Progress of Theoretical Physics Supplement No. 114, 1993

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On the Mass of 2d Quantum Black Hole*) Tsukasa TADA**l

Uji Research Center, Yukawa Institute for Theoretical Physics Kyoto University, Uji 611

We introduce a mass function for spherically symmetric gravitational systems. Our interest lies in its behavior when quantum effect is included. We present some results obtained for the CGHS model.

§ 1.

Introduction

The quantum behavior of black holes has been attracting much interest since the discovery of Hawking effect. 2 l However the difficulty of the quantum gravity prevents its study. A possible way to avoid this difficulty is to take advantage of the significant success of two-dimensional quantum gravity. One may wonder if there is any counterpart of 4d black hole in two-dimensional gravity. The answer is yes. 3 l And a model proposed by Callan et al. 4 l- 6 ) has been studied subsequently in the expectation that this model may reveal the secret of the evaporating black hole. The virtue of this model lies in the fact that we can solve it exactly although it describes the formation of a black hole. Its exact solvability may partly come from the symmetry that the model own. It was pointed out that the model can be obtained from the n---,) oo limit of the spherically symmetric n-d Einstein gravity. 7 l This interpretation may help to understand this model. Unfortunately, quantum effect breaks its exact solvability. We have troubles in following the process of black hole formation and evaporation. We need, therefore, some index to follow the black hole state. In four dimension there is a mass function introduced by Fischler, Morgan and Polchinski. 8 l This function gives a value that corresponds to black hole mass at each space-time point. Tomimatsu observed the expectation value of this mass function actually decreases. 9 l The similar mass function also exist in 2-d dilaton gravity. 10 ),ll),l) In particular Mann obtained the expression for wider class of actions. Here we follow Mann's derivation. We derive the mass function from the general n-d Einstein gravity in § 2 though we may be interested only in n =4 and n---,) oo case. In § 3 we concentrate on the case that matter shock wave forms a black hole in 2-d dilaton gravity. 4 l We summarize our result in Ref. 1). There we calculated the mass function in the case quantum effect is included. Though we could obtain the *)

**)

Based on the work with S. Uehara. 0 JSPS fellow.

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value of it for limited regions, we found a change of the mass forced by quantum effect. § 2.

Mass function

Now we introduce a function on space-time, which may be a good index for black hole state at each space-time point. Having been inspired by Tomimatu's work 9 > we derived a two-dimensional version of the mass function for spherically symmetric 4d gravity. We used Hamiltonian formal with ADM decomposition of the metric in Ref. 1) following the derivation in Ref. 8). We found a certain combination of (two) Hamiltonian constraints to be a total divergence of the mass function as in Ref. 8). In the following we rather treat the energy momentum tensor. This enables us to keep the covariance in the derivation and make it transparent. In fact Mann derived mass functions for a wider class of gravitational actions. 11 > Here we follow Mann's treatment. Let us consider the n-dimensional Einstein gravity. We take its metric as the following form: (1)

here hab is 2 X 2 matrix while rij is ( n- 2) X ( n- 2) matrix. Thus we have decomposed n-dimensional space-time into two-dimensional space-time and (n-2)-dimensional space. We regard Q as a function on two-dimensional space-time. What we have done in the above is reducing n-dimensional space-time into spherically symmetric one. Q is nothing but the transverse radius. We are going to integrate out Yii in the Einstein-Hilbert action. (2)

(3)

where R< 2 > and R stand for scalar curvature of 2d space-time and n-2-d space each other. We can set fY to be 1 without loosing the generality. After pursuing the partial integral in terms of two-dimensional space-time we find

It was pointed out by Soda 7 > that n--" oo limit of the above action gives the two-dimensional dilaton gravity. 8 >' 4 > Rewriting Q as ¢= -n-2 --logQ, 2

(5)

On the Mass of 2d Quantum Black Hole

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one finds

In fact, n ~co limit yields the following action: (7) where we have put f dn- 2 x= V and f dn- 2 xR xo +) one finds a black hole with mass M. Therefore an observer at each space-time point may know whether a black hole has formed or not. We expect that this mass function could be a good index for the state of the black hole.

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Our main motivation is to study the black hole evaporation through this mass function. It is apparent from Eq. (20) that this black hole does not change its mass once it forms. Therefore we should include the quantum effect to make it evaporate. Concerning the one-loop approximation, the quantum effect corresponds to the following term;

(21) Unfortunately, once we include the above term in the action, we cannot solve the model exactly. Therefore our analysis is limited to certain regions. Let me summarize the results obtained in Ref. 1). First at past null infinity (x~ - oo) .we find 5Jilx-~-oo . M .

(22)

This matches with the result of the other calculation of the ADM mass 4 > because the asymptotic value of the mass function at infinity (x+-x-~oo) may give the ADM mass. 10> And the derivative of the mass function with respect to x- proved to be zero, (23)

Next we turn to the matter shock wave line (x+=xo+) where we find (24)

The divergence of (24) at x-= -(K/A 2 xo+) shows the break down of the semiclassical analysis. As for its derivative along x+ direction, it has quite complicated form. However its value remains negative at least till the apparent horizon, which is defined by the following on the matter shock wave, 6 > K

(25)

(see Fig. 1). The apparent horizon is the point where the light cone shrinks to the vertical line. Reminding Q ( =exp[ -2¢/(n-2)]) is the transverse radius, it is easy to understand that the apparent horizon is given by the condition f7 ¢=0, which leads to Eq. (25). While 5J1 diverges at the· singular point (xo +, - K/A 2 Xo +), its value at the apparent horizon remains finite. It was pointed out that the singular point and the apparent horizon will collide at a certain space-time point, 6> which is now beyond our analytic investigation. It is interesting to see the behavior of the mass function to these regions, hence the 0 + +oo X numerical study is now in progress. 13 > Fig. 1.

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Acknow ledemen ts The author would like to thank Soryuushi Shogakukai for financial support. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

T. Tada and S. Uehara, Phys. Lett. B305 (1993), 23. S. W. Hawking, Commun. Math. Phys. 43 (1975), 199. E. Witten, Phys. Rev. D44 (1991), 314. C. G. Callan, S. B. Giddings, ]. A. Harvey and A. Strominger, Phys. Rev. D45 (1992), R1005. T. Banks, A. Dabholkar, ·M. R. Douglas and M. O'Loughlin, Phys. Rev. D45 (1992), 3607. ]. G. Russo, L. Susskind and L. Thorlacius, Phys. Lett. B292 (1992), 13; Phys. Rev. D46 (1992), 3444; Phys. Rev. D47 (1993), 533. ]. Soda, Prog. Theor. Phys. 89 (1993), 1303. W. Fischler, D. Morgan and ]. Polchinski, Phys. Rev. D42 (1990), 4042. A. Tomimatsu, Phys. Lett. B289 (1992), 283. V. P. Frolov, Phys. Rev. D46 (1992), 5383. R. Mann, Phys. Rev. D47 (1993), 4438. Y. Oshito et al., Nagoya Preprint DPNU-92-46 (1992). T. Tada and S. Uehara, in preparation.