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From wormhole to time machine:

arXiv:hep-th/9202090v2 8 Oct 1992From wormhole to time machine:Comments on Hawking’s Chronology Protection ConjectureMatt Visser∗Physics Department, Washington University, St. Louis, Missouri 63130-4899(Received 26 February 1992; Revised September 1992)The recent interest in “time machines” has been largely fueled by the apparentease with which such systems may be formed in general relativity, given relativelybenign initial conditions such as the existence of traversable wormholes or of infinitecosmic strings. This rather disturbing state of affairs has led Hawking to formulatehis Chronology Protection Conjecture, whereby the formation of “time machines” isforbidden.

This paper will use several simple examples to argue that the universeappears to exhibit a “defense in depth” strategy in this regard. For appropriate pa-rameter regimes Casimir effects, wormhole disruption effects, and gravitational backreaction effects all contribute to the fight against time travel.

Particular attentionis paid to the role of the quantum gravity cutoff. For the class of model problemsconsidered it is shown that the gravitational back reaction becomes large before thePlanck scale quantum gravity cutoffis reached, thus supporting Hawking’s conjec-ture.04.20.-q, 04.20.Cv, 04.60.+n; hepth@xxx/9202090Typeset Using REVTEX1

I. INTRODUCTIONThis paper addresses Hawking’s Chronology Protection Conjecture [1,2], and some ofthe recent controversy surrounding this conjecture [3].The recent explosion of interest in “time machines” is predicated on the fact that itappears to be relatively easy to construct such objects in general relativity. Time machineshave been based on the traversable wormholes of Morris and Thorne [4,5], and on Gott’scosmic string construction [6].These results have lead to a flurry of papers aimed at either making one’s peace withthe notion of time travel [7–9], or of arguing that — despite prima facie indications — theconstruction of time machines is in fact impossible.

Papers specifically addressing Gott’scosmic string construction include [10,11]. In a more general vein Hawking’s ChronologyProtection Conjecture [1,2] is applicable to both wormhole constructions and cosmic stringconstructions.This paper will focus on Hawking’s Chronology Protection Conjecture specifically as ap-plied to traversable wormholes.

The discussion will be formulated in terms of a simple modelfor a traversable wormhole which enables one to carefully explain the seemingly trivial ma-nipulations required to turn a traversable wormhole into a time machine. To help illustratethe manner in which nature might enforce the Chronology Protection Conjecture the utilityof two particular approximation techniques will be argued.

Firstly, it is extremely usefulto replace the original dynamical system with a quasistationary approximation where oneconsiders adiabatic variation of parameters describing a stationary spacetime. It shall beargued that this approximation captures essential elements of the physics of time machineconstruction while greatly simplifying technical computations, pedagogical issues, and con-ceptual issues.

Secondly, for a particular class of wormholes, once one is sufficiently closeto forming a time machine, it shall be shown that it is a good approximation to replace thewormhole mouths with a pair of planes tangent to the surface of the mouths. This approx-imation now reduces all computations to variations on the theme of the Casimir effect and2

permits both simple and explicit calculations in a well controlled parameter regime.The reader’s attention is particularly drawn to the existence and behaviour of a one–parameter family of closed “almost” geodesics which thread the wormhole throat.Thewormhole is deemed to initially be well behaved so that the closed geodesics belongingto this one–parameter family are initially spacelike. The putative construction of a timemachine will be seen to involve the invariant length of the members of this family of geodesicsshrinking to zero — so that the family of geodesics, originally spacelike, becomes null, andthen ultimately becomes timelike.

Not too surprisingly, it is the behaviour of the geometryas the length of members of this family of closed geodesics shrinks to invariant length zerothat is critical to the analysis.It should be emphasised that the universe appears to exhibit a “defense in depth” strategyin this regard.For appropriate ranges of parameters describing the wormhole (such asmasses, relative velocities, distances, and time shifts) Casimir effects (geometry inducedvacuum polarization effects), wormhole disruption effects, and gravitational back reactioneffects all contribute to the fight against time travel.The overall strategy is as follows: start with a classical background geometry on whichsome quantum fields propagate — this is just the semi–classical quantum gravity approxima-tion. In the vicinity of the aforementioned family of closed spacelike geodesics the vacuumexpectation value of the renormalized stress–energy tensor may be calculated approximately— this calculation is a minor variant of the standard Casimir effect calculation.

The associ-ated Casimir energy will diverge as one gets close to forming a time machine. Furthermore,as the length of the closed spacelike geodesics tends to zero the vacuum expectation value ofthe renormalized stress–energy tensor itself diverges — until ultimately vacuum polarizationeffects become larger than the exotic stress–energy required to keep the wormhole throatopen — hopefully disrupting the wormhole before a time machine has a chance to form.Finally, should the wormhole somehow survive these disruption effects, by considering lin-earized gravitational fluctuations around the original classical background the gravitationalback reaction generated by the quantum matter fields can be estimated.

For the particular3

class of wormholes considered in this paper this back reaction will be shown to become largebefore one enters the Planck regime. It is this back reaction that will be relied upon asthe last line of defence against the formation of a time machine.

In this matter I am inagreement with Hawking [1,2].Units: Adopt units where c ≡1, but all other quantities retain their usual dimensionali-ties, so that in particular G = ¯h/m2P = ℓ2P/¯h.4

II. FROM WORMHOLE TO TIME MACHINEThe seminal work of Morris and Thorne on traversable wormholes [4] immediately ledto the realization that, given a traversable wormhole, it appeared to be very easy to build atime machine [4,5].

Indeed it so easy is the construction that it seemed that the creation ofa time machine might be the generic fate of a traversable wormhole [12].In this regard it is useful to perform a gedankenexperiment that clearly and cleanlyseparates the various steps involved in constructing a time machine:— step 0: acquire a traversable wormhole,— step 1: induce a “time shift” between the two mouths of the wormhole,— step 2: bring the wormhole mouths close together.It is only the induction of the “time shift” in step 1 that requires intrinsically relativisticeffects.It may perhaps be surprising to realise that the apparent creation of the timemachine in step 2 can take place arbitrarily slowly in a non–relativistic manner. (Thisobservation is of course the ultimate underpinning for one’s adoption of quasistationaryadiabatic techniques.)A.

Step 0: Acquire a traversable wormholeIt is well known that topological constraints prevent the classical construction of worm-holes ex nihilo [13–15]. The situation once one allows quantum gravitational processes is lessclear cut, but it is still quite possible that topological selection rules might constrain [16–19]or even forbid [20] the quantum construction of wormholes ex nihilo.

