AI 中文总结
该研究推广了Hawking辐射的Fredenhagen-Haag推导，在一类Vaidya时空上构造了探测器响应不等式，得到了有限参数下的相关估计并分析了Hadamard态的构造及渐近蒸发情形的结果。
AI 中文摘要
我们开发了一种类似Fredenhagen-Haag的定量方法，用于在可控的球对称Vaidya时空上用无质量标量场描述Hawking辐射。系统的局域热特性由外捕获视界处Hadamard两点函数的普适标度极限提供[KurpiczPinamontiVerch2021]。在规则的探测器-视界窗口上，我们构造了整体双曲发展，并在Sobolev能量空间上将非自治Vaidya演化与冻结的Schwarzschild传播子进行比较。我们推导了一个对角动量一致的显式Duhamel估计，计算了精确的线性零剥系数，并对射线映射的二次余项进行了界定。将这些估计与正性相结合，得到了相对于局域热参考的双边探测器响应不等式。其误差项量化了算子变化、定态散射尾、有限Hadamard标度、视界局域化和出射通道。由于保留了所有这些贡献，该不等式在有限参数下有效。在衰减假设下，探测器响应收敛到对应于渐近定态吸积和蒸发-吸积转变剖面的Fredenhagen-Haag形式。我们还通过从最终定态Unruh协方差的柯西传输构造了一个Hadamard态，并表明有限蒸发板在没有规定未来扩展的情况下无法确定后期响应。对于每个有限时间m(u)>0且m(u)→0的渐近蒸发，质量重标度的共形公式为随标度变化的探测器产生了有限窗口的标度协变估计。
英文摘要
We develop a quantitative Fredenhagen--Haag like approach for describing Hawking radiation using massless scalar fields on controlled spherically symmetric Vaidya space-times. The local thermal character of the system is supplied by the universal scaling limit of Hadamard two-point functions at an outer trapping horizon \cite{KurpiczPinamontiVerch2021}. On regular detector--horizon windows, we construct globally hyperbolic developments and compare the nonautonomous Vaidya evolution with a frozen Schwarzschild propagator on Sobolev energy spaces. We derive an explicit Duhamel estimate that is uniform in angular momentum, calculate the exact linear null-peeling coefficient, and bound the quadratic remainder of the ray map. Combining these estimates with positivity gives a two-sided detector-response inequality relative to the local thermal reference. Its error terms quantify operator variation, stationary scattering tails, finite Hadamard scaling, horizon localisation, and the outgoing channel. The inequality is valid at finite parameters because each of these contributions is retained. Under the decay hypotheses, the detector response converges to the corresponding Fredenhagen--Haag form for asymptotically stationary accretion and for evaporation--accretion turnaround profiles. We also construct a Hadamard state by Cauchy transport from an eventually stationary Unruh covariance and show that a finite evaporating slab does not determine a late-time response without a prescribed future extension. For asymptotic evaporation with $m(u)>0$ at every finite time and $m(u)\to0$, a mass-rescaled conformal formulation yields a scale-covariant finite-window estimate for scale-following detectors.
Comments35 pages, 4 figures
