Abstract
We show within the usual two-dimensional approximation that unitarity and the restoration of Minkowski vacuum correlations at the end of black hole evaporation impose unexpected constraints on its mass loss rate: before disappearing the black hole emits one or more negative energy burst, leading to a temporary increase of its mass.
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Notes
It is immediate to estimate the size of the core of a black-hole-like object with the singularity resolved at a Planck-scale energy density: if we concentrate all the mass \(M\) of the black hole in a core of size \(\ell \) we find a density \(\varrho \sim M/\ell ^3\). A Planck scale upper bound \(\varrho \sim \hbar ^{-1}\) results in a core of size \(\ell \sim (M\,\hbar )^{1/3}\). For a solar mass black hole this size is \(10^{12}\) times larger that the Planck length \(\ell _P=\hbar ^{\,1/2}\) and can be described by a classical effective metric.
We partially fix the ambiguity in the choice of the affine parameter \(u\) by demanding that the future-pointing null vectors \(l=\partial _v\) and \(n=\partial _u\) have scalar product \(l\cdot n=-1\) at spatial infinity.
In terms of the ray-tracing function this condition corresponds to \(\dot{p}(u)\rightarrow 1\) smoothly as \(u\rightarrow \pm \infty \), i.e. no redshift or blueshift at initial and late times.
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Acknowledgments
We thank Abhay Ashtekar and Ted Jacobson for useful discussions on energy and unitarity in black hole evaporation.
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Bianchi, E., Smerlak, M. Last gasp of a black hole: unitary evaporation implies non-monotonic mass loss. Gen Relativ Gravit 46, 1809 (2014). https://doi.org/10.1007/s10714-014-1809-9
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DOI: https://doi.org/10.1007/s10714-014-1809-9