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arXiv:h

ep-th/9303023v13Mar1993

Hawkings chronology protection conjecture:

singularity structure of the quantum

stressenergy tensor

Matt Visser[*]Physics Department

Washington UniversitySt. Louis

Missouri 63130-4899

November 1992

Abstract

The recent renaissance of wormhole physics has led to a very disturbingobservation: If traversable wormholes exist then it appears to be rathereasy to to transform such wormholes into time machines. This extremelydisturbing state of affairs has lead Hawking to promulgate his chronologyprotection conjecture.

This paper continues a program begun in an earlier paper [Physical Re-

viewD47

, 554565 (1993)]. An explicit calculation of the vacuum expec-tation value of the renormalized stressenergy tensor in wormhole space-times is presented. Pointsplitting techniques are utilized. Particularattention is paid to computation of the Green function [in its Hadamardform], and the structural form of the stress-energy tensor near short closedspacelike geodesics. Detailed comparisons with previous calculations arepresented, leading to a pleasingly unified overview of the situation.

PACS: 04.20.-q, 04.20.Cv, 04.60.+n. hep-th/9303023

1

http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023v1http://arxiv.org/abs/hep-th/9303023http://arxiv.org/abs/hep-th/9303023http://arxiv.org/abs/hep-th/9303023v1

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1 INTRODUCTION

This paper addresses Hawkings chronology protection conjecture [1, 2], andsome of the recent controversy surrounding this conjecture [3, 4]. The qualitativeand quantitative analyses of reference [5] are extended. Detailed comparisons aremade with the analyses of Frolov [6], of Kim and Thorne [3], and of Klinkhammer[4].

The recent renaissance of wormhole physics has led to a very disturbing ob-servation: If traversable wormholes exist then it appears to be rather easy to totransform such wormholes into time machines. This extremely disturbing stateof affairs has lead Hawking to promulgate his chronology protection conjecture[1, 2].

This paper continues a program begun in an earlier paper [5]. In particular,this paper will push the analysis of that paper beyond the Casimir approxima-

tion. To this end, an explicit calculation of the vacuum expectation value of therenormalized stressenergy tensor in wormhole spacetimes is presented. Pointsplitting regularization and renormalization techniques are employed. Particularattention is paid to computation of the Green function [in its Hadamard form],and the structural form of the stress-energy tensor near short closed spacelikegeodesics.

The computation is similar in spirit to the analysis of Klinkhammer [ 4], butthe result appears at first blush to be radically different. The reasons forthis apparent difference (and potential source of great confusion) are trackeddown and examined in some detail. Ultimately the various results are shown toagree with one another in those regions where the analyses overlap. Further-more, the analysis of this paper is somewhat more general in scope and requiresfewer technical assumptions.

Notation: Adopt units where c 1, but all other quantities retain their usualdimensionalities, so that in particular G = h/m2P =

2P/h. The metric signature

is taken to be (, +, +, +).

2 THE GREEN FUNCTION

2.1 The Geodetic Interval

The geodetic interval is defined by:

(x, y) 1

2[s(x, y)]

2. (1)

Here we take the upper (+) sign if the geodesic from the point x to the pointy is spacelike. We take the lower () sign if this geodesic is timelike. In eithercase we define the arc length s(x, y) to be positive semi-definite. Note that,

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provided the geodesic from x to y is not lightlike,

x(x, y) = s(x, y) xs(x, y), (2)

= +s(x, y) t(x; ; x y). (3)

Here t(x; ; x y) xs(x, y) denotes the unit tangent vector at the pointx pointing along the geodesic away from the point y. When no confusionresults we may abbreviate this by t(x y) or even t(x). If the geodesic fromx to y is lightlike things are somewhat messier. One easily sees that for lightlikegeodesics x(x, y) is a null vector. To proceed further one must introduce acanonical observer, characterized by a unit timelike vector V at the point x.By parallel transporting this canonical observer along the geodesic one can setup a canonical frame that picks out a particular canonical affine parameter:

x(x, y) = + l; lV = 1; = Vx (x, y). (4)

Note that this affine parameter can, crudely, be thought of as a distance alongthe null geodesic as measured by an observer with fourvelocity V.

