Generalized Misner-Sharp quasilocal mass in Einstein-Gauss-Bonnet gravity

We investigate properties of a quasilocal mass in a higher-dimensional spacetime having symmetries corresponding to the isomertries of an ($n\ensuremath{-}2$)-dimensional maximally symmetric space in Einstein-Gauss-Bonnet gravity in the presence of a cosmological constant. We assume that the Gauss-Bonnet coupling constant is non-negative. The quasilocal mass was recently defined by one of the authors as a counterpart of the Misner-Sharp quasilocal mass in general relativity. The quasilocal mass is found to be a quasilocal conserved charge associated with a locally conserved current constructed from the generalized Kodama vector and exhibits the unified first law corresponding to the energy-balance law. In the asymptotically flat case, it converges to the Arnowitt-Deser-Misner mass at spacelike infinity, while it converges to the Deser-Tekin and Padilla mass at infinity in the case of asymptotically anti-de Sitter. Under the dominant energy condition, we show the monotonicity of the quasilocal mass for any $k$, while the positivity on an untrapped hypersurface with a regular center is shown for $k=1$ and for $k=0$ with an additional condition, where $k=\ifmmode\pm\else\textpm\fi{}1$, 0 is the constant sectional curvature of each spatial section of equipotential surfaces. Under a special relation between coupling constants, positivity of the quasilocal mass is shown for any $k$ without assumptions above. We also classify all the vacuum solutions by utilizing the generalized Kodama vector. Lastly, several conjectures on further generalization of the quasilocal mass in Lovelock gravity are proposed.