On the Dirac operator for a test electron in a Reissner--Weyl--Nordström black hole spacetime
Michael K.-H. Kiessling, A. Shadi Tahvildar-Zadeh, Ebru Toprak
aa r X i v : . [ m a t h - ph ] S e p ON THE DIRAC OPERATOR FOR A TEST ELECTRON IN AREISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIME
MICHAEL K.-H. KIESSLING, A. SHADI TAHVILDAR-ZADEH, EBRU TOPRAK
Abstract.
The present paper studies the Dirac Hamiltonian of a test electron with adomain of bi-spinor wave functions supported on the static region inside the Cauchy hori-zon of the subextremal RWN black hole spacetime, respectively inside the event horizonof the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian isnot essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a suf-ficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restoresessential self-adjointness; the empirical value of the electron’s anomalous magnetic momentis large enough. The spectrum of the subextremal self-adjoint Dirac operator with anoma-lous magnetic moment is purely absolutely continuous and consists of the whole real line; inparticular, there are no eigenvalues. The same is true for the spectrum of any self-adjointextension of the Dirac operator without anomalous magnetic moment interaction, in thesubextremal black hole context. In the extremal black hole sector the point spectrum, ifnon-empty, consists of a single eigenvalue, which is identified.
Date : Version of September 14, 2020; printed: September 17, 2020c (cid:13) (2020) The authors. Reproduction of this preprint is permitted for noncommercial purposes. Introduction
It is well-known that if
M >
Q > G denotes Newton’s constant ofuniversal gravitation, then the RWN spacetime features a naked singularity when GM < Q and a black hole when GM ≥ Q ; in the borderline case GM = Q one speaks of the extremal RWN black hole, while GM > Q is called the subextremal black hole parametersector, cf. [17]. For the original publications, see [26], [34], and [25].In their paper “The general-relativistic hydrogen atom” [10] Cohen and Powers rigorouslystudied the general-relativistic Dirac operator H ([35], [28], [6]) for a test electron in the RWNspacetime of a point nucleus for both the naked singularity sector and the subextremal blackhole sector. They made the startling discovery that in the naked singularity sector H isnot well-defined, while for the black hole sector there is a well-defined H but its essentialspectrum is the whole real line, void of any eigenvalues.The truly startling part of the discoveries of Cohen and Powers [10] concerns the nakedsingularity sector, for it means that ‘switching on relativistic gravity’ destroys the well-defined (i.e. essentially self-adjoint) special-relativistic purely electrical hydrogenic ion prob-lem for all parameter values which correspond to empirically known nuclei (1 ≤ Z ≤ m p ≤ M < m p ; here m p denotes the proton mass). In more technical language,general-relativistic gravity is not at all a ‘weak perturbation’ (see [22]) of special-relativisticelectricity in the atomic realm, notwithstanding the general folklore that ‘the gravitationalinteraction between an electron and a nucleus is too weak to be significant,’ cf. [12]. One may be tempted to consider this result as a vindication for the widespread opinion that “nakedsingularities are considered unphysical” (cf. [16], p.562). However, this opinion propagates an unfortunatemyth. It is based on a misunderstanding of Penrose’s weak cosmic censorship hypothesis, which surmisesthat gravitational collapse of cosmic matter does not form a naked singularity. In its strict sense the surmiseis wrong, as shown first by Christodoulou [7], [8] for spherically symmetric collapse of scalar matter, and mostrecently by Rodnianski and Shlapentokh-Rothman [27] for collapsing gravitational waves without symmetryassumption; yet it is expected that these scenarios are not generic (this was confirmed for the sphericallysymmetric scalar case, also by Christodoulou [9]), and that generically (or: typically) a gravitational collapseof cosmic ‘matter’ will not form a naked singularity. However, the point nuclei used in quantum-mechanicalmodels of hydrogenic ions, whether of the kind created in our laboratories, or the hypothetical ‘hyper-heavy’type “out there” in space, are not assumed to have formed through gravitational collapse of charged matterin cosmic proportions. In short, the weak cosmic censorship hypothesis, even if generically true, is entirelyirrelevant to the problem of general-relativistic hydrogenic ions.
IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 3
Also the special-relativistic Dirac Hamiltonian for hydrogenic ions with purely electri-cal Coulomb interactions is not always essentially self-adjoint [on the minimal domain C ∞ c ( R \{ } ) ]. We recall that when Z ∈ N counts the number of elementary charges inthe nucleus, then the Dirac Hamiltonian is essentially self-adjoint for Z ≤
118 [24, 30]. For119 ≤ Z ≤
137 it has a distinguished self-adjoint extension, yet for Z >
137 nobody seemsto know which one of uncountably many self-adjoint extensions is physically distinguished.The heuristic explanation for the breakdown of analytical self-adjointness in the special-relativistic purely electrical hydrogenic ion problem is that the electrical Coulomb attractionbetween nucleus and electron becomes too strong for the angular momentum barrier to sta-bilize, and a collapse of the ground state ensues. Since gravity is generally attractive, onewould have expected a worsening of the self-adjointness properties in the general-relativisticproblem, but a complete wipeout was presumably not expected by anyone!The problem with the lack of essential self-adjointness of the special-relativistic hydro-genic Dirac Hamiltonian goes away, however, if one takes the anomalous magnetic moment µ a of the electron into account. Indeed, as shown in [2, 15] for the special-relativistic hydro-genic problem, adding an anomalous magnetic moment operator to the Dirac Hamiltonianof a test electron with purely electrostatic interactions produces an essentially self-adjointHamiltonian for the electron of any hydrogenic ion, independently of the strength of itsnon-vanishing anomalous magnetic moment; see [30, 31] for numerically computated eigen-values as functions of Z beyond Z = 137. More recently Belgiorno, Martellini, and Baldicchi[4] showed that the Dirac operator for a test electron with anomalous magnetic moment isessentially self-adjoint in the naked RWN geometry (only) if | µ a | ≥ √ G ~ c . The empirical | µ a | ≈ µ class := π e m e c , which we call the classical magnetic moment of the electron . Since √ G ~ /c ≈ . · − µ class , (1)the hurdle for essential self-adjointness, | µ a | ≥ √ G ~ c , is easily cleared with the empiricalelectron data. Here, m e is the empirical mass of the electron and − e its charge, c is thespeed of light in vacuum, and ~ is Planck’s constant divided by 2 π , as usual.With the Dirac operator for an electron in the naked singularity sector of the RWNspacetime basically understood, in this paper we will revisit the problem of the Dirac operatorfor an electron in the black hole sector of the RWN spacetime family; we will also includesome comparative remarks concerning electrons in the naked singularity sector, though. The distinguished self-adjoint extention is defined by allowing Z ∈ C and demanding analyticity in Z .The real threshold values then become Z = √ / α S instead of Z = 118, and Z = 1 /α S instead of Z = 137.Here, α S := e / ~ c ≈ / .
