sean a. hayward et al- dilatonic wormholes: construction, operation, maintenance and collapse to...

8/3/2019 Sean A. Hayward et al- Dilatonic wormholes: construction, operation, maintenance and collapse to black holes

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arXiv:g

r-qc/0110080v118Oct2001

Dilatonic wormholes: construction, operation, maintenance and collapse to black holes

Sean A. Hayward, Sung-Won Kim, and Hyunjoo Lee

Department of Science Education, Ewha Womans University, Seoul 120-750, Korea

(Dated: February 7, 2008)

The CGHS two-dimensional dilaton gravity model is generalized to include a ghost Klein-Gordonfield, i.e. with negative gravitational coupling. This exotic radiation supports the existence of statictraversible wormhole solutions, analogous to Morris-Thorne wormholes. Since the field equations

are explicitly integrable, concrete examples can be given of various dynamic wormhole processes,as follows. (i) Static wormholes are constructed by irradiating an initially static black hole withthe ghost field. (ii) The operation of a wormhole to transport matter or radiation between the twouniverses is described, including the back-reaction on the wormhole, which is found to exhibit atype of neutral stability. (iii) It is shown how to maintain an operating wormhole in a static state,or return it to its original state, by turning up the ghost field. (iv) If the ghost field is turned off,either instantaneously or gradually, the wormhole collapses into a black hole.

PACS numbers: 04.70.Bw, 04.20.-q, 04.50.+h

I. INTRODUCTION

The theoretical existence of space-time wormholes has

intrigued experts and public alike. Wheeler [1] specu-lated that quantum fluctuations in space-time topologycan occur, so that the smooth space-time of classical Ein-stein gravity becomes, at the Planck scale, a continualfoam of short-lived interconnections. If such a worm-hole could be expanded to macroscopic size, it could pro-vide a short cut between otherwise distant regions, like abookworm tunnelling between different pages of an atlas.Certainly all the standard stationary black-hole solutionshave wormhole (RS2) spatial topology, describing twoAlice-through-the-looking-glass universes joined by thefamous Einstein-Rosen bridge [2], or wormhole throat.In such cases, the two universes are not causally con-

nected, so that it is impossible to travel between them,any attempt leading instead into the black hole [ 3]. How-ever, this is only just so; a light ray can be sent along thepast boundary of one universe to the future boundary ofthe other, just escaping the black hole. Thus it is not dif-ficult to imagine a small modification which would yielda traversible wormhole.

Morris & Thorne [4] popularized traversible wormholesas a respectable theoretical possibility, studying static,spherically symmetric cases in detail. The spatial topol-ogy is the same as in the black-hole cases, but the throator minimal surface is preserved in time, so that observerscan pass through it in either direction, travelling freely

between the two universes. According to the Einsteinequation, such a geometry requires matter which doesnot satisfy a classical positive-energy property, the weakenergy condition. However, this condition can be vio-lated quantum-mechanically, e.g. in the Casimir effect,

Electronic address: [email protected] address: [email protected] address: [email protected]

or in alternative gravity theories. Such negative-energymatter was dubbed exotic matter by Morris & Thorne.This provoked renewed interest in traversible wormholes,

as reviewed by Visser [5]. The possible astrophysical ex-istence of wormholes has been taken seriously enough forsearches of observational data [6].

The Morris-Thorne framework begs development in atleast two important respects, quite apart from generaliz-ing beyond static, spherically symmetric cases. Firstly,the exotic matter was not modelled, but simply assumedto exist in exactly the configuration needed to supportthe wormhole. Secondly, the back-reaction of the trans-ported matter on the wormhole was ignored. If the worm-hole turns out to be unstable to such back-reaction, itwould be useless for actual transport. Many physicistsinstinctive reaction is that negative energy, unbounded

below, will lead to instability. Another practical ques-tion is: if the negative-energy source fails, i.e. the exoticmatter is not maintained, what happens to the worm-hole? Again, a pessimistic reaction is that the negativeenergy densities are likely to create naked singularities.An alternative prediction was that the wormhole wouldcollapse to a black hole [7]. The same reference intro-duced a framework for studying dynamic wormholes, inwhich both wormholes and black holes are locally definedby the same geometrical objects, trapping horizons, withone key difference: black holes have achronal horizonsand wormholes have temporal horizons, so that they arerespectively one-way and two-way traversible. This alsoindicated that a wormhole could be constructed from a

black hole using exotic matter.

