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

Understand

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).