Imaginary wormholes

NUCLEAR P H VS IC S B

Nuclear Physics B 278 (1992) 247—287 North-Holland

_________________

Imaginary wormholes J. Twamley

*

and D.N. Page

**

Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 211 Received 19 August 1991 Accepted for publication 2 December 1991

Using a Friedman—Robinson--Walker minisuperspace model with a minimally coupled homogeneous scalar field we search for, and discover, wormhole-type solutions, connect two 2~2which +~ For these asymptotically flat euclidean spaces, when (a) V = ~ and (b) V ~m potentials, stationary configurations satisfying the boundary conditions of asymptotic flatness necessitate that the scalar field be imaginary. Each solution found can be labeled by an asymptotic constant; however, in distinction from all previously found wormhole solutions, none possess a conserved charge. In the case of potential (a), all solutions found have negative actions, whereas for potential (b), there exist regions of the (in, A) parameter space for which the solutions have negative, zero, and positive action. Major ramifications implied by the discovery of these new wormholes solutions arc the following: Firstly, the existence of such solutions dispels a longstanding conjecture that euclidean wormholes must possess a conserved charge. Secondly, their existence also dispels a conjecture made by Halliwell and Hartle regarding the behaviour of the real part of the action for wormholes possessing complex geometries. As a result, the criterion proposed by 1-lalliwell and Hartle, for choosing thosc wormhole saddle-points of thc action which should be included into the complex contour of integration, must he augmented to include the effects of complex matter fields. Thirdly, these new wormholes may also seriously undermine current arguments concerning the resolution of the “largc-wormhole problem” of Fischler and Susskind in field theories that allow such wormholes to occur.

1. Introduction In this article we present new asymptotically flat euclidean wormhole solutions in the context of a minisuperspace model consisting of a euclidean FRW cosmology minimally coupled to an imaginary scalar field with a non-trivial self-interaction ~iotentia1. In distinction from all previously known wormhole solutions, the solutions presented here do not possess a conserved charge and may have positive, zero, or negative action. There exist three primary reasons why such solutions are of interest. Firstly, their existence dismisses the pre-supposed belief that euclidean *

**

Present address: Department of Physics and Mathematical Physics, University of Adelaide, GPO Box 498, Adelaide, South Australia 5001, Australia; E-mail jtwamleycà physics.adclaide.edu.au CIAR Cosmology Program; E-mail pagefypage.phys.ualberta.ca

0550-3213/92/$O5.O() © 1992

—

Elsevier Science Publishers By. All rights reserved

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wormholes must possess a conserved charge [1]. Secondly, they dispel a conjecture by Halliwell and Hartle stating that wormhole configurations with Re(~/~) > 0 possess positive euclidean actions and are thus suppressed in the euclidean path-integral formulation of quantum cosmology [2]. Thirdly, these new wormholes may seriously undermine current arguments attempting to resolve the “LargeWormhole Problem” of Fischler and Susskind [31.in what follows we elaborate on the above mentioned motivations. In sects. 2 and 3 we describe the appearance of wormholes in the two models studied. In sect. 4 we discuss the presence of the imaginary scalar field and the stability of the solutions. It has been a longstanding belief, both in the literature and without, that a euclidean wormhole possessing a non-zero throat size can only occur if there is present in the lagrangian, a cyclic quantity, the conservation of which “holds open” the wormhole throat [1]. The solutions described below do not possess a conserved charge. Instead the throat is “held open” by the nonlinearity of the dynamics of the models. A key ingredient in all previously studied wormhole solutions has been the appearance of such a conserved charge. However, (as pointed out by Polchinski [4]), the tractability of such models via a routhian reduction may have misled workers in this field to believe that all wormhole solutions would possess a conserved charge. The second motivation for this work concerns recent research on the “problem of the contour of integration” in the path-integral approach to quantum gravity [2,51.This problem occurs when one attempts to construct a sum over histories formulation for the transition amplitude between two three-surfaces with matter fields S, S’ defined on them, in an effort to create a Feynman-like quantisation scheme for gravity. One essentially sums over all four-geometries .4’, and matter fields 5”, that interpolate between (~,S) and (.~‘, S’), weighted by exp(—IE), where ‘E is the total euclidean action of the interpolating geometry 4’ and matter fields 5”. However, this path integral is ill defined, as the euclidean gravitational action is unbounded below under conformal variations of the geometry. Thus, it becomes necessary to rotate the contour of integration in the path integral to make it complex. One thus sums over complex intermediate geometries and matter fields. For the case where the four-space is asymptotically flat, a prescription for the rotation of the contour has been advanced by Gibbons et al. [6]. This leads to convergent meaningful results in a number of examples [7]. If the four-geometry is closed, as in the case of quantum cosmology, no prescription exists. A program to search quite generally for convergent contours has been carried out by Halliwell, Hartle and Louko [2,5]. In this program, the emphasis is on finding a suitable complex contour which is capable of describing the wave function of the Universe. Having examined the no-boundary and tunnelling proposals for the initial state of the Universe in a number of minisuperspace models without matter content, Halliwell and Hartle proposed a list of five criteria that the contour should obey in ~,

~‘

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order to yield a physically appealing amplitude [2]. It is their fifth criterion that is of interest to us. Their fifth criterion states that to the extent that wormholes make the cosmological constant dependent on initial conditions, the wave function should predict its vanishing. Preliminary analysis by Halliwell and Hartle [2], of semi-classical solutions of the complex Einstein field equation satisfying the no-boundary boundary conditions, indicated that for every solution possessing Re(~/j)>0 with action I~,there existed a complementary solution with Re(~/~) <0 having an action ‘2’ whose real part satisfied Re(12) Re(11). [Later analysis by Halliwell and Louko has shown =

that in more complicated models, the topology of the space of geometries of the particular minisuperspace in question may conspire to restrict one to either the positive or negative sector of Re(~/~)> 0 within the model.] It was then conjectured in ref. [2], that whereas Re(%/~)>0 implies negative action for four-spheres, Re(~/~)> 0 implies positive action for wormholes. Similarly, wormhole configurations possessing Re(\/~)<0 would have negative action. Thus, wormholes with Re(~/~)> 0 would be suppressed and such configurations would included 1~) <0,then andbe therefore into the contour of integration. Wormholes with Re(~ Re(IE) <0, would not be suppressed and, thus, if one wishes large wormholes to be suppressed, the dominant contribution to the contour should not arise from such saddle points with negative action. The discovery of these new wormhole solutions shows that this conjecture is not valid when complex matter is included. Specifically, we have found wormhole solutions with Re(%/~)>0 which possess negatil’e actions. Thus the conjecture that Re(~/~)> 0 implies positivity of the euclidean action for wormholes is disproved. However, we feel that the issue of whether they should still be included into the contour of integration cannot be resolved solely by the sign of the action. For example, large four-spheres play a vital role in the recent arguments concerning the vanishing of the cosmological constant while still possessing a negative euclidean action. To resolve whether a certain saddle point should be included into the contour, one must consider the fourth criterion proposed by Halliwell and Hartle [2], i.e. a consistent quantum field theory in curved space-time (QFTICS) must be recovered on the background configuration represented by the saddle point in question. We do not address this here. Thus, when the matter action is included and one allows fully complex matter configurations, Re(V~)ceases to be a good indicator concerning the treatment of a particular stationary point of the action as the dominant contributor to the path integral if one wishes the suppression of large wormholes. We note that the negative actions in wormholes I and II are not due to the unboundedness of the conformal mode of the gravitational action, but rather due to the kinetic term of the scalar field lagrangian having the “wrong sign”. We note that in both models the gravitational action is nonnegative. In the other known wormhole solution possessing an imaginary scalar field, the Giddings—Strominger

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solution [8], the action is identically zero by virtue of scale invariance and the equations of motion. The “wrong sign” for the kinetic energy of the scalar field is necessary for the construction of a wormhole solution in the minisuperspace ansätze we consider. It appears quite likely that utilisation of imaginary matter fields will generically lead to negative actions except in special cases of high symmetry in the lagrangian. The third motivation concerns the inclusion of topologically non-trivial fourgeometries into the euclidean path-integral formulation of quantum gravity. Recent work has shown that such inclusions may have severe effects on the low-energy effective coupling constants [9]. Essentially, the inclusion of planckian wormholes which either connect large disconnected four-geometries, or connect separate regions of the same large four-geometry, may cause the effective low-energy coupling constants to “float” unpredictably. To outline one of the major pitfalls encountered in this work, namely the “large-wormhole problem”, we now adopt the rules of wormhole calculus (though questionable) and assume (a) the euclidean path integral method is a viable formulation for quantum gravity, and (b) wormholes connect large four-geometries together that are smooth on the scale of the wormhole. Having done this, one can evaluate a probability distribution function for the coupling constants through a semi-classical approximation of the partition function. In the approximation that the large four-geometries are large fourspheres, this probability distribution function takes the form~[3] P(a)

exp[

—

~D11a~a1]exp[

exp[e~’c1~”A ~

~‘d]

(1.1)

where the a’s are the low-energy corrections to the coupling constants and I~1(g,A + a) is the action of a four-sphere with the altered couplings. We see that the effective low-energy coupling “constants” are now functions of the a’s. The most probable configuration is that which maximizes the argument of the double exponential in (1.1). the first three couplingofconstants -y (y is the 2 Keeping term), semi-classical evaluation the actionA, of w,a four-sphere coefficient of the R [3] gives the argument of the second exponential to be

[A(p)

+ ai][K2(p)

+ a 7]

~

[y(p)

+

a3].

