Hawking Radiation Without Transplanckian Frequencies

TL;DR

Brout, Massar, Parentani, and Spindel analyze how Hawking radiation emerges when transplanckian frequencies are truncated near a black hole horizon. Using momentum-space Damour–Ruffini formalism and two truncation schemes (a hydrodynamic Unruh model and a regulated dispersion g(p)), they show that the thermal flux is preserved provided in-modes have positive momentum and cisplanckian physics dominates the late-time emission. The results argue for a universal robustness of Hawking radiation against Planck-scale physics and offer a framework for connecting near-horizon kinematics to observable spectra without requiring a full quantum gravity theory.

Abstract

In a recent work, Unruh showed that Hawking radiation is unaffected by a truncation of free field theory at the Planck scale. His analysis was performed numerically and based on a hydrodynamical model. In this work, by analytical methods, the mathematical and physical origin of Unruh's result is revealed. An alternative truncation scheme which may be more appropriate for black hole physics is proposed and analyzed. In both schemes the thermal Hawking radiation remains unaffected even though transplanckian energies no longer appear. The universality of this result is explained by working in momentum space. In that representation, in the presence of a horizon, the d'Alembertian equation becomes a singular first order equation. In addition, the boundary conditions corresponding to vacuum before the black hole formed are that the in--modes contain positive momenta only. Both properties remain valid when the spectrum is truncated and they suffice to obtain Hawking radiation.

Figures (5)

• Figure 1: The two curves $-\omega \pm \vert F(p)\vert$ (solid curves) and the family of lines $-V(\xi) p$ (doted lines) are plotted as functions of $p$ for some representative values of $\xi$. Their intersections give the trajectories $\xi(p)$. Their are three trajectories labeled I, II, III. The set I corresponds to a $v=const$ trajectory. The set II crosses the horizon when $\xi =0$ and then starts to propagate for $\xi < O(-4 M \omega)$ whereupon it corresponds to a $u=const$ trajectory. The points in class III never reach the horizon: there are no solutions in this class for $\xi < \xi_{min}(\omega)=O(4 M \omega)$ but their are two solutions for $\xi < \xi_{min}$ corresponding to a trajectory which bounces.
• Figure 2: The classical trajectories of outgoing null geodesics (thin dotted curves) given by the Hamiltonian eq. \ref{['E18']}) are displayed in the Eddington-Finkelstein coordinate system. The light ray which generates the horizon, the infalling shell, the origin $r=0$ and the singularity are also represented.
• Figure 3: The trajectories of outgoing light rays given by the truncated Hamiltonian eq. \ref{['E23']}) are displayed in the Eddington-Finkelstein coordinate system. In this case the trajectories no longer approach the horizon exponentially but rather they stick at a planckian distance.
• Figure 4: The trajectories of light rays in Unruh's truncated hydrodynamic model are displayed in the Eddington-Finkelstein coordinate system. Only class II and III light rays have been displayed as it is these which are responsible for production. Note that since $v$ and $\eta$ differ by a regular function of $x$, the trajectories in the coordinate system $\eta , \xi$ given by the Hamiltonian eq. \ref{['E4.1']}) would be very similar to those depicted in the figure.
• Figure 5: The trajectories of light rays in the Eddington-Finkelstein coordinate system if the truncation $g(p)$ where complex. In this case the trajectories disapear into some quantum fuzz which is represented by some shading in the figure.