3+1 decomposition in modified gravities within the Palatini formalism and some applications

TL;DR

This paper develops a 3+1 spacetime decomposition framework for modified gravity theories in the Palatini formalism, deriving key relations and boundary terms, and applies it to ADM foliation and junction conditions.

Contribution

It provides a systematic 3+1 decomposition method for Palatini-modified gravities, including Gauss-Codazzi relations and junction conditions, extending previous approaches.

Findings

Derived Gauss-Codazzi relations for Palatini theories.

Established boundary terms for gravitational action.

Applied decomposition to ADM foliation and junction conditions.

Abstract

In the present paper, the 3+1 decomposition of the spacetime onto hypersurface(s) is analysed and established for theories within the Palatini formalism by considering a general function of the Ricci scalar in the gravitational action. The corresponding Gauss-Codazzi relations are obtained and the boundary term that has to be subtracted in the gravitational action is easily deduced. Then, these relations are applied to the so-called ADM decomposition to describe the foliation of the spacetime onto hypersurfaces of constant time within these theories. Finally, the junction conditions are also obtained by using a decomposition in Gaussian normal coordinates, which coincide with the conditions deduced previously through different approaches.