(1.2)

Maximizing (1.2) with respect to the a’s gives us Act1 —s 0. This process is most efficient with small wormholes. However, to study the behaviour of K (the gravitational constant) and y one must introduce infrared cutoffs [10,11]. Naively, one would expect y~ and all higher-order effective coupling constants to be forced to their maximum allowable values. However, Preskill [11] has advocated that the determination of the unique minimum of l/ic~ffwill fix all other couplings through renormalization effects. Either way, the processes effecting shifts in all coupling

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other than A are most efficient if large wormholes are dense in spacetime on all possible scales. This is the infamous large-wormhole problem. A number of authors [1,4,11—14]have suggested mechanisms attempting to resolve the large-wormhole problem. The presence of a conserved charge plays a key role in the operation of many of these mechanisms ~. However, as pointed out by Polchinski [4], the large-wormhole problem will occur for any classical wormhole solution. Below, we present new classical wormhole solutions to the euclidean Einstein field equations which connect two asymptotically flat regions. In distinction from all other previously known solutions, these wormholes do not possess any conserved charge. It is not clear that the mechanisms which have been suggested with reference to charged wormholes [1,12—14]will still apply to these new solutions. Thus, for theories admitting such non-charge conserving wormholes, the largewormhole problem may reappear, causing wormholes to become dense in spacetime on the scales that they are predicted to exist irrespective of their actions! To summarise, we show that euclidean wormholes can exist while not possessing a conserved charge. We disprove a conjecture of Halliwell and Hartle concerning the positivity of the action of wormholes with Re(~I~)> 0. Thus, large wormholes are not suppressed by the criterion Re(~/~)> 0. If one requires that large wormholes be suppressed, then additional criteria must be imposed which recognises the complex nature of the matter field. Finally, the existence of these new solutidns casts serious doubts on current proposals to resolve the “large-wormhole problem”.

2. Model I In this section we describe the wormholes found when the scalar field potential is of the form V(4) ~A44. For simplicity we will adopt the k + 1 euclidean Robertson—Walker 0(3)-invariant metric ansatz, appropriately scaled for later convenience, =

=

ds~

=

2G —

[N2(t)

dt2

+

a2(t) dflfl

(2.3)

,

where dt2~is the standard 0(3)-invariant metric on the unit S3, and h c 1. The scalar field will be taken to be ~ (3/4~~G)t/2~(t), with a self-interaction potential V(~) (9/8G2)V(~). The euclidean Einstein—Hubert action, with the =

=

=

=

*

Preskill [11] did not utilize this charge but instead advocated that small wormholes will crowd out large wormholes. However, Polchinski showed [4J that this can only lead to a finite suppression of large wormholes and consequently cannot beat the infinite enhancement caused by (1.2).

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York—Gibbons—Hawking boundary term [15] at the S3 boundaries at

t

~,

is

~

=

~Jt~N dt(—aá2+a3~2—a+2a3V)

+

=

~ft~N dt(GAB~~8+ 2U)

~

+ ~a~+

~a2(t~)

+

~a2(t_)

(2.4)

where K is the trace of the actual extrinsic curvature of the boundaries in the metric (2.3), K 0 is that of the corresponding flat metric inside the same boundary, an overdot denotes N~ d/dt, 2=GAB dXA dX’~=—a da2+a3 d~2=e3~(—da2+d~2) (2.5) ds is the metric on the (a, 4) minisuperspace, a ln a, and U -~a+ a3V(~)is the minisuperspace potential. The classical euclidean equations of motion are =

2Z

=

GABXAXB

—

2U

=

a(—á2 + a242

+

1

—

2a2V)

a.

dV

a

dq~

a + 2a~2+ giving the trajectories of a particle of mass-squared spacelike geodesics in the conformally-related metric d~2 2U ds2 =

=

(a2

—

2a4V)(da2

—

—

—

=

0,

(2.7)

2aV= 0,

(2.8)

2U in the metric (2.5), or

a2 d~2) e4~(1 2 e2*V)(da2 =

(2.6)

—

—

d42). (2.9)

We are looking for wormhole solutions connecting two asymptotically flat regions, so a ±1as t —s ~ and ~1 ~± with V(çb~)=0 and dV(~~)/d~ 0. We will primarily be interested in solutions with finite action (2.4) as t ± ±~, though wormholes with infinite action may also be relevant in certain circumstances [1,10]. It is well known that no real wormhole solutions exist for pure gravity [16,17]. It is essentially necessary for the Ricci tensor, and hence for the matter stress tensor, to have at least one negative eigenvalue [18]. This does not occur for a real scalar —~

—‘

=

—~

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field with a non-negative potential V(4). In our 0(3)-symmetric model, this is illustrated by the fact that eq. (2.8) does not allow a wormhole throat with a > 0 if a’ 0 and J/ a 0. This restriction can be circumvented by going to imaginary itp, which makes 42 ~2 <0. For a massless imaginary scalar field (V= 0), wormhole solutions have been found [8]. These have a conserved charge, the value of ~, which is the momentum conjugate to 4. Solutions exist for any value of ~ and have a linear throat size proportional to We are interested in discovering solutions with no conserved charge and so consider non-constant V(~).Before choosing a suitable form for V(~),we remind the reader of a theorem of Jungman and Wald [19], which shows that no finite-action wormholes are possible for a real scalar field with 4 dV/dq~~ 0, and their argument applies virtually unchanged for an imaginary scalar field with ~pdV/d~<0. Choosing the potential to be of the form =

=

A V(~)

2p

(2.10)

~2p

we see, from Jungman and Wald’s theorem, no wormholes are allowed for 11 real. This is in accord with the comments in the above paragraph. For ~ icc’ imaginary (~real) and p odd, finite action wormholes are also disallowed by this theorem. The massive field case with m2> 0 for either real or imaginary scalar fields is thus ruled out as admitting complete, finite-action wormhole solutions. We wish to consider the simplest non-trivial potential which is non-negative for real ~ and which is not ruled out by the above reasoning. We therefore choose =

J/= ~

or

V= ~Açb4= ~A~4

(2.11)

with A=A/2~-2. It is convenient to use the conformal radial coordinate =

N dt f—

=

JN di e~’,

(2.12)

with a prime denoting d/dr 1

=

aN

d/dt. One may again derive the action to be 4~4)+~a~+

1~—~

~

(2.13)

~Aa

the classical solutions being spatial geodesics of d~2 e4’~(1—2 e2aV(~’))(da2+ d~2) e4”(1 =

=

=

F(a, cc’)(da2 + d~2),

—

-~Ae2~4)(da2+ d~2) (2.14)

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Imaginary wormholes

obeying the single second-order equation [201

=

—(1— ~A e2~4)1{1

+

(~)2][

e2~)~ +A

~

e2~3].

(2.15)

Another coordinate system which allows a more descriptive portrayal of the dynamics are the polar coordinates r=a2/2,

O=2~.

(2.16)

Rescaling the lapse to be N ~/a and denoting u~ d/dt by a hat, the action and equations of motion become =

_~ft~dt~(~2+r2~2+1_r~O4)

‘E=

rAG4

i~2+r2O2=1—4rV=1—

F’— r02

rO

2P0

+

=

—2V=

=

dV —2r-— dO

+r÷+r,

(2.18)

—

16

AG4

—

—i-—,

(2.19)

rAG3 =

—

(2.17)

(2.20)

—.

16

The classical solutions are spatial geodesics of A r04

d~2 (1 =

=

—

4rV(O))(dr2

P(r, O)(dr2

+

+

r2 do2)

=

1

—

—i--—--

(dr2

+

r2 do2)

r2 dO2),

(2.21)

obeying the single second-order ordinary differential equation d2r

2

—=r+—

do2

r

dr —

dO

Ar04 1————— 16

2 —

~‘

r2+

dr —

dO

2

4 dr 0—— r dO —

A03 —.

32

(2.22)

For large a eq. (2.15) has the asymptotic form d2z —z +Az3

=

dz 0(e_2~)0[z3, z2~,

dz Z(~)

(~)j, dz ~

2 ,

(2.23)

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Imaginary wormholes

255

with z e’~ a~,so when we can neglect the right-hand side we get an asymptotically conserved quantity E

1 2

=

—

dz da

2

—

—

~z2

+

~Az4,

(2.24)

which is the energy of a unit-mass particle in the potential ~-z2+ ~Az4, if the time is taken to be a. If we eliminate the arbitrary constant associated with the zero of s~,eqs. (2.6)—(2.8) or eq. (2.15) has a two-parameter set of solutions. In one asymptotic region, the two parameters may be taken to be E and an associated phase angle of the oscillations of z, the integration constant a 1) obtained in solving eq. (2.24) for —

a

—

E

=

f (1

cn1

2

dz 2 ~Az4 V2E + z —

+

=

~‘~

z I

=

+ e

~

1

+

—

+a 0

-

4AE)~2.

(2.25) (2.26)

Alternatively, the two parameters may be taken to be the minimum value of a and the value of ~ (or ~) there. Although all of the solutions are asymptotically flat, when E ± 0 the scalar field ~ =z/a undergoes oscillations of amplitude decreasing only as a1, which is too slow for the action to remain finite as ~ ±—s ± For a complete finite-action wormhole solution, we need E~=E= 0 in both asymptotic regions. These two conditions on the two-parameter set of generic solutions may be expected to lead to a discrete set of finite-action solutions. To obtain an asymptotic solution we assume E 0 in one asymptotic region, say E= 0 at ri Then in that region one can find a large-time asymptotic solution for ~ as a function of a by assuming the asymptotic form for ~ and a to be ~.

