On general-relativistic hydrogen and hydrogenic ions
Michael K.-H. Kiessling, A. Shadi Tahvildar-Zadeh, Ebru Toprak
aa r X i v : . [ m a t h - ph ] A ug ON GENERAL-RELATIVISTIC HYDROGEN AND HYDROGENIC IONS
MICHAEL K.-H. KIESSLING, A. SHADI TAHVILDAR-ZADEH, EBRU TOPRAK
Abstract.
This paper studies how the static non-linear electromagnetic-vacuum space-time of a point nucleus with negative bare mass affects the self-adjointness of the general-relativistic Dirac Hamiltonian for a test electron, without and with an anomalous magneticmoment. The study interpolates between the previously studied extreme cases of a test elec-tron in (a) the Reissner–Weyl–Nordstr¨om spacetime (Maxwell’s electromagnetic vacuum),which sports a very strong curvature singularity with negative infinite bare mass, and (b)the Hoffmann spacetime (Born or Born–Infeld’s electromagnetic vacuum) with vanishingbare mass, which features the mildest possible curvature singularity. The main conclusionreached is: on electrostatic spacetimes of a point nucleus with a strictly negative bare mass(which may be −∞ ) essential self-adjointness fails unless the radial electric field divergessufficiently fast at the nucleus and the anomalous magnetic moment of the electron is takeninto account. Thus on the Hoffmann spacetime with (strictly) negative bare mass the DiracHamiltonian of a test electron, with or without anomalous magnetic moment, is not essen-tially self-adjoint. All these operators have self-adjoint extensions, though, with the usualessential spectrum ( −∞ , − m e c ] ∪ [ m e c , ∞ ) and an infinite discrete spectrum located inthe gap ( − m e c , m e c ). Introduction
State of Affairs.
In non-relativistic physics, whether Newtonian mechanics or quan-tum mechanics, the gravitational and the electrical attraction between a point electron anda point proton obey the same mathematical force law and only their coupling strengths differ— though by a lot: If (in Gaussian units) e denotes the elementary charge, m e the empiricalmass of the electron and m p the one of the proton, and G is Newton’s constant of universalgravitation, then Gm p m e e =: γ pe ≈ . · − . Thus, in such theories gravity is an extremelyweak pair interaction between electron and proton, compared to electricity, indeed so weakthat it is hard to imagine how any experimental study of the hydrogen atom’s spectrum couldpossibly reveal its effects — assuming that non-relativistic quantum mechanics predicts theeffect accurately enough for all practical purposes. Explicitly, the bound state spectrum ofhydrogen in non-relativistic QM is readily obtained from the familiar Bohr formula through Date : Corrected version of Aug. 28, 2020; Printed: September 1, 2020c (cid:13) (2020): The authors. Reproduction of this article for non-commercial purposes is permitted. the replacement e e + Gm e m p , viz.1 m e c E Bohr n ( Z, N ; γ pe ) (cid:12)(cid:12)(cid:12) Z =1 ,N =0 = − α (1 + γ pe ) ǫ n , n ∈ N ; (1)here, ǫ := m e /m p ≈ / α S := e / ~ c ≈ / .
036 is Sommerfeld’s fine structureconstant, where ~ is the Planck constant divided by 2 π , and c the speed of light. Note that m e ǫ = m e m p m e + m p is the reduced mass of the electron-proton system. We see that each Bohrlevel is lowered by a factor ≈ − compared to the result for purely electrical Coulombinteraction, E Bohr n (1 ,
0; 0). This effect is almost 30 orders of magnitude smaller than the bestspectral resolution achieved today.The non-relativistic gravitational effects on the spectrum of a hydrogenic ion are slightlymore pronounced, though not in any significant way. To obtain the Bohr spectrum of ahydrogenic ion, replace the proton charge e Ze and the proton mass m p A ( Z, N ) m p ,with Z ∈ N , N ∈ { , , , ... } , and A ( Z, N ) ≥ Z (N.B.: Z ≤
118 and
N <
200 in thecurrently known chart of the nuclids, and Z ≤ A ( Z, N ) ≤ Z for the known long-livednuclei; for hydrogen: A (1 ,
0) = 1). Thus (1) is the Z = 1 & N = 0 special case of1 m e c E Bohr n ( Z, N ; γ pe ) = − α (cid:0) Z + γ pe A ( Z, N ) (cid:1) ǫ/A ( Z, N ) 1 n , n ∈ N . (2)So E Bohr n ( Z, N ; γ pe ) differs from E Bohr n ( Z, N ; 0) by not more than 3 · − E Bohr n ( Z, N ; 0).
Remark 1.1.
We recall that the Bohr model yields the same energy spectrum for hydrogenicatoms / ions as does the Schr¨odinger Hamiltonian. We also recall that the spectrum in theBorn–Oppenheimer approximation (the electron is treated as a test particle in the Coulombfield of a fixed nucleus) is recovered by letting m p → ∞ , equivalently ǫ → in (1), (2). The anticipated tininess of the gravitational effect in the hydrogen spectrum is also thereason why Sommerfeld [27] did not generalize his special-relativistic calculations of thehydrogen spectrum (in the Born–Oppenheimer approximation) to the freshly created general-relativistic setting. In fact, Sommerfeld had consulted with Einstein prior to publication of[27] whether it would be advisable to include the general-relativistic effects, but Einsteinadvised against it [10], stating that the quantitative results would essentially agree withSommerfeld’s fine structure formula obtained by invoking only special relativity (for thekinetic energy of the electron), and only Coulomb electricity for the interaction betweenelectron and proton; see also [28]. Here, A ( Z, N ) ≈ Z + N , roughly the number of nucleons in a nucleus. ENERAL-RELATIVISTIC HYDROGEN 3
However, relativistic electricity (read: electromagnetism) and gravity (read: spacetimecurvature) are no longer mathematically identical structures, and so their relative contribu-tions to the atomic spectra cannot obviously be estimated merely in terms of a comparisonof their coupling constants. Sure enough, not long after Sommerfeld published his workon the relativistic hydrogen fine structure he was criticized by Wereide [35] for not havingmathematically demonstrated that general-relativistic effects were indeed so tiny as to benegligible. Eventually, Vallarta in his MIT Ph.D. thesis (the main results are published in[31]) supplied mathematically definitive estimates of the general-relativistic effects in theBohr–Sommerfeld-type spectrum of hydrogen. Vallarta considered a test electron in theReissner–Weyl–Nordstr¨om (RWN) spacetime with a naked timelike singularity, equippedwith the electric charge of the proton, and the ADM mass equated with the empirical protonmass, and applied the Bohr–Sommerfeld quantization rules to the bound electron orbits. Heconcluded that the relativistic gravitational effects were immeasurably tiny, and so he did noteven bother to actually compute their corrections to Sommerfeld’s fine structure spectrum,although he could have done so with the help of perturbation theory. Such computationswere done recently, for circular orbits, by Dreifus in her honors thesis at Rutgers [8].Even though Vallarta’s estimates and Dreifus’ perturbative computations have producedquantitatively tiny general-relativistic corrections to the special-relativistic Sommerfeld fine-structure spectrum of hydrogen using Bohr–Sommerfeld-type quantization, it would be quitea mistake to now conclude from this that general relativity would always manifest itself onlyin form of a tiny perturbation of special-relativistic atomic spectral results. As emphasizedalready, the general theory of relativity reveals that gravity is not a weaker attractive ‘clone’of electromagnetism, but a completely different ‘force of nature.’ There is little doubt nowa-days that general relativity correctly predicts that nature is capable of forming black holeswhich can swallow unlimited amounts of matter as long as supplies will last. Intuitively,therefore, one would be inclined to suspect that general relativity should have a destabiliz-ing effect in the theory of large- Z atoms. At the very least one might expect a worseningof the spectral ‘large- Z catastrophe’ in the special-relativistic ( G = 0) Bohr–Sommerfeldtheory of hydrogenic ions, where it occurs when the nuclear charge number Z exceeds 1 /α S and the bottom drops out from under the energy functional because the electrical attractionoverpowers the angular momentum barrier of the circular motion; see our Appendix A.Curiously, ‘switching on relativistic gravity’ instead removes this ‘large- Z catastrophe’ ofthe special-relativistic Bohr–Sommerfeld-type model for hydrogenic ions. Namely, in our KIESSLING, TAHVILDAR-ZADEH, TOPRAK
Appendix A we show that for each Z ∈ N there is a unique Bohr–Sommerfeld-type spectrumof the general-relativistic hydrogenic ion obtained from a minimum-energy variational prin-ciple. By contrast, in the special-relativistic Bohr–Sommerfeld-type model of a hydrogenicion, the pertinent minimum-energy variational principle has no lower bound when Z > /α S .The just mentioned catastrophe at Z = 1 /α S in the special-relativistic Bohr–Sommerfeldmodel of hydrogenic ions has a counterpart in the spectral theory of the special-relativisticDirac Hamiltonian for a hydrogenic atom/ion [25, 30, 14], where there is also an earliercatastrophe at Z = √ / α S . We recall that this Dirac operator is essentially self-adjointonly if Z ≤ Z ∈ C ) which is self-adjointalso when Z ∈ { , ..., } , but the analytical extension is no longer self-adjoint when Z > Z ≥
119 the deficiencyindices of the Dirac operator restricted to a fixed angular momentum subspace are (1 , Z >
137 it is not clear which one, if any, is physically distinguished.
Remark 1.2.
We recall that Sommerfeld’s fine-structure formula for the energy spectrumof hydrogen agrees with the hydrogen spectrum obtained with Dirac’s special-relativistic waveequation for an electron in the Coulomb field of a fixed proton. The assignment of angularmomentum quantum numbers in Sommerfeld’s calculations of course does not agree withthe spectral formula of the special-relativistic Dirac Hamiltonian, for Sommerfeld did notincorporate any form of electron spin. The subtle reason for this remarkable coincidence ofthe Sommerfeld and Dirac energy spectra for hydrogen is nicely explained in [20] . Since the Bohr–Sommerfeld theory of the spectra of hydrogenic ions exactly captures theirquantum-mechanical energy spectra in both the non-relativistic (Schr¨odinger) setting andin the special-relativistic (Dirac) setting for Zα S ≤
1, and since general relativity has aregularizing effect on the Bohr–Sommerfeld theory, at this point it certainly would seemreasonable to expect that general relativity will have a regularizing effect also in the Diractheory of hydrogenic spectra. However, the opposite is true!Namely, as discovered by Cohen and Powers [7], the general-relativistic Dirac Hamiltonian[26], [6] for hydrogen differs dramatically from the familiar special-relativistic Dirac Hamil-tonian for hydrogen. More precisely, like Vallarta, so also Cohen and Powers modelled generalrelativistic hydrogen as consisting of a test electron in the static Reissner–Weyl–Nordstr¨om Allowing Z ∈ R + essential self-adjointness holds for Z ≤ √ / α S . Allowing Z ∈ R + the analytical extension is self-adjoint for Z ≤ /α S ≈ . ENERAL-RELATIVISTIC HYDROGEN 5 spacetime of a fixed point proton. While the special-relativistic Dirac Hamiltonian for hydro-genic ions is essentially self-adjoint (on the domain C ∞ c ( R \{ } ) ) for all Z ≤
118 [22, 30],Cohen and Powers discovered that the general-relativistic hydrogen Hamiltonian is not —it has uncountably many self-adjoint extensions; the same conclusion holds for all
Z > − m e c , m e c ) of the essential spectrum. As far as we are aware, it is not known whetherany of these point spectra converges to the Sommerfeld fine structure spectrum when G ց Remark 1.3.