If such proves to bethe case then even an arbitrarily advanced civilization would be reduced to the mining ofprimordial traversable wormholes — assuming such primordial wormholes to have been builtinto the multiverse by the “first cause”. Nevertheless, to get the discussion started, assume(following Morris and Thorne) that an arbitrarily advanced civilization has by whatevermeans necessary acquired and continues to maintain a traversable wormhole.5

In the spirit of references [20–22] it is sufficient, for the purposes of this chapter, to take anextremely simple model for such a traversable wormhole — indeed to consider the wormholeas being embedded in flat Minkowski space and to take the radius of the wormhole throat tobe zero. Thus our wormhole may be mathematically modelled by Minkowski space with twotimelike world lines identified.

For simplicity one may further assume that initially the twomouths of the wormhole are at rest with respect to each other, and further that the wormholemouths connect “equal times” as viewed from the mouth’s rest frame. Mathematically thismeans that one is considering (3+1) Minkowski space with the following two lines identified:ℓµ1(τ) = ¯ℓµ0 + 12δµ + V µτ, ℓµ2(τ) = ¯ℓµ0 −12δµ + V µτ,(2.1)here V µ is an arbitrary timelike vector, δµ is perpendicular to V µ and so is spacelike, and ¯ℓµ0is completely arbitrary.

The center of mass of the pair of wormhole mouths follows the line¯ℓµ(τ) = ¯ℓµ0 + V µτ,(2.2)and the separation of the wormhole mouths is described by the vector δµ.The present construction of course does not describe a time machine, but one shall soonsee that very simple manipulations appear to be able to take this “safest of all possiblewormholes” and turn it into a time machine.B. Step 1: Induction of a time–shiftAs the penultimate step in one’s construction of a time machine it is necessary to inducea (possibly small) “time shift” between the two mouths of the wormhole.

For clarity, workin the rest frame of the wormhole mouths. After a suitable translation and Lorentz transfor-mation the simple model wormhole of step 0 can, without loss of generality, be described bythe identification (t, 0, 0, 0) ≡(t, 0, 0, δ).

One wishes to change this state of affairs to obtaina wormhole of type (t, 0, 0, 0) ≡(t+T, 0, 0, ℓ), where T is the time–shift and ℓis the distancebetween the wormhole mouths. Mathematically one wishes to force the vectors V µ and δµ6

to no longer be perpendicular to each other so that it is possible to define the time–shift tobe T = V µδµ, while the distance between the mouths is ℓ= ∥(δµν + V µVν)δν∥. Physicallythis may be accomplished in a number of different ways:1.

The relativity of simultaneityApply identical forces to the two wormhole mouths so that they suffer identical accelera-tions (as seen by the initial rest frame). It is clear that if the mouths initially connect equaltimes (as seen by the initial rest frame), and if their accelerations are equal (as seen by theinitial rest frame), then the paths followed by the wormhole mouths will be identical up to afixed translation by the vector δµ.

Consequently the wormhole mouths will always connectequal times — as seen by the initial rest frame. Mathematically this describes a situationwhere δµ is a constant of the motion while V µ is changing.

Of course, after the applied ex-ternal forces are switched offthe two mouths of the wormhole have identical four–velocitiesV µf not equal to their initial four–velocities V µi . And by the relativity of simultaneity equaltimes as viewed by the initial rest frame is not the same as equal times as viewed by thefinal rest frame.

If one takes the center of motion of the wormhole mouths to have beenaccelerated from four–velocity (1, 0, 0, 0) to four–velocity (γ, 0, 0, γβ) then the relativity ofsimultaneity induces a time–shiftT = γβδ,(2.3)while (as seen in the final rest frame) the distance between the wormhole mouths becomesℓ= γδ.2. Special relativistic time dilationAnother way of inducing a time shift in the simple model wormhole of step 0 is to simplymove one of the wormhole mouths around and to rely on special relativistic time dilation.This can either be done by using rectilinear motion [5] or by moving one of the mouths7

around in a large circle [23]. Assuming for simplicity that one mouth is un-accelerated, thatthe other mouth is finally returned to its initial four–velocity, and that the final distancebetween wormhole mouths is the same as the initial distance, then the induced time–shiftis simply given by:T =Z fi (γ −1)dt.

(2.4)Mathematically, V µ and ℓare unaltered, while T and hence δµ are changed by this mecha-nism.3. General relativistic time dilationAs an alternative to relying on special relativistic time dilation effects, one may insteadrely on the general relativistic time dilation engendered by the gravitational redshift [12].One merely places one of the wormhole mouths in a gravitational potential for a suitableamount of time to induce a time–shift:T =Z fi (√g00 −1)dt.4.

CommentOf course, this procedure has not yet built a time machine. The discussion so far hasmerely established that given a traversable wormhole it is trivial to arrange creation ofa traversable wormhole that exhibits a time–shift upon passing through the throat — thistime–shift can be created by a number of different mechanisms so that the ability to producesuch a time–shift is a very robust result.C.

Step 2: Bring the wormhole mouths close togetherHaving constructed a traversable wormhole with time–shift the final stage of time ma-chine construction is deceptively simple: merely push the two wormhole mouths towards one8

another (this may be done as slowly as is desired). A time machine forms once the physicaldistance between the wormhole mouths ℓis less than the time–shift T. Once this occurs,it is clear that closed timelike geodesics have formed — merely consider the closed geodesicconnecting the two wormhole mouths and threading the wormhole throat.

As an alternativeto moving the mouths of the wormhole closer together one could of course arrange for thetime–shift to continue growing while keeping the distance between the mouths fixed — suchan approach would obscure the distinctions between steps 1 and 2.The advantage of clearly separating steps 1 and 2 in one’s mind is that it is now clear thatthe wormhole mouths may be brought together arbitrarily slowly — in fact adiabatically —and still appear to form a time machine. If one makes the approach adiabatic (so that onecan safely take the time–shift to be a constant of the motion) then the wormhole may bemathematically modeled by Minkowski space with the lines (t, 0, 0, z(t)) and (t+T, 0, 0, ℓ0 −z(t)) identified.

The physical distance between the wormhole mouths is ℓ(t) = ℓ0 −2z(t).The family of closed “almost” geodesics alluded to in the introduction is just the set ofgeodesics (straight lines), parameterized by t, connecting ℓ1µ(t) with ℓ2µ(t). Specifically,with σ ∈[0, 1]:Xµ(t, σ) = σℓ1µ(t) + {1 −σ}ℓ2µ(t)=t + {1 −σ}T, 0, 0, {1 −σ}ℓ0 −{1 −2σ}z(t).

(2.5)Here σ is a parameter along the closed geodesics, while t parameterizes the particular closedgeodesic under consideration. These curves are true geodesics everywhere except at (σ =0) ≡(σ = 1), the location of the wormhole mouths, where there is a “kink” in the tangentvector induced by the relative motion of the wormhole mouths.

