Black Hole Shadow and other Observables away from the Horizon: Extending the Effective Metric Descriptions

TL;DR

This paper extends an effective metric description of quantum black holes beyond the horizon using Padé approximants, enabling accurate computation of observables like the black hole shadow with quantum corrections.

Contribution

It introduces a method to extend the effective metric framework for quantum black holes using Padé approximants, allowing for accurate observable calculations at larger distances.

Findings

Padé approximants improve convergence of the effective metric description.

Explicit formulas for black hole shadows with quantum corrections.

Effective approach matches well with previous quantum black hole models.

Abstract

In previous work we have developed a model-independent, effective description of quantum deformed, spherically symmetric and static black holes in four dimensions. The deformations of the metric are captured by two functions of the physical distance to the horizon, which are provided in the form of self-consistent Taylor series expansions. While this approach efficiently captures physical observables in the immediate vicinity of the horizon, it is expected to encounter problems of convergence at further distances. Therefore, we demonstrate in this paper how to use Pad\'e approximants to extend the range of applicability of this framework. We provide explicit approximations of physical observables that depend on finitely many effective parameters of the deformed black hole geometry, depending on the order of the Pad\'e approximant. By taking the asymptotic limit of this order, we in particular provide a closed-form expression for the black hole shadow of the (fully) deformed geometry, which captures the leading quantum corrections. We illustrate our results for a number of quantum black holes previously proposed in the literature and find that our effective approach provides excellent approximations in all cases.