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Ultramicro Black Holes and Finiteness of the Electromagnetic Contribution to the Electron Mass

It is argued that the nonintegrably singular energy density of the electron's electromagnetic field (in both the classical point-charge model and quantum electrodynamics) must entail very strong self-gravitational effects, which, via black hole phenomena at finite radii, could well cut off the otherwise infinite electromagnetic contribution to the electron's mass. The general- relativistic equations for static, spherically symmetric stellar structure are specialized to treat the self-gravitational effects of static, spheri- cally symmetric, nonnegative, localized energy densities which may exhibit nonintegrable singularities at zero radius. It is demonstrated that in many situations, including the electromagnetic ones of interest here, such a system has a black hole whose Schwarzschild radius is that where the original energy per radial distance (the spherical shell area times the original energy density) reaches the inverse of (2G). The total mass of the system is that of this black hole (which follows in the usual way from its Schwarz- schild radius) plus the integrated original energy density outside this black hole. These results produce, for the classical point-charge model of the electron, an electrostatic contribution to its mass which is many orders of magnitude larger than its measured mass. For quantum electrodynamics, how- ever, the result is an electromagnetic mass contribution which is approxi- mately equal to its bare mass -- thus about half of its measured mass.


I. Introduction and Discussion of Results
Notwithstanding the great success of the renormalization program in quantum electrodynamics, the electromagnetic contribution to the electron mass cannot be related to the bare mass and charge parameters of the theory, as it comes out infinite. Such an infinity is already present in the simple classical point-charge model of the electron, making the problem a venerable one. Indeed, the quantized theory improves the situation to a considerable extent, as virtual pair creation in effect endows the electron with some spatial extension, such as a bit of smearing of its effective charge distribution at distances smaller than its Compton wavelength [1]. This quantum structure considerably ameliorates the degree of the divergence of the electromagnetic mass contribution (from linear to logarithmic) but fails to eliminate it.
Of course, this particular difficulty with the electromagnetic contribution to the electron mass is by no means the only example of apparently well-grounded physical theory producing bewildering infinities. The frequency-divergent blackbody radiation spectrum obtained classically by Rayleigh and Jeans led Planck to essay his quantum hypothesis. Lorentz's classical demonstration that the Rutherford atomic model must result in infinite radiation and instability as the elecron spirals inward toward the nucleus motivated Bohr to enlarge Planck's quantum hypothesis to embrace atomic structure. On a less exalted level, the infinite total cross section for Coulomb scattering finds its resolution in the observation that the real world always provides other charged matter to shield the Coulomb potential at some finite distance. All of these successful resolutions of theoretical infinities were due to recognition that the theoretical framework which produced them involved some type of critical overidealization of the problem being treated-the neglect of certain physical effects which lay outside the utilized theoretical framework and happened to be crucial to the problem which generated the infinity. Thus in retrospect we can assert that the Rayleigh-Jeans and Lorentz classical divergences are due to a crucial neglect of quantum effects, while the infinite Coulomb cross section arises from neglecting the inevitable eventual shielding of any Coulomb potential. These "infinity-killing" precedents unequivocably suggest that the resolution of the infinite electromagnetic contribution to the electron mass should be sought outside the context of the classical or quantum electromagnetic theory which produces it. In particular, the renormalization program, which operates entirely within the context of quantum electrodynamics, cannot itself be expected to fully resolve the infinity problems of that theory-indeed, although it very usefully "quarantines" the infinities into just a few locations such as the electromagnetic contribution to the electron mass, it does not banish them.
We are thus now compelled to ask what nonelectromagnetic phenomena might place a finite limit on the electromagnetic contribution to the electron mass. For the strong and weak nuclear forces, no obvious mechanism which could accomplish this suggests itself. Indeed, these forces are thought to be described by renormalizable gauge theories with divergences similar to those of quantum electrodynamics. Thus, rather than curing the infinite electromagnetic contribution to the electron mass, the theories of the strong and weak forces would appear to contribute additional infinities of their own. The gravitational force, as it exists between two electrons, is so utterly minute relative to the corresponding electrostatic force that conventional thinking dismisses any significant role for it in quantum electrodynamics without further consideration. However, this ostensible logic overlooks the very infinity in the electromagnetic contribution to the electron mass with which we are wrestling here! A mass contribution which purely electromagnetic theory says is infinite could hardly fail to overwhelmingly engage self-gravitational effects. Moreover, bringing gravity into the picture lends encouragement that the infinity may be resolvable because (1) gravity contributes negative potential energy which could conceivably tend in the direction of cancelling the positive electromagnetic field energy, and (2) very strong stimulation of the gravitational force is known to produce finite Schwarzschild radius black-hole effects which could very possibly serve to cut off the infinity in the electromagnetic field contribution to the electron mass (the mass of a black hole isn't infinite; it is essentially determined by (proportional to) its Schwarzschild radius). The thinking which we essay here may strike the reader as unorthodox, but it is in fact profoundly conservative is spirit. We take very seriously the theoretical electromagnetic field contribution to the electron mass (does the physicist exist who would make light of the result of a calculation in electromagnetic theory?), and then ask whether that result itself implies that known nonelectromagnetic phenomena must figure prominently in the total physical picture. For self-gravitational effects the answer is overpoweringly in the affirmative.
