fulltext

Effects of quantum fields on singularity

in spherical gravitational collapse (*)

M. SERRA(**)

ENEA, C. R. Clementel, Centro Dati Nucleari Via Martiri di Monte Sole 4, 40129 Bologna, Italy (ricevuto il 23 Luglio 1996; approvato il 4 Febbraio 1997)

Summary. — Using a semiclassical model for the formation and evaporation of a 4D black hole, we examine if the presence of a quantum matter source of the gravitational field can modify the occurrence of the classical singularity r 40. In the limit r K0, we solve analytically the equations of the back-reaction for a massless scalar quantum field propagating in the space-time. We find two solutions and both are singular at r 40.

PACS 04.20.Cv – Fundamental problems and general formalism. PACS 04.60 – Quantum gravity.

1. – Introduction

The theorems of Hawking and Penrose [1] demostrate that space-time singularities are expected to develop inside black holes. From these theorems that tell us almost nothing about the nature and the location of singularities, it follows that the geodesic incompleteness of a space-time is mainly due to the conditions imposed on the classical stress-energy tensor Tmn, source of the gravitational field. The presence of singularities

is usually interpreted as the sign that General Relativity, a classical theory, is no more valid since quantum effects have to be taken into account when the curvature reaches the Planck limit [2].

A further demostration that quantum fields have important effects on black holes space-times is Hawking’s discovery [3] of quantum creation of particles induced by a collapsing object. A knowledge of the energy-momentum tensor of the quantum field “near the object” may clarify the creation process.

In order to take quantum effects into account in the description of the gravitational collapse, we need to know how to describe the interaction between quantum matter and the gravitational field. Although an exact description of this interaction would require a

(*) The author of this paper has agreed to not receive the proofs for correction. (**) Guest researcher.

full theory of quantum gravity, quantum matter effects on the gravitational field may be studied using the semiclassical approach to the quantized space-time geometry. In this approach [4], using the renormalized expectation value of the stress-energy tensor aTmnbren of a quantum field propagating in the space-time as the source term in the semiclassical Einstein equations, it is possible to calculate the back-reaction of the quantum field on the geometry.

In this paper, since aTmnbren does not satisfy all the classical conditions taking to the geodesic incompleteness of space-times, we examine the possibility that the presence of quantum matter modifies the classical singularity of the gravitational collapse at r 40 using a self-consistent 4D semiclassical model describing the formation and the evaporation of a spherically symmetric black hole [5]. Here, self-consistency implies that aTmnbren is expressed in function of the unknown metric gmn. Moreover, because of

the complexity of the back-reaction equations, it is assumed that the internal region to the event horizon of the black hole is homogeneous. In the limit r K0, two analytical solutions are found and the analysis of these solutions shows that both are singular at r 40.

This paper is arranged as follows. In sect. 2 the model is presented and the field equations for the gravitational collapse are derived. Since they are singular at small radii a regularizing prefactor is introduced. In sect. 3, the field equations are solved in the limit r K0 and the properties of the two solutions are discussed. Section 4 contains some discussions of the results.

2. – Model and the field equations

To solve the back-reaction problem, we use the spherically symmetric metric ds2

4 exp [ 2 f (h) ](2dh21 dx2) 1r2_{(h) dV}2_, (1)

where exp [ 2 f (h) ] is the conformal function to the 2D Minkowski space-time and r (h) is defined in such a way that 4 pr2_{(h) constitutes the area of a 2-sphere. The assumption} that the metric functions f and r depend only on the timelike variable h, corresponds to the hypothesis that the internal region to the event horizon is homogeneous.

As matter source of the gravitational field, we consider the renormalized expectation value of the stress-energy tensor aTmnbren of massless scalar fields propagating in the space-time described by the unknown metric gmn. This quantity is

the cause of the Hawking flux and of the black-hole evaporation. In the absence of quantum constraints on the boundaries it is entirely determined by the geometry. On the other hand, in 4D the analytical expression of aTmnbren is still unknown for an arbitrary spherically symmetric space-time. As a model for the quantum source of the gravitational field we take the expectation value of the tensor [6] arising from the quantization of a 2D massless field in the 2D curved background described by ds2₄ exp [ 2 f (h) ](2dh2

1 dx2). It is divided by 4 pr2_{to represent the four-dimensional radial} dependence. Thus, the nonzero components of this tensor are given by (1)

aThxb 42 a 16 pr2 d2_f dh2 , (2) (1_{) We use G 4ˇ4c41.}

aThhb 4 a 16 pr2

y

d2_f dh2 2

g

df dh

h

2

z

, (3) aTxxb 4 a 16 pr2

y

d2f dh2 2

g

df dh

h

2

z

, (4)

where the constant a depends linearly on the number of massless fields.

