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
Edit 1: ADDITIONAL COMMENTS
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.
Best Answer
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
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
appreciate some kind of energy conditions that prohibits or the negative energy densities
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
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