casimir energy of a long wormhole throat

Casimir energy of a long wormhole throat

Luke M. Butcher*

Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue,Cambridge CB3 0HE, United Kingdom

Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom(Received 6 May 2014; published 7 July 2014)

We calculate the Casimir energy-momentum tensor induced in a scalar field by a macroscopic ultrastaticspherically symmetric long-throated traversable wormhole, and examine whether this exotic matter issufficient to stabilize the wormhole itself. The Casimir energy-momentum tensor is obtained (within theR × S2 throat) by a mode sum approach, using a sharp energy cutoff and the Abel-Plana formula; Lorentzinvariance is then restored by use of a Pauli-Villars regulator. The massless conformally coupled case isfound to have a logarithmic divergence (which we renormalize) and a conformal anomaly, thethermodynamic relevance of which is discussed. Provided the throat radius is above some fixed length,the renormalized Casimir energy density is seen to be negative by all timelike observers, and almost all nullrays; furthermore, it has sufficient magnitude to stabilize a long-throated wormhole far larger than thePlanck scale, at least in principle. Unfortunately, the renormalized Casimir energy density is zero for nullrays directed exactly parallel to the throat, and this shortfall prevents us from stabilizing the ultrastaticspherically symmetric wormhole considered here. Nonetheless, the negative Casimir energy does allow thewormhole to collapse extremely slowly, its lifetime growing without bound as the throat length is increased.We find that the throat closes slowly enough that its central region can be safely traversed by a pulseof light.

DOI: 10.1103/PhysRevD.90.024019 PACS numbers: 04.62.+v, 04.20.Gz

I. INTRODUCTION

The idea of a “bridge” of curved space, linking twootherwise distant regions, has served as a rich basis forthought experiments, and a valuable test bed for questionsat the interface of gravitational and quantum theory. Thesewormholes have found varied applications, from models offundamental particles [1] to ingredients of a mechanismthat would supposedly suppress the cosmological constant[2,3]. More recently, a fascinating connection betweenwormholes and quantum entanglement has been conjec-tured [4,5] which has played a key role in the ongoingdebate over the existence of a “firewall” behind a black holeevent horizon [6]. Lastly, and most provocatively of all,there is the question of whether stable traversable worm-holes can exist, and if so, whether anything prevents theirbeing used as time machines [7–9].In this paper we will focus on traversable wormholes,

and explore a mechanism which might allow them to exist,at least in principle. As is well known, the key impedimentto their stability is the need for exotic matter: negativeenergy is required, as averaged along a null geodesic thatthreads the throat and escapes to infinity [7]. The onlyexperimentally verified phenomenon expected to producenegative energy is the Casimir effect [10], wherein con-ductive plates are introduced to empty space, and theseplates impose boundary conditions on the vacuum state of a

quantum field. In many cases, the new ground state energyis less than that of the original (zero-energy) vacuum,leading to the conclusion that a negative energy has beenachieved. Adapting this phenomenon to the problem athand, one would hope to induce a negative energy vacuumin the throat of a wormhole, presumably by capping itsmouths with conductive plates (as in [11], for example).However, the plates themselves will inevitably possesssome mass, and under reasonable assumptions1 this pos-itive energy will outweigh the negative energy between theplates when averaged along a null geodesic that escapes toinfinity.Fortunately, there remains a plausible route around this

obstacle, which we shall presently explore. The idea is this:discard the conductive plates altogether, and ask whetherthe wormhole itself, by virtue of its curvature and topology,can generate the negative Casimir energy it requires.Now, cursory dimensional analysis would suggest that

this mechanism can only stabilize a Planckian wormhole,2

in which case the semiclassical approach (quantum fieldpropagating on classical spacetime) would be expected to

*[email protected]

1The plates should have a mass-to-charge ratio no less than theelectron, and should be further apart than the electron’s Comptonwavelength [7].

2If it were possible to describe the wormhole/field system by asingle characteristic length (the “size” of the wormhole) then itfollows that ðsizeÞ ∼ ðPlanck lengthÞ as there is no other quantityavailable with the correct dimensions. Of course, the wormhole/field system need not be characterized by a single length.

PHYSICAL REVIEW D 90, 024019 (2014)

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break down anyway.3 To avoid this pitfall, it is thereforedesirable to optimize the shape of the wormhole so as to(a) increase the magnitude of the (negative) Casimir energydensity it generates, and (b) decrease the magnitude of thenegative energy density it requires.One very simple way of achieving this is to make the

wormhole much longer than it is wide. For the purpose ofexplaining this claim, let us consider the following spheri-cally symmetric static traversable wormhole4:

ds2 ¼ −dt2 þ dz2 þ A2ðdθ2 þ sin2θdϕ2Þ;A≡

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ z2

p− Lþ a: ð1Þ

As Fig. 1 illustrates, this spacetime is a smooth realizationof a simple surgically constructed wormhole which con-nects two flat regions with a spherically symmetric throat oflength 2L and constant radius a.The Einstein tensor for the spacetime (1) is straightfor-

ward to calculate, and reveals the energy-momentum tensorrequired by the wormhole:

T μ ν ¼ Gμ ν=κ

¼ L2

ðL2 þ z2ÞA2κdiag

�1;−1;

AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ z2

p ;Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L2 þ z2p

�

þ 2L2

ðL2 þ z2Þ3=2Aκ diagð−1; 0; 0; 0Þ; ð2Þ

where the hats over indices indicate that components havebeen expressed in the orthonormal basis along theft; z; θ;ϕg coordinate lines. Let us assume L ≥ a, andhence A=


p≤ 1. Consequently, the first tensor on

the right-hand side of (2) obeys all four energy conditions(null, weak, strong and dominant)5 and the second tensorcan be interpreted as the exotic matter required to stabilizethe wormhole. Note that the magnitude of this secondtensor is greatest at z ¼ 0, where it takes the value 2=Laκ.This will serve as an adequate measure of the negativeenergy density required by the wormhole (1).Now we turn to the Casimir energy density generated by

the spacetime (1). Near the center of the wormhole, wherethe negative energy-density requirements are greatest, thethroat radius is

A ¼ aþ z2

2LþOðz4=L3Þ; ð3Þ

and on scales much smaller than the throat length, A takesthe constant value a to a good approximation: jdA=dzj≈jzj=L ≪ 1. Hence, near the center of the wormhole,quantum field modes with wavelengths much smaller thanL may as well be propagating in a throat of constant radiusa, and can be expected to produce a Casimir energy densityof order ℏ=a4. Clearly, if we hold a constant and increase L,then (i) this approximation will improve, with the Casimirenergy density tending to a fixed value Oðℏ=a4Þ, and(ii) the negative energy density Oð1=LaκÞ required by thewormhole will decrease in magnitude. Ignoring the non-exotic matter, then, the Einstein equations (2) take the formℏ=a4 ∼ 1=Laκ, from which it follows that

a2 ∼ ðlpÞ2ðL=aÞ; ð4Þ

where lp is the Planck length. This suggests that a and Lcan both be much larger than the Planck length, providedL ≫ a.What remains is to actually calculate the Casimir

energy-momentum tensor, and to check it possesses therequired structure (in particular, negative energy density) toallow this rough argument to carry through. To simplify thecalculation, we shall focus on the limit L → ∞, in whichthe spacetime (1) becomes

ds2 ¼ −dt2 þ dz2 þ a2ðdθ2 þ sin2 θdϕ2Þ; ð5Þ

and the Ricci tensor is

Rμ ν ¼ a−2diagð0; 0; 1; 1Þ: ð6Þ

This will provide us with a good approximation to theCasimir energy-momentum generated by a wormhole with

FIG. 1. (i) Spatial profile of the two-parameter wormhole (1).(ii) Spatial profile of a simple surgically constructed wormholewith throat length 2L and throat radius a.

3This is the main criticism one can levy at the self-sustainingwormhole solution obtained in [12] by numerical techniques.Besides the presence of Planck-scale structure, this solution isalso asymptotically ill behaved: the time-directed killing vectordiverges.

4We set c ¼ 1, write κ ≡ 8πG, and adopt the sign conventionsof Wald [13]: ημν ≡ diagð−1; 1; 1; 1Þ, ½∇μ;∇ν�vα ≡ Rα

βμνvβ, andRμν ≡ Rα

μαν.

5It is a simple matter to prove that an energy-momentum tensorT μ ν ¼ diagð1;−1; p; pÞ obeys the null, weak and dominantenergy conditions if and only if jpj ≤ 1. Furthermore, the strongenergy condition is obeyed if and only if p ≥ 0. Consequently,this energy-momentum tensor (and any positive multiple thereof)will obey all the energy conditions if and only if 0 ≤ p ≤ 1.

LUKE M. BUTCHER PHYSICAL REVIEW D 90, 024019 (2014)

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L ≫ a, at least in the vicinity of the center point z ¼ 0.Calculating the Casimir energy-momentum tensor inducedby (5) will be the main task this paper,6 followed by anassessment of wormhole stability in Sec. IV.

