david hochberg and matt visser- dynamic wormholes, anti-trapped surfaces, and energy conditions

8/3/2019 David Hochberg and Matt Visser- Dynamic wormholes, anti-trapped surfaces, and energy conditions

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a r X i v : g r - q c / 9 8 0 2 0 4 6 v 2 1 8 J u n 1 9 9 8

Dynamic wormholes, anti-trapped surfaces, andenergy conditions

David Hochberg + and Matt Visser ++

+ Laboratorio de Astrofsica Espacial y Fsica FundamentalApartado 50727, 28080 Madrid, Spain

++

Physics Department, Washington UniversitySaint Louis, Missouri 63130-4899, USA

February 7, 2008

Abstract

It is by now apparent that topology is too crude a tool to accuratelycharacterize a generic traversable wormhole. In two earlier papers wedeveloped a complete characterization of generic but static traversablewormholes, and in the present paper extend the discussion to arbitrarytime-dependent (dynamical) wormholes. A local denition of wormholethroat, free from assumptions about asymptotic atness, symmetries, fu-ture and past null innities, embedding diagrams, topology, and eventime-dependence is developed that accurately captures the essence of whata wormhole throat is, and where it is located. Adapting and extending asuggestion due to Page, we dene a wormhole throat to be a marginallyanti-trapped surface, that is, a closed two-dimensional spatial hypersur-face such that one of the two future-directed null geodesic congruencesorthogonal to it is just beginning to diverge. Typically a dynamic worm-hole will possess two such throats, corresponding to the two orthogonalnull geodesic congruences, and these two throats will not coincide, (thoughthey do coalesce into a single throat in the static limit). The divergenceproperty of the null geodesics at the marginally anti-trapped surface gen-eralizes the are-out condition for an arbitrary wormhole. We derivetheorems regarding violations of the null energy condition (NEC) at andnear these throats and nd that, even for wormholes with arbitrary time-dependence, the violation of the NEC is a generic property of wormholethroats. We also discuss wormhole throats in the presence of fully anti-symmetric torsion and nd that the energy condition violations cannot be dumped into the torsion degrees of freedom. Finally by means of a concrete example we demonstrate that even temporary suspension of

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energy-condition violations is incompatible with the are-out property of dynamic throats.

PACS: 04.40.-b; 04.20.Cv; 04.20.Gz; 04.90.+e

e-mail: [email protected]; [email protected]

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1 Introduction

Traversable Lorentzian wormholes [ 1, 2, 3] have often been viewed as intrinsi-cally topological objects, with the topological nature of their spatial sectionsrevealed graphically by means of embedding diagrams and shape functionsas either handles in spacetime (intra-Universe wormholes joining two distantregions of the same Universe) or as bridges (inter-Universe wormholes linkingtwo distinct spacetimes ). Both of these types of wormhole give rise to the no-tion of multiply-connected Universes and spatio-temporal networks possessinga non-trivial topology [ 4]. More often than not, global geometric constraintsare imposed on the wormhole, as well as symmetry properties. For example,the static Morris-Thorne inter-Universe wormhole is an example of this morerestrictive class in that it requires exact spherical symmetry and the existenceof two asymptotically at regions in spacetime [ 1]. As we have previously ar-gued [5, 6] there are many other spacetime congurations and geometries thatone might still quite reasonably want to classify as wormholes, that either donot possess any asymptotically at regions, or have trivial topology, or exhibitboth these features. An example of the former is provided by the Hochberg-Popov-Sushkov self-consistent semi-classical wormhole (which is a wormhole of the inter-Universe type joining up two spacetimes with no asymptotically atspatial regions) [ 7]. Examples of the topologically trivial wormholes [ 3] are pro-vided by a closed Friedmann-Robertson-Walker (FRW) spacetime joined to anordinary Minkowski spacetime by a narrow neck or two closed FRW spacetimes joined by a bridge [5, 6].

The only difference between these two classes of wormholes ( i.e. bridges andhandles versus topologically trivial) arises at the level of global geometry andglobal topology. This suggests that it is important to identify a fundamental lo-cal property that can be used to characterize what one means by a wormhole, anintrinsic property to be abstracted from the broad phylum of wormholes whichcan then be used to unambiguously dene what is meant by a wormhole. Indeed,the local physics, that which is operative near the throat of the wormhole,is insensitive to global properties and indicates that a local denition of whatis meant by a wormhole throat is called for. This denition should be basedsolely on local properties and be free from technical assumptions about asymp-totic atness, future and past null innities, global hyperbolicity, symmetries,embeddings and topology.

In two previous papers [ 5, 6] we have performed such an analysis for statictraversable wormholes. In this paper, we lift the static restriction and shallinvestigate the generic (not necessarily static) traversable wormhole. We make

no assumptions about symmetries, spherical or otherwise, nor do we assumethe existence of asymptotically at regions. To proceed, we rst have to de-ne exactly what we mean by a wormhole and we nd, just as in the treat-ment of the generic static case [5, 6], that there is a natural local geometric(not topological) characterization of the existence and location of a wormhole

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throat. This characterization is developed in the language of the expansion of null geodesic congruences propagating outward from, and orthogonal to, closedtwo-dimensional spatial hypersurfaces (denoted ). The congruence is subjectto a are-out condition that suitably generalizes that of the Morris-Thorneanalysis. But, unlike that earlier denition [ 1, 2], ours makes no reference toembeddings or shape-functions. In this language, the spatial hypersurface inquestion will be a wormhole throat provided the expansion of one of the twoorthogonal null congruences vanishes on that surface: + = 0 and/or = 0,and if the rate-of-change of the expansion along the same null direction ( u )is positive-semi-denite at the surface: d /du 0. This latter constraint isprecisely the are-out condition generalized to an arbitrary wormhole. Thesetwo conditions on the expansion dene the throat to be a minimal hypersur-face, i.e. , an extremal surface of minimal area (with respect to deformations inthe appropriate u null direction). Thus, a wormhole throat is a marginally anti-trapped surface . Historically, Page [ 8] was the rst to suggest that undersuitable circumstances a wormhole throat could be viewed as an anti-trappedsurface in spacetime, and we shall soon see that this denition promises to bethe most efficient and most physical framework for generalizing the concept of throat to the fully arbitrary and dynamic case.

While this denition captures the intuitive concept of throat admirably,there can be cases calling for slight denitional renements, for example, whend /du > 0 is strictly positive on the throat, in which case we are dealing witha strongly anti-trapped surface , as well as other cases for which weaker, averagednotions of are-out will suffice.

In general, the vanishing of the independent expansions + = 0 and = 0will take place on two distinct hypersurfaces. Thus, (dynamical) wormholes

generally possess two throats provided each hypersurface is individually ared-out: d+ /du + 0 on u + , and d /du 0 on u . Of course, the twothroats must (and they do) coincide in the static limit.With these denitions in place, we move on to develop a number of theo-

rems about the existence of matter at or near the throat(s) violating the nullenergy condition (NEC). These theorems make repeated use of the Raychaud-huri equation for the expansions . These results are local and pointwise, indistinction to energy conditions obtained by averaging over inextendible nullgeodesics, which are global in nature. These energy theorems generalize theoriginal MorrisThorne result by demonstrating unequivocally that the NEC isgenerically violated at some points on or near the two-dimensional hypersur-face comprising the throat(s). This is an important result since these theoremshold for an arbitrary dynamic or static wormhole irrespective of symmetries or

other global concerns and demonstrate that the energy condition violations aretruly generic. Our results are (of course) also completely in accord with thetopological censorship theorem of Friedman, Schleich, and Witt [ 9].

