EDIT: Further clarification in the context of answers/comments received to 20 Jan has been appended

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. Block Universe Wormhole

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)

alt text

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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    $\begingroup$ "The question is: How can a wormhole be converted into a time machine?" -- that must be also a title for your question. $\endgroup$
    – Kostya
    Commented Jan 14, 2011 at 13:33
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    $\begingroup$ What? A question about me? Time Machines are awesome. :) $\endgroup$
    – user1313
    Commented Jan 15, 2011 at 1:10
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    $\begingroup$ 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. $\endgroup$ Commented Jan 20, 2011 at 18:54
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    $\begingroup$ @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? $\endgroup$ Commented Jan 21, 2011 at 10:58
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    $\begingroup$ @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. $\endgroup$ Commented Jan 28, 2011 at 17:31

5 Answers 5


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:



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.


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

  • $\begingroup$ 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. $\endgroup$ Commented Jan 15, 2011 at 10:46
  • $\begingroup$ 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. $\endgroup$ Commented Jan 20, 2011 at 8:40
  • $\begingroup$ Lubos, @Expansion... I'm taking that on board (wish there was a whiteboard to hand) and thinking hard... will respond more fully later. Thanks $\endgroup$ Commented Jan 20, 2011 at 16:53

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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    $\begingroup$ Right, but this still doesn't directly answer the question whether the wormhole may become a time machine, does it? $\endgroup$ Commented Jan 20, 2011 at 8:42
  • $\begingroup$ @Lubos & Peter. I've added some extensive clarification to the original question to help focus on the points at issue. $\endgroup$ Commented Jan 20, 2011 at 9:34

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.

Diffeomorphisms are about coordinate transformations, and have nothing to do with anything physical. If, in a coordinate patch, you switch to a different coordinate system, then if the transformation (of coordinates) is a diffeomorphism no physics will change. If moving from one coordinate patch to another, you need to convert coordinates in the overlap region, and in the overlap region the transformation will be a diffeomorphism. But a diffeomorphism is just about two coordinate systems describing the same physics. If has nothing to do with motion (except in as far as for a given motion there might be a naturally associated coordinate systems, but we can use any coordinate system we want, so who cares if one has a simpler form, they all work fine).

Can a wormhole be converted into a time machine?

Yes. I'll describe in detail how to make a time machine out of a wormhole, without "moving" either wormhole, and hopefully you won't be bothered by it. First note that if you arrange (positive energy density) matter in a spherical shell and send one twin to the inside of the shell and another sits on the outside of the shell (both at rest in one and the same coordinate system where the metric is static), then the twin on the outside of the shell dies first (ages more quickly). The twin inside is in a perfectly flat spacetime. The twin outside is a Schwarzschild spacetime, and we can make the shell rather thin and undense so there need not be any strong curvature anywhere (you mentioned strong curvature in your gravity dunking example, and it is unnecessary to the effect in question).

Now, imagine the sphere large enough that you could place a wormhole at the center without it changing the metric out by/near the shell very much. So make your wormhole with a pair of mouths. Make them far enough apart that there is room for you to build a shell around one with the other one being outside the shell. The key here is that I'm not asking you to move the wormholes (or to move them moving fast relative to each other), just that they have their mouths far apart because a wormhole with both mouths close can't do much at all. Given that they are far apart, then build a spherical shell (of positive energy density) around one of them. So we don't move either wormhole (so we don't need to worry about your Omega, to which I can't understand your objection in the slightest since diffeomorphisms have nothing to do with physics, the particles don't know about coordinate systems and don't know when or if we switch from one coordinate system to another).

