The Weiss Variation of the Gravitational Action

TL;DR

This paper reviews the Weiss variational principle and introduces a new geometric derivation for the gravitational action's variation, emphasizing boundary displacements and avoiding explicit 3+1 spacetime decomposition.

Contribution

It provides a novel geometric derivation of the Weiss variation for the gravitational action using area variation formulas, simplifying the analysis of boundary terms in General Relativity.

Findings

Formalizes the Weiss variation for gravitational action

Derives boundary term variations using area formulas

Facilitates time evolution discussion without 3+1 decomposition

Abstract

The Weiss variational principle in mechanics and classical field theory is a variational principle which allows displacements of the boundary. We review the Weiss variation in mechanics and classical field theory, and present a novel geometric derivation of the Weiss variation for the gravitational action: the Einstein-Hilbert action plus the Gibbons-Hawking-York boundary term. In particular, we use the first and second variation of area formulas (we present a derivation accessible to physicists in an appendix) to interpret and vary the Gibbons-Hawking-York boundary term. The Weiss variation for the gravitational action is in principle known to the Relativity community, but the variation of area approach formalizes the derivation, and facilitates the discussion of time evolution in General Relativity. A potentially useful feature of the formalism presented in this article is that it avoids an explicit 3+1 decomposition in the bulk spacetime.