036 is Sommerfeld’s fine structure constant.
KIESSLING, TAHVILDAR-ZADEH, TOPRAK
Our point of departure is the fact that Cohen and Powers [10] considered a Dirac Hamil-tonian with the minimal domain of C ∞ bi-spinor functions with compact support outside theevent horizon of a subextremal black hole. They proved that this H is essentially self-adjoint,and that it has the whole real line as its essential spectrum. Thus its discrete spectrum isempty and any eigenvalues would have to be embedded in the continuum. Yet in [10] theabsence of eigenvalues is shown altogether. Their result means that a test electron outsidethe event horizon of a subextremal RWN black hole cannot be in a stationary bound state.In concert with a result of Weidmann [32] this now implies that the essential spectrum is purely absolutely continuous , and so in fact is the spectrum of this Dirac Hamiltonian.Upon reflection, it is not too surprising not to find bound states of an electron whosewave function is supported outside the event horizon of an RWN black hole. After all, oneexpects the electron to be swallowed by the black hole unless it escapes to spatial ∞ . Thecapture of the electron by the black hole is not seen in the treatment by Cohen and Powers,who worked with a coordinate system that near the end of the first quarter of the 20thcentury gave rise to the “frozen star” scenario. The purpose of this coordinate system wasto describe the collapsing evolution of gravitating masses as seen from spatial infinity, andtherefore failed to capture the formation of a black hole. Thus, conceivably, a Dirac boundstate in the black hole sector of RWN may exist after all, but it would require the domainof the Dirac Hamiltonian to not be restricted to bi-spinor wave functions supported outsidethe event horizon. Of course, it is often argued on positivistic grounds that physics is notconcerned with what goes on inside an event horizon, but positivism is merely a form ofphilosphy which should not be confused with the foundations of physics. Also Werner Israeland his collaborators have long advocated [11] investigating what’s going on inside an eventhorizon according to general relativity theory. Finster, Smoller, and Yau [14] in particularhave inquired into “time-periodic” Dirac bi-spinor wave functions that are supported bothoutside and inside the event horizon of a RWN black hole spacetime, and found no nontrivialones in L . However, since the region between the Cauchy and the event horizon of a RWNblack hole spacetime is not static, insisting on time-periodic bi-spinors also there seems likeasking for too much. In this vein, in this paper we will investigate the Dirac Hamiltonian fora test electron in the RWN black hole spacetime with the bi-spinor wave function supportedentirely on the static part of the region inside the event horizon of the black hole spacetime,which is a static spherically symmetric spacetime with a naked singularity in its own right— it is not asymptotically flat, though. IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 5
Our results, stated informally, are:
Theorem 1 : The Dirac Hamiltonian H for a test electron in the static interior of a(sub-)extremal RWN black hole, if it interacts with the singularity only electrically and grav-itationally, is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Inthe subextremal case, each self-adjoint extension has a purely absolutely continuous spectrumthat extends over the whole real line. Theorem 2 : The Dirac Hamiltonian H for a test electron in the static interior of a (sub-)extremal RWN black hole, if it interacts with the singularity electrically, gravitationally, andthrough its anomalous magnetic moment, is essentially self-adjoint if and only if | µ a | ≥ √ G ~ c .In the subextremal essentially self-adjoint situation, the unique self-adjoint extension haspurely absolutely continuous spectrum that covers the whole real line. Thus, the singularity of the RWN spacetime causes a lack of essential self-adjointness(e.s.a.) if the electron is not shielded from it by the event horizon and is assumed to interactonly electrically and gravitationally with the singularity, but e.s.a. is restored if a sufficientlylarge anomalous magnetic moment of the electron is taken into account. The empiricalanomalous magnetic moment of the electron is about 10 times larger than the criticalvalue, the same critical value as found in [4] for the naked singularity sector.However, while in Appendix C of [4] it is shown that the general-relativistic hydrogenicDirac Hamiltonian of a test electron with anomalous magnetic moment in the naked sin-gularity sector of the RWN spacetime of a nucleus has infinitely many discrete eigenvaluesin the gap ( − m e c , m e c ) of its essential spectrum, the essentially self-adjoint operator ofan electron with anomalous magnetic moment inside the Cauchy horizon of a