At the end of the day, the findings of Cohen and Powers vindicate the earlierexpressed intuition that general relativity might worsen the spectral ‘large- Z catastrophe’in the special-relativistic ( G = 0 ) treatment of hydrogenic ions, except that this turns outto be true for the Dirac theory of the energy spectra, not for the Bohr–Sommerfeld theory.Specifically, the first ‘large- Z catastrophe’ in the special-relativistic Dirac theory of hydrogenicions (i.e. the loss of essential self-adjointness when the nuclear charge number Z exceeds thecritical value √ / α S ) is worsened, with the critical Z -value reduced to if G > . Now, the Dirac Hamiltonian for a point electron in an externally generated magnetostaticinduction field B ( s ) = ∇ × A ( s ) automatically endows the electron with a magnetic momentof magnitude µ Bohr = π hem e c and a g factor of 2. Empirically, the electron does seem to have amagnetic moment which differs slightly from the Bohr magneton, though, and the differenceis known as its anomalous magnetic moment µ a . Using perturbative QED it has beencomputed in terms of a truncated power series in powers of α S (and log α S ). Interestingly,the leading order term in the expansion of the anomalous magnetic moment µ a is independentof ~ and reads µ class = π e m e c , which we call the classical magnetic moment of the electron .It already gives a very accurate value for the anomalous magnetic moment of the electron.It has been known for a long time that the addition of an anomalous magnetic momentoperator to the Dirac Hamiltonian of a test electron with purely electrostatic interactionsremoves both of the spectral ‘large- Z catastrophes,’ in the sense that it produces an essen-tially self-adjoint Hamiltonian for the electron of any hydrogenic ion [2, 13], independently KIESSLING, TAHVILDAR-ZADEH, TOPRAK of the strength of the non-vanishing anomalous magnetic moment. More recently Belgiorno,Martellini, and Baldicchi [4] showed that the Dirac operator with anomalous magnetic mo-ment is essentially self-adjoint in the naked RWN geometry only if | µ a | ≥ √ G ~ c . Since √ G ~ c = r Gm e ~ ce e m e c = √ γ pe ǫ α s πµ class , (3)the requirement | µ a | ≥ √ G ~ c corresponds to | µ a | & . · − µ class , which is manifestlysatisfied by the empirical value | µ a | ≈ µ class of the electron’s anomalous magnetic moment.While general relativity therefore does not have a catastrophic effect in the spectral theoryof physical hydrogenic ions thanks to the sufficiently large empirical value of the anomalousmagnetic moment of the electron, it still would have a catastrophic effect if the empirical valuewere much smaller. In this mathematical sense ‘switching on general-relativistic gravity’is generally not a harmless weak perturbation [19] of the essentially self-adjoint special-relativistic Dirac operator with Coulomb electricity and anomalous magnetic moment!The RWN spacetime of a proton has a number of suspicious features, though [36]. Inparticular, it has a very strong curvature singularity at its center. Also, its electrostatic fieldenergy is infinite, but it has a finite positive ADM mass (which is identified with the mass m p of the proton). This suggests that the RWN spacetime singularity sports a negative infinitebare mass; indeed this can be computed using the Hawking mass formula for the mass inthe immediate vicinity of the naked point singularity of the RWN spacetime of the proton.The origin of the divergent field energy is long known: the same divergence occurs in flatspacetime, namely point charges in Lorentz electrodynamics have an infinite self-field energy.And since therefore the energy-momentum-stress tensor of the Maxwell–Lorentz fields witha point charge source is not locally integrable over any vicinity of the point charge, couplingit via Einstein’s equations to the Ricci curvature of spacetime will inevitably cause verystrong spacetime singularities, no matter how tiny the gravitational coupling constant is.This suggests that the problems may go away if one works with an electromagnetic fieldtheory of non-linear electromagnetic vacua which give rise to an energy-momentum-stresstensor of the electrostatic field with a point charge source which is globally integrable.Prominent examples are the Born and the Born–Infeld vacuum laws; we recall that theycoincide in the electrostatic limit. As Born found [5], the electrostatic potential field ofa point charge in a Born(–Infeld) vacuum is bounded and Lipschitz continuous. It nowfollows from quite general results about the spherically symmetric special-relativistic Dirac ENERAL-RELATIVISTIC HYDROGEN 7
Hamiltonian [18, 30] that for a test electron in the electrostatic Born field of a point nucleusof charge Ze the Hamiltonian is essentially self-adjoint for Z ∈ R .The question thus becomes whether the elimination of the infinite electrostatic self-fieldenergy problem with the help of some non-linear vacuum law such as the Born(–Infeld) lawsuffices to guarantee an essentially self-adjoint Dirac Hamiltonian for a test electron also inthe general-relativistic spacetime of a point nucleus. Since the electrostatic spacetime of astable nucleus should have an ADM mass identical to A ( Z, N ) m p , if the energy-momentum-stress tensor of the electrostatic field with a point charge source is integrable then also thebare mass of the central singularity has to be finite. To avoid a black hole, it has to be non-positive. However, a non-positive bare mass of the central singularity alone is not sufficientto avoid a black hole; further conditions need to be met, but they can.Balasubramanian in his Ph.D. thesis [1] showed that the Dirac Hamiltonian for an electronin the Hoffmann spacetime [17] of a point nucleus with zero bare mass is essentially self-adjoint for all Z ∈ N . He actually showed it for a larger class of similar black-hole-freeelectrostatic spacetimes [29], all having zero bare mass .1.2. Terra incognita.
The works [7], [4], and [1] concern two ‘opposite’ endpoints of alarge multi-parameter family of black-hole-free electrostatic spacetimes which in a senseinterpolate between the two extreme cases. Thus their results have left open the question ofwhat happens in black-hole-free electrostatic spacetimes with either integrable field energy-momentum-stress tensor, yet with a finite strictly negative bare mass at their center, or withnon-integrable field energy density function ε ( r ), and thus with a negative infinite bare massat their center, yet with a radial mass function m ( r ) := M ADM − c Z ∞ r ε ( s )4 πs d s (4)which diverges to −∞ as r ↓ m RWN ( r ) := M ADM − Z e c r ; (5)recall that M ADM = A ( Z, N ) m p for a nucleus of charge Ze . Will the Dirac operator of a testelectron in any of these spacetimes be more similar to the RWN case or to the Hoffmanncase with vanishing bare mass? Or will there be a critical strictly negative borderline valueof the bare mass where a switch-over happens? And what is the influence of the electron’sanomalous magnetic moment? These questions do not have an obvious answer. KIESSLING, TAHVILDAR-ZADEH, TOPRAK
Remark 1.4.
We remark that a strictly negative bare mass of the nucleus seems hard toavoid theoretically. Recall that in non-perturbative renormalized QED, which needs an UVcutoff, the bare mass of the electron is strictly negative [and even −∞ in perturbative QED],and similar conclusions are to be expected for nuclei due to their electric charges. This Paper.
In this paper we study the Dirac operator on a class of electrostatic space-times which includes those studied in [29] as well as the RWN spacetime with naked singu-larity. We show that whenever the bare mass of the central singularity of the electrostaticblack-hole-free spacetime of a point nucleus is strictly negative, possibly negatively infinite,then the Dirac Hamiltonian for hydrogen / hydrogenic ions without anomalous magnetic mo-ment of the electron has uncountably many self-adjoint extensions. Any of these self-adjointextensions has purely absolutely continuous spectrum in ( −∞ , − m e c ) ∪ ( m e c , + ∞ ), itsclosure being the essential spectrum, plus a discrete spectrum with infinitely many eigen-values located in the gap ( − m e c , m e c ) of the essential spectrum. Which one of these, ifany, is the physically correct one is an open question. This demonstrates that the localnon-integrability of the field energy-momentum-stress tensor over any neighborhood of thepoint charge, a feature of the RWN spacetime, is not the only source of trouble for the DiracHamiltonian of general-relativistic hydrogen.We also address the question whether the addition of a sufficiently large anomalous mag-netic moment operator to the Dirac Hamiltonian for such hydrogenic ions will result in anessentially self-adjoint operator in all cases studied here where essential self-adjointness failswithout such an anomalous magnetic moment. In particular, in all spacetimes studied herethe curvature singularity is milder than the one of the RWN spacetime, so that one might ex-pect a lowered threshold value for the strength of the electron’s anomalous magnetic moment.Interestingly, the situation is more complicated!Namely, while we find that there is a family of electromagnetic vacuum laws for whichessential self-adjointness of the Dirac operator on the pertinent spacetime of a nucleus holdswhen the test electron exhibits any anomalous magnetic moment, no matter how small, therealso is another family — which includes the Born and Born–Infeld vacuum laws — for whichthe addition of an anomalous magnetic moment operator of any strength, no matter howlarge, is not sufficient to obtain an essentially self-adjoint Dirac Hamiltonian. Explicitly, thismeans that the Dirac operator for a test electron in the Hoffmann spacetime of a nucleuswith (inevitably finite) negative bare mass is not essentially self-adjoint, with or without theanomalous magnetic moment of the electron. ENERAL-RELATIVISTIC HYDROGEN 9
The rest of the paper is structured as follows:In section 2 we stipulate the class of electrostatic spacetimes considered in this paper; withsome technical details relegated to Appendix B.In section 3 we discuss the Dirac operator for a test electron in the type of electrostaticspacetime defined in section 2. The section is devided into two subsections, one devoted totest electrons without, and one to test electrons with anomalous magnetic moment.In section 4 we offer an outlook on open questions to be addressed in some future work.In Appendix A we explain the generally regularizing effect of general relativity in theBohr–Sommerfeld type theory of quantized circular orbits.2.