Now for the standard argument, you can have a region of wormhole, you can use Visser's metric $$ds^2=-e^{2\phi(l)}dt^2+dl^2+dr^2(l)\left[d \theta^2+\sin^2\theta d\varphi^2\right]$$ and have $\phi(l)$ approach a constant as $l\rightarrow +\infty$, a possibly different constant as $l\rightarrow -\infty$. It sounds like you are fine with this if the $\pm l$ correspond to "different universes." Let's explicitly ask for more though, let's also insist (like page 101 of Visser) that $\lim_{l\rightarrow +\infty}r(l)/|l|=1=\lim_{l\rightarrow -\infty}r(l)/|l|$. So we have a single coordinate system $(t,l,\theta, \phi)$ for both sides of the wormhole (as a technicality the coordinate system is bad at the north and south poles and can't go all the way around, this happens even for spherical coordinates in a totally flat spacetime and I bring this up not to be pedantic but because the exact same issue will come up when Visser makes this an intrauniverse wormhole).

Define $\phi_+=lim_{l\rightarrow +\infty}e^{\phi(l)}$ and $\phi_-=lim_{l\rightarrow -\infty}e^{\phi(l)}$. Even though we have one coordinate system for everything, imagine that far from the throat, people prefer to use $T_+=e^{\phi_+}t$ or $T_-=e^{\phi_-}t$, then we could always use three coordinate systems, because coordinate systems are up to us. The point is that when $|l|$ is large, the new coordinate systems have a metric that numerically looks very very close to that of the flat spacetime coordinate system. So you can imagine taking two flat empty universes, cutting out a giant ball from each and putting our wormhole in between and not much geometry needs to change.

Simply do the same with one flat spacetime, except cut two giant balls out that are far from each other. And while Visser didn't say this, I'm saying that you can do this even when $\phi_+=\phi_-$. Hopefully you agree here, because we haven't made a time machine yet. OK. Now we cut those two giant balls out of a flat empty spacetime, but those balls themselves are inside an even larger ball, and outside that ball there could have been a giant shell of positive energy density matter (that would leave a flat spacetime metric inside, and that flat sapcetime metric inside is all that we needed to patch in our wormhole), so why not assume the entire wormhole and both mouths were not inside a flat empty spacetime but where inside a spherical shell of matter. We can do that, since the math inside isn't any different. Still no time machine.

Now, let's make a time machine. We have the wormholes, with equal time rate around each mouth and far from each other. Then we go all the way out to the shell of mater and remove a thin layer and take it over to the region outside one mouth and build a thin shell there. Now there is a difference in time flow from right inside the shell and right outside the shell (this already happens without wormholes, so again there shouldn't be any controversy, and all we moved was ordinary positive energy density matter, so hopeful that is unproblematic as well).

The secret to the time machine now is to wait. How long we wait depends on how far (through external space and through the wormhole) the mouths are from each other, and how much matter we put in the shell. And when we place the two mouths far apart, we need them to also be far apart relative to the curvature that will be induced by the spherical shell we later place about one of the mouths. But before we do make the time machine, I want to talk about what Visser did since I think your question might actually be more about that than how to make a time machine.

Where have I gone wrong?

Since you seem extremely worried about a coordinate discontinuity without citing any reason to be worried, that's probably at least one conceptual error. Imagine the spherical coordinate system, when you move in the $\varphi$ direction the coordinate changes smoothly and, say, increases but eventually you end up back where you started. How can it always increase but yet you end up at an earlier/lower value. An elementary differential geometry text will tell you that you need two coordinate systems, for instance one that doesn't include the international date line and one that excludes a line that is twelve time zones away, and your elementary textbook will tell you to switch coordinate systems from one to the other before reaching the place your coordinate system fails. This works. However it is a bit overkill for experienced practitioners. Instead you could use the one coordinate system and simply identify $\varphi=2\pi$ and $\varphi=0$ and accept that there is a coordinate discontinuity but that it means nothing except that your skill always you to be frugal with coordinate patches at the expense of a minor hassle with coordinate discontinuities.

So now lets redo our intra-universe wormhole before we turn it into a time machine. Unlike Visser we will avoid discontinuities by using many coordinate systems. So first we take a giant ball of flat spacetime (later this will be the inside of the original big shell). Inside it we remove two huge balls, but first we imagine a spherical coordinate system centered about the center of each ball. So we have two spherical coordinate systems in a regular flat spacetime. Nothing weird or problematic. And you can clearly switch from one to the other.