=

=

—~.

a(t)

~t+

~

~(t)

~

~

(2.27)

~..

This form yields the dominant behaviour as —s ~ Solving for the constants b 1, c1 in terms of the arbitrary parameter c_= c1, we invert the a(t) dependence and substitute back into ~(t) to obtain t

2— ~Ac3 a4+(~c3+ ~A2c5 )a6+(—~Ac5— ~(a) =c_a

3c7)a~ 4-~sA

+(~c5+-~)A2c7+~-~

4c9)a~°+0(a~2). 5A

(2.28)

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Imaginary wormholes

4 wormhole solution (i) in the throat region where the abscissa is t in the Fig. 1. Behaviour of V ~A4 gauge N= A~1~2 with the origin of t at the point of symmetry of a(t). This solution corresponds to 0.486421 A’. The separate graphs refer to (a) scale factor A~’2a(t),(b) behaviour of ~ (which does not scale with A) and (c) Aa3~.

We integrated the equations of motion by a fourth-order Runge—Kutta ODE solver from one asymptotic region (large a_) where E= 0 and ~ had the asymptotic form (2.28), through the throat, and out to the other asymptotic region (large at), where E~was evaluated. The function E~(c_)showed two zeros for positive c corresponding to two discrete wormholes. Using Brent’s root-finding algorithm, we found the two solutions occurred at —‘

(i) c_=c 1~0.486421A’~9.6015A~, (2.29) 1. (2.30) (ii) c=c2~6.0055A’=118.5438A The symmetry ~-s implies that there are two other wormholes with the same geometry but opposite values of given by c= —C 1 and c_= —c2. If we choose the gauge N 1 and set t 0 and ij 0 at the throat, then for both wormholes a is an even function of t and ij. Solution (i) has ~(t) and ~ also even, but solution (ii) has ~ an odd function of t or i~(see figs. (1) and (2)). The values of a at the throat are 2~4.63434A1”2, (2.31) (i) a=a1~ 1.04307A” (ii) a a 2 33577~l/2 (2.32) 2 0.75575A’~ For solution (i), ii —a 1’ 2, <0 at t 0, and the minimum value of a occurs at where t ±t0 ±0.650888A’~ a(±t 2 4.2199A1~2. (2.33) 0) amjnl 0.94981131A~~ ~

—~

~,

=

=

=

.~

=

=

=

=

=

/

J. Twamley, DIV. Page

Imaginary wormholes

257

‘N/a

~6~.4’~O3.46.8 4 wormhole solution (ii) in the throat region. This solution corresponds to Fig. 2. Behaviour of V= ~A4 c. 6.0055 A’. The separate graphs refer to (a) A”2a(t), (b) ~, and (c) Aa3çb. Note that we have compressed t by a factor of 5 for curve (c) so that the large t behaviour is apparent.

For solution (ii),

a

2a~t> 0 at

=

t

=

0, and a

2 is the global minimum of a, Remembering that a is a dimensionless quantity in the metric (2.3), the dimensionful three-volumes of the minimal surfaces in the two cases are thus 2L~ 2~~22G =

—

3IT

(i) 2~(ii) 2~2L~ 2~~2 =

2G —

3ir

3/2 a~

3”2G3”2,

*

(2.34)

151 145.0022A 3/2

-

a~ 73.0499A35’G3~2,

(2.35)

and the linear sizes of the three-surfaces of circumference 2irL are (i) L

2G’~2, 1

(2.36)

12.2141A ‘~

(ii) L

2G1~2, (2.37) 2 9.7187A~~ where we have expressed these quantities in terms of the coupling constant A in the original potential (2.11). Note that G~2 M~ L~ 1.616 x iO~°cm is the Planck length. The euclidean action of the complete wormhole solution may be split up into a gravitational part, a kinetic matter part and a potential matter part, i.e. ‘E ‘0 + ‘K + I~By considering the constant or scaling conformal transformation of the =

=

=

258

/

J. Twamley, D.N. Page

metric, g,~., to zero gives

—~ ~,,

=

Q2g~where £2

=

1

+

Imaginary wormholes

setting the first variation of the action

,

—2I~,

‘G~’K

(2.38)

or ~‘v~ Referring to eq. (2.13) we see that l~~ 0, and thus the total euclidean action must be negative. By performing the transformation and setting the first variation of the action to zero, we see that ‘K —2 I~,and thus from (2.38), J~ 0 identically. When we take 1~ ±~ or ~ ±—s ±~ to obtain the complete wormhole solutions, the numerical results of the euclidean action ‘E (2.4) or (2.13) are =

~

—*

=

=

=

—*

(i) I~ —0.479314A1 (ii)

—2.26035A~

‘2

—9.46128A1,

(2.39)

—44.6175A.1.

(2.40)

Since the actions (2.39) and (2.40) are negative, it appears that these wormholes are enhanced rather than suppressed. The action for a single wormhole is bounded below, however. The possible consequences of a negative action for these wormholes are discussed in sect. 3. Because the scalar field c1 has a nontrivial potential J(c1) the conjugate momentum =

~

~

=f~d3~= ~

a(a~/at)

(2.41)

(defined using the lorentzian action ‘L fL dt iJ~and the choice of gauge N i(3~/2G)”2 so that t becomes the proper lorentzian time) is not conserved but rather obeys the equation =

=

=

dir,, =

—/~Aa~çc’~.

However, the right-hand side tends to zero sufficiently rapidly so that toward the constant values

—~)

(i)

ir,,,(t

(ii)

ir~,(t = —cc)

=

=

—ir,,,(t

=

+3c)

=

=

(2,42) ir,,,

tends

—4V~~c

2At, (2.43) 1 l8.1049G~ —223.5339G~’2A~,(2.44) —

=

lTçj,(t

=

+cc)

=

=

—

in each asymptotic region of the two wormhole solutions. These may be viewed as the asymptotic charges of the wormhole. Note that the asymptotic charges are opposite for solution (i), so that the wormhole effectively adds asymptotic charge to both flat regions it connects (or drains charge if one takes the sign-reversed solution c c 1). Solution (ii) connects two flat regions where the asymptotic .

=

—

/

J. Twamley, D.N. Page

imaginary wormholes

259

C 2 Cos(2~)

5a

Fig. 3. Behaviour of types (i) and (ii) V= ~A~4 wormhole solutions in the polar coordinates (2.16). Graph (a) shows the type (ii) wormhole which is antisymmetric and passes through 0 = 0. Graphs (b) show the type (i) wormhole which is symmetric in t. Because of the 0 — 0 symmetry two solutions of this type are shown. Also shown are lines of constant conformal factor P = (I — r04/16) ranging from P=Oto P=0.9instepsof0.l.

charges are equal. Here the wormhole appears as normal, acting as a sink for asymptotic charge on one side and as a source on the other side, but again the charge ‘n-~is not conserved within the throat. A more pictorial description of the two wormhole solutions can be obtained by going to the polar coordinate representations (2.21). Using the polar coordinates (2.16), the two solutions are plotted in fig. 3. The dynamics is that of a particle with unit mass moving in a potential U (~Ar64 1)/2. The scalar curvature of the auxiliary two-dimensional minisuperspace (2.21) is given by =

R

—

A02(Ar04 +

4fl2 +

48)

=

(2.45)

.

64r(1 —ArO~/16)3 This diverges for 0 ±2/(Ar)~4,that is, when P(r, 0), the conformal factor in the minisuperspace metric (2.21), is zero. The loci of various values of P(r, 0) (including zero) are shown in fig. 3. In this auxiliary minisuperspace, all the dynamics occurs within the region P s 0, and the typical trajectory will avoid the infinite barrier P(r, 0) 0 (see for example the behaviour of the type (ii) wormhole solution). However, on the surface P 0 we see from (2.18) that F’ 0 a 0, indicating a complete time reversal of the dynamics. In eq. (2.22) we have eliminated the time dependence, and so the behaviour of d2r/d02 is regular at P 0. For this to be so we must have =

=

=

=

=

=

=

=

dr d0p=o

=

rO 4 —.

This is demonstrated in the type (i) wormhole behaviour.

(2.46)

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Imaginary wormholes

Solution (ii) also possesses at least one lorentzian section. This occurs at the time symmetric surface at the throat’s center where ~ 0 and the extrinsic curvature vanishes. The lorentzian section represents the lorentzian evolution of a 0(3)-symmetric three-geometry from a singularity to a maximum size (that of the time symmetric three-geometry of the euclidean wormhole) and recollapse to a singularity. In this evolution the lorentzian extrinsic curvature is zero uniquely at the point of maximum expansion. We note that there may exist other lorentzian sections which may be obtained through complex transformations. We have not, however, succeeded in discovering such transforms. For further information concerning solutions consisting of a real euclidean geometry connected to a real lorentzian geometry, see Gibbons and Hartle in ref. [21]. For a fixed A there exists a maximum size for a classical wormhole solution with finite action. Thus naively it appears that the unbridled fury of the large wormhole problem is somewhat abated and only the two finite action solutions are dense in space-time. It could well prove, however, that the relaxing of the dilute gas approximation will allow the solutions with infinite action (if evaluated out to infinity) to be patched together, thus allowing wormholes of arbitrary size. =

3. Model II In this section we describe the wormhole solutions found in the case where ~m2~2 + ~A~4 and where ~ is taken to be imaginary. The analysis proceeds much the same as in model I. However, the dynamics is more complicated, because 4 may cross zero more than once, and because the solution depends nontrivially upon the parameter ,a m/ The description is split into a number of subsections, the first of which motivates the above-started form for the potential. Subsect. 3.2 describes preliminary analytical analysis, much akin to that performed in model I. Subsect. 3.3 outlines the results of the numerical analysis, while subsect. 3.4 contains an analytical approximation to the dynamics which corroborates the numerical results. To summarise the results briefly, an elaborate spectrum of wormhole solutions is found which can be labeled by the asymptotic value of a quantity x (to be defined later) and the value of ~c. The wormholes do not possess a conserved charge. Wormholes only exist for ,tc
V(cb)

=

=

~.

