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

TL;DR

This paper analyzes how the stress-energy tensor of a scalar field approaches the Unruh state after gravitational collapse, establishing a power-law decay rate and its underlying mathematical structure.

Contribution

It provides the first detailed derivation of the power-law decay rate for the stress-energy tensor in a collapsing spacetime, linking it to branch-point singularities in the wave equation.

Findings

The stress-energy tensor approaches the Unruh state as u_s^{-3} at null infinity.

The decay rate is determined by the ^2\,ln branch-point singularity in the wave equation.

Numerical data supports the sign and power-law behavior of the approach.

Abstract

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 \[ |\Delta\langle T_{\mu\nu}\rangle|\leq C(r)\,t_s^{-3} \] 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 \[ \Delta\langle T_{uu}\rangle\sim C_{uu}\,u_s^{-3} \] 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 \[ \Delta\langle T_{uu}\rangle\big|_{\Iscr^+}(u_s) \sim C_{uu}\,u_s^{-3}, \qquad u_s\to\infty, \] with $C_{uu}\neq 0$. The exponent is determined by the $\omega^2\ln\omega$ branch-point singularity in the Wronskian of the $\ell=0$ 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 ($\ell_{\min}=2$), though the extension to the spin-2 case involves gauge issues not addressed here.