Boundary conditions from boundary terms, Noether charges and the trace K lagrangian in general relativity

TL;DR

This paper introduces a Lagrangian for General Relativity based on the trace K action, analyzing boundary conditions, Noether charges, and recovering the ADM energy in asymptotically flat spacetimes.

Contribution

It develops a variational principle for second order Lagrangians with boundaries and derives conditions for conserved Noether charges, including the ADM energy.

Findings

The trace K Lagrangian has the same asymptotic behavior as other GR Lagrangians.

A pre-symplectic form with bulk and boundary terms is obtained.

Diffeomorphisms satisfying boundary conditions yield conserved charges, including ADM energy.

Abstract

We present the Lagrangian whose corresponding action is the trace K action for General Relativity. Although this Lagrangian is second order in the derivatives, it has no second order time derivatives and its behaviour at space infinity in the asymptotically flat case is identical to other alternative Lagrangians for General Relativity, like the gamma-gamma Lagrangian used by Einstein. We develop some elements of the variational principle for field theories with boundaries, and apply them to second order Lagrangians, where we stablish the conditions -- proposition 1 -- for the conservation of the Noether charges. From this general approach a pre-symplectic form is naturally obtained that features two terms, one from the bulk and another from the boundary. When applied to the trace K Lagrangian, we recover a pre-symplectic form first introduced using a different approach. We prove that all diffeomorphisms satisfying certain restrictions at the boundary -- that keep room for a realization of the Poincar\'e group -- will yield Noether conserved charges. In particular, the computation of the total energy gives, in the asymptotically flat case, the ADM result.