The striking nature of the violations of the null energy condition rst dis-covered for the Morris-Thorne wormhole [ 1, 2, 3], has led numerous authors to

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try and nd ways of evading or minimizing these violations. Most of these at-tempts focus on alternative gravity theories, be they Brans-Dicke, dilaton grav-ity, higher-derivative theories, etc. What all these extensions of Einstein gravityaccomplish from a practical point of view is to provide one with additionaldegrees of freedom (beyond the metric), which under certain circumstances canbe coerced into absorbing the energy condition violations (leaving the remainingordinary-matter sector free to satisfy the classical energy conditions). Neverthe-less, the total effective stress energy tensor will violate the null energy conditionat or near the throat, so sweeping the unavoidable energy condition violationsinto a specic sector does not make the problem go away. This important butoft overlooked point has been treated in some detail in [ 6]. (We would be re-miss in not warning the reader that a sizable fraction of the published papersclaiming to build wormholes without violating the energy conditions suffer fromsevere technical problems, and are often internally inconsistent.)

Similar comments apply of course to gravity plus torsion, although theorieswith torsion are distinguished from other variants of gravity by the fact thatnon-zero torsion gives rise to a non-trivial contribution to the Raychaudhuriequation which cannot be absorbed into an effective total stress energy tensor.Moreover, torsion appears naturally (and unavoidably) in theories of gravitybased on low-energy closed string theories. These facts make it of interestto treat the torsion case separately and in some detail to assess the ability of torsion to defocus (null) geodesics and to check the status of the NEC for throatsin the presence of torsion. We nd that totally antisymmetric torsion actuallypromotes the energy condition violation at the throat (but helps to lessen it awayfrom the throat by generating twist). Other attempts to get around the energy-condition violations have led to considerations of time-dependent wormholes.

In this domain, it is indeed possible to temporarily suspend the violations, butonly at the heavy expense of totally destroying the are-out properties of thethroat.

Since the Raychaudhuri equation with torsion is not standard textbook fare,we include a brief resume of torsion in Section II to establish the notationused in the rest of the paper and provide a simple derivation of the generalizedRaychaudhuri and the companion twist equations corresponding to the twoindependent null congruences in Section III. We then dene wormhole throatsin terms of the expansions in Section IV and prove the coalescence of the twothroats in the static limit. Armed with these denitions, we go on to derive theenergy condition theorems for wormholes in normal spacetime as well as in thepresence of torsion in Section V. Worked examples of dynamic wormholes areprovided in Section VI where, among other things, we show how the temporal

suspension of energy-condition violations eradicates the throat. Conclusionsand a discussion of our results are collected in Section VII.

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2 Geometric preliminaries: spacetimes with

torsion

In preparation for the derivation of the Raychaudhuri equation governing theexpansion in the presence of torsion, and to establish the notation to be usedthroughout, we collect here a few basic denitions and identities which will proveuseful later on. (Basic denitions regarding torsion can be gleaned from [ 10, 11,12], while an overview of torsion in the string theory context can be extractedfrom [13].)

As we are interested in keeping our discussion as general as possible, weendeavour to work in a coordinate-free language and to this end, shall make useof the abstract index notation, later specializing when and if needed, to explicitcoordinate systems. For the time being then, the use of lower case latin lettersdesignates abstract indices: ( a,b,c,... ) and run from 0 to 3. (See Wald [ 14] fora discussion of the subtleties associated with the use of abstract indices.) Letva be a covariant vector, its covariant derivative is

a vb = a vb C cab vc , (1)where C bac denotes the connection of the underlying four-dimensional spacetime.In principle, the connection can be any tensor eld guaranteeing that thecovariant derivative ( 1) based upon it satises all the usual properties (linear,Leibnitz, etc.) [ 14]. However, we will not impose the torsion-free condition,which means that the (total) connection can be decomposed as

C cab = cab + H

cab , (2)

where C c(ab ) =12 (C

cab + C cba ) = cab is the ordinary symmetric Christoffel connec-

tion, depending on the metric in the usual way, while C c[ab ] =12 (C

cab C cba ) = H cabdenes the torsion, which is manifestly anti-symmetric in its two lower indices.

Due to the mixed-symmetry of the connection, the commutator of the co-variant derivative, which is used to dene the curvature tensor, works out tobe

[a ,b]vc = ( 2 [a C db]c + 2 C e[a | c C db]e )vd 2C e[ab ]e vc= R ab,c d (C )vd 2H eabe vc , (3)

whereR ab,c

d

(C ) = 2 [a C d

b]c + 2 C e

[a | c C d

b]e , (4)is the associated curvature tensor. The vertical bar within the antisymmetriza-tion brackets indicates that one is to antisymmetrize over the pair a and b,but not c. We have distinguished the curvature with an overbar in order to

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emphasize that this tensor is not the ordinary Riemann tensor, unless the tor-sion vanishes identically. It is however the curvature associated with a generalconnection C . We note that the derivative of a vector couples directly to thetorsion, as evidenced by the second term in the above identity ( 3). The tor-sion also shows up explicitly (and implicitly in the covariant derivatives) in thecommutator of two vector elds:

[v, w]b = vaa wb waa vb 2va wc H bac . (5)

Although ( 4) is not the standard Riemann tensor, it is related to it as follows:

R ab,c d (C ) = R ab,c d () ( a H dbc bH dac ) + 2 H e[a | c H db]e , (6)where the covariant derivatives of the torsion are calculated with respect to thesymmetric (Christoffel) part of the connection only; that is,

a H

dbc = a H

dbc eab H dec eac H dbe + dae H ebc . (7)

This identity ( 6) suggests that the torsion can be regarded as a dynamic eldpropagating over a normal Riemannian spacetime, i.e. , may either be regardedas fundamentally geometric, as part and parcel of the connection ( 2), or as amatter tensor eld in a spacetime with a conventional symmetric connection.We can make this latter association more precise by writing the action fromwhich we will infer the corresponding equations of motion. We form the equiv-alent of the Einstein-Hilbert action for the generalized curvature and allow forthe presence of ordinary matter (every other dynamical eld imaginable exceptfor the metric and torsion):

S = 1

16 d4x g R(C ) + d

4x gLmatter , (8)where the generalized scalar curvature is R (C ) = gac R ab,c b(C ) and is related tothe scalar of Riemannian curvature via

R(C ) = R() gbc bH aac H abc H abc , (9)which follows from (6) and using the covariant constancy of the metric a gbc =0, with respect to . (Mathematically, it is possible to consider even moregeneral affine connections for which the covariant derivative of the metric is notzero. The most general such affine connection is then a linear combination of theChristoffel connection, the torsion tensor, and a non-metricality tensor. Wewill not generalise our analysis to this level of abstraction as little seems to be

gained, and there are good physics reasons for keeping the covariant derivativeof the metric zero.)Thus far, we have kept the treatment of the torsion part of the connec-

tion completely general. If we now identify the torsion with the totally anti-symmetric rank-three eld strength H = dA , where A is a two-form potential,

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or in terms of components

H abc = a Abc + bAca + c Aab , (10)

then we have an explicit realization of torsion that is known to arise naturallyin closed string-theoretic low energy gravity [ 13, 15, 16]. In this particularincarnation as an antisymmetric rank-three tensor, the torsion is also known asthe Kalb-Ramond eld. From here on, when we refer to torsion, it will be of this form.

