Entropy of black holes
285
The most elegant derivation of the quantum result utilizes Feynman's pathintegral formulation of quantum field theory [9]. In this, the generating function
W[J] for the Green functions of the quantum field theory is a functional integral
over classical fields l/J
W[J] = f Vl/Jexp{iS[l/J. J]}
(10.50)
where
S[l/J. J] = f d 4 x [C(l/J, B/Ll/J) + Jl/J]
(10.51)
is the action integral with C the Lagrangian density. and J a source current.
In the present context. the functional integral is over both matter fields l/J and
metrics 8/Lv. The latter should include both metrics that can be continuously
deformed to the flat metric as well as homotopically disconnected metrics such
as those of black holes. The evaluation of the action integral is problematic for
black-hole metrics because of the spacetime singularities they contain. However,
this difficulty can be surmounted by complexifying the metric and evaluating the
integral over a contour that avoids the singularities [10].
As an example. we take the Schwarzchild metric (10.2) which. as already
noted. has a coordinate singularity at r = 2M and a curvature singularity at
r = O. The former can be removed by transforming to Kruskal coordinates in
which the line element has the form
-r/2M
ds 2 = 32M 3 e _ _ (dz2 _ dy2) _ r2 dn~
(10.52)
r
where
-Z2 + i = (2~ - I) e r / 2M
(10.53)
y +z = e,/2M.
(10.54)
y -z
The singularity at r = 0 is now on the surface Z2 - y2 = 1 but it can be avoided
by defining a new coordinate ~ = iz. Then the metric has the Euclidean form
e-r/ 2M
_ds 2 = 32M3 _ _ (d~2 +di) +r2dn~
(10.55)
r
where now
~2 + y2 = (2~ - I) e r / 2M
(10.56)
y - i~ = e,/2M.
(10.57)
y +i~
Thus. on the contour where ~ and y are real. r is real and r > 2M. Further. on this
contour we define an imaginary time t" by 1" = it and then (10.57) shows that t" =