At any "ordinary" distance from the electron, the energy density of its electromagnetic field as calculated from purely electromagnetic theory must be extremely accurate; it is only at extraordinarily short distances that self-gravitational effects can significantly modify this electromagnetic field energy density. In this regard gravitational resolution of the electron's infinite electromagnetic mass contribution would parallel the resolution of the other infinities mentioned earlier. The Rayleigh-Jeans blackbody spectrum is adequate at low frequencies; the Planck quantum modification only makes a difference (albeit an "infinity-killing" one) at sufficiently high frequencies. The classical Lorentz model for the electron radiatively spiraling in toward the nucleus is accurate at large radii, with the "infinity-killing" energy level quantization only entering prominently at sufficiently small radii. The differential cross section for Coulomb scattering (which leads to a divergent total cross section) is accurate for impact parameters up to where shielding of the Coulomb potential sets in. An electromagnetic field energy density for the electron which is accurately given by purely electromagnetic theory down to radii small enough to provoke significant self-gravitational effects is thus deeply consonant in spirit with the above "infinity-killing" precedents of the past.
In Section II we specialize the general-relativistic equations for static, spherically symmetric stellar structure [2] to treat the self-gravitational effects flowing from an arbitrary static, spherically symmetric, nonnegative localized energy density which has been specified in a "world" where G = 0 (e.g., a static field energy density calculated in the quantum electrodynamicist's "world" of flatspace electromagnetic theory). We give particular attention to the case where this G = 0 static, spherically symmetric, nonnegative localized enegy density tends toward a nonintegrable singularity at the origin, as is the case for the electomagnetic field energy density of the electron, and discuss locating (when G is "switched on") the resulting black-hole Schwarzschild radius and obtaining the now finite total energy. For many situations, an adequate rendition of our results for the black-hole Schwarzschild radius in such static, spherically symmetric cases is that it is located where the G = 0 energy per radial distance (namely 4πr 2 times the G = 0 energy density at r) attains the the limiting value (2G) −1 (we use units where the speed of light c = 1). An effective energy (or mass) contribution from this black hole may be obtained directly from this radius by using the familiar relation that a black hole's mass equals its Schwarzschild radius divided by (2G). The remaining effective mass contribution, which may be thought of as coming from that part of the energy density which lies outside the Schwarzschild radius, can be calculated by simply integrating the G = 0 energy density over this external region. This simple, intuitively appealing procedure is fully adequate for obtaining the electromagnetic contribution to the electron's mass.
In Section III we apply this procedure to the classical point-charge model of the electron; the result for the electrostatic mass contribution is unacceptably large-many orders of magnitude larger than the measured electron mass. We then extract an effective quantum electromagnetic field energy density in ordinary space from the lowest order Feynman diagram for the electromagnetic field contribution to the electron's mass [3]. This effective quantum electromagnetic field energy density is less singular as r → 0 than is that of the classical point-charge model, but is still not quite integrable there (the G = 0 effective total electromagnetic energy comes out logarithmically divergent due to this singularity). In addition, it differs, by a constant multiplicative factor whose value is smaller than unity, from the classical point-charge model in the limit r → ∞ as well. This large r deviation represents a finite charge renormalization effect which arises from partial shielding of the bare charge by the spontaneous (out of the vacuum) production of a virtual pair and photonan intrinsic aspect of this Feynman diagram (this spontaneously produced virtual pair is responsible as well for the above-described amelioration of the singularity as r → 0, relative to the classical point-charge case). Use of this effective quantum electromagnetic energy density yields a much smaller Schwarzschild radius than does the classical point-charge model, and, after taking account of the accompanying finite (but not negligible!) charge renormalization, yields an electromagnetic mass contribution which is approximately equal to the bare mass (i.e., approximately half of the measured mass). This quantum electrodynamics cum gravity result is certainly not implausible, and is a pleasantly far cry from the unresolved infinite electromagnetic mass contribution of the renormalization program.
One can question the appropriateness of combining quantum electrodynamics with a completely classical approach to the consequent gravitational self-energy in the manner just described, but, practically speaking, there is no viable alternative at this time. One wouldn't know how to tackle quantized gravitational theory other than by perturbation expansions in nonnegative powers of G, which is hopelessly inappropriate to the ultrastrong gravitational forces involved with black holes-the domain in which the above-described physics lies. We can take some comfort from the thought that for such very strong gravitational forces in a static situation, the classical limit may very well be an adequate description.
In Section IV we entertain the speculation that gravitation can as well resolve the remaining infinities of the quantum electrodynamics renormalization program. For the lowest order divergent Feynman diagram which contributes to charge renormalization, we can at least adduce something of a plausibility argument. That diagram describes the photon's dissociation into and recombination from a virtual electron-positron pair. We identify two successive subdiagrams of this diagram, which may be interpreted as the four-dimenensional Fourier transforms of, respectively, the stressenergy contribution to the electromagnetic field from the above-mentioned photon dissociationrecombination to a virtual pair, and the stress-energy of the virtual pair itself. Both of these subdiagrams carry the logarithmic divergence of the parent diagram, but the convolution character of the second one permits calculation of its inverse Fourier transform to obtain the abovementioned virtual pair stress-energy-which must be highly localized in space-time and have a nonintegrable singularity that produces the divergence in its Fourier transform. Self-gravitational effects should, however, modify this nonintegrable G = 0 virtual pair stress-energy to make it everywhere integrable, as occurs with the electromagnetic field energy density of the electron. Thus, with G "switched on", its Fourier transform will become convergent, and with it all the above-mentioned diagrams for photon dissociation-recombination to a virtual pair, as well as their contribution to charge renormalization.
