How to calculate the free energy in curved space?

Question

To study the Hagedorn temperature of string near a black hole, we need to calculate the free energy in curved space. This is can be done calculating a torus path integral, but I want to know if an alternative way possible.

The free energy of a single bosonic degree of freedom in flat space is

$$ F=\frac V \beta \int \frac{d^{d-1}k}{(2\pi)^{d-1}} \ln (1- \exp(-\beta E(\mathbf k))).$$

What is the generalization if there is a metric $g_{\mu\nu}$?

I guess one would need to recalculate the spectrum $E$ and then substitute in an expression of the type

$$ F=\frac V \beta \int \frac{d^{d-1}k}{(2\pi)^{d-1}} \sqrt{g}\ln (1- \exp(-\beta E(\mathbf k))).$$

Answer 1

I think there are a number of generalizations. The first is the partition function should be considered quantum mechanically. The reason is that spacetime can contribute to entropy, such as with Hawking radiation. To start the partition function is the trace $$ Z[\phi] = \sum_n\langle\phi_n|e^{-H(\phi)\beta}|\phi_n\rangle $$ Now, if one is interested in the case of a black hole there are then two Hilbert spaces for states on either side of the event horizon. Consequently our field is expanded according to modes in the region I outside the black hole and region II inside the black hole $\phi = \phi_I + exp(-\pi\omega/g)\bar\phi_{II}$, where $g$ is the gravity computed from the Killing vectors $$ g^2 = -\frac{1}{2}\nabla_\mu\xi_\nu\nabla^\mu\xi^\nu $$ The summation over states is then going to include modes for the fields on either side of the horizon. The summation is replaced with integration and from there one performs standard statistical mechanical calculations.

The need to do this stems from the fact that the vacuum with the appearance of the event horizon is transformed into vacuum plus a thermal bath of radiation. Hence the spacetime will contribute to the free energy.