subextremalblack hole has no eigenvalues at all. We will also show that in the extremal case there canbe at most one eigenvalue, possibly infinitely degenerate, which we identify.In the remainder of this paper we make all this precise.In section 2 we explain that normal nuclei are associated with the naked singularity sectorof the RWN spacetime, while hypothetical ‘hyper-heavy nuclei’ have to be associated withthe RWN black hole sector. We also stipulate our dimensionless notation for discussing boththe spacetime and the Dirac operators.Section 3 is the main technical section. We define the Dirac operators, state our theoremsprecisely, then present their proofs, using strategies of [33], [21], and [10]. Some of our proofsare overall very similar to proofs in [23] for naked-singularity spacetimes, yet details vary.We conclude in Section 4 and emphasize open problems. KIESSLING, TAHVILDAR-ZADEH, TOPRAK The Reissner–Weyl–Nordstr¨om spacetime of a point nucleus
In order to facilitate the comparison of our results with those for hydrogenic ions, includingsome speculative hyper-heavy ones defined by the inequality
GM m e > Ze (here, m e is theempirical rest mass of the electron, and the inequality means that the gravitational attractionof a positron (!) to the nucleus overcomes their electrical repulsion), from now on we think ofthe central timelike singularity of the RWN spacetime as a proxy for the worldline of a pointnucleus at rest. Thus for the charge parameter Q of the RWN spacetime we set Q = Ze ,where e > Z ∈ N counts thenumber of elementary charges carried by the nucleus. We let the ADM mass of the RWNspacetime be the nuclear mass, M ADM = M = A ( Z, N ) m p , where m p is the proton mass, N ∈ { , , , ... } is the number of neutrons in the nucleus, and A ( Z, N ) ≥ A ( Z, N ) ≈ Z + N to within 1% accuracy.All empirically known long-lived nuclei are far away from the black hole regime GM ≥ Z e . This conclusion extends to hypothetical nuclei with arbitrary large Z if they obey thebounds Z ≤ A ( Z, N ) ≤ Z known empirically to hold for all long-lived nuclei with Z ≤ chart of the nuclids . Assuming these empirical bounds, essentially N ≤ Z towith 1% accuracy, one finds GM Z e < Gm e , and since Gm e ≈ · − ≪ GM Z e ≪
1, thus
GMm p Ze ≪
1, and so
GMm e Ze ≪ GM m e > Ze , are as-sociated with the black hole sector of the RWN spacetime. For suppose not. Then both GM m e > Ze and GM < Z e (the latter condition means we are in the RWN naked sin-gularity regime). Since M = A ( N, Z ) m p for all nuclei, we have M ≥ Zm p , and since m p =1836 m e , we find that GM < Z e implies that 1836 AGM m e < Z e , while Ze < GM m e implies 1836 AZe < AGM m e . And so, by transitivity, we have 1836 AZe < Z e ,hence 1836 A < Z , which is impossible with the empirical A ( Z, N ) ≈ Z + N .Thus, the assumption N ≤ Z cannot be imposed as a condition when inquiring intohyper-heavy hydrogenic ions. Fortunately, neutron stars are in a fair sense examples ofgravitationally bound nuclei with M ≈ ( Z + N ) m p and Z very small while N is very large.Of course, neutron stars are not point-like, yet they are only mentioned as an example ofgigantic nuclei in nature not obeying the N ≤ Z rule. Hyper-heavy nuclei would not onlynot obey the N ≤ Z rule, they would have to be associated with the black hole sectorand therfore de-facto be point singularities covered by an event horizon — as per Einstein’sgeneral relativity theory. IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 7
The naked singularity regime.
The electrostatic Reissner–Weyl–Nordstr¨om (RWN)spacetime of a naked point nucleus is spherically symmetric, static, asymptotically flat andtopologically identical to ‘ R , \ a timelike line,’ equivalently R × ( R \ { } ), covered by asingle global chart of ‘spherical coordinates’ ( t, r, ϑ, ϕ ) ∈ R × R + × [0 , π ] × [0 , π ). Here, r is the so-called area radius: every point in the RWN spacetime is an element of a uniqueorbit of the Killing vector flow corresponding to its SO (3) symmetry, and this orbit is ascaled copy of S with area A =: 4 πr , defining r >
0. Moreover, the variables ϑ and ϕ are the usual polar and azimuthal angles on S . In dimensionless units where r is measuredin multiples of the electron’s reduced Compton wave length ~ /m e c , and t in multiples of ~ /m e c , its metric has the line element ds = − f ( r )d t + f − ( r )d r + r dΩ , (2)where dΩ = d ϑ + sin ϑ d ϕ is the line element on S , and where f ( r ) = 1 − GM m e ~ c r + Gm ~ c Z e ~ c r . (3)Here, M = A ( Z, N ) m p , and we note that Gm ~ c ≈ . · − and Gm p m e ~ c ≈ . · − ;incidentally, Gm ~ c ≈ . · − . Also, e ~ c ≈ / .
036 is Sommerfeld’s fine structure constant.The known long-lived nuclei, for which A ( Z, N ) ≈ Z + N and Z ≤ A ( Z, N ) ≤ Z , areassociated with the naked singularity sector of the RWN spacetimes, i.e. f ( r ) > ∀ r > The black hole regime.