The equations of motion now follow immediately upon varying the full action(8) with respect to the metric, torsion, and what ever other matter elds maybe present. The equation of motion for the metric is given by

Gab () = R ab

12

gab R = 8 T ab + 3 H ade H deb

12

gab H cde H cde , (11)

where T ab is the complete stress-energy tensor for the matter elds. We see thatalthough H originates from the connection, it can also be treated as simplyan additional species of matter and can therefore be shifted into an effectivematter stress tensor. However, it is of more than academic interest not to doso at this stage. When we come to consider the expansion and twist of (null)geodesic congruences in spacetimes with torsion, we will nd that the torsionmakes explicit non-dynamic contributions to the differential equations for theexpansion and twist that cannot be re-dened away, as it were, by invoking theequations of motion, or by redening the total effective stress energy tensor.Thus, it will be of interest to see what inuence the torsion may have to focusand defocus bundles of null geodesics. The equation of motion for the torsionthat follows from varying ( 8) is simply that

a H

abc = 0 . (12)

Using the metric equation ( 11), it follows that the Ricci tensor obeys the equa-tion

R ab = 8 [T ab 12

gab T ] + 3H ade H deb gab (H ade H ade ), (13)while the scalar curvature is

R = 8T H ade H ade . (14)

3 Null geodesic congruences

We start by considering a compact two-dimensional hypersurface that is bothorientable and embedded into spacetime in a two-sided manner in such a waythat the induced two-metric is spacelike. To discuss the null geodesic congru-ences orthogonal to this surface, we shall, following the description of Carter [ 17]

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begin by introducing a future-directed outgoing null vector la+ , a future-directed ingoing null vector la and a spatial orthogonal projection tensor ab satisfying the following relations:

la+ l+ a = la l a = 0

la+ l a = la l+ a = 1

la ab = 0

ac cd = ad . (15)

In terms of these null vectors and projector, we can decompose the full spacetimemetric (indeed, any tensor) uniquely:

gab = ab l a l+ b l+ a l b. (16)Physically, this decomposition leads to a parameterization of spacetime points interms of two spatial coordinates (typically denoted x) plus two null coordinates[u , or sometimes ( u, v )]. (We do not want to prejudice matters by taking thewords outgoing and ingoing too literally, since outside and inside do notnecessarily make much sense in situations of nontrivial topology. The criticalissue is that the spacelike hypersurface must have two sides and + and are just two convenient labels for the two null directions.)

We consider the tensor elds dened by the covariant derivative of the future-directed null vectors (there is one such tensor eld for each null congruence)

B ab bl a , (17)and ask for their rate of change along the corresponding null geodesic parame-terized with affine parameter u :

dB abdu l

c c B

ab = l

c cbl a

= lc bc l a + lc [c ,b]l a

= blc c l a + lc [c ,b]l a= B

cb B

ac + R cb,a

d (C )l d lc 2lc H cb B ae . (18)This uses the fact that the parallel transport of a tangent vector along its corre-sponding geodesic vanishes: lcc l b = 0 (see technical comment below dealingwith non-affine parameterizations), plus the commutator identity in ( 3).

In contrast to the case of timelike geodesics, the tensor eld B ab is not purelyspacelike but has in addition, mixed null-spacelike components:

a l+ b = ca

db c l+ d l+ b da lc d l+ c

= v+ab l+ b da lc d l+ c , (19)

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and

a l b = ca

db c l d l b da lc+ d l c

= vab l b da lc+ d l c . (20)which dene the purely spatial tensors vab =

ca dbc l d , which admit the fur-ther decomposition as follows ( ab ba = 2):

vab =12

ab + ab + ab (21)

= ab vab = gab

a l b (22)

ab = v(ab )

12

ab (23)

ab = v[ab ], (24)

where is the trace of vab and provides the measure of the instantaneousexpansion of the cross-sectional area of a bundle of null geodesics, while ab andab denote the shear and twist, respectively, and are also purely spatial tensors.

From these relations one may derive rate-of-change equations for the expan-sion, shear and twist with respect to the corresponding affine parameters ustarting from ( 18), though we shall be primarily interested in the rate of changeof the expansion as this equation will play a fundamental role later on whenwe come to dene a generic wormhole throat. So, taking the trace of ( 18) yieldsa generalized version of the Raychaudhuri equation (generalized as it containsthe effects of torsion) for the two expansions (one for the (+)-congruence, theother for the ( )-congruence):

ddu = 12 2 ab ab + ab ab Rdc () lc l d

2H dcb B bd l

c + H eac H

ead lc l d . (25)

With a view to applications for deriving the energy conditions associated withgeneric wormhole throats, it is useful to have at hand the companion equationgoverning the rate of change of the twist along null geodesics. This is derivedby going back to ( 18), antisymmetrizing on the free indices and projecting outthe purely spatial part of the resulting equation. These two operations yielda generalization of the twist equation [again, one for the (+)congruence, theother for the ( )congruence]:

dba

du=

ba

2

c[a

b]c +

c H dab lc l d + H

ec[a H

db]e l

c l d

2lc H ec [bB

a ]e . (26)

The term linear in H that appears in both the expansion and twist equationsis purely geometrical in origin, arising as it does, from the commutator of two

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torsion-bearing covariant derivatives ( 3). The other torsion contributions aredynamic in origin, as these arise instead directly from the action and equationsof motion. These features distinguish the torsion from all other elds. Of course,in the absence of torsion, these reduce to the standard Raychaudhuri and twistequations, for and , respectively [ 18, 14].

Technical aside: if one is working with a non-affine parameterization forthe null congruences, then the parallel transport equation becomes lcc l b =

K l b where K = la lbbl a . The expansion is still given by the trace of the spatial part of a l b and we have that = ab vab = gaba l b K . TheRaychaudhuri equation ( 25) will then pick up an extra factor of K [17].

4 Denition of generic wormhole throats

Our aim is to provide a precise, local, and robust geometric denition of a(traversable) wormhole throat, equally valid for static as well as time-dependentwormholes. As a guide, we recall that in the generic but static case, the throatwas dened as a two-dimensional hypersurface of minimal area [ 5, 6]. The timeindependence allows one to locate that minimal hypersurface entirely withinone of the constant-time three-dimensional spatial slices, and the conditionsof extremality and minimality can be applied and enforced within that singletime-slice. For a static throat, variational principles involve performing arbi-trary time-independent surface deformations of the hypersurface in the remain-ing spatial direction orthogonal to the hypersurface, which can always be takento be locally Gaussian. By contrast, in the time-dependent case, it may notbe possible to locate the entire throat within one time slice, as the dynamic

throat is an extended object in spacetime, and the variational principle mustbe carried out employing surface deformations in the two independent null di-rections orthogonal to the hypersurface: say, u+ and u . This, by the way,suggests why it is that the embedding of the spatial part of a wormhole space-time in an Euclidean R 3 is no longer a reliable operational technique for deningare-out in the time-dependent case. Of course, in the static limit these twovariations will no longer be independent and arbitrary deformations in the twonull directions reduce to a single variation in the constant-time spatial direc-tion (see below). Realizing that the time-dependent wormhole typically has twonon-coincident throats was perhaps the major conceptual stumbling block toovercome in developing this formalism.