Electrostatic spacetimes with negative bare mass and no horizon
The electrostatic spacetimes discussed in this paper are equipped with an electromagneticvacuum law derived from a Lagrangian density which is a function of the two invariantsof the Faraday field tensor F . As shown already in [29], the spherically symmetric, static,asymptotically flat ones among them which are topologically identical to ‘ R , minus a time-like line,’ equivalently R × ( R \ { } ), and covered by a single global chart of ‘sphericalcoordinates’ ( t, r, ϑ, ϕ ) ∈ R × R + × [0 , π ] × [0 , π ), have a metric given by the line element ds = − f ( r ) c d t + 1 f ( r ) d r + r dΩ , f ( r ) = 1 − Gc m ( r ) r . (6)Here, r is the so-called area radius of a spherical orbit; i.e., every point in the stipulatedspacetime is an element of a unique orbit under a Killing vector flow corresponding to the SO (3) symmetry, and this orbit is a scaled copy of S with area A =: 4 πr , defining r > = d ϑ + sin ϑ d ϕ is the line element on S . Moreover, m ( r ) c = M c − E ( r ) (7)is the radial mass function, where M is the ADM mass M ADM and E ( r ) is the electrostaticfield energy outside a ball of surface area 4 πr . The field energy function r
7→ E ( r ) is strictlypositive and monotone decreasing to 0.For the class of models studied in [29] and here, E ( r ) turns out to be independent of G and, hence, identical to the corresponding flat-space formula. Thus, for instance, inMaxwell–Lorentz electrodynamics, if s ∈ R is a point in flat space and s := | s | , and E ( s ) ∈ R denotes the electric field strength vector at s of a point nucleus located at , then E ( r ) = π R ∞ r | E ( s ) | πs d s with | E ( s ) | = | E | ( s ) = Ze/s . Here are two well-known examples of such spacetimes.First, for the RWN spacetime of a nucleus of charge Ze , we have E ( r ) = 18 π Z ∞ r Z e s πs d s = 12 Z e r . (8)Clearly, m ( r ) ց −∞ as r ց
0, but we also want to have a spacetime without a black hole.The RWN spacetime features a black hole if there is at least one value of r > f ( r ) = 0. Since f ( r ) is a quadratic polynomial in 1 /r , its zeros are formally given by r ± = GMc (cid:18) ± q − Z e GM (cid:19) , (9)and this is real if and only if Z e GM ≤
1. However, for the known nuclei M ADM = A ( Z, N ) m p ,with Z ≤ A ( Z, N ) ≤ Z , and e Gm ≈ · , so f ( r ) is never zero and we are deep in thenaked singularity sector of the RWN spacetimes.Second, for the Hoffmann spacetime of a nucleus of charge Ze , one has E ( r ) = b π Z ∞ r (cid:16)q b Z e s − (cid:17) πs d s ∼ Z e r as r → ∞ ,b ( Ze ) (cid:16) B (cid:0) , (cid:1) − (cid:0) bZe (cid:1) r (cid:17) as r → , (10)where B( x, y ) is Euler’s Beta function, and b > m ( r ) = M − c E ( r ) must have a non-positivelimit when r ց
0. For assume that m (0) >
0, then f ( r ) ց −∞ as r ց
0, while f ( r ) → r ր ∞ , which means that there is at least one real r > f ( r ) vanishes. Thisimplies a lower bound on b , namely ∀ Z : b ≥ A ( Z, N ) m c Z e (cid:0) B (cid:0) , (cid:1)(cid:1) , (11)which is a necessary condition for not to have a black hole in the spacetime whatever thevalue of Z . Since Z ≥ A ( Z, N ) ≤ Z , replacing A ( Z, N ) /Z by 9 at r.h.s.(11) yieldsa lower bound on b , uniformly in Z . A sufficient condition would guarantee that as long as m ( r ) ≥
0, then 2 Gc m ( r ) r <
1. When m (0) = 0 we can state the following sufficient criterionfor not having a black hole in the spacetime, based on the fact that one can show that m ( r ) /r > r ց bZe , with b given by r.h.s(11). Thus, if m (0) = 0 then we have no black hole if bZe < c / G , but b isgiven by r.h.s(11) when m (0) = 0, and this yields the necessary and sufficient condition1 < (cid:0) B (cid:0) , (cid:1)(cid:1) Z A ( Z,N ) e Gm (12) ENERAL-RELATIVISTIC HYDROGEN 11 for the absence of a black hole when m (0) = 0, given Z . For all the known nuclei thecondition is clearly met, i.e. we are once again deep in the naked singularity sector, this timeof the Hoffmann spacetimes. Lastly, f ( r ) increases when b increases from r.h.s(11) and allother parameters are kept fixed, so it follows that we stay in the naked singularity sector ofthe Hoffmann spacetimes if the central singularity has negative bare mass m (0) < The Dirac Hamiltonian for hydrogen and hydrogenic ions
Test electron without anomalous magnetic moment.
Due to the spherical sym-metry and static character of the spacetimes, the Dirac operator H of a test electron in thecurved space whose line element d s is given by (6) separates in the spherical coordinates andtheir default spin frame [7]. More precisely, H is a direct sum of so-called partial-wave Diracoperators H rad k which act on two-dimensional bi-spinor subspaces. This reduces the spectralproblem to studying the family of radial Dirac operators H rad k := m e c K k , k ∈ Z \{ } , with K k := " f ( r ) − em e c φ ( r ) ~ m e c (cid:2) kr f ( r ) − f ( r ) ∂ r (cid:3) ~ m e c (cid:2) kr f ( r ) + f ( r ) ∂ r (cid:3) − f ( r ) − em e c φ ( r ) , (13)acting on g ( r ) := (cid:0) g ( r ) , g ( r ) (cid:1) T , with g and g C ∞ functions compactly supported awayfrom r = 0, equipped with a weighted L norm given by k g k := Z ∞ f ( r ) (cid:16) | g ( r ) | + | g ( r ) | (cid:17) d r. (14)Note that K k is physically dimensionless.To state and prove our theorems we make a few assumptions on f ( r ) and φ ( r ) whichare satisfied by the finite-bare-mass electromagnetic spacetimes in [29], but also by somemore general spacetimes with negative infinite bare mass, of which the RWN spacetime inits naked singularity sector is but one member (See Appendix B). By (6), assumptions on f ( r ) are equivalent to assumptions on m ( r ). Assumptions 3.1. • m ( r ) is continuous; • m ( r ) /r < c / G ; • m ( r ) ∼ − C α r − α as r ց , where α ≥ and C α > ; • m ( r ) → M > as r → ∞ . With the above specification of the radial mass function, one has f ( r ) = 0 for all r , and f ( r ) ∼ Gc C α r α as r → + ; f ( r ) → r → ∞ . (15) Remark 3.2.
Our first two assumptions on m ( r ) are equivalent to ruling out black holes inspacetimes with the line element (6). So our spacetimes feature a charged naked singularity.By the third and fourth assumptions on m ( r ) , the naked singularity has a negative bare mass( lim r ↓ m ( r ) ), which is finite only for α = 0 , in which case m (0) = − C . The function φ ( r ) is the potential of the electrostatic field generated by the nucleus, in thesense that the radial component E ( r ) = − ∂ r φ ( r ). We will make the following assumptions. Assumptions 3.3. • φ ( r ) is continuously differentiable; • φ ( r ) ∼ Ze/r as r → ∞ ; • φ ( r ) ∼ C ′′ β + C ′ β r − β around zero, with β ≤ . Remark 3.4.
The second assumption on φ ( r ) expresses Gauss’ law in our asymptoticallyflat spacetimes. Remark 3.5.
We note that β = 1 for the RWN spacetime. The following is the first main theorem of this section.
Theorem 3.6.
Under the stated assumptions on m ( r ) and φ ( r ) the operator H rad k has un-countably many self-adjoint extensions, ∀ k ∈ Z \{ } . Remark 3.7.
Inspection of our proof will reveal that we can generalize our Theorem 3.6and still conclude, with the same proof, that the operator H radk has multiple self-adjointextensions if we allow α > − and β < α . However, for α < the mass function m ( r ) is monotone decreasing in a right neighborhood of r = 0 , which disqualifies it from the rosterof mass functions for the electrostatic spacetimes considered in [29] , and their generalizationconsidered here. Also, β > does not occur in our electrostatic spacetimes; see Appendix B. ENERAL-RELATIVISTIC HYDROGEN 13
Remark 3.8.
Since the third bullet point in Assumption 3.3 covers electric potentials whichare bounded at the origin, as well as those which diverge like an inverse power law when r ց , it is natural to suspect that the conclusion of our Theorem 3.6 will also hold if weallow φ ( r ) to diverge, but less strongly than an inverse power law, e.g. logarithmically, when r ց . The proof of the so-modified Theorem 3.6 requires only minor adjustments.Proof of Theorem 3.6. Under our assumptions on the mass function, we can change variables r x as follows, drdx = ~ m e c f ( r ( x )) , (16)and study e K k = " f ( r ( x )) − em e c φ ( r ( x )) ~ m e c kr ( x ) f ( r ( x )) − ddx ~ m e c kr ( x ) f ( r ( x )) + ddx − f ( r ( x )) − em e c φ ( r ( x )) (17)=: " a ( x ) − b ( x ) kc ( x ) − ddx kc ( x ) + ddx − a ( x ) − b ( x ) (18)with the inner product h g, h i := Z ∞ (cid:16) g ( r ( x ))¯ h ( r ( x )) + g ( r ( x ))¯ h ( r ( x )) (cid:17) dx. (19)Let C denote a generic constant. One can show that as r → + ( x → + ) we have r ∼ Cx α , and as r → ∞ ( x → ∞ ), we have r ∼ x . Therefore, a ( x ) ∼ Cx − α α and c ( x ) ∼ Cx − α α as x → + , and a ( x ) ∼ c ( x ) ∼ x − as x → ∞ . One further has b ( x ) ∼ Cx − β α as x → + , and b ( x ) ∼ x − as x → ∞ .Let e K ∗ k be the adjoint operator of e K k . Recall that D ( e K k ) is all C ∞ functions of compactsupport in (0 , ∞ ), and D ( e K ∗ k ) includes the functions f so that f and f ′ are integrable inany compact subset of [0 , ∞ ), see Section XIII in [9]. We define the following sesquilinearform on D ( e K ∗ k ): [ g, h ] := h e K ∗ k g, h i − h g, e K ∗ k h i , (20)where h· , ·i is defined as in (19). By Theorem 4.1 in [34], e K ∗ k | D is a self-adjoint extension of e K k iff i ) D ( e K k ) ⊂ D ⊂ D ( e K ∗ k ) ii ) [ g, h ] = 0 for all g, h ∈ D iii ) if g ∈ D ( e K ∗ k ) and [ g, h ] = 0 holds for every h ∈ D then g ∈ D . We first start considering the spaces in which [ g, h ] = 0. Take g ∈ D ( e K ∗ k ), i.e. e K ∗ k g = ψ forsome ψ ∈ L ([0 , ∞ ). Since D ( e K k ) ⊂ C c ([0 , ∞ )), g ∈ AC ([ x , x ]) for each 0 ≤ x < x < ∞ ,and therefore we can integrate to get g ( x ) = e − µ ( x ) (cid:16) g (0) + Z x e + µ ( y ) [( a ( y ) + b ( y )) g ( y ) + ψ ( y )] dy (cid:17) , (21) g ( x ) = e + µ ( x ) (cid:16) g (0) + Z x e − µ ( y ) [( a ( y ) − b ( y )) g ( y ) − ψ ( y )] dy (cid:17) (22)for each x ≤ x < ∞ , where µ ( x ) = R x kc ( y ) dy ∼ Cx α α → x →
0. To have g , g tobe defined we also need a ( x ) , b ( x ) ∈ L ([0 , x ]), x < ∞ . Remark 3.9.
For our electrostatic spacetimes, α ≥ and β ≤ , so we do have a ( x ) , b ( x ) ∈ L ([0 , x ]) , x < ∞ . If we drop the ‘electrostatic’ requirement and also allow negative α > − , then this requires β < α . Cf. our earlier remark. By integration by parts, we have[ g, h ] = lim x → lim x →∞ Z x x (cid:16) ( e K ∗ k g ) h − ( e K ∗ k g ) h + g ( e K ∗ k h ) − g ( e K ∗ k h ) i (cid:17) dx (23)= lim x → lim x →∞ h g ( x ) h ( x ) − g ( x ) h ( x ) − g ( x ) h ( x ) + g ( x ) h ( x ) i (24)= g (0) h (0) − g (0) h (0) . (25)We used (21), (22), and the fact that g, h ∈ L ([0 , ∞ )) to obtain the last equality.This suggests that any symmetric extension requires g (0) h (0) − g (0) h (0) = 0. Taking g = h , one can see this is true if g (0) is a real multiple of g (0), or vice versa. Therefore,for any 0 ≤ θ < π , e K k ; θ = e K ∗ k |D θ , where D θ = { g ∈ D ( e K ∗ k ) : g (0) sin θ + g (0) cos θ = 0 } (26)gives a symmetric extension, cf. [7]. Note that D θ satisfies both the conditions i ) and ii ).Condition iii ) is also clearly satisfied. Let h ∈ D θ [ g, h ] = 0 ⇐⇒ g (0) h (0) − g (0) h (0) = 0 ⇐⇒ h (0) h (0) = g (0) g (0) = − tan θ ∈ R . (27)This finishes the proof. (cid:3) Remark 3.10.