The RWN spacetime features a black hole if there is at leastone value of r > f ( r ) = 0. Since f ( r ) is a quadratic polynomial in 1 /r , viz. f ( r ) = r ( r − r + )( r − r − ), with the zeros formally given by r ± = GMm e ~ c (cid:18) ± q − Z e GM (cid:19) , (4)and those are real if and only if Z e GM ≤
1. If Z e GM = 1, one says the asymptotically flatspacetime contains an extremal black hole; if Z e GM <
1, the asymptotically flat spacetimecontains a subextremal black hole. In the extremal case, r + = r − = GMm e ~ c =: r , and then f ( r ) = (cid:18) − GM m e ~ c r (cid:19) = 1 r ( r − r ) . (5)Continuing an asymptotically flat RWN black-hole spacetime analytically, one finds twostatic regions: either r > r + or r < r − ; this is true even for the extremal case when r + = r − .The maximal analytically extended spacetime even has infinitely many copies of such regions.We will be concerned with spacetimes given by one copy of the inner static region. KIESSLING, TAHVILDAR-ZADEH, TOPRAK The Dirac operators
In this section we formulate the Dirac operator for a test electron with or without anoma-lous magnetic moment in the RWN spacetime of a naked point nucleus. For the sake ofdefiniteness, we will define the electrons’ anomalous magnetic moment as identical to itshighly accurate approximation µ a = − µ class . However, we multiply µ a by an ‘amplitude’ a :for a = 0 we obtain the Dirac operator for a point electron without anomalous magneticmoment, whereas a = 1 if the electron’s anomalous magnetic moment is taken into account.By varying a continuously we can inquire into the threshold for essential self-adjointness.Electrons with wave function restricted to the region r > r + on the subextremal RWNblack hole spacetime were studied in [10]; no bound states exist, then. We will investigateelectrons with wave function restricted to the region r < r − on the subextremal RWN blackhole spacetime, i.e. to the spacetime given by (2), (3), with ( t, r, ϑ, ϕ ) ∈ R × (0 , r − ) × [0 , π ] × [0 , π ). We will also study wave functions on the extremal RWN black hole spacetime,supported inside the event horizon at r (= r − = r + ). For the purpose of comparison, wewill also recall the Dirac operator defined on the naked singularity sector.Due to the spherical symmetry and static character of the spacetimes, the Dirac operator H of a test electron in the curved space whose line element d s is given by (2) separates inthe spherical coordinates and their default Cartan frame [10]. More precisely, H is a directsum of so-called radial partial-wave Dirac operators H rad k := m e c K a k , k ∈ Z \{ } , with K a k := " f ( r ) − Zα S 1 r (cid:2) kr − Zα S2 a π r (cid:3) f ( r ) − f ( r ) ddr (cid:2) kr − Zα S2 a π r (cid:3) f ( r ) + f ( r ) ddr − f ( r ) − Zα S 1 r , (6)which act on two-dimensional bi-spinor wave function subspaces. The spectrum of H is theunion of the spectra of the H rad k . This reduces the problem to studying the spectrum of K a k .3.1. Point nucleus as naked singularity of static spacetime.
In this case the bi-spinorwave functions are supported on R minus a point. The ‘radial Hilbert space’ consists ofpairs g ( r ) := (cid:0) g ( r ) , g ( r ) (cid:1) T equipped with a weighted L norm given by k g k := Z ∞ f ( r ) (cid:16) | g ( r ) | + | g ( r ) | (cid:17) d r. (7)As mentioned in the introduction, Cohen and Powers [10] proved that for a = 0 the DiracHamiltonian is not essentially self-adjoint, but has uncountably many self-adjoint extensions.Belgiorno et al. [4] subsequently showed that H is essentially self-adjoint on the domain of C ∞ bi-spinor wave functions which are compactly supported away from the singularity at r = 0 whenever a is large enough, viz. if a | µ class | ≥ √ G ~ c . IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 9
Point nucleus as singularity in static interior of black hole spacetime.
In thiscase the bi-spinor wave functions are supported on S × (0 , r − ). The ‘radial Hilbert space’consists of pairs g ( r ) := (cid:0) g ( r ) , g ( r ) (cid:1) T equipped with a weighted L norm given by k g k := Z r − f ( r ) (cid:16) | g ( r ) | + | g ( r ) | (cid:17) d r. (8)In the extremal case, r − = r .We change variables r x such that f ( r ) ddr = ddx , (9)with x = 0 when r = 0, which for the subextremal sector yields x = r + r r + − r − ln (cid:18) − rr + (cid:19) − r − r + − r − ln (cid:18) − rr − (cid:19) ; r < r − ; (10)in the extremal limit r − ր r & r + ց r this becomes x = r " − rr + 2 ln (cid:18) − rr (cid:19) − (cid:18) − rr (cid:19) , r < r . (11)Note that x → ∞ when r ր r − , respectively when r ր r . This maps K a k into e K a k = f ( r ( x )) − Zα S 1 r ( x ) h kr ( x ) − Zα S2 a π r ( x ) i f ( r ( x )) − ddx h kr ( x ) − Zα S2 a π r ( x ) i f ( r ( x )) + ddx − f ( r ( x )) − Zα S 1 r ( x ) , (12)=: " a ( x ) − b ( x ) kc ( x ) − a d ( x ) − ddx kc ( x ) − a d ( x ) + ddx − a ( x ) − b ( x ) (13)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. (14)3.2.1. Electron without anomalous magnetic moment.
Theorem 3.1.
