The cosmological constant and general covariance

Abstract

In the standard gravitational action without a cosmological constant, one may consider the determinant of the metric as an external field. One then extremizes the action only with respect to variations of the metric that do not change the local volume. The resulting field equations are Einstein's equations with a cosmological constant, which appears as a constant of integration. The dynamics of that theory is analyzed from the point of view of constrained hamiltonian systems. It is observed that contrary to what one might think, the theory is fully covariant and contains only one overall degree of freedom (the cosmological constant) in addition to the two degrees of freedom per point of ordinary Einstein's theory. In the hamiltonian formalism the missing coordinate invariance re-emerges through a tertiary constraint. A Yang-Mills analog is pointed out. The theory is then made manifestly generally covariant by introducing auxiliary fields that are pure gauge except for one overall zero mode. This global mode is a “cosmic time” canonically conjugate to the cosmological constant.