The trace-free Einstein tensor is not variational for the metric as field variable
General Relativity and Quantum Cosmology
2026-03-02 v2 High Energy Physics - Theory
Mathematical Physics
Analysis of PDEs
math.MP
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.
Cite
@article{arxiv.2509.02490,
title = {The trace-free Einstein tensor is not variational for the metric as field variable},
author = {Arian L. von Blanckenburg and Domenico Giulini and Philip K. Schwartz},
journal= {arXiv preprint arXiv:2509.02490},
year = {2026}
}
Comments
4+1 pages (main text + references), to appear in Classical and Quantum Gravity. v2: extended discussion, added references