Introducing the prefactor P f 1 2a/r2_{, the independent semiclassical Einstein} equations read P d 2_f dh2 2 1 r2

g

dr dh

h

2 2 exp [ 2 f ] r2 4 0 , (5) 1 r

g

1 r2 2 1

h

d2_r dh2 2 1 r2

g

1 r2 1 P

h

g

dr dh

h

2 2 exp [ 2 f ] r2 4 0 . (6)

We can note that, as in dilatonic gravity [7], the inclusion of the quantum matter leads to singular equations at the critical radius ra4k1 /a. In 4D, the presence of the

singularity is caused by the assumption of the validity of eqs. (2)-(4) in the Planckian domain. Moreover, eqs. (5) and (6) are a good approximation of a black-hole evaporation description when the prefactor P is still close to 1.

To solve the dynamical equations in the Planckian domain, one may regularize the prefactor P at small radii as follows:

Pn4 1 2 a r2 1 1 1 (a/r2₎n , (7)

then, the metric functions f and r will depend on the value of the parameter n. Introducing the variables x and z defined as

x(r) 42dr dh , (8) z(r) 4 exp [ 2 f ] xr2 (9)

and the prefactor Pn, eqs. (5) and (6) become

Pn 2 d dr

k

2 x r 1 dx dr 1 xz dz dr

l

2 x r2 2 z 4 0 , (10) 1 r

g

1 r2 2 1

h

2 1 r2

g

Pn1 1 r2

h

x 2z40 . (11)

3. – Solutions of the dynamical equations

Since our interest lies in studying the classical singularity r 40 of a spherically symmetric metric, we consider the dominant divergent terms in eqs. (10)-(11) in the

limit r K0. In particular, (n, nF1, we can observe that eq. (11) does not depend on the parameter n. Thus, this equation becomes

rdx

dr 2 x 4 r 4_{z .} (12)

Moreover, assuming that in the limit r K0 the term r4

z is negligible, we find the following solution for x:

x 4r . (13)

In the following, we have substituted this solution into eq. (10) and solved the resulting differential equation for the function z(r), distinguishing for different values of the parameter n.

Two analytical solutions have been found in the limit r K0. The first (A) exists for each value of the parameter n, n F1, and does not depend on a, that is on the number and kind of fields which are in the space-time. The second solution (B) exists for n 41 and a G0. The constant a derives from the trace anomalies calculation [4], which is positive in the region outside the event horizon. On the other hand, it is not possible to estimate its value in the limit r K0 precisely. In fact, as we will see, in this limit, the components of the curvature tensor Rmnand its norm assume infinite values.

Having found the solutions A and B, we have verified their consistency with the hypothesis r4

z A0. To examine if these solutions lead to a space-time singularity at r 40, we have evaluated the components of the tensor Raband the Ricci scalar R in the

limit r K0. Finally, we have examined the character of the singularity by calculating timelike geodesics.

The nonzero components of Raband the expression of R for the line element (1), in

function of the variables z and x defined in eqs. (8)-(9), can be written as

R004 22 x r dx dr 1 3

g

x r

h

2 2 x 2 d2_x dr2 2 1 2 x z dx dr dz dr 2 x2 2 d2 dr2ln z 1 x2 rz dz dr , (14) R114 3 x r dx dr 1

g

x r

h

2 1 x 2 d2_x dr2 1 1 2 x z dx dr dz dr 1 x2 2 d2 dr2ln z 1 x2 rz dz dr , (15) R334 sin2uR224 1 1 x r2z 1 1 rz dx dr , (16) R 4 1 zxr3(2R001 R11) 1 2 r2R22. (17)

3.1. Solution A. – Introducing eq. (13) in eq. (10), we obtain Pn 2 d dr

k

r z dz dr

l

2 1 r 2 z 4 0 . (18)

This equation is always verified if z(r) has the following form: z(r) 421

r . (19)

In this case, the solution of the semiclassical eqs. (10)-(11) is x 4r ,

(20)

z 421 r . (21)

Integrating eq. (8) and substituting the result in eq. (9), we obtain for the metric functions the following expressions:

r(h) 4exp [2h] , (22)

exp [ 2 f (h) ] 42exp [2h] . (23)

In the coordinates (h , x , u , f), it corresponds to h K1Q. Thus, in the limit rK0, the line element (1) becomes

ds2

4 exp [22 h](2dx21 dh21 dV2) , (24)

where we can note that, now, x is the timelike coordinate, while h has become the spatial one.

