Thanks! I read the Susskind papers a long time ago (though with limited comprehension given the QM perspective). Loved the self-deprecation. Not many papers make one laugh aloud. I agree in general with the difficulties re WEC etc. but the problem is not whether wormholes are feasible, it's about my lack of understanding of the MTY/Visser arguments & math. Your penultimate para seems to be a restatement of MTY, treating a mouth as an object in spacetime rather than a feature of it. Grateful for the input but still feel stupid. –  Julian Moore Jan 15 '11 at 10:46
Dear Julian, this is a good point whether wormholes are "objects with many features such as the mouths" or "paired objects in spacetime". In linearized GR, it's possible to look at it from the latter perspective, especially when the radius of the mouth is very small - relatively to other length scales in the problem, such as the distance between the mouths. Of course, for this "much greater" inequality to hold, one needs to fine-tune positive and negative energies across the solution in a rather unnatural way. –  Luboš Motl Jan 20 '11 at 8:40
Lubos, @Expansion... I'm taking that on board (wish there was a whiteboard to hand) and thinking hard... will respond more fully later. Thanks –  Julian Moore Jan 20 '11 at 16:53

{\bf There is a more complete answer appended here}

The problem with the conversion is at the point the two worm hole openings are separated by a lightlike interval. This is a Cauchy horizon and there is this winding of paths, traversed by particles or photons, which pile up at the horizon. This would include virtual particles or the vacuum as well as I see it. The worm hole has a Lansczoc junction with a shell of negative mass-energy that permits the wormhole to exist. The winding up of vacuum modes on the Cauchy horizon would introduce enormous fluctuation of positive energy which I think would destroy the wormhole. I think this would destroy the solution, so that even if a wormhole can exist any attempt to convert it to a time machine would destroy it.

The other problem is that in QM we have that momentum is the generator of a position change. Yet with multiply connected spacetimes there is a funny problem of an amgibuity. A particle can travel from $x$ to $y$ by two types of momentum generators or transformation $e^{ipx}$. So it is not entirely clear whether the wormhole is consistent with quantum mechanics.

This is just a brief answer at this point. I will try to work out a more complete version in a few days.

addendum: This is not airtight, but it is worth a couple of evening’s work. I think this gives a pretty good argument for why you can’t transform a wormhole into a time machine. I will have to confess that I am a big enemy of these quirky spacetimes which give faster than light or time reversal results. They are mathematical result, but quantum mechanics kills them in physics. Further, the averaged weak energy condition is violated $T^{00}~<~0$, which means the quantum field which acts as this source has no lower bounded eigenvalue. That is a complete disaster.

A worm hole has a membrane or surface with a field that has an energy density. This starts with a static wormhole as a necessary starting point. Start with the Reissner-Nordstrom metric $$ds^2~=~-F(r)dt^2~+~{1\over {F(r)}}dr^2~+~d\Omega^2,$$ for $F(r)~=~1~-~r_0/r$. The event horizon with radius $r_0$ is replaced by a junction or thin shell at $r_0(t)~\rightarrow~r_0$ $+~\delta r(t)$. This defines the openings of the wormhole in two regions. The normal vectors to this sphere are $$n^\mu~=~\pm\Big({{\dot r_0}\over{F(r_0)}},~\sqrt{F(r_0)~-~{\dot r_0}},~0,~0\Big), n_\mu~=~\pm\Big({\dot r_0},~{{\sqrt{F(r_0)~-~{\dot r_0}^2}\over {F(r_0)}}}, ~0, ~0\Big).$$ The sign is an indication of the direction of the normal on the sphere. The extrinsic curvature is computed as $K_{\mu\nu}~=~{1\over 2}n^\sigma\partial_\sigma g_{\mu\nu}$ and the components are $$K_{\theta\theta}~=~\pm {1\over r_0}\sqrt{F(r_0)~-~{\dot r_0}^2}$$ $$K_{tt}~=~\mp{1\over 2}{1\over{\sqrt{F(r_0)~-~{\dot r_0}^2}}}\Big({{\partial F(r_0)}\over{\partial r_0}}~-~{\ddot r_0}\Big).$$

The energy density $\rho~=~G^{00}$ $=~1/4\pi K_{\theta\theta}$ is integrated on a pillbox configuration to find the jump in the surface energy on the membrane, which is the jump in extrinsic curvature at $r~=~r_0(t)$. For a static membrane the tension equals the energy density so there is an overall conservation of the stress-energy on the membrane. The energy density is found from the ADM tensors $$G^{00}~=~\rho~=~{1\over{2\pi r_0}}\sqrt{F(r_0)~-~{\dot r_0}^2}$$ and the tension is $$\tau~=~-{1\over{4\pi}}(K_{\theta\theta}~-~K_{tt}).$$ The static membrane then infers the following equation $${{F(r_0)}\over{r_0^2}}~-~\Big( {{\dot r_0}\over {r_0}}\Big)^2~=~4\pi^2\tau^2.$$ The conservation of energy $\partial\rho /\partial t$ applied to this constraint equation results in the evolution equation $${\ddot r_0}~+~{1\over {r_0}}\Big(F(r_0)~-~{\dot r_0}^2\Big)~-~{{\partial F(r_0)}\over {\partial r_0}}~=~0.$$ This describes the dynamical evolution of the wormhole in a purely classical setting.