The invariant length of themembers of this family of closed geodesics is easily seen to be:δ(t) = ∥ℓ1µ(t) −ℓ2µ(t)∥= ∥(T, 0, 0, ℓ(t))∥=qℓ(t)2 −T 2. (2.6)Once ℓ(t) < T a time machine has formed.

Prior to time machine formation the vec-tor (T, 0, 0, ℓ(t)) is, by hypothesis, spacelike.Therefore it is possible to find a Lorentz9

transformation to bring it into the form (0, 0, 0, δ) with δ =√ℓ2 −T 2.A brief cal-culation shows that this will be accomplished by a Lorentz transformation of velocityβ = T/ℓ(t) = T/qT 2 + δ2(t), so that γ = ℓ/δ and γβ = T/δ. In this new Lorentz frame thewormhole connects “equal times”, and this frame can be referred to as the frame of simul-taneity (synchronous frame).

Note in particular, that as one gets close to building a timemachine δ(t) →0, that β →1, so that the velocity of the frame of simultaneity approachesthe speed of light.D. DiscussionThe construction process described above clearly separates the different effects at workin the putative construction of time machines.

Given a traversable wormhole arbitrarilysmall special and/or general relativistic effects can be used to generate a time–shift. Givenan arbitrarily small time shift through a traversable wormhole, arbitrarily slow adiabaticmotion of the wormhole mouths towards each other appears to be sufficient to construct atime machine.

This immediately leads to all of the standard paradoxes associated with timetravel and is very disturbing for the state of physics as a whole.As a matter of logical necessity precisely one of the following alternatives must hold:1: the Boring Physics Conjecture,2: the Hawking Chronology Protection Conjecture,3: the Novikov Consistency Conspiracy,4: the Radical Rewrite Conjecture.The Boring Physics Conjecture may roughly be formulated as: “Suffer not traversablewormholes to exist”. Merely by asserting the nonexistence of traversable wormholes all timetravel problems of the particular type described in this paper go quietly away.

However,to completely forbid time travel requires additional assumptions – such as the nonexistenceof (infinite length) cosmic strings and limitations on the tipping over of light cones. Forinstance, requiring the triviality of the fundamental group π1(M) (the first homotopy group)10

implies that the manifold possesses no closed noncontractible loops. Such a restriction wouldpreclude both (1) traversable wormholes (more precisely: that class of traversable wormholesthat connect a universe to itself), and (2) cosmic strings (both finite and infinite).

But eventhis is not enough — time travel can seemingly occur in universes of trivial topology [1,2,24],so even stronger constraints should be imposed.In this regard Penrose’s version of theStrong Cosmic Censorship Conjecture [25] implies the Causality Protection Conjecture viathe equivalence: (Strong Cosmic Censorship) ⇔(global hyperbolicity) ⇒(strong causality)⇔(∃global time function). The “Boring Physics Conjecture” might thus best be formulatedas Strong Cosmic Censorship together with triviality of the fundamental group π1(M).There is certainly no experimental evidence against the Boring Physics Conjecture, butthis is a relatively uninteresting possibility.

In particular, considering the relatively benignconditions on the stress–energy tensor required to support a traversable wormhole it seemsto be overkill to dispose of the possibility of wormholes merely to avoid problems with timetravel.Less restrictively the Hawking Chronology Protection Conjecture allows the existence ofwormholes but forbids the existence of time travel [1,2]: “Suffer not closed non-spacelikecurves to exist”. If the Chronology Protection Conjecture is to hold then there must besomething wrong with the apparently simple manipulations described above that appearedto lead to the formation of a time machine.

In fact, that is the entire thrust of this paper— to get some handle on precisely what might go wrong. It proves to be the case that theproblem does not lie with the induction of a time–shift — as previously mentioned this isa robust result that can be obtained through a number of different physical mechanisms.Rather, it is the apparently very simple notion of pushing the wormhole mouths togetherthat is at fault.

A number of physical effects seem to conspire to prevent one from actuallybringing the wormhole mouths close together. This will be discussed in detail in subsequentchapters.More radically, the Novikov Consistency Conspiracy is willing to countenance the ex-istence of both traversable wormholes and time travel but but asserts that the multiverse11

must be be consistent no matter what — this point of view is explored in [7–9]: “Suffer notan inconsistency to exist”.More disturbingly, once one has opened Pandora’s box by permitting closed timelikecurves (time travel) I would personally be rather surprised if something as relatively mildas the Novikov Consistency Conspiracy would be enough to patch things up. Certainly thesometimes expressed viewpoint that the Novikov Consistency Conspiracy is the unique an-swer to the causality paradoxes is rather naive at best.

The Novikov Consistency Conspiracyis after all still firmly wedded to the notion of spacetime as a four–dimensional Hausdorffdifferentiable manifold.For a rather more violently radical point of view permit me to propound the RadicalRewrite Conjecture wherein one posits a radical rewriting of all of known physics from theground up. Suppose, for instance, that one models spacetime by a non-Hausdorffdifferen-tiable manifold.

What does this mean physically? A non-Hausdorffmanifold has the bizarreproperty that the dimensionality of the manifold is not necessarily equal to the dimensional-ity of the coordinate patches [26].

From a physicist’s perspective, this idea has been exploredsomewhat by Penrose [25]. Crudely put: while coordinate patches remain four dimensionalin such a spacetime, the dimensionality of the underlying manifold is arbitrarily large, andpossibly infinite.

Local physics remains tied to nicely behaved four dimensional coordinatepatches. Thus one can, for instance, impose the Einstein field equations in the usual man-ner.

Every now and then, however, a passing wave front (generated by a “branching event”)passes by and suddenly duplicates the whole universe. It is even conceivable that a branch-ing non-Hausdorffspacetime of this type might be connected with or connectable to theEverett “many worlds” interpretation of quantum mechanics [27,28].If one wishes an even more bizarre model of reality, one could question the naive notionthat the “present” has a unique fixed “past history”.

After all, merely by adding a timereversed “branching event” to our non-Hausdorffspacetime one obtains a “merging event”where two universes merge into one. Not only is predicibility more than somewhat dubiousin such a universe, but one appears to have lost retrodictability as well.

Even moreso than12

time travel, such a cognitive framework would render the universe unsafe for historians, asit would undermine the very notion of the existence of a unique “history” for the historiansto describe!Such radical speculations might further be bolstered by the observation that if one takesFeynman’s “sum over paths” notion of quantum mechanics seriously then all possible pasthistories of the universe should contribute to the present “state” of the universe.I raise these issues, not because I particularly believe that that is the way the universeworks, but rather, because once one has opened Pandora’s box by permitting time travel,I see no particular reason to believe that the only damage done to our notions of realitywould be something as facile as the Novikov Consistency Conspiracy.13

III. QUANTUM EFFECTSA.

Vacuum polarization effectsAttention is now drawn to the effect that the wormhole geometry has on the propagationof quantum fields. Generically, one knows that a non–trivial geometry modifies the vacuumexpectation value of the renormalized stress–energy tensor.