4.1 PreliminariesIn the following, we set up and dene the properties of throats in terms of thenull congruences. Bear in mind that a throat will be characterized in termsof the behavior of a single set of null geodesics orthogonal to it. We dene

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a wormhole throat u + (there is also one for the other null congruence) tobe a closed 2-dimensional hypersurface of minimal area taken in one of theconstant- u+ slices, where u+ is an affine parameter suitable for parameterizingthe future-directed null geodesics l+ orthogonal to u + . All this means is thatwe imagine starting off a collection of light pulses along the hypersurface andwe can always arrange the affine parameterizations of each pulse to be equal tosome constant on the hypersurface; we take this constant to be zero. We wishto emphasize that there is a corresponding denition for the other throat u .In the following, we dene and develop the conditions that both hypersurfacesmust satisfy individually to be considered as throats, and shall do so in a uniedway by treating them together by employing the -label. Our next task is tocompute the hypersurface areas and impose the conditions of extremality andminimality directly and to express these constraints in terms of the expansionof the null geodesics. The area of u is given by

A( u ) = u d2 x. (27)An arbitrary variation of the surface with respect to deformations in the null

direction parameterized by u is

A( u ) = u d du u (x) d2 x.= u 12 ab d abdu u (x) d2 x. (28)

If this is to vanish for arbitrary variations u (x), then we must have that

12

abd abdu

= 0 , (29)

which expresses the fact that the hypersurface u is extremal.This condition of hypersurface extremality can also be phrased equivalently

and directly in terms of the expansion of the null congruences. The simplestway to do so is to consider the Lie derivative Ll acting on the full spacetimemetric:

Ll gab = lcc gab + gcba lc + gacblc= a l b + bl a= B ba + B ab = 2 B

(ab

), (30)

with the second equality holding provided the metric is covariantly constantwith respect to the full covariant derivative, which is in fact the case, even inthe presence of arbitrary torsion. We now use the decomposition ( 16) of the

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spacetime metric and work out the Lie derivative using the Leibnitz rule:

B (ab ) =12L

l gab

=12L

l ( ab l a l+ b l+ a l b),

=12L

l ab

12

(l a Ll l+ b + l+ bLl l a + ( a b)), (31)from which, and using the properties in ( 15), implies

= gab B (ab ) = ab B (ab ) = ab vab

=12

ab Ll ab= 12

ab d abdu. (32)

So the condition that the area of the hypersurface be extremal is simply thatthe expansion of the null geodesics vanish at the surface: = 0. To ensurethat the area be minimal , we need to impose an additional constraint and shallrequire that 2A( u ) 0. By explicit computation,

2 A( u ) = u 2 + ddu u (x) u (x)d2 x= u ddu u (x) u (x) d2 x 0, (33)

where we have used the extremality condition ( = 0) in arriving at this lastinequality. For this to hold at the throat for arbitrary variations u (x) , andsince (u (x))2 0, we must have

ddu 0, (34)

in other words, the expansion of the cross-sectional area of the future-directednull geodesics must be locally increasing at the throat. This is the precisegeneralization of the Morris-Thorne are-out condition to arbitrary wormholethroats. This makes eminent good sense since the expansion is the measure of the cross-sectional area of bundles of null geodesics, and a positive derivativeindicates that this area is locally increasing or aring-out as one moves along

the null direction. Note that this denition is free from notions of embeddingand shape-functions. So in general, we have to deal with two throats: u +such that + = 0 and d+ /du + 0 and u such that = 0 and d /du 0.We shall soon see that for static wormholes the two throats coalesce and thisdenition automatically reduces to the static case considered in [ 5, 6]. The

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logical development in the present paper closely parallels that of the static casethough there are many technical differences.

The conditions that a wormhole throat be both extremal and minimal arethe simplest requirements that one would want a putative throat to satisfy andwhich may be summarized in the following denition (in the following, the hy-persurfaces are understood to be closed and spatial). Since these denitions holdof course for both throats, we momentarily drop the distinction and suppressthe label.

4.1.1 Denition: Simple are-out condition

A two-surface satises the simple are-out condition if and only if it is ex-tremal, = 0 , and also satises d/du 0. The characterization of a genericwormhole throat in terms of the expansion of the null geodesics shows that anytwo-surface satisfying the simple are-out condition is a marginally anti-trapped surface , where the notion of trapped surfaces is a familiar concept that arisesprimarily in the context of singularity theorems, gravitational collapse and blackhole physics [14, 18]. We hasten to point out however, that in the present con-text, identifying a wormhole throat as a marginally anti-trapped surface in noway, shape or form is meant to convey that we are dealing with horizons, ap-parent horizons, or singularities. Nor should this nomenclature suggest thatwormholes are somehow allied with or are analogous to black holes or whiteholes. (For some special cases where wormholes do have applications in blackhole physics see [6]).

Generically, we would expect the inequality 2A( u ) > 0 to be strict, sothat the surface is truly a minimal (not just extremal) surface. This will pertain

provided the inequality d/du > 0 is a strict one for at least some points on thethroat. This suggests the following denition.

4.1.2 Denition: Strong are-out condition

A two-surface satises the strong are-out condition at the point x if and only if it is extremal, = 0 , satises ddu 0 everywhere on the surface and if at thepoint x, the inequality is strict:

ddu

> 0. (35)

If the latter strict inequality holds for all x u in the surface, then thewormhole throat is seen to correspond to a strongly anti-trapped surface . Again,

this terminology is not intended to convey any relation between wormholes andblack holes. The physical distinction between simple and strong are-out willbecome evident when we come to explore the consequences these denitions haveon the energy conditions required to maintain a generic traversable wormholethroat.

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It is sometimes sufficient and convenient to work with a weaker, integratedform of the are-out condition.

4.1.3 Denition: Averaged are-out condition

A two-surface satises the averaged are-out condition if and only if it isextremal, = 0 , and

u sgn ddu d2x > 0, (36)where sgn( x) is the sign of x. This averaged are-out condition places a con-straint on the putative throat by asking that the extremal surface be outward-aring over at least half its area before one can be justied in calling it awormhole throat. This denition has been carefully constructed to remaininvariant under arbitrary affine reparameterizations of the null geodesic con-gruence. An apparently plausible alternative to the above, using the integral

I u (d/du )d2 x, is decient in that if the integrand d/du changes

sign anywhere on the surface then by appropriate affine reparameterizationsof the null geodesic congruence the integral may be made arbitrarily positiveor arbitrarily negative [ 19]. (Thus if one were to require the integral I to bepositive for all affine parameterizations, one would simply recover the strongare-out condition, while if we were to merely require that the integral I bepositive for at least one choice of affine parameterization we would have theextremely weak constraint that d/du be positive for at least one point on thesurface . Either option though mathematically consistent is physically unrea-sonable, and the denition in terms of the sgn function is the best intermediatestrength denition we have found. This comment also implies that constraintson weighted averages of the form u f (x) (d/du )d

2 x are too subject toreparameterization effects to be useful.)

The conditions under which the average are-out are appropriate arise forexample for situations with multiple throat wormholes. Indeed, suppose wehave a double throat wormhole where each of the two throats are ared-out inthe strong sense. Then the spacetime between the throats contains an extremalhypersurface which is not minimal, but which can be minimal in the integrated,averaged sense. (See, e.g. , [5, 6]). Independently from this, averaged are-out conditions of various types crop up in energy conditions averaged over thehypersurface [ 5, 6].

