Axisymmetric black holes allowing for separation of variables in the Klein-Gordon and Hamilton-Jacobi equation

TL;DR

This paper characterizes axisymmetric, asymptotically flat black-hole spacetimes that permit variable separation in Klein-Gordon and Hamilton-Jacobi equations, including known solutions like Kerr and Kerr-Newman, and explores their approximations and parametrizations.

Contribution

It identifies the class of black-hole metrics allowing variable separation and discusses their relation to known solutions and parametrizations, including approximate metrics in alternative theories.

Findings

Known black-hole solutions are within this class.

Metrics in some theories can be approximated by these solutions.

The paper provides a continued fraction expansion for the metric.

Abstract

We determine the class of axisymmetric and asymptotically flat black-hole spacetimes for which the test Klein-Gordon and Hamilton-Jacobi equations allow for the separation of variables. The known Kerr, Kerr-Newman, Kerr-Sen and some other black-hole metrics in various theories of gravity are within the class of spacetimes described here. It is shown that although the black-hole metric in the Einstein-dilaton-Gauss-Bonnet theory does not allow for the separation of variables (at least in the considered coordinates), for a number of applications it can be effectively approximated by a metric within the above class. This gives us some hope that the class of spacetimes described here may be not only generic for the known solutions allowing for the separation of variables, but also a good approximation for a broader class of metrics, which does not admit such separation. Finally, the generic form of the axisymmetric metric is expanded in the radial direction in terms of the continued fractions and the connection with other black-hole parametrizations is discussed.