Canonical General Relativity on a Null Surface with Coordinate and Gauge Fixing

TL;DR

This paper develops a canonical formalism for general relativity on a null surface with specific gauge fixing, reducing the phase space to key variables and relating it to established methods, with implications for gravitational radiation and quantization.

Contribution

It introduces a gauge-fixed canonical formalism on null surfaces that simplifies the phase space and connects to Bondi-Sachs and Newman-Penrose frameworks.

Findings

Phase space reduced to one pair of conjugate variables.

Asymptotic solutions exclude logarithmic behavior with specific fall-off conditions.

Logarithmic behavior affects the peeling theorem and gravitational radiation analysis.

Abstract

We use the canonical formalism developed together with David Robinson to st= udy the Einstein equations on a null surface. Coordinate and gauge conditions = are introduced to fix the triad and the coordinates on the null surface. Toget= her with the previously found constraints, these form a sufficient number of second class constraints so that the phase space is reduced to one pair of canonically conjugate variables: $\Ac_2\and\Sc^2$. The formalism is related to both the Bondi-Sachs and the Newman-Penrose methods of studying the gravitational field at null infinity. Asymptotic solutions in the vicinity of null infinity which exclude logarithmic behavior require the connection to fall off like $1/r^3$ after the Minkowski limit. This, of course, gives the previous results of Bondi-Sachs and Newman-Penrose. Introducing terms which fall off more slowly leads to logarithmic behavior which leaves null infinity intact, allows for meaningful gravitational radiation, but the peeling theorem does not extend to $\Psi_1$ in the terminology of Newman-Penrose. The conclusions are in agreement with those of Chrusciel, MacCallum, and Singleton. This work was begun as a preliminary study of a reduced phase space for quantization of general relativity.