发表机构
School of Physics and Optoelectronics, South China University of Technology; Faculty of Arts and Sciences, Beijing Normal University（华南理工大学物理与光电学院; 北京师范大学艺术与科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
以四维史瓦西黑洞为例，通过角分裂重整化方案和高阶大\(\ell\)渐近减法，首次计算出无质量最小耦合标量场在史瓦西黑洞内部的完整重整化应力 - 能量张量，填补相关空白。
AI 中文摘要
我们计算了无质量最小耦合标量场在四维史瓦西黑洞内部，处于安鲁（Unruh）态和哈特尔 - 霍金（Hartle--Hawking）态下的完整重整化应力 - 能量张量（RSET）。四维黑洞内部的完整RSET长期以来一直无法得到，使得半经典反作用所需的局部源项未知。在类空奇点附近，这一差距更为明显。以史瓦西黑洞内部为例，我们首次填补了这两个空白。通过角分裂重整化方案和高阶大\(\ell\)渐近减法，我们确定了从事件视界到\(r/M\simeq10^{-4}\)范围内\(\langle T^a{}_{b}\rangle_{\rm ren}\)的所有独立分量，并同时得到了相应的真空极化\(\langle\Phi^2\rangle_{\rm ren}\)。该张量通过了协变守恒和迹恒等式的交叉检验。在类空奇点附近，安鲁态和哈特尔 - 霍金态趋近于相同的守恒标度解，\(\begin{align}M^4\langle T^a{}_{b}\rangle_{\rm ren} &\simeq \left(\frac{r}{M}\right)^{-6}\tau^a{}_{b}\end{align}\)，而与态相关的安鲁通量相对于对角混合分量被\((r/M)^4\)抑制。因此，主导的紫外源是局部真空极化应力而非传输的霍金通量。极限张量违反了主导能量条件，但满足零能量条件。所以，在完整的固定背景史瓦西RSET层面，主导的半经典源并不支持量子散焦使奇点平滑的直觉；相反，它在局部瑞查德胡里（Raychaudhuri）方程中提供了一个发散的聚焦源。然而，一个真正的全局结论需要求解反作用的半经典几何。
英文摘要
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.
Comments7 pages, 2 figures; includes Supplemental Material. Revision adds a leading-RSET rigidity theorem for arbitrary Hadamard states and a null-projection phase diagram for nonminimal coupling
