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

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 $f(\mathbb{Q})=\mathbb{Q}+α~\mathbb{Q}^2$ with a small parameter $α$. Using a perturbative approach, we derive asymptotically flat, analytical solutions up to second order in $α$. 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.

Figures (1)

• Figure 1: The real branches of the Lambert function, $\mathcal{W}(y)$ and $\mathcal{W}(-1,y)$. $\mathcal{W}(y)$ is indicated by the blue solid line, defined for $-e^{-1} \leq y < +\infty$. The red dashed line corresponds to $\mathcal{W}(-1,y)$, defined on the interval $-e^{-1} \leq y < 0$.