The trace-free Einstein tensor is not variational for the metric as field variable

Abstract

It is well-known that the trace-free Einstein tensor of a pseudo-Riemannian metric cannot arise by variation of a local diffeomorphism-invariant action functional with the (inverse) metric as field variable. We show that this statement remains true even for general local actions, without the assumption of diffeomorphism invariance.

4+1 pages (main text + references), to appear in Classical and Quantum Gravity. v2: extended discussion, added references