The operator e K k given by (12) with a = 0 has uncountably many self-adjointextensions for both the subextremal and the extremal black-hole sector.Proof. We use the strategy of [10], [23] for the naked singularity spacetimes. We start withthe subextremal case. Note that, with the change of variable (10) one has ( r − r − ) ∼ e − κ x where κ = r + − r − r − as x → ∞ and r ∼ x / as x →
0. Therefore, in the subextremal case theoperator (13) features a ( x ) ∼ x − / , b ( x ) ∼ x − / and c ( x ) ∼ x − / as x →
0. Furthermore,as x → ∞ , we have f ( r ( x )) ∼ e − κ x as well as a ( x ) , c ( x ) ∼ e − κ x , and b ( x ) → Zα s r − . Here, “ f ( x ) ∼ g ( x ) as x → x ∗ ” means ∃ C > f ( x ) /g ( x ) → C as x → x ∗ , where x ∗ = 0 or ∞ . Let e K ∗ k be the adjoint operator of e K k . The domain D ( e K k ) comprises all C ∞ functions ofcompact support in (0 , ∞ ), and D ( e K ∗ k ) includes the functions f which together with f ′ areintegrable in any compact subset of [0 , ∞ ). On D ( e K ∗ k ) we now define the sesquilinear form[ g, h ] := h e K ∗ k g, h i − h g, e K ∗ k h i , (15)with h· , ·i defined in (14). By Theorem 4.1 in [33], 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 .Now consider the spaces in which [ g, h ] = 0. Take g ∈ D ( e K ∗ k ) so that e K ∗ k g = ψ for some ψ ∈ L ([0 , ∞ ). Since D ( e K k ) ⊂ C c ([0 , ∞ )), g ∈ AC ([ x , x ]) for each 0 ≤ x < x < ∞ , andso we can integrate to obtain 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) , (16) 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) (17)for each x ≤ x < ∞ , where µ ( x ) = R x kc ( y ) dy ∼ x / → x →
0. We also need b ( x ) , a ( x ) ∈ L ([0 , x ]), x < ∞ , for g , g to be defined. Integration by parts yields[ 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 (18)= lim x → lim x →∞ h g ( x ) h ( x ) − g ( x ) h ( x ) − g ( x ) h ( x ) + g ( x ) h ( x ) i (19)= g (0) h (0) − g (0) h (0) . (20)To obtain the last equality we used (16), (17), and the fact that g, h ∈ L ([0 , ∞ )).Thus any symmetric extension requires g (0) h (0) − g (0) h (0) = 0. By taking g = h onesees that this is possible iff one of g (0) and g (0) is a real multiple of the other. Therefore, 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 } , (21)gives a symmetric extension for any 0 ≤ θ < π , cf. [10]. Note that D θ satisfies both theconditions i ) and ii ). To see that condition iii ) is also satisfied, let h ∈ D θ , then[ g, h ] = 0 ⇐⇒ g (0) h (0) − g (0) h (0) = 0 ⇐⇒ h (0) h (0) = g (0) g (0) = − tan θ ∈ R . (22)This completes the proof for the subextremal case (cf. the proof of Thm.3.6 in [23]). IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 11
For the extremal case, we consider the change of variable (11) and consider the operator(13). Note that the above argument is valid if µ ( x ) = R x kc ( y ) dy → x → b ( x ) , a ( x ) ∈ L ([0 , x ]), x < ∞ .One can easily see that c ( x ) ∼ x − / as x → b ( x ) , a ( x ) ∼ x / as x →
0, and both are continuous in [0 , b ]. Hence, the proof appliessimilarly. (cid:3)
Remark 3.2.
We remark that the deficiency indices of e K k are (1 , , for [ g, h ] is the differ-ence of two positive rank-one bilinear forms. This already implies that an orbit of self-adjointextensions must exist. The proof of Thm.3.1 identifies these. Our next theorem identifies the essential spectrum of any self-adjoint extension of the Diracoperator acting on bi-spinor wave-functions supported inside the inner horizon of either thesubextremal and the extremal black-hole spacetime.
Theorem 3.3.
For each θ , one has σ ess ( e K k ; θ ) = R . To prepare the proof of Theorem 3.3, as in [23] we recall the following lemma from [10].
Lemma 3.4.
Let D : = " − ddxddx (23) 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 ) , (24) where the a ij are functions in L ([0 , b ]) for all < b < ∞ and a ij ( x ) → as x → ∞ . Then A is D θ compact.Proof of Theorem 3.3. We prove the theorem explicitly for the subextremal case. Yet notethat the extremal case follows verbatim after setting r − → r .We split the operator e K k in (12) as e K k = " − Zα s r − kc ( x ) − ddx kc ( x ) + ddx − Zα s r − + a ( x ) − h b ( x ) − Zα s r − i − a ( x ) − h b ( x ) − Zα s r − i (25)=: e K k + V. (26) Note that Theorem 3.1 is valid when a ( x ) = 0 and b ( x ) = Zα s r − . 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 , cf. (21). We define the following Weyl sequencefor e K k ; θ : let w = − Zα s r − − λ , with any λ ∈ R , then f n,λ ( x ) = 12 n xe − x n + ixw (cid:20) − i (cid:21) ; n ∈ N . (27)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 ∈ R is in the essential spectrum of e K k ; θ , and so σ ess ( e K k ; θ ) = R .Next we will show that V is e K k ; θ compact. This is done essentially verbatim to thepertinent part in the proof of Lemma 3.12 in [23]. We define ξ ( x ) = − R x kc ( y ) dy for0 ≤ x ≤ ξ ( x ) = ξ (1) for x >
1. Then the following matrix is bounded, S = " e − ξ ( x ) e ξ ( x ) . (28)Assume that k g ( n ) k ( L ( R + )) , k e K k ; θ g ( n ) k ( L ( R + )) , n ∈ N , 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 boundedand the latter one 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 3.4. Hence, V S − Sg ( n ) = V g ( n ) (29)has a convergent subsequence. This proves that V is D θ , and e K k,θ compact. (cid:3) We next show that the essential spectrum has no singular continuous part. As in [23], forthis we will need the following Theorem from [32, 33].
Theorem 3.5. (Weidmann) Let τ := " − ddxddx + P ( x ) + P ( x ) (30) 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 + . (31) Then every self-adjoint realization A of τ has purely absolutely continuous spectrum in ( −∞ , µ − ) ∪ ( µ + , ∞ ) . IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 13
Proposition 3.6.