To find definite answers to the above questions, it isnecessary to specify the exotic matter model. Variousstudies have been based on alternative gravity theoriesor semi-classical quantum field theory, e.g. [8, 9]. How-ever, the difficulty of solving the field equations tendsto obscure issues of principle. Here we propose a toymodel, intended to describe the essential dynamics of awormhole, but explicitly integrable, so that concrete an-swers to the above questions are easily found. A simi-
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lar approach in the context of black-hole evaporation in-volved the CGHS two-dimensional dilaton gravity model[10], which was expected to share similar features withspherically symmetric black-hole evaporation by Hawk-ing radiation. Certainly classical black-hole dynamics isqualitatively similar, for instance in regard to cosmic cen-sorship, but such properties are much easier to prove inthe toy model [11] than in the corresponding realistic

model, the spherically symmetric Einstein-Klein-Gordonsystem [12]. Here we propose a simple generalization ofthe CGHS model to include a ghost Klein-Gordon field,i.e. with the gravitational coupling taking the oppositesign to normal. This is a specific model for the exoticmatter, or more accurately exotic radiation, as we takethe massless field for simplicity.

The article is organised as follows. II describes themodel and how its general solution can be constructedfrom initial data. III reviews the static black-hole so-lution and introduces the static wormhole solution, eachof which depend on one parameter, mass or horizon ra-dius. IV describes what happens to an initially staticwormhole when the ghost field supporting it is turnedoff and allowed to disperse. V shows how to constructa static wormhole by irradiating an initially static blackhole with the ghost field. VI describes the operation ofa wormhole for transport or signalling, using a normalKlein-Gordon field to model the matter or radiation, in-cluding the back-reaction of the field on the wormhole.VII shows how to maintain the operating wormhole ina static state, or return it to its original state. VIIIconcludes.

II. DILATON GRAVITY

The CGHS two-dimensional dilaton gravity of Callanet al. [10] is generalized by the action

S

e2

R + 4()2 + 42 1

2(f)2 + 1

2(g)2

(1)

where S is a 2-manifold, , R and are the area form,Ricci scalar and covariant derivative of a Lorentz 2-metricon S, represents a negative cosmological constant, is ascalar dilaton field, f is a scalar field representing matterand g is a ghost scalar field with negative gravitationalcoupling. The last term is added to the CGHS actionin order that g provides the negative energy densities

needed to maintain a wormhole [4, 5, 7, 13, 14, 15, 16].By choosing future-pointing null coordinates (x+, x),

the line element may be written as

ds2 = 2e2dx+dx (2)where the conventional factor of 2 differs from that ofearlier references [10, 11]. One component of the fieldequations is

+ = + (3)

FIG. 1: Initial data, taking x0 = 0.

where = /x, so the coordinate freedom x x(x) can be used to take

= . (4)

The remaining coordinate freedom is just

x ebx + c (5)

where the constants c fix the origin and the constantb refers to relative rescalings of x. It is convenient totransform the dilaton field to

r = 2e2. (6)

Then the remaining field equations are the evolutionequations

+f = 0 (7)+g = 0 (8)

+r = 42 (9)

and the constraints

r = (g)2 (f)2. (10)

The evolution equations (7-9) have the general solutions

f(x+, x) = f+(x+) + f(x

) (11)

g(x+, x) = g+(x+) + g(x

) (12)

r(x+, x) = r+(x+) + r(x

)

42x+x. (13)

The constraints (10) are preserved by the evolution equa-tions in the directions, and so may be reduced to

r = G2 F2 (14)

where the null derivatives

F = f (15)

G = g (16)


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are convenient variables. The constraints are manifestlyintegrable for r given initial data

(f+, g+) on x = x0 (17)

(f, g) on x+ = x+0 (18)

(r, +r, r) at x+ = x+0 , x

= x0 (19)

for constants x0 (Fig. 1). Then the general procedure is

to specify this initial data, integrate the constraints (14)for r, then the general solution follows as (11-13).

Finally, we note from the vacuum constraints (10) thatr is analogous to the areal radius in spherically symmet-ric Einstein gravity [11, 17], correctly reproducing theexpansion of a light wave in flat space-time, r = 0.