Finally, it is also useful to dene a weighted are-out condition.

4.1.4 Denition: Averaged f -weighted are-out condition

A two-surface satises the f -weighted are-out condition if and only if it is

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extremal, = 0 , and

u f (x)sgn ddu d2 x > 0, (37)where f is a positive denite function dened on the two-surface.

Note that the strong are-out condition implies both the simple are-outcondition and the averaged are-out condition, but the simple are-out condi-tion does not necessarily imply the averaged are-out condition (the integralmight vanish). However, we see that if the averaged f -weighted are-out con-dition is satised for all positive denite f , then it implies the simple are-outcondition, which follows from identifying f (x) = u(x)2 0 and using theminimality constraint ( 33).

4.1.5 Technical aside: degenerate throatsA class of wormholes for which we have to extend these denitions arises whenthe wormhole throat possesses an accidental degeneracy in the expansion of thenull geodesics at the throat. The above discussion has been tacitly assumingthat in the vicinity of the throat we can Taylor expand the expansion

(x, u ) = (x, 0) + ud(x, u )

du u =0+ O(u2 ), (38)

with the constant term vanishing by the extremality constraint and the rstderivative term being constrained by the are-out conditions.

Now if the extremal two-surface has an accidental degeneracy with the rstderivative term (and possibly higher-order terms) vanishing identically, then we

would have to develop the above expansion further out to the rst non-vanishingterm. This would mean we would have to re-phrase the are-out in terms of these higher-order derivatives of the null geodesic expansion. In fact, the rstnon-vanishing term would appear at odd order in u:

(x, u ) =u2N 1

(2N )!d2N 1(x, u )

du2N 1 u =0+ O(u2N ), (39)

since the surface is by denition extremal. It must be odd in u otherwise thethroat would be a point of inection and not a true minimum of the area.Simply put, even-order surface deformations involve odd-order derivatives of the expansion. We can see this in another way by computing higher-ordervariations in the area. The condition that it be a minimum is

2N

A( u ) = u d2N 1du2N 1 (u(x))

2N

d2

x > 0, (40)

which leads to the are-out condition being stated in terms of the (2 N 1)-thderivative of the expansion. Note: for N = 1, this reduces to the minimalityconstraint in ( 33). This motivates the following denition.

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4.1.6 Denition: N -fold degenerate are-out condition

A two-surface satises the N -fold degenerate are out condition if and only if it is extremal, = 0 , the rst (2N 2) u-derivatives of vanish, (d2N 1(x, u )/du 2N 1) 0 everywhere on the surface and if nally, for at least some point x on the surface, the inequality is strict:

d2N 1du2N 1

> 0. (41)

Physically, at an N -fold degenerate point, the wormhole throat is seen to beextremal up to order 2 N 1 with respect to the derivatives of the expansion, i.e. ,the are-out condition is delayed in the (outgoing) null direction with respectto throats in which the are-out occurs at N = 1, which (by the way we haveset up the denition) corresponds to the strong are-out condition.

These considerations bring us to the following surprising result already al-luded to above: namely, there is no a priori reason for the two independent nullvariations u+ , u to single out the same minimal hypersurface. That is, ingeneral

u + = u , (42)

and we must conclude that generic time-dependent wormholes possess two throats.If these hypersurfaces are in causal contact then it will be possible to enter thewormhole via one throat and exit through the other. If the two throats are notin causal contact then the wormhole is not two-way traversable, and you have atbest two one-way traversable wormholes with no way of getting back to whereyou started from.

4.2 Static limitIn a static spacetime, a wormhole throat is a closed two-dimensional spatialhypersurface of minimal area that, without loss of generality, can be locatedentirely within a single constant-time spatial slice [ 5, 6]. Now, for any staticspacetime, one can always decompose the spacetime metric in a block-diagonalform as

gab = V a V b + (3) gab , (43)where V a = exp[ ]( t )

a is a timelike vector eld orthogonal to the constant-time spatial slices and is some function of the spatial coordinates only. In thevicinity of the throat we can always set up a system of Gaussian coordinates nso that

(3) gab = n a n b + ab , (44)

where n a = ( n )a , n a n a = +1, and ab is the two-metric of the hypersurface.Putting these facts together implies that in the vicinity of any static throat wemay write the spacetime metric as

gab = V a V b + n a nb + ab . (45)

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Table 1: Summary of the are out conditions for wormhole throats; all quantitiesare evaluated on the throat. The are-out conditions are understood to apply

to both throats, and we drop the label.

Flare-out condition Expansion Constraints on the throat

simple = 0 ddu 0

strong = 0 ddu 0, and x u ddu > 0

strongly anti-trapped = 0 x u ,ddu > 0

averaged = 0 u sgnddu d

2 x > 0

f -averaged = 0 u f (x)sgnddu d

2x > 0, for an f (x) 0

N -fold degenerate = 0 dm

du m = 0, for m = 1 , , 2N 2 and d2 N 1

du 2 N 1 0

But ( 16) holds in general, so comparing both metric representations yields theidentity

la lb+ la+ lb = V a V b + n a nb , (46)and the following (linear) transformation relates the two metric decompositionsand preserves the inner-product relations in ( 15):

la =12

(V a + n a ), la+ =12

(V a n a ). (47)Since the throat is static, ab is time-independent, hence when we come to vary

the area ( 27) with respect to arbitrary perturbations in the two independentnull directions we nd that

abu +

u+ =12

exp[] ab

tt +

abn

n =12

abn

n,

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abu

u

=1

2exp[]

abt

t

abn

n =

1

2

abn

n. (48)

Thus the variations are no longer independent, and reduce to taking a singlesurface variation in the spatial Gaussian direction. So, + = 0 = 0at the same hypersurface, proving that u + = u in the static limit, and sostatic wormholes have only one throat. An exhaustive analysis of the geometricstructure of the generic static traversable wormhole may be found in [ 5, 6].

5 Constraints on the stress-energy

With the denition of wormhole throat made precise we now turn to deriveconstraints that the stress energy tensor must obey on (or near) any wormholethroat. The constraints follow from combining the Raychaudhuri equation ( 25)with the are-out conditions, and using the Einstein equation ( 11). It is clearthat these constraints apply with equal validity at both the + and throats,and in the following we cover both classes simultaneously and without risk of confusion by dropping the -labels. We rst treat the zero-torsion case.

5.1 Zero torsion

Since all throats are extremal hypersurfaces ( = 0) the Raychaudhuri equationat the throat ( 25) reduces to

d

du+ ab ab =

8T ab la lb, (49)

where we have used the Einstein equation ( 11) after setting the torsion terms tozero and the fact that the null geodesic congruences are hypersurface orthogonal,so that the twist ab = 0 vanishes identically on the throat. We make no claimregarding the shear, except to point out that since ab is purely spatial, itssquare ab ab 0 is positive semi-denite everywhere (not just on the throat).Consider a marginally anti-trapped surface, i.e. , a throat satisfying the simpleare-out condition. Then the stress energy tensor on the throat must satisfy

T ab la lb 0. (50)The NEC is therefore either violated, or on the verge of being violated ( T ab la lb 0), on the throat. Of course, whichever one of the two null geodesic congruences(l+ or l ) you are using to dene the wormhole throat (anti-trapped surface),you must use the same null geodesic congruence for deducing null energy con-dition violations.

