Unimodular relativity and cosmological constant

TL;DR

Unimodular relativity introduces a fixed volume element in spacetime, maintaining covariant conservation laws and treating the cosmological constant as an integration constant, differing from general relativity in its gauge conditions.

Contribution

The paper clarifies that unimodular relativity preserves covariant conservation and treats the cosmological constant as an integration constant, countering claims of variable cosmological constant.

Findings

Covariant continuity holds in unimodular relativity.

The cosmological constant remains a constant of integration.

Unimodular relativity differs from general relativity in gauge conditions.

Abstract

Unimodular relativity is a theory of gravity and space-time with a fixed absolute space-time volume element, the modulus, which we suppose is proportional to the number of microscopic modules in that volume element. In general relativity an arbitrary fixed measure can be imposed as a gauge condition, while in unimodular relativity it is determined by the events in the volume. Since this seems to break general covariance, some have suggested that it permits a non-zero covariant divergence of the material stress-energy tensor and a variable cosmological ``constant.'' In Lagrangian unimodular relativity, however, even with higher-derivatives of the gravitational field in the dynamics, the usual covariant continuity holds and the cosmological constant is still a constant of integration of the gravitational field equations.