II. CASIMIR EFFECT

When one naively calculates the vacuum energy of aquantum field, one finds that it is infinite. The canonicalremedy for this is normal ordering, which subtracts thisinfinite constant and essentially defines the vacuum energyto be zero. However, the ground state of a quantum field isdependent on the field’s environment: the presence ofconductive surfaces will impose boundary conditions,spacetime curvature will alter the field equations, andnontrivial topology will introduce additional constraints.Hence, even after one has fixed the vacuum energy ofempty Minkowski spacetime at zero, energy differencespersist between vacuum states of different environments,and one is forced to admit that the vacua of nontrivialenvironments have nonzero energy.Let us formulate this symbolically for the case at hand:

let j0i be the vacuum state of a quantum field φ in theinfinite throat spacetime (5), and let j0Mi be the vacuumstate of the same quantum field in empty Minkowskispacetime

ds2 ¼ −dt2 þ δijdxidxj; ð7Þ

then the Casimir energy-momentum tensor of φ in thethroat is

TCasimirμν ≡ h0jTμνj0i − h0MjTμνj0Mi; ð8Þ

that is, TCasimirμν is the vacuum energy-momentum that

remains once we have accounted for the spurious vacuumenergy-momentum of φ in flat empty space. Unlikeh0jTμνj0i, we expect TCasimir

μν to be observable, giving riseto measurable forces on physical objects, and acting as asource of gravity in the Einstein field equations.There still remains the technical issue of regularizing the

two infinite expectation values on the right-hand side of (8),and the question of whether their difference remains finiteonce the regulator is sent to infinity; for the sake ofexpediency, however, let us postpone this discussion fornow, and take this formal definition of TCasimir

μν as sufficientfor the time being.As is typical, we will choose the quantum field φ to be a

free real scalar field. Although correct physical predictionsmay ultimately require the full complement of standardmodel fields, it clearly serves no purpose to burden thepresent abstract investigation with such a detailed and

realistic model. Rather, it is hoped that the results of thescalar case will accurately portray the flavor of a morecomplete calculation. In the interest of generality, we willinitially proceed without fixing the field’s mass; however,as the Casimir effect is exponentially suppressed forsystems much larger than a field’s Compton wavelength([15], Sec. 4.2), the massless case will be our primaryinterest. The most physically pertinent case will then be theconformally coupledmassless scalar field, due to the stronganalogy with electromagnetism; however, again for thesake of generality, we will leave the curvature couplingparameter arbitrary for now. We begin by summarizing thebasic ingredients of field theory that we require.

A. Basic formalism

The action for the free real scalar field φ is

Sφ ¼ 1

2

Zdx4

ffiffiffiffiffiffi−g

p ðð∇φÞ2 þ ðm2 þ ξRÞφ2Þ; ð9Þ

where ξ is the curvature coupling parameter. For theconformally coupled scalar field, ξ ¼ 1=6. This actiongives rise to the classical field equation,

0 ¼ −1ffiffiffiffiffiffi−gp δSφδφ

¼ ð∇2 −m2 − ξRÞφ; ð10Þ

and the classical energy-momentum tensor,

Tμν ≡ 2ffiffiffiffiffiffi−gp δSφδgμν

¼ ∇μφ∇νφþ ξðRμνφ2 −∇μ∇νðφ2ÞÞ

− gμν1 − 4ξ

2ðð∇φÞ2 þ ðm2 þ ξRÞφ2Þ; ð11Þ

where we have used (10) to simplify the last line. Note thatit is only for the conformally coupled field that Tμν agreeswith the “new improved” energy-momentum tensor whichbehaves well in the renormalized quantum theory [16]. Itwill also be convenient to define a symmetric bilinear formTμνf·; ·g based on the classical energy-momentum tensor:

Tμνfφ1;φ2g≡∇ðμjφ1∇jνÞφ2 þ ξðRμνφ1φ2 −∇μ∇νðφ1φ2ÞÞ

− gμν1 − 4ξ

2ð∇αφ1∇αφ2 þ ðm2 þ ξRÞφ1φ2Þ: ð12Þ

To quantize φ, let us set ℏ ¼ 1 and specialize to ultra-static spacetimes:

ds2 ≡ gμνdxμxν ¼ −dt2 þ hijð~xÞdxidxj: ð13Þ

Under canonical quantization, φ is replaced by theoperator

6Note that because curvature coordinates are degenerate in theinfinite throat (5) the treatment of vacuum energies by Andersonet al. [14] cannot be applied.

CASIMIR ENERGY OF A LONG WORMHOLE THROAT PHYSICAL REVIEW D 90, 024019 (2014)

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φ ¼Xn

ðφ−n a−n þ φþ

n aþn Þ; ð14Þ

where the aþn ¼ ða−n Þ† are creation/annihilation operators:

½aþn ; aþm� ¼ ½a−n ; a−m� ¼ 0; ½a−n ; aþm� ¼ δnm: ð15Þ

In this generic treatment, the mode index n is discrete; foreach continuous index k taking values in R, the sum in (14)should be augmented by

Rdk=2π, and the Kronecker delta

in (15) should be multiplied by 2πδðk − k0Þ.The modes fφ�

n g are an orthogonal basis of solutions tothe field equation (10); they are required to have definiteenergy,

∂tφ�n ¼ �iωnφ

�n ; ωn ≥ 0; ð16Þ

and also obey

φþn ¼ ðφ−

n Þ�; ð17Þ

so that φ is Hermitian (corresponding to φ ∈ R).The modes are normalized such that

½φðt; ~xÞ; Πðt; ~x0Þ� ¼ ih−1=2δð~x − ~x0Þ; ð18Þ

where Π≡ ∂tφ is the conjugate momentum of φ, andh≡ detðhijÞ. Substituting (14) and using Eqs. (15)–(17),this condition becomesX

n

2ωnℜfφþn ðt; ~xÞφ−

n ðt; ~x0Þg ¼ h−1=2δð~x − ~x0Þ: ð19Þ

The Fock space is constructed in the usual fashion, withthe vacuum state j0i defined by a−n j0i ¼ 0 for all n.Consequently, the vacuum energy-momentum tensor is

h0jTμνj0i ¼ h0jTμνfφ; φgj0i¼

Xn;m

Tμνfφ−n ;φþ

mgh0ja−n aþmj0i

¼Xn

Tμνfφ−n ;φþ

n g: ð20Þ

Typically this sum will diverge, so some form of regulari-zation is required to render it meaningful. The simplestapproach is to introduce an energy cutoff:

h0jTμνj0i ¼Xn

Tμνfφ−n ;φþ

n gfðωn=ΩÞ; ð21Þ

where fðxÞ is a monotonically decreasing function of x,such that fð0Þ ¼ 1, which vanishes fast enough as x → ∞to render the sum finite. Based as it is on the energy of themodes, this scheme can be expected to break Lorentzinvariance; as such it will be a temporary measure,

necessary at this stage to prevent us from deriving nonsensefrom infinite expressions. In Sec. III, we will replace itwith a Lorentz invariant regularization scheme and sendΩ → ∞.

B. Modes in the infinite throat

Fixing the metric to be that of the infinite throat (5), themost convenient set of orthogonal field modes becomes

φþklm ¼ ðφ−

klmÞ� ¼1

affiffiffiffiffiffi2ω

p eiðωt−kzÞYlmðθ;ϕÞ; ð22Þ

where Ylm are spherical harmonics, k ∈ R, l ∈ N,m ∈ f−l;−lþ 1;…; lg, and

ω≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þm2 þ lðlþ 1Þ þ 2ξ

a2

rð23Þ

ensures that the field equation (10) is satisfied. Note that if2ξ < −m2a2, then we must disregard modes for whicha2ðk2 þm2Þ þ lðlþ 1Þ þ 2ξ < 0, as ω is imaginary in thiscase. For now, let us proceed under the assumption that2ξ ≥ −m2a2; it will be trivial to deal with 2ξ < −m2a2 inSec. II E, by taking the real part of our results.To ensure the modes are correctly normalized, we must

check that they are in agreement with (19):Zdk2π

Xl;m

2ωℜfφþklmðt; ~xÞφ−

klmðt; ~x0Þg

¼ 1

a2

Zdk2π

Xl;m

eikðz−z0ÞYlmðθ;ϕÞY�lmðθ0;ϕ0Þ

¼ 1

a2 sin θδðz − z0Þδðθ − θ0Þδðϕ − ϕ0Þ

¼ h−1=2δð~x − ~x0Þ: ð24Þ

Thus the canonical commutation relation (18) is obeyed.

C. Vacuum energy-momentum

To calculate the vacuum energy-momentum tensor, wesubstitute the modes (22) into Eq. (21):

h0jTμνj0i ¼Z

dk2π

Xl;m

fðωΩÞ2ωa2

× ½∂ðμjðeiðωt−kzÞYlmÞ∂ jνÞðe−iðωt−kzÞY�lmÞ

þ ξðRμνjYlmj2 −∇μ∇νðjYlmj2ÞÞ

− gμν1 − 4ξ

2ðð−ω2 þ k2 þm2 þ ξRÞjYlmj2

þ ∂αYlm∂αY�lmÞ�: ð25Þ

We can perform the sum over m by use of theidentities


024019-4

Xl

m¼−ljYlmj2 ¼

2lþ 1

4π;

Xl

m¼−l∂μYlm∂νY�

lm ¼ 2lþ 1

4π·lðlþ 1Þ2a2

Θμν; ð26Þ

where we have introduced the tensor

Θμ ν ≡ diagð0; 0; 1; 1Þ ð27Þ

to represent the angular part of the metric. The result is

h0jTμνj0i¼Z

∞

−∞

dk16π2a2

X∞l¼0

ð2lþ1ÞfðωΩÞω

×

�ω2ð∂μtÞð∂νtÞþk2ð∂μzÞð∂νzÞ

þΘμνlðlþ1Þþ2ξ

2a2

−gμν1−4ξ

2

�−ω2þk2þm2þ lðlþ1Þþ2ξ

a2

��:

ð28Þ

Applying (23) this becomes

h0jTμνj0i ¼Z

∞

−∞

dk16π2a2

X∞l¼0

ð2lþ 1ÞfðωΩÞω

�ω2ð∂μtÞð∂νtÞ

þ k2ð∂μzÞð∂νzÞ þ Θμνω2 − k2 −m2

2

�

¼Z

∞

−∞

dk32π2a2

X∞l¼0

ð2lþ 1ÞfðωΩÞω

× ½ω2Aμν þ k2Bμν −m2Θμν�; ð29Þ

in which we have introduced the tensors

Aμ ν ≡ diagð2; 0; 1; 1Þ;Bμ ν ≡ diagð0; 2;−1;−1Þ: ð30Þ

Lastly, we define the dimensionless quantities

u≡ ka; v≡ ωa; λ≡ Ωa; μ≡ma; ð31Þ

and use them to write

h0jTμνj0i ¼Z

∞

0

du8π2a4

X∞l¼0

ðlþ 12ÞfðvλÞv

× ½v2Aμν þ u2Bμν − μ2Θμν�; ð32Þ

wherein

v≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ ðlþ 1=2Þ2 þ α

q;

