Extracting the expression for the field equations of a diffeomorphism invariant theory of gravity from surface term

TL;DR

This paper demonstrates how the field equations of diffeomorphism invariant gravity theories can be derived from surface terms, linking surface variations to the equations of motion and Noether charges.

Contribution

It provides a detailed derivation of field equations from surface terms using Lie derivatives, unifying the description of equations and charges in gravity theories.

Findings

Field equations derived from surface terms match Euler-Lagrange equations.

Anti-symmetric tensor identified as Noether charge two-form.

Surface term approach applies broadly to diffeomorphism invariant theories.

Abstract

As a contribution towards the understanding for the field equations of diffeomorphism invariant theories of pure gravity, we demonstrate in great detail that the expression for the field equations of such theories can be derived within the perspective of the surface term coming from the variation of the Lagrangian. Specifically, starting with the surface term, we extract a symmetric rank-two tensor together with an anti-symmetric one out of this term with the variation operator replaced with the Lie derivative along an arbitrary vector field. By utilizing an equality stemming from the Lie derivative of the Lagrangian density along an arbitrary vector field, it is proved that the resulting symmetric rank-two tensor is identified with the functional derivative of the Lagrangian density with respect to the metric. Such a result further brings forth the expression for the field equations constructed from the symmetric rank-two tensor, which naturally rules out the derivative of the Lagrangian density with respect to the metric and coincides with the one for the Euler-Lagrange equations of motion. Furthermore, it is illustrated that the construction of the expression for the field equations from the symmetric rank-two tensor must be feasible as long as the variation operator in the variation equation of the Lagrangian is allowed to be substituted by the Lie derivative along an arbitrary vector field. On the other hand, as a byproduct, the anti-symmetric rank-two tensor turns out to be the Noether charge two-form. Our results offer a straightforward support on the proposal in our previous work that the surface term gives a unified description for field equations and Noether charges in the context of theories of gravity admitting diffeomorphism invariance symmetry.