7 September 1995

ELSEVIER

PHYSICS LETTERS B

Physics Letters B 357 ( 1995) 295-299

Entropy for extremal Reissner-Nordstrom black holes

Amit Ghosh ‘, P. Mitra2 Saha lnstirute of Nuclear Physics. Block AE Bidhannagar; Calcutta 700 064, India

Received 1 June 1995; revised manuscript received 20 July 1995 Editor: H. Georgi

Abstract

We find in the brick wall model that the entropy of scalar matter in an extremal Reissner-Nordstrom background does not vanish and in fact has a stronger divergence than usual. We also interpret earlier results on the classical action of these black holes to argue that the mass may be a measure of the gravitational entropy, somewhat like in string models.

A black hole has a horizon beyond which no mat-

ter or information can leak out. The absence of in- formation about the region inside the horizon mani-

fests itself in an entropy. There have been different

approaches to the evaluation of this thermodynamic

quantity, and different expressions have appeared in

the literature in different cases. The area of the hori- zon was first interpreted as an entropy mainly on the basis of analogy [ 11. It satisfies all the thermodynam-

ical laws. The analogy was made quantitative after the discovery of Hawking radiation [2] and the associ-

ation of the Hawking temperature with a black hole.

The entropy is not understood in terms of the usual formulation of entropy as a measure of the number of

states available, but the functional integral does lead to a partition function from which the area formula for

entropy can be obtained [3] by neglecting quantum fluctuations.

That formula is supposed to describe the gravita-

tional entropy corresponding to a black hole. A re- lated object is the entropy of quantum fields in black hole backgrounds [ 4-61. This possesses divergences,

’ E-mail: amit@saha.emet.in.

*E-mail: mitra@saha.emet.in.

Ekvier Science 5.V.

SffiDf 0370-2693(95)00922-l

but the area of the horizon appears as a factor. The

values thus obtained are usually thought to be addi- tional contributions to the entropy of the black hole-

field system. Then the interpretation is that the gravi-

tational constant gets renormalized in the presence of

quantum fields [ 51. A view held by some people is

that the gravitational entropy itself can be imagined to arise from contributions of this type.

Extremal black holes often have peculiarities not

present in the corresponding non-extremal cases [ 71.

For extremal dilatonic black holes, where the tempera-

ture is nonzero, the area vanishes. It has recently been

found that the entropy of a scalar field in this back- ground does not vanish, [ 61. For extremal Reissner-

Nordstrom black holes, the temperature is zero, but the area is nonzero. However, topological arguments

in the context of Euclidean quantum gravity suggest

that these black holes do have zero gravitational en- tropy in spite of the nonvanishing area [ 8-101. What

is actually known relates to the classical action for the Euclidean black hole configuration, and it has been argued to lead semiclassically to a vanishing gravita- tional entropy.

In this note we shall examine the two different kinds

296 A. Ghosh. P. Mitra / Physics Letters B 357 (1995) 295-299

of entropy for Reissner-Nordstrom black holes. We shall use the brick wall mode1 [ 41 to investigate what

happens to the entropy of a scalar field in the back-

ground of such a black hole. A nonzero result will

emerge. Thereafter, we shall study the connection be-

tween the action and the gravitational entropy and try to see if a nonzero entropy can be accommodated.

The metric of the Reissner-Nordstrom spacetime is

given by

+ r2dCl’ (1)

in general, with M and Q denoting the mass and the charge respectively. There are apparent singularities at

r’* = M & d/M2 - Q2, (2)

provided M > Q. Cosmic censorship dictates that this

inequality holds and then there is a horizon at r+. The limiting case when Q = M and r+ = r_ is referred to

as the extremal case. The standard formula for the Hawking temperature

of such a black hole is

(3)

Clearly this expression vanishes in the extremal case.

We shall study scalar matter in this spacetime. We

employ the brick wall boundary condition [4]. Then the wave function is cut off just outside the horizon.