One way of proceeding is torecall that the wormhole is modelled by the identification(t, 0, 0, 0) ≡(t + T, 0, 0, ℓ). (3.1)This formulation implies that one is working in the rest frame of the wormhole mouths.

Fordefiniteness, suppose one is considering a quantized scalar field propagating in this spacetime.One could then, in principle, investigate solutions of d’Alembert’s equation in Minkowskispace subject to these identification constraints, find the eigenfunctions and eigenvalues,quantize the field and normal order, and in this manner eventually calculate < 0|T µν|0 >.Alternatively, one could adopt point–splitting techniques. Such calculations are decidedlynon–trivial.

For the sake of tractability it is very useful to adopt a particular approximationthat has the effect of reducing the problem to a generalized Casimir effect calculation.Perform a Lorentz transformation of velocity β = T/ℓ, so that in this new Lorentz framethe wormhole connects “equal times”. In this frame of simultaneity (synchronous frame)the wormhole is described by the identification(γt, 0, 0, γβt) ≡(γt, 0, 0, γβt + δ).

(3.2)The discussion up to this point has assumed “point like” mouths for the wormhole throat.To proceed further one will need to take the wormhole mouths to have some finite size, andwill need to specify the precise manner in which points on the wormhole mouths are tobe identified. Two particularly simple types of pointwise identification are of immediateinterest.14

(1) Synchronous identification:∀x1 ∈∂Ω1, x2 ∈∂Ω2, xµ1 ≡xµ2 = xµ1 + sˆδµ, where ˆδµ is a fixed (spacelike) unit vector and sis a parameter to be determined. A particular virtue of synchronous identification is thatonce one goes to the synchronous frame, all points on the wormhole mouths are identifiedat “equal times”, that is: x1 = (t,⃗x1) ≡(t,⃗x2) = (t,⃗x1 + sˆz) = x2.

(2) Time-shift identification:∀x1 ∈∂Ω1, x2 ∈∂Ω2, x1 = (t,⃗x1) ≡(t + T,⃗x2) = x2, with ⃗x1, ⃗x2 ranging over the mouths ofthe wormhole in a suitable manner, This “time–shift” identification is the procedure adoptedby Kim and Thorne and is responsible for many of the technical differences between thatpaper and this. A particularly unpleasant side effect of “time–shift” identification is thatthere is no one unique synchronous frame for the entire wormhole mouth.Though these identification schemes apply to arbitrarily shaped wormhole mouths, onemay for simplicity, take the wormhole mouths to be spherically symmetric of radius R asseen in their rest frames.

The resulting spacetime is known as a capon spacetime. Adopting“synchronous identification”, in the frame of simultaneity the wormhole mouths are oblatespheroids of semi–major axis R, and semi–minor axis R/γ = Rδ/ℓ.

Let us work in theparameter regime δ << ℓand δ << R, then one may safely approximate the wormholemouths by flat planes. One is then seen to be working with a minor generalization of theordinary Casimir effect geometry — a Casimir effect with a time–shift.

(For a nice surveyarticle on the Casimir effect see [29]. See also the textbooks [30,31].

)Alternatively, one could work with the cubical wormholes of reference [21]. When twoof the flat faces of such a wormhole are directly facing each other, the technical differencesbetween synchronous and time shift identifications vanish for those faces.

With this un-derstanding the following comments can also be applied to cubical wormholes with eitheridentification scheme.The model for the wormhole spacetime is now Minkowski space with two planes identified:(γt, x, y, γβt) ≡(γt, x, y, γβt + δ),(3.3)15

or, going back to the rest frame(t, x, y, 0) ≡(t + T, x, y, ℓ). (3.4)The net effect of (1) allowing for a finite size for the wormhole throat, of (2) either adoptingsynchronous identification or of restricting attention to flat faced wormholes, combined with(3) the approximation δ << R, has thus been to replace the identification of world–lineswith the identification of planes.

The same effect could have been obtained by staying in therest frame, giving the wormhole mouths finite radius R, displacing the wormhole mouths toz = −R and z = ℓ+ R respectively, and letting R →∞. The advantage of the argumentas presented in the frame of simultaneity is that it makes clear that for any synchronouslyidentified traversable wormhole close enough to forming a time machine this approximationbecomes arbitrarily good.Continuing to work in the frame of simultaneity, the manifest invariance of the boundaryconditions under rotation and/or reflection in the xy plane restricts the stress–energy tensorto be of the form< 0|Tµν|0 >=Ttt00Ttz0Txx0000Txx0Ttz00Tzz(3.5)By considering the effect of the boundary conditions one may write eigenmodes of thed’Alambertian in the separated formφ(t, x, y, z) = e−iωteikxxeikyyeikzz,(3.6)subject to the constraint φ(t, x, y, z) = φ(t, x, y, z + δ).

Therefore the boundary conditionon kz implies the classical quantization kz = ±n2π/δ, while kx and ky are unquantized (i.e.arbitrary and continuous). The equation of motion constrains ω to beω =q(n2π/δ)2 + k2⊥.

(3.7)16

Now, recalling that < 0|Ttz|0 >∝< 0|∂tφ∂zφ|0 >, and performing a mode sum over n, it isclear that the positive values of kz exactly cancel the negative values so that < 0|Ttz|0 >= 0.Furthermore, one may note that the quantization conditions on ω and kz depend only on δ— not on β or γ. Thus, without loss of generality one may immediately apply the result forβ = 0 to the case β ̸= 0 and obtain< 0|Tµν|0 >= −¯hkδ4 (ηµν −4nµnν).

(3.8)Here nµ = δµ/δ is the tangent to the closed spacelike geodesic threading the wormhole throat.In the frame of simultaneity nµ = (0, 0, 0, 1), while in the rest frame nµ = (−T/δ, 0, 0, ℓ/δ).The constant k is a dimensionless numerical factor that it is not worth the bother to com-pletely specify. In a more general context the argument given above applies not only toscalar fields but also to fields of arbitrary spin that are (approximately) conformally cou-pled.

In general k will depend on the nature of the applied boundary conditions (twistedor untwisted), the spin of the quantum field theory under consideration, etc. Furthermore,as δ decreases, massive particles may be considered as being approximately massless once¯h/δ >> m. Thus k should be thought of as changing in stepwise fashion as δ →0.

