Non-stationary Energy in General Relativity

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.

16 pages,dedicated to the memory of Rahmi Guven (1948-2019) who spent a life in gravity research in a region quite timid about science

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