Universal Leading Quantum Stress Near the Schwarzschild Singularity

Shun Jiang; Jie Jiang

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arXiv · 2607.04386

Universal Leading Quantum Stress Near the Schwarzschild Singularity

Shun Jiang·Jie Jiang

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Abstract

Determining the local quantum source near a spacelike singularity is essential for assessing semiclassical backreaction inside black holes. By standard point splitting, we provide, to our knowledge, the first complete numerical determination of the renormalized stress-energy tensor (RSET) of a massless scalar field at arbitrary curvature coupling in the four-dimensional Schwarzschild interior. Calculations in the Unruh and Hartle--Hawking states reach $r/M\simeq10^{-4}$, yielding a common leading $r^{-6}$ coefficient tensor and coupling-dependent anisotropic pressure ratios. Uniform estimates of the complete mode sums then establish a rigidity theorem: at each fixed coupling, any Hadamard-state change on the common interior evolution domain contributes only subleading stress. The leading tensor is therefore insensitive to time and directional dependence of the quantum environment, in contrast with state-dependent leading Cauchy-horizon coefficients in asymptotically flat Reissner--Nordström spacetime. Its directional null projections define positive, mixed-sign, and negative regimes in coupling and direction, specifying the local mean focusing source; at conformal coupling, all leading projections are positive.

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Shun Jiang, Jie Jiang. 2026-09-20. Universal Leading Quantum Stress Near the Schwarzschild Singularity. https://arxiv.org/abs/2607.04386

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