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This question is based on page 68 of Thomas Hartman's notes on Quantum Gravity and Black Holes.

To evaluate a path integral in ordinary quantum field theory, we integrate over fields defined on a fixed spacetime manifold.

In quantum gravity, however, we integrate over both the (non-gravitational) fields and the geometry. The (Euclidean) gravitational path integral is therefore

$$\int \mathcal{D}g\mathcal{D}\phi\ e^{-S_{E}[g,\phi]},$$

with the boundary conditions

$$t_{E} \sim t_{E} + \beta, \qquad g_{tt} \to 1\ \text{as}\ r \to \infty.$$

How would you explain these boundary conditions without alluding to finite-temperature quantum field theory?

Qmechanic
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nightmarish
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1 Answers1

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Also on page 68 of the same notes, Tom Hartman explains that this is how we choose the boundary conditions for a particular path integral, i.e. the one that should compute the thermal partition function. Therefore we have to allude to finite-temperature quantum field theory, as that is exactly what we'd like to be doing.

The $g_{tt} \to 1$ as $r \to \infty$ condition just says that we want our space to be asymptotically flat. For example we can compute the thermal partition function in AdS space where we have different boundary conditions.

I don't know of a way to understand Euclidean QFT with periodic time other than as being at a finite temperature.

If you'd like a purely gravitational motivation for that boundary condition we can work backwards. We'd like the Euclidean Schwarzschild metric to be a saddle point for that path integral, and therefore we need time to have periodicity $\beta$. See this question for a purely graviational explanation of that condition.

jswien
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