Dirac s Electron via General Relativity

Transcription

1 PACS : 3.5.De; 14.6.C University of Michigan Ann Arbor, MI , USA sblinder@umich.edu Contents 1. Introduction 6 2. Electron without Spin 7 3. Kerr-Newman Geometry 8 4. Modified Kerr-Newman 8 5. Conclusion 1 Abstract The Kerr-Newman solution to the Einstein-Maxwell equations in General Relativity describes the behavior of a charged, spinning black hole. We show that an appropriate modification of this result can be interpreted as a classical model for the electron as a point charge with the correct rest mass, angular momentum and magnetic moment. The Coulomb singularity causes a warping of spacetime around the point charge and a resulting polarization of the vacuum analogous to what happens, according to Hawking, in the vicinity of a black hole. The rest-energy and angular momentum of the electron can be attributed to its associated electromagnetic field. This provides a classical model mirroring essential features of Dirac s relativistic quantum theory of the electron, including the g-factor of Introduction This issue of Electromagnetic Phenomena commemorates the 75th anniversary of P.A.M. Dirac s 1928 publication of the relativistic quantum theory of charged elementary particles, which accounted for the spin and fine structure of the electron, as well as prefiguring the existence of antimatter. From the viewpoint of Dirac s theory, as well as its later elaboration into quantum electrodynamics, the electron is an elementary point charge. This is, in fact, borne out by recent high-energy scattering experiments, which establishes an upper limit on the electron s radius of the order of 1 16 cm. In quantum electrodynamics, the infinite Coulomb self-energy of a point charge, as well as infinities from interactions with the transverse radiation field, can be handled by gauge-invariant renormalization techniques. In this paper we consider a classical analog of Dirac s theory of the electron. Apart from its intrinsic academic interest, a classical model might represent the limit of the corresponding quantum theory. Since it is by no means settled that the current formalism of quantum electrodynamics is the final theory of the electron, it is worthwhile to explore the classical limit that some successor theory might also exhibit. In a recent paper [1], we have shown that a modification of the Kerr-Newman solution to the Einstein-Maxwell equations in general relativity can be interpreted as a classical model for the electron with spin. According to our model, the electron s selfenergy remains finite despite the Coulomb singularity. This can be attributed to deformation of the electromagnetic field in the ultramicroscopic vicinity of the singularity, analogous to the warping of spacetime around a black hole [2]. Our solution also implies the existence of an imponderable fluid concentrated within a radius of r 1 13 cm around the electron. We conjecture that this is a classical simulation of vacuum polarization, which produces a tenuous cloud of virtual particles and antiparticles 6

2 flickering in and out of existence around the electron. 2. Electron without Spin As a preliminary étude in the development of our model, we consider a hypothetical electron with mass m and charge e without spin or magnetic moment. This is described in General Relativity by a static, spherically-symmetrical metric of the form ds 2 = f(r)dt 2 1 f(r) dr2 r 2 dθ 2 r 2 sin 2 θ dφ 2. (1) (We have taken the Minkowski signature {+1, 1, 1, 1}, preferred by quantum theorists.) Reissner [3] and Nordstrøm [4] considered a generalization of the Schwarzschild metric with f RN (r) = 1 2M r + Q2 r 2 (2) where M and Q are mass and charge in geometrized units: M = Gm/c 2, Q = Ge/c 2. (3) Later we will also need the corresponding relations for spin angular momentum S = Gs/c 3 and for magnetic moment M = QS/M = Gµ/c 2. The Einstein-Maxwell equations [5] R λµ 1 2 R g λµ = 8πG c 4 T λµ (4) for the metric of Eqs (1) and (2) can be solved exactly to give the stress-energy