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For throats satisfying the strong are-out condition, we have instead thestronger statement that for all points on the throat,

T ab la lb 0, and x u suchthat T ab la lb < 0, (51)so that the NEC is indeed violated for at least some points lying on the throat.By continuity, if T ab la lb < 0 at x, then it is strictly negative within a nite openneighborhood of x: B(x). For throats that are strongly anti-trapped surfaces,we derive the most stringent constraint stating that

T ab la lb < 0 x u , (52)

so that the NEC is violated everywhere on the throat.Weaker, integrated energy conditions are obtained for throats satisfying the

averaged are-out conditions. For a throat that is ared-out on the average,integrating the Raychaudhuri equation ( 49) over the throat implies

u sgn(T ab la lb) d2 x < 0, (53)indicating that the NEC, when averaged over the throat, is strictly violated(Warning: this has nothing to do with the violation of the averaged null energycondition, or ANEC. In the ANEC, the averaging is dened to take place alonginextendible null geodesics. See in particular [ 9].) By the same token, throatssatisfying the f -weighted averaged are-out condition imply that

u f (x)sgn( T ab la lb) d2x < 0, (54)indicating that the sign of the NEC, when weighted with the positive denitefunction f (x) is strictly violated on the average over the throat.

What can we say about the energy conditions in the region surroundingthe throat? This requires knowledge of the expansion, shear and twist in theneighborhood of the throat. Luckily, we can dispense with the twist immedi-ately. Indeed, the (torsion-free) twist equation ( 26) is a simple, rst-order lineardifferential equation:

dbadu

= ba 2c[a b]c , (55)whose exact solution (if somewhat formal in appearance) is

ab (u) = exp

u

0

(s)ds U a c (u) U bd (u)cd (0), (56)where the quantity U (u) denotes the path-ordered exponential

U a c (u) = P exp u

0 ds a c . (57)

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So, an initially hypersurface orthogonal congruence remains twist-free every-where, both on and off the throat: ba (0) = 0

ba (u) = 0. Then theequation

ddu

+12

2 + ab ab = 8T ab la lb, (58)is seen to be valid for all u. Coming back to simply-ared throats, we havetwo pieces of information regarding the expansion: namely that (0) = 0 and(d(u)/du )u =0 0, so that if we expand in a neighborhood of the throat asin (38), then we have that

d(u)du

=d(u)

du u =0+ O(u), (59)

so over each point x on the throat, there exists a nite range in affine parameter

u (0, u

x ) for whichd (u )

du 0. Since both 2

and ab

ab are positive semi-denite, we conclude that the stress-energy is either violating, or on the verge of violating, the NEC along the partial null curve {x} (0, ux ) based at x. If thethroat is of the strongly-ared variety, then we see that the NEC is denitelyviolated at least over some nite regions surrounding the throat: x {x} (0, ux ), and including the base points x. For strongly anti-trapped surfaces, theNEC is violated everywhere in a nite region surrounding the entire throat, andincluding the throat itself.

Finally, if the throat is N -fold degenerate (and N > 1), then there existpoints x on the throat for which ( d2N 1(x, u )/du 2N 1)|u =0 > 0. This impliesthat the rst derivative

d(x, u )du

=(2N 1)u2N 2

(2N )!d2N 1 (x, u )

du2N 1u =0

+ O(u2N 1), (60)

is positive along a partial null curve {x} (0, ux ) based at x and it follows by(49) that the NEC is violated along the nite bristles x {x} (0, ux ).

5.2 Non-zero H-torsion

Torsion, although contributing additional terms to the Einstein ( 11) and Ray-chaudhuri equations ( 25) does not necessarily alleviate the problem of the vio-lation of the NEC on or near wormhole throats. This state-of-affairs holds atboth throats so without loss of generality, take (+)-throats and consider theterm linear in H that appears in ( 25). This can be simplied as follows:

lc

+ H d

cb B+ b

d = lc

+ H d

cb v+ b

d lb

+ e

d lc

e l+ c ,= lc+ H

dcb

+ bd , (61)

since the mixed spatial-null components of B + bd are orthogonal to H dcb , and byvirtue of the latters antisymmetry, projects out the twist from the purely spatial

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tensor v+ bd . Now consider an initially hypersurface orthogonal null congruence,then at the throat of the wormhole we have

d+du+

+ +ab

+ ab = 8T ab la+ lb+ 2H ade H deb la+ lb+ , (62)after using the expression for the Ricci tensor in ( 13).

We could now run through the list of are-out conditions (see Table I) asbefore and we would obtain, as expected, constraints on the combination of stress-energy and torsion appearing on the right hand side of ( 62). Thus, for asimply-ared throat, or marginally anti-trapped surface, we must have

4T ab la+ lb+ + H ade H

deb l

a+ l

b+ 0, (63)

at the throat and one might propose sweeping the violations into the torsionsector. We will nd that this is not possible. For illustrative purposes, supposewe consider the ansatz

H abc =1

gabce we (x), (64)

for any vector eld we . Then the combination

H ade H deb la+ l

b+ = +2( w

a l+ a )2 0, (65)is positive denite for all w. Such a torsion-eld aggravates the violation of theNEC and all of the above constraints on the stress-tensor derived at and nearthe throat in the zero-torsion case apply as well to throats in the presence of thisclass of non-zero torsion. Actually, with a little more work, it is possible to relaxthe assumption of total antisymmetry and demonstrate that all torsion leadsto enhanced violation of the NEC! To see why this comes about rst considerthe general decomposition of an arbitrary antisymmetric rank-two tensor Aab =

Aba in terms of null vectors and spatial projector. We nd that we can writeAab = al [a l+ b] + c[a

db]Acd

+ 2 l [a lc+ db]Adc + 2 l+[ a l

c

db]Adc , (66)

where the coefficient a = 2lc ld+ Acd . Now evaluate this for Ade = la+ H ade . Onends that a = 2la lb+ lc+ H cab = 0. The third term above also vanishes sincel [a db]lc+ le+ H edc = 0, which leaves us withAab = Aab + 2 l+[ a lc

db]Acd , (67)

where Aab = c[a db]Adc is a purely spatial tensor. Now, the square of ( 67)

involves only the purely spatial components:

Ade Ade la+ H ade lb+ H deb = Ade Ade 0, (68)

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and this is precisely the combination appearing in ( 63). Thus, the torsion termscannot be made to absorb any energy violations. On the contrary, torsion tendsto focus null geodesics. While the normal stress-energy must continue toviolate (or be on the verge of violating) the NEC on the throat, the presenceof any non-zero torsion does act to lessen the violation off the throat. This issimply because torsion acts as a source of twist, and even if the twist vanisheson the throat, nonvanishing twist is eventually generated in the neighborhoodsurrounding the throat, as can be appreciated by examining Eq. ( 26), and twistcomes in with the just the right sign in the Raychaudhuri equation. Of course,without further input, we have no way of knowing if this happens in the regionnear the throat or far away from the throat. If it occurs near the throat, thenthe energy violations in that region might be (partially) absorbed into the twist,but the violation persists nonetheless.

