We study the Misner-Sharp mass for the $f(R)$ gravity in an $n$-dimensional $(\mathrm{n}\ge 3)$ 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, which uses a generalized Kodama vector. In the first approach, we assume the unified first law still holds in $n$-dimensional $f(R)$ gravity, which requires a quasilocal 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, which are bridged by a constraint, are equivalent. This constraint determines the existence of a well-defined Misner-Sharp mass. In an important special case, we present the explicit form for the static space, and we calculate the Misner-Sharp mass for the Clifton-Barrow solution as an example.

• Received 3 June 2014

DOI:https://doi.org/10.1103/PhysRevD.90.024062