Misner-Sharp Mass in $N$-dimensional $f(R)$ Gravity

TL;DR

This paper extends the concept of Misner-Sharp mass to n-dimensional $f(R)$ gravity, deriving it via two methods and applying it to specific solutions, thereby broadening understanding of quasi-local mass in modified gravity theories.

Contribution

It introduces a generalized Misner-Sharp mass in n-dimensional $f(R)$ gravity using two approaches and establishes their equivalence through a key constraint.

Findings

Derived the generalized Misner-Sharp mass using inverse unified first law.

Established the equivalence of two methods for defining the mass.

Calculated the mass explicitly for the Clifton-Barrow solution.

Abstract

We study the Misner-Sharp mass for the $f(R)$ gravity in an $n$-dimensional (n$\geq$3) spacetime which permits three-type $(n-2)$-dimensional maximally symmetric subspace. We obtain the Misner-Sharp mass via two approaches. One is the inverse unified first law method, and the other is the conserved charge method by using a generalized Kodama vector. In the first approach, we assume the unified first still holds in the $n$-dimensional $f(R)$ gravity, which requires a quasi-local mass form (We define it as the generalized Misner-Sharp mass). In the second approach, the conserved charge corresponding to the generalized local Kodama vector is the generalized Misner-Sharp mass. The two approaches are equivalent, which are bridged by a constraint. This constraint determines the existence of a well-defined Misner-Sharp mass. As an important special case, we present the explicit form for the static space, and we calculate the Misner-Sharp mass for Clifton-Barrow solution as an example.