In both the subextremal and the extremal case, R \ {− α s Zr − } ⊂ σ ac ( e K k ; θ ) .Proof. Recall that e K k,θ is in the form of τ , with P ( x ) = 0, and with P ( x ) = P a ( x ) for a = 0, where P a ( x ) = f ( r ( x )) − Zα S 1 r ( x ) h kr ( x ) − Zα S2 a π r ( x ) i f ( r ( x )) h kr ( x ) − Zα S2 a π r ( x ) i f ( r ( x )) − f ( r ( x )) − Zα S 1 r ( x ) ; (32)here, both f ( r ( x )) and r ( x ) are continuously differentiable and hence of bounded variation.Furthermore, lim x →∞ f ( r ( x )) = 0 , and lim x →∞ r ( x ) = r ∗ , (33)where r ∗ = r − in the subextremal, and r ∗ = r in the extremal case. This implieslim x →∞ P a ( x ) = " − α s Zr ∗ − α s Zr ∗ , ∀ a ≥ . (34)Hence, the spectrum of e K k ; θ is purely absolutely continuous on R \ {− α s Zr ∗ } . (cid:3) Corollary 3.7.
The singular continuous spectrum σ sc ( K k ; θ ) = ∅ in both the subextremal andthe extremal case.Proof. By Proposition 3.6 and Theorem 3.3 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 single point − α S Z/r ∗ , which is impossible. (cid:3) The results obtained so far show the absence of a discrete spectrum, but not the absence ofpoint spectrum. Obviously, any point spectrum would have to consist of a single eigenvalue, − α S Z/r ∗ , which could be infinitely degenerate. We next show that in the subextremal case, − α S Z/r ∗ is not an eigenvalue. Theorem 3.8.
In the subextremal case e K k ; θ has no eigenvalues. For the proof of Theorem 3.8 we will utilize the following Lemma of Cohen & Powers [10].
Lemma 3.9.
Let H be a Hilbert space. Let V t for t > a be a bounded linear operator on H so that V t f is continuous in t for each f ∈ H . Suppose f ( t ) solves the differential equation df ( t ) dt = V t f ( t ) , for t > a (35) where the derivative exists in the strong sense. Suppose R ∞ a k V t k dt = C < ∞ . Then the lim t →∞ f ( t ) exists, and if this limit is zero then f ( t ) = 0 for all t . Proof of Theorem 3.8.
We recall that by Proposition 3.6, λ = − Zα s r − is the only possible valuewhich may be an eigenvalue. Hence, it is enough to show that if ( e K k ; θ + Zα s r ∗ I ) g = 0 and g ∈ L , then g is identically zero.Now note that e K k ; θ + Zα s r − I = " − b ( x ) + Zα s r ∗ − ddxddx − b ( x ) + Zα s r ∗ + " a ( x ) kc ( x ) kc ( x ) − a ( x ) . (36)Recall that a ( x ) , c ( x ) → x → ∞ . More specifically, a ( x ) , c ( x ) ∼ e − κ x in the subextremalcase, whereas in the extremal case a ( x ) , c ( x ) ∼ x − , as x → ∞ .Let η ( x ) = R xA ( − b ( y ) + Zα s r ∗ ) dy for some A >
0. Recall that b ( y ) is continuous away fromzero and therefore, η ′ is defined. Next, note that ignoring the second matrix at r.h.s.(36),the so truncated eigenvalue problem is locally solved by g ± ( x ) = A ± e ± iη ( x ) γ ± , where γ ± =[1 , ∓ i ] T ; yet note that g ± ( x ) is not in L .The full eigenvalue problem ( e K k ; θ + Zα s r ∗ I ) g = 0, and for g ∈ L , can now be addressedwith the help of the method of variation of constants. Thus we make the ansatz g ( x ) = " u ( x ) e iη ( x ) + v ( x ) e − iη ( x ) − iu ( x ) e iη ( x ) + iv ( x ) e − iη ( x ) . (37)Inserting (37) into ( e K k ; θ + Zα s r ∗ ) g = 0 we obtain " iu ′ ( x ) e iη ( x ) − iv ′ ( x ) e − iη ( x ) u ′ ( x ) e iη ( x ) + v ′ ( x ) e − iη ( x ) = " u ( x ) e iη ( x ) [ a ( x ) − ikc ( x )] + v ( x ) e − iη ( x ) [ a ( x ) + ikc ( x )] u ( x ) e iη ( x ) [ ia ( x ) + kc ( x )] + v ( x ) e − iη ( x ) [ − ia ( x ) + kc ( x )] . (38)Written in terms of ddx ( u, v ) T this yields ddx " u ( x ) v ( x ) = " e − iη ( x ) [ ia ( x ) + kc ( x )] e iη ( x ) [ − ia ( x ) + kc ( x )] 0 u ( x ) v ( x ) . (39)Recall that in the subextremal case a ( x ) , c ( x ) ∼ e − κ x and therefore, by Lemma 3.9, weobtain u = 0 = v . This finishes the proof. (cid:3) Remark 3.10.
Lemma 3.9 can be applied only to the subextremal case, where we have r − r − ∼ e − κ x as x → ∞ , and e − κ x is integrable at ∞ . On the other hand, in the extremal case, r − r ∼ x − as x → ∞ and this does not satisfy the integrability condition in Lemma 3.9. As an immediate consequence of our Proposition 3.6, and Theorem 3.8, we have
Corollary 3.11.
In the subextremal case the essential spectrum, given by the whole real line,is purely absolutely continuous: σ ac ( K k ; θ ) = R . IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 15
Electron with anomalous magnetic moment.