tensor T = T 1 1 = T 2 2 = T 3 3 = e2 8πr 4. (5) This is consistent with a point charge since T λ µ can be equated to the Maxwell tensor for an electromagnetic field, such that 4πT λ µ = F λν F µν gλ µf νσ F νσ. (6) This implies that the only non-zero field tensor components are F 1 = F 1 = E r = e/r 2. (7) The electromagnetic self-energy for the Reissner- Nordstrøm solution, given by W = T 4πr 2 dr (8) remains divergent, retaining the Coulomb singularity of euclidean electrodynamics. For the case Q > M, which obtains here, g in the metric (1) has no real roots, indicating the absence of a horizon. The electron is thus not a conventional black hole, despite the singularity at r =. We can obtain a modified solution with finite electromagnetic self energy by reverse engineering on the Reissner-Nordstrøm metric, replacing the function (2) by f(r) = 1 2M r exp( Q2 /2Mr). (9) This is actually a very small change, differing from the original form by less than 1 percent down to the Planck length ( G/c cm). With removal of the singularity at r =, the metric now becomes Lorentz flat as r (as well as r ). Note that Q 2 /2Mr = e 2 /2mc 2 r = r /2r (1) where r = e 2 /mc cm, the Thomson or classical electron radius. The modified metric in the Einstein-Maxwell equations can be solved for a stressenergy tensor with the nonvanishing components: T = T 1 1 = e2 e r/2r 8πr 4, T 2 2 = T 3 3 = e2 e r/2r 8πr 4 ( 1 r 4r ). The electromagnetic energy is now calculated to be W = T 4πr 2 dr = (11) e 2 e r/2r 8πr 4 4πr 2 dr = e2 = mc 2. (12) r We obtain, as advertised, a finite electron self-energy. The parameters have in fact been adjusted so that the electron rest mass is entirely electromagnetic in origin, the result originally sought by Lorentz and Abraham a century ago [6]. We should remark on the similarity our metric containing the exponential function (9) to the Yilmaz- Rosen metric, [7] in which f(r) = e 2M/r = e 2Gm/c2r. (13) This was proposed as an alternative to the original Schwarzschild metric, in order to give the gravitational self-energy of a point mass a finite value, mc 2 in fact. As we deduced in an earlier paper, [8] the electromagnetic self-energy (12) could also be obtained by assuming an electric permittivity in euclidean space given by ( ) e 2 ɛ(r) = exp 2mc 2 r ( r ) = exp. (14) 2 r "Electromagnetic Phenomena", V.3, 1 (9), 23 7

3 The net charge density associated with the electron the original point charge plus the polarization density is then given by ρ(r) = 1 4π E = er 8πr 4 e r/2 r. (15) This is a charge distribution highly localized within a radius of r. Although the apparent charge appears to approch zero as r decreases, the effective fine structure constant e 2 ɛµ/ c actually increases, in agreement with quantum electrodynamics. The detailed behavior of this running electromagnetic coupling constant are, of course, outside the scope of the present computation. The trace of Tµ λ from Eq (11) is not equal to zero, which indicates the presence of a source of stressenergy in addition to the electromagnetic field. The deviation can be neatly represented as a contribution from a perfect fluid, whereby {T λ µ } = with ϱ =, p r = but ϱ p θ = p φ = Q4 e Q2 /2Mr 32πMr 5 p r p θ p φ (16) = e2 r e r/2r 32πr 5. (17) The fluid is evidently imponderable (ϱ = ) with zero radial pressure (p r = ) but a spherically-symmetrical tangential pressure p θ = p φ, much like the surface tension of a bubble. It can be conjectured that this is a simulation for the cloud of virtual particles produced by vacuum polarization. An appropriate designation for the nonelectromagnetic contributions to Tµ λ might be Hawking stresses, in analogy with the Faraday (or Maxwell) stresses which are sometimes ascribed to the electromagnetic field. 