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Imaginary wormholes

261

3.1 CHOICE OF POTENTIAL

We are interested in euclidean wormhole solutions which connect two asymptotic regions which are asymptotically flat. Our model consists of a euclidean FRW cosmology minimally coupled to a scalar field ~ with a self-interaction potential l/(4.). For a realistic self-interaction potential we impose the following natural restrictions: V(q~)E11,

V(~)~0 Vq!efl.

(3.1)

With these restrictions Jungman and Wald’s theorem [19] shows that no finite-action wormhole solutions are possible for a real scalar field. We must again consider imaginary scalar fields 4 = ip (~real). Examining the dynamical equations of motion dV(icp) (3.2)

d~

a

=

2a(~

-

V(i~)),

(3.3)

if ~ —s ~ ± as t ±cc then the recovery of asymptotic flatness requires that V(itp ~) = 0, dV(i~÷)/d~= 0. If we also restrict the potential to be such that d2V(i~)/d~2<0 at ~ = ~ then from (3.2) the behaviour of near ~ = ~ ± will be exponential. Restricting to the exponentially damped mode ensures that the action will be finite. Note, however, that finite actions are also possible for potentials going as a higher power of ~ ± near ~ = ~ We choose the quadratic power case for simplicity. A suitable potential which satisfies all the above criteria is —~

~,

~‘

~

=

—

~th2I2+

~.

~

(3.4)

giving the resealed potential m2

A

A (3.5)

with 2G m2= —th2 3ir

1 —A. 2ir2 -

A

=

(3.6)

A phenomenological description of the solution can be found as follows: Generically, the imaginary scalar field executes damped oscillations about zero in model I. The two wormhole solutions correspond to critical damping. All other solutions are asymptotically flat, but because of the oscillatory behaviour, are unbounded in

262

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Imaginary wormholes

action. Phenomenologically, adding a small ~m2ip2 barrier near ~ = 0 will eventually halt the scalar field from crossing zero. It will either execute damped oscillations about either of the “true” vacua at the minima of V(ip) or (if we are careful), asymptotically approach the “false” vacuum at ~ = 0, where we have V = 0 for asymptotic flatness. To obtain a wormhole, we start the scalar field at the “false” vacuum such that, subsequent to a number of oscillations about ~ = 0, it finishes back at the “false” vacuum. For a finite action we set d2V((i~)/d~2 <0 at = 0 for the reasons stated above. —

3.2. PRELIMINARY ANALYTICAL ANALYSIS

3.2.1. Classical equations and auxiliary minisuperspace We begin by noting that under a constant conformal transformation gd,,,. = ~ the classical equations of motion (2.6)—(2.8) are left unchanged except for a constant rescaling of the potential J/ —s V = ~ Using this freedom to scale out A from V, and letting —

4irGm—2

m

the action and equations of motion become

f

AIE=

Ndt(aa+

1/2

(3.7)

~

+a+a[~~_

+~(a~+a~)

(3.8)

~

+ ~2

=

—2V(i~)+ e2~*= dV(iqi) —

=

d~

3~2 — e2”

/j2ç02

—

~

(3.9)

+ e2”,

~2cc”~3 =

(3.10)

_~2 + 2~2 —

2V(i~)

~a2+ 2~2+ ~2~2

—

~

(3.11)

For ease of visualization and future analytical approximations, we again introduce the polar coordinates r = a2/2, 0 = 2c0. The action in these polar coordinates is AIE=

=

where N =

v/a

lft*dt(P2+r2ë2+I4rV(O))+(r+r)

—~f’odt

F’2

+

r202 + 1—

and a hat denotes v

r02

_~~~(O2 — 8~c2)

d/dt.

+

(r~+r_),

(3.12)

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Imaginary worinholes

263

As noted in model I, the solutions describe the dynamics of a particle of mass-squared —2U moving in the flat minisuperspace metric ds2 = ~e3~c(da2 + d~2).The solutions also correspond to the geodesics of a particle of mass-squared F(a,

~)

=

e4”(2 e2”V(i~)

1)

—

moving in the alternate flat metric da2

e4~(e2~*[~2~2— ~4]

=

—

1)

(3.13)

d~2,or of mass-squared

+

r02 P(r, 0)

=

1

—

4rV(O)

=

1

—

_~-~-(O28~.c2),

(3.14)

—

in the third flat metric ds2

=

dr2

+

r2 dO2.

(3.15)

Alternatively, the solutions are spatial geodesics of d~2=F(a, ~)(da2+dçc2) =P(r, O)(dr2

+

r2 dO2)

(3.16)

in (a, cc’) and (r, 0) coordinate system. Another description is that the solutions are the trajectories in the flat space (3.15) of a nonrelativistic constant-mass particle of zero energy moving under the influence of the potential —F(r, 0). Eliminating the time parameter, one finds that the evolution is governed by a second-order ordinary differential equation, given in the polar coordinates by d2r 2 =r+ do2 r

—

—

dr dO

2

+

—

1 r2+ P —

dr dO

—

2

—2V+

2 dr dV rdOdO

,

———

(3.17)

or, more explicitly, by d2r 2 ~=r+_(~~)

dr

r02

2

+(1_~(o2_8~2))

X[O(O2_8~L2)_

dr

2

r2+(~)

±(o2_4~2)(~)}~.

(3.18)

The two-dimensional scalar curvature of the conformally flat metric (3.16) is

r02(04

—

4~.c2O2+ 32~c4)+ 4(O~+ 1202_ 8~i/O2— 161L2) 64rP3(r, 0)

(3.19)

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Imaginary wormholes

This diverges when P = 0, which occurs on the curve r = 16/02(0 2 8~.c2).However, eq. (3.17) does not obstruct particular trajectories from crossing P = 0. In fact, eq. (3.17) is insensitive to a change in signature of the metric (3.16) from (+, +) to (—, —). This makes sense, as we have eliminated the time in obtaining (3.17). Euclidean time dynamics is restricted to those regions of configuration space where P(r, 0) 0 (or F(a, ~) ~ 0), while lorentzian time dynamics is restricted to those regions where P(r, 0) < 0 (or F(a, ~) < 0). Trajectories in the auxiliary minisuperspace can cross the P(r, 0) = 0 (or F(a, 0) = 0) curve and will be regular there. However, re-inserting the time dependence, one sees that on the curve P(r, 0) = 0, ci = = 0, and the euclidean time dynamics of any trajectory in the (r, 0) auxiliary minisuperspace that reaches the P = 0 curve from the P> 0 region will have a time-symmetric bounce in both the geometry and matter and hence return to the P> 0 region. Crossing P = 0 would correspond to analytically continuing the euclidean time (or the lapse function N) to imaginary values, and would give a transition to a lorentzian geometry (see ref. [21] for more information concerning the euclidean/lorentzian transition). Regularity of d2r/d02 at P = 0 gives —

~‘

dr —

dO and the tangent vector at P

=

j’—~~

(3.20)

(dV/dO)

0 is proportional to

dr T=

rV =

—

F’+rO=r

dO

Since the gradient of P, in the metric dr2

dV

VP= —4—

do

+

V dV/dO

r+O

(3.21)

.

r2 do2, is given by

V 1+0 dV/dO

(3.22)

we see, from eq. (3.21), that these trajectories meet the curve P

=

0 normally.

3.2.2. Phenomenology of the dynamics Eliminating the arbitrary constant associated with the origin of time, eqs. (3.9)— (3.11), or eqs. (3.17) and (3.18), have a two-parameter set of solutions. Since the form of V(O) was chosen to yield an exponential solution for ~ at large a, choosing the decaying mode of ~ leaves us with a one-parameter set of solutions. We then integrate in the + t direction, through the wormhole throat and out to the other side. Imposing the boundary condition that ~ —s 0 as t + cc yields a wormhole configuration connecting two asymptotically flat regions. Imposing a single boundary condition on a one-parameter set of solutions should give a discrete “spectrum” of allowable wormhole configurations. However, since we are also treating ~c as a —,

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Imaginary wormholes

265

free parameter we will finally obtain a discretely labeled continuous family of wormhole solutions. A intuitive understanding of what we expect to occur can be obtained through consideration of the actual form of V(icp). The potential V has the standard Mexican-hat shape. If, for example, one begins with ci = = 0 at a throat where = one needs V(~11)>0 or I > ‘~/~c. Then the scalar field ~ will roll down V and will generically execute damped oscillations about either of the two true vacua at the minima of V. Naively, if one increases I I, one expects ~ to make a larger number of oscillations crossing ~ = 0 before setting in either true vacuum. By continuity, there exist discrete where ~(t = ±cc) = 0, and the field ends up in the unstable “false vacuum” state at this local maximum of V. However, because of the friction term in eq. (3.2), the dynamics is rather more complicated and the dependence of N (the number of zero crossings of ~) on p~is quite involved. We now proceed much the same as in model I. We obtain a large-time asymptotic solution which yields asymptotic flatness in one asymptotic region. We then integrate through the throat both by numerical and analytical means and evaluate a test function, the zeros of which indicate the attainment of asymptotic flatness on the other side of the throat. The asymptotic solution, will, as before, depend on one arbitrary parameter x. The test function .f will depend Ofl x and ~i. The zeros of ~7 in the two-dimensional parameter space (x, .c) indicate the existence of wormholes with finite action.

