Power-Law Approach of the Stress-Energy Tensor to the Unruh State after Gravitational Collapse

We establish the rate at which the renormalized stress–energy tensor of a massless minimally coupled scalar field in the in-vacuum state of a collapsing null-shell spacetime approaches the corresponding Unruh-state value. At finite exterior radius, we establish the upper bound $$\mathrm{\mid }\mathrm{\Delta }⟨T_{\mu \nu }⟩\mathrm{\mid }\le C(r)t_s^{-3}$$ from the Cauchy-surface decomposition of the Hadamard difference and the branch-cut structure of the retarded Green function. At future null infinity, we show that the leading coefficient in the late-time expansion $$\mathrm{\Delta }⟨T_{uu}⟩\sim C_{uu}{u}_s^{-3}$$ is nonzero, by computing the branch-cut residue explicitly at small frequency and using the Planck suppression of the thermal spectrum at large frequency to show that the dominant contribution to $C_{uu}$ has a definite sign. The result gives $$\mathrm{\Delta }⟨T_{uu}⟩{\mid }_{{\text{Iscr}}^{+}}(u_s)\sim C_{uu}{u}_s^{-3},u_s\to \mathrm{\infty },$$ with $C_{uu}\mathrm{\ne }0$. The exponent is determined by the ${\omega }^2\ln \omega$ branch-point singularity in the Wronskian of the $\mathrm{\ell }=0$ radial wave equation, the same structure responsible for Price’s law. The sign $C_{uu}<0$ is supported by a physical argument and by the numerical mode data of Gholizadeh Siahmazgi, Anderson, and Fabbri. The result confirms their conjecture that the approach is a power law. We conjecture that the same mechanism gives an analogous $t_s^{-7}$ bound for gravitational perturbations (${\mathrm{\ell }}_{\min }=2$), though the extension to the spin-2 case involves gauge issues not addressed here.