We proceed now to blithely speculate that taking proper account of gravitation replaces all the unresolved ultraviolet divergences of quantum electrodynamics with finite G-dependent quantities (which, of course, must diverge in the limit that G → 0, and thus cannot be calculated in the context of a perturbative gravitational development in nonnegative powers of G, such as is normally used in quantized gravitational theory). Indeed, we extend this speculation to all of quantum field theory, and ask what its implications are vis-à-vis the conventional insistence that any acceptable quantum field theory needs to be renormalizable. We conclude that under such circumstances the renormalizability requirement becomes a physically sensible constraint, rather than a calculational necessity, for narrowing the field of candidate quantum field theories which one might try to postulate for almost any force-with one transcendent exception: there is no longer any compelling argument as to why quantized gravitation theory itself must be renormalizable. This would be a delightful dénouement indeed, as it has long been a bewilderment that quantized general relativity alone, among the quantum field theories which one is compelled to take seriously, turns out to be nonrenormalizable and thus ostensibly calculationally senseless.

II. Self-Gravitational Corrections to Static, Spherically Symmetric, Nonnegative
Localized Energy Densities Obtained from G = 0 Theories From symmetry considerations it can be shown that the general static, spherically symmetric metric has a proper time interval of the form [4], Because general relativity is a gauge theory in which the metric plays the role of a tensorial gauge potential, the metric is not unique. This gauge freedom turns out to permit one to fix the metric function C(r) above at its flat-space value, C(r) = 1, resulting in the static, spherically symmetric metric of "standard form" ("standard gauge" would seem to be a more appropriate terminology), In our problem the source of this static, spherically symmetric curvature (i.e., the departure of the metric functions B(r) and A(r) from their flat-space value of unity) is, in the context of an idealized flat-space (G = 0) calculation, a static, spherically symmetric, nonnegative localized energy density ε(r). In the G = 0 Minkowskian "world", the corresponding complete stress-energy tensor may be formally written as, where U µ is the velocity four-vector, which satisfies, where η µν is the Minkowski (flat-space) metric. In our particular frame of reference, where T µν reduces to just its T 00 component, which is the static energy density ε(r), we have the spatial components U i (i = 1, 2, 3) of U µ vanishing and U 0 = 1. It is easily verified that T µν meets all the flat-space criteria for a stress-energy tensor-namely Lorentz covariance, symmetry in its indices, and divergencelessness in the ordinary (Minkowskian) sense. We note in passing that the radius argument r of ε(r) has the Lorentz-invariant representation, In the general-relativistic case, it would seem natural to try to keep Eq. (3) as the stress-energy tensor which serves as the "source" term in the Einstein equation for the metric (i.e., the functions A(r) and B(r)). Of course, the four-velocity U µ must then satisfy the fully covariant generalization of Eq. (4), which, for our static reference frame, now implies that U t = (B(r)) −1/2 instead of unity, while the spatial components U r , U θ , and U φ of course still vanish. However, a problem now arises. While this generalization of Eq. (3) continues to be divergenceless in the ordinary (Minkowskian) sense, it fails to be divergenceless in the general covariant sense with the metric of Eq. (2). Indeed, it turns out that, which fails to vanish unless B ′ (r) = 0 (as is the case for flat space) or ε(r), the G = 0 energy density itself, vanishes. We thus cannot use the T µν of Eq. (3) in the Einstein equation, which is a gauge equation that is inconsistent unless T µν ;µ vanishes. To obtain a T µν which is divergenceless in the generally covariant sense, we need to generalize Eq. (3). The other symmetric tensor besides U µ U ν which is natural to the problem is the metric tensor g µν itself. So we follow the precedent of static, spherically symmetric stellar structure theory [2] and introduce a static pressure term into the T µν of Eq. (3), Note that the pressure p is introduced in such a way that we still have T 00 = (ε(r)/B(r)) in our static reference frame, exactly the same as was the case for Eq. (3). For our purposes the static pressure p (r) is meant to have no existence independent of that of ε(r) and nonflatness of the metric-it is introduced only to ensure that T µν is divergenceless in the generally covariant sense, and we thus shall require it to vanish if ε(r) does, or if the metric functions B(r) and A(r) go to unity. As p (r) is purely a creature of the metric and ε(r), we shall require that it be localized, as ε(r) is. With p (r) in place, the offending (radial) component of the generally covariant divergence of T µν becomes, This can be made to vanish if p (r) is required to satisfy the linear inhomogeneous first-order differential equation, which may readily be reduced to quadrature after multiplying through by the integrating factor B .
We enforce the requirement that p (r) be localized, as ε(r) is, by choosing the integration constant r 0 to be ∞. The resulting p (r), is localized and vanishes if ε(r) does or the space becomes flat, as we have required. The T µν of Eqs. (6) and (8) is now divergenceless in the generally covariant sense and, of course, has all of its Cartesian components localized in our static reference frame. It vanishes identically when ε is put to zero, and, in the flat-space limit in our static reference frame, satisfies T 00 = ε, with all other components zero.
