Noether symmetries, conservation laws and geometric properties of the Banados-Teitelboim-Zanelli black hole metric

Abstract

This study provides a comprehensive analysis of the Banados-Teitelboim-Zanelli (BTZ) black hole, a significant solution in three-dimensional gravity. We begin with a detailed exploration of the BTZ black hole metric, highlighting its derivation and physical properties, such as its horizon structure and thermodynamic characteristics. The thermodynamics of BTZ black holes, including temperature, entropy, and free energy, are analyzed to understand their stability and behavior. A central focus of this work is the application of Noether symmetry principles to the BTZ black hole. We systematically compute the Noether symmetries of the BTZ metric and discuss their physical implications, including their role in conservation laws and stability analysis. This includes an examination of perturbations and their impact on the stability of the black hole, as well as a detailed geodesic analysis that leverages Noether symmetries to simplify the study of particle motion in this spacetime. Additionally, we investigate the first integrals associated with the BTZ black hole metric, elucidating their significance for the dynamics and conservation laws within this framework. The geometric analysis is further expanded through the Cartan structure equations, providing insight into the intrinsic geometric properties of the BTZ black hole and associated curvature tensors. Overall, this work integrates thermodynamic, symmetric, and geometric analyses to offer a unified understanding of the BTZ black hole’s structure and behavior, contributing to the broader field of black hole physics in lower-dimensional spacetimes.