To properly place these concepts within the context of the chronology protec-tion conjecture, see, for example, reference [5]. Consider an arbitrary Lorentzianspacetime of nontrivial homotopy. Pick an arbitrary base point x. Since, byassumption, 1(M) is nontrivial there certainly exist closed paths not homo-topic to the identity that begin and end at x. By smoothness arguments therealso exist smooth closed geodesics, not homotopic to the identity, that connectthe point x to itself. However, there is no guarantee that the tangent vector iscontinuous as the geodesic passes through the point x where it is pinned down.If any of these closed geodesics is timelike or null then the battle against time

travel is already lost, the spacetime is diseased, and it should be dropped fromconsideration.

To examine the types of pathology that arise as one gets close to buildinga time machine, it is instructive to construct a one parameter family of closedgeodesics that captures the essential elements of the geometry. Suppose merelythat one can find a well defined throat for ones Lorentzian wormhole. Considerthe world line swept out by a point located in the middle of the wormhole throat.At each point on this world line there exists a closed pinned geodesic threadingthe wormhole and closing back on itself in normal space. This geodesic willbe smooth everywhere except possibly at the place that it is pinned down bythe throat.

If the geodetic interval from x to itself, (x, x), becomes negative thena closed timelike curve (a fortiori a time machine) has formed. It is thisunfortunate happenstance that Hawkings chronology protection conjecture ishoped to prevent. For the purposes of this paper it will be sufficient to considerthe behaviour of the vacuum expectation value of the renormalized stress energytensor in the limit (x, x) 0+.

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2.2 The Hadamard Form

The Hadamard form of the Green function may be derived by an appropriateuse of adiabatic techniques [7, 8, 9, 10, 11, 12]:

G(x, y) < 0|(x)(y)|0 >

=

(x, y)1/2

42

1

(x, y)+ (x, y) ln |(x, y)| + (x, y)

.

(5)

Here the summation,

, runs over all distinct geodesics connecting the point xto the point y. The symbol (x, y) denotes the van Vleck determinant [13, 14].The functions (x, y) and (x, y) are known to be smooth as s(x, y) 0[7, 8, 9, 10, 11, 12].

If the points x and y are such that one is sitting on top of a bifurcation

of geodesics (that is: if the points x and y are almost conjugate) then theHadamard form must be modified by the use of Airy function techniques in amanner similar to that used when encountering a point of inflection while usingsteepest descent methods [8].

3 RENORMALIZED STRESSENERGY

3.1 Point splitting

The basic idea behind point splitting techniques [7, 8, 9, 10, 11, 12] is to defineformally infinite objects in terms of a suitable limiting process such as

< 0|T(x)|0 >= limy

x

< 0|T(x, y)|0 > . (6)

The pointsplit stressenergy tensor T(x, y) is a symmetric tensor at the pointx and a scalar at the point y. The contribution to the pointsplit stressenergytensor associated with a particular quantum field is generically calculable interms of covariant derivatives of the Green function of that quantum field.Schematically

< 0|T(x, y)|0 >= D(x, y){Gren(x, y)}. (7)

Here D(x, y) is a second order differential operator, built out of covariantderivatives at x and y. The covariant derivatives at y must be parallel prop-agated back to the point x so as to ensure that D(x, y) defines a propergeometrical object. This parallel propagation requires the introduction of the

notion of the trivial geodesic from x to y, denoted by 0. (These and subsequentcomments serve to tighten up, justify, and make explicit the otherwise ratherheuristic incantations common in the literature.)

Renormalization of the Green function consists of removing the short dis-tance singularities associated with the flat space Minkowski limit. To this end,

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consider a scalar quantum field (x). Again, let 0 denote the trivial geodesic

from x to y. Define

Gren(x, y) G(x, y) 0(x, y)

1/2

42

1

0(x, y)+ 0(x, y) ln |0(x, y)|

.

(8)

These subtractions correspond to a wavefunction renormalization and a massrenormalization respectively. Note that any such renormalization prescription isalways ambiguous up to further finite renormalizations. The scheme describedabove may profitably be viewed as a modified minimal subtraction scheme. Inparticular, (for a free quantum field in a curved spacetime), this renormalizationprescription is sufficient to render < 0|2(x)|0 > finite:

< 0|2(x)|0 >ren= Gren(x, x),

= 0(x, x)1/2 0(x, x)

42

+

(x, x)1/2

42

1

(x, x)+ (x, x) ln |(x, x)| + (x, x)

.