Putting this satisfactory T µν of Eqs. (6) and (8) into the Einstein equation together with the metric form of Eq. (2), results, in our static reference frame, in the following three equations for the metric functions B(r) and A(r) [2], where p is, of course, given by Eq. (8). If we multiply Eq. (9a) through by (r 2 /A), we find that we can re-express it as, which can be solved for A, Since ε(r) is a localized energy density, we can see from Eq. (10) that A(r) → 1 as r → ∞ regardless of the value of the integration constant a 0 . This physically proper behavior of A(r), tending toward the flat-space value of unity at large distances from the localized energy density ε, leaves us without a straightforward criterion at this point to pin down the value of the integration constant a 0 . We shall return to the issue of determining a 0 below-indeed it will turn out that the proper determination of a 0 , together with the consequences which flow from that determination, comprise the main result of this section. First, however, we show that Eqs. (9a), (9b), and (9c) contain redundancy-as well they ought, since there are only two unknown functions, B(r) and A(r), with which to solve these three equations. If we multiply Eq. (9b) through by (B 1 2 /A), differentiate both sides with respect to r, and then multiply the result by (A/B 1 2 ), we obtain, Using Eq. (7), we see that we can replace the right hand side of Eq. (11) by −4πGAε(B ′ /B).
Having done this, we multiply both sides of Eq. (9a) by (B ′ /(2B)) and add this to the just described modified form of Eq. (11) to obtain, Now we multiply Eq. (12) by (r/2) and rearrange the terms to obtain, We have derived Eq. (13) from Eqs. (9b) and (9a). However, Eq. (13) follows as well from Eqs. (9b) and (9c)-these have identical right hand sides, so we may equate their left hand sides, which also yields Eq. (13).
We have thus demonstrated that only two of the three equations (9a), (9b), and (9c) are independent. (As we mentioned above, this is fortunate, as we deal with only two metric functions B(r) and A(r).) In practice we shall be using Eq. (13) above and Eq. (9a) (whose solution is given by Eq. (10)) as the two equations which determine B and A. The redundancy we have demonstrated is a consequence of the gauge ambiguity of the Einstein equation-this is what permits us to set the third metric function C(r) to unity in the general static, spherically symmetric metric given by Eq. (1), producing the "standard form" (or "standard gauge") given by Eq. (2).
It is interesting to note that if we multiply Eq. (13) through by B Since the energy density ε(r) and stress-energy tensor T µν are localized, we expect the gravitational forces to be weak at large r, and for B 1 2 to asymptotically approach a Newtonian limiting form [5], where M is the effective total gravitating mass of the system. Also at large r we have from Eq. (10) that the asymptotic form of A is, If we multiply Eq. (14) through by r 2 , and insert into the resulting equation the large r asymptotic forms of Eqs. (15) and (16), the terms through first order in r −1 yield, We thus see that the total gravitating mass M of our system depends on the constant of integration a 0 . In the case where ε(r) is everywhere sufficiently small that the whole problem may be treated throughout in the weak-gravitational Newtonian limit, we know from Newtonian gravity theory that the total gravitating mass comes out to be simply the G = 0 energy density ε(r) integrated over all space, Thus, in the weak-gravitational limit, we may conclude that a 0 = 0. Can we now conclude that a 0 is always equal to zero? We can see from Eq. (10) that if ε(r) is not everywhere sufficiently small, it is possible for the metric function A to develop singularities. They occur at values of r where 8πG r r a 0 is satisfied. Since ε(r) is localized and nonnegative, choosing a 0 sufficiently large ensures that there exists no r where the singularity condition (18) can be satisfied. So if we want to suppress singularities in the metric function A, it is not always possible to stick with the weak-gravitational prescription that a 0 be set to zero. Of course, because of the gauge ambiguity, metric functions such as A are not unique, which makes their physical meaning somewhat obscure. Before we try to draw definite conclusions on the matter of how a 0 is to be chosen, we first need to see if there are quantities having clearcut physical meaning which as well become singular when Eq. (18) is satisfied. An ideal such quantity would be an energy density which can be interpreted as properly incorporating the self-gravitational corrections to our G = 0 energy density ε(r). Because ε(r) and our source stress-energy tensor T µν are localized, it happens to indeed be possible to construct such a full energy density. When the Einstein equation's source stress-energy tensor T µν is localized, and the metric tensor is constrained to approach the Minkowskian flat-space value at large distances from that source (the situation for our problem), it is legitimate to view that equation as describing the dynamics of an ordinary classical field (a Minkowskian tensor one, which is the metric minus the Minkowski flat-space metric) in the context of Minkowskian flat space. The dynamics of this field are, to be sure, highly nonlinear, and have a gauge character, but the well-known classical non-Abelian gauge theories share these particular features as well, and they are always viewed as playing out their dynamics in the context of Minkowskian flat space. Field theories in flat space generally possess a total stress-energy tensor which is divergenceless in the ordinary (Minkowskian) sense, a property which allows the normal definition of conserved total energy, momentum, and angular momentum. This turns out to be the case as well for asymptotically flat gravity theory, where this ordinary total stress-energy tensor τ µν is constructed from the linear, purely second-order derivative part of the Cartesian components of the Einstein tensor [6], an object which is divergenceless in the ordinary, Minkowskian, sense (the full nonlinear Einstein tensor is, of course, only divergenceless in the generally covariant sense, which does not produce a conserved total energy, momentum, and