Analytic semiclassical backreaction of a Schwarzschild black hole in a finite cavity: horizon shift, temperature renormalization, and canonical stability in the Hartle-Hawking State

We construct an analytic model of static semiclassical backreaction for a Schwarzschild black hole in the Hartle–Hawking state enclosed within a finite spherical cavity. Using a minimal renormalized stress–energy tensor consistent with conservation, thermal asymptotics, and horizon regularity, we integrate the reduced semiclassical Einstein equations under Dirichlet boundary conditions at the cavity wall. This yields explicit expressions for the corrections to the mass function, redshift factor, horizon location, and surface gravity. We obtain a closed-form first-order correction to the Hawking temperature in terms of a dimensionless backreaction parameter and the cavity radius. The temperature shift decomposes into redshift renormalization, geometric horizon displacement, and a local energy-density contribution at the horizon. The perturbative expansion is controlled by a parameter of order $M_P^2\mathrm{/}{M}^2$, ensuring validity for macroscopic black holes. The near-horizon geometry retains its universal Rindler${}^2\times S^2$ structure, indicating that semiclassical effects renormalize rather than modify the geometric origin of Hawking radiation.