Computing the spectrum of black hole radiation in the presence of high frequency dispersion: an analytical approach

TL;DR

This work analyzes how high-frequency dispersion alters the spectrum of black-hole radiation for a scalar field, by solving modified wave equations with a $k_0$-scaled fourth-derivative term using an analytical blend of Laplace transforms and WKB. It treats both subluminal and superluminal dispersion and shows that, to leading order in $1/k_0$, the Hawking flux remains thermal at $T_H = \kappa/(2\pi)$ and the out-state observed at infinity is thermal as well. The results reinforce the robustness of Hawking radiation against ultraviolet modifications and provide a detailed, controllable analytic framework for horizon scattering and mode mixing, while clarifying the parameter ranges and boundary conditions under which the conclusions hold. Potential extensions include higher-order corrections, backreaction effects, and non-stationary spacetimes to address remaining conceptual issues like the stationarity puzzle.

Abstract

We present a method for computing the spectrum of black hole radiation of a scalar field satisfying a wave equation with high frequency dispersion. The method involves a combination of Laplace transform and WKB techniques for finding approximate solutions to ordinary differential equations. The modified wave equation is obtained by adding a higher order derivative term suppressed by powers of a fundamental momentum scale $k_0$ to the ordinary wave equation. Depending on the sign of this new term, high frequency modes propagate either superluminally or subluminally. We show that the resulting spectrum of created particles is thermal at the Hawking temperature, and further that the out-state is a thermal state at the Hawking temperature, to leading order in $k_0$, for either modification.

Figures (6)

• Figure 1: Plot of the dispersion relations for the ordinary wave equation and its subluminal and superluminal modifications. The intersection points with the $(\omega - v k)$ line are the possible wavevector roots. For the particular $v$ shown, there are four real wavevector roots for the subluminal dispersion relation, but only two real roots for the superluminal case. The other two roots in this case are complex.
• Figure 2: Diagram of the steepest descent contour $C_{0}$. The unmarked regions are directions in which the contour must asymptote for the integral to converge. The $\times$'s are singularities of the integrand and the wavy line is a branch cut.
• Figure 3: Diagram of the steepest descent contours $C_{1}$, $C_{2}$, and $C_{3}$. $C_{1}$ and $C_2$ pass through the saddle points $s_+$ and $s_-$ respectively. The unmarked regions are directions in which the contour must asymptote for the integral to converge. The $\times$'s are singularities of the integrand and the wavy line is a branch cut.
• Figure 4: Diagram of the steepest descent contours $C_{2}$, $C_{4}$, and $C_{5}$. $C_{2}$ and $C_5$ pass through the saddle points $s_+$ and $s_-$ respectively, the solutions for these contours are valid for $x>0$. The solution corresponding to the contour $C_4$ is valid for $x<0$. The unmarked regions are directions in which the contour must asymptote for the integral to converge. The $\times$'s are singularities of the integrand and the wavy line is a branch cut.
• Figure 5: Diagram of the steepest descent contour $C_{6}$. The unmarked regions are directions in which the contour must asymptote for the integral to converge. The $\times$'s are singularities of the integrand and the wavy line is a branch cut.