angular momentum). We are particularly interested in the ordinary total energy density τ 00 (which is the net physical energy density resulting from our G = 0 input energy density ε plus that from its self-gravitational interaction), since the total gravitating mass M must be the total localized physical energy, i.e., the integral over all space of τ 00 . This, together with Eq. (17), implies that, Since τ 00 is constructed from part of the Einstein tensor (the linear part), and thus from the metric, it will be interesting to see whether it shares the singularities which occur in our metric function A when Eq. (18) is satisfied. Such singularities in τ 00 , if they are nonintegrable, could disrupt the physically required relationship given by Eq. (19), and thus could constrain a 0 to values sufficiently large so that the singularity condition of Eq. (18) cannot be satisfied. However, we are getting ahead of ourselves: we now need to calculate τ µν and then investigate the properties of τ 00 . In the flat-space approach to gravity theory, the dynamic gravitational gauge field h µν is given by the metric tensor g µν in Cartesian coordinates, minus the flat-space (Minkowski) metric η µν , This flat-space approach only turns out to be fully viable in situations where h µν is localized [6], as is the case for our problem. Following reference [6], we exhibit the part of the Ricci curvature tensor which is linear in h µν , where we adopt the useful convention that indices on h µν , ∂/∂x λ , and R (1) µκ are raised and lowered with the Minkowski metric η µν rather than g µν . Using R (1) µκ , we obtain G (1) µκ , the part of the Einstein tensor which is linear in h µν (and, like R (1) µκ , purely second-order in the derivatives of h µν ), It can be verified that G (1) µκ is symmetric in its indices and divergenceless in the ordinary (Minkowskian) sense. Thus we adopt as the definition of τ µν , the ordinary total stress-energy tensor [6], If h µν is localized, this divergenceless (in the Minkowskian sense) τ µν is localized as well, and permits construction in the usual flat-space way of conserved total energy, momentum, and angular momentum. For our purposes, the physical quantity of commanding interest is the total energy, which is obtained by integrating the localized τ 00 over all space, as in Eq. (19). For our problem, with the metric of Eq. (2), h µν is given by, As Eqs. (15) and (16) imply that B(r) → 1 and A(r) → 1 as r → ∞, our h µν is indeed localized, and the resulting ordinary total stress-energy tensor τ µν will be fully physically meaningful in the familiar flat-space sense [6]. Proceeding now to apply the above-given formulas for calculating τ µν from h µν , we obtain from Eq. (20), after many tedious differentiations and index contractions, The static stress tensor τ ij , i, j = 1, 2, 3, is symmetric in its two indices and identically divergenceless in three-dimensional Euclidean space, as indeed it must be. The three-momentum density τ i0 , i = 1, 2, 3, vanishes, as it physically ought to in such a static situation. The energy density τ 00 of Eq. (21a) turns out to depend only on the metric function A(r), which we know explicitly from Eq. (10). Thus we can write, or, explicitly, It is interesting to note, from Eq. (22b), that if a 0 > 0, then any nonnegative G = 0 energy density ε(r) which is nonintegrably singular as r → 0 results in the universal asymptotic behavior τ 00 ∼ −(8πGr 2 ) −1 as r → 0, which is integrable as r → 0, and negative there to boot! We thus see that self-gravitational effects are indeed in principle capable of overwhelming intractable positive singularities of G = 0 energy densities with negative gravitational energy that renders such singularities harmless.
There still remains the question of whether strong self-gravitational effects compel a non-Newtonian positive choice for a 0 . Examination of Eq. (22a) shows that τ 00 (r) is nonintegrably singular at positive values of r where Eq. (18) is satisfied, just as the metric function A(r) is singular at such positive values of r. As we have already stated, the fact that ε(r) is localized and nonnegative ensures that for sufficiently large a 0 there exists no r which satisfies the singularity condition of Eq. (18). Thus to keep τ 00 integrable, we can indeed sometimes be compelled to choose a 0 > 0.
Provided that we have now chosen a 0 so as to render τ 00 (r) free from nonintegrable singularities, we may use Eq. (22a) to evaluate the total energy (mass) of the system, Thus we have indeed arrived at Eq. (19) (for which we had previously given a nonformal, physically motivated argument), but with the previously anticipated caveat that a 0 must have been chosen sufficiently large that the τ 00 nonintegrable singularity condition given by Eq. (18) has no solutions. The bottom-line question now presents itself: how is a 0 to be uniquely determined in all situations? The two constraints which we have so far obtained on the choice of a 0 is that it must be zero (its smallest possible value, since, of course, r ≥ 0) in the weak-gravity Newtonian limit, and large enough to avoid the occurrence of nonintegrable singularities in τ 00 (r) when gravitational effects become strong. We may obtain yet another constraint on the choice of a 0 from the general principle of continuity in classical physics. Thus, we expect a 0 (and, perhaps more to the point, the total mass M of the system, which depends on a 0 , as Eq. (17) demonstrates) to vary in a continuous way if ε(r) is varied continuously. If ε(r) is sufficiently small everywhere, we will be in the weakgravitational Newtonian regime, where a 0 must be zero. Now, as we continuously vary ε(r) to take on larger and larger values, we must eventually reach a point where the choice a 0 = 0 results in τ 00 (r) developing a nonintegrable singularity, which then forces us to make a 0 sufficiently positive. But just how positive? The only choice which can be consistent with the continuity requirement is that a 0 must be the smallest positive value for which the singularity condition of Eq. (18) has no solution. Only such a minimum recipe permits a 0 to fall continuously from positive values to the required Newtonian limit value of zero as ε(r) is continuously weakened. We may thus