The Absorption Horizon: A Critical Radius of the Schwarzschild Geometry Derived from Radiation Equilibrium, with Extensions to Reissner-Nordstrom, Kerr, and Kerr-Newman Black Holes

A critical radius of the Schwarzschild black hole geometry is derived analytically from a radiation equilibrium argument. Photons propagating in the Schwarzschild metric carry two complementary gravitational energy fractions determined by the local redshift factor: a transmission fraction F_trans measuring the energy that escapes to infinity, and a trapping fraction F_trap measuring the energy retained by the gravitational field, which sum to unity by energy conservation. The net outgoing radiation flux Phi(r) = F_trans · F_trap vanishes at both the event horizon and spatial infinity, and is maximised at the unique radius where F_trans = F_trap, yielding the Absorption Horizon at r_ah = (4/3)r_s, located strictly between the event horizon and the photon sphere. Below this radius gravitational trapping dominates transmission and the net radiation field decays monotonically to zero. The singularity energy within the Absorption Horizon admits a complex decomposition via analytic continuation through the horizon; the factor of -i arises exactly from the ie prescription, placing the result in the Breit-Wigner form E_0 - iGamma/2 and identifying the imaginary part with the Hawking evaporation rate Gamma proportional to M^{-1}. The framework extends analytically to the Kerr-Newman family and to arbitrary spacetime dimension d, with no free parameters.