ADM Pseudotensors, Conserved Quantities and Covariant Conservation Laws in General Relativity

TL;DR

This paper reviews the ADM formalism in general relativity, demonstrating how covariant conservation laws can be used to derive conserved quantities and highlighting limitations of standard ADM methods in non-Cartesian coordinates.

Contribution

It introduces techniques to decompose covariant conservation laws and compares their effectiveness to standard ADM quantities, especially in non-Cartesian coordinate systems.

Findings

Standard ADM quantities fail in non-Cartesian coordinates.

Covariant conservation laws provide consistent results regardless of coordinate system.

Explicit conditions are identified under which ADM and covariant methods agree.

Abstract

The ADM formalism is reviewed and techniques for decomposing generic components of metric, connection and curvature are obtained. These techniques will turn out to be enough to decompose not only Einstein equations but also covariant conservation laws. Then a number of independent sets of hypotheses that are sufficient (though non-necessary) to obtain standard ADM quantities (and Hamiltonian) from covariant conservation laws are considered. This determines explicitely the range in which standard techniques are equivalent to covariant conserved quantities. The Schwarzschild metric in different coordinates is then considered, showing how the standard ADM quantities fail dramatically in non-Cartesian coordinates or even worse when asymptotically flatness is not manifest; while, in view of their covariance, covariant conservation laws give the correct result in all cases.