(9)

For the particular case of a conformally coupled massless scalar field [ 4]:

D(x, y) 1

6

x g

(x, y)y + g(x, y)y

x

1

12g

(x)g(x, y)

x

y

1

12

x

x + g

(x, y)y g(x, y)y

+

1

48g(x)

g(x)x

x + g

(y)yy

R(x) +

1

4g(x)R(x). (10)

As required, this object is a symmetric tensor at x, and is a scalar at y. Thebi-vector g

(x, y) parallel propagates a vector at y to a vector at x, the parallelpropagation being taken along the trivial geodesic 0. (The effects of this parallelpropagation can often be safely ignored, vide reference [4] equation (3), andreference [6] equation (2.40).)

3.2 Singularity structure

To calculate the renormalized stressenergy tensor one merely inserts theHadamard form of the Green function (propagator) into the point split formal-

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ism [3, 4, 6].

< 0|T(x)|0 >=

(x, x)1/2

42limyx

D(x, y)

1

(x, y)

+O((x, x)

3/2).

(11)Observe that

x y

1

(x, y)

=

1

(x, y)2

4xs(x, y) ys(x, y)

x

y(x, y)

.

(12)Similar equations hold for other combinations of derivatives. Let y x, anddefine tx t

1; t

y t

2. Note that g(x)t

xt

x = +1, since the selfconnecting

geodesics are all taken to be spacelike. Then, keeping only the most singularterm

< 0|T(x)|0 >=

(x, x)1/2

42(x, x)2(t(x; ) + s(x; )) + O((x, x)

3/2).

(13)Here the dimensionless tensor t(x; ) is constructed solely out of the metricand the tangent vectors to the geodesic as follows

t(x; ) =2

3

t1t

2 + t

2t

1

1

2g(t

1 t2)

1

3

t1t

1 + t

2t

2

1

2g

.

(14)

The dimensionless tensor s(x; ) is defined by

s(x; ) limyx

D(x, y) {(x, y)} . (15)

In many cases of physical interest the tensor s(x; ) either vanishes identicallyor is subdominant in comparison to t(x; ). A general analysis has so farunfortunately proved elusive. This is an issue of some delicacy that clearlyneeds further clarification. Nevertheless the neglect of s(x; ) in comparisonto t(x; ) appears to be a safe approximation which shall be adopted forthwith.

Note that this most singular contribution to the stress energy tensor is infact traceless there is a good physical reason for this. Once the length of theclosed spacelike geodesic becomes smaller than the Compton wavelength of theparticle under consideration, s =

(x, x)1/2

2s(x, x)4t(x; ) + O(s(x, x)

3). (16)

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A formally similar result was obtained by Frolov in reference [6]. That result

was obtained for points near the Nth polarized hypersurface of a locally staticspacetime. It is important to observe that in the present context it has notproved necessary to introduce any (global or local) static restriction on thespacetime. Neither is it necessary to introduce the notion of a polarized hyper-surface. All that is needed at this stage is the existence of at least one short,nontrivial, closed, spacelike geodesic.

To convince oneself that the apparent s4 divergence of the renormalizedstressenergy is neither a coordinate artifact nor a Lorentz frame artifact con-sider the scalar invariant

T =

< 0|T(x)|0 > < 0|T(x)|0 >. (17)

By noting that

t(x; )t(x; ) = 1

3

3 4(t1 t2) + 2(t1 t2)2

(18)

one sees that there is no accidental zero in t , and that T does in fact divergeas s4. Thus the s4 divergence encountered in the stressenergy tensor associ-ated with the Casimir effect [5] is generic to any multiply connected spacetimecontaining short closed spacelike geodesics.