The evolution of the vacuum is governed by the discontinuity in the Einstein field at $r~=~r_0$. This discontinuity is then given by $$\lim_{\epsilon~\rightarrow~0}\int_{-\epsilon}^{+\epsilon} G^{00}dn~=~\delta\rho~=~-{1\over{2\pi r_0}}\sqrt{F(r_0)~-~(\dot r_0~+~U)^2},$$ where $U~=~\sqrt{U^\mu U_\mu}$ is the speed of the $r_1$ opening. The value of this change in energy density is positive. To induce the wormhole a field of some sort of field $\phi$ with a negative energy density $T^{00}(\phi)~<~0$. This discontinuity is determined by this field, ie $\delta\rho~=~T^{00}(\phi)$.

With quantum mechanics we have a vacuum which winds through the wormhole. The creation and annihilation operators $a^\dagger$ $a$ for this field are boosted relative to the moving opening. This results in a Bogoliubov transformed operator for these operators with $$b~=~a~cosh(g(s))~+~a^\dagger sinh(g(s)),~b^\dagger~=~ a^\dagger~cosh(g(s))~+~a sinh(g(s)).$$ The term $g(s)~=~gs$ for a constant acceleration. We interpret this as a rapidity angle which varies with respect to the proper time $s$ and which diverges when the two openings have a lightlike separation. The Hamiltonian is found to be $$b^\dagger b~=~a^\dagger a~+~\big((a^\dagger)^2~+~a^2\big)cosh(g(s))sinh(g(s))$$ The expectation of the field remains the same for $E_n~=~\langle n|b^\dagger b|n\rangle$. The off diagonal term is a form of the squeezed state operator, which removes the vacuum state of photons off quadrature. The uncertainty in the momentum and positions are evaluated so that $\langle\Delta x\Delta p\rangle~=~(1/2)sinh(2gs)$ which is the evaluation of off diagonal term with completeness. The entropy can be evaluated from $${k\over 2}ln\big(\langle\Delta x\rangle\langle\Delta p\rangle~-~\langle\Delta x\Delta p\rangle^2 \big)~=~k~ln(cosh(2g(s)).$$ This entropy is associated with the generation of photons from the vacuum, or as a form of Hawking radiation. The energy of this radiation is positive. As the opening of the wormhole approach a lightlike separation this radiation will demolish the wormhole by overwhelming the negative energy at the junction.

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Dear Lawrence, you've put so much effort into this I'm embarrassed to say I don't understand it beyond the generalities - but I'm sure many others will benefit from the detail. However, insofar as I think I understand the general thrust and one or two details: you put a wormhole mouth in motion and say "The creation and annihilation operators... are boosted relative to the moving opening", which is at the heart of my general difficulty. How is the movement of a feature of spacetime defined? I think - see also the answer to Peter Shor re the Twin's paradox - that everything is locally at rest –  Julian Moore Jan 21 '11 at 10:19
(cont) in which case (in this particular case) would the operators in fact be boosted? More generally. the fundamental issue remains with the mechanism for the creation of the time difference between the ends of the wormhole and not what would happen as a result. [I'm sorry my physics is rusty, and this is way beyond a lowly BSc... ignorance is not bliss!] –  Julian Moore Jan 21 '11 at 10:20
The transformation of the operators is somewhat qualitative at this point. A part of the problem is that the twin paradox argument involves a variable acceleration. A derivation of Unruh radiation invokes an “eternal acceleration,” and general physics for a variable acceleration is not known. There is a bit of an physical assumptions here. From the perspective of an exterior observer the acceleration varies in a way so that spacelike separated fields on the wormhole openings are transformed into lightlike separated fields on the Cauchy horizon. –  Lawrence B. Crowell Jan 21 '11 at 12:24
Continued: However, from the perspective of traversing the wormhole the fields retain a spacelike separation across the junction or wormhole membrane demarking the openings. So depending upon which path you take fields are lightlike separated or spacelike separated. Further, there must be some spacetime transformation between these two. The rapidity $\gamma$ for such a transformation is “$\infty$” and the boost function is $\sim~cosh(\gamma)$ diverges. –  Lawrence B. Crowell Jan 21 '11 at 12:25
continued: As I said this is a few hours effort and the argument is not airtight. The transformation argument needs to be put on more solid ground. It is an imposition of a local transformation rule on fields that puts sharp separation between timelike, spacelike and lightlike intervals onto a multiply connected spacetime. The multiply connected spacetime intertwines spacelike and timelike intervals in a way which is not possible by Lorentzian transformation. –  Lawrence B. Crowell Jan 21 '11 at 12:26
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