Boundary terms and on-shell action in Ricci-based gravity theories: the Hamiltonian formulation

TL;DR

This paper develops a Hamiltonian formulation for Ricci-based gravity theories, decomposing the connection and curvature to derive boundary terms, constraints, and energy expressions, exemplified with Schwarzschild spacetime.

Contribution

It introduces a Hamiltonian framework for Ricci-based gravity theories, including boundary terms and ADM energy, extending the analysis beyond General Relativity.

Findings

Derived the 3+1 decomposition and Gauss-Codazzi relations for Ricci-based theories.

Constructed the Hamiltonian and ADM energy for these theories.

Applied the formalism to Schwarzschild spacetime.

Abstract

Considering the so-called Ricci-based gravity theories, a family of extensions of General Relativity whose action is given by a non-linear function of contractions and products of the (symmetric part of the) Ricci tensor of an independent connection, the Hamiltonian formulation of the theory is obtained. To do so, the independent connection is decomposed in two parts, one compatible with a metric tensor and the other one given by a 3-rank tensor. Subsequently, the Riemann tensor is expressed in terms of its projected components onto a hypersurface, allowing to construct the $3+1$ decomposition of the theory and the corresponding Gauss-Codazzi relations, where the boundary terms naturally arise in the gravitational action. Finally, the ADM decomposition is followed in order to construct the corresponding Hamiltonian and the ADM energy for any Ricci-based gravity theory. The formalism is applied to the simple case of Schwarzschild space-time.