III. STATIC BLACK-HOLE AND WORMHOLE

SOLUTIONS

In vacuum, f = g = 0, the general solution to the fieldequations is [10]

r = 2m 42x+x (20)where the origin has been fixed using (5). The constant mmay be interpreted as the mass of the space-time, whoseglobal structure has been described previously [11]. Form > 0 there is a static black hole, analogous to theSchwarzschild black hole (Fig. 2(a)). The case m = 0is flat and the case m < 0 contains an eternal naked sin-gularity, analogous to the negative-mass Schwarzschildsolution. Henceforth we take m > 0. Note that the orig-inal reference [10] defined mass as m, in which case rwould be analogous to radius, but for > 0 this makesno essential difference in the following.

Next, we consider the case that f = 0 but g = 0,finding the solution

r = a + 22(x+ x)2 (21)g = 2(x+ x) (22)

where the origin has again been fixed. A coordinatetransformation x = (t z)/2 shows that the solutionis static:

ds2 =2(dz2 dt2)

a + 42z2. (23)

If a > 0, this represents a traversible wormhole with

analogous global structure to a Morris-Thorne wormhole,with a throat r = a at z = 0, joining two regions withr > a, a z > 0 universe and a reflected z < 0 universe(Fig. 2(b)). Ifa < 0, there is an eternal naked singularityr = 0, while if a = 0, the space-time has constant neg-ative curvature, as can be seen by calculating the Ricciscalar [11]

R = r1+rr +r. (24)Henceforth we take a > 0.

0

0

0

0

FIG. 2: Penrose conformal diagrams of (a) the static blackhole, (b) the static wormhole. Both space-times are di-vided into two universes (unshaded regions), but observerscan travel freely between them via the wormhole, whereas theblack hole (upper shaded region) swallows any such attempt.

In summary, the model naturally contains both staticblack holes and static traversible wormholes. Before pro-ceeding, we note an important feature of both cases, thetrapping horizons, defined by r r = 0, or equivalently+r = 0 or r = 0 [7, 11, 17, 18]. In the black hole,they coincide with the event horizons r = 2m at x = 0and x+ = 0 respectively. In the wormhole, there is adouble trapping horizon, +r = r = 0, at the throatr = a. This illustrates how trapping horizons of differenttype may be used to locally define both black holes andwormholes [7]. Locating the trapping horizons, or equiv-alently the trapped regions where r r < 0, is a keyfeature of the following analysis of dynamic situations.

IV. WORMHOLE COLLAPSE

One can now study what happens to a static worm-hole if its negative-energy source fails. We consider firstthat the supporting ghost field g is switched off suddenlyfrom both sides of the wormhole, then that g is insteadgradually reduced to zero.


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0

0 0

FIG. 3: (a) The ghost field g is switched off suddenly fromboth sides of the wormhole. (b) Conformal diagram: when thewormholes negative-energy source fails suddenly at x = 0,it immediately collapses into a black hole.

A. Sudden collapse

We set the initial data so that there is a static worm-hole, with g then switched off suddenly from both sidesof the wormhole (Fig. 3(a)):

G =

2 , x < 00 , x 0. (25)

Taking f = 0, the constraints (14) are

r =

42 , x < 00 , x 0. (26)

Integrating twice, we obtain

r =

22(x)2 + a

2, x < 0

a2

, x 0 (27)

where the constants of integration are determined by con-tinuity at x = 0. The resulting solution from (13) is

r =

a + 22(x+ x)2 , x+ < 0 and x < 0a + 22x+(x+ 2x) , x+ < 0 and x 0a 22x(2x+ x) , x+ 0 and x < 0a 42x+x , x+ 0 and x 0.

(28)

One may recognize the solution in the final region as astatic black hole (20) with mass m = a/2. The doubletrapping horizon of the static wormhole bifurcates intothe event horizons of the black hole, all with r = a = 2m(Fig. 3(b)). Thus the wormhole immediately collapsesinto a black hole.

The Schwarzschild-like relationship between mass andthroat radius is no accident; there is a definition of active

gravitational mass-energy [11]

E =r

2

1 r r

42r2

(29)

which evaluates as r/2 on any trapping horizon. Thusa wormhole with a throat r = a has an effective massa/2. Here and elsewhere, it is useful to recall the anal-ogy with spherically symmetric Einstein gravity, where rcorresponds to areal radius and there is a similar defini-tion of active gravitational mass-energy [17].

