Ruppeiner thermodynamic geometry, microstructure, quasinormal modes and greybody factors of the Einstein-Skyrme black hole

We investigate the thermodynamic microstructure, quasinormal mode spectrum, and greybody factors of the exact, static, spherically symmetric black hole in Einstein-Skyrme theory, building on the recently established first law that promotes the Skyrme couplings K and λ to extensive variables. Using Ruppeiner geometry, we construct the scalar curvature from the Hessian of the mass. The curvature remains finite at the second-order (heat-capacity) phase transition but diverges in the extremal, zero-temperature limit. Throughout the physically admissible region the curvature retains a single sign, indicating microscopic interactions of one dominant (attractive) type whose strength grows towards extremality. We perform a Joule-Thomson-like isenthalpic expansion and prove analytically that the corresponding coefficient is strictly negative across the entire physical parameter space, implying the black hole always cools as λ increases at a fixed mass, with no inversion temperature. Turning to perturbations, we derive the effective potential for massless scalar fields. The solid-angle-deficit coupling K controls its shape-substantially lowering the barrier-while at fixed K the quartic coupling λ produces only minor changes. Rigorous lower bounds on greybody factors are obtained in closed form using Visser's method: increasing K raises the bound, since a larger horizon lowers the centrifugal barrier, whereas increasing λ mildly suppresses low-frequency transmission. Our results provide the first comprehensive study of the thermodynamic microstructure and perturbative spectroscopy of this rare analytic hairy black hole, complementing and extending the existing thermodynamic analysis into previously unexplored territory.

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