On uniqueness of static asymptotically anti-de Sitter black hole

In this paper, the asymptotically anti-de Sitter spacetime with a hypersurface orthogonal Killing vector field is discussed near the conformal boundary by using Newman–Penrose formalism. We first investigate the multipole structure of a static, asymptotically anti-de Sitter spacetime. We note that, different to asymptotically flat case in which the multipoles are contained in , the multipoles of a static, asymptotically anti-de Sitter spacetime are all contained in . We then show that the Schwarzschild anti-de Sitter spacetime is the only asymptotically anti-de Sitter, vacuum, static, axisymmetric, type-D analytic solution of vacuum Einstein equation with a negative cosmological constant in a Bondi coordinate neighborhood of the conformal boundary if the spherical harmonic expansion of is finite.