α≡ μ2 þ 2ξ − 1=4: ð33Þ

D. Minkowski vacuum energy-momentum

To complete the calculation of TCasimirμν , we also require

the vacuum energy-momentum of φ in Minkowski space-time (7), evaluated according to the same regularizationscheme. The Minkowski modes are of course

φþ~k¼ ðφ−

~kÞ� ¼ 1ffiffiffiffiffiffi

2ωp eiðωt−~k·~xÞ;

ω≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij~kj2 þm2

q; ð34Þ

and lead to a regularized vacuum energy

h0MjT μ νj0Mi ¼Z

d3~kð2πÞ3

fðωΩÞ2ω

diagðω2; ðk1Þ2; ðk2Þ2; ðk3Þ2Þ:

ð35Þ

To rewrite this integral in a way the resembles the throatresult (32) let us parametrize ~k ¼ ðk; q cos ϑ; q sin ϑÞ andperform the integral over ϑ:

h0MjT μ νj0Mi

¼Z

∞

−∞

dkð2πÞ2

Z∞

0

dqqfðωΩÞ2ω

diagðω2; k2; q2=2; q2=2Þ

¼Z

∞

−∞

dk16π2

Z∞

0

dqqfðωΩÞω

½ω2Aμ ν þ k2Bμ ν −m2Θμ ν�;

ð36Þ

where q2 ¼ ω2 − k2 −m2 was used in the last line. Writingq ¼ l=a (with l a continuous variable) we express every-thing in terms of the dimensionless variables (31):

h0MjTμνj0Mi

¼Z

∞

0

du8π2a4

Z∞

0

dllfðvλÞv

½v2Aμν þ u2Bμν − μ2Θμν�; ð37Þ

wherein

v≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ l2 þ α

p; α≡ μ2: ð38Þ

E. Casimir energy-momentum

Finally, we subtract the Minkowski energy-momentum(37) from the throat energy-momentum (32) to arrive at theCasimir energy-momentum tensor:

TCasimirμν ¼ 1

8π2a4ðIAμν þ JBμν − αKΘμνÞ; ð39Þ


024019-5

where

I ≡Z

∞

0

du

�X∞l¼0

�lþ 1

2

�vf

�vλ

�−Z

∞

0

dllvf

�vλ

��;

J ≡Z

∞

0

du

�X∞l¼0

ðlþ 12Þu2fðvλÞv

−Z

∞

0

dllu2fðvλÞ

v

�;

K ≡Z

∞

0

du

�X∞l¼0

ðlþ 12ÞfðvλÞv

−Z

∞

0

dllfðvλÞv

�;


q; α≡ μ2 þ 2ξ − 1=4;


p; α≡ μ2: ð40Þ

Recall that this result is only valid for 2ξ ≥ −m2a2

(equivalently, α ≥ −1=4) and that if 2ξ < −m2a2 we must

be careful to remove any modes for which ω is imaginary.Fortunately, these modes produce a purely imaginarycontribution to I, J and K, so they are easily removedsimply by taking the real part of the above expression.Hence,

TCasimirμν ¼ 1

8π2a4ℜfIAμν þ JBμν − αKΘμνg ð41Þ

is now valid for all α ∈ R.In Appendix A, we use the Abel-Plana formula to

evaluate ℜfIg, ℜfJg and ℜfKg when f enacts a sharpcutoff at energy Ω ¼ λ=a; the results can be found inEqs. (A23)–(A25). Consequently, the Casimir energy-momentum tensor (expressed in the orthonormal basis)is given by

TCasimirμ ν ¼ 1

8π2a4½ℜfI þ Jgdiagð1; 1; 0; 0Þ þℜfI − Jgdiagð1;−1; 1; 1Þ − αℜfKgdiagð0; 0; 1; 1Þ�

¼ 1

192π2a4

��ð1þ 12ðα − αÞÞλ2 − 1

2

�αþ 6ðα2 − α2Þ − 7

40

��diagð1; 1; 0; 0Þ

þ��

αþ 6ðα2 − α2Þ − 7

40

�lnð2λÞ þ 3ðα2 ln jαj − α2 ln jαjÞ − 3

2ðα2 − α2Þ þ 48XðαÞ

�diagð1;−1; 1; 1Þ

− α

�ð1þ 12ðα − αÞÞ lnð2λÞ þ 6ðα ln jαj − α ln jαjÞ þ 6ðα − αÞ − 48YðαÞ

�diagð0; 0; 1; 1Þ

�þOðλ−2Þ; ð42Þ

as λ → ∞, where the functions XðαÞ and YðαÞ are definedin Eqs. (A16) and (A26).At this stage, the key fact to recognize is that, unlike in

Casimir’s original calculation [10], we cannot send λ → ∞and recover a finite result: although the subtraction of theMinkowski vacuum has removed a singularity Oðλ4Þ fromthe expansion, there remains singularities Oðλ2Þ andOðln λÞ.7 With finite λ, Lorentz invariance remains broken,and this is manifest in two ways. First, TCasimir

μν is dependenton the cutoff energy Ω ¼ λ=a, which is obviously a frame-dependent quantity. Second, the piece of TCasimir

μν propor-tional to diagð1; 1; 0; 0Þ is not invariant under boosts in thez direction, and so does not respect the Lorentz symmetryof the throat metric (5). The purpose of the followingsection is to remedy these failings with the introduction of aLorentz-invariant regularization scheme.

III. RESTORING LORENTZ INVARIANCE

The simplest way to restore Lorentz invariance to TCasimirμ ν

is to introduce a Pauli-Villars regulator [17]. The regulator

is a fictitious scalar field φ� (with a very large mass m�)the energy-momentum of which we subtract from thatof φ:

TPVμν ≡ TCasimir

μν ½φ� − TCasimirμν ½φ��: ð43Þ

Following this scheme, the low-energy modes of φcontribute to TPV

μν as usual, with negligible subtractionfrom φ�; for modes with energies far above m�, however,the contributions from the two fields almost exactlycancel. Consequently, high-energy modes are suppressedin a smooth and Lorentz-invariant fashion. Once thisregulator has been added, it will hopefully be possible tosend our original cutoff λ → ∞, with TPV

μν remaining finite,and m� retained as a Lorentz-invariant regularizationscale.Although Pauli-Villars regularization is rarely used for

Casimir energy-momentum calculations, it has a number ofadvantages over the alternative schemes (dimensionalregularization [18,19], point splitting [20], and zeta-functionregularization [21,22]) at least for the case at hand. First,this approach follows very easily from the energy cutoffresult (42), requiring only elementary algebra, with no needfor additional mathematical tools or formalism. Second, thePauli-Villars approach has no additional ambiguities orfreedoms, beyond the energy-scale m� that is present in

7We can nullify the λ2 divergence by setting α − α ¼ 1=12,which corresponds to conformal coupling ξ ¼ 1=6; however, thecoefficient of the remaining lnðλÞ term is then αþ 6ðα2 − α2Þ−7=40 ¼ −2=15, independent of the value of μ2, and so cannotalso be set to zero.


024019-6

all methods.8 Lastly, Pauli-Villars has an attractive “toymodel” physical interpretation: one can think of the regulatorfield as representing the appearance of new particle species(at an energy scale m�) which suppress the energy-momentum of high-energy modes of φ. This behaviorwould be expected from spontaneously broken supersym-metry: as energies exceed the symmetry-breaking scale,superpartner fields would appear that exactly cancel theenergy-momentum contribution of the fields present atlow energy. Of course, this is only a toy model, and onemay prefer to default to the minimal interpretation, whereinregularization is a purely mathematical device, devoid ofphysical meaning. As we will see in Sec. III B, this morerigorous approach (in which m� is eventually sent to infinityand divergences are absorbed via renormalization) givesessentially the same results as the toy model.

A. Pauli-Villars regularization

Employing the Pauli-Villars regularization scheme issimply a matter of inserting (42) into (43). Although it iscertainly possible to proceed without fixing the mass of φ, it

will streamline our analysis to now focus specifically on themassless case. Recall that this was our original intention: theCasimir effect of the massive case is expected to diminish ase−2ma for ma ≫ 1 ([15], Sec. 4.2), and so would be unableto support a macroscopic wormhole. Of course, we will stillrequire the formula (42) for m ≠ 0 in order to calculate thecontribution from the regulator field.Referring to definitions (31), (33) and (38), we see that

the massless field φ requires α ¼ 2ξ − 1=4 and α ¼ 0, andthe regulator φ� requires α ¼ 2ξ − 1=4þ ðm�aÞ2 andα ¼ ðm�aÞ2. Hence, the Pauli-Villars regularized Casimirenergy-momentum (43) can be written as

TPVμν ¼ TCasimir

μν jα¼ζ; α¼0− TCasimir

μν jα¼ζþμ2�; α¼μ2�; ð44Þ

where

ζ ≡ 2ξ − 1=4; μ� ≡m�a ð45Þ

have been introduced as a convenient shorthand. We cannow substitute (42) into (44) and arrive at

TPVμ ν ¼ 1

192π2a4

�μ2�2ð1 − 12ζÞdiagð1; 1; 0; 0Þ þ

�μ2�ð12ζ − 1Þ lnð2λÞ þ 3ζ2ðln jζj − 1=2Þ − 3ðμ2� þ ζÞ2ðln jμ2� þ ζj − 1=2Þ

þ 3μ4�ðln jμ2�j − 1=2Þ þ 48XðζÞ − 48Xðμ2� þ ζÞ�diagð1;−1; 1; 1Þ þ μ2�

�ð1 − 12ζÞ lnð2λÞ þ 6ðμ2� þ ζÞ ln jμ2� þ ζj

− 6μ2� ln jμ2�j − 6ζ − 48Yðμ2� þ ζÞ�diagð0; 0; 1; 1Þ

�þOðλ−2Þ: ð46Þ

Notice that, as a consequence of the new regularization, theOðλ2Þ divergence has vanished entirely. In general, though,two pathologies still remain: first, a divergenceOðlnðλÞÞ, andsecond, the frame-dependent tensor diagð1; 1; 0; 0Þ. Fortu-nately, bothof these features canbe removed simplybysettingζ ¼ 1=12; a glance at (45) confirms that this is the same as