Thisbehaviour is entirely analogous to the behaviour of the R parameter in e+ e−annihilation.A complete specification of k would thus require complete knowledge of the asymptotic be-haviour of the spectrum of elementary particles, and it is for this reason that one declinesfurther precision in the specification of k.The form of the stress–energy tensor can also be determined by noting that the iden-tifications (3.3) and (3.4) describing the wormhole can be easily solved by considering anotherwise free field in Minkowski space subject to the constraintφ(xµ) ≡φ(xµ ± nδµ)n = 0, 1, 2, ...(3.9)Since these constraints do not depend on V , the four velocity of the wormhole mouth, onemay directly apply the results obtained for the ordinary Casimir effect [29–31].At this stage it is also important to realise that the sign of k is largely immaterial. If kis positive then one observes a positive Casimir energy which tends to repel the wormhole17

mouths, and so tends prevent the formation of a time machine. On the other hand if k isnegative one finds a large positive stress threading the wormhole throat.

This stress tendsto act to collapse the wormhole throat, and so also tends to prevent the formation of a timemachine.Note the similarities — and differences — between this result and those of Hawking[1,2] and Kim and Thorne [3].The calculation presented here is in some ways perhapsmore general in that it is clear from the preceding discussion that this calculation is capableof applying to the generic class of “synchronously identified” wormholes once δ << R.Furthermore, instead of the value of the stress–energy tensor being related to the temporaldistance to the “would be Cauchy horizon”, it is clear in this formalism that it is theexistence and length of closed spacelike geodesics that controls the magnitude of the stress–energy tensor.The limitations inherent in the type of wormhole model currently under considerationshould also be made clear: the model problem is optimally designed to make it easy to thwartthe formation of a time machine (i.e. closed timelike curves).

It is optimal for thwartingin two senses: (1) the choice of synchronous identification prevents defocussing of light rayswhen they pass through the wormhole, (i.e. it makes f = 0 in the notation of Hawking[1,2]).

If an advanced civilization were to try to create a time machine, that civilizationpresumably would optimize their task by choosing f arbitrarily large, and not optimizeNature’s ability to thwart by choosing f = 0. (2) the adiabatic approximation requiresthe relative motion of the wormhole mouths to be arbitrarily slow, (i.e.

h arbitrarily closeto zero, in the notation of Hawking [1,2]). This is an idealized limit; and again, it is thecase that this idealization makes it easier for Nature to thwart the time machine formation,because it gives the growing vacuum polarization an arbitrarily long time to act back on thespacetime and distort it.18

B. k positive — repulsive Casimir energyFor the synchronously identified model wormholes we have been discussing one maycalculate the four–momentum associated with the stress–energy tensor byP µ =I< 0|T µν|0 > dΣν =< 0|T µν|0 > π R2 δ nµ⊥. (3.10)Here nµ⊥= (1, 0, 0, 0) in the frame of simultaneity, so that n⊥is perpendicular to n. ThusP µ = ¯hkδ4 π R2 δ nµ⊥.

(3.11)By going back to the rest frame one obtainsP µ = ¯hkδ4 πR2δ ℓδ, 0, 0, Tδ!,(3.12)so that one may identify the Casimir energy asECasimir = P µVµ = ¯hkπR2ℓδ4. (3.13)Take k > 0 for the sake of discussion, then this Casimir energy would by itself give aninfinitely repulsive hard core to the interaction between the wormhole mouths.

However oneshould exercise some care. The calculation is certainly expected to break down for δ < ℓP,so that one may safely conclude only that there is a finite but large potential barrier toentering the full quantum gravity regime.

Fortunately this barrier is in fact very high. ForT >> TP one can estimateEbarrier ≈EP RℓP2 TTP,(3.14)while for T << TP (i.e.

T ≈0) one may estimateEbarrier ≈EP RℓP2. (3.15)For macroscopic wormholes these barriers are truly enormous.

For microscopic worm-holes (R << ℓP) one really doesn’t care what happens. Firstly because it is not too clearwether wormholes with R << ℓP can even exist.

Secondly, because even if such wormholes19

do exist, they would in no sense be traversable [20], and so would be irrelevant to the con-struction of usable time machines. The non–traversability of such microscopic wormholes iseasily seen from the fact that they could only be probed by particles of Compton wavelengthmuch less that their radius, implying thatEprobe > ¯h/R >> EP.

(3.16)On the other hand, for macroscopic wormholes one may safely assert thatR/ℓp > 2m/mP,(3.17)since otherwise each wormhole mouth would be enclosed by its own event horizon and thewormhole would no longer be traversable. Combining these results, it is certainly safe toassert that the barrier to full quantum gravity satisfiesEbarrier >> EP2mmP2≈m2mP.

(3.18)To get a better handle on the parameter regime in question, consider the combined effectsof gravity and the Casimir repulsion. In linearized gravity the combined potential energy isV = − mmP2 ¯hℓ+¯hkπR2ℓ(ℓ2 −T 2)2.

(3.19)The Casimir energy dominates over the gravitational potential energy onceδ4 ≈kπR2ℓ2(m/mP)2 > kπℓ2Pℓ2, i.e.δ ≥qℓPℓ. (3.20)One may safely conclude: (1) for k positive there is an enormous potential barrier totime machine formation, (2) for “reasonable” wormhole parameters the repulsive Casimirforce dominates long before one enters the Planck regime.C.

k negative — wormhole disruption effectsAt first glance, should the constant k happen to be negative, one would appear to havea disaster on one’s hands. In this case the Casimir force seems to act to help rather than20

hinder time machine formation. Fortunately yet another physical effect comes into play.Recall, following Morris and Thorne [4] that a traversable wormhole must be threaded bysome exotic stress energy to prevent the throat from collapsing.

In particular, at the throatitself (working in Schwarzschild coordinates) the stress–energy tensor takes the formTµν =¯hℓ2PR2ξ0000χ0000χ0000−1(3.21)On general grounds ξ < 1, while χ is unconstrained. On the other hand, the vacuum polar-ization effects just considered contribute to the stress–energy tensor in the region betweenthe wormhole mouths an amount< 0|Tµν|0 >= ¯hkδ44(T/δ)2 + 1004Tℓ/δ20−10000−104Tℓ/δ2004(ℓ/δ)2 −1(3.22)In particular, the tension in the wormhole throat, required to prevent its collapse is τ =¯h/(ℓ2PR2), while the tension contributed by vacuum polarization effects isτ = −¯hkδ44(ℓ/δ)2 −1≈−4¯hkℓ2δ6(3.23)Vacuum polarization effects dominate over the wormhole’s internal structure onceδ3 < ℓPℓR/q|k|,(3.24)and it is clear that this occurs well before reaching the Planck regime.By looking at the sign of the tension it appears that for negative values of k these vacuumpolarization effects will tend to decrease R, that is, will tend to collapse the wormhole throat.Conversely, for positive values of k these vacuum polarization effects would appear to tendto make the wormhole grow.21

Concentrate on negative values of k. Well before δ shrinks down to the Planck length,the vacuum polarization will have severely disrupted the internal structure of the wormhole— presumably leading to wormhole collapse.It should be pointed out that there is noparticular need for the collapse to proceed all the way down to R = 0 and subsequenttopology change. It is quite sufficient for the present discussion if the collapse were to haltat R ≈ℓP.

