Intrinsic time gravity and the Lichnerowicz-York equation

TL;DR

This paper explores a reformulation of general relativity using intrinsic time, leading to a simplified Hamiltonian structure and a generalized Lichnerowicz-York equation with good mathematical properties.

Contribution

It introduces an intrinsic time approach that replaces the Hamiltonian constraint with a dynamical equation, allowing for a simpler local Hamiltonian and generalizations including Cotton-York tensor terms.

Findings

Maintains 3-covariance with momentum constraint.

Derives a generalized Lichnerowicz-York equation with existence and uniqueness.

Facilitates initial data construction and inclusion of matter fields.

Abstract

We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a dynamical equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalised; in particular, the square of the Cotton-York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended theory; we get a generalised Lichnerowicz-York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.