B. Gradual collapse

Again starting with a static wormhole, we now reducethe ghost field gradually to zero in the simplest way, lin-early (Fig. 4(a)):

G =

2 , x < 0x 2 , 0 x < x00 , x0 x

(30)

where is a constant and x0 = 2/. Here 0recovers the static wormhole and recovers suddencollapse. By similar calculations, again taking f = 0,

r =

42 , x < 0

2x2 4x + 42 , 0 x < x0

0 , x0 x(31)

integrate to

r =

2

2

x2

+

a

2 , x

< 0212

x4 2

3x

3+ 22x

2+ a

2, 0 x < x0

83

3x 44

32+ a

2, x0 x.

(32)

and the solution


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and

r =

2m 42x+x , x+ < x0 and x < x02m 42x+x + 222 (x+ x0)2 + (x x0)2 , x0 x+ < x1 and x0 x < x122(x+ x)2 + 2m 42x0x1 , x1 x+ and x1 x

(38)

where we have omitted the less relevant regions. Onemay recognize the solution in the final region as a staticwormhole (21) with throat radius 2m 42x0x1. Thuswe require 22x0x1 < m. By choice of the parameters(x0, x1), we have constructed a static wormhole with anythroat radius less than 2m, the radius of the originalblack hole. The trapping horizons r = 0 are locatedat

x =

0 , x < x02(x

x0) , x0

x < x1

x , x1 x(39)

which are straight line segments. Thus the ghost radia-tion causes the trapping horizons of the initial black holeto shrink towards each other, eventually merging to formthe throat of the final static wormhole (Fig. 5(b)). Thetrapped region composing the black hole simply evapo-rates. This classically unexpected behaviour is, of course,due to the negative energy densities. As in the wormholecollapse case, the trapping horizons can be smoothed offby taking smoother profiles for G, but the details arenot particularly illuminating. In summary, static worm-holes have been constructed by irradiating a black hole

with ghost radiation.One can also regard this as analogous to black-holeevaporation, with the ghost radiation modelling the ingo-ing negative-energy Hawking radiation, suggesting thatthe final state of black hole evaporation might be a sta-tionary wormhole [7, 8, 9]. This would naturally resolvethe information-loss puzzle, as there is no singularity inwhich information is lost; everything that fell into theblack hole eventually re-emerges.

VI. WORMHOLE OPERATION

If a wormhole is actually used to transport a parcel orperson between the two universes, the transported mat-ter will affect the wormhole by changing the gravitationalfield. In principle this occurs even if the wormhole ismerely used for signalling. We will study the dynamicaleffects of such back-reaction by using the field f to modelthe matter or radiation. The use of Klein-Gordon radi-ation rather than more realistic matter is for simplicityonly; any source of mass-energy would have more or lesssimilar gravitational effects. In the current model, theconstraints (14) show explicitly that increasing F

2 has

0

0 0

FIG. 5: (a) Irradiating a vacuum black hole with the ghostfield g. (b) Conformal diagram: the initially static black hole,as in (Fig. 2(a)), becomes a dynamic wormhole, eventuallyreaching a static state as in (Fig. 2(b)). The black hole hasbeen converted into a traversible wormhole.

an equivalent gravitational effect to reducing G2, which

we have already shown can cause collapse to a black hole.Thus it is to be expected that too much transport woulddestroy the wormhole. A worse possibility is that thewormhole might be unstable to the slightest perturba-

tion and start to collapse immediately. We investigatethis below, giving the first concrete examples of worm-hole operation including back-reaction.

Again taking the simplest case, we consider a step pulseof positive-energy radiation (Fig. 6(a)):

F+ =

0 , x+ < 0 , 0 x+ < x00 , x0 x+

(40)

F = 0


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0

0

FIG. 6: (a) A step pulse of positive-energy radiation isbeamed through the wormhole. (b) Conformal diagram: thewormhole becomes non-static but, for a small-energy pulse,remains traversible for a long time.

with G = 2. The energy of the pulse may be definedas the change in the gravitational energy (29) due to thepulse, which evaluates at x = 0 as

=1

4(x0)

2. (41)

In the following, we assume that the energy of the pulseshould not be too large, corresponding to the anticipatedlimit on how much mass-energy can be sent through thewormhole without causing it to collapse into a black hole.Specifically we find < a/2 to avoid an r = 0 singularityin the middle region. Thus the pulse energy should beless than the effective mass of the wormhole. Insertingthe conversion factor c2/G from length to mass, a one-metre wormhole could transport over a hundred Earthmasses. This is hardly a practical limitation, once weestablish stability.

