Entropy of black holes
287
and we may use the remaining coordinates for the ~a. Then the unit outgoing
normal is
1 r",
I" -
-~Or
(10.63)
n -~
and
Kab = !nl" agab .
(l0.64)
2
axl"
For the Schwarzchild metric (10.2). this gives (exercise 8)
1/2
2R-3M
and
(
2M) R2 sin(J
K(g) = R2(l _ 2M/ R)I/2
A = -i I - If
(10.65)
and then
f.M H[K (g) - K(q)) d' . ~ -i4"P [2R - 3M - 2R (1 -2: f2].
(10.66)
Thus, in the limit R -+ 00, the Schwarzchild action is
I.
ill'
S[
0]
g(S),
= -lfJM = -M.
(10.67)
2
K
The general result [ 10] for Kerr-Newman metrics g(K N) of the form (10.35)
is that the action integral has the value
iJr
S[g(KN), O] = -(M - H Q).
(10.68)
K
(The rotation does not affect the evaluation of the action.) The dominant
contribution to the path integral (10.50) comes from fields (in our case metrics
g) with the correct periodicity that minimize the action. Such fields are solutions
of the classical equations of motion and. in the present context. are the KerrNewman metrics g(KN). Thus.
In W[O] = In Z ~ is[g(KN). 0].
( 10.69)
In a thermodynamic system, the partition function Z for a grand canonical
ensemble at temperature T = fJ- 1 with chemical potentials I L; associated with
conserved charges Ni is defined as
Z=Trexp [ -fJ(H- ~IL;N;)]
(10.70)
I
and its logarithm is related to the free energy F by
InZ = -fJF = -fJ( E - TS - ~lljNj),
(10.71)
I