ξ ¼ 1=6; ð47Þwhich of course ensures that φ is conformally coupled.Recall that this value for ξ was physically well motivateda priori, as the conformal invariance of φ makes it mostclosely analogous to the electromagnetic field, and thus agood model for the only massless field in the standardmodel. Moreover, (47) also defines the new improvedenergy-momentum tensor [16] of Callan, Coleman and

Jackiw,which is the formof energy-momentum tensormostsuitable for renormalized quantum theory. Thus, althoughthe other possibilities ξ ≠ 1=6 can presumably be dealt withusing a more complicated regularization scheme, thereseems little to be gained in pursuing these results, consid-ering that they were less well motivated in the first place.With conformal coupling (47) fixed, there is nothing to

stop us from sending λ → ∞ and recovering Lorentzinvariance. The final step of the calculation is then to letour Lorentz-invariant regulator m� become very large;specifically, we insist that the wormhole radius shouldbe much greater than the regulator’s Compton wavelength,so m�a ¼ μ� ≫ 1. We can then use

ln jμ2� þ ζj ¼ ln jμ2�j þζ

μ2�−

ζ2

2μ4�þOðμ−6� Þ; ð48Þ

and the following asymptotic expansions,

Xðμ2� þ ζÞ¼ 1

96

��7

40−ζ−μ2�

�ln jμ2�j−ζþ 7

40

�þOðμ−2� Þ;

Yðμ2� þ ζÞ¼ 1

96

�ln jμ2�jþ

1

μ2�

�ζ−

7

40

��þOðμ−4� Þ;

ð49Þ

8For instance, there is the question of how, in detail, one shouldperform dimensional regularization: the wormhole throat (5) has a(wick-rotated) geometry R2 × S2 which generalizes to Rd1 × Sd2with two degrees of freedom. Similarly, point splitting requires achoice of splitting direction, and the subtleties in application ofthe zeta function give rise to other ambiguities (see p. 167 of [23]).Even if these ambiguities can be fixed post hoc (e.g. by insisting onsome property of Tμν, or by averaging over splitting direction [20])or can be absorbed in the process of renormalization, it will clearlybe advantageous to avoid these additional complications.


024019-7

which can be derived by the same methods as used in thesteps between (A10) and (A17). Inserting these into (46)with ζ ¼ 1=12, we finally obtain (after considerablecanceling) the Pauli-Villars regularized Casimir energy-momentum tensor:

TPVμ ν ¼ 1

2880π2a4½diagð−1; 1;−1;−1Þðln jμ2�j þ ΔÞ

þ diagð0; 0; 1; 1Þ� þOðμ−2� Þ; ð50Þ

where

Δ≡ 37þ 10 lnð12Þ32

− 360

Z∞

0

dttðt2 − 1

12Þ ln jt2 − 1

12j

e2πt þ 1

≅ 2.2325 ð51Þ

is a numerical factor.As far as the toy model is concerned, we can end the

calculation here. In (50) we have μ� ¼ m�a, where m� is avery large, unknown but finite mass, quantifying the energyat which new particle species arise and suppress the energy-momentum of φ. Our present analysis cannot predict m�,but it could be determined experimentally, at least inprinciple. Fortunately, as TPV

μν is only logarithmicallydependent on m�, this uncertainty will have little impacton our conclusions. In particular, we know that μ� ≫ 1, soln jμ2�j is positive, and thus the Casimir energy density TPV

00

is negative, just as we had hoped. However, negativeenergy density is not sufficient in itself to guarantee thestability of the wormhole; we will examine this subjectproperly in Sec. IV.Before this, it will be useful to briefly review how

renormalization allows us to take the regulator m� toinfinity, while retaining a finite energy-momentum tensor.In truth, this more rigorous treatment has little effect on theimportant features of TPV

μν , so a reader who is happy toaccept the toy model picture of Pauli-Villars regularizationmay wish to skip to Sec. IV at this point. In Sec. III C wewill also discuss the conformal anomaly displayed by (50).

B. Renormalization

Prior to renormalization, the semiclassical Einstein fieldequations are

Gμν þ gμνΛB ¼ κBhTμνi; ð52Þ

with “bare” cosmological and gravitational constants ΛBand κB. Utiyama and DeWitt [24] proved that the expect-ation value of the energy-momentum tensor will genericallytake the form

hTμνi ¼ c1gμνm4� þ c2Gμνm2� þ c3Hμν lnðm�bÞ þ Trenμν ;

ð53Þ

where m� is a Lorentz-invariant regulator, fc1; c2; c3g arenumerical constants, Hμν is a tensor composed of R2 and∇2R terms,9 and Tren

μν is finite asm� → ∞. Notice that it hasbeen necessary to introduce an arbitrary length scale b,without which the logarithm would have a dimensionfulargument. Because hTμνi does not actually depend on b,any change b → b0 must produce a compensating changein the finite part of the energy-momentum tensor:ΔTren

μν ¼ c3 lnðb=b0ÞHμν.Substituting (53) into (52), grouping terms and dividing

by ð1 − c2κBm2�Þ, we arrive at the following field equations:

Gμν þ gμνΛB − c1m4�κB1 − c2κBm2�

¼ κB1 − c2κBm2�

½Trenμν þ c3Hμν lnðm�bÞ�: ð54Þ

Ignoring the logarithmic divergence for the moment, we seethat neither the bare constants ΛB; κB, nor the m2�; m4�divergences, can be observed directly: one can onlymeasure the renormalized quantities,

Λ≡ ΛB − c1m4�κB1 − c2κBm2�

; κ ≡ κB1 − c2κBm2�

: ð55Þ

These quantities have been measured experimentally, andare known to be finite.10 Consequently, we can infer thebehavior of ΛB and κB as m� → ∞.This deals with the quadratic and quartic divergences:

they simply produce an unobservable shift in the cosmo-logical and gravitational constants.11 The interesting physi-cal behavior is then confined to Tren

μν and the logarithmicdivergence. To absorb this divergence, one must posit theexistence of extra R2 terms in the gravitational action, withan (unobservable) bare coupling parameter σB. Thesecontributions produce a term σBHμν=ð1 − c2κBm2�Þ onthe left-hand side of the field equations (54) whichcombines with the logarithmic divergence to give

Gμν þ gμνΛþ σHμν ¼ κTrenμν ; ð56Þ

where the renormalized coupling

9In fact, there are two linearly independent terms of this sort,so c3Hμν should really be replaced by c3H

ð1Þμν þ c4H

ð2Þμν . This

complication is irrelevant to the schematic explanation givenhere, so we will ignore it.

10The empirical value of the cosmological constant is so smallthat it is unlikely to have a significant effect on the wormhole; assuch, we set Λ ¼ 0 outside this section of the paper.

11As it happens, neither of these divergences are present in(50): the quadratic divergence does not appear when the field isconformally coupled, and the quartic divergence was removedby subtracting the Minkowski energy-momentum in Eq. (8). Thislatter process is essentially a renormalization of Λ.


024019-8

σ ≡ σB1 − c2κBm2�

− c3κ lnðm�bÞ ð57Þ

can once again be determined by experiment. Astrophysicalobservations provide a stringent upper bound for σ, so itgoes without saying that its value must be finite; indeed,σ ¼ 0 remains a possibility, in which case Einstein’s theorysurvives in spite of the R2 counterterms in the action.Notice that the definition of σ depends on the arbitrary

length scale b. Changing b allows us to add any finiteamount to σ, with an equal and opposite change in Tren

μν .A particularly convenient way to remove this ambiguity isto choose b such that σ ¼ 0. This convention promotes b toa physically meaningful quantity (determined by experi-ment) and ensures the field equations take on their usualEinsteinian form,

Gμν þ gμνΛ ¼ κTrenμν : ð58Þ

Let us now apply this process to the Casimir energy-momentum of the long wormhole throat (50). We write thedivergent logarithm as

ln jμ2�j ¼ 2 lnðm�aÞ ¼ 2 lnðm�bÞ þ 2 lnða=bÞ;

with the first piece identified as producing the logarithmicdivergence in (53), and the second piece to be included inTrenμν . Following the scheme above, R2 terms are added to

the gravitational action, the logarithmic divergence isabsorbed into σHμν, and we fix b by insisting that σ ¼ 0.Consequently, we arrive at the renormalized Casimirenergy-momentum tensor

Trenμ ν ¼ 1

2880π2a4½diagð−1; 1;−1;−1Þ2 lnða=a0Þ

þ diagð0; 0; 1; 1Þ�; ð59Þ

where

a0 ≡ be−Δ=2 ð60Þis a fixed length that can only be determined by experiment.Equation (59) is then our final result. As previously

advertised, the renormalized energy-momentum tensordisplays much of the same structure as the Pauli-Villarsregularized tensor (50), with the unknown length scale a0replacing 1=m�. The parameter a0 has a straightforwardinterpretation: it is the throat radius of a wormhole forwhich the Casimir energy density vanishes. Provided thewormhole has a throat radius greater than a0, the Casimirenergy density will be negative.

