Weiss variation for general boundaries

TL;DR

This paper extends the Weiss variation of the Einstein-Hilbert action to general boundary surfaces, deriving boundary contributions, conjugate momenta, and a covariant energy-momentum tensor, with implications for quantum gravity and the problem of time.

Contribution

It introduces a generalized Weiss variation applicable to all boundary types, deriving boundary variables, a covariant energy-momentum tensor, and a gravitational Schrödinger equation.

Findings

Derivation of boundary contributions for spacelike, timelike, and null surfaces.

Identification of conjugate momenta and boundary variables.

Formulation of a covariant Einstein energy-momentum pseudotensor.

Abstract

The Weiss variation of the Einstein-Hilbert action with an appropriate boundary term has been studied for general boundary surfaces; the boundary surfaces can be spacelike, timelike, or null. To achieve this we introduce an auxiliary reference connection and find that the resulting Weiss variation yields the Einstein equations as expected, with additional boundary contributions. Among these boundary contributions, we obtain the dynamical variable and the associated conjugate momentum, irrespective of the spacelike, timelike or, null nature of the boundary surface. We also arrive at the generally non-vanishing covariant generalization of the Einstein energy-momentum pseudotensor. We study this tensor in the Schwarzschild geometry and find that the pseudotensorial ambiguities translate into ambiguities in the choice of coordinates on the reference geometry. Moreover, we show that from the Weiss variation, one can formally derive a gravitational Schr{\"o}dinger equation, which may, despite ambiguities in the definition of the Hamiltonian, be useful as a tool for studying the problem of time in quantum general relativity. Implications have been discussed.