In fact, there is evidence, based on minisuperspace calculations [32–34], thatthis is indeed what happens. If indeed collapse is halted at R ≈ℓP by quantum gravityeffects then the universe is still safe for historians since there is no reasonable way to get aphysical probe through a Planck scale wormhole [20].D.

SummaryVacuum polarization effects become large as δ →0. Depending on an overall undeter-mined sign either (1) there is an arbitrarily large force pushing the wormhole mouths apart,or (2) there are wormhole disruption effects at play which presumably collapse the wormholethroat down to the size of a Planck length.

Either way, usable time machines are avoided.Unfortunately, because the class of wormholes currently under consideration is optimallydesigned to make it easy to thwart the formation of a time machine, it is unclear to whatextent one may draw generic conclusions from these arguments.22

IV. GRAVITATIONAL BACK REACTIONThere is yet another layer to the universe’s “defense in depth” of global causality.

Con-sider linearized fluctuations around the locally flat background metric describing the syn-chronously identified model wormhole.gµν = ηµν + ¯hµν −12¯hηµν. (4.1)By going to the transverse gauge ¯hµν,ν = 0, one may write the linearized Einstein fieldequations as∆¯hµν = −16πℓ2P¯h< 0|Tµν|0 > .

(4.2)Now, working in the rest frame of the wormhole mouths, the time translation invarianceof the geometry implies ∂t¯hµν = 0. Similarly, within the confines of the Casimir geometryapproximation, the translation symmetry in the x and y directions implies ∂x¯hµν = 0 =∂y¯hµν.

Thus the linearized Einstein equations reduce to∂z2¯hµν = 16πkℓ2Pδ4(ηµν −4nµnν) . (4.3)Note that the tracelessness of Tµν implies the tracelessness of ¯hµν so that ¯hµν = hµν.

Usingthe boundary condition that hµν(z = 0, t = 0) = hµν(z = ℓ, t = T) the linearized Einsteinfield equations integrate tohµν(z) = 16π2 (z −ℓ/2)2 kℓ2Pδ4(ηµν −4nµnν) . (4.4)To estimate the maximum perturbation of the metric calculateδgµν = hµν(z = 0) = hµν(z = ℓ) = 16πkℓ2Pℓ28δ4(ηµν −4nµnν) .

(4.5)In particularδgtt = −16πkℓ2Pℓ28δ4 1 + 4T 2δ2!. (4.6)23

Now if T >> TP, which is the case of interest for discussing the putative formation oftime machines, this back reaction certainly becomes large well before one enters the Planckregime, thus indicating that the gravitational back reaction becomes large and importantlong before one needs to consider full quantum gravity effects.Even if T = 0 one has δgtt = −(16π/8)k(ℓ2P/ℓ2). While the back reaction is in this casesomewhat smaller, it still becomes large near the Planck scale.It might be objected that the frame dependent quantity δgtt is not a good measure ofthe back reaction.

Au contraire, one need merely consider the manifestly invariant objectϕ = hµνV µV ν. (4.7)Here V µ is the four velocity of the wormhole throat, and ϕ is the physical gravitational po-tential governing the motion of the wormhole mouths.

For instance, the geodesic accelerationof the wormhole mouths is easily calculated to beaµ = V µ;νV ν = ΓµαβV αV β = −12∇µϕ. (4.8)Because of the delicacy of these particular issues, it may be worthwhile to belabor thepoint.

Suppose one redoes the entire computation in the frame of simultaneity. One stillhas∆¯hµν = −16πℓ2P¯h< 0|Tµν|0 > .

(4.9)Where now the boundary conditions are hµν(z = βt, t) = hµν(z = βt+δ, t). This immediatelyimplies that hµν is a function of (z −βt).

Indeed the linearized Einstein equations integratetohµν(z, t) = 16π2(z −βt −δ/2)2(1 −β2)kℓ2Pδ4(ηµν −4nµnν) . (4.10)So that at the mouths of the wormhole, (z = βt) ≡(z = βt + δ), factors of δ and γ =1/√1 −β2 combine to yieldδgµν = hµν(z = βt, t) = hµν(z = βt + ℓ, t) = 16πkℓ2Pℓ28δ4(ηµν −4nµnν) .

(4.11)24

Which is the same tensor as the result calculated in the rest frame, (as of course it mustbe).The only real subtlety in this case is to realise that, as previously argued, δgtt as measuredin the frame of simultaneity is not a useful measure of the gravitational back reaction.Yet another way of seeing this is to recall that in linearized Einstein gravity the gravita-tional potential of a pair of point particles is determined by their masses, relative separation,and their four velocities by [35,36]V =Gm1m2r (2(V1 · V2)2 −1γ1γ2). (4.12)Applied to the case presently under discussion, one immediately infers that the differencebetween the synchronous frame and the rest frame induces an enhancement factor of γ2.While the four velocity of the wormhole throat does not enter into the computation of thestress energy tensor it is important to realise that the four velocity of the throat does, via thenaturally imposed boundary conditions, have an important influence on the gravitationalback reaction.In any event, the gravitational back reaction will radically alter the spacetime geometrylong before a time machine has the chance to form.

Note that for k negative, so that theCasimir energy is negative and attractive, the sign of δgtt obtained from this linearizedanalysis indicates a repulsive back reaction and furthermore hints at the formation of anevent horizon should δ become sufficiently small. Unfortunately a full nonlinear analysiswould be necessary to establish this with any certainty.25

V. GENERALITIESTo put the analysis of this paper more properly in perspective it is useful to abstractthe essential ingredients of the calculation. Consider a stationary, not necessarily static, butotherwise arbitrary Lorentzian spacetime of nontrivial topology.

Specifically, assume thatπ1(M) ̸= e, that is, that the first homotopy group (the fundamental group) is nontrivial. Bydefinition the non–triviality of π1(M) implies the existence of closed paths not homotopicto the identity.

This is the sine qua non for the existence of a wormhole. By smoothnessarguments there also exist smooth closed paths not homotopic to the identity.

Take one ofthese smooth closed paths and extremize its length in the Lorentzian metric. One infers theexistence of at least one smooth closed geodesic in any spacetime with nontrivial fundamentalgroup.

If any of these closed geodesics is timelike or null then the spacetime is diseased andshould be dropped from consideration. Furthermore, since the metric can be expressed in at independent manner one may immediately infer the existence of an infinite one–parameterfamily of closed geodesics parameterized by t.A slow adiabatic (i.e.quasi–stationary)variation of the metric will preserve this one parameter family, though the members of thisfamily may now prove to be only approximately geodesics.

