General parametrization of axisymmetric black holes in metric theories of gravity

TL;DR

This paper introduces a new parametric framework for describing axisymmetric black holes in various metric theories of gravity, using a double expansion approach that improves convergence and accurately models known solutions.

Contribution

The paper develops a novel double expansion method for axisymmetric black hole metrics, extending previous spherical symmetry work to more general cases with better convergence properties.

Findings

Accurately models Kerr, dilaton, and Einstein-dilaton-Gauss-Bonnet black holes.

Provides a convergent parametrization that improves with higher-order terms.

Matches known solutions well at lowest order, with increasing accuracy as more terms are added.

Abstract

Following previous work of ours in spherical symmetry, we here propose a new parametric framework to describe the spacetime of axisymmetric black holes in generic metric theories of gravity. In this case, the metric components are functions of both the radial and the polar angular coordinates, forcing a double expansion to obtain a generic axisymmetric metric expression. In particular, we use a continued-fraction expansion in terms of a compactified radial coordinate to express the radial dependence, while we exploit a Taylor expansion in terms of the cosine of the polar angle for the polar dependence. These choices lead to a superior convergence in the radial direction and to an exact limit on the equatorial plane. As a validation of our approach, we build parametrized representations of Kerr, rotating dilaton, and Einstein-dilaton-Gauss-Bonnet black holes. The match is already very good at lowest order in the expansion and improves as new orders are added. We expect a similar behavior for any stationary and axisymmetric black-hole metric.