Pointlike Electric Charge in Gravitational Field Theory

The existence of charged elementary ’point particles’ still is a basically unsolved puzzle in theoretical physics. The present work takes a fresh look at the problem by including gravity—without resorting to string theory. Using Einstein’s equations for the gravitational fields in a general static isotropic metric with the full energy-momentum tensor (for the charged material mass and the electromagnetic fields) as the source term, a novel exact solution with a well-defined characteristic radius emerges where mass and charge accumulate: —with being the ’classical’ radius associated with the total charge and where is the Schwarzschild radius belonging to the observable mass (for the electron one has m and m). The resulting ’Einstein-Maxwell’ gravitational electron radius can also be written as , where m is the fundamental Planck length and the fine-structure constant, which yields m


I. Introduction
Modern theoretical physics is essentially based on the existence of a finite set of elementary 'point particles'-leptons and quarks-and their electromagnetic, gravitational, and weak or strong interactions (see, e.g., Refs.[1][2][3][4][5]).Apart from the neutrino's, all fundamental particles carry an electric charge.However, the very concept of a stable 'point charge'-such as the electron-is an old and as yet basically unsolved problem.Namely, why should it be possible to accumulate a finite amount of electric charge in an infinitely small volume?What internal force does the work against the repulsive self-interaction?In fact, without such a force the charged particle should immediately explode.
Historically, the problems with a point charge were already recognized in classical physics (see, e.g., Refs.[6][7][8][9][10][11][12]). For instance, in Poincaré's 'electron model' [7] the electric force on the charged sphere was counteracted by an elastic force of unspecified, non-electromagnetic and non-gravitational nature in order to define a total energy-momentum tensor satisfying the condition characteristic of a closed system (see, e.g., Ref. [13], Ch. 7.3).In fact, even more than a decade after the advent of general relativity, during a visit to Leiden University in 1925, Einstein asked Lorentz' opinion on a purely electromagnetic model for the electron-i.e., without gravitational forces.Lorentz, however, rejected the idea (Ref.[12], Letter 398).
In any case, to quote from Feynman's Lectures (Ref.[10] Vol.I, p. 32-4): "The classical electron radius m no longer has the significance that we believe that the electron really has such a radius".More recently, based on state-of-the-art precision measurements of the electron's gyromagneticfactor (and using a simple 'electron model' due to Brodsky and Drell [14]), 1989 Nobel laureate H.G. Dehmelt has pointed out that "Today everybody 'knows' the electron is a Dirac point particle with radius and .But is it?The value m given here constitutes an important new upper limit.[..] Thus, the electron may have size."(Ref.[15]; see also, e.g., Refs.[16,17]).
Nevertheless, in non-gravitational quantum theory the electron can be treated successfully as a structureless point particle-at least, if the problem of its infinite self-energy is being 'swept under the carpet'.Indeed, as is well-known from the very beginning of quantum electrodynamics (see, e.g, Ref. [18]), this 'success' is only possible at a price.Namely, handling the infinite self-energy of a point charge requires an infinite mass renormalization to yield results in terms of the observed mass (see, e.g., Refs.[3,4,[18][19][20][21][22][23]).Unfortunately, this fundamentally hampers the unification with quantum gravity-as the latter has been found to be non-renormalizable.
For instance, to quote from Ref. [3], p. 568: "The definition of quantum gravity beyond the formal level leads to a number of unsolved problems.[...]Even pure quantum gravity is non-renormalizable in four dimensions.In fact, it is commonly believed that the theory remains non-renormalizable, a property which would indicate the breakdown of local quantum theory at Planck's length scale m."It is further worth noticing that modern string theory has been proposed-and is still under development-inter alia to cope with the problem of point-like particles, replacing them by tiny one-dimensional loops of Planck size (see, e.g., Refs.[4,5]).
The present work takes a fresh look at the problem by including gravity-as yet without resorting to string theory, ignoring the weak interaction and without a priori assuming the mass-charge density to be rigorously zero outside some (as in the usual Reissner-Nordström analysis; see, e.g., Refs.[13,24,25] and Appendix A).Namely, the enormous amount of electrostatic energy compressed into an infinitely small volume must-according to Einstein's general theory of relativity-give rise to huge local gravitational effects.Therefore, in this article the gravitational field equations for the Ricci tensor are studied for a classical selfgravitating charged mass, i.e., with the full energy-momentum tensor for the material mass and the electromagnetic fields as the source of gravitational energy-in a static isotropic metric (Sec.II and III).The analysis involves an 'electrostatic equilibrium' condition (Sec.IV), and rigorously yields a well-defined novel mass-charge distribution with (in fact, for all charged leptons) a characteristic size m (Sec.V and VI).