.~,

3.2.3. Asymptotic solution and test function We now briefly describe the derivation of the asymptotic solution. The dominant behaviour near small ~ (large t) may be uncovered by noting 2z

3á2

3ä

d

(3.23)

where z

large

t,

= a3~2~and where we have discarded non-linear terms in assuming dl —si, ä—sOas t—s ±cc,

~(t)~t~3~2(A

~.

e~+B e~’).

Thus, for

(3.24)

Retaining the first two terms on the right-hand side of eq. (3.23), we get

-

[~

~}z=0.

(3.25)

266

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I Twamley, D. N. Page

The decaying solution to eq. (3.25)

Imaginary wormholes

is

=AVTK1(~~t) -~A1/2,a e~’k=0 ~ —

y

T(1+k+~) k+

(2tLt) k k!F(l

—

-

(3.26)

,

where K 1 is the first-order modified Bessel function. Thus a first approximation gives a t, ~ —~AK1(,at)/t.One can re-iterate these equation to obtain further refinements. Doing this results in

a

2ir e2~’~ 3 I 8t-

7 4ti.r

+

t +A

75 96t2j.c2

+

+

...

~

,

—A K(,at) t

.

(3.27)

However, this method tends not to give uniformly convergent results, and the estimation of the degree of approximation is difficult. Instead, we first assumed the following forms for a, ~: a

t + e2~t2

~ b,,r’,

~

e~t3”2 ~ c,,r”,

n=O

(3.28)

n=0

where T = jit. Inserting these into the equations of motion and solving for the coefficients b,,, c,, gives

a ~t+e~2Tt2~[1

~ —c

where c

=

+

e_rt_3/2[l

+

—

—

128T2

32~2 +O(T_3)}~

(3.29)

0(r~3)}~

(3.30)

+

c 0. Next, we made the more elaborate trial expansions a

_

2Tt2 t +

e

et3~2

~

b,,r”

~ c,,~” n—t)

~ d~,r’,

+ e4TtS

(3.31)

n—U

11:—u

+

e_3Tt_Y/2

~

e,,r~,

(3.32)

n=0

and wrote a symbolic computer code to solve for the coefficients b,,, c,,, d,,, e,,. Other than the expressions appearing in (3.31) and (3.32), no combinations of exponentials and powers of t and T of lower power give consistent equations for

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J. Twamley, D.N. Page

Imaginary wormholes

the coefficients. With these forms for a and (3.10). The resulting solution is 2 3 1+ 4 4~

a~t+e_2Tt2_3c

4645

—

61305

2048r4

—

8192T5 45c4

+

e4~t5

/.*‘2c

~

one can include the q~term in eq.

103 128T3

+

1855455

+

—

11 32r2

—

~,

267

[1 +

65536T6 456

+O(T8)

33 —

7

40T2 +

(3.33)

+ 0(T_4)J~

[~ V_K1(T

15

9c3 —

e3Tt9/Z2

—

16

1

+

—

107

ST

—

18632

128T2

+

8417r3

205005 —

37768r4

+

0(r5)

.

(3.34)

Thus ~p(a)

x

ca3~’2e~”’

—

x

—3/2

e~x,

(3.35)

where x=~ca and

~+~ln

x=ln(ci~2)w~.

(3.36)

We now obtain an expression for a test function, the zeros of which indicate the presence of asymptotically flat space. Integrating through the wormhole and beyond, ~ will generically undergo decaying oscillations towards ±~c, the minimum of the potential. Assuming ~ = +~.c and neglecting ~, eq. (3.9) may be integrated to give a~H’ sinh Ht,

ci”Hcoth Ht-~-~-~-+H,

where H = ,a2/ = 2V(i,a). For small perturbations in ~ about ,a, eq. (3.10) becomes ~

—

dV —

~I+

~

= p.

+

(3.37)

x, say (not the same x as above),

3 -~=~p.2.~+2p.2x=0.

(3.38)

J. Twamley, DIV. Page

268

Taking x =A c”t

+

/

c.c., we obtain for p.

=

±

=

±WOV1Y /w

+

Imaginary wormhole.s

<

i ~ = ±~

+

(3.39)

3~p.2

0 +iy~ ±w0+iy, where w1~= I

I

=

Vip.,

y

LW0.

=

If we define S =A

x=S+S*, which can be solved for S(x, 2~’ yields IA 12 e~

(3.40)

e”~’, then

~=iciiS_i~*S*,

.s~)and

(3.41)

S*(x, .x~). Computing the norm SS’~=

2yx.~+~2

w~x2 +

IAI2=e2~

2

4(w

(3.42)

2

)

0—y

2~, and using the definitions of From eq. (3.37) we can eliminate and y we finally obtain

t

=

w,

w 11

a/y~i

(3.43) (Note: if, instead, ~ tends towards —p., we replace p. with —p. in eq. (3.43)). Defining ~ = lime ~ this ~ will be larger if the trajectory spends more time near ~ = 0 before rolling down into the trough. To obtain the desired test function, we calculate

9’= lim

sign(~)

(3.44)

.

3.2.4. Analytical considerations of the action Before discussing the results of the numerical and analytical analysis, let us first consider the total action of an asymptotically flat euclidean wormhole, ‘E

f~d4x[_ =

‘0

=

—

16~G

—

~(V~)2+

~(_~p.2~2+

~4)]

+ ‘K + ‘v

3(a ~JN

dt a

+

2d2

+

e_2a + ~2

+

p.2~2

—

~

(3.45)

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Itnaginary wormholes

269

For the wormhole to be a solution of the Einstein field equations, the action must be stationary with respect to small perturbations in the metric and matter field. Making the scaling or constant conformal transformation, g~ Q2g,~,the action becomes —~

112 Assuming 12

=

1

+

+

+

~

(3.46)

e, setting the first variation of the action to vanish gives l(j+IK

_21V,

(3.47)

~fN

dt a3(p.2~2

or I=

=

—

~qc4).

(3.48)

Integrating ‘K by parts and using the equations of motion and boundary conditions at infinity, one can show ‘K

that is obtain

‘K <

(3.49)

~

‘v’ irrespective of the sign of ‘v~Inserting this into eq. (3.47) we

=

1v +

~f~N

dt a3~4=

N dt a3p.2~2.

(3.50)

This agrees with model I, since for p. = 0 we get I~,= 0. With the purely quartic potential Iv > 0 and by (3.49), ‘K = ~21v (This can also be obtained through an infinitesimal variation of the action with respect to ~.) The sign of the present potential term, and hence the total action, cannot be determined, as the potential is the difference of two positive quantities. From eq. (3.48) one sees that for ~2 <2p.2 the p.~2term dominates and makes the integrand positive, so wormholes with a small value of ~ throughout the evolution would have a positive action. However, there are regions in the evolution of each wormhole where ~2> 2p.2, for instance near the turning points of If these regions are too large, the action will be negative. One could, for instance, take the maximum value of I I as an indicator for the sign of the action. This general argument is born out in the numerical analysis. ~.

3.3. NUMERICAL RESULTS

To integrate the equations of motion, a fourth-order Runge—Kutta algorithm was used. To allow the possibility of fairly general throat configurations, we impose

1 Twamley, D.N. Page

270

40

/

Imaginary wormholes

~

Fig. 4. Bold curves are the numerical calculation of the loci of the N = 0 5 V = ~,m2~2 + wormhole solutions in the ~ x) parameter space with j.c = A — ‘/2~~ and x defined by the asymptotic form (3.35). The bold squares are the numerical estimates of the zero-action wormholes.

the boundary conditions at large t through the use of the asymptotic solutions (3.33) and (3.34), with t replaced by —t. Examining the zeros of 9(x) for fixed p. yields all the possible wormholes occurring at this value of p.. Brent’s algorithm was used to pin-point the roots of ~‘(~) [22]. The zeros of ~ p.) corresponding to the first six wormholes are shown in fig. 4. We now partially summarize the numerical and analytical results: (a) Wormholes only occur for p.
—

=

(1.2 +

V(i~)).

