Extracting Dynamical Degrees of Freedom From the Quasi-Local Energy Term in the Gravitational Action

TL;DR

This paper demonstrates that the Hamiltonian and gravitational action can be expressed in terms of the Brown-York quasi-local energy, revealing the dynamical content of gravity and enabling derivation of uncertainty relations relevant for quantum gravity and black hole physics.

Contribution

It shows how the gravitational action can be written in terms of quasi-local energy, highlighting its role as the dynamical core of general relativity.

Findings

Hamiltonian and action expressed via Brown-York energy

Derived uncertainty relations for gravity and quantum state reduction

Application to modified Vaidya metric for black hole analysis

Abstract

It is shown that under proper conditions in an appropriate coordinate system with a suitable time slicing the Hamiltonian and the Einstein-Hilbert action including all necessary boundary terms can be written on shell in terms of the Brown-York quasi-local energy in the absence of matter. If matter is present the non-vanishing bulk term only consists of stress-energy. It is argued that the dynamical content of general relativity is stored in the quasi-local energy term. The results underscore the interpretation of the Brown-York quasi-local energy as the field energy of the gravitational field plus stress-energy. As an application we derive uncertainty relations of the time-energy kind which may be useful in the understanding of gravity induced quantum state reduction and the more conventional kind for conjugate variables. The latter is computed for a modified Vaidya metric which may be used in the investigation of black hole radiance. The boundary terms expressed as quasi-local energy cancel second derivatives in the action leaving only a square of a first derivative term in the chosen gauge which is desirable for a quantization of the action.