aa r X i v : . [ g r- q c ] J a n A ﬁnite model of an elementary charge
Suvikranth Gera ∗ and Sandipan Sengupta † Department of Physics, Indian Institute ofTechnology Kharagpur, Kharagpur-721302, India
We set up a model of an elementary electric charge where the noninvertible metric phase of ﬁrstorder gravity supercedes the standard point singularity. A topological interpretation of the electriccharge is provided in terms of an index deﬁned for the degenerate spacetime solution, being closelyrelated to the Euler characteristic. The gravitational equations of motion at this phase are foundto be equivalent to the laws of electrostatics. The associated ﬁeld energy is ﬁnite and the geometrysourcing the charge is regular. ∗ [email protected] † [email protected], [email protected] . INTRODUCTION A point particle model for electric charge is inherently singular, and is unsatisfactory forseveral reasons. In Einstein’s general relativity, the point charge leads to divergences bothin the ﬁeld energy and the geometry. Attempts towards alternative constructions that arefree of the inﬁnity in the self-energy of the electron have a long history, highlighted by theworks of Abraham, Lorentz, Lees, Dirac, Born-Infeld[1–4] and so on.The present work is concerned with developing a ﬁnite model of an elementary charge.We adopt a geometric perspective, rooted ﬁrmly within classical ﬁrst order gravity theorywhich can exhibit invertible as well as noninvertible metric phases.Firstly, we show that an (apparent) electric charge can be described by a non-invertiblemetric phase of ﬁrst order gravity. The solution to the ﬁeld equations is worked out in detail.The geometry outside this phase is locally equivalent to the Reissner-Nordstrom metric uptoa coordinate transformation, and hence corresponds to the Einstein phase.Within this framework, the gravitational equations of motion provide a natural deﬁnitionfor an emergent electric ﬁeld at the zero-determinant phase. These equations are equivalentto the laws of electrostatics. The associated three-geometry is shown to be regular. We alsoanalyze the ﬁeld energy in the Hamiltonian form for the full spacetime. In contrast to thepoint charge model, it is ﬁnite.Finally, we unravel a topological interpretation of the electric charge in the ﬁrst orderformulation of gravity. It is shown to have a one to one correspondence with a topologicalinvariant deﬁned for degenerate spacetime solutions. This index is integer valued and isclosely related to the Euler characteristic, the latter being well-deﬁned only for a non-invertible geometry.Let us now present our main ideas and results in some detail.
II. A CHARGED SPACETIME SOLUTION IN EINSTEIN GRAVITY
In Einsteinian gravity theory, the presence of an electric charge curves the spacetime.For spherical symmetry, the relevant geometry is given by the electrically charged Reissner-Nordstrom solution. To begin with, let us consider a closely related metric outside a radius2 > Q : ds = − (cid:20) − QR ( u ) (cid:21) dt + R ′ ( u ) du h − QR ( u ) i + R ( u )( dθ + sin θdφ ) (1)where Q is a constant and R ( u ) is any smooth, monotonically increasing function satisfyingthe properties below: R ( u ) → Q, R ( n ) ( u ) → u → u ( R → Q ) ,R ( n ) ( u ) being the n-th derivative for any arbitrary integer n .For the special parametrization R ( u ) = u , this geometry formally reduces to the extremalReissner-Nordstrom spacetime at R > Q , Q being the electric charge. However, such acoordinate transformation does not satisfy the properties above. In other words, the chargedspacetime (1) is locally equivalent to the Reissner-Nordstrom solution everywhere except at u = u ( R = Q ).The curvature 2-forms ﬁelds read: R = QR (cid:18) − QR (cid:19) R ′ dt ∧ du, R = − QR (cid:18) − QR (cid:19) dt ∧ dθ,R = − QR (cid:18) − QR (cid:19) sin θdt ∧ dφ, R = − QR R ′ du ∧ dθ,R = QR (cid:18) − QR (cid:19) sin θdθ ∧ dφ, R = − QR R ′ sin θdφ ∧ du (2)The only component that has a non-trivial limit as u → u is R → sin θdθ ∧ dφ . In termsof the radial coordinate R , the Electric ﬁeld is manifestly Coulombic: E R ≡ F tR = QR (3)The determinant of the metric above goes to zero smoothly as the surface u = u isapproached. Thus, the natural continuation that