The weight of matter

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.