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

TL;DR

This work argues that gravitational path integrals should be understood as coarse-grained, fluctuating gravito-hydrodynamics rather than fundamental analytic continuations of quantum gravity. It builds on Jacobson's Covariant Entropy Principle to derive Einstein's equations as emergent hydrodynamics of causal diamonds and interprets bulk energy as an entropy deficit with fluctuations encoded on diamond boundaries. It further connects these hydrodynamic fluctuations to spectral form-factor corrections via Lorentzian wormhole-like contributions and discusses when higher-genus topology sums may be relevant (primarily in AdS contexts). The result is a framework that rationalizes path-integral computations in gravity without appealing to a universal topology sum and clarifies the special role of AdS and tensor-network realizations of the vacuum state.

Abstract

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.

Figures (1)

• Figure 1: Nested Causal Diamonds Separated by Stretched Horizon