C. Conformal anomaly and wormhole thermodynamics

Being largely irrelevant to the stability of the wormhole,we have thus far paid little attention to the diag(0, 0, 1, 1)part of the Casimir energy-momentum tensor (59). However,

this part of the tensor plays a key role in generating theconformal anomaly, and ensuring the self-consistent thermo-dynamic behavior of the wormhole. We shall quickly coverthese details here, for the sake of completeness, beforefinally examining the energy conditions violated by Tren

μν .The presence of a conformal anomaly is evidenced by

the trace of the renormalized energy-momentum tensor,

Tren ¼ 1

1440π2a4;

which classically would be expected to vanish for aconformally coupled massless scalar field. This anomalycould have been anticipated from general considerations ofquantum fields in curved backgrounds; for example, onemight have used Eq. (6.114) of Birrell and Davies [23].Accounting for the difference in metric sign convention,this gives

Tren ¼ 1

2880π2ðRαβγδRαβγδ − RαβRαβ −∇2RÞ

¼ 1

1440π2a4; ð61Þ

in agreement with the result above. This trace, which arisesfrom the diag(0, 0, 1, 1) part of Tren, is intimately connectedto the logarithmic dependence of the traceless part [propor-tional to diagð−1; 1;−1;−1Þ] as we will see by examiningthe thermodynamical behavior of the wormhole.Consider a section of throat (5) of length l. From (59) we

see that Casimir energy contained within is

E ¼ 4πa2lρ ¼ −l

360πa2lnða=a0Þ: ð62Þ

Thus, if the throat radius undergoes a change da (with a0held constant) the Casimir energy is altered by

dE ¼ −l

360πa3ð−2 lnða=a0Þ þ 1Þda; ð63Þ

where theþ1 arises from differentiating the logarithm. Theenergy-momentum tensor (59) also reveals the pressureacting in the angular directions:

P ¼ 1

2880π2a4ð−2 lnða=a0Þ þ 1Þ; ð64Þ

where the þ1 arises from the diag(0, 0, 1, 1) part of Trenμν .

Consequently, under a change in radius, the work done bythe throat (on the field φ) is

PdV ¼ 1

2880π2a4ð−2 lnða=a0Þ þ 1Þdð4πa2lÞ

¼ l360πa3

ð−2 lnða=a0Þ þ 1Þda: ð65Þ


024019-9

Hence the logarithm and the diag(0, 0, 1, 1) tensor conspireto ensure that the first law of thermodynamics,

dE ¼ −PdV; ð66Þis obeyed when the throat radius expands or contracts.

IV. WORMHOLE STABILITY

When the throat radius is sufficiently large (a > a0) therenormalized Casimir energy density Tren

00 is negative (inviolation of the weak and dominant energy conditions) andhas a magnitude of order a−4. Hence, the required exoticenergy density −2=Laκ [obtained from the second term onthe right of (2) in the limit that the throat length L becomesextremely large] can be supplied by the Casimir energy-momentum tensor (59) if

−2Laκ

¼ − lnða=a0Þ1440π2a4

; ð67Þ

or equivalently,

a2 ¼ ðlpÞ2ðL=aÞlnða=a0Þ360π

; ð68Þ

where lp is the Planck length: κ ¼ 8πðlpÞ2. Thus, ashypothesized in the Introduction, the Casimir effect of avery long wormhole (L ≫ a) is capable of generatingexotic matter in quantities that could, in principle, allow forthe stability of a wormhole with a macroscopic throatradius a ≫ lp.However, thus far we have satisfied only the first

component of the Einstein field equation (ignoring non-exotic matter). Unfortunately, the positive Casimir pressureTren11 ¼ −Tren

00 will prove problematic when extending thisprocedure to the other components. To understand thisissue, let us relax our focus on the single component Tren

00 ,and consider the inequalities obeyed by the tensor Tren

μν asa whole.

A. Energy conditions

As previously mentioned, the weak and dominant energyconditions are clearly violated if a > a0; indeed, we showin Appendix B that if this inequality is strengthened veryslightly, so that

a > a0e1=4; ð69Þthen Tren

μν will violate all four energy conditions: weak,strong, dominant and null. Furthermore, the stipulation (69)ensures that, for all null or timelike vectors vμ,

vμTrenμν vν ≤ 0; ðTren

μν vνÞ2 ≤ 0; ð70Þ

with equality if and only if vμ ∝ ð1;�1; 0; 0Þ. The firstinequality reveals that all four-velocities vμ define negative

energy densities, with the sole exception being null vectorsrunning directly parallel to the throat, for which the energydensity is zero. The second inequality reassures us that theenergy flux is always causal: Tren

μν vν is never spacelike.Thus Tren

μν is impressively exotic (defining negative energydensities in almost all directions) but does not violate themore fundamental expectation that energy (whether pos-itive or negative) should never flow faster than light.With regards to the stability of the wormhole, the

problem arises from those particular null directions vμ ∝ð1;�1; 0; 0Þ which define vanishing energy density. Wewould like to be able to solve the Einstein equations withthe addition of some ordinary matter Tord

μν :

Gμν ¼ κðTrenμν þ Tord

μν Þ; ð71Þ

where Gμν is given by (2) in the large L limit. Let uscontract this equation with vμvν, where vμ is timelike ornull:

vμGμνvν ¼ κvμðTrenμν þ Tord

μν Þvν: ð72Þ

For all timelike vμ, and almost all null vμ, we havevμTren

μν vν < 0, sowe should be able to accommodate negativevalues for vμGμνvμ. However, for vμ ∝ ð1;�1; 0; 0Þ, we havevμTren

μν vμ ¼ 0 and vμGμνvμ < 0, leaving us with

vμTordμν vν < 0; ð73Þ

which requires the ordinary matter to violate the null energycondition—clearly a contradiction. Thus it is impossible tosolve the Einstein equations for the static wormhole (1) inthe large L limit, using only its Casimir energy-momentum(59) and ordinary matter. Unfortunately, the exotic energy-momentum generated is not quite of the right form tostabilize the wormhole.The root cause of this obstacle is the symmetry of the

throat metric (5) under Lorentz boosts in the z direction.Consistency with this symmetry (and the spherical sym-metry of the cross sections) guarantees that Tren

μ ν ¼diagðρ;−ρ; p; pÞ, for some ρ and p, with vμ ∝ð1;�1; 0; 0Þ ⇒ vμTren

μν vμ ¼ 0 an inevitable consequence.Considering that all other four-velocities define negativeenergy density, it is conceivable that some symmetry-breaking deformation of the spacetime may alleviate thisproblem. In particular, noting that null rays with nonzeroangular momentum do see negative energy density, onemight hope that introducing a “twist” to the throat wouldmix the angular and longitudinal behavior in a profitableway. Alternatively, one could examine wormholes withvery short throats: their extreme aspect ratios leave themopen to the method of attack described in the Introduction,but unlike the L → ∞ limit considered here, the L → 0limit is not invariant under longitudinal boosts. We shallleave these possibilities for separate investigations.


024019-10

It should be stressed, however, that although the Casimireffect cannot fully stabilize the wormhole, this does notpreclude some very interesting behavior. In particular, itseems that Casimir effect can slow down the collapse of thewormhole, so that it can have an arbitrarily long lifetime,and moreover, that the collapse may be slow enough toallow a light pulse to traverse the throat before closing. Weconclude the paper with an analysis of this phenomenon.

B. Slow collapse

To understand the collapse of the wormhole, let us beginwith the static wormhole spacetime considered in theIntroduction (1) and promote the radius a to a timedependent quantity aðtÞ:

ds2 ¼ −dt2 þ dz2 þ A2ðdθ2 þ sin2θdϕ2Þ;A≡


p− Lþ aðtÞ: ð74Þ

The Einstein tensor of this spacetime is then

Gμ ν ¼L2

ðL2 þ z2ÞA2diag

�1;−1;

AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ z2

p ;Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L2 þ z2p

�

þ _a2

A2diagð1;−1; 0; 0Þ

þ diag

�−2L2

AðL2 þ z2Þ3=2 ;−2aA

;−aA

;−aA

�: ð75Þ

Taking the large L limit (or equivalently, focusing on thecenter of the throat: z ¼ 0) we find that

Gμ ν ¼1

a2diag

�1;−1;

aL;aL

�þ _a2

a2diagð1;−1; 0; 0Þ

þ diag�−2La

;−2aa

;−aa

;−aa

�: ð76Þ

To calculate the Casimir energy-momentum tensorgenerated by this nonstatic spacetime, one can proceedin a similar fashion to the static case, representing theL → ∞ limit of (74) as an infinite throat,

ds2 ¼ −dt2 þ dz2 þ a2ðdθ2 þ sin2θdϕ2Þ; ð77Þ

with a time-dependent radius a. As this spacetime isindependent of L, the Casimir energy-momentum mustbe independent of L also, and so we can perform thefollowing Taylor expansion in f _a; a; a:::;…g about the staticcase:

Trenμν ¼ Tren

μν ½a; _a; a; a:::;…�;¼ Tren

μν ½a; 0; 0; 0;…�× ð1þOð _aÞ þOðaaÞ þOða:::a2Þ þ � � �Þ; ð78Þ

where the factors of a in each term follow from dimensionalanalysis. Thus, if the expansion/collapse of the wormhole issufficiently slow, that is

j _aj; jaaj; ja:::a2j;… ≪ 1; ð79Þthen we can neglect the extra terms and use the result (59)that was derived for the static case. Once we havedetermined the behavior of aðtÞ, we will be able to confirmthe validity of (79); for now, let us take it as given, andneglect the time-derivative terms in Eq. (78). Hence Tren

μν

is simply given by Eq. (59) with the radius a now timedependent; let us write this as

Trenμ ν ¼ lnða=a0Þ

1440π2a4diagð−1; 1; β − 1; β − 1Þ; ð80Þ

where

β≡ 1

2 lnða=a0Þ: ð81Þ

Let us also introduce some ordinary matter to the worm-hole, with the following energy-momentum tensor:

κTordμ ν ¼

�κ lnða=a0Þ1440π2a4

−2

La

�diagð1;−1;1−β;1−βÞ

þ 1

a2diag

�1;−1;

aLþ aað2β−3Þ; a

Lþ aað2β−3Þ

�

þ _a2

a2diagð1;−1;0;0Þ: ð82Þ

The precise form of this tensor has been chosen to simplifyour analysis; for the purposes of this discussion, we needonly check that it obeys all the energy conditions, but let uspostpone this task until we have found the form of aðtÞ.12Substituting (76), (80) and (82) into the Einstein field

equations (71) and simplifying, we arrive at the following:

2ð1þ LaÞLa

diagð0; 1; β − 1; β − 1Þ ¼ 0; ð83Þ

the solution of which is of course

a ¼ −1=L: ð84Þ

Hence, the throat radius accelerates towards closure at aconstant rate, and fixing t ¼ 0 to be the time at which aachieves its maximal value amax we conclude that

12One should also check that the tensor obeys the conservationlaw ∇μTord

μν ¼ 0, where ∇μ is the covariant derivative of thetime-dependent throat metric (77). Fortunately, this follows fromthe field equations (71) and the identities ∇μTren

μν ¼ 0 and∇μGμν ¼ 0, which one can check hold true without restrictionon aðtÞ.