The constraints (14)

++r+ =

42 , x+ < 042 2 , 0 x+ < x042 , x0 x+

(42)

r = 42

integrate to

r+ =

22

x+2

+

a

2 , x+

< 022x+2 2

2x+

2+ a

2, 0 x+ < x0

22x+2

+ 2x01

2x0 x+

+ a

2, x0 x+

(43)

r = 22x

2+

a

2

and the solution is

r =

a + 22 (x+ x)2 , x+ < 0a + 22 (x+ x)2 2

2x+

2, 0 x+ < x0

a + 22 (x+ x)2 + 2x01

2x0 x+

, x0 x+.

(44)

The locations of the trapping horizons +r = 0 andr = 0 are given respectively by

x =

x+ , x+ < 0

x+ 242

x+ , 0 x+ < x0x+ 2

42x0 , x0 x+

(45)

x+ = x .

Thus the double trapping horizon of the initiallystatic wormhole bifurcates when the radiation arrives

(Fig. 6(b)). After the pulse has passed, the two trap-ping horizons run parallel in the x coordinates, forminga non-static traversible wormhole. If the pulse energy issmall, a, the wormhole persists in an almost staticstate for a long time, t x0a/. Nevertheless, even foran arbitrarily weak pulse, eventually a spatial r = 0 sin-gularity develops, similar to that of the static black hole.Observers close enough to the singularity can no longertraverse the wormhole, so it constitutes a black hole with


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two event horizons. Thus the static wormhole exhibits atype of neutral stability, neither strictly stable in that itdoes not return to its initial state, nor strictly unstable inthat there is no sudden runaway. Note that black holesare also neutrally stable in this sense; perturbing a blackhole by dropping positive-energy matter into it increasesthe area of the trapping horizon finitely, by the first andsecond laws of black-hole dynamics [11, 17, 18].

VII. WORMHOLE MAINTENANCE

Keeping an operating wormhole viable indefinitely, de-fying its natural fate as a black hole, requires additionalnegative energy to balance the transported matter. Thesimplest way to maintain the wormhole is just to set ini-tial data

G =

42 + F2 (46)

so that the source terms in the constraints (14) cancel

to the static wormhole values. Thus the wormhole re-mains static. The additional negative-energy radiationhas balanced the positive-energy radiation, leaving thegravitational field unchanged.

Alternatively, the wormhole may be maintained bybeaming in additional negative-energy radiation beforeor after the positive-energy pulse. We take the same steppulse in F+ (40) and precede it with a compensating pulsein the ghost field (Fig. 7(a)):

G+ =

2 , x+ < x04

2

+ 2

, x0 x+

< 02 , 0 x+ (47)G = 2 .

Again we require small pulse energy, < a/2, to avoid asingularity. The constraints (14) become

++r+ =

42 , x+ < x042 + 2 , x0 x+ < 042 2 , 0 x+ < x042 , x0 x+

(48)

r = 42

which integrate to

r+ =

22x+2

+ a2

, x+ < x022x+

2+ 2x+

1

2x+ + x0

12

2x02 +a2

, x0 x+ < 022x+

2 2x+ 12

x+ x0 1

22x02 +

a2

, 0 x+ < x022x+

2+ a

2, x0 x+

(49)

r = 2

2

x

2

+

a

2

and the solution follows as

r =

a + 22 (x+ x)2 , x+ < x0a + 22 (x+ x)2 + 1

22

x+

2+ 2x0x+ x02

, x0 x+ < 0

a + 22 (x+ x)2 12

2

x+2 2x0x+ + x02

, 0 x+ < x0

a + 22 (x+ x)2 , x0 x+.

