Generalized Misner-Sharp Energy in f(R) Gravity

TL;DR

This paper investigates the conditions under which a generalized Misner-Sharp energy can be defined in f(R) gravity, revealing dependencies on specific constraints and providing explicit forms in certain symmetric spacetimes.

Contribution

It introduces a generalized Misner-Sharp energy in f(R) gravity, highlighting the dependency on a constraint condition and deriving explicit forms in FRW and static solutions.

Findings

Existence of generalized Misner-Sharp energy depends on a constraint condition.

Explicit quasi-local form obtained in FRW universe and static solutions with constant scalar curvature.

In FRW universe, the energy equals the total matter energy inside a sphere.

Abstract

We study generalized Misner-Sharp energy in $f(R)$ gravity in a spherically symmetric spacetime. We find that unlike the cases of Einstein gravity and Gauss-Bonnet gravity, the existence of the generalized Misner-Sharp energy depends on a constraint condition in the $f(R)$ gravity. When the constraint condition is satisfied, one can define a generalized Misner-Sharp energy, but it cannot always be written in an explicit quasi-local form. However, such a form can be obtained in a FRW universe and for static spherically symmetric solutions with constant scalar curvature. In the FRW universe, the generalized Misner-Sharp energy is nothing but the total matter energy inside a sphere with radius $r$, which acts as the boundary of a finite region under consideration. The case of scalar-tensor gravity is also briefly discussed.