Thermodynamic and observational implications of black holes in toroidal geometry
Usman Zafar, Kazuharu Bamba, Abdul Jawad, Tabinda Rasheed, Sanjar Shaymatov
TL;DR
The study addresses how torus-like charged black holes in AdS behave thermodynamically under different entropy formalisms, specifically Hawking–Bekenstein, Rényi, and exponential-corrected entropies, within the extended black hole thermodynamics framework where the cosmological constant is treated as pressure. It develops comprehensive mass–entropy relations, conjugate temperatures, volumes, and potentials for each entropy model, enforcing the first law and revealing how entropy corrections modify P–V behavior and stability. A key finding is that exponential corrected entropy exhibits multiple phase transitions, supported by divergences in the Ruppeiner Ricci scalar, indicating enhanced sensitivity to microstructure. Observationally, the work analyzes frequency shifts (redshift, blueshift, gravitational shift) of photons in torus-like spacetimes and shows consistency with megamaser measurements from NGC 4258 and UGC 3789, highlighting the potential to constrain non-spherical horizon geometries with strong-field data.
Abstract
We investigate the thermodynamic and observational implications for the charged torus-like black holes, a class of solutions distinct from the classical Schwarzschild black holes. We explicitly derive the fundamental thermodynamic properties, such as heat capacity, P-V diagram, isothermal compressibility, Helmholtz free energy, and Gibbs free energy, under different entropy models. We find that only the exponential corrected entropy demonstrates multiple phase transitions, which we validate with the Ricci Scalar divergence obtained from the Ruppeiner formalism. This indicates that exponential corrected entropy is more sensitive to BH's microstructure as compared to the Hawking-Bekenstein and Rènyi entropy models. In addition, we study the sparsity and emission rates of Hawking radiation, demonstrating that exponential correction entropy yields more consistent and stable behavior. In our observational analysis, we graphically demonstrate the behavior of redshift, blueshift, and gravitational shift, and identify specific conditions where the photon sphere radius exceeds the innermost stable circular orbit radius, which depends on the values of parameters such as electric charge and cosmological constant. The novel insight of this work is that despite this violation, our computed redshift, blueshift, and gravitational shifts fall within the range of the observational data of NGC 4258 and UGC 3789.