Quantum fields during black hole formation: How good an approximation is the Unruh state?

TL;DR

This study analyzes quantum effects during black hole formation using a 2D model, comparing the Unruh state approximation with actual collapsing spacetime, revealing its limitations and accuracy over time.

Contribution

It provides a detailed comparison of the Unruh state with the true quantum state during black hole formation in a collapsing null shell scenario.

Findings

Unruh state accurately predicts Hawking radiation at late times.

Back-reaction estimates based on Unruh state may overestimate pre-Hawking radiation.

The Unruh state approaches the actual state exponentially fast over time.

Abstract

We study the quantum effects of a test Klein-Gordon field in a Vaidya spacetime consisting of a collapsing null shell that forms a Schwazschild black hole, by explicitly obtaining, in a $(1+1)$-dimensional model, the Wightman function, the renormalised stress-energy tensor, and by analysing particle detector rates along stationary orbits in the exterior black hole region, and make a comparison with the folklore that the Unruh state is the state that emerges from black hole formation. In the causal future of the shell, we find a negative ingoing flux at the horizon that agrees precisely with the Unruh state calculation, and is the source of black hole radiation, while in the future null infinity we find that the radiation flux output in the Unruh state is an upper bound for the positive outgoing flux in the collapsing null shell spacetime. This indicates that back-reaction estimates based on Unruh state calculations over-estimate the energy output carried by so-called pre-Hawking radiation. The value of the output predicted by the Unruh state is however approached exponentially fast. Finally, we find that at late times, stationary observers in the exterior black hole region in the collapsing shell spacetime detect the local Hawking temperature, which is also well characterised by the Unruh state, coming from right-movers. Early-time discrepancies between the detector rates for the Unruh state and for the state in the collapsing shell spacetime are explored numerically.