# Polymer Schwarzschild black hole: An effective metric

## TL;DR

This paper derives an effective metric for a polymer-inspired quantum black hole, revealing a regular interior with a quantum-generated inner horizon, extending classical Schwarzschild solutions within Loop Quantum Gravity frameworks.

## Contribution

It provides an explicit solution for the interior black hole geometry incorporating anomaly-free holonomy quantum corrections, accounting for the ambiguity in polymer regularization parameters.

## Key findings

- The interior metric is regular and resembles Reissner-Nordström with a quantum inner horizon.
- The exterior Schwarzschild metric remains unchanged for the simple case considered.
- The approach offers an alternative to existing models by emphasizing covariance and anomaly-free regularization.

## Abstract

We consider the modified Einstein equations obtained in the framework of effective spherically symmetric polymer models inspired by Loop Quantum Gravity. When one takes into account the anomaly free point-wise holonomy quantum corrections, the modification of Einstein equations is parametrized by a function $f(x)$ of one phase space variable. We solve explicitly these equations for a static interior black hole geometry and find the effective metric describing the trapped region, inside the black hole, for any $f(x)$. This general resolution allows to take into account a standard ambiguity inherent to the polymer regularization: namely the choice of the spin $j$ labelling the SU$(2)$-representation of the holonomy corrections. When $j=1/2$, the function $f(x)$ is the usual sine function used in the polymer litterature. For this simple case, the effective exterior metric remains the classical Schwarzschild's one but acquires modifications inside the hole. The interior metric describes a regular trapped region and presents strong similarities with the Reissner-Nordstr\"om metric, with a new inner horizon generated by quantum effects. We discuss the gluing of our interior solution to the exterior Schwarzschild metric and the challenge to extend the solution outside the trapped region due to covariance requirement. By starting from the anomaly free polymer regularization for inhomogeneous spherically symmetric geometry, and then reducing to the homogeneous interior problem, we provide an alternative treatment to existing polymer interior black hole models which focus directly on the interior geometry, ignoring covariance issue when introducing the polymer regularization.