(3.51)

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The friction term, 3ci~,appears to dominate the dynamics if I ~Pmax I > I ~crit I where, for small p., ~p,,~~1/p. is a decreasing function of p., and in particular, is roughly 6.0 near p. = 0.2. For I ~ I > I ~crit I, numerical simulations showtmax thatI ~ does not cross zero but tends to settle in the nearest false vacuum. For I ~ ‘Pcrjt I, ~ decreases and the friction losses are overcome by the initial energy of the scalar field, thus enabling ~ to cross zero. Decreasing I increases the margin of p over the friction losses, thus allowing ~ to cross zero more than once. Decreasing ~max still further will eventually reduce this margin as p also decreases. Thus N decreases. Decreasing I I to i/~p.will not result in a wormhole as p = 0 there, while the friction will still be present. Thus N decreases to zero. From fig. 5 we note that large x corresponds to smaller I ~max I. [Note: in fig. 5 —

x increases from graph (A) to (B) to (C) to (D) as can be seen from fig. 4. Also

note that I I > V~p. 0.283 at the turning points of However, I c’max decreases monotonically towards ~/~Jp.as increases. I ~max I approaches y~p. closest in graph (D) in fig. 5. If K wormholes occur at a particular p., then the K/2 wormholes with larger x represent those wormholes referred to in our previous heuristic model with low-energy density p, increasing friction and N decreasing to zero in unit steps. The K/2 wormholes with lower x are those wormholes possessing an increasing margin of p over friction as max I I decreases. From fig. 4, for an N-type wormhole to exist, p. ~ For N = 0, ~max~, 0.402. An empirical estimate of ~.

~‘

0.402 ~max~

N+1

(3.52)

models the results quite well. Defining oi ~p. (not the same as in (3.40)), the results show that when W ~, for fixed p., N is maximized, whereas for fixed N, p. is maximized, that is d —(p.N) dW

~0.

(3.53)

The large x asymptotic behaviour of these curves is also well modelled by the formula X

1 2(N+1)p.2~

(3.54)

Information concerning the minimum scale factor, the matter action and the gravitational action along each N-type wormhole curve in the (x, p.) parameter space was also gathered by the computer program. Information concerning the minimum scale factor, and hence, the volume of the minimal hypersurface ~ is presented in fig. 6. As a function of p., amjn is

J. Twamley, D.N. Page

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Imaginary wormholes

0.0

10

-0.4 —0.2

\\\\\\\\\\\~\\,,,//////////

6. 9-

Cl

<4.

—0.5

2.

-1.0

0.

—~

—5

I

I

5

—1.2

1)

(A)

—10

—~

5

ID

—ID

10

20

—10

10

20

N=O

0~

.~0.00

-0.25

4. —20

—10

I

10

20

(B) 30.

—0.50 —20

N=1 0.50

I

~-0.00o.25

26

-0.25

14. 10.

—20

—10

ID

20

(C)

-0.50

—20

N=1

Li 00

65

9-0.2

1<

60.

55. —20

I

—10

10

20

(D)

Fig.

5.

Behaviour of a(t) and

q~(t)for

—0.3 —20

I

—ID

_____________ 10 20

N=O

the four wormhole solutions occurring at j.c labeling.

=

0.2. See table 1 for

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Imaginary wor,nholes

273

1OO.jN~

Fig. 6. Dependence of the minimal scale factor a,.,~,on p. and on

x~for the N

=

0

5 wormholes.

double valued, while as a function of x it is single valued. For fixed p., amjn monotonically increases as N ranges from zero to a maximum and back to zero again. Thus, amin may be used to label the “spectrum” of wormholes for a fixed p~. Of more interest, however, is the behaviour of the action. In figs. 7 and 8 we plot the matter action, gravitational action and total action suitably scaled in order to enhance the behaviour at small values of the action. As we predicted in subsect. 3.2.4, 1~~ 0, whereas ‘E = ‘~. + ‘M = — I~changes sign when

1~=_~fNdta3(p.2~2_~4)=0.

(3.55)

1e = ‘v = 0 are For N = 0 5, the approximate parameters (x11’ p.05) where shown in fig. 4. For a specific N, when x
J. Twamley, D.N. Page

274

/

Imaginary wormholes

E’~:: Fig.

Dependence of the gravitation and matter actions I~and I~ on x for the N = 0 wormholes. The action has been scaled by sinh — to magnify the behaviour at small actions.

7.

5

that XON> l/2p.05,. Thus there exist certain small ranges of p. where the number of negative action wormholes (N1) exceeds the number of positive action wormholes (N1.~).However, for most of the range of p., N1~=N1_. For a more concrete visualization of the various wormhole configurations, we plot the solutions for p. = 0.2 in the cartesian coordinates (a, ~) and polar coordinates (r, 0) in fig. 9 and 10. See table I for the roots of .~(x,p. = 0.2). Also plotted in fig. 9 is a non-wormhole solution where ~ falls into one trough. Clearly visible are the two types of wormholes, with arnie monotonically increasing and I ~Pmax I decreasing with each new wormhole solution. Also featured in fig. 10 are the constant contour lines of P, given by eq. (3.14), beginning with P = 0. From

0~1~N73~I0 t to the the behaviour actions. The action has been Fig. 8. Dependence of the total action ‘E magnify on x for N = 0 at small 5 wormholes. scaled by sinh

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Imaginary wormholes

275

O2a3~5

Fig. 9. Graphical presentation of the four p. = 0.2 wormholes in the (a, ~) auxiliary minisuperspace. The separate curves (A), (B), (C) and (D) are the four wormhole solutions referred to in table 1, (E) is a zero of F(a, ~,) (3.13), (F) is a minimum of V(~),i.e. a “true vacuum”, and (G) is the trajectory of a non-wormhole solution which falls into a true vacuum. Due to the symmetry g -. — g, images of curves (A), (D), (E) and (F), mirrored in the cc = 0 axis, are also wormhole solutions, a zero of F, and a true vacuum respectively.

0

50

100

150 200 r Cos(O)

250

300

Fig. 10. Graphical presentation of the four p. 0.2 wormholes as bold lines (except for D, which does not enter the region shown) in the (r, 0) auxiliary minisuperspace. Shown also, as lighter lines, are the contours of constant P in z~P= 0.2 intervals beginning at P = 0.

1 giving four wormhole solutions

TABLE

Table of roots of

~

p. =

0.2)

Label

N

x

A

0 1

0.382392 2.226829

B C D

1

3.597605

0

11.881919

J. Twamley, D.N. Page

276

9 it is clear that type factor a or a — one at trajectory is vertical (see point in a, that is, where fig.

3.4.

/

Imaginary wormholes

(i) wormholes possess three turning points in the scale P = 0 (ci = = 0) and two where the tangent of the also fig. 5). Type (ii) wormholes have only one turning ~ = 0 at a = arnie.

ANALYTICAL APPROXIMATIONS

In this subsection we briefly outline the key assumptions and calculational steps needed to arrive at an approximate analytical form for the curves displayed in fig. 4. We also derive an approximate analytical expression for the action. The central premise of the approximation scheme is that I Lla/a I << 1 and I ~ld/d I << 1 during a single oscillation in ~, which we may expect to be valid for large N. We are thus able to calculate the adiabatic change in various quantities, and in particular, the change in a, during a quarter cycle of We can then integrate in/LI I a I as if it were dn/d I a I, where n is the number of quarter cycles, from I a I = 1 to I a I = 0, to obtain an estimate for the total number of quarter cycles of ~ occurring in the evolution of ~ as I a I ranges from 1 to 0. (A quarter cycle is the evolution between successive zeros of q~b.) The number of zeros of ~ throughout the whole evolution of the wormhole is then given by N = 1. We begin by noting that asymptotically ~ 0 as a —s cc~ However, as the evolution proceeds towards the throat, ~ will deviate from zero and will roll down into the trough. It will then execute a number of quarter oscillations. If one uses the asymptotic expression (3.35) and sets ~.

.~,

—

—,

(3.56) for

a1 or x = ~ (but see below for a small correction). For future reference, we now obtain the approximate behaviour of là I for x ~ the equations Introducing the variables 2)t~2~, and eliminating time from of motion, we canz =obtain y = (2p. dqs d~ 1/2 a~—=~——=F(—2a2Vz+z—1) , (3.57) ~,

given by eq. (3.36), large, then ~ will cross the trough roughly at a

dz a— da For x

=

p.a

>>

dz =

—

da

=

(3.58)

—4z(—3a2Vz+z—1).

1 and ~ <
=

—~=-

(~) x

~,

or

—3/2

ex~,

which is obtainable from (3.57) and (3.58) in the approximation z

(3.59) —

1

-‘~

—

2a2Vz

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277

-~v1, i.e. at large a and small q,. With these same approximations, eq. (3.57) yields z—1~3x~2.

(3.60)

We now find an approximate solution for ~ which is valid up to a point near the end of the second quarter cycle, where ~ crosses zero for the first time. Using the variables x, y and z, eqs. (3.57), (3.58) and (3.60) become x~

z

—

=

~

—y2)x2z +

=

_4z[3p.2y2(1

~(z

—

—y2)x2z+z—

1)]~

(3.61)

1],

(3.62)

1 — 6p.2xy2.

(3.63)

Inserting eq. (3.63) into (3.61) and using the fact that z dy 3 —~+y 1+——y2 dx x

1 gives

1/2

(3.64)

.

The approximation (3.35) or (3.59) implies that y crosses the trough at y2 = roughly at x = x>> 1, so for a more accurate approximation near there we may set x = ~z in eq. (3.64), integrate and match to (3.35) at x >> x to get y~sech[p(x—x+~ln2)J,

~

(3.65)

Inserting eq. (3.63) into (3.62), we get _12p.2z2y2{(1 —y2)x+ which, using eq. (3.65) with ~P= —--z

/3(x

—

2dV~~I’+f3x+2/3/z cosh 11’

=

cc

(where 11’

—

=

(3.66)

x) as a new integration variable, becomes

5-~12p.

