2009/12/19 by Hai Siong Tan, H. S. Tan, Tan, H. S.
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematics and Applications #Quantum chaos and dynamical systems #gr-qc
paper · pdf · doi:10.48550/arxiv.0912.3875
Part of author's B.Sc. Honors Thesis (NUS, 2003)
arxiv created 2009/12/19 · openalex publication_date 2009/12/19 · arxiv updated 2010/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Static, axisymmetric solutions form a large class of important black holes in classical GR. In four dimensions, the existence of their most general metric ansatz relies on the fact that two-dimensional subspaces of the tangent space at each point spanned by vectors orthogonal to the time-translation and rotation Killing fields are integrable. This was first proved by Wald via an application of Frobenius theorem. In this note, we furnish an elementary proof for this theorem by Wald in arbitrary dimensions which yields the metric ansatz for the most general solution of the D-dimensional vacuum Einstein equations that admits D-2 orthogonal and commuting Killing vector fields.