Constraints on the Lepton Charge, Spin and Mass from Quasi-Local Energy
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Constraints on the Lepton Charge, Spin and Mass from Quasi-Local Energy
Bjoern S. Schmekel
Department of Physics, College of Studies for Foreign DiplomaRecipients at the University of Hamburg, 20355 Hamburg, Germany ∗ The masses of the elementary particles as well as their charges and spins (herein expressed in termsof the elementary charge and Planck’s constant, respectively) belong to the fundamental physicalconstants. Presently, no fundamental theory describing them is available, so their values remainmysterious. In this work we offer an approach based on the Brown-York quasi-local energy whichincludes the self-energy of an object. In order to compute this energy we model the spacetime ofthe renormalized electron (and other leptons) by the Kerr-Newman metric. Placing conditions onthe associated energies at different radii we arrive at various constraints on the mass, charge andspin.
I. INTRODUCTION
Even though a unified theory combining quantum fieldtheory and general relativity is still unavailable ulti-mately our conception of elementary particles will haveto be compatible with general relativity. As part of suchan effort the question of the spacetime structure of ele-mentary particles will have to be resolved as well.It has been known since a landmark paper by Carter[1] that the Kerr-Newman metric possesses many inter-esting characteristics including a gyromagnetic ratio of g = 2 which matches the value predicted for a fermiondescribed by the Dirac equation. Indeed there appearsto be a deep relation between the Kerr-Newman met-ric and the Dirac equation which has been thoroughlyinvestigated by Burinskii [2–5] and others, cf. e.g. [6].Apart from such deep interconnects on a more ”macro-scopic level” it should be possible to describe a particleviewed from some distance as a tiny rotating chargedsphere at least to some approximation. According to theno-hair conjecture any black hole solution of the Einstein-Maxwell equations can be completely characterized byits mass, charge and angular momentum with the space-time being given by the Kerr-Newman metric. The latterargument is not completely valid, though, because theKerr-Newman metric describes a real black hole with anevent horizon only if m > a + Q , i.e. when the masssquared is much larger than the sum of the squares ofthe charge and the rotation parameter a = J/m where J is the angular momentum. However, for the electron wehave in geometrized units with G = c = 1 used through-out the entire text m = 6 . · − cm, a = 1 . · − cmand Q = 1 . · − cm, i.e. the rotation parameter aloneexceeds the mass by a factor of 2 . · . Still, theKerr-Newman metric naturally extends into this param-eter space.It has been argued that an electron cannot be a blackhole because its Schwarzschild radius would be many or-der of magnitude smaller than its Compton length thusnot fitting inside its own Schwarzschild radius. There are ∗ [email protected] two problems with this argument. First, it completelydisregards possible effects from the angular momentumand the charge which in our used unit system are muchlarger than the mass. Therefore, these effects can almostcertainly not be neglected. In fact, the Kerr-Newmanmetric used for the present purpose does not describeblack holes with an event horizon since the condition m > a + Q which is necessary for the formation of anevent horizon is violated by a wide margin as mentionedbefore. Second, the notion of mass and energy is a com-plicated one in general relativity and one has to settleon a suitable definition of energy before proceeding witharguments based on energy and mass. The notion is diffi-cult because the influence of gravity can always be gaugedaway at a single point in spacetime. Fortunately, a suit-able definition is available. The Brown-York quasilocalenergy (”QLE”) [7] satisfies all important properties anenergy is supposed to possess. In particular it satisfiesa conservation law [8] which attributes a change in QLEto a flux of ordinary stress-energy and a flux of a secondquantity into or out of the region of interest. There iscompelling evidence [9, 10] that the latter is to be consid-ered a flux of gravitational field energy. Furthermore, theQLE is additive in the sense that the QLE of two disjointregions is the sum of the QLEs contained in the regions.It is defined quasilocally, i.e. in a finite region larger thana point thus avoiding the problem mentioned above andcan be derived naturally from an action principle whereall applicable boundary terms [11] have been added tothe Einstein-Hilbert action. As such the concept is fullycovariant even though the exact value will depend on achosen time slicing and the size and shape of the bound-ary since the region cannot be point-like. Like potentialenergy in classical mechanics the QLE allows to set areference energy adding an arbitrary functional S to theaction which only depends on the induced metric of thechosen boundary. Other than that the functional is arbi-trary and has no influence on the conservation law andthe equations of motion derived from