So now we put in our wormhole. It has it's own coordinate system as Visser gave (and as you and I both wrote up). We could cutout a large positive value of $l$ and a very negative value of $l$ and sew it to the spherical boundaries of where those balls used to be. But the elementary textbook way would be to include values in the wormhole coordinate patch that are slightly more positive and more negative, so that when you arrive outside the former holes you are still in the wormhole coordinate system for a bit then switch to the spherical coordinate system of that mouth of the wormhole. Then later later when you get closer to the other mouth than the first mouth you switch to that other spherical coordinate system, then when you get super close to the other mouth you switch to the coordinate system of the wormhole again this time for a very negative value of $l$ then you move in more and eventually are inside the wormhole again. This could have been done all entirely inside the one wormhole coordinate system if you just identified some points with very positive $l$ with some points of very negative $l$. But if that advanced technique confuses you don't do it.

But it has nothing to do with one universe or two, it's just about making one coordinate system work when an elementary textbook would say you need to use more than one.

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.

So far we could start with a super giant shell of matter, inside have a flat spacetime. Inside that cut out two balls very large and very far from each other. Then we can take our wormhole and sew it up together with time on both ends ticking at the same rate, have three coordinate systems (one for the throat, one for the region right outside one mouth, one for the region outside the other mouth, note that the wormhole coordinates could have worked for everything with an identification, and that having both coordinates outside the mouths is redundant to each other, but that then at least the transitions between each coordinate system is standard and not confusing). What Visser calls the $l=\pm L/2$ identification is what I call switching from the spherical coordinate system about one mouth to the spherical coordinate system about the other mouth in that region far from the throat (far from either mouth). And in fact it is easier to just stay with the one wormhole metric.

Hopefully you and I and everyone agree up to this point, and that I've addressed any confusions you have. Which means any problems with the next part are actual problems with the physics.

Next we steal matter from the big shell and place it around one of the mouths. If the amount of matter we moved is small compared to how far apart the mouths are, then after a time we might expect that there is a Schwarzschild type metric around the outside of the shell around mouth one, and that the flow of time near the other mouth is hardly affected at all because it is so very far away. So the time flow right near both mouths is unaffected by the shell. But now mouth one opens up in a spacetime region that is inside a shell of matter. If you placed that shell far enough from the throat then out there the metric of the wormhole looked almost flat spherical with time ticking at a normal constant rate.

So imagine you start right inside the shell and move out. You move in an approximately Schwarzschild metric so you use Schwarzschild coordinates, originally your coordinate time ticked much faster than your proper time. But eventually get so far away that things are pretty flat and proper time here now ticks much like coordinate time. Then you switch to the spherical coordinate system for the other mouth, and the rate of time ticking is also in line with your coordinate time. Then before you enter the mouth of the wormhole you switch to the wormhole coordinate system and now you clock ticks at a rate of $dT=e^{\phi_+}dt=e^{\phi_-}dt$ where $dt$ is the rate of coordinate time. You then traverse the throat using the metric:

$$ds^2=-e^{2\phi(l)}dt^2+dl^2+dr^2(l)\left[d \theta^2+\sin^2\theta d\varphi^2\right],$$

all the way through the throat of the wormhole, eventually getting to a region of large $l$ where again your proper time ticks at $dT=e^{\phi_+}dt=e^{\phi_-}dt$ where $dt$ is the rate of coordinate time. Then you switch to the original spherical coordinates around that mouth where proper time and coordinate time tick together. You go out some more, get to the shell and then switch back to the original Schwarzschild coordinate system, where proper time now ticks much slower than coordinate time.