3.3 Wormhole disruption

To get a feel for how this divergence in the vacuum polarization back reactson the geometry, recall, following Morris and Thorne [15, 16] that a traversablewormhole must be threaded by some exotic stress energy to prevent the throatfrom collapsing. In particular, at the throat itself (working in Schwarzschild

coordinates) the total stressenergy tensor takes the form

T =h

2PR2

0 0 00 0 00 0 00 0 0 1

(19)

On general grounds < 1, while is unconstrained. In particular, to preventcollapse of the wormhole throat, the scalar invariant T must satisfy

T =h

2PR2

1 + 2 + 22

h

2PR2

. (20)

On the other hand, consider the geodesic that starts at a point on the throatand circles round to itself passing through the throat of the wormhole exactly

once. That geodesic, by itself, contributes to the vacuum polarization effects just considered an amount

< 0|T(x)|0 >=(x, x)1/2

2s(x, x)4t(x; ) + O(s(x, x)

3). (21)

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So the contribution of the single pass geodesic to the invariant T is already

T =(x, x)1/2

2s(x, x)4

1

4

3(t1 t2) +

2

3(t1 t2)2

(x, x)1/2

2s(x, x)4. (22)

Therefore, provided that there is no accidental zero in the van Vleck de-terminant, vacuum polarization effects dominate over the wormholes internalstructure once

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x. Then the geodesic from x to x0 is by construction timelike. One defines

t = s(x, x0), and V = x(x, x0). One interprets these definitions asfollows: a geodesic observer at the point x, with four-velocity V, will hit thefountain after a proper time t has elapsed. One now seeks a computationof the stressenergy tensor at x in terms of various quantities that are Taylorseries expanded around the assumed impact point x0 with t as the (hopefully)small parameter.

Consider, initially, the geodetic interval (x, y). Taylor series expand thisas

(x, x) = (x0, x0) + (t V)

x(x, y) + y(x, y)

(x0,x0)

+ O(t2).

(24)Firstly, by definition of the fountain as a closed null geodesic, (x0, x0) = 0.Secondly, take the vector V, defined at the point x, and parallel propagateit along to x0. Then parallel propagate it along the fountain . This nowgives us a canonical choice of affine parameter on the fountain. Naturally thiscanonical affine parameter is not unique, but depends on our original choice ofV at x, or, what amounts to the same thing, depends on our choice of x0 as areference point. In terms of this canonical affine parameter, one sees

x(x, y)(x0,x0)

= n l; (25)

y(x, y)(x0,x0)

= n l. (26)

Here the notation n denotes the lapse of affine parameter on going around thefountain a total of n times in the left direction, while n is the lapse of affineparameter for n trips in the right direction. The fact that these total lapses

are different is a reflection of the fact that the tangent vector to the fountainundergoes a boost on travelling round the fountain. Hawking showed that [1, 2]

n = enhn . (27)

So, dropping explicit exhibition of the ,

(x, x) = +t n

enh 1

+ O(t2). (28)

This, finally, is the precise justification for equation (5) of reference [4].In an analogous manner, one estimates the tangent vectors t1 and t2 in terms

of the tangent vector at x0:

x(x, y)yx = x(x, y)(x0,x0) + O(t) (29)

= nl + O(t). (30)

This leads to the estimate

t1 enh t2 (n/s) l. (31)

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Warning: This estimate should be thought of as an approximation for the domi-

nant componentsof the various vectors involved. If one takes the norm of thesevectors one finds

1 enh 0. (32)

This is true in the sense that other components are larger, but indicates force-fully the potential difficulties in this approach.

One is now ready to tackle the estimation of the structure tensor t(x; ).Using equations (14) and (31) one obtains

t(x; ) 2n

3s2

1 + 4enh + e2nh

ll . (33)

Pulling the various estimates together, the approximation to the Hadamardstressenergy tensor is seen to be

< 0|T(x)|0 >=

(x, x)

1/2

242n

1 + 4enh + e2nh

[enh 1]

3

ll

(t)3+ O(t2).

(34)This, finally, is exactly the estimate obtained by Klinkhammer [4] his equa-tion (8). Furthermore this result is consistent with that of Kim and Thorne [3] their equation (67). The somewhat detailed presentation of this derivationhas served to illustrate several important points.