II. The gravitational field equations
Throughout the present article, it is chosen to keep both the speed of light and the gravitational constant explicitly in the formula-rather than using 'geometrized units' where .The notation closely follows that of Weinberg's book [1].The field equations of general relativity may then be written as where, in this paper, time will be labelled by .
For the problem of a charged mass, the energy-momentum tensor consists of two contributions, viz., for the-electrically charged-matter and the electromagnetic field itself.In the standard 'ideal fluid' form (without internal pressure), the energy-momentum tensor for the mass reads where is the determinant of the metric tensor, so that the mass is given by where is the determinant of the three-dimensional metric tensor -see, e.g., Refs.[13,26].
The system is closed (so that ) by including the electromagnetic field energy-momentum, given by (2.7) where the antisymmetric electromagnetic field tensor is defined by for the electric field and for the magnetic field, representing the usual three-dimensional Levi-Civita symbol and being the vacuum permittivity.It is useful to note that, since and , Eq. (2.7) implies that .
The electromagnetic fields satisfy Maxwell's equations (2.8) where is the current four-vector-with being the proper charge density, which in the Dirac representation reads (2.9) Since the current satisfies the conservation law , the charge is conserved and given by (2.10) Note that both and transform like a scalar.

III. The static isotropic case
The static isotropic metric may be written in the 'standard' form in spherical coordinates , so that the only nonvanishing components of the metric tensor are .Note that , as it should be.

IV. The equilibrium condition
Einstein's gravitational field equations (2.1) for the static isotropic mass-charge system may thus be written as For a structureless charged mass the intrinsic charge-to-mass ratio should be anindependent constant, i.e., , which according to Eqs. (2.5) and (2.9) implies the equation of state for the proper density.Without loss of generality, one may put so that-by virtue of Eq. (2.10) for -the as yet undetermined mass satisfies Eq. (2.6).Hence, the equilibrium equation (4.8) may be rewritten as (4.9) with .Now using the Newtonian limit and the Poisson limit for , one obtains (4.10)By Eq. (4.9) the problem of the charged mass is reduced to finding the temporal metric function .

V. The temporal metric function
Consider Eq. (4.5) for , and using Eq.(4.9) for write it as (5.1) where and .Now again take , collect the terms and once more use Eq.(4.9) for .After a somewhat laborious but otherwise elementary calculation this leads to (5.2) Substituting and from Eq. (5.1), and combining similar terms-some of which add up to zero-Eq.(5.2) is found to factorize such that for it becomes a trivial zero identity (see Appendix A and, e.g., Ref. [27]) while for it either leads to the trivial solution or to a nontrivial metric function satisfying which is akin to the prototype equation for the temporal evolution of so-called 'finite-time blowup' processes in, e.g., chemistry and hydrodynamic turbulence (see, e.g., Ref. [28], p. 353).
Noticing that , using Eq.(5.3) for and defining the auxiliary variable , one thus rigorously obtains (5.4) The exact solution of Eq. (5.4) is given by (5.5) where the integration constant has been set equal to the Schwarzschild radius belonging to the observed mass m in order to satisfy the Newtonian limit for large values of .
For , one has as well.On the other hand, for .Namely, expanding Eq. (5.5) in powers of yields , with (5.6) Using [as given below Eq. (4.9)] in the definition of [as given below Eq. (5.1)] and invoking Eq. (4.10) for , one obtains .Once more using (4.10), this becomes which leads to -with being the 'classical' radius belonging to the charge and where is the Schwarzschild radius belonging to the mass -so that , which explicitly amounts to (5. 7) where the exponential factor from Eq. (5.6) has been omitted as it is always of the order of unity.Actually, with m and m for the electron, one has so that the exponential correction tends to .Finally, one obtains m-which may be called the 'Einstein-Maxwell' gravitational electron radius.

