Field equations from a surface term

TL;DR

This paper investigates the derivation of Einstein field equations from boundary terms in the action, revealing that the approach works for pure gravity but fails when matter is included, due to fundamental reasons.

Contribution

It demonstrates that deriving field equations from boundary invariance applies to more general actions in pure gravity but encounters limitations with matter.

Findings

Boundary-based derivation is valid for pure gravity with general actions.

The approach breaks down when matter is incorporated into the action.

The paper clarifies the fundamental reasons for this breakdown.

Abstract

As is well known, in order for the Einstein--Hilbert action to have a well defined variation, and therefore to be used for deriving field equation through the stationary action principle, it has to be amended by the addition of a suitable boundary term. It has recently been claimed that, if one constructs an action by adding this term to the matter action, the Einstein field equations can be derived by requiring this action to be invariant under active transformations which are normal to a null boundary. In this paper we re-examine this approach both for the case of pure gravity and in the presence of matter. We show that in the first case this procedure holds for more general actions than the Einstein-Hilbert one and trace the basis of this remarkable attribute. However, it is also pointed out the when matter is rigorously considered the approach breaks down. The reasons for that are thoroughly discussed.