Primus, the present result is a special case of the more general result (16),the present result being obtained only at the cost of many additional technicalassumptions. The previous analysis has shown that the singularity structure ofthe stressenergy tensor may profitably be analysed without having to restrict

attention to regions near the fountain of a compactly generated chronologyhorizon. The existence of at least one short, closed, nontrivial, spacelike geodesicis a sufficient requirement for the extraction of useful information.

Secundus, the approximation required to go from (16) to (34) are subtle andpotentially misleading. For instance, calculating the scalar invariant T from(34), the leading (t)3 term vanishes (because l is a null vector). The potentialpresence of a subleading (t)5/2 cross term cannot be ruled out from the presentapproximation, (34). Fortunately, we already know [from the original generalanalysis, (16) ] that the dominant behaviour of T is T 2 s4. In view ofthe fact that, under the present restrictive assumptions, t, one sees thatT (t)2. The cross term, whatever it is, must vanish.

Tertius, a warning this derivation serves to expose, in excruciating detail,that the calculations encountered in this problem are sufficiently subtle that two

apparently quite different results may nevertheless be closely related.

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5 DISCUSSION

This paper has investigated the leading divergences in the vacuum expectationvalue of the renormalized stressenergy tensor as the geometry of spacetime ap-proaches time machine formation. Instead of continuing the defense in depthstrategy of the authors previous contribution [5], this paper focuses more pre-cisely on wormhole disruption effects. If one wishes to use a traversable worm-hole to build a time machine, then one must somehow arrange to keep thatwormhole open. However, as the invariant distance around and through thewormhole shrinks to zero the stressenergy at the throat diverges. Vacuumpolarization effects overwhelm the wormholes internal structure once

s(x, x)2 > P, which fact I interpret as supporting Hawk-ings chronology protection conjecture. This result is obtained without invokingthe technical requirement of the existence of a compactly generated chronologyhorizon. If such additional technical assumptions are added, the formalism maybe used to reproduce the known results of Kim and Thorne [3], of Klinkhammer[4], and of Frolov [6].

Moving beyond the immediate focus of this paper, there are still some techni-cal issues of considerable importance left unresolved. Most particularly, compu-tations of the van Vleck determinant in generic traversable wormhole spacetimesis an issue of some interest.

Acknowledgements:I wish to thank Carl Bender and Kip Thorne for useful discussions. I also

wish to thank the Aspen Center for Physics for its hospitality. This research

was supported by the U.S. Department of Energy.

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References

[*] Electronic mail: visser@kiwi.wustl.edu

[1] S. W. Hawking,The Chronology Protection Conjecture,in Proceedings of the 6th Marcel Grossmann Meeting,Kyoto, Japan, June 23-29, 1991,edited by H. Sato, (World Scientific, Singapore, 1992).

[2] S. W. Hawking, Phys. Rev. D46, 603 (1992).

[3] S. W. Kim and K. S. Thorne, Phys. Rev. D43, 3929 (1991).

[4] G. Klinkhammer, Phys. Rev. D46, 3388 (1992).

[5] M. Visser, Phys. Rev. D47, 554 (1993).

[6] V. P. Frolov, Phys. Rev. D43, 3878 (1991).

[7] B. S. De Witt and R. W. Brehme, Ann. Phys. NY 9, 220 (1960).{see esp. pp. 226235}

[8] B. S. De Witt, Dynamical theory of Groups and Fields,in: Relativity, Groups, and Topology [Les Houches 1963]edited by: C. De Witt and B. S. De Witt,(Gordon and Breach, New York, 1964).{see esp. pp. 734745}

[9] B. S. De Witt, Phys. Rep. 19, 295 (1975).{see esp. pp. 342345}

[10] B. S. De Witt, Spacetime approach to quantum field theory,in: Relativity, Groups, and Topology II [Les Houches 1983]edited by: B. S. De Witt and R. Stora,(North Holland, Amsterdam, 1984).{see esp. pp. 531536}

[11] N. D. Birrell and P. C. W. Davies,Quantum fields in curved spacetime,(Cambridge University Press, Cambridge, 1982).

[12] S. A. Fulling,

Aspects of quantum field theory in curved spacetime,(Cambridge University Press, Cambridge, 1989).

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[14] C. Morette, Phys. Rev. 81, 848 (1951).

[15] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).

[16] M. S. Morris, K. S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61, 1446(1988).

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