write down a technical definition of a 0 as follows, a 0 (ǫ) = min a | a ≥ 0 and max and For every a 0 (ǫ) where ǫ > 0, τ 00 (r) has no nonintegrable singularities. For sufficiently strong ε(r), as ǫ gets smaller and smaller the singularity condition of Eq. (18) will come closer and closer to being satisfied. The radius r = r s for which Eq. (18) tends toward being satisfied as ǫ → 0+ in Eqs. (23), can be regarded as the location of an "incipient" singularity of τ 00 (r), and interpreted as the Schwarzschild radius of a black hole. In particular, we can rewrite Eq. (17) for the total mass M of the system to refer only to this critical radius r s instead of to a 0 itself. First we split up the integration interval of Eq. (17) as follows, and thus, We may give a loose intuitive interpretation of Eq. (25) by saying that the total mass of the system comes from the G = 0 contribution for the region outside the critical radius r s , plus the standard expression for the mass of a black hole of Schwarzschild radius r s (the mass of a black hole equals its Schwarzschild radius divided by (2G)). In the weak-gravitational Newtonian limit, of course a 0 = 0, Eq. (18) is not satisfied (even "incipiently") for any r ≥ 0, and the critical radius r s is properly taken to be zero in Eq. (25), making it consistent with Eq. (17) in this weak-gravitational case. In many situations it is possible to obtain the critical radius r s directly from ε(r), without reference to a 0 . To demonstrate this we define, because of the content of Eqs. (23), It will frequently be the case that the maximum in r of f (r; a), which is required in Eq. (23a), occurs where the derivative with respect to r of f (r; a) vanishes. The radius r max , where the maximum of f (r; a) occurs, will in that case satisfy, In general, r max is a function of a, and may be written r max (a). When a = a 0 , r max (a = a 0 ) = r s and, from Eqs. (26) and (24), Thus, putting a to a 0 in Eq. (27) yields, Eq. (29) has given us the condition for the critical radius r s that was mentioned in Section I, namely that it occurs where the G = 0 energy per radial distance attains the limiting value (2G) −1 . Eq. (25) may then be further loosely interpreted as saying that the energy per radial distance in effect "freezes" at this (2G) −1 limiting value within the black hole of radius r s , while it takes on its G = 0 value outside the black hole-up to the point that it equals (2G) −1 , which defines the black-hole boundary radius r s . Of course there exists no black-hole interior region to be concerned with if the G = 0 energy per radial distance is always less than the limiting value (2G) −1 . In that case, Eqs.
(23) imply that a 0 = 0 (of course r s = 0 as well for consistency between Eqs. (25) and (17)), so the total mass M of the system is equal to its G = 0 value, just as in the Newtonian limit. Eqs. (29) and (25) will be all that we require for our discussion of the electromagnetic field contribution to the electron's mass in the next section. It does seem disappointing that such very simple and intuitively appealing results have entailed such a lengthy, ponderous, and sometimes subtle derivation.
Before we go on to the discussion of the electromagnetic field contribution to the electron mass, it is interesting to ask how our stress-energy tensor τ µν looks in the weak-gravitational Newtonian limit. Of course we know that we must take a 0 = 0 in that situation. We also need to approximate the metric functions B(r) and A(r) through just first order in G. In the Newtonian limit, we further know that [5], where Thus, To first order in G with a 0 = 0 (Newtonian limit), Eq. (10) becomes, which, together with the previous equation, yields, If we put the Newtonian limiting forms for B(r) and A(r) given by Eqs. (30) and (32) into Eqs.
Thus the Newtonian limit of τ µν reduces to its G = 0 case. We see that spherically symmetric Newtonian gravity theory entails no self-gravitational correction effects whatsoever.

III. The Electromagnetic Field Contribution to the Electron Mass
The classical point-charge model for the electron has the familiar electrostatic field, which produces the static energy density, Eq. (29) for the critical radius r s thus reads, Eq. (25) for this electrostatic energy contribution to the electron mass then becomes, which, unfortunately, is about 2 × 10 21 times the measured electron mass! Clearly the classical point-charge model is far out of its depth in this issue. In quantum electromagnetic field theory, however, charge smearing and shielding arising from spontaneous vacuum virtual pair and photon production reduces ε(r) well below the value given by Eq. (33) at r smaller than an electron Compton wavelength. In this case, δm (for G = 0) is supposed to be given by the Feynman diagram of Fig. 1, with the external electron line taken to be on shell and at rest. The calculation diverges, but only logarithmically, which is a far weaker divergence than the linear divergence of the G = 0 classical point-charge model.
Since we don't know how to do our ultrastrong, black-hole gravity theory in the context of fourmomentum space, which is the natural environment for Feynman diagrams, we need to transform the integrand of the δm Feynman diagram into an energy density in configuration space. The first order of business will be to carry out the formal integration over the k 0 component of the virtual photon four-momentum in the loop part of the Feynman diagram for δm. The result of this, of course, is a d 3 k integral expression for δm, whose integrand turns out to be nonnegative and spherically symmetric (i.e., it depends only on | k|). We interpret this integrand as an energy density in k-space, and find that it lends itself to transformation to a nonnegative, spherically symmetric energy density in r-space by the appropriate Fourier technique. Of course both the k-space and the r-space energy densities yield a logarithmically divergent δm (G = 0 theory), but we readily obtain the self-gravitational critical cutoff radius r s for the r-space energy density, and then proceed to calculate a finite δm in terms of G and the bare mass and charge of the quantized electromagnetic field theory. We seth as well as c to unity in the calculations which follow.