024019-11

aðtÞ ¼ amax −t2

2L: ð85Þ

We are now in a position to confirm the validity of ourearlier assumptions. Excluding the unphysical times

jtj >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Lamax

p; ð86Þ

for which a < 0, we see that

j _aj ¼ jtj=L ≤ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2amax=L

p≪ 1; ð87Þ

as L ≫ a. Furthermore, it follows from (84) that jaaj ¼a=L ≪ 1 and that all higher time derivatives vanish. Thisvalidates the conditions for slow collapse (79).To prove that Tord

μν obeys the energy conditions, we shallconsider each term of (82) individually. Referring to foot-note 5, it is clear that the first term on the right-hand side of(82) will obey all four energy conditions if and only if

0 ≤ β ≤ 1 andκ lnða=a0Þ1440π2a4

≥2

La; ð88Þ

or equivalently,

a ≥ a0e1=2 and a−3 lnða=a0Þ ≥360π

LðlpÞ2: ð89Þ

As can be seen from Fig. 2, both these inequalities can besatisfied if and only if

L ≥ a0

�a0lp

�2

720πe3=2; ð90Þ

in which case they are equivalent to

a0e1=2 ≤ a ≤ aþ; ð91Þ

where a ¼ aþ is the larger solution of

a−3 lnða=a0Þ ¼360π

LðlpÞ2; ð92Þ

and can thus be expressed in closed form using the LambertW function [25]:

aþ ¼�−LðlpÞ21080π

W−1

�−1080πa30LðlpÞ2

��1=3

: ð93Þ

Note that both conditions (90) and (91) are consistent withthe long-throat assumption L ≫ a, and a > a0 > lp.Furthermore, we have

aþ ¼�LðlpÞ2360π

lnðaþ=a0Þ�

1=3

≥�LðlpÞ2720π

�1=3

;

by virtue of (92) and then (91); hence the wormhole canbe much larger than the Planck scale: aþ ≫ lp as L → ∞.With (90) and (91) taken as given, the other two terms in

Tordμν are essentially trivial. With (84) the second term on the

right of (82) becomes

1

a2diag

�1;−1;

2aL

ð2 − βÞ; 2aL

ð2 − βÞ�; ð94Þ

recalling that a=L ≪ 1, and that 0 ≤ β ≤ 1 as a conse-quence of (91), we refer to footnote 5 to verify that (94)obeys all the energy conditions. The last term in (82) alsosatisfies the specifications of footnote 5, and so is similarlynonexotic.We have thus justified our earlier claims: (i) that the

collapse of the wormhole is so slow (79) that one canneglect the time-derivative terms from the renormalizedCasimir energy-momentum tensor Tren

μν , and (ii) that theEinstein equations (71) have been solved by adding onlyordinarymatter, Tord

μν . As long as L is sufficiently large (90)the throat of the wormhole will therefore collapse accordingto (85) with a constant acceleration a ¼ −1=L that can bearbitrarily small.This picture breaks down if ever the throat radius exits

the range (91), as Tordμν is then required to be exotic, at least

for the particular form (82) considered here. If we consideronly wormholes for which

amax ≤ aþ; ð95Þ

then we need not worry about exceeding the top limit of(91). However, as the wormhole collapses there willinevitably be a time t ¼ tclose when the bottom limit of(91) is reached,

aðtcloseÞ ¼ a0e1=2 ≡ amin; ð96Þ

after which the slow collapse can no longer be supported bythe Casimir effect and ordinary matter. At this point, thecollapse presumably becomes much more rapid, and withina very short time the throat radius falls to zero, splitting the

FIG. 2. Plot of y ¼ a−3 logða=a0Þ, indicating the regiona0e1=2 ≤ a ≤ aþ in which both inequalities (89) are satisfied.It is clear from the graph that aþ exists and is greater than a0e1=2

if and only if 360π=LðlpÞ2 ≤ 1=2a30e3=2; this condition can be

written as (90).


024019-12

wormhole (74) into two disconnected spacetimes.13

Treating this process as instantaneous in comparison toperiod of slow collapse, one can easily solve (85) and (96)to obtain the closure time for the throat:

tclose ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Lðamax − aminÞ

p: ð97Þ

As L has no upper bound, we conclude that a longwormhole can exist intact for an arbitrarily long time.Indeed, if we consider the optimal case, where the inequal-ity (95) is saturated, we can use the asymptotic behavior ofthe Lambert W function,

W−1ðxÞ ∼ lnð−xÞ as x → 0−; ð98Þ

to obtain

tclose ∼ ðL2lpÞ1=3�

1

135πln

�LðlpÞ21080πa30

��1=6

; ð99Þ

as L → ∞. Thus, under optimal conditions, the closuretime grows as L2=3ðlnLÞ1=6.Considering that the Casimir effect failed to fully

stabilize the wormhole, it is exciting to find that it none-theless allows the wormhole a remarkable longevity.Indeed, this unexpectedly long lifetime poses an interestingquestion: can the wormhole remain open long enough for apulse of light to travel through, thus permitting the trans-mission of information across the throat? To answer thisquestion definitively lies beyond the scope of this article;however, we shall attempt a preliminary analysis of trans-mission near the center of the wormhole.

C. Transmission of information

The possibility of light traversing the closing throat cannotbe dismissed a priori: the wormhole avoids the topologicalcensorship theorem [26] because the renormalized Casimirenergy-momentum tensor Tren

μν violates the averaged nullenergy condition, albeit in a manner that does not allow forabsolute stability. Likewise, one cannot immediately con-clude that because tclose ∼OðL2=3ðlnLÞ1=6Þ grows moreslowly than the throat crossing time OðLÞ, so must trans-mission fail in the large L limit. The flaw in this reasoninglies in the fact that tclose is only the time taken for thewormhole to close at z ¼ 0: we must also account for thepropagation of closure outwards from the center.To proceed we therefore need to consider the approx-

imately quadratic profile of the wormhole (74) near z ¼ 0:

A ¼ aþ z2

2LþOðz4=L3Þ: ð100Þ

Note that in Eq. (84) the acceleration a is independent ofthe throat radius a; thus we do not expect the accelerationto change as the throat widens out, at least to a firstapproximation for jzj ≪ L. We therefore model the col-lapse as uniform in close vicinity to the center, with thetime-dependent profile obtained simply by inserting (85)into (100):

A ¼ amax þz2

2L−

t2

2LþOðz4=L3Þ: ð101Þ

Discarding the negligible terms, it follows that the worm-hole throat is open everywhere for jtj < tclose, and thatwhen jtj ≥ tclose the wormhole closes (A ¼ amin) at z ¼�zcloseðtÞ, where

zclose ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 − 2Lðamax − aminÞ

q< jtj: ð102Þ

Thus, as illustrated in Fig. 3, the null geodesics

z ¼ �t; θ ¼ const; ϕ ¼ const; ð103Þ

lie entirely within the allowed region jzj > zclose, and hencesafely thread the collapsing throat. We conclude that, in thevicinity of the center of the throat, the wormhole collapsesslowly enough to let a pulse of light pass through.Extending this idea very slightly, we can consider the

timelike geodesics,

z ¼ �tð1 − ϵÞ; θ ¼ const; ϕ ¼ const; ð104Þ

with 0 < ϵ ≪ 1, and observe that these worldlines remainwithin the allowed region out to a distance of jzj ≈ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLðamax − aminÞ=ϵ

p. A massive particle, moving suffi-

ciently close to the speed of light (ϵ ∼Oða=LÞ), cantherefore safely traverse the regime of validity of thepresent analysis jzj ≪ L.These are intriguing results: although the wormhole is

not stable, it seems that it may be traversable for particlesmoving at (or near) the speed of light. Depending on how

FIG. 3. Spacetime plot (with angular directions suppressed)illustrating that the null geodesics given in (103) avoid the regionsjzj ≤ zclose where the throat has closed.

13More accurately, the throat radius approaches the Planckscale, wherein quantum effects allow for the change in spacetimetopology.


024019-13

the two ends of the wormhole connect to the externalspacetime, this could allow for communications that exceedthe speed of light (from the point of view of externalobservers) or even closed causal curves. However, it mustbe stressed that we do not currently know the behavior ofthe spacetime far from the center, so it remains to be seenwhether the light pulse can actually escape the wormholecompletely. To tackle this problem properly would require,first, an extension of the Casimir energy-momentum (59)beyond infinite throat approximation (77), and second, amore general model of collapse than the metric (74).We leave these developments for future work.

V. CONCLUSION

We have obtained the renormalized energy-momentumtensor (59) induced in the vacuum state of the masslessconformally coupled scalar field by the geometry and top-ology of a long wormhole throat (5). The tensor describeshighly exotic matter (70) and is sufficiently large that it could,in principle, stabilize a macroscopic wormhole (68).Unfortunately, the energy density vanishes along null vectorsrunning parallel to the throat, and this prevents the Casimireffect from stabilizing the wormhole in this particular case(Sec. IVA). Nonetheless, the exotic matter provides partialsupport to the wormhole, allowing it to collapse extremelyslowly (Sec. IV B) and remain open for an arbitrarily longtime (99). Moreover, near the center of the throat, the collapseis sufficiently slow that a pulse of light can be safelytransmitted (Sec. IVC), although it is currently unknownwhether this light pulse can escape the wormhole completely.These results tentatively suggest that a macroscopic

traversable wormhole might be sustained by its ownCasimir energy, providing a mechanism for faster-than-light communication and closed causal curves. To obtain amore definitive assessment of this possibility, the presentresearch suggests two main avenues of investigation forfuture work. First, we should explore other traversablewormhole metrics [e.g. (1) in the short-throat limit: L ≪ a]with the hope of finding a spacetime which avoids thethroat-parallel null-vector energy-density problem, and cantherefore be fully stabilized by its own Casimir energy.Second, we should seek to better understand the dynamicsof the long-throated wormhole, with a view to establishingwhether Casimir-supported collapse does indeed allowinformation to be transmitted from one end of the worm-hole to the other. If either approach succeeds, it would thenbe important to determine whether the solutions can survivethe introduction of symmetry-breaking perturbations.