We now address the Dirac Hamiltonian for an electron with anomalous magnetic momentin the static interior of a RWN black-hole spacetime.
Theorem 3.12.
In both the subextremal and the extremal case, the operator e K a k given by(12) is essentially self-adjoint iff a e πm e c ≥ √ G ~ /c .Proof. We will show that the limit point case (LPC) is verified both in the right neighborhoodof x = 0, and in the left neighborhood of x = ∞ iff a e πm e c ≥ √ G ~ /c , i.e. then there is atleast one non-square integrable solution to e K a k g = λg for each λ ∈ C , or equivalently for afixed λ , see [33, Theorem 5.6].We start with the left neighborhood of x = ∞ . As x → ∞ , the operator e K a k approaches K ∗ := " − Zα s r ∗ − ddxddx − Zα s r ∗ , (40)where again r ∗ = r − in the subextremal case and r ∗ = r in the extremal case. Clearly, g ± = ( e ± i Zαsr ∗ x , ∓ ie ± i Zαsr ∗ x ) T are solutions to K ∗ g = 0, and g ± is not square integrable at ∞ .Hence, the LPC is satisfied in the left neighborhood of x = ∞ .Next, we address the right neighborhood of x = 0 ( r = 0), and consider the solutions to h f ( r ) − Zα S r i g + h kf ( r ) r − Zα S a f ( r )4 πr − f ( r ) ddr i g = 0 , (41) h − f ( r ) − Zα S r i g + h kf ( r ) r − Zα S a f ( r )4 πr + f ( r ) ddr i g = 0 . (42)Recall that g = ( g , g ) T is square integrable in the right neighborhood of r = 0 with theinner product associated with (8) iff for each 0 < R < r − , Z R f ( r ) (cid:16) | g ( r ) | + | g ( r ) | (cid:17) dr < ∞ . (43)Therefore, we aim to find solutions to (41), (42) such that (43) does not hold. Note that as r → f ( r ) ∼ ar − , where a = ( r − r + ) in the subextremal case and a = r in the extremalcase, when r − = r + (= r ).Hence, around zero equations (41), (42) become g ′ + Zα S a πar g − ka g = O ( r ) , (44) g ′ − Zα S a πar g + ka g = O ( r ) . (45) The above equations imply that in a right neighborhood of r = 0 we have g ∼ r Zα S a πar and g ∼ r − Zα S a πar . Note that (43) implies that local square integrability holds for g and g if Z R r ± Zα S a πa +2 dr < ∞ . (46)Recalling the definition of r ± from (4), and r from (5) we see that a = G / m e Ze ~ c in boththe subextremal and extremal case, so that Z cancels out in the power of r . Therefore, inboth the subextremal and extremal case the LPC is satisfied iff − π α S a ~ cG / m e e + 2 ≥ −
1, whichtranslates into a e πm e c ≥ √ G ~ /c . (cid:3) Inserting numerical values for the physical and mathematical constants, we conclude that a ≥ . · − implies essential self-adjointness. Therefore we arrive at Corollary 3.13. e K a k is essentially self-adjoint if the empirical value of the electron’s anoma-lous magnetic moment is used, in which case a = 1 to three significant digits. In the rest of this section we characterize spec e K a k when a e πm e c > √ G ~ /c . Theorem 3.14.
The essential spectrum σ ess ( e K a k ) = R for both the subextremal and theextremal case.Proof. We define the operators e K a k ([0 , b ]) and e K a k ([ b, ∞ )) as the restriction of e K a k to L ([0 , b ])and L ([ b, ∞ ]) respectively. Then by Theorem 11.5 in [33], we have σ ess ( e K a k ) = σ ess ( e K a k ([0 , b ])) ∪ σ ess ( e K a k ([ b, ∞ ))) . (47)Instead of (27) we now use the following Weyl sequence, f n,λ ( x ) = 1 √ n e − ( x − b )2 n + iw (cid:20) − i (cid:21) ; n ∈ N . (48)Then ξ ( x ) = − R xb [ kc ( y ) − a d ( y )] dy for b ≤ x ≤ b + 1 in (28), and one can show that σ ess ( e K [ b, ∞ ) µ a ,k ) = R in a similar way as in the proof of Theorem 3.3.On the other hand, the operator e K a k (0 , b ]) can only have discrete spectrum. To see that, weuse Theorem 2 in [18]. In particular, since the limit point case holds, e K a k ([0 , b ]) has discretespectrum if also Z b | kc ( x ) − a d ( x ) | dx = ∞ . (49)Notice that d ( x ) ∼ x − as x →
0, which is not locally integrable around zero. See (13) forthe definitions of c ( x ) and d ( x ). (cid:3) IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 17
Proposition 3.15.
In both the subextremal and the extremal case, R \ {− α s Zr − } ⊂ σ ac ( e K a k ) and σ sc ( e K a k ) = ∅ .Proof. We first note that the fact that σ sc ( e K a k ) = ∅ follows from the claim on the absolutelycontinuous spectrum, see Corollary 3.7. Therefore, we only prove that R \{− α s Zr − } ⊂ σ ac ( e K a k ).Similarly to the proof of Proposition 3.6, we use Theorem 3.5. In particular, we now needto consider the operator P a ( x ) defined in (32), with a >
0. We already proved the limitproperty (34) for P a . And so, our proof of Proposition 3.6 in concert with Corollary 3.7 alsoproves Proposition 3.15. (cid:3) Theorem 3.16.