The Feynman diagram of Fig. 1 for δm is [3], where the external electron of momentum p is "on shell" (i.e., satisfies the Dirac equation, so that p / may be set to m) and at rest, i.e., p = m, 0 . The upshot of both of these things together is that γ 0 may be set to unity. Standard Dirac matrix identities [7] imply that, Since we can put p / to m and γ 0 to unity, the above may be simplified further, We can clearly drop the γ · k term from the numerator of the integrand of Eq. (37), as it has negative parity in k and the integrand factors which multiply it are of positive parity in k (they depend only on | k|). The next step is to fully factorize the denominator of the integrand, The k o integration is now carried out by using the residue calculus. If, for example, the k o contour is closed with a semicircle above the real axis, then one calculates the residues at the two roots which have the negative real parts, i.e., −| k| and m − | k| 2 + m 2 , as these are the ones with infinitesimal positive imaginary parts. The upshot of the k o integration and gathering of terms turns out to be, where, We note that f (x) increases monotonically from unity at x = 0 toward its asymptotic value of 3/2 as x → ∞. As was mentioned above, the integrand in Eq. (39a) is nonnegative and depends only on | k| (i.e., is spherically symmetric). It may be interpreted as the electron's electromagnetic field energy density in k-space. The integral over k-space of a nonnegative, spherically symmetric integrand can be reexpressed as the integral over r-space of an integrand which is also nonnegative and spherically symmetric. The r-space integrand is obtained by squaring the Fourier transform of the square root of the k-space integrand. Thus it may be shown to formally follow from Eq. (39a) that, where, We note here in passing that although Eq. (39a) for δm is logarithmically divergent, and Eq. (40a) for δm must thus prove to be so as well, there is no convergence problem for the Fourier transform expression in Eq. (40b) for the configuration space energy density ε(r)-the object we are actually interested in. We could have consistently inserted cutoff functions to avoid dealing with divergent integrals, but ε(r) is clearly finite and well-defined, independent of cutoff technique. It is possible to work out the asymptotic behaviors for small and large r of the energy density ε(r) of Eq. (40b). The results are, and ε(r) ∼ (2e/π) 2 8πr 4 for r ≫ 1 m .
Eq. (41a) shows that this quantum field theoretic energy density is less singular as r → 0 than that of the classical, point-charge model, as given by Eq. (33). Its r −3 asymptotic behavior in this region makes for the logarithmic divergence of δm mentioned above. Effective smearing and shielding of the electron's charge due to spontaneous vacuum production of a pair and photon, as shown in Fig. (2a), which is one of the two "time-ordered" versions of Fig. (1), is responsible for this amelioration of the r → 0 singularity. Eq. (41b) shows that at large r, the behavior of ε(r) is very similar to that of the classical point-charge model, Eq. (33), with the exception of an overall constant factor which is less than unity. This represents a finite renormalization of the bare charge (by a factor of (2/π), which is less than unity) that is also due to effective charge shielding by the spontaneously produced virtual pair.
We now obtain the critical radius r s by putting the small r asymptotic form of Eq. (41a) for ε(r) into Eq. (29), 3 2 e 2 m πr s = 1 2G , or r s = 3Ge 2 m π .
As the expression (40b) for ε(r) is not very tractable, we make an approximate model for ε(r) by piecing together its two asymptotic forms given by Eqs. (41a) and (41b) at the point where these cross. This results in the continuous approximation, Putting Eqs. (42) and (43) into Eq. (25), we arrive at, At this stage it is convenient to define the dimensionless quantity ∆, and rewrite Eq. (44b) as, The quantities e and m in our results of Eqs. (42), (44), and (46) are the "bare" mass and charge parameters of the theory. We have already seen from the comparison of the large r asymptotic behavior of ε(r) given by Eq. (41b) with the expected behavior given by Eq. (33), that the charge actually measured at large distance, e m , will be less than the bare charge e by a factor of 2/π, Also, of course, the measured mass m m of the electron will not be the bare mass m, but the bare mass plus the electromagnetic field contribution δm, In terms of the measured quantities and ∆, our bare charge and mass become, Our electromagnetic contribution to the electron mass is likewise, which, iterated a couple times, yields, Thus, from Eq. (50), we have that the electromagnetic field contribution is nearly equal to the bare mass, and about half of the measured mass, From Eq. (51) we obtain the critical radius r s , r s ≈ 5.99 × 10 −45 fm, which is an ultramicro black hole indeed! It is so small, in fact, that we need to ask whether the black-hole radius due to the bare mass m doesn't seriously compete. The negative answer to the question is due to the fact that spontaneous virtual pair production and annihilation will also cause the bare mass to be slightly smeared out on the scale of a Compton wavelength, rather than its being a point mass (this is an aspect of the zitterbewegung effect). For the electron at "rest", Weisskopf [1] gives the following bare charge density,ρ, where d(r) is the spherically symmetric probability density, which has the small r asymptotic behavior, We remark here in passing that a charge density which behaves as Cr − 5 2 as r → 0, as does that of Eqs. (56), produces an electrostatic field whose modulus ∼ 8π|C|r − 3 2 as r → 0, which it turn implies an electrostatic energy density ∼ 8πC 2 r −3 as r → 0, rather similar to what is found in Eq. (41a)-the latter, however, includes magnetic (spin) effects as well as electrostatic ones.
In view of the bare charge density of Eqs. (56) for the electron at "rest", it is reasonable to as well model its distribution of bare mass by the energy density, which, of course, has the small r asymptotic behavior, Putting Eq. (57b) into Eq. (29), we have that the bare mass distribution critical radiusr s satisfies, We can thus see thatr s ≪ r s . Indeed, so, notwithstanding the ultrasmall radius r s of the electron's electromagnetic field induced black hole, the black-hole radiusr s associated with its bare mass distribution is vastly smaller, and doesn't interfere with our calculation of δm.