Mathematically,

p(x) = 0 at r = r+ + 6, (4)

where E is a small, positive, quantity and signifies an

ultraviolet cut-off. There is also an infrared cut-off:

P(X) = 0 at r = L, (5)

with the box size L > r+. The wave equation for a scalar is

-a,( &gp”d,q) - m2q = 0. 4-G A solution of the form

cp = e-iE’fE1hn, (7)

satisfies the radial equation

2M Q2 (I - 7 + -)-YfEI +

r2 $$[(r2-2Mr

1(1+ I) +Q’$$ - [----

r2 +m21fEl =O. (8)

An r-dependent radial wave number can be introduced

from this equation by

k2(r,Z,E) = (l-~+~)-‘[(1-~+~)-‘E2

l(lS 1) - r2 -m2]

Only such values of E are to be considered here as

make the above expression nonnegative. The values are further restricted by the semiclassical quantization

condition

L

II,??- = s

drk(r,l,E), (10)

r,.+c

where nr has to be a nonnegative integer.

To find the free energy F at inverse temperature p one has to sum over states with all possible single-

particle combinations:

PF = c log( I - e-PE> n, ,l.lni

s dl (21+ 1) s

dn, log( 1 - e+)

y-i,, (21+ 1) jd(j?E) (epE - 1)-h,

=-g/dl(2Z+l) dE(ePE-I)-’ 7r s L

X s dr (I - E +$)-I r

r++c

x F-(,_x+$)(y+m*) J r L

w =-- 3r s

dr (1 - y + $)-2r2

r++E

X s

2

dE(e~E-1)-‘[E2-(l-~+~)m2]3/2.

(11)

A. Ghosh, P. Mitra/ Physics Letters B 357 11995) 295-299 297

Here the limits of integration for 1, E are such that the arguments of the square roots are nonnegative. The 1

integration is straightforward and has been explicitly

carried out. The E integral can be evaluated only ap-

proximately.

The contribution to the r integral from large values of r yields the expression for the free energy valid in

flat spacetime because of the asymptotic flatness:

We ignore this part [ 3,4]. The contribution of the hole is singular in the limit E --+ 0. The leading singularity

for a non-extremal black hole is linear:

(13)

where the lower limit of the E integral has been ap- proximately set equal to zero. If the proper value is

taken, there are corrections involving m2p2 which will

be ignored here. This result reduces to the formula [ 41 for the Schwarzschild black hole and more generally

yields an entropy proportional to the area if ,L? is re-

placed by the reciprocal of the Hawking temperature and the cutoff E in r replaced by one in the “proper”

radial variable defined by

(14)

(cf. [ 41) . The divergence becomes quadratic as usual

in terms of the new cutoff Z = 2

The result changes if the black hole is extremal:

F 2dr2, r+ 4

e”=-~(p). (15)

The contribution to the entropy due to the presence of

the black hole can be obtained from the formula

This gives

(16)

(17)

This expression diverges cubically in the cutoff in r (cf. [ 111). However, as mentioned above, one should

use a cutoff in the “proper” radial variable. This is

defined by

df* = dr*

(I- z&)2’ (18) r

i.e.,

r”=r+r+log( r - r+ -).

rt

The horizon is at i: = --oo. Thus the cutoff is at a large

negative distance P = --A, where

-A =r+(l +logt) +E. (20)

Consequently, the divergence is exponential:

s ex x g(:)3exp[c]. (21)

If the usual vanishing expression for the tempera-

ture is accepted, the expression (2 1) for the entropy

also vanishes and this is one way of understanding

this entropy. However, the temperature may be nonva-

nishing and even arbitrary, because [ 91 the Euclidean solution can be identified with an arbitrary period j?.

For a general /3 (21) is nonzero and in fact expo-

nentially divergent, whereas non-extremal black holes have only a quadratic divergence.

This is the main result of our paper. Whereas the

gravitational entropy seems to be zero [ 9, lo], the en-

tropy of the scalar field is even more singular than in the non-extremal case. (Cf. however the claim made in

[ 121 about Pauli - Villars regularization.) The back-

ground of an extreme dilatonic black hole also gave

a nonzero result when a zero was expected (for a

different reason), but there the quadratic singularity did drop out and only a logarithmic term remained.