VI. Results
The temporal metric function now follows from .Using Eq. (5.4) for , this yields , so that the function becomes .Fig. 1 shows the exact solution of Eq. (5.3) for a few values of .Further, using Eq.(5.5) for , Eq. ( 6.1) can also be written as .Hence, while a simple calculation using Eqs.(5.4) and (5.5) yields .
Similarly, the radial metric function follows from Eq. (5.1) as (6.2) while .For finite and , one gets .Next, the radial electrostatic field is obtained from Eq. (4.9), which yields  The singular part of the density follows from the Poisson equation rather than from the gravitational field equations per se.Namely, from Eq. (6.5) one has , so that by virtue of one has .Of course, the continuous part of the density obeys the Einstein equations for (for all values of ).E.g., consider Eq. (4.4) and note that its right-hand side amounts to , so that it remains to show that -which is easily done since [by Eqs.(5.4) and (6.2)] and by definition.

VII. Final remarks
The mass-charge distribution (6.5) is an exact particle-like solution of the classical Einstein-Maxwell equations (for all values of the parameter ).It emerges from a rigorous balance between electrostatic self-repulsion and gravitational self-attraction for all values of the radius (see Sec. IV), which is part of its novelty-see, e.g., Refs.[13,24,25].It is further worth noticing that it appears to be the only solution for a point-like charge which correctly satisfies the observable Newtonian and Poisson limits for large tensor, the covariant energy-momentum tensor, the metric tensor, and (with ).In mixed components (see, e.g., Sec.IV), Eq. (2.1) becomes with and .The covariant tensor defines the Riemannian space-time geometry by means of the proper time , such that (2.3) (2.4)    where is the proper mass density.In the Dirac representation one has (2.5)

3
nonzero component of the material energy-momentum tensor now reads (the metric (3.1),Eq. (2.8) leads to the Poisson equation (3.7) for the only nonzero component of the electric field, while for the contributions from Eq. (2.7) one obtains (3.8) while equation (3.7) can be rewritten as (4.8) which represents the balancing of the repulsive electrostatic self-force and the attractive gravitational selfforce.It is the electrostatic counterpart of the usual 'hydrostatic equilibrium' condition for ideal fluids.In fact, by virtue of the Bianchi identities, Eq. (4.8) is a direct consequence of the conservation law .Namely, one gets with , while by means of the Poisson equation (3.7) one obtains .
. (5.5) and (5.6), while .Notice that for finite values of and one has

Figure 1 .
Figure 1.The temporal metric function [for (bottom line), (middle), (top)].Shown is the numerical solution of the exact Eq. (5.3), as a function of the non-dimensional radial variable .
distribution and which rigorously satisfies Eq. (2.10) for the total charge -recalling that [see its definition below Eq. (4.9)] with [from Eq. (4.10) and ] and using Eq.(5.6) for , so that(6.6)For finite values of and , the second part of the density (6.5) now becomes .Clearly, with [see below Eq. (5.7)] charge and mass almost completely accumulate at the Einstein-Maxwell radius -the tail in Eq. (6.5) containing only some .

Figure 2 .
Figure 2. The radial electrostatic field [for (bottom line), (middle), (top)].Shown is [with ], using the numerical solution of the exact Eq. (5.3), as a function of the nondimensional radial variable .