Non-stationary Energy in General Relativity

TL;DR

This paper introduces an integral measure of non-stationary energy in general relativity using the Fischer-Marsden form, which vanishes for stationary spacetimes and relates to Dain's invariant under certain conditions.

Contribution

It constructs a new integral invariant for non-stationary energy in general relativity based on the first order Fischer-Marsden form and Einstein constraints.

Findings

The integral vanishes for stationary spacetimes.

Under additional assumptions, it reduces to Dain's invariant.

Provides a new tool to quantify non-stationary energy in GR.

Abstract

Using the time evolution equations of (cosmological) General Relativity in the first order Fischer-Marsden form, we construct an integral that measures the amount of non-stationary energy on a given spacelike hypersurface in $D$ dimensions. The integral vanishes for stationary spacetimes; and with a further assumption, reduces to Dain's invariant on the boundary of the hypersurface which is defined with the Einstein constraints and a fourth order equation defining approximate Killing symmetries.