The formation of a time machineis then signalled by this one parameter family of closed “almost” geodesics switching overfrom spacelike character to null and then timelike character.If one is desirous of proceeding beyond the adiabatic approximation this can be done atthe cost of additional technical machinery. Consider now a completely arbitrary Lorentzianspacetime of nontrivial topology.

Pick an arbitrary base point x. Since π1(M) is nontrivialthere certainly exist closed paths not homotopic to the identity that begin and end at x. Bysmoothness arguments there also exist smooth closed paths not homotopic to the identity —though now there is no guarantee that the tangent vector is continuous as the path passesthrough the point x where it is pinned down.

Take one of these smooth closed paths andextremize its length in the Lorentzian metric. One infers the existence of many smooth(except at the point x) closed geodesics passing through every point x in any spacetime26

with nontrivial fundamental group. Again, if any of these closed geodesics is timelike or nullthen the spacetime is diseased and should be dropped from consideration.To get a suitable one parameter family of closed geodesics that captures the essentialelements of the geometry, suppose merely that one can find a well defined throat for one’sLorentzian wormhole.

Place a clock in the middle of the throat. At each time t as measuredby the wormhole’s clock there exists a closed “pinned” geodesic threading the wormhole andclosing back on itself in “normal space”.

This geodesic will be smooth everywhere exceptpossibly at the place that it is “pinned” down by the clock. This construction thus providesone with a one parameter family of closed geodesics suitable application of the precedinganalysis.Pick one of these closed geodesics.

Let σ ∈[0, 1] be a parameterization of the geodesic.Extend this to a coordinate patch {t, x, y, σ} covering a tube–like neighbourhood surroundingthe geodesic. By adopting the spacelike generalization of comoving coordinates one maywithout loss of generality write the metric in the formds2 = gij(t, x, y, σ)dxidxj + δ2(t, x, y)dσ2.

(5.1)Here dxi ∈{dt, dx, dy}, while the fact that the curve x = y = t = 0 is a geodesic is expressedby the statement ∂iδ(0, 0, 0) = 0. By considering the three dimensional Riemann tensor onslices of constant σ one may define a quantityR−2 = maxσ∈[0,1]qRijklRijkl.

(5.2)This quantity R estimates the minimum “radius of curvature” of the constant σ hypersur-faces and so may usefully be interpreted as a measure of the minimum radius of the wormholethroat.For a traversable wormhole, precursor to a traversable time machine, one wishes R tobe macroscopic and to remain so. Now consider what happens as δ shrinks.

Eventuallyδ << R, at which stage the length of the closed geodesic is much less than the lengthscale of the transverse dimensions of the wormhole throat. By going to Riemann normalcoordinates one may now approximate the metric by27

ds2 = ηijdxidxj + δ2(t, x, y)dσ2, (x, y << R),(5.3)which is just a generalization of the Casimir geometry expressed in the synchronous frame.Nasty “fringe” effects occur for x, y ≥R, but these may be quietly neglected as is usualand reasonable within the confines of the Casimir approximation. Of greater significance isthe behaviour of δ(t, x, y) as a function of t, x, and y.

If the variation of δ(t, x, y) is notparticularly rapid one may safely make the further approximation of modelling the situationby the ordinary Casimir effect. In that case the conclusions of the previous chapter thenfollow in this more general context.This happy circumstance is automatically fulfilled if if the wormhole mouths are syn-chronously identified, since then δ(t, x, y) ≤δ(t, 0, 0) + O(R/γ) = δ(t, 0, 0)[1 + O(R/ℓ)].

Onthe other hand, the time shift identification adopted by Kim and Thorne leads to a veryrapid growth of δ(t, x, y). This rapid variation of δ(t, x, y) with position makes analysis con-siderably more difficult.

Fortunately, it is still relatively easy to convince oneself that thevacuum expectation value of the renormalized stress energy tensor along the central geodesicmust be proportional to δ−4. Therefore, at least along the central geodesic, the exotic mat-ter supporting the wormhole throat is overwhelmed by the renormalized stress energy.

Theextent to which this disruption along the central geodesic might lead to disruption of thetraversability of the wormhole is unfortunately not calculable by the present techniques.The whole point of this aspect of the discussion is, of course, to see what exactly isspecial about the simple model wormholes considered earlier — the analysis presented inthis paper is seen to be limited to adiabatic (spatial and temporal) changes in δ(t, x, y).28

VI. COMPARISONS WITH PREVIOUS WORKThere are several manners in which one might attempt to parameterize the strengthof the divergences in the stress–energy tensor and the gravitational back reaction.Theestimates of Kim and Thorne [3] make extensive use of the “proper time to the Cauchyhorizon” as measured by an observer who is stationary with respect to one of the wormholemouths.

Hawking [1,2] advocates the use of an “invariant distance to the Cauchy horizon”,which is equal to the proper time to the Cauchy horizon as measured by an observer whois stationary with respect to the frame of simultaneity. On the other hand, this paper hasfocussed extensively on the invariant length of closed geodesics.The calculations of this paper suggest that the use of the “proper time to the Cauchyhorizon”, whether measured by an observer in the rest frame or in the synchronous frame isnot a useful way of parameterizing the singularities in the stress–energy tensor.

To see this,transcribe the “order of magnitude estimates” of Kim and Thorne [3] into the notation ofthis paper (∆t is the proper distance to the Cauchy horizon as measured in the rest frame).Consider a pair of slowly moving wormhole mouths with relative velocity βrel (not to beconfused with the totally different β ≡T/ℓassociated with the transformation from the restframe to the synchronous frame). For a small relative velocity, the wormhole is modeled bythe identification of world–lines(t, 0, 0, 0) ≡(t + T, 0, 0, ℓ−βrel t),(6.1)so that δ2(t) = (ℓ−βrel t)2 −T 2.

(There is nothing particularly sacred about taking a smallrelative velocity — it just simplifies life in that one only has a simple linear equation to solveinstead of a quadratic.) The Cauchy horizon forms when δ2(∆t) = 0, that is, when∆t = (ℓ−T)βrel.

(6.2)Resubstituting into δ2 ≡δ2(t = 0), noting that ℓ+ T ≈2ℓ, and being careful to retain allfactors of βrel, one obtains for the Kim-Thorne estimates29

δ2≈2ℓ(ℓ−T) = 2ℓβrel ∆t. (6.3)< 0|T µν|0 > ≈(R/ℓ)ζ1ℓ(βrel∆t)3,(6.4)= (R/ℓ)ζ1ℓ(ℓ−T)3,(6.5)δgµν≈(R/ℓ)ζℓ2Pℓβrel∆t,(6.6)= (R/ℓ)ζℓ2Pℓ(ℓ−T).

