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Lattice Black Holes

Steven Corley, Ted Jacobson

TL;DR

The paper addresses how Hawking radiation emerges when short-distance physics is modeled with a lattice cutoff in a freely falling frame around a static black hole, using a two-dimensional scalar field discretized along a falling coordinate.A lattice with a remnant time-translation symmetry reproduces the continuum Hawking spectrum under the regime $κ \ll ω^{1/3} \ll 1$, with outgoing modes arising from ingoing high-$k$ modes that refract at the horizon; an alternative discretization with constant spacing at infinity breaks this mechanism. Through mode equations, near-horizon analysis, and WKB matching, the authors show that the Hawking occupation numbers are thermal with $T_H = κ/(2π)$ in the preserved-symmetry model, while the symmetry-breaking discretization prevents such mode conversion, highlighting the role of short-distance time-translation invariance in Hawking radiation.These results illuminate the robust features of Hawking radiation under high-frequency modifications and point to the need for physically well-behaved lattice models (potentially dynamical lattices) that maintain a consistent short-distance structure in curved spacetimes.

Abstract

We study the Hawking process on lattices falling into static black holes. The motivation is to understand how the outgoing modes and Hawking radiation can arise in a setting with a strict short distance cutoff in the free-fall frame. We employ two-dimensional free scalar field theory. For a falling lattice with a discrete time-translation symmetry we use analytical methods to establish that, for Killing frequency $ω$ and surface gravity $κ$ satisfying $κ\llω^{1/3}\ll 1$ in lattice units, the continuum Hawking spectrum is recovered. The low frequency outgoing modes arise from exotic ingoing modes with large proper wavevectors that "refract" off the horizon. In this model with time translation symmetry the proper lattice spacing goes to zero at spatial infinity. We also consider instead falling lattices whose proper lattice spacing is constant at infinity and therefore grows with time at any finite radius. This violation of time translation symmetry is visible only at wavelengths comparable to the lattice spacing, and it is responsible for transmuting ingoing high Killing frequency modes into low frequency outgoing modes.

Lattice Black Holes

TL;DR

The paper addresses how Hawking radiation emerges when short-distance physics is modeled with a lattice cutoff in a freely falling frame around a static black hole, using a two-dimensional scalar field discretized along a falling coordinate.A lattice with a remnant time-translation symmetry reproduces the continuum Hawking spectrum under the regime $κ \ll ω^{1/3} \ll 1$, with outgoing modes arising from ingoing high-$k$ modes that refract at the horizon; an alternative discretization with constant spacing at infinity breaks this mechanism. Through mode equations, near-horizon analysis, and WKB matching, the authors show that the Hawking occupation numbers are thermal with $T_H = κ/(2π)$ in the preserved-symmetry model, while the symmetry-breaking discretization prevents such mode conversion, highlighting the role of short-distance time-translation invariance in Hawking radiation.These results illuminate the robust features of Hawking radiation under high-frequency modifications and point to the need for physically well-behaved lattice models (potentially dynamical lattices) that maintain a consistent short-distance structure in curved spacetimes.

Abstract

We study the Hawking process on lattices falling into static black holes. The motivation is to understand how the outgoing modes and Hawking radiation can arise in a setting with a strict short distance cutoff in the free-fall frame. We employ two-dimensional free scalar field theory. For a falling lattice with a discrete time-translation symmetry we use analytical methods to establish that, for Killing frequency and surface gravity satisfying in lattice units, the continuum Hawking spectrum is recovered. The low frequency outgoing modes arise from exotic ingoing modes with large proper wavevectors that "refract" off the horizon. In this model with time translation symmetry the proper lattice spacing goes to zero at spatial infinity. We also consider instead falling lattices whose proper lattice spacing is constant at infinity and therefore grows with time at any finite radius. This violation of time translation symmetry is visible only at wavelengths comparable to the lattice spacing, and it is responsible for transmuting ingoing high Killing frequency modes into low frequency outgoing modes.

Paper Structure

This paper contains 14 sections, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Falling lattice models
  3. Lattice dispersion relation
  4. Origin of the outgoing modes
  5. Hawking radiation
  6. Mode equation
  7. Near horizon approximation
  8. Match to the far zone
  9. Kinematic factors
  10. Models with finite lattice spacing at spatial infinity
  11. Origin of the outgoing modes revisited
  12. Discussion
  13. Free-fall coordinates
  14. WKB solutions to the DDE

Figures (3)

  • Figure 1: Painlevé-Gullstrand coordinates and ingoing light rays. The trajectory of a wavepacket that is outgoing with low wavevector at late times is sketched.
  • Figure 2: Graphical representation of the dispersion relation (\ref{['dr']}).
  • Figure 3: Plot of the wavevector trajectories as a function of the background free-fall velocity function $v(x)$.