Gravity and Strings

208 The Schwarzschild black hole
As we are going to see, string theory allows us to calculate the entropy and temperature of
certain BHs for which this theory provides quantum-mechanical models, from the density
of the associated microstates. In this way string theory seems to solve (at least to some
extent) the BH entropy and information problems by treating BHs as ordinary quantummechanical
systems.
7.4 The Euclidean path-integral approach
It is desirable to have an independent and more direct calculation of the BH entropy and
temperature. This can be achieved by using the Euclidean path integral as suggested by
Gibbons and Hawking [436, 514].
The thermodynamical study of a statistical-mechanical system starts with the calculation
of a thermodynamical potential. If there are certain conserved charges Ci (their related
potentials being µi), it is convenient to work in the grand canonical ensemble, where the
fundamental object is the grand partition function
and the thermodynamic potential
is related to the grand partition function by
Z = Tr e −β(H−µi Ci ) , (7.55)
W = E − TS− µiCi
(7.56)
e − W T = Z. (7.57)
All thermodynamic properties of the system can be obtained from knowledge of Z. In
particular, the entropy is given by
S = (E − µiCi)/T + ln Z. (7.58)
The idea is to calculate the thermal grand partition function of quantum gravity through
the path integral of a Euclidean version of the Einstein–Hilbert action Eq. (4.26), ˜SEH,
Z =
Dg e − ˜S EH
, (7.59)
where one has to sum over all metrics with period 26 β = c/T . The only modification that
has to be made to the Einstein–Hilbert action is the addition of a surface term to normalize
the action so that the on-shell Euclidean action vanishes for flat Euclidean spacetime (the
vacuum). The Einstein–Hilbert action becomes [436]
SEH[g] =
c3 16πG (4)
N
M
d 4 x |g| R + c3
26 β has dimensions of length if T has dimensions of energy.
8πG (4)
N
d
∂M
3 (K − K0), (7.60)