ACKNOWLEDGMENTS

The author is supported by a research fellowship at JesusCollege, Cambridge, and wishes to also thank AnthonyLasenby and Mike Hobson for helpful advice.

APPENDIX A: EVALUATION OF I, J AND K

Let us begin by focusing on I with α ≥ 0 and α ≥ 0. Thedefinitions we will need are

I ≡Z

∞

0

du

�X∞l¼0

�lþ 1

2

�vf

�vλ

�−Z

∞

0

dllvf

�vλ

��;


q;


p: ðA1Þ

Using the (half-integer) Abel-Plana formula ([27], Sec. 2.2),

X∞l¼0

F

�lþ 1

2

�−Z

∞

0

dlFðlÞ ¼ iZ

∞

0

dtFð−itÞ − FðitÞ

e2πt þ 1;

ðA2Þthe bracketed quantity in (A1) becomes

X∞l¼0

�lþ 1

2

�vf

�vλ

�−Z

∞

0

dllvf

�vλ

�

¼ iZ

∞

0

dte2πt þ 1

�ð−itÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ ð−itÞ2 þ α

qf

− ðitÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ ðitÞ2 þ α

qf

�þZ

∞

0

dllffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ l2 þ α

pf

−Z

∞

0


pf; ðA3Þ

where, in each case, the argument of f is the adjacent squareroot, divided by λ. Care must be taken with the first twoterms in (A3) as the square roots must be evaluated byanalytically continuing

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ l2 þ α

pfrom l ≥ 0 to l ¼ �it,

t ≥ 0. Following this stipulation, one finds that

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ ðitÞ2 þ α

q¼

( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ ð−itÞ2 þ α

pt2 ≤ u2 þ α

−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ ð−itÞ2 þ α

pt2 > u2 þ α

:

ðA4Þ

Furthermore, let us assume that the cutoff function fðxÞ is ananalytic function of x2, so that it takes the same value in boththe first and second terms of (A3), regardless of the value oft. Hence,

X∞l¼0

�lþ 1

2

�vf

�vλ

�−Z

∞

0

dllvf

�vλ

�

¼Z ffiffiffiffiffiffiffiffi

u2þαp

0

dt2t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 − t2 þ α

p

e2πt þ 1f

þZ

∞

0


pf

−Z

∞

0


pf; ðA5Þ


024019-14

and consequently, I can be split into three pieces:

I ¼ Ið1Þα þ Ið2Þα − Ið2Þα ; ðA6Þ

where

Ið1Þα ≡Z

∞

0

duZ ffiffiffiffiffiffiffiffi

u2þαp

0

dt2t


p

e2πt þ 1f;

Ið2Þα ≡Z

∞

0

duZ

∞

0


pf: ðA7Þ

Concentrating on Ið1Þα to begin with, we first swap theorder of integration, and then substitute u2 ¼ x2 þ t2 − α,giving

Ið1Þα ¼�Z ffiffi

αp

0

dtZ

∞

0

duþZ

∞ffiffiα

p dtZ

∞ffiffiffiffiffiffiffit2−α

p du

�

×2t

e2πt þ 1


pf

¼�Z ffiffi

αp

0

dtZ

∞ffiffiffiffiffiffiffiα−t2

p dxþZ

∞ffiffiα

p dtZ

∞

0

dx

�

×2t

e2πt þ 1

x2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ t2 − α

p fðx=λÞ: ðA8Þ

Notice that the assumption α ≥ 0was necessary at this step.At this point, we shall take f to be a sharp cutoff

[i.e. fðxÞ≡Hð1 − x2Þ, H being the Heaviside stepfunction]14 and perform the x integration using hyperbolicsubstitutions:

Ið1Þα ¼Z ffiffi

αp

0

dt2t

e2πt þ 1

Zλffiffiffiffiffiffiffiα−t2

p dxx2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ t2 − αp

þZ

∞ffiffiα

p dt2t

e2πt þ 1

Zλ

0

dxx2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ t2 − αp

¼Z ffiffi

αp

0

dt2tðα − t2Þe2πt þ 1

Zarcoshðλ=

ffiffiffiffiffiffiffiα−t2

pÞ

0

dycosh2y

þZ

∞ffiffiα

p dt2tðt2 − αÞe2πt þ 1

Zarsinhðλ=

ffiffiffiffiffiffiffit2−α

pÞ

0

dysinh2y

¼Z

∞

0

dttðα − t2Þe2πt þ 1

�ln ðλþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 þ t2 − α

pÞ

−1

2ln jt2 − αj þ λ

α − t2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 þ t2 − α

p �: ðA9Þ

We are now in a position to expand the integrand as aTaylor series in λ−1:

Ið1Þα ¼Z ffiffiffiffiffiffiffiffi

λ2þαp

0

dttðα − t2Þe2πt þ 1

�ln ð2λþOðλ−1ÞÞ

−1

2ln jt2 − αj þ λ2

α − t2−1

2þOðλ−2Þ

�þ P; ðA10Þ

where

P≡Z

∞ffiffiffiffiffiffiffiffiλ2þα

p dttðα − t2Þe2πt þ 1

�ln�λþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 þ t2 − α

p �

−1

2ln jt2 − αj þ λ

α − t2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 þ t2 − α

p �ðA11Þ

is the part of the integral where λ2 < jt2 − αj and so theexpansion cannot be performed. Fortunately, P is expo-nentially suppressed as λ → ∞, and so can safely beneglected:

P ¼ O

�Z∞

λdtt3e−2πt ln t

�¼ Oðe−2πλλ3 ln λÞ: ðA12Þ

Hence, Eq. (A10) becomes

Ið1Þα ¼Z

∞

0

dtt

e2πt þ 1

�λ2 þ ðα − t2Þ lnð2λÞ

þ t2 − α

2ð1þ ln jt2 − αjÞ

�−QþOðλ−2Þ; ðA13Þ

where

Q≡Z

∞ffiffiffiffiffiffiffiffiλ2þα

p dtt

e2πt þ 1

�λ2 þ ðα − t2Þ lnð2λÞ

þ t2 − α

2ð1þ ln jt2 − αjÞ

�¼ Oðe−2πλλ3 ln λÞ ðA14Þ

is also negligible.Using the standard results,Z

∞

0

dtt

e2πtþ1¼ 1

48;

Z∞

0

dtt3

e2πtþ1¼ 7

1920; ðA15Þ

and defining

XðαÞ≡ 1

2

Z∞

0

dttðt2 − αÞ ln jt2 − αj

e2πt þ 1; ðA16Þ

we conclude that

14We have written Hð1 − x2Þ rather than Hð1 − xÞ to beconsistent with our previous specification that fðxÞ be ananalytic function of x2. Of course, the Heaviside step func-tion is not analytic, but it can be thought of as the limit ofan analytic sigmoid function as the width of its step is takento zero.


024019-15

Ið1Þα ¼ 1

48

�λ2 þ

�α −

7

40

��lnð2λÞ − 1

2

��þ XðαÞ

þOðλ−2Þ: ðA17Þ

All that remains is to calculate the integral Ið2Þα thatappears in Eq. (A6). Swapping the order of integration,substituting u2 ¼ x2 − l2 − α, and taking f to be a sharpcutoff, we have

Ið2Þα ¼Z

∞

0

dlZ

∞

0

dulffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ l2 þ α

pf

¼Z

∞

0

dlZ

∞ffiffiffiffiffiffiffil2þα

p dxlx2fðx=λÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 − l2 − α

p

¼Z ffiffiffiffiffiffiffiffi

λ2−αp

0

dlZ

λffiffiffiffiffiffiffil2þα

p dxlx2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 − l2 − αp : ðA18Þ

Swapping the order of integration once more,

Ið2Þα ¼Z

λffiffiα

p dxx2Z ffiffiffiffiffiffiffiffi

x2−αp

0

dllffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 − l2 − αp

¼Z

λffiffiα

p dxx2ffiffiffiffiffiffiffiffiffiffiffiffiffix2 − α

p

¼ −α2

8

�ln ðλþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 − α

pÞ − 1

2ln α

�

þ λ

8ð2λ2 − αÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 − α

p: ðA19Þ

Thus, as λ → ∞,

Ið2Þα ¼ λ4

4−α

4λ2 −

α2

8lnð2λÞ þ α2

32ð2 ln αþ 1Þ þOðλ−2Þ;

ðA20Þ

and

Ið2Þα − Ið2Þα ¼ α − α

4λ2 þ α2 − α2

8lnð2λÞ

þ α2

32ð2 ln αþ 1Þ − α2

32ð2 ln αþ 1Þ

þOðλ−2Þ: ðA21Þ

We now have all the results we need. Inserting (A17) and(A21) into Eq. (A6), we arrive at

I ¼ 1þ 12ðα − αÞ48

λ2 þ 1

48

�αþ 6ðα2 − α2Þ − 7

40

�lnð2λÞ

−1

96

�α −

7

40

�þ 1

16ðα2 ln α − α2 ln αÞ

þ 1

32ðα2 − α2Þ þ XðαÞ þOðλ−2Þ; ðA22Þ

which, according to our original assumption, is valid forα ≥ 0 and α ≥ 0. To obtain the result for α ≤ 0, we cananalytically continue the above expression, moving α tonegative values while avoiding the branch point α ¼ 0 inthe complex plane. The only subtlety to this step is that theln α term in (A22) will produce�iπ in the process, the signdepending on whether one moves clockwise or anticlock-wise. Clearly, the same can be said of α. Fortunately, we areonly interested in the real part of I, so this ambiguity is ofno concern:

