Hawking Spectrum and High Frequency Dispersion

TL;DR

This work investigates Hawking radiation in a two-dimensional black hole with Lorentz-non-invariant, high-frequency dispersion implemented in the free-fall frame. It identifies two particle-creation channels: a horizon-driven thermal flux produced by mode conversion at the horizon, and a non-thermal flux from scattering off background curvature into negative free-fall frequency modes. For smooth metrics, the horizon component is nearly perfectly thermal with tiny deviations of order ${\cal O}((T_H/k_0)^3)$ at $\omega\approx T_H$ and persists up to $\omega/T_H\approx 45$, while kinked metrics exhibit substantial non-thermal flux at high $\omega$, with oscillatory spectra and contributions reaching up to ~10% of the total luminosity. The results demonstrate the robustness of the Hawking effect to high-frequency dispersion in smooth geometries and reveal a secondary, geometry-induced particle production channel tied to static curvature, with implications for backreaction and fundamental short-distance physics.

Abstract

We study the spectrum of created particles in two-dimensional black hole geometries for a linear, hermitian scalar field satisfying a Lorentz non-invariant field equation with higher spatial derivative terms that are suppressed by powers of a fundamental momentum scale $k_0$. The preferred frame is the ``free-fall frame" of the black hole. This model is a variation of Unruh's sonic black hole analogy. We find that there are two qualitatively different types of particle production in this model: a thermal Hawking flux generated by ``mode conversion" at the black hole horizon, and a non-thermal spectrum generated via scattering off the background into negative free-fall frequency modes. This second process has nothing to do with black holes and does not occur for the ordinary wave equation because such modes do not propagate outside the horizon with positive Killing frequency. The horizon component of the radiation is astonishingly close to a perfect thermal spectrum: for the smoothest metric studied, with Hawking temperature $T_H\simeq0.0008k_0$, agreement is of order $(T_H/k_0)^3$ at frequency $ω=T_H$, and agreement to order $T_H/k_0$ persists out to $ω/T_H\simeq 45$ where the thermal number flux is $O(10^{-20}$). The flux from scattering dominates at large $ω$ and becomes many orders of magnitude larger than the horizon component for metrics with a ``kink", i.e. a region of high curvature localized on a static worldline outside the horizon. This non-thermal flux amounts to roughly 10\% of the total luminosity for the kinkier metrics considered. The flux exhibits oscillations as a function of frequency which can be explained by interference between the various contributions to the flux.

Figures (12)

• Figure 1: A patch of spacetime showing a free-fall trajectory and some $t$ and $x$ (Lemaı̂tre-like) coordinate lines. $u$ and $s$ are orthonormal vectors, and the derivative along $s$ is modified, while that along $u$ is just the partial derivative. The notations $\delta_t$ and $\delta_x$ denote $\partial/\partial t$ and $\partial/\partial x$ respectively, and $\delta_t$ is the Killing vector.
• Figure 2: Curve $a$ is the standard dispersion relation for the massless wave equation, curve $b$ is the type used by Unruh, and curve $c$ is the one used in this paper (\ref{['F']}).
• Figure 3: Graphical solution of the position-dependent dispersion relation (\ref{["disp'"]}), with $F(k)$ given by (\ref{['F']}), in units where $k_0=1$. The line labeled $v_{0}$ corresponds to a position far from the hole. The other line corresponds to the classical turning point. The $k$ values of the intersections of the straight and curved lines are the solutions to the dispersion relation for fixed $\omega$ and $v$. For the $v_{0}$ line these are denoted from left to right by $k_{-}$, $k_{-s}$, $k_{+s}$, and $k_{+}$ in the text. The filled arrowheads indicate the direction of propagation of wavepackets, in momentum space, as discussed in the text.
• Figure 4: Schematic representation of the history of an outgoing low frequency positive wavevector wavepacket. The solid vertical line is the horizon, and the dashed line is the "kink" where the scattering takes place. The $+$ and $-$ signs indicate the sign of the wavevector.
• Figure 5: Plot of the real and imaginary parts of the solution to eqn. (\ref{['ode']}) for a free-fall velocity $v_{\rm kink}$ (\ref{['vkink']}), temperature $T_{H}$ = 0.003, and a frequency of $\omega$ = 0.01, in units where $k_0=1$. The horizon is located at $x=0$ and the kink is located at $x\simeq 26$. Note how the solution tunnels out across the horizon, growing exponentially, and then begins oscillating. Both the short and long wavelength components are clearly visible.