Stable models in lower dimensions within the quadratic form of the non-metricity theory

Abstract: This work investigates exact charged and uncharged black hole solutions in a (2+1)-dimensional spacetime within the framework of the quadratic form of the symmetric teleparallel theory, known as $f(\mathbb{Q})$ gravity, where $\mathbb{Q}$ is the non-metricity scalar. By employing a spherically symmetric ansatz and considering both vanishing and non-vanishing electromagnetic fields, we derive new classes of black hole solutions and analyze their geometric and physical properties. The study demonstrates that the inclusion of the quadratic correction term in the gravitational Lagrangian significantly modifies the structure of the solutions, producing deviations from the standard BTZ geometry. Invariants such as curvature and non-metricity scalars are calculated to classify the singularity structure and spacetime behavior. Thermodynamic quantities, including Hawking temperature, entropy, and heat capacity, are computed, showing consistency with the first law of black hole thermodynamics. Furthermore, we examine the geodesic motion of test particles and derive the effective potential to explore the stability of photon orbits. A notable outcome is the identification of weaker black hole singularities in comparison to general relativity, attributed to the non-metricity corrections. The possibility of multi-horizon configurations is also explored. This study provides a comprehensive analysis of the gravitational, thermodynamic, and dynamical features of lower-dimensional black holes in $f(\mathbb{Q})$ gravity and highlights their distinct characteristics relative to general relativity.