Integrating in from x

—],

cc,

—

l~2~”-~3x cosh ~1’

(3.67)

z = 1), we get to leading order in x~

1— 4p.2x[1 ±(1 _y2)3/2].

(3.68)

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Imaginary wormholes

The upper sign holds during the first quarter cycle, as y increases from 0 to 1 (where ~ = 0), and the lower sign holds during the second cycle, when y decreases back to 0. Eq. (3.65) breaks down near the end of the second quarter cycle, as y actually crosses the ridge at y = 0 rather than simply asymptotically approaching it as it would if z stayed at unity. However, eq. (3.68) should be valid through most of the second quarter cycle, as long as I x x I <> 1, and so eq. (3.65) is valid with 1~ 1. We now begin the calculation of the change in various quantities over a quarter cycle of During the analysis we will make several assumptions, some of which will be recursively justified by the results. We first calculate I LIx/x I over a quarter cycle. We make the assumption that z is approximately constant during a cycle. This will be shown to hold for ~ large. Using eq. (3.61) with —

~.

-____ (z-1)

Z-

21

2

~

i\

1

2p. 2 x 2 z

we can obtain —

1 2(y~ z’~

r

x=+-~-

—

J

1)1/2

[(1

—

dt t2)(1 +y~t2/(y~

—

1))11/2

,

(3.70)

where y = Ymt and where Z and hence Ym is assumed to be nearly constant during each cycle. Integrating between the limits of t = 0 (at which ~ = 0), and t = 1 (at which ~ = 0), we obtain (2k2

1)Y2K(k2)x_1z1/2,

—

(3.71)

where K(k2) is the complete elliptic integral of the first kind, k2=

1

+

(3.72)

2Vf~4Z

and m

2=1—k2=

2~

1~k

,

(3.73)

or k2(1

—

k2) 2~

(2k2

—

1)

(3.74)

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279

To estimate I LIx/x I, note from eq. (3.65) that ~ just crosses the trough at 2 or y2 = ~ for the first time when x x + ln(V~+ 1) ~ln 2, and ~ = p. reaches its maximum (the completion of the first quarter cycle) at x x ~ln 2. Thus, for x>> 1, the second part of the first quarter cycle (y2 = to y2 = 1) has I LIx/x I 1. Eq. (3.65) would give an infinite LIx for the completion of the second quarter cycle (to y = 0), but that formula breaks down and should be replaced by eq. (3.71), which gives ILIx/x I In x/2x ~ 1 for 1 <“zx ~ using the expansion K(m 1 <<1) ~ln(16/m~). We now calculate I LIz/z I over a quarter oscillation. Eqs. (3.61) and (3.62) may be divided to give —

—

-‘~

dz dy

=

—

2) + 2z] ~/[y2(1—y2) +z}

+12p.2z3~2x[y2(1 —y —

(3.75)

.

Letting y =y,~(1—X2) with Z ~ 1), we can integrate (3.75) from y y = y~,taking x and Z to be approximately constant, giving —

LIz z

2(2k2_1)’/2 (z—1) k2(1

0 to

4p.2xz~2

~

— k2)

=

F(k2)

=

(2k2

—

~/2()

(3.76)

where F(k2)

(1 _k2)2K(k2) + (2k2

—

1)E(k2),

(3.77)

and where E(k2) is the complete elliptic integral of the second kind. Since k = 1 for z = 1, and F(l) = 1, we see that I LIz I 4p.2x where z — 1, which agrees with (3.68) where x x. From the definitions of k2 and Z we can also compute LI(k2)

—2(2k2

LIZ

2(2k2—

Z

—

l)3/2G(k2)x_!z_~2,

(3.78)

l)h/~2

k2(l —k2)

(3.79)

G(k2),

where G(k2)

(1

—

k2)K(k2)

+

(2k2

—

flE(k2).

(3.80)

Since G(1)= 1, ILI(k2)I <> 1, though LIZ/Z is large because Z is small.

280

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4p.2x. Thus, our initial assumption that z changed Now, for k2 1, I LIz/z I little during an oscillation implies p.2x << 1. We now adopt the new variable v là I =z”2, and find that LIv~—F(k 2 )

2p.2x (2k2— 1)3/2

—2p.2x when Using ILIx/xI ~<1,

LIv/vI

4~C1

k2— 1.

(3.81)

we canwrite

dv

2p.2x

—

dn

(1—2m1) 3/2F(ml),

(3.82)

dZ dn

2G(m1)v x(1 2m1) 3/2’

(3.83)

—

—

dx

12

—(1— 2m1)

~ ~ 2 G(m1)v

dm1 ~2(I

—

(3.84)

/ K(m1)v,

(3.85)

,

—2m1)”

where n is the number of quarter cycles, and eq. (3.73) gave m 2. We are 1 = for 1 kn. now assuming ~ is large and taking the continuum approximation By integrating (3.82) we can get an estimate for ~, —

1

dv 4

0

[i

+

2(1

—

.

(3.86)

v2)/p.2x2~/ xp.2F(k2)

To integrate (3.86) we must have x(v) and F(v). To find x(v) let us call u 1 i,2 We see that —

dlnZ dIn u

u =

—

dZ

2Zv dv

G(k2) F(k2)

=

(3.87)

and thus, dZ F lnu—_f_~_~.

(3.88)

From eq. (3.73), we have 1 Z=—n ‘~

1 2

(1—2m 1)

—1

(3.89)

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Imaginary wormholes

281

yielding dZ

dm1

Z

m1(1—m1)(1—2m~)~

=

(3.90)

Inserting eq. (3.90) into (3.88) we have dmi

2

ln(1

—

v

)

—

In

U

f m1(l

—

m~K(m1)+ (I

m~)(1— 2m1)

m1K(m1)

+

(1

—

2m1)E(m1)

—

2m1)E(m1) (3.91)

Now, to provide a rough estimate of (3.91), we make the approximation of the square bracket above, F/G, as F (3.92) This is not as crude as it may first appear, as F 16 —~l—~m1ln—, m1<<1. G m1

F =~,

—

G

mj—1/2

(3.93)

2x2 will be small for large x the maximum possible m Since Z < ~p. 1 Z will also be small. A more accurate estimate is F/G 1 ~ Inserting the approximation (3.92) into (3.91), integrating, and matching to (3.68) at the end of the first quarter cycle, where —-,~,

—

2l—la2I”=1——~4/L2X, 1

(3.94)

mi{i+~2]~.

(3.95)

1—L’ we get

Inserting eq. (3.95) into I finally obtain

+

4Z = (1

—

2m

2, and using the definition of Z, we 1Y

x~x

1+

1—v2 2

2p.

—1/2

(3.96)

/

J. Twamley, D.N. Page

282

Using the approximation F

I

—

Imaginary wormholes

m1 together with eqs. (3.96) and (3.95), we can

now evaluate the integral (3.86), 2 _

~x(l

+ p.2~2)_l/2(1 —

v2

+ 2p.2~2)~2

v

—

+ 2p.2~2~)/2F(cot_1

p.x I k2

=

(1

+

p.2x2)/(1 + 2p.2x2)), (3.97)

where 6 ln(cp.1~2) x + ~ln x or x 6 ~ln 6 (for 6>> 1), and F is the incomplete elliptic integral of the first kind. Letting U) p.~we can obtain expressions for N and ~ for 0)>> 1, —

1 2p.W

1 or p.~—-—,

(3.98)

2W

and for ~ -ccz 1, ~ =N+ 1

~[W(ln_)(1

-

-W

~~2)

1n(~ + l)~.

(3.99)

We note that d 0,

(3.100)

0.39p.’ —1.

(3.101)

=

dW

w0-5l

giving Nmax =.m,x

—

1

In fig. 11 we compare ~Xx, p.), as predicted by eq. (3.97), with the numerical results. The fit is surprisingly good. Thus, we can alternately label the curves by N, the number of zeros of ~ throughout the whole evolution, or by ~, the number of quarter oscillations of ~pon one side of the throat. This again indicates that only two types of wormholes exist, types (1) and (ii) referred to in subsect. 3.3. The results also show that adiabaticity condition for z is well satisfied for ~ >~ 1, and the results are even reasonably good for all ~ ~ 1. The estimates (3.100) and (3.101) also match well with the empirical estimates (3.52)—(3.54). 3.5.

APPROXIMATION TO THE ACTION

From subsect. 3.2.4, ‘E = ~‘v = —Ja3NV dt. Introducing v = a, the action can be written as ‘E = fa3V da/v, where the potential v = + —

—

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Imaginary wormholes

283

Fig. 11. Comparison between numerical and analytical results. The solid curves are the numerical results of fig. 4. The long-long dashed curves are the analytical approximations given by eq. (3.97) for N 0 5. The bold squares are the numerical estimates for the zeros of the action. The bold curve is 75/p. while the long-short dashed curve is x w~~/p., the analytical estimate of the zero-action x 0. solutions [see eq. (2.144)].

==

=

p.4y2(y2 obtain =

—

1). Using the variable x

AlE

=

and referring to subsect. 3.2.1, we

p.a

2J(Xmin),

=

where ocx3

J(X)

dxy2(1 —y2)

f

=

v

(3.102)

Differentiating J with respect to x and using eq. (3.61), dJ —

x3y2(l —y2) =

+

dy

(3.103)

.