6 Worked examples

6.1 Conformally expanding Morris-Thorne wormhole

We shall illustrate these basic concepts and constructs with the following explicitexample. Consider the time-dependent spherically symmetric inter-Universewormhole described by a pair of coordinate patches in which the metric takesthe form

ds2 = 2(t) dt2 +dr 21,2

1 b( r 1 , 2 )

r 1 , 2

+ r 21,2 [d2 + sin 2 d2] . (69)

This metric is conformally related to a zero-tidal force inter-Universe Morris-Thorne wormhole by a simple time-dependent but space-independent conformalfactor [20, 21, 22]. (Other versions of time-dependent wormholes are discussedin [23, 24, 25].) Each coordinate system used to exhibit the metric given abovecovers only half the wormhole spacetime, and there are two radial coordinates,r1 and r2 , each of which runs only from r 0 to innity, where r 0 is obtained bysolving the implicit equation b(r 0) = r0 . See [1, 3]. The two radial coordinatescover two distinct universes and overlap only at r1 = r 0 = r 2 which denes thecenter of the wormhole (we will nd that the center coincides with the throatonly in the static limit). For simplicity this wormhole is taken to be symmetricunder interchange of the two asymptotically at regions but this is not essential

to the analysis.It should be clear that we look for throats within each coordinate patchseparately. We will see below that for suitable energy conditions, the abovemetric corresponds to a wormhole with two time-dependent throats, each throatresiding in one of the two Universes joined by the wormhole.

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6.1.1 First coordinate patch

The throats, when and if they exist, will be located on spheres of (instantaneous)radii (t)r 1 (where r 1 r 0) possessing the spatial metric (written in block-diagonal form)

1ab = 2 r 210 0

0 1 00 sin2 . (70)

We can easily nd a set of two independent null vectors orthogonal to the spheresin this patch; they are given by

la =1

2 1, 1 b(r 1)

r 1

12

, 0, 0 , (71)

and it is easy to verify that all the inner-product relations ( 15) are satised andthat the metric ( 69) in this patch can be decomposed in terms of l+ a , l a , 1ab just as in (16). The expansions of these null geodesics are calculated in astraightforward manner:

= ab1 a l b =2

2 2r 1

1 b(r 1)

r1

12

, (72)

where the overdot stands for the derivative with respect to (conformal) time t.The derivatives, taken with respect to the affine parameter, used for testing forare-out are ( d+ /du + = lt+ + /t + lr+ + /r ), etc. ,

ddu =

12

2

2

2

r 1 1

b(r 1)r 1

12

1r 21 1

b(r 1)r 1

+1

2r 21 b (r 1 ) +

b(r 1)r 1

. (73)

Now we can search for throats in this patch. First we locate the extremalhypersurfaces; these coincide with the zeroes of the expansions:

= 01r 1

1 b(r 1)

r 1

12

=

, (74)

which denes the time-dependent throat radius r1 (t) implicitly. We note thatthe factor involving the square-root is always positive semi-denite, hence we

nd that (in the r 1 coordinate patch) it is only that can vanish for anexpanding ( > 0) background while it is + that can vanish for a collapsing( < 0) background. There is therefore, always only one extremal hypersurfacein the rst patch.

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Irrespective of expansion or collapse, the are-out evaluated on that extremalhypersurface works out to be

ddu =0

=1

2 2

2

2+

12r1 (t)2 b

(r1 (t)) +b(r1 (t))

r1 (t). (75)

The are-out of the hypersurface is a function of time. Note that the secondgrouped term on the right hand side is always greater than or equal to zerowhile the rst grouped term can in principle, have any sign, depending on thenature of the background expansion (or contraction). This observation was rstproposed in [21, 22] as a means of temporarily suspending the energy conditionviolations for dynamic wormholes. However, the Einstein tensor associated withthe above metric ( 69) can be easily worked out [ 21, 22] and taking its projectionalong the radial null direction yields the combination

G t t + G r r = 8 (1 1) = 2 b(r 1)

r 31+

b (r 1)r 21 2

+ 42

2, (76)

where 1 and 1 denote the energy density and radial tension as seen by anobserver in the proper reference frame. Evaluate this at r1 = r1 (t) and compareit to ( 75) to conclude that any conformal factor that is chosen so as to suspendthe violation of the NEC, will at the same time eradicate the are-out condition:

(1 1 ) 0ddu |r 1 ( t ) 0, (77)

and the hypersurface at r1 (t) will not be ared-out! In other words, the extremal

hypersurface will be a throat of the simply ared-out variety if and only if theNEC is violated or on the verge of being violated there.

This is completely compatible with the topological censorship theorem [ 9]. If one picks an ingoing radial null geodesic along which the NEC is always satised,then by the above argument the expansion can never are out, one is forced tocontinue moving inward, and so one cannot pass through a wormhole throat.

6.1.2 Second coordinate patch

Many of the results from the rst coordinate patch can be carried over to thesecond coordinate patch with a few key ips in signs.

The throats in this second patch, when and if they exist, will be locatedon spheres of (instantaneous) radii ( t)r 2 (with r2

r 0) possessing the spatial

metric (written in block-diagonal form)

2ab = 2 r 220 0

0 1 00 sin2 . (78)

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We can easily nd a set of two independent null vectors orthogonal to the spheresin this patch; they are given by

la =1

2 1, 1 b(r 2)

r 2

12

, 0, 0 . (79)

It is easy to verify that the key sign ip above guarantees that the vector eldsl dened patch one connect smoothly with their denitions on patch two.Furthermore all the inner-product relations ( 15) are satised and that the metric(69) in this patch can still be decomposed in terms of l+ a , l a , and 2ab just asin (16). Their respective expansions are calculated in a straightforward manner:

= ab2

a l b = 2

2

2r 2

1

b(r 2)

r2

12

. (80)

The search for throats in this second patch proceeds just as above. For thelocation of the extremal hypersurfaces we now have

= 01r 2

1 b(r 2)

r 2

12

=

, (81)

which now denes the throat radius r2 (t) implicitly. We again note that theleft hand side is always positive semi-denite, hence we nd that it is now +that vanishes for an expanding background while it is that vanishes for acollapsing background (in this patch!). Therefore, there is again exactly oneextremal hypersurface in this patch. Note that because of the crucial sign ip,whichever of the two expansions it is that vanishes in coordinate patch one, itis the other expansion that will now vanish in patch two.

Because of the assumed symmetry between the two patches the rest of theanalysis follows through without difficulty and we can again see that any con-formal factor that is chosen so as to suspend the violation of the NEC, willat the same time eradicate the are-out condition at this second throat:

(2 2) 0ddv |r 2 ( t ) 0. (82)

Once again, this extremal hypersurface will be a throat of the simply ared-outvariety if and only if the NEC is violated or on the verge of being violated there.

(As indicated previously, the assumption that the wormhole is symmetricunder interchange of the two asymptotically at regions is not essential to theanalysis. To generalise this point one just needs to choose two un-equal shapefunctions b1(r 1 ) and b2(r 2) that need be linked only by the fact that they simul-taneously satisfy b1(r 0) = r 0 = b2(r 0 ). It is now a simple exercise to go throughthe preceding formulae making minor changes as appropriate.)

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6.1.3 Static limit

In the static limit, we have = 0 and the simultaneous vanishing of the expan-sions now occurs at the unique point where the two coordinate patches overlap:b(r0 ) = r 0 , this value being none other than the center of the wormhole: there-fore, the static wormhole has only one throat, and the throat coincides withthe center of the wormhole. We thus recover the zero-tidal force Morris-Thornewormhole. Reality of the expansions further restrains the b-function to satisfyb(r ) r so that b (r 0) 1. The are-outs of this unique throat with respect toeither coordinate patch are

ddu r 0

=1r 20

(b (r 0) + 1) 0, (83)

so that the sphere of constant radius r0 is a throat satisfying the simple are-outcondition and is therefore a marginally anti-trapped surface. It follows imme-diately from the above theorems, and in complete agreement with the standardanalyses, that the NEC is either violated, or on the verge of being violated,at the throat. Note of course, that if these inequalities are strictly positive atany point on the throat, then these derivatives are strictly positive everywhereon the the throat (by spherical symmetry) and the throat satises the strongare-out condition everywhere and is therefore a strongly anti-trapped surface.The NEC is strictly violated in this case.

