2015/01/31 by Joshua S. Schiffrin, Joshua S Schiffrin, Robert M. Wald +1
Mathematics · Physics and Astronomy · #Action (physics) #Black Holes and Theoretical Physics #Black hole (networking) #General relativity #Geometric Analysis and Curvature Flows #Isometry (Riemannian geometry) #Isometry group #Killing vector field #Quantum Electrodynamics and Casimir Effect #Reflection (computer programming) #Spacetime #Stationary spacetime #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/0264-9381/32/10/105005
published as Class. Quantum Grav. 32 (2015) 105005 · 31 pages. v2: minor corrections; remark added to section VI; section VII (generalizations) rewritten. v3: minor corrections
arxiv created 2015/04/15 · openalex publication_date 2015/04/23 · arxiv updated 2015/04/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
In four spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein’s equation there exists a ‘ t - ϕ ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in spacetime dimensions such that the action of the isometry group is trivial. (Here ‘trivial’ means that if the ‘axes’—i.e., the points where the axial Killing fields are linearly dependent—are removed, the action of the isometry group is that of a trivial principal fiber bundle. This excludes actions like that found in the Sorkin monopole.) We prove that there exists a ‘ t - ϕ ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of each axial Killing vector field. The proof relies in an essential way on the first law of black hole mechanics.