It is worth mentioning that, since [ g, h ] is the difference of two positiverank-one bilinear forms, e K k , and thus K k , has deficiency indices (1 , . ENERAL-RELATIVISTIC HYDROGEN 15
Recall that in the partial wave decomposition the Dirac Hamiltonian H is a direct sum ofoperators H rad k = m e c K k which act on two-dimensional bi-spinor subspaces. Having shownthat these have self-adjoint extensions K k,θ for 0 ≤ θ < π , we now define H θ , as the directsum of the H rad k ; θ := m e c K k,θ . We are ready to state our next main theorem of this section. Theorem 3.11.
Under Assumptions 3.1 and 3.3, for hydrogenic ions we have (a)
The essential spectrum σ ess ( H θ ) = ( −∞ , − m e c ] ∪ [ m e c , ∞ ) ; (b) H θ has purely absolutely continuous spectrum in ( −∞ , − m e c ) ∪ ( m e c , ∞ ) ; (c) the singular continuous spectrum σ sc ( H θ ) = ∅ .Proof. We prove Theorem 3.11 by a series of Lemmas, Corollaries, and Remarks below.In particular, Lemma 3.12 establishes part ( a ); this together with Lemma 3.15 and Corol-lary 3.17 establishes parts ( b ) and ( c ). (cid:3) Lemma 3.12. σ ess ( K k ; θ ) = ( −∞ , − ∪ [1 , ∞ ) . To prove Lemma 3.12 we recall the following lemma from [7].
Lemma 3.13.
Let D : = " − ddxddx (28) be defined on the C ∞ two-component functions of compact support in the positive realhalf-line. Now take the closure of this operator in ( L ( R + )) with the boundary condition f (0) sin θ + f (0) cos θ = 0 at x = 0 , denoted D θ . Let A be the operator A = " a ( x ) a ( x ) a ( x ) a ( x ) , (29) where the a ij are functions in L ([0 , x ]) for all < x < ∞ and a ij ( x ) → as x → ∞ .Then A is D θ compact. Remark 3.14.
The value x = 0 in this lemma plays no role in its proof. In particular, onecan pick x = x , < x < x < ∞ , and consider the operator D with boundary condition f ( x ) sin θ + f ( x ) cos θ = 0 in L ([ x , x ]) .Proof of Lemma 3.12. We split the operator e K k in (17) as e K k = " kc ( x ) − ddx kc ( x ) + ddx − + h a ( x ) − i − b ( x ) 00 − h a ( x ) − i − b ( x ) =: e K k + V. (30) Note that Theorem 3.6 is valid when a ( x ) = 1 and b ( x ) = 0. Therefore, e K k has deficiencyindices (1 ,
1) and has multiple self-adjoint extensions similar to e K k . We define these self-adjoint extensions as e K k ; θ similar to e K k ; θ , see (26). We define the following Weyl sequencefor e K k ; θ , with any λ ∈ R , | λ | > f n,λ ( x ) = 12 n xe − x n + ix √ λ − q λ i q − λ ; n ∈ N . (31)We have that k f n,λ k ( L ( R + )) = 1, f n,λ ( x ) → k ( e K k ; θ − λ ) f n,λ ( x ) k ( L ( R + )) → n → ∞ . Hence, any | λ | > e K k ; θ . Further, since the essentialspectrum is a closed subset of R , one has ( −∞ , − ∪ [1 , ∞ ) ⊂ σ ess ( e K k ; θ ).For the reverse inclusion, we consider the operator [ e K k ; θ ] . Let g ∈ D θ , then one has h [ e K k ; θ ] g, g i = h e K k ; θ g, e K k ; θ g i (32)= Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ddx + kc ( x ) (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) dx + Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − ddx + kc ( x ) (cid:19) g (cid:12)(cid:12)(cid:12)(cid:12) dx (33)+ k g k L ( R + )) + sin(2 θ ) (cid:0) | g (0) | + | g (0) | (cid:1) ;For the boundary term at the r.h.s. of this equality, recall the boundary conditions in D θ .Clearly if 0 ≤ θ ≤ π , then h [ e K k ; θ ] g, g i ≥ h g, g i . Therefore, if 0 ≤ θ ≤ π then σ ( e K k ; θ ) = σ ess ( e K k ; θ ) = ( −∞ , − ∪ [1 , ∞ ). On the other hand, all self-adjoint extensions of e K k havethe same essential spectrum, cf. [34], p.163. Therefore, σ ess ( e K k ; θ ) = ( −∞ , − ∪ [1 , ∞ ) forall θ ∈ [0 , π ).Next we will show that V is e K k ; θ compact. We define ξ ( x ) = R x kc ( y ) dy for 0 ≤ x ≤ ξ ( x ) = ξ (1) for x >
1. Then the matrix S = " e − ξ ( x ) e ξ ( x ) (34)is bounded.Assume that k g n k ( L ( R + )) , k e K k ; θ g n k ( L ( R + )) are bounded sequences. Then k Sg n k ( L ( R + )) and k D θ Sg n k ( L ( R + )) are also bounded, the first one is because S is bounded and the latterone is by the fact that D θ S = S − SD θ S = S − ( e K k ; θ + W ) for some bounded W . Moreover,one can check that V S − is D θ compact by Lemma 3.13. Hence, V S − Sg n = V g n (35)has a convergent subsequence. This proves that V is D θ compact. (cid:3) ENERAL-RELATIVISTIC HYDROGEN 17
Lemma 3.15. σ ( K k ; θ ) is purely absolutely continuous in ( −∞ , − ∪ (1 , ∞ ) . For the proof we utilize the following Theorem from [33, 34].
Theorem 3.16. (Weidmann) Let τ := " − ddxddx + P ( x ) + P ( x ) (36) be defined on ( a, ∞ ) . Further assume that | P ( x ) | ∈ L ( c, ∞ ) for some c ∈ ( a, ∞ ) , and P ( x ) is of bounded variation in [ c, ∞ ) with lim x →∞ P ( x ) = " µ + µ − for µ − ≤ µ + . (37) Then every self-adjoint realization A of τ has purely absolutely continuous spectrum in ( −∞ , µ − ) ∪ ( µ + , ∞ ) .Proof of Lemma 3.15. Recall that e K k,θ is in the form of τ with P (0) = 0 and P ( x ) = " f ( r ( x )) − em e c φ ( r ( x )) ~ m e c kr ( x ) f ( r ( x )) ~ m e c kr ( x ) f ( r ( x )) − f ( r ( x )) − em e c φ ( r ( x )) . (38)By hypothesis, both φ ( r ( x )) and m ( r ( x )) are continuously differentiable and hence ofbounded variation. Furthermore, we can see thatlim x →∞ φ ( r ( x )) = 0 = lim x →∞ kr ( x ) f ( r ( x )) , and lim x →∞ f ( r ( x )) = 1 , (39)or, equivalently, lim x →∞ P ( x ) = " − . (40)Hence the spectrum of e K k ; θ is purely absolutely continuous in ( −∞ , − ∪ (1 , ∞ ). (cid:3) The two lemmas imply the following.
Corollary 3.17.
The singular continuous spectrum σ sc ( K k ; θ ) = ∅ .Proof. By Lemma 3.15 and Lemma 3.12 the essential spectrum is the closure of σ ac ( K k,θ ).Since the singular continuous spectrum is a subset of the essential spectrum, and since theinterior of the essential spectrum here is purely absolutely continuous, a non-empty σ sc ( K k ; θ )would have to consist of the discrete set {− , } , which is impossible. (cid:3) Next we turn to the discrete spectrum.
Theorem 3.18.
Under Assumptions 3.1 and 3.3, the eigenvalues of any self-adjoint exten-sion H θ of H form a countably infinite set located in the gap of the essential spectrum. The setof accumulation points of this discrete spectrum σ disc ( H θ ) is either { m e c } or {− m e c , m e c } ;for the empirically known hydrogenic ion parameters, only m e c is an accumulation point ofthe discrete spectrum. To validate this theorem we utilize Theorem 2.3 from [15].
Theorem 3.19. (Hinton et al.) Let Ly := " −
11 0 y ′ − " p ( x ) c + V ( x ) c − V ( x ) − p ( x ) y ) =: J y ′ − P y. (41)
Assume that d > . Let g be a nontrivial positive linear functional, and assume P is locallyabsolutely continuous. Let L be any self-adjoint extension of L . Then σ ( L ) ∩ ( − d, d ) isinfinite if the scalar differential equation, − g ( I ) z ′′ + g (cid:16) P − d I + [ P ′ J − J P ′ ]2 (cid:17) z = 0 (42) is oscillatory either at or at ∞ . In (42), g is a nontrivial linear positive functional defined on the real n × n matri-ces. Therefore, for any symmetric and positive semidefinite operator B , one has g ( B ) = P ni =1 δ i h Bu i , u i i , where the δ i ≥ u i are non-zero orthonormal n -vectors.In our case, n = 2, with u = (1 , T and u = (0 , T , and we will consider the “+ case”with δ +1 = 1 and δ +2 = 0, and the “ − case” with δ − = 0 and δ − = 1. Proof of Theorem 3.18.