In the subextremal case e K a k has no eigenvalues.Proof. The proof follows similarly to the proof of Theorem 3.8. One needs to consider theoperator in (36) with kc ( x ) replaced by kc ( x ) − a d ( x ). Note that d ( x ) ∼ c ( x ) at ∞ , andhence the integrability condition in Lemma 3.9 is satisfied. This concludes that g has to beidentically zero, if g ∈ L and ( e K a k + Zα s r − I ) g = 0. (cid:3) Corollary 3.17.
In the subextremal case the continuous spectrum is purely absolutely con-tinuous and given by the whole real line, σ ac ( K a k ) = R . Summary and outlook
Hypothetical ‘hyper-heavy nuclei,’ which by definition obey
GM m e > Ze , also obey GM > Ze (because Zm e < Zm p ≤ M = A ( Z, N ) m p with A ( Z, N ) ≈ Z + N ), and thusare associated with the black hole sector of the Reissner–Weyl–Nordstr¨om (RWN) spacetime.This means that the number of neutrons N ≫ Z , as we have shown in section 2. In thispaper we have investigated the Dirac Hamiltonian with and without anomalous magneticmoment of the electron when the electron is assumed to reside in the static subregion of theinterior of an RWN black-hole spacetime of a hyper-heavy point nucleus.Using the partial wave decomposition the Dirac Hamiltonian becomes a direct sum ofso-called radial Dirac operators e K a k , where a = 0 amounts to an electron without, and a ≈ e K a k is essentially self-adjoint if and only if a ≥ a crit ≈ . · − , and has infinitely many self-adjoint extensions when a = 0. (We expectinfinitely many self-adjoint extensions for all 0 ≤ a < a crit , but we haven’t verified this.) Sowhen working with the empirical value of the electron’s anomalous magnetic moment, i.e. a = 1 to several significant digits, this Dirac operator is well-defined and generates a unitarydynamics for the electron on the static subregion inside the RWN black hole.We have characterized the spectrum of any self-adjoint extension in all subextremal caseswhere we showed they exist, and we found the essential spectrum is the whole real line,consisting of purely absolutely continuous spectrum. So there is no gap in the continuum,and therefore no discrete hyper-heavy hydrogenic ion spectrum in the subextremal blackhole sector of RWN, unlike the situation in the naked singularity sector. Worse, the absolutecontinuity result for the spectrum means the complete absence of point spectrum for an elec-tron in the static interior region of a subextremal RWN black-hole spacetime. An analogousresult was proved by Cohen and Powers [10] for an electron outside the event horizon of asubextremal RWN black hole, so therefore we can now conclude that there is no hyper-heavyhydrogenic ion point spectrum at all in the subextremal RWN black-hole sector.It still remains to discuss the Cohen-Powers setup for the extremal sector, i.e. electronoutside of the horizon — but in this paper we were only concerned with electrons in the staticpart of the interior region. It also remains to settle the issue of the point spectrum in theextremal black hole case, when the electron spinor wave function is supported inside the eventhorizon. We have shown that the only possible eigenvalue is − Zα S /r , where r is the arearadius at which the horizon is located, but it is not clear whether this value is an eigenvalue,and if so, whether it is simple, finitely degenerate, or even infinitely degenerate. The extremalRWN black hole sector is not generic in the RWN spacetime family, but the open questionsare technically challenging, and it is curious to contemplate that this exceptional black holesetting is so far the only one which has not been ruled out of permitting bound states.It also remains to be seen whether the presence of a horizon generically causes absence ofeigenvalues for the Dirac operator, as conjectured in [10] for electrons with wave functionssupported outside the event horizon, and which may now be conjectured to be true also forelectrons in the static interior of other subextremal black hole spacetimes. To prove sucha conjecture in all generality, if indeed true, is a challenging project. Yet there are severalfeasible generalizations of our study which are worthy of pursuit, and which could cementthe conjecture further or, possibly, disprove it.One further direction of inquiry could be an investigation of the Dirac operator for a testelectron in generalizations of the RWN black hole spacetime of a single point nucleus thatobey other electrostatic vacuum laws. The naked singularity sector of such spacetimes hasbeen described in [29], and generalized in appendix B of [23]. Such a study has the technical IRAC OPERATOR ON REISSNER–WEYL–NORDSTR ¨OM BLACK HOLE SPACETIMES 19 advantage that the spherical symmetry of the spacetime allows one to work with the partialwave decomposition of the Dirac Hamiltonian, as done in the present paper.For the naked singularity sector such a study has recently been carried out in [1] forsingularities with zero bare mass, and in [23] for spacetimes with naked singularities of strictlynegative bare mass, with some surprising results. The perhaps most surprising result of [23](to its authors at least) is that the Dirac operator for an electron in the naked singularitysector of the Hoffmann spacetime [20, 29] (Born [5] or Born–Infeld vacuum law) of a pointnucleus is not essentially self-adjoint, with or without anomalous magnetic moment, unlessthe bare mass of the singularity vanishes [1]. A vanishing bare mass is not typical, though,and so the upshot is that in the naked singularity sector of the Hoffmann spacetime familythe Dirac Hamiltonian of a test electron is typically not well-defined even if the anomalousmagnetic moment is taken into account.We suspect that the same conclusion will hold for the Dirac operator of a test electron inthe static part of the interior region of a Hoffmann black hole spacetime.Another generalization of the present work is to study the Dirac equation for a test electronin the multi-black hole spacetime family of Hartle and Hawking [19], obtained by analyticalcompletion of the asymptotically flat, static, Majumdar–Papapetrou metrics. These are veryspecial spacetimes, but there are not many explicit representations of spacetimes with severalblack holes in them. Each black hole of the Hartle–Hawking family obeys the RWN extremalcondition GM = Q . 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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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