Reflection symmetry in higher dimensional black hole spacetimes

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-phi' 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 d >= 4 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-phi' 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.