Quantum optics of an oscillator falling into a black hole

Derek Raine,Paul G Abel

doi:10.1088/1361-6382/abc8c8

Abstract

We present a quantum optics treatment of the near horizon behaviour of a quantum oscillator freely-falling into a pre-existing Schwarzschild black hole. We use Painlevé–Gullstrand coordinates to define a global vacuum state. In contrast to an accelerated oscillator in the Minkowski vacuum, where there is no radiation beyond an initial transient, we find that the oscillator radiates positive energy to infinity and negative energy into the black hole as it attempts to come into equilibrium with the ambient vacuum. We discuss the relationship of the model to Hawking radiation.

Highlights

Hawkings original paper [11] showed that when quantum effects are considered, black holes radiate a thermal flux of particles
Grove [9] was the first to object to this interpretation and suggested instead that the accelerating oscillator emits negative energy with respect to the Minkowski vacuum, which balances out the positive energy emitted by the oscillator as it makes a downward transition, and overall there is no net energy flux in the Minkowski vacuum, thereby breaking the analogy
We show that the oscillator emits a negative energy flux into the hole

Summary

Introduction

Hawkings original paper [11] showed that when quantum effects are considered, black holes radiate a thermal flux of particles. We show that as well as emitting to infinity, the infalling oscillator emits negative energy into the black hole.The Hawking radiation proper arises only when we consider the collapse phase in the formation of a black hole. This phase is nothing other than the in-fall of a collection of radiating atoms (or oscillators). The energy going into the hole will perturb the hole and induce it to emit further radiation which will be correlated with the outgoing flux from the infalling matter If this all happens on an infall timescale, or on the relaxation timescale of the event horizon, this is a transient from the collapse that just happens to have a blackbody form at the Hawking temperature and has little directly to do with Hawking radiation. To decide we need a more detailed model of collapse and evaporation that will allow us to calculate the relaxation time in the presence of the external radiation

Gravitational Collapse in Painleve-Gullstrand

The Quantum Langevin Equation

The Solution to the Field Equation

The Energy Flux at the Detector

Discussion