IV. Speculations about Future Developments Having adduced from self-gravitational effects a finite electromagnetic contribution, of order of the bare mass, to the measured mass of the electron, we can ask what implications self-gravitational effects might hold for the remaining ultraviolet divergences of quantum electrodynamics. The key remaining one is associated with the photon's virtual pair dissociation and recombination Feynman diagram of Fig. (3a) (usually referred to as "the vacuum polarization diagram" despite the fact that such an appellation may be more appropriate to the diagram of Fig. (2a)), which diverges logarithmically and makes an infinite contribution to charge renormalization [8]. We recall, of course, that Eq. (41b) implies that the spontaneous vacuum virtual pair and photon production diagram of Fig. (2a) makes a finite (but not small!) contribution to charge renormalization.
We now proceed to speculate about the possibility of gravitationally resolving the ultraviolet divergence in the diagram of Fig. (3a). If we remove one of the external photon propagators from Fig. (3a), we produce Fig. (3b), which lends itself to the interpretation of being the fourdimensional Fourier transform of the stress-energy contribution to the electromagnetic field of the photon's dissociation into and recombination from the virtual pair. This subdiagram carries the logarithmic divergence of its parent diagram, but it is also a second-rank tensor object which is symmetric in its two indices and has vanishing four-divergence (in the algebraic sense which applies to Fourier transforms), because of the gauge character of electromagnetism [8]. These properties, along with dimensional consistency and physical plausibility, support the above-stated interpretation of Fig. (3b) as being the four-dimensional Fourier transform of the stress-energy contribution to the electromagnetic field due to the photon's dissociation into and recombination from the virtual pair. If we now multiply this subdiagram of Fig. (3b) by k 2 (2em) −2 , we sever the remaining external photon propagator and remove all charge dependence, arriving at the lesser subdiagram depicted in Fig. (3c), which still carries the logarithmic divergence and has all the properties required of a four-dimensional Fourier transform of a stress-energy tensor-we can only interpret it as the four-dimensional Fourier transform of the stress-energy of the transient virtual pair itself. This Fig. (3c) subdiagram is written as the trace of a (divergent) convolution of two Dirac matrix functions. These matrix functions are not themselves divergent (only their convolution fails to converge), and each may be individually inverse Fourier transformed back to configuration space. Then the convolution theorem may be applied to easily obtain the stress-energy tensor of which the diagram of Fig. (3c) is the Fourier transform. This stress-energy can be expected to be very localized in space-time (as befits the stress-energy of a virtual pair), and the logarithmic divergence of its Fourier transform must be reflected as a nonintegrable local singularity in this stress-energy itself. We may then expect, in analogy to what we found in Section III, that self-gravitational effects will, via black-hole phenomena, cut off the local singularity of this stress-energy, producing a modified one which is fully integrable and Fourier transformable, enabling us to successively remove the logarithmic divergence from Figs. (3c), (3b), (3a), and charge renormalization.
Let us now wildly leap to the happy conclusion that every ultraviolet divergence of not just quantum electrodynamics, but of any quantum field theory, elicits a self-gravitational response which renders it finite. How would this affect the conventional wisdom that the only admissible quantum field theories are the renormalizable ones? Nonrenormalizable theories would no longer have infinities ultimately contaminating every quantity of interest, instead the gravitational constant G would ultimately enter into every quantity of interest. For, say, every strong interaction scattering cross section to involve G hardly seems physically sensible. So the injunction against nonrenormalizable quantum field theories would still hold, with the justification for it shifted from sheer calculational necessity to considerations of physical plausibility. There is one transcendent exception to the renormalizability requirement under this scenario, however, and that is quantized gravity theory itself-there can certainly be no objection to G entering into every quantity of interest in gravity theory. This is quite pleasing, as quantized gravity theory is indeed nonrenormalizable in the context of the standard perturbation expansion in nonnegative powers of G, and it would be bewildering indeed if this had to be regarded as an insuperable pathology, given the great success and profound physical elegance of general relativity. However, the indications of this paper are that the gravitational cures for quantum field theoretic ultraviolet divergences lie with black-hole phenomena-ultrastrong gravitational effects which involve the likes of G −1 and are absolutely unamenable to sensible perturbative expansion in nonnegative powers of G. Ironically, it seems that the only acceptable nonrenormalizable quantum field theory, quantized gravitational theory, is also the only one which is, in fact, self-consistently finite in its own right. To demonstrate this beyond doubt would, of course, require the development of a powerful nonperturbative approach to quantized gravitation.
Meantime, a program that would progressively banish the ultraviolet divergences from renormalizable quantum field theories by the proper invocation of strong self-gravitational effects could well stimulate useful calculational progress in some of these theories. For the strong interactions, for example, the manifestly covariant perturbation expansion-the environment in which the eggwalk of the renormalization program is carried out-may be calculationally quite inappropriate because of slow or nonconvergence. One would like to have nonperturbative approximation techniques to deal with such strongly coupled renormalizable theories, but the development of these has been hindered by uncertainty over whether the ultraviolet divergences are properly dealt with. Physical understanding that the proper introduction of self-gravitational effects in fact eliminates these divergences could well encourage the confident development of more powerful approximation techniques for such important classes of quantum field theories.