{y2(l_y2)+Zj’/2

—

From eq. (3.79) one can show that LIZ 2/x for k2 1(Z -sz 1), and so we may treat Z as a constant in eq. (3.103) during a quarter cycle. Letting y2 =y,~(1—X2), integrating (3.103) from X= 0 to X= 1 yields dJ —

dn

k2x3

f 1)3~/2 0

(2k2

—

[(1

k2)(1

—

—

1

[k2

—

1

+

(1

3k2)K(k2)

k2X2](1

_X2)~2

_k2X2)’~~’2

+

(2k2

—

dX

(3.104)

1)E(k2)J

3/2’

3(2k2

—

1)

(3.105)

J. Twamley, D.N. Page

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Imaginary wormholes

where k2 is defined in eq. (3.72). Referring to eqs. (3.77) and (3.80) we may thus write dJ

(3F

—

dn

3(1

—

2G)x3 (3.106)

,

2m

—

2 1)~

which, together with (3.82), gives

(3.107)

6p.2[32F}.

ln order to integrate (3.107) from t’ = 1 to i. = 0, we must have x(v) and G(v)/F(v). Using the approximation (3.92) for F/G and (3.96) for x(v), and letting w p.x, we arrive at dJ

x4 —

1+v2)

(2w2_

—

3

1

(2u~ +

—

H)

dv,

(3.108)

which, when integrated, finally gives 1

2~4 AIn=2J(Xxrnin)=

(1+2w2)~2_sinh_1~

3/2

.

(3.109)

3(1+2w2)

Limiting forms of AlE are 2~4

—

V~

—ln——-3 w

for

U)

-‘<

1,

(3.110)

for

0)>>

1.

(3.111)

0)2

AlE

—i

3/.L

Eliminating (3.99) we have

0)

from eqs. (3.110) and (3.111) through the use of eq. (3.98) and

~ AlE

AlE where

E

=

2~/p.~.

—-

—

3(ln e) +

~ [ln ln

1 12p.6~2

E

—

ln p..fl

for

0) <<

1,

(3.112)

for

0)>>

1,

(3.113)

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Imaginary wormholes

I

285

I

=

Fig. 12. Comparison between numerical and analytical results for the action of wormhole solutions with N 0 5 plotted against p.. The solid curves are the numerical results, while the dashed curves are the analytical approximations given by eq. (3.109).

In fig. 12 we compare the behaviour predicted by eq. (3.109) with the numerical results. Keeping in mind the exponential compression of the action in this figure, the comparison is again remarkably good. The action (3.109) is zero when sinh~ A

=

~1

+

-~,

(3.114)

where A = 1/ %/~W, which gives w = W~ 0.46865. In fig. 11 we have shown the numerical results of fig. 4 and superimpose the curve x = W0/p.. The comparison of the analytical approximation (3.114) with the numerical results is not quite as impressive as the agreement between the loci. However, an empirical result for the zeros of the action 0.75 shows that the discrepancy is relatively small. 4. Discussion and conclusion We have explicitly shown the existence of euclidean wormhole solutions with no conserved charge. This necessitated that the scalar field be imaginary in the minisuperspace ansatz we chose. The solutions discovered possessed positive, zero, and negative actions. From the construction used, it seems likely that the negative action feature of these wormholes is not specific to these models but arises from the “wrong sign” of the matter kinetic term in the action due to the scalar field being imaginary. Presumably, such solutions with negative actions would not be included into the path integral. However, as we have noted in sect. 1, further analysis concerning the recovery of QFTICS is necessary before ruling out these saddle points. We also note that even though this analysis was performed in a minisuperspace context, the saddle points of a minisuperspace contour analysis will

286

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Imaginary wormholes

remain saddle points in the full theory. Thus, this minisuperspace analysis does yield information concerning the full theory. The stationary configurations of the action satisfying the boundary conditions of asymptotic flatness necessitate that the scalar field be imaginary. Complex scalar fields arise in the program of complexification of the contour of integration in the euclidean path-integral approach to quantum gravity [2,5,23]. However, except for ref. [23], all of the models studied have been vacuous. In this program of complexification one treats the euclidean Einstein—Hilbert action as a complex analytic functional and all stationary points, be they complex in the metric or matter fields, are treated equally. One can also imagine connecting a lorentzian evolution to these wormholes at the point of symmetry. As shown in refs. [2,21], = 0 at the connecting three-surface, where q” are all the degrees of freedom of the minisuperspace model. Thus, both the extrinsic curvature and matter conjugate momenta must vanish at this juncture hypersurface. This condition occurs in half of the wormholes discovered above. One is also interested in the stability properties of these wormholes. However, it is unclear whether the perturbations to test for such stability should be real, imaginary, or generally complex. The first variation of the action about a stationary point is quadratic in the perturbations. If, in the case of the Giddings—Strominger wormhole, one chooses the perturbations in the scale factor to be imaginary (to defeat the conformal instability), one can show that no negative modes exist if the scalar field perturbations are taken to be real. For the wormholes above, assuming 6a imaginary and 6t~real (4 = uip + 6~would then be complex), negative modes do exist. However, it is very plausible that there will exist complex 6a and 64 for which no negative modes exist. It is not apparent that any one set of phases for the perturbations of complex quantities are more fundamental than any other. A clearer definition of stability is needed for the case of complex extrema of actions of more than one variable. The discovery of these solutions shows without doubt that euclidean wormholes need not possess a conserved charge to exist. They also disprove a conjecture by Halliwell and Hartle concerning the sign of the action of such wormholes. Finally, their existence may seriously undermine current proposals concerning the resolution of the large-wormhole problem. We thank S. Davidson, J. Louko and A. Lyons for many useful discussions. This work was submitted in partial fulfillment of the doctoral degree for J. Twamley. This research was also supported in part by NSERC and by the Canadian Institute for Advanced Research. References [1] 5. Coleman and K. Lee, Nuel. Phys. B329 (1990) 387; B341 (1990) 101 [2] J.J. Halliwell and J.B. Hartle, Phys. Rev. D41 (1990) 1815

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[3] W. Fischler and L. Susskind, Phys. Lett. B217 (1989) 48 [4] J. Polchinski, Nuel. Phys. B325 (1989) 619 [5] J.B. Hartle, J. Math. Phys. 30 (1989) 452; J.J. Halliwell and R. Myers, Phys. Rev. D40 (1989) 4011; J.J. Halliwell and J. Louko, Phys. Rev. D39 (1989) 2206; D40 (1989) 1868; D42 (1990) 3997; S. Chakraborty, Phys. Rev. D42 (1990) 2924; J. Louko and P. Tuckey, submitted to Class. Quant.Grav. [6] G.W. Gibbons, SW. Hawking and M.J. Perry, Nuci. Phys. B138 (1978) 141 [71 K. Schleich, Phys. Rev. D36 (1987) 2342; J.B. Hartle, Phys. Rev. D29 (1984) 2730; J.B. Hartle and K. Schleich, in Quantum field theory and quantum statistics: Essays in honour of the sixtieth birthday of ES. Fradkin, ed. l.A. Batalin, G. Vilkovisky and C.J. Isham (Adam Hilger, Bristol, 1987); H. Arisue, T. Fujiwara, M. Kato and K. Ogawa, Phys. Rev. D35 (1987) 2308 181 SB. Giddings and A. Strominger, NucI. Phys. B306 (1988) 890; K. Lee. Phys. Rev. Lett. 61(1988) 263; C.P. Burgess and A. Kshirsagar, NucI. Phys. B324 (1989) 157; J.D. Brown. C.P. Burgess, A. Kshirsagar, B.F. Whiting and J.W. York. NucI. Phys. B328 (1989) 213 [9] SB. Giddings and A. Strominger, NucI. Phys. B307 (1988) 854; S. Coleman, NucI. Phys. B307 (1988) 867; T. Banks, NucI. Phys. B309 (1988) 493; S. Coleman, Nucl. Phys. B31t) (1988) 643; SW. Hawking. NucI. Phys. B335 (1990) 155 [10] L.F. Abbott and MB. Wise. NucI. Phys. B325 (19S9) 687 [11]J. Preskill, NucI. Phys. B323 (1989) 141 112] S. Coleman and K. Lee. Phys. Lett. B221 (1989) 242: 113] A. Iwazaki. Phys. LetI. B229 (1989) 211: 114] B. Grinstein. Nuci. Phys. B321 (1989) 439 [15] J.W. York, Pliys. Rev. Lett. 28(1972) 10S2; G.W. Gibbons and SW. Hawking. Phys. Rev. D15 (1977) 2752 [16] G.W. Gibbons and C.N. Pope. Commun. Math. Phys. 66 (1979) 267 [17]R. Schoen and ST. Yau, Phys. Rev. Lett. 42 (1979) 547 [18] J. Cheeger and D. Grommol, J. Diff. Geom. 4 (1971) 119 [19] G. Jungman and R.M. Wald, Phys. Rev. D40 (1989) 2615 [20] D.N. Page. in Gravitation: A Banff Summer Institute. ed. R. Mann and P. Wesson (World Scientific. Singapore, 1991) [21]G.W. Gibbons and J.B. Hartle, Phys. Rev. D42 (1990) 2458 [22] W.H. Press. B.P. Flannery. S.A. Teukolsky and W.T. Vetterling, Numerical recipes (Cambridge Univ. Press. Cambridge, 1990) [23] L.J. Garay. J.J. Halliwell and GA. Mena Marugln. Phys. Rev. D43 (1991) 2572