What is a Gravitational Path Integral? {\it or} Gravitational Path Integrals as Fluctuating Gravito-Hydrodynamics

We show how Gravitational Path Integral formulae for various quantities that have been computed in the literature, follow from a few coarse grained hydrodynamic assumptions about the relations between space-time geometry, entropy, and fluctuations of the modular Hamiltonian of causal diamonds. These remarks have implications for the way we think about such path integrals in relation to a more fundamental model of quantum gravity, and to questions about which space-time topologies are actually summed over in real models.

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