A generalization of the Fredenhagen-Haag derivation of Hawking radiation for a class of Vaidya space-times
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.