could be deﬁned beyond this surface (at u < u ) must involve a noninvertible metric phase [6–10]. In the next section, we ﬁnd theappropriate noninvertible metric solution of the ﬁrst order equations of motion in gravitytheory which allows such a (smooth) continuation.This should be contrasted to the case where the full spacetime exhibits an Einstein phase( det g µν = 0 ) only. Since the associated spacetime metric there does not degenerate at R = Q , it could only be continued to an invertible phase of ﬁrst order gravity. Thus,3he metric throughout must correspond to the (electrically charged) Reissner-Nordstromsolution of the Einstein-Maxwell theory. In other words, an elementary charge is necessarilyassociated with a point (curvature) singularity, or equivalently, a divergent (electrostatic)self-energy, at least within the standard formulation of Einstein gravity. III. THE NON-INVERTIBLE PHASE
This phase in vacuum is deﬁned by any degenerate tetrad solution of the ﬁrst-orderLagrangian formulation : L = ǫ µναβ ǫ IJKL e Iµ e Jν R KLαβ ( ω ) (4)The basic ﬁeld variables are the tetrad and connection, and the indices µ and I label thespacetime and local Minkowski coordinates, respectively. Given metric (1), a natural choicefor its continuation at R ≤ Q ( u ≤ u ) to this phase is the following: ds = 0 + F ( u ) du + R ( u )( dθ + sin θdφ ) (5)The internal metric is η IJ = diag ( − , , , . In general, degenerate space-time solutions ofﬁrst order gravity could exhibit torsion even in the absence of matter ﬁelds (e.g. fermions).For our purposes here, it is suﬃcient to consider the case of vanishing torsion: T Iµν ≡ D [ µ ( ω ) e Iν ] = 0 .Note that even though the tetrad ﬁelds are not invertible, the nontrivial part of the metricabove may be used to deﬁne a set of triad ﬁelds ˆ e ia ≡ e ia ( i ≡ [1 , , , a ≡ [ u, θ, φ ] ) alongwith their inverses ˆ e ai ( ˆ e ai ˆ e ja = δ ji , ˆ e bk ˆ e ka = δ ba ).The vacuum theory deﬁned by (4) corresponds to the following set of ﬁeld equations: ǫ µναβ ǫ IJKL e Iµ T Jνα = 0 , ǫ µναβ ǫ IJKL e Iν R JKαβ = 0 (6)For zero torsion, the ﬁrst set above is satisﬁed trivially. It is straightforward to see that thecorresponding general solution for the connection ﬁelds is given by: ω = (cid:18) − R ′ F (cid:19) dθ, ω = − cos θdφ, ω = R ′ F sin θdφ,ω i = E ik e k ≡ E i (7)where E kl = E lk is a symmetric matrix and we have deﬁned ω ia ≡ E ia in the last equality.Among the second set of ﬁeld equations in (6), all components are satisﬁed identically except4 = t, I = 0 , which reads: ǫ abc ǫ ijk e ia R jkbc = 0 = 2 RF (cid:18) R ′ F (cid:19) ′ + (cid:18) R ′ F (cid:19) −
12 [ E ii E kk − E ik E ki ] R − (8)Rather than working with the most general solution for E ia in eq.(7), we shall use aparticular solution in order to be explicit: E ia = λ ˆ e ia (9)where λ is a constant. Insertion of this into eq.(8) implies: RF (cid:18) R ′ F (cid:19) ′ + (cid:18) R ′ F (cid:19) − λ R − This has the following solution: F = R ′ q − QR + λ R With this, the solution for the four-metric ﬁnally becomes: ds = 0 + R ′ ( u ) du λ R ( u ) − QR ( u ) + R ( u )( dθ + sin θdφ ) . (10) IV. ELECTROSTATICS FROM DEGENERATE GEOMETRY
The nondegenerate part of the four-metric solution (10) naturally deﬁnes an emergent(spatial) three-geometry. This three-metric exhibits an R ⊗ S topology, having an innerboundary u = u ∗ ( R = R ∗ ) deﬁned by the relation − QR ∗ + R ∗ Q = 0 , where the geometryis nevertheless regular. Within such an eﬀective description, the geometric (gravitational)ﬁelds are completely given by the associated triad ˆ e ia and the (torsionless) spatial connectioncomponents ˆ ω ija (ˆ e ) ≡ ω ija . The remaining components of the spacetime connection ω i , whichare not determined by the emergent triad ﬁelds ˆ e ia , are not part of this geometry. Hence,these should be interpreted as matter ﬁelds. Clearly, these must encode the emergent electricﬁeld sourcing the apparent electric charge Q as measured at a large radial distance R > Q .To this end, we consider the equation of motion involving ω i , given by the µ = t, I = i component of the second set in eq.