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Thermodynamic Geometry and Hawking Radiation

This work explores the role of thermodynamic fluctuations in the two parameter Hawking radiating black hole configurations. The system is characterized by an ensemble of arbitrary mass and radiation frequency of the black holes. In the due course of the Hawking radiations, we find that the intrinsic geometric description exhibits an intriguing set of exact pair correction functions and global correlation lengths. We investigate the nature of the constant amplitude radiation and find that it's not stable under fluctuations of the mass and frequency. Subsequently, the consideration of the York model decreasing amplitude radiation demonstrates that thermodynamic fluctuations are globally stable in the small frequency region. In connection with quantum gravity refinements, we take an account of the logarithmic correction into the constant amplitude and York amplitude over the Hawking radiation. In both considerations, we notice that the nature of the possible parametric fluctuations may precisely be ascertained without any approximation. In the frequency domain $w \in (0, \infty)$, we observe that both the local and the global thermodynamic fluctuations of the radiation energy flux are stable in the s-channel. The intrinsic geometry exemplifies a definite stability character to the thermodynamic fluctuations, and up to finitely many topological defects on the parametric surface, the notion remains almost the same for both the constant amplitude and the York model. The Gaussian fluctuations over equilibrium radiation energy flux and fluctuating horizon configurations accomplish a well-defined, non-degenerate, curved and regular intrinsic Riemannian manifolds, for all the physically admissible domains of the radiation parameters. PACS numbers: 04.70.-s: Physics of black holes; 04.70.-Bw: Classical black holes; 04.70.Dy Quantum aspects of black holes, evaporation, thermodynamics; 04.60.Cf Gravitational aspects of string theory. Keywords: Quantum Gravity, Hawking Radiation; Horizon Perturbations; Vaidya Geometry; York Model; Statistical Fluctuations; Thermodynamic Configurations