You can place an observer at each mouth at the same wormhole coordinate time, and their clocks can tick the same number of times between two wormhole coordinate times (if $\phi_+=\phi_-$ and if $|l|$ is large enough that $\phi(l)\approx \phi_+$). But the clock by the empty wormhole ticks at the same rate as the Schwarzschild coordinate time, and the clock by the other wormhole ticks at a rate slower than the Schwarzschild coordinate time. When you wait long enough for this difference to build up you get your time machine. This example is more complicated than Visser's example since he just asserted an identification for a time origin, whereas we started with a wormhole that had synchronized ends and then we over time built up the shell, so it starts to turn into a time machine as we move the matter and we wait, but it doesn't happen all once at one easily labelled instant.

So now I'll show that a time machine is formed.

For concreteness and simplicity, first fix a length $R$, then find the parameter $l=W$ in the wormhole coordinates where $R=|r(\pm W)|$. We wanted the asymptotic $\phi$ to be equal, we'll choose that $\phi_+=\phi_-=0$ and that $\phi(\pm W)\approx 0$, $|r(\pm W)|\approx W$ and that these would have only gotten closer as $|l|$ got even bigger than $W$ if they had connected two different universe. This is what we will call the throat, and these surfaces have a surface area of $\approx 4\pi R^2$. Next we collect enough matter that it would form a Schwarzschild radius of $R$, but instead we position at a spherical shell of surface area $4\pi(10R)^2$, so there is a $9R$ proper distance between it and the $l=+W$ end of the throat. We set up the space time to have a proper distance of $10^9R$ between the shell and the $l=-W$ other end of the wormhole throat.

Now, we hang out in the flat space between the shell and the $l=+W$ throat, right near the shell. And just like you can yell at a canyon and hear the echo, you decide to send a message to yourself every second, by sending an absolutely perfect light pulse straight at the throat so that it comes out the other end of the wormhole, travels the long way through space all the way to the shell and through a small hole in the shell you placed. By a perfect pulse we just mean following a lightlike geodesic. The same effect will happen if you send the messages at sub lightspeed, but this just makes the math easier.

So you send these messages once a second, and you might have to wait a while to get them at first. You can use the wormhole coordinate system since you are inside the shell, and so your proper time is the coordinate time ($\phi_+=0$), say you send it at $t=0$. So it is wormhole coordinate time $t=9R/c+\Delta t_1= 9R/c+\int_{-W}^{+W}e^{\phi(l)}dl$ when it arrives at the other mouth (the $l=-W$ mouth), just like in 18.42 of page 241 of Visser. Over there, the wormhole coordinate time ticks at the same rate as the Schwarzschild coordinate time. This means if you are sending at one a second at your end, they are arriving a proper distance of $10^9R$ from the shell at a rate of one every Schwarzschild coordinate second. And they have $10^9R$ distance to travel, so we can compute the Schwarzschild coordinate time it take to get to the shell, by computing $\Delta t_2= \int_{10R}^{10^9R+10R}(1-R/r)^{-1})dr/c=(10^9R+Rln(\frac{10^9+9}{9}))/c$. Then it goes through the shell and gets to you.

There is something going on here that didn't go on before we made the shell. Before we made the shell we could do the same thing, send them once a second, then after $\Delta t_1$ wormhole coordinate, but since $\phi_+=\phi_-$ they would start coming out the other end at a rate of once per Wormhole coordinate second once they started coming out, and then once they travelled to you, they would continue to arrive once a second.

But after the shell we have a different story. They go into the wormhole at a rate of once per wormhole coordinate second. Since $\phi_+=\phi_-$ they come out at a rate of once per wormhole coordinate second (once they start to come out). But this is almost exactly equal to the Schwarzschild coordinate time a distance $10^9R$ from the mass (which is on a surface of surface area $4\pi (10R)^2$). And so they appear outside of the wormhole at a rate of almost once per Schwarzschild coordinate time $\sqrt{1-1/(10^9+10)}=\approx 1$. But that means once they reach the shell they start to arrive to someone just inside the shell at a rate of $\sqrt{1-1R/10R}$ seconds apart. So they arrive every $\sqrt{1-1/10}$ seconds apart but are sent every second. They are arriving faster than they are sent. The first messages could be boring, like just sending the number 1, then the number 2, then the number 3, and send a bunch before you start to get them, but once you start to get them too, and you notice that they are coming faster than you send them, you might choose to start sending yesterday's lotto numbers instead. There is a finite backlog of the old boring messages you already sent to work through and a finite number of messages in the (mental construct) queue of things you've sent but haven't gotten, but you reduce that queue relentlessly since they arrive faster than you send them. Eventually you start to get messages you haven't sent yet. That's your proof that you have a time machine.