the action. Withan appropriate reference term it is possible to recover theADM limit, though. The reference term has been sub-ject to a thorough investigation in the literature becausethe suggested procedure to determine it proposed in theoriginal article [7] is not always applicable. The develop-ment of modifications to the original Brown-York QLE[12, 13] with a well-defined ADM limit can be seen asefforts to rectify this situation. However, because of thereasons mentioned above the reference term is still arbi-trary. As in past works we set S = 0 which is a perfectlyvalid choice. We will either consider energy differences orargue that a suitable reference term will drop out in thesmall sphere limit. It is worth pointing out that the QLEis implemented as a surface integral over a contractionof a quasi-local stress-energy-momentum surface density.Although evaluating the QLE gives the total energy con-sisting of ordinary stress-energy and gravitational fieldenergy contained in a region enclosed by a boundary itonly requires knowledge of the metric and its derivativesat the boundary. Therefore, results for the QLE of theKerr-Newman metric will be valid as long as the Kerr-Newman metric as an exterior solution to the Einsteinfield equations is valid at the chosen boundary. For theinterpretation of the following results it is important tostress that we ultimately apply the Kerr-Newman metricto the renormalized particle using the renormalized val-ues for charge and mass. We view the renormalized par-ticle as surrounded by a cloud of virtual particles whichare enclosed by the boundary as well and which preservethe axial symmetry of the problem at least if the bound-ary exceeds a minimum size of the Compton length. Likein quantum field theory the effect of the vacuum polar-ization is absorbed by replacing the bare charge and thebare mass with their renormalized counterparts. II. QLE OF THE KERR-NEWMAN METRIC
The QLE of the Kerr-Newman metric given in Boyer-Lindquist coordinates ds = − (cid:18) − mr − Q r + a cos θ (cid:19) dt + r + a cos θr − mr + a + Q dr + (cid:0) r + a cos θ (cid:1) dθ +sin θ r + a + (cid:0) mr − Q (cid:1) a sin θr + a cos θ ! dφ − a (cid:0) mr − Q (cid:1) sin θr + a cos θ dφdt (1)has been computed before [9]. The result is given by − i mr − r − r (cid:0) m + 2 Q + a (cid:1) + 2 mQ + ma | a | p Q − mr + r ˜Ξ E + i mr + r − r (cid:0) m − a (cid:1) + 2 mQ + ma | a | p Q − mr + r ˜Ξ F − ( m − r ) p ( r + a ) ( Q + a − mr + r )2 Q − mr + 2 r = E (2) a/a e m/m e log ( − E / ( r c / E in the plateau region extending fromapproximately r q to r a with the time slice chosen by the unit normal vector u µ = s r + a cos θr − mr + a cos θ + Q δ µt (3)and the boundary described by the unit normal vector n µ = r r + a − mr + Q r + a cos θ δ µr (4)Here, ˜Ξ E ≡ E i (cid:12)(cid:12)(cid:12) ar (cid:12)(cid:12)(cid:12) , | r | p Q − mr + r ! (5)˜Ξ F ≡ F i (cid:12)(cid:12)(cid:12) ar (cid:12)(cid:12)(cid:12) , | r | p Q − mr + r ! (6)and the incomplete elliptic integrals are defined as E ( z, k ) ≡ Z z p − k ζ p − ζ dζ (7) F ( z, k ) ≡ Z z p − ζ p − k ζ dζ (8) III. THREE PARAMETERS, THREECONTRAINTS
Looking at the unreferenced QLE in fig. 1 we recognizea plateau extending roughly between Q = r q < r < r a .The value of E at the plateau E plateau is approximately − r a = − a = − J/m .Miscellaneous values of E plateau can be found in tableI. Demanding r a ≈ r c we obtain J ∼ h with the real valuebeing J = ~ / h/ (4 π ). This is a reasonable conditionbecause we want the ring singularity to be hidden behindthe Compton region. !" ! " $ % & ’ ( ) * + , !" +,-./$012/,345$ 6453/278$ 9:78$ ;1:$ <51=>$012/,345$ FIG. 1. log( − E / r/ J = ~ / . · − cm, | Q | = 1 . · − cm and (from above to below) m e = 6 . · − cm, m µ = 206 . · m e , m τ = 3477 . · m e , m light = 10 − · m e and finally m heavy = 10 · m e . aa A Mathematica notebook deriving and evaluating eqn. 2 can befound in [14]
In the limit r −→ m ≪ Q ≪ a the unreferencedQLE E can be approximated as E ( r = 0) ≈ − m (cid:12)(cid:12)(cid:12)(cid:12) aQ (cid:12)(cid:12)(cid:12)(cid:12) = − (cid:12)(cid:12)(cid:12)(cid:12) JQ (cid:12)(cid:12)(cid:12)(cid:12) (9)Substituting a = J/m for the rotation parameter theQLE in this limit even becomes independent of m . Weobtain E ( r = 0) = − . · E Planck which is indepen-dent of m as long as the stated conditions are met. In[9] we argue that this result is not a coincidence. In factit helps to resolve a puzzle because the Compton wave-length of the Planck energy is identical to its associatedSchwarzschild radius such that the Compton wavelengthcan be confined within the Schwarzschild radius. Elevat-ing this to a principle we have two conditions for J and Q which allows us to solve for them.The final remaining parameter we would like to deter-mine is the (ADM) mass of the particle. Classically, wecan only provide a large range of allowed values. As men-tioned above we treat the leptons as overextreme Kerr-Newman ”black holes” without event horizon for which m < J /m + Q resulting in m < J for J/m ≫ Q and m < Q if J = 0. Both conditions lead to approx-imately the same bound of m < − cm with angularmomentum J = ~ / m > < m < Q . Otherwise,the