Complete Quantum Stress Tensor Inside a Four Dimensional Schwarzschild Black Hole: A Divergent Focusing Source

Abstract

We compute the complete renormalized stress-energy tensor (RSET) of a massless minimally coupled scalar field throughout the interior of a four-dimensional Schwarzschild black hole, in both the Unruh and Hartle--Hawking states. The complete RSET inside four-dimensional black holes has long been unavailable, leaving the local source term required for semiclassical backreaction unknown. This gap is even sharper near spacelike singularities. Taking the Schwarzschild interior as a concrete example, we close both gaps for the first time. Using an angular-splitting renormalization scheme together with a high-order large-$\ell$ asymptotic subtraction, we determine all independent components of $\langle T^a{}_{b}\rangle_{\rm ren}$ from the event horizon down to $r/M\simeq10^{-4}$, and simultaneously obtain the corresponding vacuum polarization $\langle\Phi^2\rangle_{\rm ren}$. The tensor passes the cross-checks of the covariant conservation and the trace identity. Near the spacelike singularity, the Unruh and Hartle--Hawking states approach the same conserved scaling solution, \ba M^4\langle T^a{}_{b}\rangle_{\rm ren} \simeq \left(\frac{r}{M}\right)^{-6}\tau^a{}_{b},\nn \ea while the state-dependent Unruh flux is suppressed by $(r/M)^4$ relative to the diagonal mixed components. The leading ultraviolet source is therefore a local vacuum-polarization stress rather than transported Hawking flux. The limiting tensor violates the dominant energy condition but satisfy the null energy condition. Thus, at the level of the complete fixed-background Schwarzschild RSET, the leading semiclassical source does not support the intuition that quantum defocusing smooths the singularity; instead, it supplies a divergent focusing source in the local Raychaudhuri equation. A genuine global conclusion, however, requires solving the backreacted semiclassical geometry.