If one does the path integral in flat spacetime identified with period β in the imaginary time direction one gets the usual result for the partition function of black body radiation.
However, as we have just seen, the Euclidean- Schwarzschild solution is also periodic in
imaginary time with period 2 π . This means that fields on the Schwarzschild background κ
will behave as if they were in a thermal state with temperature κ .
2 π
The periodicity in imaginary time explained why the messy calculation of frequency
mixing led to radiation that was exactly thermal. However, this derivation avoided the
problem of the very high frequencies that take part in the frequency mixing approach.
It can also be applied when there are interactions between the quantum fields on the
background. The fact that the path integral is on a periodic background implies that all
physical quantities like expectation values will be thermal. This would have been very
difficult to establish in the frequency mixing approach.
One can extend these interactions to include interactions with the gravitational field
itself. One starts with a background metric g 0 such as the Euclidean-Schwarzschild metric that is a solution of the classical field equations. One can then expand the action I in a power series in the perturbations δg about g 0.
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I[ g] = I[ g 0] + I 2( δg)2 + I 3( δg)3 + …
The linear term vanishes because the background is a solution of the field equations. The
quadratic term can be regarded as describing gravitons on the background while the cubic
and higher terms describe interactions between the gravitons. The path integral over
the quadratic terms are finite. There are non renormalizable divergences at two loops in
pure gravity but these cancel with the fermions in supergravity theories. It is not known
whether supergravity theories have divergences at three loops or higher because no one
has been brave or foolhardy enough to try the calculation. Some recent work indicates
that they may be finite to all orders. But even if there are higher loop divergences they
will make very little difference except when the background is curved on the scale of the
Planck length, 10 − 33 cm.
More interesting than the higher order terms is the zeroth order term, the action of
the background metric g 0.
Z
Z
1
1
1
I = − 1
R( −g) 2 d 4 x +
K( ±h) 2 d 3 x
16 π
8 π
The usual Einstein-Hilbert action for general relativity is the volume integral of the scalar curvature R. This is zero for vacuum solutions so one might think that the action of the Euclidean-Schwarzschild solution was zero. However, there is also a surface term in the
action proportional to the integral of K, the trace of the second fundemental form of the boundary surface. When one includes this and subtracts off the surface term for flat space
one finds the action of the Euclidean-Schwarzschild metric is β 2 where β is the period in 16 π
imaginary time at infinity. Thus the dominant contribution to the path integral for the
−β 2
partition function Z is e 16 π .
X
Z =
exp( −βEn) = exp − β 2
16 π
If one differentiates log Z with respect to the period β one gets the expectation value of the energy, or in other words, the mass.
β
< E > = − d (log Z) =
dβ
8 π
So this gives the mass M = β . This confirms the relation between the mass and the 8 π
period, or inverse temperature, that we already knew. However, one can go further. By
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standard thermodynamic arguments, the log of the partition function is equal to minus
the free energy F divided by the temperature T .
log Z = − F
T
And the free energy is the mass or energy plus the temperature times the entropy S.
F = < E > + T S
Putting all this together one sees that the action of the black hole gives an entropy of
4 πM 2.
β 2
1
S =
= 4 πM 2 =