We use Theorem 3.19 for P = − P , where P is as in (38) and c = c = d = 1. In particular, we have V = em e c φ ( r ( x )) + f ( r ( x )) − , (43) V = em e c φ ( r ( x )) − f ( r ( x )) + 1 , (44) p ( x ) = − ~ m e c kf ( r ( x )) r ( x ) . (45)Note that rx → ~ m e c as x → ∞ . Hence f ( r ( x )) ∼ − GMm e ~ cx as x → ∞ .Therefore, in the “+ case”, resp. the “ − case”, equation (42) yields − z ′′ + Γ ± ( x ) z = 0 , (46) ENERAL-RELATIVISTIC HYDROGEN 19 where Γ + ( x ) = V ( x ) − V ( x ) + p + p ′ , (47)Γ − ( x ) = V ( x ) + 2 V ( x ) + p − p ′ . (48)This gives Γ ± ∼ − h GMm e ~ c ± Ze ~ c i x . (49)Clearly, lim x →∞ x Γ + ( x ) < − . Hence, equation (46) with “+” has solutions with oscillatorybehavior at infinity, see [9, Section XIII].Thus by Theorem 3.19, σ ( e K k,θ ) ∩ ( − ,
1) is infinite, so e K k,θ , and therefore K k,θ , haveinfinitely many eigenvalues in the gap of their essential spectrum.Lastly we recall that the spectrum of H θ is the union of the spectra of the radial Dirac op-erators obtained by the partial wave decomposition. This proves that the discrete spectrumof H θ is infinite and located in the gap ( − m e c , m e c ).We next prove our statement about its accumulation points. We will need the following. Lemma 3.20.
For each operator K k,θ the set of accumulation points of its discrete spectrumis { } or {− , } , depending on whether GM m e < Ze or GM m e > Ze , respectively.Proof. To determine if ± L in Theorem 3.19 for c = 1 ± ǫ , c = 1 ∓ ǫ for some 0 < ǫ <
1. In particular, ± d = 1 ± ǫ , c and c is non-oscillatory at both 0and ∞ , see [15, Theorem 4.1].We first show that 1 is an accumulation point. Plugging c = 1 + ǫ , and d = c = 1 − ǫ together with the functions in (43) into (42) we obtain (46) with Γ ± replaced by Γ ± , whereΓ := V − − ǫ ) V + p + p ′ , (50)Γ − := V + 2(1 + ǫ ) V + p − p ′ + 4 ǫ. (51)It suffices to show that equation (46) with Γ has an oscillatory solution either at ∞ or 0.An easy calculation shows that, by (49), one hasΓ ∼ − − ǫ ) h GMm e ~ c + Ze ~ c i x as x → ∞ . (52)Since 0 < ǫ <
1, one has lim x →∞ x Γ < − . Therefore, (46) with Γ has an oscillatorysolution near infinity, and thus 1 is always an accumulation point of the set of eigenvalues. Next we show that − GM m e > Ze but not if GM m e < Ze . Consider(42) with d = c = 1 − ǫ , and c = 1 + ǫ . One obtains (46) with Γ ± replaced by Γ − ± , whereΓ − := V − ǫ ) V + p + p ′ + 4 ǫ, (53)Γ − − := V + 2(1 − ǫ ) V + p − p ′ . (54)We now find Γ − − ∼ − − ǫ ) h GMm e ~ c − Ze ~ c i x as x → ∞ . (55)Thus, since 0 < ǫ <
1, one has lim x →∞ x Γ − − < − if GM m e > Ze , and then (46) with Γ − − has an oscillatory solution near infinity, so − < ǫ <
1, one has lim x →∞ x Γ − − > > − ) if GM m e < Ze .Therefore, the solution to (46) with Γ − − is non-oscillatory at ∞ if GM m e < Ze . To see thatit is also non-oscillatory at 0, we consider p ± ν := p ±
12 ( V + V + c − c ) = − ~ km e c f ( r ( x )) r ( x ) ± em e c φ ( r ( x )) − ǫ. (56)By Corollary 3.4 in [15], (42) is non-oscillatory at 0 if p ± ν ≤ − f ( r ( x )) r ( x ) ∼ x − α α , and φ ( r ( x )) ∼ x − β α around zero, with α ≥ β ≤
1. (Orif we allow also α <
0, then β < α ; cf. Remark 3.9.) Hence, p ± ν → −∞ , and so − GM m e < Ze . (cid:3) Lemma 3.20 and the fact that the spectrum of H θ is the union of the spectra of theradial partial wave Dirac operators now concludes our proof about the accumulation pointsof the discrete spectrum, for general M . For known nuclei M = M ADM = A ( Z, N ) m p , with A ≤ Z , so GMm e Ze < × − , and so − m e c is not an accumulation point of the discretespectrum of H θ for any empirical hydrogenic ions. The proof of our theorem is complete. (cid:3) Remark 3.21.
We suspect that the boundary points of the essential spectrum, − m e c and m e c , are generally not eigenvalues of H θ , but we have not tried to prove it, and the answermay depend on θ and on the value of GM m e /Ze . Remark 3.22.
We proved with the actual empirical values of
GMm e Ze for physical hydrogenicions that m e c is a limit point of the discrete spectrum while − m e c is not. The appearanceof − m e c as a limit point of the discrete spectrum for hypothetical hyper-heavy ion values GMm e Ze > can be explained in physics lingo if we recall that the negative continuum is usuallyinterpreted as being associated with positrons, which do not bind electrically to the positivelycharged nuclei, but which can be bound gravitationally if the gravitational attraction to the ENERAL-RELATIVISTIC HYDROGEN 21 nucleus overcomes the electrical repulsion. Incidentally, the same critical value
GMm e Ze = 1 features also in the non-relativistic treatment, where the Newtonian gravitational attractionbetween a positron and a nucleus overpowers their Coulomb repulsion if and only if GMm e Ze > ,in which case the Schr¨odinger Hamiltonian also has infinitely many bound states, while thereare no bound states when GMm e Ze ≤ . Our results do not reveal whether the general-relativisticDirac problem in the critical case GMm e Ze = 1 features any bound positron states; our resultsonly show that there are none if GMm e Ze < , and infinitely many if GMm e Ze > . Test electron with anomalous magnetic moment.
In the special-relativistic prob-lem of hydrogenic ions at any Z ∈ N it was found long ago [2, 13] that the addition ofan anomalous magnetic moment operator to the Dirac Hamiltonian of a test electron inthe Coulomb field of the point nucleus suffices to produce an essentially self-adjoint DiracHamiltonian. For a test electron in the RWN spacetime of a point nucleus it was found in[4] that a sufficiently large anomalous magnetic moment of the electron is required to obtainan essentially self-adjoint Hamiltonian of the hydrogenic ions; it turns out that the empiricalelectron value is large enough uniformly for all Z ∈ N .This suggests that adding an anomalous magnetic moment operator to the Dirac Hamil-tonian of a test electron may restore essential self-adjointness also in all situations discussedhere so far where essential self-adjointness fails, in particular for the Dirac Hamiltonianof a test electron in the Hoffmann spacetime of a point nucleus with negative bare mass.Interestingly, the situation is more complicated, as shown by our next theorem.The radial partial-wave Dirac operator H rad µ a ,k = m e c K µ a ,k now is given by K µ a ,k := " f ( r ) − em e c φ ( r ) ~ m e c (cid:2) kr f ( r ) − f ( r ) ∂ r (cid:3) − µ a m e c φ ′ ( r ) f ( r ) ~ m e c (cid:2) kr f ( r ) + f ( r ) ∂ r (cid:3) − µ a m e c φ ′ ( r ) f ( r ) − f ( r ) − em e c φ ( r ) . (57) Theorem 3.23.
Let f ( r ) be as in (15) with α ≥ and φ ( r ) = C ′′ β + C ′ β r − β + O ( r − β ) for β ≤ , where f ∈ O k ( g ) indicates d j dr j f = O ( d j dr j g ) for j = 0 , , ..k . Then the operator H rad µ a ,k is essentially self-adjoint if either β > α , or β = α and | µ a | ≥ α α ~ p GC α / | C ′ β | . Onthe other hand, if β < α , then H rad µ a ,k has multiple self-adjoint extension.Proof. Note that the fact that H rad µ a ,k has multiple self-adjoint extension if β < α is aconsequence of Theorem 3.6. In particular, if the change of variable as in (16) is applied to K µ a ,k , one obtains the operator e K µ a ,k = e K k + µ a " d ( x ) d ( x ) 0 , (58)where d ( x ) ∼ Cx − α +2 β α as x →
0, and d ( x ) ∼ x − as x → ∞ . Therefore, g and g in(21), (22) arises with µ ( x ) = R x [ kc ( y ) + µ a d ( y )] dy . Since, µ ( x ) → β < α , the prooffollows similar to the proof of Theorem 3.6. Therefore, it remains to prove the assertions for β ≥ α .We start the proof with the case that β > α . We will show that the limit point case(LPC) is verified in the right neighborhood of r = 0 if β > α , i.e. there is at least onenon-square integrable solution to K µ a ,k g = λg for each λ ∈ C , or equivalently for a fixed λ ,see [34, Theorem 5.6]. In particular, we will consider the solutions to h − eφ ( r ) m e c f ( r ) + 1 f ( r ) i g = h ~ m e c (cid:20) ∂ r − krf ( r ) (cid:21) + µ a φ ′ ( r ) m e c f ( r ) i g , (59) h eφ ( r ) m e c f ( r ) + 1 f ( r ) i g = h ~ m e c (cid:20) ∂ r + krf ( r ) (cid:21) − µ a φ ′ ( r ) m e c f ( r ) i g . (60)Recall that g = ( g , g ) T is square integrable in the right neighborhood of r = 0 with theinner product (14) iff for each 0 < R < ∞ , Z R f ( r ) (cid:16) | g ( r ) | + | g ( r ) | (cid:17) dr < ∞ . (61)Therefore, we aim to find solutions to (59), (60) such that (61) does not hold.Let A ( r ) = − eφ ( r ) m e c f ( r ) + f ( r ) , and use the ansatz g ( r ) = e h ( r ) . Then, by (59) we have g = A − ( r ) h ~ m e c (cid:20) h ′ − krf ( r ) (cid:21) + µ a φ ′ ( r ) m e c f ( r ) i g . (62)Recall that f ( r ) ∼ Gc C α r − − α and φ ( r ) ∼ C ′′ β + C ′ β r − β around zero with α < β ≤ g into (60) we obtain the following asympototic expansion as r → h ∂ r + ( A − ( r )) ′ A ( r ) ih h ′ − krf ( r ) + µ a φ ′ ~ cf ( r ) i + ( h ′ ) − h krf ( r ) − µ a φ ′ ~ cf ( r ) i = O ( r α ) . (63)Noting also that β > α ≥
0, we can find a solution so that h ( r ) ∼ Cr − β + α as r → + , (64)or equivalently g ( r ) ∼ e Cr − β + 1+ α as r → + . (65) ENERAL-RELATIVISTIC HYDROGEN 23
In a similar way, one can show that h ( r ) ∼ − Cr − β + α ⇒ g ( r ) ∼ e − Cr − β + 1+ α as r → + . (66)It is now clear that since β > α , (61) does not hold for g = ( g , g ) T , and the LPC issatisfied in the right neighborhood of zero.Finally, we consider the case β = α . Recall that, since α ≥
0, we have f ( r ) ∼ Gc C α r − − α as r → + . Therefore, equations (59), (60) around zero become g ′ − (1 + α ) µ a C ′ β p GC α ~ r − kc p GC α r − α = O ( r / ) , (67) g ′ + (1 + α ) µ a C ′ β p GC α ~ r + kc p GC α r − α = O ( r / ) . (68)Notice that if µ a C ′ β > r = 0 we have g ∼ r − (1+ α ) µaC ′ β √ GCα ~ ; (69)and if µ a C ′ β < r = 0 we have g ∼ r (1+ α ) µaC ′ β √ GCα ~ . (70)Note that (61) implies that local square integrability holds for g and g if Z R r ± (1+ α ) µaC ′ β √ GCα ~ +1+ α dr < ∞ . (71)Hence, the LPC is satisfied in the right neighborhood of zero if − (cid:12)(cid:12)(cid:12) (1 + α ) µ a C ′ β p GC α ~ (cid:12)(cid:12)(cid:12) + 1 + α ≤ − . (72) (cid:3) Our last theorem states that an anomalous magnetic moment can only regularize the Diracoperator for a test electron in the static spherically symmetric spacetime of a point nucleuswith negative bare mass if the electric field of the nucleus diverges sufficiently fast at r ց if and only if the baremass of the nucleus vanishes. No anomalous magnetic moment can come to the rescue if thebare mass of the nucleus is strictly negative. We end this section with the analogues of Theorems 3.11 and 3.18 for test electrons withanomalous magnetic moment. By H µ a ,θ we denote any self-adjoint extension of H µ a , whereit is understood that the subscript θ is mute in all cases where H µ a is essentially self-adjoint.For the essential spectrum we have: Theorem 3.24.