3. Kerr-Newman Geometry We extend our model to include electron spin by drawing from the theory of rotating black holes. Kerr [9] first solved Einstein s equations for a black hole with angular momentum. Newman [1] generalized this result to include electric charge. The metric in Kerr-Newman geometry, in coordinates introduced by Boyer and Lindquist, [11] can be written ds 2 = ρ 2 [ dt a sin 2 θdφ ] 2, sin2 θ ρ 2 [ (r 2 + a 2 )dφ adt ] 2 ρ 2 dr2 ρ 2 dθ 2 (18) where ρ 2 r 2 + a 2 cos 2 θ (19) and a 2 + r 2 f(r). (2) The element of 3-volume, which we will require in computations, is given by d 3 x = (r 2 + a 2 cos 2 θ) sin θdrdθdφ (21) independent of f(r). The parameter a represents the angular momentum per unit mass a S/M (22) and is equal to the radius of the Kerr-Newman ring singularity for a spinning black hole. In the present context, S can be conjectured to represent electron spin, so that a becomes equal to half the Compton radius: a = λ 2 = 2mc. (23) In the Kerr-Newman metric, f(r) has the form given by Eq (2) and KN = a 2 + r 2 2Mr + Q 2. (24) Solution of the Einstein-Maxwell equations for Kerr- Newman geometry results in a stress-energy tensor with the following nonvanishing elements (T λ µ ) KN, which we abbreviate as τ λ µ : τ = τ 3 3 = e2 (r 2 + a 2 + a 2 sin 2 θ) 8π (r 2 + a 2 cos 2 θ) 3, τ 1 1 = τ 2 2 = τ 3 = e 2 8π (r 2 + a 2 cos 2 θ) 2, ae 2 4π(r 2 + a 2 cos 2 θ) 3, τ 3 = ae2 ( a 2 + r 2) sin 2 θ 4π(r 2 + a 2 cos 2 θ) 3. The electromagnetic self energy is given by W = τ ρ 2 sin θdrdθdφ (25) and is again divergent, just as in the Reissner- Nordstrøm case. 4. Modified Kerr-Newman To obtain a finite self-energy, we modify the metric as was done in Section 2, with f(r) given by Eq. (9), so that = a 2 + r 2 2Mre Q2 /2Mr (26) in the metric (18). The solution of the Einstein- Maxwell equations (4) requires a more lengthy computation. The resulting stress-energy tensor, expressed in a compact form by incorporating the 8 "Электромагнитные Явления", Т.3, 1 (9), 23 г.

4 original Kerr-Newman analogs τ λ µ, has the following nonzero elements: T = (τ a 2 sin 2 θσ)e r/2r, T1 1 = τ1 1 e r/2r, T2 2 = [ τ2 2 + (r 2 + a 2 cos 2 θ)σ ] e r/2r, T3 3 = [ τ3 3 + (a 2 + r 2 )σ ] e r/2r, T 3 = (τ 3 aσ)e r/2r, T3 = [ τ3 + a(a 2 + r 2 ) sin 2 θσ ] e r/2r (27) where σ r e 2 32πr 3 (r 2 + a 2 cos 2 θ) 2. (28) The electromagnetic energy for modified Kerr- Newman geometry is given by the integral W = T (r 2 + a 2 cos 2 θ) sin θdrdθdφ = mc 2 (29) again in accord with the Lorentz-Abraham model of the electron as a purely electromagnetic object. The angular momentum density associated with an electromagnetic field is defined by S = 1 4πc r (E H) = 1 c T 3. (3) This is symmetrical about the z-axis. Integration gives what we interpret as the spin angular momentum s z = 1 T3 (r 2 + a 2 cos 2 θ) sin θdrdθdφ = c 2. (31) whose value was predetermined by the value of a from (23). This angular momentum resides in the electromagnetic field, which can reconcile the seeming paradox of a dimensionless particle exhibiting spin angular momentum. In this way one can circumvent the invention of distributions of orbiting charge and mass cunningly contrived to reproduce the observed electron spin and magnetic moment. In fact, one might conceptualize spin as a vortex in the polarized vacuum surrounding the electron. The nonvanishing trace of the stress-energy tensor is again explained by a contribution from Hawking stresses [cf. Eqs (16),(17)] p θ = p φ = e 2 r e r/2r 32πr 3 (r 2 + a 2 cos 2 θ) 2. (32) These are still tangential, but now have a dependence on the angle θ. The electric and magnetic fields are determined by the elements of Tµ λ in (28). Their approximate forms for large r are given by E r e r 2 er 2r 3, B r ea cos θ r 3, B θ E θ 2ea2 cos θ sin θ r 4, ea sin θ r 3. (33) Fig. 1. Contour plot of ρ(r, θ) showing the surface of constant value 1 3. The vertical axis is expanded by a factor of 1. Distance scale in units of r = cm. At distances r r and a, the electric field approaches that of a point charge. Higher-order contributions to E reveal an axially-symmetric oblate spheroidal distribution of charge. The asymptotic dependence of the magnetic field is precisely that of a point magnetic dipole of magnitude µ = e a = e /2mc. This is evidently produced by circulating