Abstract
Horizons are classical causal structures that arise in systems with sharply defined energy and corresponding gravitational radius. A global gravitational radius operator can be introduced for a static and spherically symmetric quantum mechanical matter state by lifting the classical “Hamiltonian” constraint that relates the gravitational radius to the ADM mass, thus giving rise to a “horizon wave-function”. This minisuperspace-like formalism is shown here to be able to consistently describe also the local gravitational radius related to the Misner–Sharp mass function of the quantum source, provided its energy spectrum is determined by spatially localised modes.
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Notes
Let us remark this is the frame in which the Tolman–Oppenheimer–Volkoff equation is usually derived [19].
We shall use units with \(c=1\), and the Newton constant \(G=\ell _{\mathrm{p}}/m_{\mathrm{p}}\), where \(\ell _{\mathrm{p}}\) and \({m_{\mathrm{p}}}\) are the Planck length and mass, respectively, and \(\hbar =\ell _{\mathrm{p}}\,m_{\mathrm{p}}\).
Conversely, but perhaps of less interest, the second term would be useful in order to describe states in which matter can be approximated classically but gravity remains fully quantum.
This point is purely technical in the global approach, but will become crucial in the local analysis.
Note the integration is formally extended from zero to infinity, although it will be naturally limited to a smaller range if the spectral decomposition of the source is limited above and/or below.
More technically, one can view \(4\,\pi \,\langle \, \Delta {\hat{r}}^2(r)\,\rangle \) as the uncertainty in the area of a sphere of coordinate radius r.
Of course, they might still be obtained as the limit of suitable series.
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Casadio, R., Giugno, A. & Giusti, A. Global and local horizon quantum mechanics. Gen Relativ Gravit 49, 32 (2017). https://doi.org/10.1007/s10714-017-2198-7
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DOI: https://doi.org/10.1007/s10714-017-2198-7