The exponential divergence in Reissner-Nordstrom is

therefore even more surprising. The question then is whether the gravitational en-

tropy is really zero. How compelling is the connection between the classical action and the entropy? There will be quantum corrections, naturally, but even at the semiclassical level one has to recognize the existence of an ambiguity in the definition of an entropy for a charged black hole [ 131. To analyse this issue, let

248 A. Ghosh, P. Mitra / Physics Letters B 357 (19951295-299

us recall that while for the Schwarzschild black hole, the classical action is related to the canonical partition

function by

e-l =Zcan-e-~ (22)

it is the grand partition function which is involved in

the case of charged black holes [ 31:

e -I = Zgrand = (?- M--l;--c”Q. (23)

Here @ is the analogue of the chemical potential for

the charge Q. Using the fact that the black hole is

extremal, we can write

s_ M(‘-@) -1,

T

Now [9] I cc l/T. It has to be remembered that the

temperature here is not a function of the mass but com-

pletely arbitrary. The above equation then suggests that the two terms on the right hand side with their l/T

dependence cancer out and leave S = 0. This is based

on the assumption that @ is independent of the exter-

nal temperature T. Note, however, that the first law of

thermodynamics can be written as

TdS=dM(l -Q), (25)

where it is understood that only processes which do

not destroy the extremality of the black hole are to be considered. It is to be emphasized that extremal and

non-extremal black holes are topologically dissimilar

systems, so the restriction to such processes is natural.

These two equations then imply

&M-g-I. Since S cannot depend on the arbitrary temperature to

which I is supposed to be inversely proportional, the only possible conclusion is that I = 0 (cf. [ IO] ) and

dS dM -=- S M’

i.e.,

(27)

S=kM, (28)

where k is an undetermined constant. If k is to vanish, we are back to a vanishing entropy, but (28) is a more

general possibility. Note that the potential is related to the temperature by the formula

@ = 1 - kj;, (29)

which again involves the constant k. If one were to take the limit of Q + M in non-extremal black holes, one would get Cp = 1, which would imply the vanish-

ing of k, but it has to be remembered that the definition

of @ depends on the entropy [ 131 and in any case it

is understood that extremal black holes have proper-

ties which cannot be obtained continuously from non-

extremal black holes. So k need not vanish,

Though the value zero of k is certainly appealing

in many ways, we cannot truly fix this constant. The indeterminate nature of this scale factor should not

however be surprising. It should be remembered that in the early seventies the laws of black hole physics

merely indicated the possibility of interpreting the area as an entropy without fixing the scale, The scale there

could be fixed only after the semiclassical calculation

of Hawking radiation yielded a definite scale involv- ing Planck’s constant. In the present situation, every- thing is classical, and the temperature, being arbitrary,

cannot set a scale. In fact, the Hawking temperature

vanishes. One can therefore only speculate that some

higher order quantum effect will enable the scale of

the entropy of the extremal black hole to be fixed. That is for the future. Meanwhile, it should be pointed

out that, just as for k = 0, the nondecreasing prop-

erty of the entropy of extremal black holes by them- selves is satisfied for nonzero k too, by virtue of the

conservation of energy. Consequently, fission of ex- tremal Reissner-Nordstrom black holes is still permit-

ted. Furthermore, in string theory, the logarithm of the density of states is known to be proportional to the

mass [ 141. Our result is certainly very suggestive of

a string interpretation. Our conclusions are as follows. First, the entropy

of scalar matter in the background of an extremal Reissner-Nordstrom black hole, as calculated in the brick wall model, is nonzero and in fact highly diver-

gent. If the gravitational entropy is to arise somehow from such contributions, it should not vanish. In fact, we have presented arguments to the effect that the gravitational entropy even at the classical level need not be zero, but should be proportional to the mass, as in string models. This proportionality constant can-

A. Ghosh. P. Mrra / Physics Letters B 357 (1995) 295-299 299

not be fixed at the classical level, and quantum studies seem to be called for.

PM wishes to thank SISSA, where most of this work was done, for its hospitality.

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