New Black hole Solutions in $f(\mathbb{Q})$ Gravity

TL;DR

This paper derives new static, spherically symmetric vacuum solutions in $f( ext{ extbf{Q}})$ gravity, including general relativity equivalents and novel solutions with connection hair, using analytical and perturbative methods.

Contribution

It systematically classifies affine connections in $f( ext{ extbf{Q}})$ gravity and finds exact and approximate vacuum solutions, revealing new features like connection hair and horizon corrections.

Findings

Exact Schwarzschild and (A)dS solutions for $f( ext{ extbf{Q}})=0$

Perturbative solutions with connection hair for quadratic $f( ext{ extbf{Q}})$

Horizon radius corrections expressed via Lambert W function

Abstract

We investigate static and spherically symmetric vacuum solutions in the symmetric teleparallel $f(\mathbb{Q})$ modified theory of gravity. Starting from a recently proposed classification of affine connections compatible with both the symmetries of spacetime and the constraints of symmetric teleparallel geometry, we develop a systematic approach to solve the full field equations. We first identify two distinct classes of connections that satisfy the off-diagonal metric field equations and the connection constraints. For an arbitrary $f(\mathbb{Q})$ function when the non-metricity scalar $\mathbb{Q}$ vanishes, we recover exact analytical solutions equivalent to those of general relativity, including the Schwarzschild and Schwarzschild (anti)de-Sitter metrics. We then extend our analysis beyond general relativity by considering the quadratic model $f(\mathbb{Q})=\mathbb{Q}+\alpha~\mathbb{Q}^2$ with a small parameter $\alpha$. Using a perturbative approach, we derive asymptotically flat, analytical solutions up to second order in $\alpha$. These solutions exhibit corrections to the standard Schwarzschild metric, characterized by new integration constants that can be interpreted as connection hair. We explore the asymptotic behavior of these solutions and disclose that the horizon radius receives corrections that can be expressed compactly using the Lambert $\mathcal{W}$ function. Our results provide new, non-trivial vacuum solutions within $f(\mathbb{Q})$ gravity and highlight the rich structure introduced by the non-metricity connection.