Universal Leading Quantum Stress Near the Schwarzschild Singularity
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≃10−4, 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.
Related
- Quantum features of a non-commutative Schwarzschild black hole
- Inaccessibility of the Cosmological Singularity to Quantum Probes
- Dynamical evolution of quantum mutual information in Schwarzschild spacetime
- Geometric Quantum Discord in Schwarzschild de Sitter Spacetime
- Quantum-corrected thermodynamics of Schwarzschild AdS black holes in conformal Killing gravity
BibTeX
@misc{jiang2026universal,
title = {{Universal Leading Quantum Stress Near the Schwarzschild Singularity}},
author = {Shun Jiang and Jie Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04386},
eprint = {2607.04386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04386},
note = {CC0},
}
RIS
TY - UNPB
TI - Universal Leading Quantum Stress Near the Schwarzschild Singularity
AU - Shun Jiang
AU - Jie Jiang
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2607.04386
UR - https://arxiv.org/abs/2607.04386
AB - 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≃10^−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.
ER -