Suppose Assumptions 3.1 and 3.3 hold, and furthermore assume that φ ( r ) = C ′′ β + C ′ β r − β + O ( r − β ) for β ≤ around zero. Then one has (a) The essential spectrum σ ess ( H µ a ,θ ) = ( −∞ , − m e c ] ∪ [ m e c , ∞ ) ; (b) H µ a ,θ has purely absolutely continuous spectrum in ( −∞ , − m e c ) ∪ ( m e c , ∞ ) ; (c) the singular continuous spectrum σ sc ( H µ a ,θ ) = ∅ .Proof. We use the representation (58) to validate the claims. First of all, note that the proofof ( b ) follows similarly to the proof of Lemma 3.15. In particular, we need to consider theoperator ˜ P := P ( x ) + µ a d ( x ) σ (73)instead of P in (38), and validate the limit property (40) for ˜ P . Here, σ = (cid:20) (cid:21) .However, lim x →∞ d ( x ) = 0, and hence part ( b ) of the statement holds. It is also clear thatthe claim of part ( c ) follows from part ( a ) and part ( b ). Therefore it remains to provepart ( a ).For this part, we need to analyze the operator (58) separately for β < α +12 and β > α +12 .If β < α +12 , i.e. if the operator (58) has multiple self-adjoint extensions, then the proof ofLemma 3.12 is directly applicable. In particular, writing e K µ a ,k = e K + µ a d ( x ) σ + V ( x ) (74)one can show that the functions defined in (31) form a Weyl sequence also for ˜ K + µ a d ( x ) σ .Furthermore, the operator S in (34) is bounded and therefore, V is e K + µ a d ( x ) σ compact.For the case that β ≥ α , we define the operators e K [0 ,b ] µ a ,k and e K [ b, ∞ ) µ a ,k as the restriction of e K µ a ,k to L ([0 , b ]) and L ([ b, ∞ ]) respectively. Then by Theorem 11.5 in [34], we have σ ess ( e K µ a ,k ) = σ ess ( e K [0 ,b ] µ a ,k ) ∪ σ ess ( e K [ b, ∞ ) µ a ,k ) . (75)Instead of (31) we now use the following Weyl sequence, f n,λ ( x ) = 1 √ n e − ( x − b )2 n + i ( x − b ) √ λ − q λ i q − λ ; n ∈ N . (76) ENERAL-RELATIVISTIC HYDROGEN 25
Then ξ ( x ) = R xb [ kc ( y ) + µ a d ( y )] dy for b ≤ x ≤ b + 1 in (34), and one can show that σ ess ( e K [ b, ∞ ) µ a ,k ) = ( −∞ , − ∪ [1 , ∞ ) in a similar way as in the proof of Lemma 3.15.On the other hand, the operator e K [0 ,b ] µ a ,k can only have discrete spectrum. To see that, weuse Theorem 2 in [16]. In particular, since the limit point case holds, e K [0 ,b ] µ a ,k has discretespectrum if also Z b | kc ( x ) + µ a d ( x ) | dx = ∞ . (77)Note that the above statement is true since for β ≥ α , d ( x ) is not locally integrable aroundzero. (cid:3) For the discrete spectrum we have:
Theorem 3.25.
Let Assumptions 3.1 and 3.3 be valid. Let also φ ( r ) = C ′′ β + C ′ β r − β + O ( r − β ) for β ≤ around zero. Then eigenvalues of any self-adjoint extension H µ a ,θ of H µ a form a countably infinite set located in the gap of the essential spectrum. The set ofaccumulation points of this discrete spectrum σ disc ( H µ a ,θ ) is either { m e c } or {− m e c , m e c } ,depending on whether GM m e < Ze or GM m e > Ze , respectively. In particular, for theempirically known hydrogenic ion parameters, only m e c is an accumulation point of thediscrete spectrum if β = α +12 , or if β = α +12 and α α (cid:12)(cid:12)(cid:12) µ a C ′ β ~ √ GC α (cid:12)(cid:12)(cid:12) = 1 .Proof. For the proof of the fact that the eigenvalues form a countably infinite set located inthe gap, we recall the proof of Theorem 3.18. In particular, we need to apply Theorem 3.19with the same V and V , but p ( x ) is exchanged with˜ p ( x ) = − ~ m e c kr ( x ) f ( r ( x )) + µ a m e c φ ′ ( r ( x )) f ( r ( x )) = p ( x ) + µ a d ( x ) . (78)Note that d ( x ) vanishes faster than p ( x ) at infinity. Therefore, the behavior of Γ + , see(49), around infinity remains the same and, the proof now follows similar to the proof ofTheorem 3.18.To prove the claim on accumulation points, we follow a similar method as in the proof ofLemma 3.20. In Lemma 3.20, note that 1 is accumulation point because equation (46) withΓ has oscillatory solutions at ∞ or, equivalently, lim x →∞ x Γ ( x ) < − . However, since˜ p ( x ) ∼ p ( x ) at infinity, the behavior of Γ ( x ) remains the same and, m e c is an accumulationpoint.Next, we prove the statement about − m e c . To do that, we have to consider the equation(46) with Γ − ± , where p ( x ) is replaced by ˜ p ( x ); see (53), (54). Again since ˜ p ( x ) ∼ p ( x ) at infinity, the solutions to (46) with Γ − ± are non-oscillatory if GM m e < Ze , and oscillatoryif GM m e > Ze . Thus, when GM m e > Ze , then − m e c is an accumulation point.Now we need to determine if the solutions to (46) with Γ − ± , and p ( x ) replaced by ˜ p ( x ),are non-oscillatory also around zero when GM m e < Ze . This part of the proof requiresmore care since the behavior of Γ − ± around zero is affected when p ( x ) is replaced by ˜ p ( x ).Note that if β ≤
0, then p ( x ) is more singular than d ( x ) as x →
0. Therefore, Corollary 3.14in [15] is applicable as in the proof of Theorem 3.18 if β ≤
0. On the other hand if β > − ± arises from ˜ p with singularity x − β +2 α +64+2 α if β > α ;and from ˜ p ′ with singularity x − β +3 α +74+2 α if β < α . In particular, if β > α the singularitycomes from the term ( φ ′ f ) , and if β < α then the singularity comes from ( φ ′ f ) ′ . Notingthat p has + sign in both Γ − ± , we see that if β > α then lim x → x Γ − ± ≥
0. On the otherhand, if β < α then β +3 α +74+2 α <
2, and hence lim x → x Γ − ± = 0. Therefore, the solutionsare non-oscillatory if β = α .To determine, the behavior in the case of the equality we need to track the exact coefficientof the term x − . We determine this coefficient as (cid:16) (1 + α ) µ a C ′ β α ) ~ √ GC α (cid:17) ± (1 + α ) µ a C ′ β α ) ~ √ GC α (79)Hence, lim x → x Γ − − > − / (cid:12)(cid:12)(cid:12) α α µ a C ′ β ~ √ GC α (cid:12)(cid:12)(cid:12) = 1.So for β = α then − m e c is not an accumulation point when GM m e < Ze . On theother hand, if β = α then − m e c is not accumulation point if α α (cid:12)(cid:12)(cid:12) µ a C ′ β ~ √ GC α (cid:12)(cid:12)(cid:12) = 1 holdstogether with GM m e < Ze . (cid:3) Summary and outlook
We have discussed the Dirac operator for a test electron in the static spherically sym-metric spacetime of a point nucleus with negative bare mass, allowing for a large class ofelectromagnetic vacuum laws compatible with the form of the spacetime metric given in (6).We have considered test electrons without and with an anomalous magnetic moment. Ourfindings demonstrate that the theory of the Dirac operator of a test electron in even thissimple class of spherically symmetric electrostatic spacetimes is rich and full of surprises!Different from the essentially self-adjoint situation which prevails when the bare mass ofthe nucleus vanishes, which was considered in [1], the Dirac operator is never essentiallyself-adjoint when the bare mass of the nucleus is strictly negative — unless the test electronfeatures an anomalous magnetic moment. Even then, essential self-adjointness holds only
ENERAL-RELATIVISTIC HYDROGEN 27 if the electric field of the nucleus diverges sufficiently fast at the nucleus, which is not thecase for a large subset of the electromagnetic vacuum laws considered. In particular, it isnot the case for the Born–Infeld vacuum law. Furthermore, on spacetimes of nuclei with(possibly infinite) negative bare mass and sufficiently rapidly diverging electric field, if theelectric field diverges precisely at the critical rate then the anomalous magnetic moment hasto be sufficiently strong to guarantee essential self-adjointness. In the special case of theReissner–Weyl–Nordstr¨om spacetime of a point nucleus our formula for the critical value ofthe electron’s anomalous magnetic moment coincides with the one found previously in [4].For all self-adjoint extensions of our Dirac operators we identified the essential spectrumwith the usual gap ( − m e c , m e c ) and showed that the gap contains infinitely many eigenval-ues. When GMm e Ze < m e c as clusterpoint. Yet when GMm e Ze > m e c as clusterpoint, and another one with − m e c as cluster point.However, the hyper-heavy nucleus regime GMm e Ze > M = A ( Z, N ) m p with A ≈ Z + N and N ≤ Z , continuesto hold for arbitrary Z and N . Namely, the hyper-heavy nucleus condition GMm e Ze > GMZm e Z e >
1, and since m p ≈ m e and M = Am p with A ≥ Z , this implies GM Z e > N ≤ Z continuesto hold for arbitrary N and Z , because the empirical charge to mass ratio of the protontogether with M = A ( Z, N ) m p and A ≈ Z + N and N ≤ Z implies that GM Z e < ≪ N ≤ Z (as in a neutron star), then the hyper-heavynucleus condition can be made compatible with the black hole sector condition and the massformula M = A ( Z, N ) m p with A ≈ Z + N . However, our results do not apply to the blackhole sector, and it is an interesting open question whether the Dirac Hamiltonian acting onbi-spinor wave functions of a test electron supported entirely inside the Cauchy horizon ofthe black hole spacetime of a hyper-heavy nucleus with mass formula M = A ( Z, N ) m p with A ≈ Z + N is well defined (with or without anomalous magnetic moment taken into account),or at least has self-adjoint extensions, and if so, whether there are two families of eigenvalueswith cluster points ± m e c . While this may never be of concern to experimental physicists,for the satisfaction of intellectual curiosity we have begun to investigate this problem [21].In any case the mathematical spectra of hypothetical hyper-heavy naked nuclei are notrealized in nature according to our analysis. Only if one drops the mass formula M = A ( Z, N ) m p with A ≈ Z + N completely and treats M , m e and e as parameters, then the family of ‘hyper-heavy hydrogenic ion’ eigenvalueshaving cluster point − m e c can exist on a naked singularity spacetime, mathematicallyspeaking, but it would be a bit of a stretch to refer to it as a hyper-heavy hydrogenic ion.Whether there is any physical scenario which could lead to such a situation in nature, orwhether this is pure science fiction, we don’t know, but it may be worth pondering.Beside the hyper-heavy hydrogenic ‘black hole ion’ problem mentioned above, there area number of spectral questions which we have not answered, such as whether the boundarypoints ± m e c of the essential spectrum are eigenvalues. We have also not attempted todetermine the discrete spectra in detail, which is worth the effort only if one has a compellingcandidate for the physically correct self-adjoint H .Then there are electromagnetic vacuum laws such as the one proposed by Bopp, Land´e–Thomas, and Podolsky (BLTP) which are not compatible with the form of the spacetimemetric given in (6). A similar study such as the one conducted in this paper should also becarried out for vacuum laws of the BLTP type.The test electron approximation can be expected to be very accurate for large Z butcertainly less so for hydrogen ( Z = 1). Therefore it is desirable to overcome the test electronapproximation. This has so far only been accomplished in a fully satisfactory manner in thenon-relativistic Schr¨odinger model of hydrogenic ions. We consider it to be one of the mostchallenging and important open problems of rigorous relativistic quantum mechanics. Acknowledgement : We thank Moulik Balasubramanian for interesting discussions.