(50)

The locations of the trapping horizons are

x =

x+

, x+

< x0x+ +

2

42(x+ + x0) , x0 x+ < 0

x+ 242

(x+ x0) , 0 x+ < x0x+ , x0 x+

(51)

x+ = x

which again are straight line segments. The double hori-zon bifurcates when the ghost pulse arrives, temporar-ily opening up a trapped region, but the two horizonssubsequently merge to form a static wormhole again

(Fig. 7(b)). This is not unexpected, since the energyof the ghost pulse, =

2x02/4, balances that of the

other pulse:

+ = 0. (52)

In fact, the final state is identical to the initial state inthis symmetric case. The wormhole can also be returnedto a different static state, with different throat radius, byless symmetric double pulses. The examples show thatthere is no practical problem of fine-tuning the ghost fieldto keep the wormhole static. An almost static wormholeis still traversible and can be adjusted at any time to


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0 0

0

0

FIG. 7: (a) The step pulse of positive-energy radiation isbalanced by a preceding pulse of negative-energy radiation.(b) Conformal diagram: the wormhole returns to its originalstatic state.

bring it closer to staticity, or to change its size. This isessentially due to the neutral stability.

VIII. CONCLUSION

Space-time wormholes remain in the realms of sciencefiction and theoretical physics. By the standards of eithergenre, they are not so far-fetched, differing from experi-

mentally established physics by only one step, the exis-tence of negative energy densities in sufficient concentra-tions. Here we have not addressed this issue but, assum-ing a positive answer, have investigated the behaviour ofthe resulting wormholes, evolving dynamically accordingto field equations. In particular, we have found detailedanswers to the following practical questions. (i) Howcan one construct a traversible wormhole? By bathing a

black hole in exotic radiation. (ii) Is an operating worm-hole stable under the back-reaction of the transportedmatter? In this case, neutrally stable. (iii) How can awormhole be maintained indefinitely for transport or sig-nalling? By balance of positive and negative energy. (iv)What happens if the negative-energy source fails? Thewormhole collapses into a black hole.

This was mostly predicted by a general theory of worm-hole dynamics [7], but here we have given concrete ex-amples, by virtue of the exact solubility of the field equa-tions of two-dimensional dilaton gravity. Despite the sup-posedly unphysical nature of a ghost Klein-Gordon field,fears about instability, runaway processes and naked sin-

gularities proved to be unfounded. More realistic situa-tions may differ in some respects, such as wormhole sta-bility, which may be affected by backscattering and willpresumably depend on the exotic matter model. How-ever, the same principles and methods should apply, suchas energy balance and tracking of trapping horizons. In-deed, apart from the inclusion of exotic matter or radia-tion, the methods are the same as those used to analyzeblack-hole dynamics [17, 18]. In particular, the explicitexamples of dynamic interconversion of black holes andwormholes should assuage objections that they are funda-mentally different objects. Rather, wormholes and blackholes have similar physics and a unified theory.

S.A. Hayward was supported by Korea Research Founda-tion grant KRF-2001-015-DP0095, S-W. Kim by KoreaResearch Foundation grant KRF-2000-041-D00128 andH. Lee by KOSEF grant R01-2000-00015.

[1] J A Wheeler, Ann. Phys. 2, 604 (1957).[2] A Einstein & N Rosen, Phys. Rev. 48, 73 (1935).[3] M D Kruskal, Phys. Rev. 116, 1743 (1960).[4] M S Morris & K S Thorne, Am. J. Phys. 56, 395 (1988).[5] M Visser, Lorentzian Wormholes: from Einstein to

Hawking (AIP Press 1995).[6] D F Torres, G E Romero & L A Anchordoqui, Phys. Rev.

D58, 123001 (1998).[7] S A Hayward, Int. J. Mod. Phys. D8, 373 (1999).[8] D Hochberg, A Popov & S V Sushkov, Phys. Rev. Lett.

78, 2050 (1997).[9] S-W Kim and H Lee, Phys. Lett. B458, 245 (1999).

[10] C Callan, S Giddings, J Harvey, and A Strominger, Phys.Rev. D 45, R1005 (1992).

[11] S A Hayward, Class. Quantum Grav. 10, 985 (1993).[12] D Christodoulou, Ann. Math. 149, 183 (1999).[13] D Hochberg & M Visser, Phys. Rev. D58, 044021 (1998).[14] D Hochberg & M Visser, Phys. Rev. Lett. 81, 746 (1998).[15] D Ida & S A Hayward, Phys. Lett. A260, 175 (1999).[16] S A Hayward, Wormhole dynamics (in preparation).[17] S A Hayward, Class. Quantum Grav. 15, 3147 (1998).[18] S A Hayward, Phys. Rev. D49, 6467 (1994).

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