Abstract
We study the quantum effects of a test Klein-Gordon field in a Vaidya space-time 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.
Highlights
State show that there is a negative energy flux across the Schwarzschild horizon that is the source of Hawking radiation
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
We have asked and answered the question of how good the usual folklore that treats the Unruh state in Schwazschild spacetime as the state emerging from physical black hole formation is
Summary
We briefly recall some elements of quantum field theory in Schwarzschild spacetime. This will serve the purpose of introducing the relevant notation for this work. We shall recall the geometric structure of Schwarzschild spacetime and its maximal extension, reminding ourselves the large symmetry structure of these spacetimes, which serves as a guideline for constructing the quantum theory states. We shall introduce a Klein-Gordon field in the Schwarzschild maximal extension and recall the properties of the usual states of the theory, the Boulware, Unruh and Hartle-HawkingIsrael states, emphasising the physical relevance of the Unruh state
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