Page 31 - Nature Of Space And Time
P. 31
standard thermodynamic arguments, the log of the partition function is equal to minus
the free energy F divided by the temperature T.
F
log Z = −
T
And the free energy is the mass or energy plus the temperature times the entropy S.
F = <E > + TS
Putting all this together one sees that the action of the black hole gives an entropy of
2
4 M .
2 1
S = =4 M 2 = A
16 4
This is exactly what is required to make the laws of black holes the same as the laws of
thermodynamics.
Why does one get this intrinsic gravitational entropy which has no parallel in other
quantum eld theories. The reason is gravity allows di erent topologies for the spacetime
manifold.
Boundary at infinity
1
S IDENTIFY
S 2
In the case we are considering the Euclidean-Schwarzschild solution has a boundary at
1
2
2
in nity that has topology S × S .The S is a large space like two sphere at in nity and
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