PHYSICAL REVIEW D VOLUME 48, NUMBER 10 15 NOVEMBER 1993

( #2 )reg in the electrically charged dilaton black hole

Min-Ho Lee, Hyeong-Chan Kim, and Jae Kwan Kim Department of Physics, Korea Advanced Institute of Science and Technology, 373-1, Kusung-dong, Yusung-ku, Taejon, Korea

(Received 20 May 1993)

The regularized expectation values of the field fluctuation (42),,, of an arbitrary coupled, nonconfor- ma1 massless scalar field in the spacetime of the electrically charged dilaton black hole in the Boulware, Unruh, and Hartle-Hawking vacua are evaluated. Their leading behaviors near the horizon and the spa- tial infinity are investigated. It is found that the result agrees with Page's approximation if the coupling constants are small.

PACS number(s): 03.70. +k, 1l.lO.Gh

Recently, black holes in higher-dimensional space-time have been studied, which are related to the superstring or the Kaluza-Klein theory. By introducing the dilaton fields a new black hole solution is obtained. In particular this solution is closely connected with the two- dimensional black hole which was discovered by Witten [ I ] . In four dimensions the exact solution to the electri- cally charged black hole coupled with dilaton fields was calculated by Garfinkle, Horowitz, and Strominger [2]. The electrically charged dilaton black hole contains an infinitely long throat in the extremal limit. This infinite throat can hold much information about the black hole, so it suggests another solution to the information prob- lem, which has been discussed since the discovery of Hawking radiation [ 3 ] . Therefore we need to study this metric further.

Since the discovery of Hawking radiation in black hole space-time [4], much effort has been made to evaluate various polarization effects. To determine the evolution of the black hole semiclassically, we need to calculate the regularized value for the stress energy tensor in a suitable vacuum state. Because the exact calculation of the stress energy tensor is very difficult, it is valuable to evaluate the regularized expectation value of the field fluctuation ( $2) in a suitable vacuum, which provides some insight into the content of the different vacua [ 5 ] .

In this paper we shall evaluate the regularized mean value of the fluctuation of a scalar field, satisfying the Klein-Gordon equation

in the spacetime of the electrically charged dilaton black hole, where P ( x ) = C R + 7 7 ( ~ @ ) 2 + c ~ 2 @ . g, 7, and c are coupling constants, and Q is the dilaton field. Above general coupling arises upon compactification of extra di- mensions. We will follow the same procedure as in Candelas's paper [ 5 ] .

The solution to the electrically charged dilaton black hole has the line element

r + and r - are related to the mass M and charge Q of the black hole according to

The dilaton field is given by

where cD, is the dilaton's value at r = m. The location of the event horizon is r = r + . The r = r - is another singu- larity, but we can ignore it, for r - < r + .

The important differences between the electrically charged dilaton and Reissner-Nordstrom metric are the horizon (singularity) structure and the nature of the ex- tremal limit. For a =O there are both inner and the outer horizons, at r =r,. The geometry is not singular a t ei- ther of these. However, for a > O the geometry does be- come singular a t r - because C vanishes there. In fact r = r - is a spacelike singularity. The extremal black hole occurs when r r which occurs for Q 2 = ( 1 + a ' ) M z ~ 2ad0. For a > 0 the horizon vanishes for the extremal black hole and the geometry is singular there.

The metric of a dilaton black hole is not geodesically complete. This fact can be easily checked by investigat- ing the affine parameter of the null radial geodesic near the horizon. By analytically extending the electrically charged dilaton black hole spacetime, the Kruskal-like metric is easily obtained as follows:

r / where ds2= ~ ' 4 / 3 ~ e * ' d ~ d v - ~ ~ d f i ~ , (8)

0556-2821/93/48(10)/5021(4)/$06.00 48 5021 @ 1993 The American Physical Society

5022 BRIEF REPORTS - 48

where

The Kruskal coordinate is then no longer singular at r = r + . ( 4 d ) - ' plays the role of the Hawking tempera- ture. In the extremal limit r + = r - , the Hawking tern- perature of the black hole is zero for a < 1, finite for a = 1 , and infinite for a > 1 .

Similarly to the Schwarzschild black hole, we can define the three different vacua: the Boulware vacuum, Hartle-Hawking vacuum, and Unruh vacuum [ 6 ] .