ℜfIg¼ 1þ12ðα−αÞ48

λ2þ 1

48

�αþ6ðα2−α2Þ− 7

40

�lnð2λÞ

−1

96

�α−

7

40

�þ 1

16ðα2 ln jαj− α2 ln jαjÞ

þ 1

32ðα2− α2ÞþXðαÞþOðλ−2Þ; ðA23Þ

valid for all α; α ∈ R.The calculations for J and K proceed in exactly the

same fashion, so it serves no purpose to repeat them here.The results (valid for all α; α ∈ R) are as follows:

ℜfJg¼ 1þ12ðα−αÞ48

λ2−1

48

�αþ6ðα2−α2Þ− 7

40

�lnð2λÞ

−1

96

�α−

7

40

�−

1

16ðα2 ln jαj− α2 ln jαjÞ

þ 3

32ðα2− α2Þ−XðαÞþOðλ−2Þ; ðA24Þ

and

ℜfKg ¼ 1þ 12ðα − αÞ24

lnð2λÞ þ 1

4ðα ln jαj − α ln jαjÞ

þ 1

4ðα − αÞ − 2YðαÞ þOðλ−2Þ; ðA25Þ

where

YðαÞ≡ 1

2

Z∞

0

dtt ln jt2 − αje2πt þ 1

: ðA26Þ

APPENDIX B: ENERGY INEQUALITIES

Here we derive some key inequalities obeyed by therenormalized Casimir energy-momentum tensor (59) of thelong wormhole throat (5). Considering that we are inter-ested in the creation of exotic matter, let us assume

a > a0; ðB1Þ

so that the weak and dominant energy conditions areimmediately violated by virtue of Tren

00 < 0. We can thenwrite


024019-16

Trenμ ν ∝ diagð−1; 1; β − 1; β − 1Þ; ðB2Þ

where we have introduced

β≡ 1

2 lnða=a0Þ; ðB3Þ

and use the symbol ∝ to indicate proportionality by meansof a positive constant:

Pμν ∝ Qμν ⇒ ∃s > 0 s:t: Pμν ¼ sQμν: ðB4Þ

The various energy conditions concern the energydensity and energy flux defined by a four-velocity vμ thatis either timelike or null; without loss of generality, let ustake this four-velocity to be

vμ ∝ ð1; vz; v⊥; 0Þ; v2z þ v2⊥ ≤ 1: ðB5Þ

The energy density vμTrenμν vν and energy current Tren

μν vν thenobey

vμTrenμν vν ∝ −1þ v2z þ v2⊥ðβ − 1Þ ≤ v2⊥ðβ − 2Þ;

ðTrenμν vνÞ2 ∝ −1þ v2z þ v2⊥ðβ − 1Þ2 ≤ v2⊥ðβ − 2Þβ; ðB6Þ

with equality if and only if the four-velocity is null. Thus0 < β < 2 ensures that the energy flux is always causal(Tren

μν vν is never spacelike) and every four-velocity defines anegative energy density, except for null vectors directlyparallel to the throat, which have zero energy density.That is,

0 < β < 2 ⇒ vμTrenμν vν ≤ 0; ðTren

μν vνÞ2 ≤ 0;

ðB7Þ

with equality if and only if vz ¼ �1. Note that we alreadyhad β > 0 as a consequence of the assumption (B1), andthat the full restriction 0 < β < 2 is equivalent to

a > a0e1=4: ðB8Þ

In contrast, β > 2 preserves the null energy condition

β > 2; vμvμ ¼ 0 ⇒ vμTrenμν vν ≥ 0; ðB9Þ

but allows for noncausal energy flux:

β > 2 ⇒ ∃vμ s: t: ðTrenμν vνÞ2 > 0: ðB10Þ

If β ¼ 2, then Trenμν ∝ gμν resembles a cosmological constant

term, trivially obeying the null energy condition,

β ¼ 2; vμvμ ¼ 0 ⇒ vμTrenμν vν ¼ 0; ðB11Þ

with causal energy flux:

β ¼ 2 ⇒ ðTrenμν vνÞ2 ≤ 0: ðB12Þ

Finally, to assess the strong energy condition we con-struct the trace-reverse energy-momentum tensor,

Trenμ ν ≡ Tren

μ ν − ημνTren=2 ∝ diagðβ − 1; 1 − β;−1;−1Þ;ðB13Þ

and observe that

vμTrenμν vν ∝ ðβ − 1Þ þ ð1 − βÞv2z − v2⊥ ≥ ðβ − 2Þð1 − v2zÞ;

ðB14Þ

so that if β ≥ 2 the strong energy condition is obeyed. If0 < β < 2 then we can consider a timelike four-velocitywith vz ¼ 0 and v⊥ ¼ 1 − ϵð2 − βÞ, where ϵ in an arbi-trarily small positive number; this gives

vμTrenμν vν ∝ ðβ − 1Þ − ð1 − ϵð2 − βÞÞ ¼ ðβ − 2Þð1 − ϵÞ

∝ ðβ − 2Þ < 0; ðB15Þin violation of the strong energy condition.In aggregate, these results demonstrate that the renor-

malized Casimir energy-momentum tensor (59) violates allfour energy conditions (null, weak, dominant and strong) ifand only if (B8) is obeyed. Under this restriction, theinequalities (B7) hold true, and are saturated if and onlyif vμ ∝ ð1;�1; 0; 0Þ.

[1] C. W. Misner and J. A. Wheeler, Ann. Phys. (Paris) 2, 525(1957).

[2] S. Coleman, Nucl. Phys. B310, 643 (1988).[3] I. Klebanov, L. Susskind, and T. Banks, Nucl. Phys. B317,

665 (1989).[4] M. Van Raamsdonk, Gen. Relativ. Gravit. 42, 2323

(2010).

[5] J. Maldacena and L. Susskind, Fortschr. Phys. 61, 781(2013).

[6] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, J. HighEnergy Phys. 02 (2013) 062.

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

[8] M. Visser, Phys. Rev. D 41, 1116 (1990).


024019-17

http://dx.doi.org/10.1016/0003-4916(57)90049-0

http://dx.doi.org/10.1016/0003-4916(57)90049-0

http://dx.doi.org/10.1016/0550-3213(88)90097-1

http://dx.doi.org/10.1016/0550-3213(89)90538-5

http://dx.doi.org/10.1016/0550-3213(89)90538-5

http://dx.doi.org/10.1007/s10714-010-1034-0

http://dx.doi.org/10.1007/s10714-010-1034-0

http://dx.doi.org/10.1002/prop.201300020

http://dx.doi.org/10.1002/prop.201300020

http://dx.doi.org/10.1007/JHEP02(2013)062

http://dx.doi.org/10.1007/JHEP02(2013)062

http://dx.doi.org/10.1103/PhysRevLett.61.1446



[9] J. Friedman, M. S. Morris, I. D. Novikov, F. Echeverria,G. Klinkhammer, K. S. Thorne, and U. Yurtsever, Phys.Rev. D 42, 1915 (1990).

[10] H. Casimir, Proc. R. Neth. Acad. Arts Sci. 51, 793 (1948).[11] A. R. Khabibullin, N. R. Khusnutdinov, and S. V. Sushkov,

Classical Quantum Gravity 23, 627 (2006).[12] D. Hochberg, A. Popov, and S. V. Sushkov, Phys. Rev. Lett.

78, 2050 (1997).[13] R. M. Wald, General Relativity (University of Chicago

Press, Chicago, IL, 1984).[14] P. R. Anderson, W. A. Hiscock, and D. A. Samuel, Phys.

Rev. D 51, 4337 (1995).[15] G. Plunien, B. Müller, and W. Greiner, Phys. Rep. 134, 87

(1986).[16] C. G. Callan, Jr., S. Coleman, and R. Jackiw, Ann. Phys.

(N.Y.) 59, 42 (1970).[17] W. Pauli and F. Villars, Rev. Mod. Phys. 21, 434 (1949).[18] C. G. Bollini and J. J. Giambiagi, Il Nuovo Cimento B 12,

20 (1972).

[19] G. ’t Hooft and M. Veltman, Nucl. Phys. B44, 189(1972).

[20] S. L. Adler, J. Lieberman, and Y. J. Ng, Ann. Phys. (N.Y.)106, 279 (1977).

[21] J. S. Dowker and R. Critchley, Phys. Rev. D 13, 3224(1976).

[22] S. Hawking, Commun. Math. Phys. 55, 133 (1977).[23] N. D. Birrell and P. C. W. Davies, Quantum Fields in

Curved Space (Cambridge University Press, Cambridge,1982), p. 167.

[24] R. Utiyama and B. S. DeWitt, J. Math. Phys. (N.Y.) 3, 608(1962).

[25] R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth,Adv. Comput. Math. 5, 329 (1996).

[26] J. L. Friedman, K. Schleich, and D. M.Witt, Phys. Rev. Lett.71, 1486 (1993).

[27] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M.Mostepanenko, Advances in the Casimir Effect (OxfordUniversity Press, New York, 2009).


024019-18



http://dx.doi.org/10.1088/0264-9381/23/3/006





http://dx.doi.org/10.1016/0370-1573(86)90020-7

http://dx.doi.org/10.1016/0370-1573(86)90020-7

http://dx.doi.org/10.1016/0003-4916(70)90394-5

http://dx.doi.org/10.1016/0003-4916(70)90394-5

http://dx.doi.org/10.1103/RevModPhys.21.434

http://dx.doi.org/10.1007/BF02895558

http://dx.doi.org/10.1007/BF02895558

http://dx.doi.org/10.1016/0550-3213(72)90279-9

http://dx.doi.org/10.1016/0550-3213(72)90279-9

http://dx.doi.org/10.1016/0003-4916(77)90313-X

http://dx.doi.org/10.1016/0003-4916(77)90313-X



http://dx.doi.org/10.1007/BF01626516

http://dx.doi.org/10.1063/1.1724264

http://dx.doi.org/10.1063/1.1724264

http://dx.doi.org/10.1007/BF02124750



casimir energy of a long wormhole throat

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