# An analytical approximation for the Einstein-dilaton-Gauss-Bonnet black hole metric

## TL;DR

This paper develops a compact analytical approximation for the numerical black hole metric in Einstein-dilaton-Gauss-Bonnet theory, enabling easier analysis of black hole phenomena with high accuracy outside the event horizon.

## Contribution

It introduces a continued fraction-based analytical formula for the EdGB black hole metric and dilaton field, improving computational efficiency and accuracy over previous numerical methods.

## Key findings

- Maximal relative error of a fraction of one percent within third order expansion
- Approximation is valid outside the black hole event horizon
- Suitable for analyzing particle motion, perturbations, and radiation processes

## Abstract

We construct an analytical approximation for the numerical black hole metric of P. Kanti, et. al. [PRD54, 5049 (1996)] in the four-dimensional Einstein-dilaton-Gauss-Bonnet (EdGB) theory. The continued fraction expansion in terms of a compactified radial coordinate, used here, converges slowly when the dilaton coupling approaches its extremal values, but for a black hole far from the extremal state, the analytical formula has a maximal relative error of a fraction of one percent already within the third order of the continued fraction expansion. The suggested analytical representation of the numerical black hole metric is relatively compact and good approximation in the whole space outside the black hole event horizon. Therefore, it can serve in the same way as an exact solution when analyzing particles' motion, perturbations, quasinormal modes, Hawking radiation, accreting disks and many other problems in the vicinity of a black hole. In addition, we construct the approximate analytical expression for the dilaton field.