A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian

TL;DR

This paper derives the equations of motion for an action-dependent Einstein-Hilbert Lagrangian using a variational approach, introducing dissipative dynamics into gravitational theory with a clear geometric formulation.

Contribution

It presents the first geometric derivation of Lorentz-invariant field equations for a singular, action-dependent gravitational Lagrangian, expanding the class of theories beyond standard models.

Findings

Derived Lorentz-invariant equations of motion for action-dependent gravity

Demonstrated dissipative dynamics in a gravitational context

Provided a geometric framework for higher-order action-dependent theories

Abstract

We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian, as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms we are able to obtain a Lorentz invariant set of equations, which contrasts with previous attempts.