(6): ǫ abc ǫ ijk e ja R kbc = 0 = ˆ D a (ˆ ω ) (cid:2) ǫ abc ǫ ijk e jb E kc (cid:3) (11)5here the three-covariant derivative is deﬁned with respect to the spatial connection ˆ ω ija (ˆ e ) .Let us now project this equation along a three-vector n i in the internal space, leading to: ∂ a (cid:2) ˆ e ( E ik ˆ e ak − E ll ˆ e ai ) n i (cid:3) = ˆ e (cid:2) E ik ˆ e ak − E ll ˆ e ai (cid:3) ˆ D a (ˆ ω ) n i (12)This precisely is equivalent to the Gauss’ law ∂ a [ˆ eE a ] = ˆ eρ in electrostatics, implying thefollowing deﬁnitions for the emergent electric ﬁeld and the charge density, respectively: E a = 12 ( E ik ˆ e ak − E ll ˆ e ai ) n i = 12 n [ a ˆ e b ] k ω kb ,ρ = 12 ( E ik ˆ e ak − E ll ˆ e ai ) ˆ D a (ˆ ω ) n i (13)where we have deﬁned: n a ≡ n i ˆ e ai . Note that the expression for the emergent U(1) electricﬁeld is gauge-invariant. Also, it satisﬁes ǫ abc ∂ a E b = 0 trivially, as it should.Let us now apply the general construction above to the special case where the ﬁeld isnon-trivial only along the radial direction. This may be achieved by choosing the vector n i to be normal to the internal two-sphere: n i ≡ (1 , , . Using eq.(9) and transforming tothe radial coordinate R inside the phase boundary (at u < u ), the resulting emergent ﬁeldthen reads: E R = λ r − QR + λ R , E θ = 0 = E φ (14)Continuity of the ﬁeld across the phase boundary u = u ﬁxes the dimensionful constant as λ = Q . The static ﬁeld E R ( R ) at R ≤ Q is not Coulombic, even though the ﬁeld outsidethe phase (outer) boundary is. It vanishes at the inner boundary R = R ∗ . This is a crucialdiﬀerence compared to the point charge model.In the limit u → u as one approaches the phase boundary, the electric ﬂux through thetwo sphere reads: π Z S d x E a n a = 14 πQ Z S dθdφ sin θR = Q, (15)which is the same as the apparent electric charge deﬁning the invertible phase at R > Q . V. FINITENESS OF FIELD-ENERGY
Let us consider the Hamiltonian form of the electrostatic self-energy for an electricallycharged conﬁguration in a ﬁxed curved spacetime. For the invertible phase, the relevant6art of the Lagrangian density is the Maxwell term: L m = − eF αβ F αβ . We shall use thestandard reparametrization of the metric (1) in terms of lapse ( N ) and shift ( N a ) variables,the latter being trivial in this case: e It = N M I , e Ia M I = 0 , q ab = e Ia e bI , e = N √ q (16)Using the expression for the canonical momenta π a = − eF ta = N √ qE a in terms of theelectric ﬁeld E a , the Hamiltonian for a vanishing magnetic ﬁeld ( F ab = 0 ) reads: H = N √ q q ab E a E b (17)In obtaining this expression, the Gauss’ law has already been implemented as a constraint.The electrostatic energy associated with the invertible phase at Q < R < ∞ is then equalto the Hamiltonian H upto a normalization: H = Z d x H = 12 Z dR dθ dφ sin θR ( E R ) = 2 π (cid:20) Q R (cid:21) Q ∞ = 2 πQ (18)The non-invertible phase at R ≤ Q , on the other hand, has no electrostatic energy in thesense above. This fact is reﬂected by the fact that the Hamiltonian density (17) vanishes asthe lapse N goes to zero (smoothly) with q ab and E a being ﬁnite, as is the case here. Thus,the degenerate phase with a vanishing lapse does not contribute to the self-energy in thislimiting sense.To emphasize, our geometric model of an electric charge exhibits a ﬁnite self-energy atthe classical level. This is in contrast to standard electrostatics where an inﬁnite amount ofenergy is needed to assemble a ﬁnite charge within a point. This divergence is precisely whatgets reﬂected in the Hamiltonian evaluated for a (extremal) Reissner-Nordstrom geometrydeﬁning the whole spacetime < R < ∞ : H = 2 π h Q R i ∞ → ∞ . VI. REGULARITY OF GEOMETRY
At this stage a relevant question is, whether the emergent three-geometry above couldbe singular, even though the ﬁeld-energy itself is manifestly not. An explicit evaluation ofall the three-curvature scalars, however, shows that the geometry is in fact regular. For7nstance, let us display the scalars upto quadratic order at R ≤ Q explicitly: R abab = 1 R − R √ σF (cid:18) R ′ √ σF (cid:19) ′ − (cid:18) R ′ √ σF (cid:19) + 3 λ R + 1 ! = 0 , R ab R ab = 3 Q R . Finally, since torsion is trivial, the corresponding three-scalars such as those below vanish: ǫ abc T abc = 0 = T abc T abc . Hence, there is no curvature or torsion singularity in the three-geometry.