Substituting the solutions (20)-(21) in eqs. (14)-(17), we obtain R004 0 , (25) R114 3 , (26) R224 21 , (27) R 42 1 r2 . (28)

Thus, in the limit r K0, the components of Rab are constant, but R is a divergent

quantity. It follows that the space-time is singular at r 40.

Finally, we calculate the proper time that a timelike geodesic takes to reach the singularity. In the limit r K0, we find that the proper time t is given by

t 4exp [h] (29)

from which it follows that a timelike geodesic reaches the singularity r 40 in a finite time.

Thus, the space-time described by eq. (24) presents a singularity at r 40 and, furthermore, a timelike geodesic reaches the singularity in a finite proper time.

3._{2. Solution B. – In the limit r K0 and for n41, the prefactor P}n given by eq. (5)

becomes

P14 r2

a . (30)

Introducing eq. (13) in eq. (8) with Pnreplaced by P1, the differential equation to solve for z(r) becomes d dr

g

r z

h

2 2 a r2

g

1 r 1 z

h

4 0 . (31)

Assuming r22_{z b 1, the integration of eq. (31) leads to} z 4exp

k

2 a

r2

l

. (32)

In the limit r K0, the solution (32) verifies the hypothesis for aG0. In this case the solutions of field eqs. (9)-(10) are

x(r) 4r , (33)

z(r) 4exp

k

2 a r2

l

. (34)

From eq. (9), we obtain the conformal function expression exp [ 2 f ] 4r3exp

k

2 a

r2

l

(35)

then, the line element (1) becomes ds2 4 2r2

g

exp

k

2 a r2

lh

dr 2 1 r3

g

exp

k

2 a r2

lh

dx 2 1 r2dV2. (36)

Now r (and then h) is the timelike coordinate, while x is the spacelike one. Using eqs. (14)-(17), the components of Raband the scalar R become

R004 5 2 14 a r2 , (37) R114 1 2 2 a r 2 6 a r2 , (38) R224 1 1 2 rexp

k

2 2 a r2

l

, (39) R 4 D r6exp

k

2 2 a r2

l

. (40)

In the limit r K0, they are all divergent quantities. As before, we calculate the proper time that a timelike geodesic takes to reach the singularity. In the limit r K0, t is given by t 4A21



0 r1 dr r4_exp

k

2 a r2

l

(41)

which is finite for r 40 and where A 4r2

g

exp

k

2 a r2

lh

dx dt . (42)

The space-time described by eq. (36) shows a singularity at r 40, which is reached in a finite proper time by a timelike geodesic. In the limit r K0, we can make an analogy between this solution and the Schwarzschild solution.

4. – Conclusions

Using a self-consistent 4D semiclassical model for the spherically symmetric gravitational collapse, two analytical solutions of the field equations are found in the limit r K0. For both these solutions the point r40 of the space-time is a singularity.

Thus in a semiclassical approach, the inclusion of quantum matter in the source term of gravitational field equations does not lead to avoid the classical singularity r 40 inside a black hole.

It must be noted that, in this case, the nature of the singularity cannot be discussed. In fact, to do that, exact solutions of the semiclassical Einstein equations are nedeed. It can only be said that, in the limit r K0, in the solution A, given by eq. (24), the ex-change of coordinates h and x leads to a possibility of the presence of the event horizon. Finally, we have to note that these solutions for spherically symmetric gravitational collapse are obtained in the limit r K0. Consequently, it is not possible to establish if they represent physical solutions. In order to do this, it is necessary to integrate the field equations to the black-hole event horizon where it is possible to set boundary conditions. At this moment, this result can be achieved only numerically [5].

* * *

I would like to thank Prof. R. BALBINOTfor suggestion of the problem and for many

illuminating comments during the work and Prof. S. ZERBINI and Dr. L. VANZO for

precious criticism.

R E F E R E N C E S

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