ENERAL-RELATIVISTIC HYDROGEN 29
Appendix
A.In this appendix we show that general relativity has a regularizing effect on the Bohr–Sommerfeld-type model of hydrogenic large- Z ions with Coulomb interactions. We alsodemonstrate this effect when Coulomb interactions are replaced by electric interactions in anonlinear electrostatic vacuum. Like Bohr we work for simplicity only with circular orbits.A.1. Coulomb interactions.
Following Vallarta, we here assume that the static spacetimeof a point nucleus is given by the Reissner–Weyl–Nordstr¨om solution of the Einstein–Maxwellsystem. Then the general-relativistic Bohr–Sommerfeld-type energies E GRn ( Z, N ) , n ∈ N , ofa hydrogenic ion with a nucleus of charge Ze and mass A ( Z, N ) m p , with Z ≤ A ( Z, N ) < Z for the known nuclei, is determined by finding, for each n ∈ N , the minimum w.r.t. r of U GRn ( r ) := m e c s n ~ m c r s − Gc r (cid:20) A ( Z, N ) m p c − Z e r (cid:21) − Ze r . (A.1)Switching to the dimensionless variables ρ = rm e c/ ~ and V GRn ( ρ ) = U GRn ( r ) /m e c yields V GRn ( ρ ) := r n ρ r − α S γ pe h A ( Z, N ) − ǫα S Z ρ i ρ − Zα S 1 ρ . (A.2)Recall that α S := e / ~ c ≈ / . ǫ := m e /m p ≈ / γ pe := Gm e m p /e ≈ . × − . Asymptotically for very large ρ the dimensionless general-relativistic energy function V GRn ( ρ ) ∼ − α S ( Z + A ( Z, N ) γ pe ) /ρ , just like the special-relativistic one with Newtonian gravity added to the Coulombian electricity. As ρ becomessmaller and smaller, the term h − ǫα S Z A ( Z,N ) 1 ρ i changes sign, namely for ρ < Z A ( Z,N ) ǫα S it is negative. We see that ‘overall’ the zero ρ ( Z, N ) := Z A ( Z,N ) ǫα S grows with Z ; moreprecisely, ǫα S Z ≤ ρ ( Z, N ) ≤ ǫα S Z . Now, the smallest such ρ ( Z, N ) where the sign switchhappens is a tiny dimensionless distance, and the factor − /ρ before the [ ]-bracketed termcould threaten that the whole expression under the square root becomes negative before thistiny ρ is reached (starting from large ρ and making ρ smaller and smaller). Yet, since γ pe ismuch tinier yet, the whole expression under the square root remains positive for all ρ .For very small ρ the general-relativistic gravitational square-root factor in (A.2) con-tributes a factor Z √ γ pe ǫα S /ρ while the special-relativistic square-root factor in (A.2) con-tributes a factor n/ρ to the total square-root term. So for very small ρ the asymptoticbehavior is V GRn ( ρ ) ∼ Zn √ γ pe ǫα S /ρ ր ∞ as ρ ց
0, and except for the different coefficient,this is like the behavior of the non-relativistic kinetic energy function ( ∝ n /ρ ) in Bohr’smodel with circular orbits. Thus V GRn ( ρ ) always has a minimum at a strictly positive ρ foreach Z ∈ N and n ∈ N . The special-relativistic version of this problem is qualitatively very different. Setting G ց U SRn ( r ) := m e c s n ~ m c r − Ze r , (A.3)which is the same as setting γ pe ց V SRn ( ρ ) := q n ρ − Zα S 1 ρ . (A.4)Finding for each n ∈ N the minimum w.r.t. r , respectively ρ , will produce the principalenergy values of Sommerfeld’s fine structure spectrum of a hydrogenic ion whenever Z ≤ n the bottom drops out when Z > Z ∗ ( n ) ≥ Z ∗ (1), with Z ∗ (1) = 137.A.2. Nonlinear electrostatic vacuum.
It follows from the discussion in the main textthat replacing Maxwell’s “law of the pure ether” by a nonlinear electromagnetic vacuumlaw of the type considered in [29], and in this paper, amounts to replacing Z e /r by E ( r )in (A.1) and Ze /r by eφ ( r ) in (A.1) and in (A.3). The class of nonlinear vacuum lawsconsidered in this paper weakens the Coulomb singularity to φ ( r ) ∼ C ′′ β + C ′ β r − β as r ց β <
1, and this already removes the ‘large Z catastrophe’ from the correspondingspecial-relativistic Bohr–Sommerfeld type theory. The question thus becomes whether thegeneral-relativistic square-root factor in (A.1) can now cause a spectral catastrophe, or not.Since we consider only black hole-free spacetimes of nuclei, we have lim r ց m ( r ) = A ( Z, N ) m p − c E (0) ≤
0; cf. section 2. We need to distinguish m (0) = 0 and m (0) < r ց m ( r ) < −∞ ). Then the general-relativistic square-root factor in (A.1) diverges ∝ /r κ as r ց
0, which together with the 1 /r singularity ofthe special-relativistic square-root factor in (A.1) yields an overall 1 /r κ singularity, with κ ≥ (N.B. κ = if lim r ց m ( r ) = m (0) < κ ∈ ( ,
1] if lim r ց m ( r ) = −∞ ).This overpowers the r − β singularity with 0 < β <
1. There is no ‘large- Z catastrophe’.Consider next the case where lim r ց m ( r ) = 0 for all Z . (Admittedly this is presumablya purely academic situation, but it’s feasible mathematically.) In this case we need also anassumption about how m (0) = 0 is approached. We consider power laws m ( r ) = Ar κ with κ > A >
0. Then as r ց ∝ /r (1 − κ ) / if κ ∈ (0 , ≤ κ ≥ < ” iff κ = 1. Together with the 1 /r singularity of the special-relativistic square-rootfactor in (A.1) this yields a r (min { κ , }− / singularity which once again overpowers the 1 /r β singularity with 0 < β <
1. There is no ‘large- Z catastrophe’ in this case either. ENERAL-RELATIVISTIC HYDROGEN 31
Appendix B. The family of electrovacuum spacetimes
In this appendix we present a large family of static spherically symmetric electromagneticvacuum spacetimes, one that includes both the RWN as well as Hoffmann’s with either zeroor negative bare mass, and all the members of which satisfy the assumptions 3.1 and 3.3 madein Section 3. We begin by recalling [29] that all such spacetimes are characterized by thechoice of a single C function of one variable ζ : R + → R + , called the reduced electromagneticHamiltonian that satisfies the following properties (R1) lim µ → ζ ( µ ) /µ = 1, (R2) ζ ′ > ζ ( µ ) − µζ ′ ( µ ) ≥ ∀ µ > (R3) ζ ′ ( µ ) + 2 µζ ′′ ( µ ) ≥ ∀ µ > ζ ( µ ) = µ , while the onecorresponding to Born’s law is ζ ( µ ) = √ µ −
1; in the electrostatic special case thiscoincides with the one from Born–Infeld’s vacuum law.It was shown in [29] that for every such choice of ζ , and parameters M > , Q ∈ R , thereis a corresponding static, spherically symmetric, asymptotically flat solution of the Einstein–Maxwell system with ADM mass M ADM equal to M , and total charge Q , as described inSec. 2, with metric line element (6) defined in terms of the radial mass function m ( r ) as in(4), where m ( r ) = M − c Z ∞ r ζ (cid:16) Q s (cid:17) s ds (B.1)and the electrostatic potential φ ( r ) = Q Z ∞ r ζ ′ (cid:16) Q s (cid:17) s ds. (B.2)It was further shown in [29] that under additional assumptions on ζ , one could make surethat the singularity present at the center of these spacetimes (which is not shielded by ahorizon) is of the mildest form possible, namely a conical singularity, with zero bare mass m (0) = 0, and that for these spacetimes the ADM mass M ADM = m ( ∞ ) is equal to the totalelectrostatic energy. The prime example of these is the Hoffmann spacetime discussed insection 2. Here we introduce a larger class of such spacetimes that includes, in addition to mildlysingular manifolds like Hoffmann’s, also those with much more severe singularity at thecenter, such as the RWN, which has negative infinite bare mass. We begin by introducing aone-parameter family of reduced Hamiltonians ζ , parametrized by a positive number µ > ζ ( µ ) := min { µ, √ µ µ } . (B.3)We note that ζ is Lipschitz continuous and satisfies assumptions (R1–R3) away from itskink at µ = µ . The mass function corresponding to ζ is denoted by m ( r ). It is a C function, and can be computed from (B.1): m ( r ) := ( M − Q / / c µ / if µ < µ , M c / Q / µ − / if µ > µ ; with µ ( r ) := Q r . (B.4)Next we show that for a particular choice of the parameter µ this becomes a model for thevacuum spacetime outside a point charge of mass M = M ADM and charge Q : Let r := Q M c (B.5)denote the “classical radius” of this point charge (i.e. the distance at which its electrostaticself-energy equals the rest energy of the particle), and set µ = µ ( r ). We then obtain m ( r ) := ( M c Q r for r < r ,M − Q c r for r > r . (B.6)It is easy to verify that the above mass function satisfies the assumptions 3.1 that were madein Section 3, provided that the mass M and charge Q of the particle satisfy the “no horizon”condition GM Q < . (B.7)We are now ready to define a whole family of electrovacuum spacetimes with mass functionsand electrostatic potentials that satisfy the assumptions 3.1 and 3.3 and can serve as modelsfor the vacuum outside a point charge of mass M ADM and charge Q and arbitrary finite orinfinite negative bare mass: Proposition B.1.
