Intrinsic Time in Geometrodynamics of Closed Manifolds

TL;DR

This paper defines a covariant global intrinsic time in Geometrodynamics for closed manifolds, using Hamiltonian reduction and deparametrization to analyze gravitational dynamics in a novel way.

Contribution

It introduces a new covariant intrinsic time based on metric determinants and develops Hamiltonian equations for gravitational fields in this framework.

Findings

Global intrinsic time is identified with the mean logarithm of metric determinant ratios.

Hamiltonian reduction and deparametrization procedures are successfully implemented.

Relations between coordinate, intrinsic, and proper time are derived.

Abstract

The global time in Geometrodynamics is defined in a covariant under diffeomorphisms form. An arbitrary static background metric is taken in the tangent space. The global intrinsic time is identified with the mean value of the logarithm of the square root of the ratio of the metric determinants. The procedures of the Hamiltonian reduction and deparametrization of dynamical systems are implemented. The explored Hamiltonian system appeared to be non-conservative. The Hamiltonian equations of motion of gravitational field in the global time are written. Relations between different time intervals (coordinate, intrinsic, proper) are obtained.