Emergent Time in Hamiltonian General Relativity

TL;DR

This paper proposes a new way to define time as an emergent property from the geometry of configuration space, applying it to systems like particle mechanics, gauge theories, and general relativity, with detailed examples.

Contribution

It introduces a novel geometric definition of emergent time applicable to diverse physical theories, including general relativity, using the Hamilton-Jacobi framework.

Findings

Emergent time can be derived from configuration space geometry.

The approach applies to particle mechanics, gauge theories, and GR.

Illustrations include de Sitter and AdS spacetimes.

Abstract

In this paper we introduce a definition of time that emerges in terms of the geometry of the configuration space of a dynamical system. We illustrate this, using the Hamilton-Jacobi equation, in various examples: particle mechanics on a fixed energy surface; non-Abelian gauge theories for compact semi-simple Lie groups where the Gauss law presents new features; and General Relativity in $d+1$ dimensions with $d$ the dimension of space. The discussion in General Relativity is like the non-abelian gauge theory case except for the indefiniteness of the de Witt metric in the Einstein-Hamilton-Jacobi equation, which we discuss in some detail. We illustrate the general formula for the emergent time in various examples including de Sitter spacetime and asymptotically AdS spacetimes.