Let M ∈ R + and Q ∈ R be given, subject to (B.7). Let ζ : R + → R beany C function satisfying assumptions (R1–R3) above, and in addition assume ζ ( µ ) > min { µ, √ µ µ } , µ := M c Q . (B.8) ENERAL-RELATIVISTIC HYDROGEN 33
Then the corresponding static, spherically symmetric, asymptotically flat solution of theEinstein–Maxwell equations, with vacuum law given by ζ , is characterized by the mass func-tion m ( r ) as in (B.1) and electrostatic potential φ ( r ) as in (B.2) that satisfy the assumptions3.1 and 3.3 made in Section 3 of this paper.Proof. We first verify assumptions 3.1. From (B.1) m is clearly a C function of r . Moreover m ( r ) = M − c Z ∞ r ζ ( µ ( s )) s ds ≤ M − c Z ∞ r ζ ( µ ( s )) s ds = m ( r ) , (B.9)so that the mass function of this manifold sits below the mass function m ( r ) of the modelspacetime we constructed in the above, and hence satisfies the no-black-hole condition m ( r ) r ≤ c G , since m ( r ) is seen to satisfy this condition.We note that m ( r ) ≤ m ( r ) allows these spacetimes to have negative bare mass that canbe finite or infinite.Consider now the function ζ . It is a smooth and by (R1,R2) positive and increasingfunction of its argument. By integrating the differential inequalities in (R2,R3) one obtainsthat ∀ < µ ′ < µ : µ ′ µ ≤ ζ ( µ ′ ) ζ ( µ ) ≤ s µ ′ µ . (B.10)Suppose now that a, b ∈ R and c a , c b > ζ ( µ ) ∼ ( c a µ a for µ → ,c b µ b for µ → ∞ . (B.11)Then by (B.10) and (R1) we have that a = 1, c a = 1, and ≤ b ≤ m (0) = M − c Z ∞ ζ ( µ ( s )) s ds. (B.12)Consider the following two cases: (F) The integral in (B.12) is finite, and (I) that integral isinfinite.If that integral is finite, then by the additional assumption we have made about ζ , namely(B.8), we have m := m (0) ≤ m (0) = 0 . (B.13)Moreover, since Z ∞ ζ ( µ ( s )) s ds = Q / / Z ∞ µ ′− / ζ ( µ ′ ) dµ ′ , (B.14)it is clear that Case (F) corresponds to b < and Case (I) to b ≥ . In Case (F), we can express the mass function in an alternative way, m ( r ) = m + 1 c Z r ζ ( µ ( s )) s ds, (B.15)where m := m (0) ≤ bare mass of the central singularity. The asymptotics we haveestablished for ζ then imply that, with λ := 3 − b , m ( r ) ∼ ( m + A λ r λ as r → ,M − Q c r as r → ∞ , λ > . (B.16)In Case (I), on the other hand, we obtain m ( r ) ∼ ( B λ r λ as r → ,M − Q c r as r → ∞ , λ < . (B.17)(The borderline case b = 3 / α = 0 in Assumptions 3.1 while(B.17) corresponds to α > λ = − α then.Having established that Assumptions 3.1 are satisfied for this family of spacetimes, wemove on to the analysis of the electrostatic potential φ . First, we observe that by (R2,R3) and (B.10), ζ ′ ( µ ) r µ µ ≤ ζ ′ ( µ ) ≤ , for all 0 < µ < µ. (B.18)It follows that ζ ′ inherits the following asymptotics from ζ : ζ ′ ( µ ) ∼ ( µ → ,bc b µ b − as µ → ∞ , (B.19)with ≤ b ≤ φ satisfies, once again with λ := 3 − b , φ ( r ) ∼ ( e C ′′ λ − e C ′ λ r λ as r → , Qr as r → ∞ , λ > , (B.20)with e C ′ λ > e C ′′ λ > Q >
0, while in Case (I) we have φ ( r ) ∼ ( b C ′ λ r λ as r → , Qr as r → ∞ . λ < . (B.21)with b C ′ λ > Q >
0. We note that (B.20) corresponds to β ≤ β > λ = − β then. Note that λ ≥ − (cid:3) ENERAL-RELATIVISTIC HYDROGEN 35
Data availability statement :Data sharing is not applicable to this article as no new data were created or analyzed inthis study.
References [1] Balasubramanian, M.K.,
Scalar fields and spin-half fields on mildly singular spacetimes , Ph.D. thesis,Rutgers Univ. (2015).[2] Behncke, H.
The Dirac equation with an anomalous magnetic moment , Math. Z. :213–225 (1980).[3] Belgiorno, F.,
Massive Dirac fields in naked and in black hole Reissner–Nordstr¨om manifolds , Phys.Rev. D :084017 (1998).[4] Belgiorno, F., Martellini, M., and Baldicchi, M., Naked Reissner–Nordstr¨om singularities and theanomalous magnetic moment of the electron field , Phys. Rev. D :084014, (2000).[5] Born, M., Modified field equations with a finite radius of the electron , Nature :282 (1933).[6] Brill, D.R., and Cohen, J.M.,
Cartan frames and the general relativistic Dirac Equation , J. Math. Phys. :238–243 (1966).[7] Cohen, J.M. and Powers, R.T., The general-relativistic hydrogen atom , Commun. Math. Phys. :96–86(1982).[8] Dreifus, E., Semi-classical calculations of general-relativistic corrections to the Sommerfeld fine structurespectrum of the hydrogen atom to second order in powers of G , Honors Thesis, Rutgers Univ. (2019).[9] Dunford, N., and Schwarz, J.T. Linear operators II : Spectral theory of self-adjoint operators in Hilbertspace , Interscience Publishers, New York (1963).[10] Einstein, A., letter to Arnold Sommerfeld, Dec. 9, 1915.[11] Esteban, M.J., and Loss, M.,
Self-adjointness of Dirac operators via Hardy–Dirac inequalities , J. Math.Phys. :112107(8)(2007).[12] Finster, F., Smoller, J., and Yau, S.T., Non-existence of time-periodic solutions of the Dirac equationin a Reissner–Nordstr¨om black hole background , J. Math. Phys. :2173–2194 (2000).[13] Gesztezy, F., Simon, B., and Thaller, B., On the self-adjointness of Dirac operators with anomalousmagnetic moment , Proc. AMS :115–118 (1985).[14] Greiner, W., M¨uller, B., and Rafelski, J., Quantum electrodynamics of strong fields , Springer, Berlin -Heidelberg - New York - Tokyo (1985).[15] Hinton D.B., Mingarelli A.B., Read T.T. and Shaw J.K.,
On the number of eigenvalues in the spectralgap of a Dirac system , Proc. Edinburgh Math. Soc. :367–378 (1986).[16] Hinton D.B., Mingarelli A.B., Shaw J.K., Dirac systems with discrete spectra , Can. J. Math.
XXXIX :100–122 (1987).[17] Hoffmann, B.,
Gravitational and electromagnetic mass in the Born–Infeld electrodynamics , Phys. Rev. :877–880 (1935). [18] Kalf, H., Schmincke, U.-W., Walter, J, and W¨ust, R., On the spectral theory of Schr¨odinger and Diracoperators with strongly singular potentials , Lect. Notes Math. :182–226 (Springer, Berlin - Heidelberg- New York, 1975).[19] Kato, T.:
Perturbation theory for linear operators , Springer, New York (1966).[20] Keppeler, S.,
Semi-classical quantization rules for the Dirac and Pauli equations , Annals Phys. (NY) :40–71 (2003).[21] Kiessling, M. K.-H., Tahvildar-Zadeh, A. S., and Toprak, E.,
On the Dirac operator for a test electronin a Reissner–Weyl–Nordstr¨om black hole spacetime , to be submitted to Gen. Rel. Grav. (2020).[22] Narnhofer, H.,
Quantum theory for /r potentials , Acta Phys. Austr. :306–322 (1974).[23] Parker, L., One-electron atom as a probe of spacetime curvature , Phys. Rev. D :1922–1934 (1980).[24] Reed, M. and Simon, B., Fourier analysis, Self-adjointness,
London, Academic Press (1975).[25] Rose, M.E.,
Relativistic Electron Theory , Wiley, New York (1961).[26] Schr¨odinger, E.
Diracsches Elektron im Schwerefeld I , Sitzungsber. Preuss. Akad. Wiss. Phys.-Math.Kl. 1932, pp.436–460; Verlag Akad. Wiss. (1932).[27] Sommerfeld, A.,
Zur Quantentheorie der Spektrallinien , Annal. d. Phys. :1–94 (1916).[28] Sommerfeld, A.,
Atombau und Spektrallinien , F. Vieweg Verlag, 1st ed. (1919); 4th ed. (1924).[29] Tahvildar-Zadeh, A.S.,
On the static spacetime of a single point charge , Rev. Math. Phys. :309–346(2011).[30] Thaller, B., The Dirac equation , Springer (1992).[31] Vallarta, M.,
Sommerfeld’s theory of fine structure from the standpoint of General Relativity , J. Math.& Phys. :66–83 (1924).[32] Weidmann, J., Oszillationsmethoden f¨ur Systeme gew¨ohnlicher Differentialgleichungen , Math. Z. :349–373 (1971).[33] Weidmann, J.,
Absolut stetiges Spektrum bei Sturm–Liouville-Operatoren und Dirac-Systemen , Math.Z. :423–427 (1982).[34] Weidmann, J.,
Spectral Theory of Ordinary Differential Operators , Springer (1987)[35] Wereide, T.,
The general principle of relativity applied to the Rutherford–Bohr atom-model , Phys. Rev. :391–396 (1923).[36] Weyl, H., Space, Time, Matter , 4th. ed., Dover (1952).
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, U.S.A.
E-mail address : [email protected] Department of Mathematics, Rutgers University, Piscataway, NJ 08854, U.S.A.
E-mail address : [email protected] Department of Mathematics, Rutgers University, Piscataway, NJ 08854, U.S.A.
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