The Hydrodynamic Approach to Quantum Gravity

Abstract

Several papers from the mid to late 1990s suggest that Einstein's equations should be thought of as the hydrodynamic equations of a special class of quantum systems. A classical solution defines subsystems by dividing space-time up into CAUSAL DIAMONDS and Einstein's equations are the hydrodynamics of a system that assigns a density matrix to each diamond whose modular Hamiltonian K has expectation value and fluctuation both given by A/4G. A is the maximal d-2 volume on the boundary of the diamond and G is Newton's constant. These properties define the EMPTY DIAMOND STATE, the analog of the quantum field theory (QFT) vacuum, in the background geometry. The assignment of density matrices to each diamond enables one to define the analog of half sided modular flow along geodesics in the background manifold, as a unitary embedding of the Hilbert space of a given diamond into the next one in a nesting with Planck scale time steps. We conjecture that this can be enhanced to a full set of compatible unitary evolutions on a Hilbert bundle of the space of time-like geodesics, using a QUANTUM PRINCIPLE OF EQUIVALENCE defined in the text. The compatibility of this formalism with the experimental success of QFT is discussed, as well as the theoretical mechanism by which QFT emerges from this version of quantum gravity. This is a slightly expanded version of an essay that won Honorable Mention in the Gravitation Research Essay Contest for 2025.