Separation of the field equation ( 1 ) with the ansatz

=(4aw)-1/2 -iwtR dm e i ( r ,w )Y im( i3 ,$ ) (10)

yields the radial equation in the form

where

(16)

The regularized mean value of ( $2) is then given by

where

~ = ( 2 / 3 ) - ' , which is the surface gravity of the black hole. Now we determine the value of x?=, (21 + 111 R i l 2 near

the horizon. The radial equation is written as

The shape of V ( r , l ) suggests that the radial functions near the horizon and the spatial infinity have the asymp- totic form

The regularized expectation of the field fluctuation is defined as

where u ( x , x l ) is half the square of the geodesic distance between x and x ' , and o p = V p o [7 ] . Taking the time separated points x = ( t , r, 8 ,$ ) and x' = ( t + 6, r , B,$ ), the geodesic interval is expanded in powers of 6:

where

Let us define a new radial variable by

Near the horizon, the Eq. (21) can be approximated as

In the above we have replaced I ( I + 1 ) by the asymptotic form for large I, and dropped PC', which is small in com- parison with x -I . The solution of Eq. (24) is then

where Ki, and ILiq are modified Bessel functions and q =2aB. Comparing the asymptotic form of R y t with Eq. ( 1 3) near r = r + , we found that

48 - BRIEF REPORTS 5023

a/=2IpiqIY - i q ) - ' c - . ' ( r + ) ( r + - r - )lq . (26) 5 (21 + I ) I R ; " ' - ~

/ =o (21 + l ) l B / 1 2 .

It follows then that near the event horizon we have c 2 ( r ) I=o

m

4 w 2 ~ -2 ( r ) , (27) Inserting ~ q s . (27) , (28) , into ~ q s (181, (191, (201, we have, 2 (21 + l ) I ~ ~ ~ ~ - 4 w ~ - - - . - - - / =o f l r - r + near the horizon,

In the high-frequency limit the leading behavior of the regularized expectation value of the field fluctuation in the Boulware vacuum is determined by the last two terms in Eq. (29). In other vacuum cases, is finite in the limit r - r + .

For the Hartle-Hawking vacuum, it is convenient to follow the Candelas's paper [ 5 ] . In the vicinity of the event hor- izon, the potential term PC' is very small if (2c-77) and [ are small. For such (26-77) and [, the Hartle-Hawking Green function takes the form

The regularized value of 4' in the limit r - r + is then

where T , , , = K / ( ~ T A A and T:,, =( A - 4 ~ ; a ~ ; a ) / ( 4 ~ ) ~ . This is the same as the regularized mean value of the field fluc- tuation obtained by Page when (26-77)=+ and c = O [8] . , This implies that the effect of the potential P ( x ) C 2 can be

neglected near the event horizon if (26-77) and c are small. The value of the potential p ( x ) c 2 at r = r + is <( [a /( 1 +a ) ] r - / r + ), for the scalar curvature vanishes at the horizon.

At the spatial infinity using the asymptotic forms

we can easily obtain

5024 BRIEF REPORTS 48

These are exactly of the same form as the equations describing vacuum polarization effects obtained in the space-time of the Schwarzschild black hole in Ref. [ 5 ] . I t is because the dilaton black hole metric is asymptotically flat. In particular, the potential PC' vanishes at r = a.

In summary, we evaluate the approximated expectation values of the field fluctuation ( + 2 ) for the various vacua in the space-time of the electrically charged dilaton black hole. Even though the existence of the dilaton fields changes the coupling term, the result is not affected by the dilaton fields if (6-71) and are small. I t is noted that the result has physical meaning only near the event horizon and spatial infinity.

This work was partially supported by KOSEF (Korea Science and Engineering Foundation).

[I] E. Witten, Phys. Rev. D 44, 314 (1991). [5] P. Candelas, Phys. Rev. D 21, 2185 (1980). [2] D. Garfinkle, G . T. Horowitz, and A. Strominger, Phys. [6] D. G. Boulware, Phys. Rev. D 11, 1404 (1975); J. B. Hartle

Rev. D 43, 3140 (1991); C. F. E. Holzhey and F. Wilczek, and S. W. Hawking, ibid. 13, 2188 (1976); W. G. Unruh, Nucl. Phys. B380, 447 (1992). ibid. 14, 870 (1976).

[3] T. Banks, M. O'Loughlin, and A. Strominger, Phys. Rev. [7] B. S. DeWitt, The Dynamical Theory of Groups and Fields D 47, 4476 (1993); J. Preskill, Report No. CALT-68-1819, (Gordon and Breach, New York, 1965). hep-th/9209058, 1992 (unpublished). [8] D. N. Page, Phys. Rev. D 25, 1499 (1982).

[4] S. Hawking, Commun. Math. Phys. 43, 199 (1975).