For the numbers, if you measure the time $$9R/c+\Delta t_1= 9R/c+\int_{-W}^{+W}e^{\phi(l)}dl(10^9R+Rln(\frac{10^9+9}{9}))/c$$ in seconds, it tells you numerically how many messages you sent before you got the first one back. Then you can switch to sending lottery numbers (or flipping a coin and giving the coin toss a unique id number and recording the outcome). And your queue gets exhausted in 18.48683298 ($1/(1/\sqrt{1-(R/10R)}-1)$) times as long as it took you to start getting your messages back.

You don't have to have discontinuities to make a time machine out of a wormhole. And you don't have to move a wormhole to make a time machine out of it. And you can even start with a wormhole where both the ends age at the same rate, and still make a time machine if you have some ordinary positive energy density matter to hand, and a whole lot of time to wait for it to get sufficiently out of sync.

And I tried to use all the required math, and explain the math I thought you might misunderstand.

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.

I could never figure out why you said that.

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.

Done, build a shell around one of the ends, wait a long time, you have a time machine.

I'm saying, "My application of the expert's (Visser's) math says it doesn't - how is my application in error?"

I can't tell why you think Visser's math doesn't allow you to make a time machine, but hopefully my example is more clear since it restricts itself to describing actual actions, and using multiple coordinate systems and only using each coordinate systems locally.

  • $\begingroup$ Quick acknowledgement: I'm sorry I missed this earlier; that is very extensive and much appreciated. More later. NB since the main discussion was 4 years ago things have moved on, following further research etc.: I have written it up in Phys Rev D style. I will recheck what I have written in the light of your comments - if there is still a gap to bridge, would you be willing to take a look at said "paper"? $\endgroup$ Commented May 13, 2015 at 9:30

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.

  • $\begingroup$ 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 $\endgroup$ Commented Jan 21, 2011 at 10:19
  • $\begingroup$ (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!] $\endgroup$ Commented Jan 21, 2011 at 10:20
  • $\begingroup$ 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. $\endgroup$ Commented Jan 21, 2011 at 12:24
  • $\begingroup$ 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. $\endgroup$ Commented Jan 21, 2011 at 12:25
  • $\begingroup$ 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. $\endgroup$ Commented Jan 21, 2011 at 12:26

Let's consider the case of a thin-shell wormhole in Minkowski space $M = \Bbb R^{2,1}$ (We need at least two spatial dimensions for a proper wormhole without getting topology change involved).

First we need to define two timelike hypersurfaces $S_1$ and $S_2$. To simplify things, we'll consider that

  1. The intersection of $S_i$ with the Cauchy hypersurface is always $S^1$
  2. That circle is of constant radius
  3. The first hypersurface will be at rest while the second will move with some velocity $\dot{q}_2(t)$. If you wish to define the velocity of the hypersurface itself properly, you can do it by consider a foliation by timelike geodesics in $S_i$ (which is guaranteed to exist by global hyperbolicity) and then considering their acceleration in $M$. It can be described by the second fundamental form of those surfaces.

