Abstract
The present paper studies the Dirac Hamiltonian of a test electron with a domain of bi-spinor wave functions supported on the static region inside the Cauchy horizon of the subextremal RWN black hole spacetime, respectively inside the event horizon of the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a sufficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restores essential self-adjointness; the empirical value of the electron’s anomalous magnetic moment is large enough. In the subextremal case the spectrum of the self-adjoint Dirac operator with anomalous magnetic moment is purely absolutely continuous and consists of the whole real line; in particular, there are no eigenvalues. The same is true for the spectrum of any self-adjoint extension of the Dirac operator without anomalous magnetic moment interaction, in the subextremal black hole context. In the extremal black hole sector the point spectrum, if non-empty, consists of a single eigenvalue, which is identified.
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Notes
One may be tempted to consider this result as a vindication for the widespread opinion that “naked singularities are considered unphysical” (cf. [16], p.562). However, this opinion propagates an unfortunate myth. It is based on a misunderstanding of Penrose’s weak cosmic censorship hypothesis, which surmises that gravitational collapse of cosmic matter does not form a naked singularity. In its strict sense the surmise is wrong, as shown first by Christodoulou [7, 8] for spherically symmetric collapse of scalar matter, and most recently by Rodnianski and Shlapentokh-Rothman [27] for collapsing gravitational waves without symmetry assumption; yet it is expected that these scenarios are not generic (this was confirmed for the spherically symmetric scalar case, also by Christodoulou [9]), and that generically (or: typically) a gravitational collapse of cosmic matter will not form a naked singularity. However, the point nuclei used in quantum-mechanical models of hydrogenic ions of the kind created in our laboratories are not assumed to have formed through gravitational collapse of charged matter in cosmic proportions. In short, the weak cosmic censorship hypothesis, even if generically true, is entirely irrelevant to the problem of general-relativistic hydrogenic ions.
The distinguished self-adjoint extention is defined by allowing \(Z\in {\mathbb {C}}\) and demanding analyticity in Z. The real threshold values then become \(Z =\sqrt{3}/2{\alpha ^{}_{{{\text {S}}}}}\) instead of \(Z=118\), and \(Z=1/{\alpha ^{}_{{{\text {S}}}}}\) instead of \(Z=137\). Here, \({\alpha ^{}_{{{\text {S}}}}}:= {e^2}/{\hbar c} \approx 1/137.036\) is Sommerfeld’s fine structure constant.
Here, “\(f(x) \sim g(x)\) as \(x\rightarrow x_*\)” means \(\exists C>0\) such that \(f(x)/g(x) \rightarrow C\) as \(x\rightarrow x_*\), where \(x_*=0\) or \(\infty \).
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Kiessling, M.KH., Tahvildar-Zadeh, A.S. & Toprak, E. On the Dirac operator for a test electron in a Reissner–Weyl–Nordström black hole spacetime. Gen Relativ Gravit 53, 15 (2021). https://doi.org/10.1007/s10714-021-02789-0
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DOI: https://doi.org/10.1007/s10714-021-02789-0