# Saddle point contributions to the gravitational path integral

In his lectures on black holes and quantum information, Tom Hartman states that the gravitational path integral can be approximated as $$Z(\beta) \approx \sum_{g_\text{cl}} e^{-I_E[g_\text{cl}, \phi]},$$ where $$g_\text{cl}$$ are the classical solutions ("saddles") that satisfy the boundary conditions $$t_E \sim t_E + \beta, \quad g_{tt} \to 1 \text{ at infinity}$$ and $$\phi$$ represents the matter fields. Hartman considers the case where there are no matter fields and says that the solution satisfying the boundary conditions is the Euclidean Schwarzschild metric. One can show that (defining $$I_E=0$$ for flat spacetime) the Euclidean action of this metric is $$I_E = \frac{\beta^2}{16 \pi}.$$ Using the standard statistical mechanic formula for the entropy $$S=(1-\beta\partial_\beta)\log Z(\beta)$$, one finds that the entropy of the black hole is the Bekenstein-Hawking entropy $$S=A/4$$, where $$A$$ is the surface area of the black hole.

However, one of the students points out that another saddle point that satisfies the boundary is the flat cylinder spacetime. Adding this to the partition function, we get (since we defined $$I_E = 0$$ for flat spacetime) that $$Z(\beta) = e^{-\beta^2/16\pi} + 1,$$ which gives $$S \neq A/4$$, in disagreement with the Bekenstein-Hawking formula. Hartman states that by tweaking the temperature, we can make it such that the dominating contribution comes from the Schwarzschild term, fixing the issue. However, we clearly have that $$e^{-\beta^2/16\pi} \leq 1$$, meaning that the Schwarzschild term will never dominate.

The flat spacetime contribution is also commented in the Jerusalem lectures, section 4.8. Harlow states that this is the contribution from the gas of radiation we know to be present. In order to find the black hole entropy he therefore considers only the contribution to the partition function from the Schwarzschild metric. This is a rather vague argument to me.

Are there any more rigorous or physically motivated reasons to ignore the contribution from the flat spacetime?

• Minute in video? Aug 1, 2022 at 14:43
• Semiclassical approximation starts around 44:00. Schwarzschild term from around 57:00, question from student around 1:03:00. Aug 1, 2022 at 14:47
• Tom Hartman's homepage Aug 17, 2022 at 12:18
• I've also wondered about this - did you find an answer? Dec 30, 2022 at 16:23