The weight of matter

TL;DR

This paper analyzes Einstein's 1919 traceless gravitational theory, revealing its equivalence to a transverse, Weyl-symmetric unimodular gravity theory, and discusses how the equivalence principle is preserved despite symmetry constraints.

Contribution

It demonstrates the equivalence of Einstein's traceless theory to a transverse, Weyl-invariant unimodular gravity framework from a variational perspective.

Findings

The theory is equivalent to a transverse, Weyl-symmetric unimodular gravity.

The gauge fixing recovers the equivalence principle despite symmetry restrictions.

The analysis clarifies the role of symmetries in gravitational theories.

Abstract

Einstein's traceless 1919 gravitational theory is analyzed from a variational viewpoint. It is shown to be equivalent to a transverse (invariant only under diffeomorphisms that preserve the Lebesgue measure) theory, with an additional Weyl symmetry, in which the gauge is partially fixed so that the metric becomes unimodular. In spite of the fact that this symmetry forbids direct coupling of the potential energy with the gravitational sector, the equivalence principle is recovered in the unimodular gauge owing to Bianchi's identities.