electric charge of angular momentum /2. Remarkably, this corresponds to an electron-spin g- factor of 2, its value in Dirac s relativistic theory of the electron, as was first noted by Carter. [12] Recent analyses by Newman [13] and by Lynden- Bell [14] have shown that the the source of the Kerr- Newman metric can be simulated by a point charge e displaced from the origin to r = ia in complex Minkowski space. This provides a simple classical geometric representation for the Dirac values for the spin angular momentum and magnetic moment. However, the physical significance of this equivalence remains to be explored. The net charge density for modified Kerr-Newman geometry is strongly localized within r < r and near the medial plane θ = π/2, as shown in Figure 1. This is very close to what would be obtained by projecting the spin-zero charge distribution (15) onto the plane θ = π/2. One might imagine a hypothetical scenario in which the sphericallysymmetrical vacuum-polarization charge distribution is set into rapid rotation, causing parallel current elements to attract and thereby becoming compressed into a Kerr-Newman disk. What is remarkable is that the rotating charge distribution is localized within a disk of approximate radius r (around 1 13 cm). All previous classical models for the spinning electron have assumed a rotating disk of the order of the Compton wavelength λ, two orders of magnitude larger. A nice by-product of this compact structure is that one need not worry about complications from electron spin in refinements of the Lorentz-Dirac equation for radiation reaction. [8] The sources corresponding to our modified [using "Electromagnetic Phenomena", V.3, 1 (9), 23 9

5 Eq (9)] Reissner-Nordstrøm and Kerr-Newman metrics are localized but finite charge distributions. We conjecture that these sources can be interpreted as the sum of a point charge plus a charge density due to vacuum polarization. In the K-N case, the polarized charge is also rotating. The properties of the electron s weak isospin partner the neutrino can also be rationalized by the model. Clearly, as the charge of the particle approaches zero, so does the rest mass. If one imagines e and m both approaching zero, while maintaining a constant value of e 2 /mc 2, then the neutrino s angular momentum of /2 (with zero magnetic moment) can likewise be accomodated. 5. Conclusion Of course, the real physical electron must ultimately be described by quantum mechanics or quantum field theory. Still, a fully consistent classical model can provide a useful starting point. [15] And classical results do (usually) represent limits in quantum theory. Although the infinities associated with transverse radiation fields do remain, we have succeeded in eliminating those of classical origin for a point charge. Online version: au:+blinder//1//past//1 [9] R.P. Kerr // Phys. Rev. Lett V. 11. P [1] E.T. Newman et al // J. Math. Phys V. 6. P [11] R.H. Boyer and R. W. Lindquist // J. Math. Phys V. 8. P [12] B. Carter // Phys. Rev V P [13] E.T. Newman // Phys. Rev. D. 22. V. 65. P [14] D. Lynden-Bell // arxiv:astro-ph/2764 v1 2 Jul 22 [15] The connection between classical and quantum theories of the electron is discussed by P. Pearle, Classical electron models, in Electromagnetism: Paths to Research, ed D. Teplitz (Plenum, New York, 1982) P References [1] // Repts. Math. Phys. 21. V. 47. P Online version: 1/au:+Blinder//1//past//1 [2] S. Hawking // Commun. Math. Phys V. 43. P [3] H. Reissner // Ann. Phys. (Germany) V. 5. P. 16. [4] G. Nordstrøm // Proc. Kon. Ned. Akad. Wet V. 2. P [5] All relevant results in general relativity can be found in C.W. Misner, K. S. Thorne and J. A. Wheeler Gravitation San Francisco: W.H. Freeman ( The Telephone Book ). [6] A definitive review of classical electron theories is given by F. Rohrlich Classical Charged Particles MA: Addison-Wesley, Reading [7] H. Yilmaz // Phys. Rev V P N. Rosen // Gen. Rel. Grav V. 4. P [8] // Repts. Math. Phys. 21. V. 47. P "Электромагнитные Явления", Т.3, 1 (9), 23 г.