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Reflection Symmetry in Higher Dimensional Black Hole Spacetimes · Around

2015

Reflection Symmetry in Higher Dimensional Black Hole Spacetimes

Schiffrin, Joshua S., Wald, Robert M.

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In 4 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 $d \geq 4$ 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.

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