(6.7)Here ζ is an integer exponent that depends on the homotopy class of the closed geodesic,and on whether or not one is close to either mouth of the wormhole. (The factors of βrelwere unfortunately omitted in the discussion portion of reference [3].

)To see the inadequacy of the use of ∆t at a conceptual level, observe that the use of ∆tintrinsically “begs the question” of the creation of a time machine by explicitly asserting theexistence of a Cauchy horizon and then proceeding to measure distances from that presumedhorizon. In particular, one may consider a wormhole in which one keeps the distance betweenthe mouths fixed (i.e.

set the relative velocity of the mouths to zero). By definition, thisimplies that the Cauchy horizon never forms and that ∆t = +∞.

(If one really wishes tobe technical, consider a collection of wormholes of the type considered by Kim and Thorneand take the limit as the relative velocity goes to zero).To get a better handle on the actual state of affairs, go to the Casimir limit by adoptingsynchronous identification and taking δ << R.In this case one may safely neglect thegeometrical factors (R/ℓ)ζ. By adopting a point spitting regularization the (renormalized)propagator may easily be seen to be approximated by< 0|φ(x)φ(x)|0 >≈δ−2.

(6.8)The stress–energy tensor is then computable by taking two derivatives of the propagator,yielding< 0|T µν|0 >≈¯hδ−4tµν. (6.9)Here tµν is a dimensionless tensor built up out of the metric and tangent vectors to the closed30

spacelike geodesic. While the components of tµν may be large in some Lorentz frames, theonly sensible invariant measure of the size, tµνtµν, is of order one.

In particular, looking atthe tt component, as viewed from the rest frame of one of the wormhole mouths, yieldsttt ≈ℓ2δ−2 ⇒< 0|T tt|0 >≈¯hℓ2δ−6 ≈¯hℓ−1(ℓ−T)−3. (6.10)This is the analogue of the estimate of Kim and Thorne, (including insertion of the appro-priate factor of βrel).

However, one sees that this estimate is somewhat misleading in thatit is a componentwise estimate that is highly frame dependent.Returning to the tensorial formulation, a double spatial integration gives an estimate forlinearized metrical fluctuationsδgµν ≈ℓ2Pℓ2δ−4tµν. (6.11)This estimate (which is backed up by the earlier explicit calculation) is radically differentfrom that of Kim and Thorne.

The difference can be traced back to the choice of synchronousidentification versus time shift identification, and the effect that these different identificationprocedures have on the van Vleck determinant.Turning to other matters: To understand Hawking’s “invariant” distance to the Cauchyhorizon, go to the synchronous frame.One requires βsynchrel<< β ≡T/ℓ, so that thewormhole may be described by the identificationt, 0, 0, βt≡t, 0, 0, (β −βsynchrel)t + δ,(6.12)then δ(t)2 = (δ −βsynchrelt)2. (Warning: βsynchrelis now the relative velocity as measured in thesynchronous frame.) The Cauchy horizon forms att = δ/βsynchrel.

(6.13)Hawking now takes this object t = δ/βsynchrel, which is in fact an invariant measure of thedistance to the Cauchy horizon, as his parameter governing the strength of the singularitiesencountered when trying to build a time machine. Of course, this parameter suffers from31

deficiencies analogous to the Kim–Thorne parameter in that it is intrinsically incapable ofproperly reflecting the divergence structure that is known to occur in the simple stationary(or indeed quasistationary) models considered in this paper.Turning to the question of the quantum gravity cutoff, Kim and Thorne assert that thiscutoffoccurs at∆t = (ℓ−T)βrel≈ℓP ⇒ℓ−T ≈βrel ℓP. (6.14)Hawking claims that the cutoffoccurs att = δ/βsynchrel≈ℓP ⇒δ ≈βsynchrelℓP.

(6.15)I beg to differ.Both of these proposed cutoffs exhibit unacceptable dependence on therelative motion of the wormhole mouths.As an improved alternative cutoff, consider the following: Pick a point x in spacetime.Since, by hypothesis, the spacetime has nontrivial topology there will be at least one closedgeodesic of nontrivial homotopy that runs from x to itself. If the length of this geodesic is lessthan a Planck length, then the region surrounding the point x should no longer be treatedsemiclassically.

Presumably, one should also supplement this requirement by a bound on thecurvature: If RαβγδRαβγδ > ℓ−4P then the region surrounding the point x should no longer betreated semiclassically.The advantage of the cutoffexpressed in this manner is that it appears to be eminentlyphysically reasonable. This cutoffgive meaningful answers in the case of a relatively sta-tionary pair of wormhole mouths, does not beg the question by requiring the existence of aCauchy horizon to formulate the cutoff, and furthermore can be concisely and clearly statedin complete generality for arbitrary spacetimes.32

VII. DISCUSSIONThis paper has examined at some length a series of physical effects that lend support toHawking’s Chronology Protection Conjecture.

Explicit calculations have been exhibited forsimple models in suitable parameter regimes. In particular much has been made of the useof adiabatic techniques combined with minor variations on the theme of the Casimir effect.These calculations suggest that the universe exhibits a defense in depth strategy withrespect to global causality violations.Effects contributing to the hoped for inability tomanufacture a time machine include vacuum polarization effects, wormhole disruption ef-fects, and the gravitational back reaction induced by the vacuum expectation value of therenormalized stress–energy tensor.Though the calculations in this paper have been phrased in terms of the traversablewormhole paradigm, the general result to be abstracted from this analysis is that: (1) anyspacetime of nontrivial topology contains closed (spacelike) geodesics.

(2) It appears thatthe universe reacts badly to closed spacelike geodesics attempting to shrink to invariantlength zero.I would conclude by saying that the available evidence now seems to favour Hawking’sChronology Protection Conjecture.Both the experimental evidence (the nonappearanceof hordes of tourists from the future) and the theoretical computations now support thisconjecture. Paraphrasing Hawking, it seems that the universe is indeed safe for historians.Perhaps most importantly, the single most serious objection to the existence oftraversable wormholes has always been the apparent ease with which they might be con-verted into time machines.

Thus adopting Hawking’s Chronology Protection Conjectureimmediately disposes of the single most serious objection against the existence of traversablewormholes. This observation should be interpreted as making the existence of traversablewormholes a much more reasonable hypothesis.

Investigations of questions such as the Av-eraged Weak Energy Condition, and all other aspects of traversable wormhole physics, thustake on a new urgency [37–41].33

ACKNOWLEDGMENTSI wish to thank Carl Bender, Leonid Grischuk, Mike Ogilvie, Donald Petcher, TomRoman, and Kip Thorne for useful discussions. I also wish to thank the Aspen Center forPhysics for its hospitality.

This research was supported by the U.S. Department of Energy.34

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출처: arXiv:9202.090 • 원문 보기