Stationary Point of the Hilbert Action Principle

TL;DR

This paper derives the boundary terms needed to make the Hilbert action principle for Einstein's equations stationary, clarifying the conditions for a true variational principle in general relativity.

Contribution

It explicitly computes both boundary terms required for the Hilbert action to be stationary, improving understanding of the variational principle in gravitational theories.

Findings

Derived the first boundary term for the Hilbert action.

Derived the second boundary term for the Hilbert action.

Clarified the conditions for stationarity in Einstein's equations.

Abstract

The Hilbert action principle for the Einstein equations has two boundary terms that must be subtracted in order to obtain a true stationary point. In this paper, I obtain both of them.