Black holes from a nonlocal equation of state: horizon geometry, singularity structure and thermodynamics

Abstract

We show that a nonlocal equation of state for an anisotropic fluid reduces to a linear equation for the enclosed matter and uniquely selects a Yukawa-type density profile in a static, spherically symmetric Kerr--Schild geometry. The resulting neutral solution has finite-range hair, exponentially screened at large distances, with the asymptotic mass as its only conserved charge, and admits an exact horizon classification: horizonless configurations, Schwarzschild-like black holes, two-horizon geometries with a Cauchy horizon, and an extremal configuration whose mass and radius are obtained in closed form. We prove that the magnitude of the inner surface gravity exceeds that of the outer horizon throughout the two-horizon regime. When the central mass vanishes, the geometry approaches a global monopole in the static branch and a Kantowski--Sachs-type region in the nonstatic branch. The singularity satisfies Królak's strong-curvature criterion and is Tipler-strong along non-radial geodesics in the spacelike branch. For vanishing central mass, the Ricci scalar and the square root of the Kretschmann scalar are locally integrable with respect to the invariant spacetime volume, despite diverging at the origin. A power-law generalization of the nonlocal closure yields solutions expressed through incomplete gamma functions, interpolating between the singular core and a regular black hole with a de~Sitter core, without invoking nonlinear electrodynamics. We obtain closed-form surface gravities, an extended first-law identity and a Smarr relation, and locate the heat-capacity divergence using the lower branch of the Lambert function. Finally, thermodynamic topology distinguishes a Schwarzschild-like regime with one defect from an extremal regime with two opposite-winding defects that merge at the maximum Hawking temperature.