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Dynamical Black Hole Emission

David A. Lowe, Larus Thorlacius

TL;DR

Problem: how Hawking radiation turns on dynamically in 4D and how back-reaction alters spacetime. Approach: solve time-dependent, anomaly-induced semiclassical equations in a collapsing spacetime using auxiliary scalar fields that render the conformal anomaly local. Key results: Hawking flux is activated after horizon formation, grows to a plateau, and a late-time solution with φ ~ αu reproduces the expected Unruh-like flux; pre-Hawking radiation is not present in the dynamical evolution. Significance: provides a controlled 4D dynamical framework for black hole evaporation and sets the stage for including back-reaction and endpoint physics.

Abstract

Semiclassical black hole emission in four spacetime dimensions is studied using a non-local effective action. The field equations that determine the time-dependent renormalized stress tensor are solved numerically for a black hole formed by an ingoing null shock wave, and otherwise smooth initial data. We find that Hawking radiation is generated dynamically near the black hole horizon and freely propagates out to null infinity, resulting in an outgoing energy flux that builds up from zero at early retarded times before the black hole forms. This resolves the long-standing issue of pre-Hawking radiation, suffered by calculations based on a static approximation to the stress tensor in an Unruh state, and paves the way towards four-dimensional black hole evolution with semiclassical back-reaction included.

Dynamical Black Hole Emission

TL;DR

Problem: how Hawking radiation turns on dynamically in 4D and how back-reaction alters spacetime. Approach: solve time-dependent, anomaly-induced semiclassical equations in a collapsing spacetime using auxiliary scalar fields that render the conformal anomaly local. Key results: Hawking flux is activated after horizon formation, grows to a plateau, and a late-time solution with φ ~ αu reproduces the expected Unruh-like flux; pre-Hawking radiation is not present in the dynamical evolution. Significance: provides a controlled 4D dynamical framework for black hole evaporation and sets the stage for including back-reaction and endpoint physics.

Abstract

Semiclassical black hole emission in four spacetime dimensions is studied using a non-local effective action. The field equations that determine the time-dependent renormalized stress tensor are solved numerically for a black hole formed by an ingoing null shock wave, and otherwise smooth initial data. We find that Hawking radiation is generated dynamically near the black hole horizon and freely propagates out to null infinity, resulting in an outgoing energy flux that builds up from zero at early retarded times before the black hole forms. This resolves the long-standing issue of pre-Hawking radiation, suffered by calculations based on a static approximation to the stress tensor in an Unruh state, and paves the way towards four-dimensional black hole evolution with semiclassical back-reaction included.

Paper Structure

This paper contains 7 sections, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Exact results in a two-dimensional model
  3. Basic setup in four-dimensional spacetime
  4. Scaling relations
  5. Numerical Evaluation
  6. Semi-Analytic Late-Time Solution
  7. Discussion

Figures (2)

  • Figure 1: The scalar field is plotted versus the null coordinates $u,\,v$. The black hole is formed by a shock at $v=0$ the scalar field is evolved out to $v_{max}=300$. The black hole horizon corresponds to $u=+\infty$ at fixed $v$. The parameters are $M=1,\,a=1,\,b=1$.
  • Figure 2: The outgoing flux $T_{uu}$ is computed at $v=v_{max}$ to indicate the flux of energy reaching $\mathscr{I}^{+}$. This does not turn on until $u\approx0$ indicating emission from the region of horizon formation. The parameters are $M=1,\,a=1,\,b=1$. The undashed line shows the results for a fixed size grid with $(1.5\times10^{4})^{2}$ points, while the dashed line is for a grid with $(10^{4})^{2}$ points.