Kodama-like vector fields in axisymmetric spacetimes

We extend the concept of the Kodama symmetry, a quasi-local time translation symmetry for dynamical spherically symmetric spacetimes, to a specific class of dynamical axisymmetric spacetimes, namely the families of Kerr–Vaidya and Kerr–Vaidya–de Sitter spacetimes. We study some geometrical properties of the asymptotically flat Kerr–Vaidya metric, such as the Brown–York mass and the Einstein tensor. Furthermore, we propose a generalization of the Kerr–Vaidya metric to an asymptotic de Sitter background. We show that for these classes of dynamical axisymmetric black hole spacetimes, there exists a timelike vector field that exhibits similar properties to the Kodama vector field in spherical symmetry. This includes the construction of a covariantly conserved current and a corresponding locally conserved charge, which in the Kerr–Vaidya case converges to the Brown–York mass in the asymptotically flat region.