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.

中文翻译：

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关于静态渐近反德西特黑洞的唯一性

在本文中，使用Newman-Penrose形式主义讨论了在共形边界附近具有超曲面正交Killing矢量场的渐近反德西特时空。我们首先研究静态的、渐近反德西特时空的多极结构。我们注意到，与多极点包含在 中的渐近平坦情况不同，静态渐近反德西特时空的多极点都包含在 中。然后，我们证明 Schwarzschild 反德西特时空是真空爱因斯坦方程的唯一渐近反德西特、真空、静态、轴对称、D 型解析解，在共形边界的邦迪坐标邻域中具有负宇宙学常数，如果的球谐展开是有限的。