Then we can define the following embeddings :

\begin{eqnarray} f_1(t, \theta) &=& (t, r\cos \theta - q^x_1, r \sin \theta - q^y_1)\\ f_2(t, \theta) &=& (t, r\cos \theta - q^x_2(t), r \sin \theta - q^y_2(t)) \end{eqnarray}

Their first fundamental form $f_{i*} g = \bar g^i$ is then just the metric tensor of a sphere for spacelike components $(\bar g^i_{\theta\theta} = r^2)$, and

\begin{eqnarray} \bar g^i_{tt} &=& -1 + \dot{\vec{q}} \cdot \dot{\vec{q}}\\ \bar g^i_{t\theta} &=& r (\dot q^y \cos\theta -\dot q^x \sin \theta) \end{eqnarray}

Next we need to perform some manifold surgery on this. We remove the interior of $S_1$ and $S_2$, giving us the manifold with boundaries $\partial S_1$ and $\partial S_2$. If we define some homeomorphism

$$h : S_1 \to S_2$$

We can define the manifold $$(M \setminus (S_1 \cup S_2)) / \sim_h $$ where $p \sim q$ if $p \in S_1$, $q \in S_2$ and $q = h(p)$. You can check the literature (like Wall on differential topology) to assure yourself that this is a manifold.

If we have a metric tensor defined on a manifold with boundaries, the requirement for having a metric tensor defined on its gluing is that $h$ be an isometry (cf Clarke and Dray "Junction conditions for null hypersurfaces"). Our gluing function $h$ will be something like

$$h(t_1, \theta_1) = (b(t_1, \theta_2), \xi(t, \theta_2))$$

We'll simplify to $b(t)$ and $\xi(\theta_2)$, corresponding to the association of the time $t_1$ to another time $t_2$ upon crossing the boundary, and the identification of the wormhole mouth itself. Additionally we'll suppose that $\dot b > 0$ (this is to preserve time orientability).

The requirement that $h$ be an isometry corresponds to

$$\bar g^1_{\mu\nu} = \bar g^2_{\alpha\beta} \frac{\partial h^\alpha}{\partial x^\mu} \frac{\partial h^\beta}{\partial x^\nu}$$

or in other words

\begin{eqnarray} \bar g^1_{tt} &=& -1 = [-1 + \dot{\vec{q}}_2 \cdot \dot{\vec{q}}_2] \dot{b}^2\\ \bar g^1_{t\theta} &=& 0 = \dot b \xi' r (\dot q^y \cos\theta -\dot q^x \sin \theta)\\ \bar g^1_{\theta\theta}&=& r^2 = r^2 (\xi')^2 \end{eqnarray}

The second term disappears due to simply being $\propto(|\dot q| \cos \theta \sin \theta - |\dot q| \cos \theta \sin \theta)$. Then we have $\xi' = \pm 1$, or $\xi(\theta) = \pm \theta + \Delta \theta$ (it's a $O(2)$ rotation), and

\begin{equation} \dot{b}^2 = \frac{1}{1 - \dot{\vec{q}}_2 \cdot \dot{\vec{q}}_2} \end{equation}

as $q^2 \in [0, 1)$, we have that $\dot{b}^2 \in (0, 1]$, and since we specified that $\dot{b} > 0$, $\dot{b} \in (0, 1]$. The exact solution can be found as

\begin{equation} b(t) = \int \frac{dt}{\sqrt{1 - \dot{\vec{q}}_2(t) \cdot \dot{\vec{q}}_2(t)}} \end{equation}

To get some actual solutions, let's suppose that $\dot q$ moves only in a single direction, and pick the old Langevin trip, the bogus one where the initial leg is just $\dot q_x = +v$ and the return trip is $\dot q_x = -v$. Then for the whole trip, we have

\begin{equation} b(t) = \int \frac{dt}{\sqrt{1 - v^2}} \end{equation}

If the two wormholes are originally "in synch", that is, $b(0) = 0$, then we can just write

$$b(t) = \frac{t}{\sqrt{1 - v^2}} = \gamma t$$

The time-shift between the two mouthes is

$$b(t) - t = t (\gamma - 1)$$

You can check that this is a constant for $v = 0$. This means that the mouth on the spacelike hypersurface $t$ is identified with the mouth on the hypersurface $\gamma t$, and vice versa : the second mouth at time $t$ is identified with the mouth at time $t / \gamma$.

From there it's not terribly hard to construct a closed timelike curve. Just connect two points identified by the gluing function via a timelike curve (if the distance $q_1 - q_2$ isn't too large, that should be possible).


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