VII. TOPOLOGICAL INTERPRETATION OF ELECTRIC CHARGE
For vanishing spatial contortion K ija at the degenerate core, the parity odd topologicalinvariants, given by Nieh-Yan and Pontryagin numbers, are both trivial. The only othergravitational invariant is the Euler index (parity even), which is not well-deﬁned for a de-generate spacetime.Here we deﬁne a topological index for the degenerate geometry. This is necessarily non-trivial for a nonvanishing electric charge at the Einstein (Reissner-Nordstrom) phase.From the ﬁeld-strength components associated with the noninvertible phase, the Eulerdensity is found to vanish (since R i = λD ( ω ) e i = 0 ): I E = 132 π ǫ IJKL R IJ ∧ R KL = 0 . (19)This, however, must be considered along with the possible boundary corrections, for whicha suitable prescription is required. For the degenerate solution discussed earlier, g tu = 0 at R ≤ Q and the time coordinate t ceases to evolve. This implies that we could consider aspacetime which allows a smooth limit g tt → and is associated with a trivial (bulk) Eulerdensity as above. It is then possible to deﬁne the boundary correction to Euler invariantbased on this spacetime, before taking the degenerate limit g tt → at the end. The resultinggeometric (topological) invariant, if nontrivial, could be seen as a genuine characteristic ofthe degenerate spacetime.To this end, we deﬁne the following spacetime metric which reduces to the degenerategeometry (10) as ǫ ( t → t ∗ ) → for some t ∗ : ds = − ǫ ( t ) dt + dR − QR + R Q + R ( dθ + sin θdφ ) ¯ ω ij = ω ij , ¯ ω i = 0 . where ω ij are given by eq.(7). Note that the associated Euler density vanishes exactly as in(19), since ¯ R i (¯ ω ) = 0 .For evaluating the contribution of the t = t ∗ boundary to the Euler number, a relevantvariable is θ IJ = ¯ ω IJ − ω IJ : θ ij = 0 , θ i = λe i Using the appropriate normalization associated with the radial direction, the boundarycorrection (in analogy to the Euler boundary correction) may be deﬁned as: χ B = − V R Z ∂V ǫ IJKL (cid:20) θ IJ ∧ ¯ R KL − θ IJ ∧ θ KM ∧ θ ML (cid:21) = 2 Q V R Z π dφ Z π dθsinθ Z QR ∗ dR R q − QR + R Q = 2 (20)where V R = πQ R R dR r − QR + R Q is the (dimensionless) proper volume of the three-manifold inquestion. Importantly, this result is independent of ǫ , which may now be taken to be zero.Finally, the sum of the volume and boundary contributions to χ leads to: χ = χ V + χ B = 0 + 2 = 2 . (21)It is interesting to note that this index is the same as the Euler characteristic of thecompact even-dimensional subspace without boundary, which happens to be a two-spherein this special case.Importantly, the emergent electric ﬁeld vanishes for λ = 0 , which implies a trivial χ -number. Thus, a nonvanishing (apparent) charge Q at the Einsteinian phase ( R > Q )essentially has a topological origin in the degenerate phase ( R ≤ Q ). VIII. CONCLUDING REMARKS
We have set up an alternative to the point model of an elementary electric charge. Such aconﬁguration is described by a noninvertible solution of ﬁrst order ﬁeld equations in gravitytheory. The emergent electrostatic energy as well as the geometry is ﬁnite, superceding thegeneric divergences of a pointlike model. 9e demonstrate that a nontrivial electric charge in the Einsteinian phase is in fact equiva-lent to a nonvanishing value for a geometric invariant χ analogous to the Euler index. Thesetwo phases together, alongwith the associated solutions to the ﬁrst order ﬁeld equations,deﬁne the full spacetime which is smooth everywhere. Thus, the electric charge acquires atopological meaning in gravity theory. This conﬁguration has vanishing torsion and does notinvolve any magnetic charge or current [5, 12]. It could be interesting to investigate a pos-sible realization of dyons or a (topological) quantization of charge within this formulation,something that is beyond the scope of the present work.It should be emphasized that the framework presented here is diﬀerent from other in-stances where charge could be an artefact of multiply connected geometries such as geonsor wormholes [13–15]. In general, these hardly have much to do with a degenerate tetradphase, and mostly thrive on nontrivial quantum physics.To conclude, these new features mentioned above, alongwith the manifest role of degen-erate spacetime solutions as a possible regulator of divergences, are intriguing enough asthey are. It seems to be an open question whether a generic connection between ﬁrst ordergravity and elementary particles could be envisaged along these lines. IX. ACKNOWLEDGMENTS:
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