6.1.4 Summary

This worked example shows how important it is to distinguish the center of the wormhole, dened by looking at the spatial behaviour of a xed time-slice,from the throat of the wormhole, dened by the are-out condition applied tonull geodesics that are actually trying to traverse the wormhole.

If the null geodesics ever succeed in getting through the traversable worm-hole, into the other universe, then they must at some stage have passed aregion where their expansion satised the are-out condition, and this region iswhat we dene to be the throat of the wormhole. By the analysis of this paper,the NEC must be violated at or near this throat. The center of the wormholeis the wrong place to look for NEC violations, except in the static limit wherethe two throats coalesce trapping the center between them.

6.2 General time-dependent spherically symmetric

traversable wormhole

The most general metric describing a time-dependent spherically symmetricspacetime can (with appropriate choice of an atlas of coordinate patches) be

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written as

ds2 = e2 1 2mr

dv2 + 2 e dv dr + r 2(d2 + sin 2 d2). (84)

Here (v, r ) and m(v, r ) are two independent functions of the radial coordinate rand an advanced time-parameter v (v t+ r at large r ) [26]. This metric can alsobe adapted to describe an inter-Universe wormhole. As in the previous example,the coordinate system employed covers only half the wormhole spacetime and sotwo patches will be required and the radial coordinate r [r 0 , ), where r 0 (v)is again the center of the wormhole. We should then introduce four independentfunctions: 1,2 and m1,2 where the labels refer to the two coordinate patches.These functions must satisfy a smoothness condition at r = r 0(v) if there is to beno delta function material concentrated on the throat (the extrinsic curvaturesshould match across the center of the wormhole, see [ 3]).

In the interest of brevity and notational economy, we will focus on one of the two coordinate patches only. So consider one of the Universes joined bythe wormhole. A throat, when it exists, will be a sphere of radius r r 0 withspatial metric given by ( 70) with = 1. The two independent sets of nullvectors orthogonal to the sphere are found to be given by

la+ = 1,12

e 1 2mr

, 0, 0 , la = (0 , e , 0, 0). (85)The expansions of the associated two sets of null rays are

+ = 2 l+ =12

e 1 2mr

, (86)

and = 2 l =

2r

e , (87)

respectively. Provided (r, v ) is non-singular (a good idea if there are to beno horizons!), the only expansion which can have zeros is + and + = 0 2m(r,v ) = r , so that r = r (v) gives the time-dependent radius of the extremalsphere.

The are-out evaluated at this hypersurface is readily calculated to be ( ddu + =la+a = l

v+

v + l

r+

r )

d+du+ + =0

= 2

r 2(v)e

m (r,v )v r (v )

. (88)

The Einstein equations are easy to work out in this metric. At this throat of the wormhole, the null-null component yields

m (r, v )v + =0

= 4 r 2 T vv , (89)

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so thatd+du+ + =0 0T vv | + =0 0. (90)

Once again, this throat will be simply-ared if and only if the null energy condi-tion is violated, or on the verge of being violated, at the throat. If the violationsare suspended at the throat, the hypersurface will not satisfy any are-out con-dition, and so ceases to be a throat. (For instance, this is what occurs inRefs. [21, 22, 23, 24, 25].) An entirely similar analysis can be carried out forthe other coordinate patch. Again, there are are total of two time dependentthroats and again, they coalesce into a single throat located at r 0 in the staticlimit.

7 DiscussionWe have presented a local geometric denition of a wormhole throat that gen-eralizes the notion of are-out to an arbitrary time-dependent wormhole andis free from technical assumptions about global properties. Flare-out is mani-fested in the properties of light rays (null geodesics) that traverse a wormhole:bundles of light rays that enter the wormhole at one mouth and exit from theother must have cross-sectional area that rst decreases, reaching a true min-imum at the throat, and then increases. These properties can be quantiedprecisely in terms of the expansion of the (future-directed) null geodesicstogether with its derivative d /du , where all quantities are evaluated at thetwo-dimensional spatial hypersurface comprising the throat. Strictly speaking,this aring-out behavior of the outgoing null geodesics ( l+ ) denes one throat:the outgoing throat. But one can also ask for the aring-out property tobe manifested in the propagation of the set of ingoing null geodesics ( l ) asthey traverse the wormhole, and this leads one to dene a second, or ingoingthroat. In general, these two throats need not be identical, but for the staticlimit they do coalesce and are indistinguishable.

The aring-out property implies that all wormhole throats are in fact anti-trapped surfaces, an identication that was anticipated some time ago by Page [ 8].With this denition and using the Raychaudhuri equation, we are able to placerigorous constraints on the Ricci tensor and the stress-energy tensor at thethroat(s) of the wormhole as well as in the regions near the throat(s). We nd,as expected, that wormhole throats generically violate the null energy conditionand we have provided several theorems regarding this matter.

The nature of the energy-condition violations associated with wormholethroats has led numerous authors to try to nd ways of evading or minimiz-ing the violations. Most attempts to do so focus on alternative gravity theoriesin which one may be able to force the extra degrees of freedom to absorb theenergy-condition violations (some of these scenarios are discussed in [ 6], see also

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[27, 28]). But the energy condition violations are still always present, as sweep-ing the energy condition violations into a particular sector surely does not makethe problem go away. As a striking case in point, we have treated in detailthe case of gravity plus torsion. If we identify the torsion with that appearingnaturally in the spectrum of closed strings, then we nd it actually worsens theviolations of the NEC at the throats. More recently it has been realized thattime-dependence lets one move the energy condition violating regions around intime [21, 22, 23, 24, 25]. Temporary suspension of the violation of the NEC at atime-dependent throat also leads to a simultaneous obliteration of the are-outproperty of the throat itself, so this strategy ends up destroying the throat andnothing is to be gained. (See also [ 6].) In arriving at this conclusion it is crucialto note that we have dened are-out in terms of the expansion properties of light rays at the throat and not in terms of shape functions or embedding dia-grams. While the latter can certainly be used without risk for detecting are-outin static wormholes, they are at best misleading if applied to dynamic worm-holes. This is simply because the embedding of a wormhole spacetime requiresselecting and lifting out a particular time-slice and embedding this instanta-neous spatial three-geometry in a at Euclidean R 3 . For a static wormhole, anyconstant time-slice will suffice, and if the embedded surface is ared-out in thespatial direction orthogonal to the throat, then it is ared-out in spacetime aswell. But if the wormhole is dynamic, are-out in the spatial direction does notimply are-out in the null directions orthogonal to the throat.

Acknowledgements

The work of M.V. was supported by the US Department of Energy. Addition-ally, M.V. wishes to thank the members of LAEFF (Spain) for their hospitalityduring early phases of this work and acknowledges interesting and constructivecomments made by Sean Hayward [ 19].

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david hochberg and matt visser- dynamic wormholes, anti-trapped surfaces, and energy conditions

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