Phase space for gravity with boundaries

TL;DR

This paper constructs the reduced phase space for gravity theories with boundaries using geometric methods, providing insights into initial conditions and symplectic structures, with applications to four-dimensional Palatini--Cartan gravity.

Contribution

It introduces a geometric approach to construct the reduced phase space for gravity with boundaries, including detailed analysis of the Palatini--Cartan formalism.

Findings

Explicit construction of the phase space for gravity with boundaries

Application to four-dimensional Palatini--Cartan gravity

Clarification of the symplectic structure in gravitational theories

Abstract

This explanatory note, based on the geometrical method by Kijovski and Tulczyjew, describes the construction of the reduced phase space of Lagrangian field theories, i.e., the correct space of initial conditions with its symplectic structure. Several examples and, in particular, the case of four-dimensional gravity in the coframe formalism (Palatini--Cartan theory) are analyzed.