Black hole solution in non-metricity gravity coupled with scalar field

Abstract

We present an exact static and spherically symmetric black hole configuration in the framework of symmetric teleparallel gravity, where non-metricity is coupled to an axion field. Starting from the modified field equations, we obtain analytical solutions characterized by the mass parameter M and an axion–geometry coupling constant α . The resulting metric exhibits a regular horizon structure with well-behaved curvature invariants. We analyze the thermodynamic quantities including the Hawking temperature, entropy, and heat capacity, and show that the system maintains thermodynamic consistency across the allowed parameter range. As for the analysis of the heat capacity we show that it remains negative for all admissible horizon radii, indicating that the black hole is thermodynamically unstable and does not undergo any phase transition. Furthermore, we study the motion of massive and massless test particles, identifying the conditions for circular orbits and the photon sphere. The axion–non-metricity coupling modifies both the effective potential and the shadow radius, producing observable deviations from the Schwarzschild limit. Using the photon-sphere/eikonal correspondence, the quasinormal mode (QNM) frequencies are computed, revealing that an increase in α enhances both the oscillation frequency and damping rate, ensuring linear stability. Overall, the axion-non-metricity interaction imprints coherent modifications on the black hole s thermodynamics, shadow, and ringdown spectrum, offering potential signatures for testing deviations from general relativity.