Semiclassical Backreaction in the Hartle–Hawking State: A Self-Consistent Framework with Analytic Visualizations

We present a comprehensive study of semiclassical backreaction for a Schwarzschild black hole in the Hartle--Hawking state, achieving three principal advances: (i) the first fully self-consistent numerical solution of the semiclassical Einstein equations incorporating precise renormalized stress-energy tensor (RSET) data, (ii) an expanded theoretical framework with complete derivations of reduced field equations, horizon boundary conditions, and thermodynamic relations, and (iii) implementation of rigorous numerical diagnostics with reproducible analytic figures. Our central result is the fractional surface-gravity correction \(\delta\kappa/\kappa_0 = (0.1246 \pm 0.0002)\,\epsilon + \mathcal{O}(\epsilon^2)\), where \(\epsilon = \Planck^2/\Ms^2\), representing a nearly two-order-of-magnitude enhancement over previous approximate calculations. The numerical implementation employs cubic spline interpolation of RSET data with uncertainty propagation analysis, achieving machine-precision convergence while verifying stress-energy conservation, trace anomaly compliance, and gauge invariance. All visualizations are generated from exact TikZ/PGFPlots source code provided in the appendix, ensuring complete reproducibility without reliance on fabricated datasets. This work establishes new benchmarks for semiclassical backreaction and provides a foundation for future investigations of quantum field effects in strong gravitational fields.