Noether Charges, Brown-York Quasilocal Energy and Related Topics

TL;DR

This paper compares different Lagrangians in General Relativity to establish a covariant framework for conserved quantities and quasilocal energy, emphasizing the advantages of the covariant first order Lagrangian in non-asymptotically flat spacetimes.

Contribution

It demonstrates the superiority of the covariant first order Lagrangian for defining conserved quantities and quasilocal energy in General Relativity, especially with reference backgrounds.

Findings

The covariant first order Lagrangian is better suited for non-asymptotically flat spacetimes.

All computed quantities are manifestly covariant using this Lagrangian.

A correspondence among Noether charges, quasilocal energy, and Hamiltonian is established.

Abstract

The Lagrangian proposed by York et al. and the covariant first order Lagrangian for General Relativity are introduced to deal with the (vacuum) gravitational field on a reference background. The two Lagrangians are compared and we show that the first one can be obtained from the latter under suitable hypotheses. The induced variational principles are also compared and discussed. A conditioned correspondence among Noether conserved quantities, quasilocal energy and the standard Hamiltonian obtained by 3+1 decomposition is also established. As a result, it turns out that the covariant first order Lagrangian is better suited whenever a reference background field has to be taken into account, as it is commonly accepted when dealing with conserved quantities in non-asymptotically flat spacetimes. As a further advantage of the use of a covariant first order Lagrangian, we show that all the quantities computed are manifestly covariant, as it is appropriate in General Relativity.