size of the ring singularity r a = a = J/m would beinfinite.It may appear that it would be hard to lower the elec- tron mass a lot further below its actual value because theCompton length r C and the size of the ring singularity r a are already roughly in the order of 10 − cm. Low-ering the value even further the electron would becomealmost ”macroscopic” putting it in severe unbalance withother subatomic lengthscales, e.g. the proton radius atapproximately 10 − cm. However, one may regard sucharguments as ”anthropic”, and we attempt to come upwith another argument constraining the lepton masses.Continuing with our analysis we consider a small ballwith mass M = E ( r = 0) trapped on the left side inthe potential depicted in fig. 1. The ball is trappedsince both M and the potential E ( r ) are negativewith | M | < | E ( r ) | . The uniformity of E ( r ) below r < r s leads us to regard M as due to pure stress-energy located at r = 0 classically because the Kerr-Newman metric is an electrovacuum solution and grav-itational field energy vanishes in the small sphere limit.(The picture is slightly more involved because for theKerr-Newman metric T µν = 0 even when r > π R R dr R π dθ √ hκ − u µ u ν G µν ≪ E ( r = 0) in the re-gion R < r s . ) Quantum mechanically this small pitof stress-energy at r = 0 is now being spread out withinthe potential well. For simplicity we approximate the po-tential for our initial attempt by a one-dimensional po-tential well with infinitely high walls ignoring the spher-ical character of them problem since for the time beingwe are only interested in order of magnitude estimates.For its width we consider the two cases d = d min and d = r Planck with the wavefunction of the ball penetratingsomewhere in between these two values whre d min is theminimum width required for an energy eigenvalue smallerthan E ( d min . A. Case I: d = r Planck
For a potential well with infinitely high walls the en-ergy of the ground state is given by E = ~ π M (2 d ) (10)Assuming the ball can move freely around within thepotential well and setting d = r Planck ∼ r q |E | = (cid:12)(cid:12)(cid:12)(cid:12) ~ π M m (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) ~ m (cid:12)(cid:12)(cid:12)(cid:12) (11)we obtain m < . · − cm. However, this upper boundis too large because the wavefunction will not penetratethrough the whole barrier. Note that we have obtainedthe similar bound m < Q before requiring the absence ofan event horizon.There is yet a third way of receiving this upper limit.If we keep raising the ADM mass until m ∼ Q the upperand the lower plateau in fig. 1 become equal and the po-tential well vanishes. For the last inequality we assumed |E | < | E plateau | since the well is actually not infinitelyhigh and we are interested in a bound state. B. Case II: d = d min We fit the slope region of E between the upper andthe lower plateau by the following function E , slope ( d ) = − d · Jm · cm − (12)Looking for the ground state we require |E | ≤ | E , slope | which yields an effective minimum size of the potentialwell d ≥ ~ π M · Jm · cm − (13)For the three charged leptons we obtain d e = 1 . · − cm, d µ = 7 . · − cm and d τ = 2 . · − cmwhich are all very similar due to the cube root involved.Spotting these distances in fig. 1 they seem to approachthe upper edge of the well, but they are all well below d < r q as required. The ratio of the the potential energyat this location to the energy of the upper plateau is inthe order of 10 − which may be a sensible value becausewe do not want the ball to tunnel out of the well.In light of these result there is hope that the allowedvalues of m will ultimately be in a reasonable mass rangeonce the wave-equation is solved for the full potentialgiven by eqn. 2. IV. NEUTRINOS
The expression for E in eqn. 2 diverges in the limit Q = 0, and it is presently not seen how the Kerr metriccould be applied to the three generations of neutrinos. The space-time structure of the Kerr metric may not evenbe applicable to uncharged particles even with spin J = ~ /
2. Maybe this is a hint that the neutrino should notbe treated as a Dirac spinor but as a Majorana spinorinstead.
V. CONCLUSIONS
Modeling the spacetime of the charged leptons by theKerr-Newman metric we came up with two simple condi-tions constraining their angular momentum and chargeto be J ∼ ~ and Q ∼ √ ~ , respectively, ignoring smallfactors. Since ~ is the only scale in our model with m having the unit of length in our chosen unit system con-straining the lepton masses is harder with the exceptionof the inequality 0 < m < Q which has to hold such thatthe Kerr-Newman metric is overextreme and the size ofthe ring singularity is smaller than infinity.Our crude model of the potential well only results ina mass range which is too broad to be useful. In futurework it is planned to solve the full eigenvalue problem forthe potential given in eqn. 2 with the additional require-ment of making the particle stable against the tunneleffect. Alternatively, it also may be possible to constructa potential from the expected eigenvalues and comparethe resulting potential with eqn. 2 using a procedureoutlined in [15]. ACKNOWLEDGMENTS