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Classical Electrodynamics of Extended Bodies

P.D. Flammer1, ∗

1Colorado School of Mines, Golden, Colorado, USA

(Dated: January 28, 2021)

We study the classical electrodynamics of extended bodies. Currently in the literature, no self-consistent dynamical theory of such bodies exists. Causally correct equations can be producedonly in the point-charge limit; this has the unfortunate result of infinite self-energies, requiringsome renormalization procedure, and perturbative methods to account for radiation. We review thehistory that has led to the understanding of this fact.

We then investigate possible self-consistent, non-point-charge, classical electrodynamic theories.We do this by first removing point charges from the Lagrangian, and then including all first-orderderivative terms of the electromagnetic field tensor, while fully accounting for space-time curvature.This results in direct current-current interactions, as well as curvature-mediated, short-range inter-actions, one of which violates parity. These could be interpreted as non-electromagnetic short-rangeforces between small, stable objects. For the simplest possible theory (which does not include thecurvature mediated terms), we find a stable spherical solution, which due to an integrable singularityat the solution’s center, has quantized mass and charge. Adding the curvature mediated interactionsappears to preclude spherically, charged solutions in the more general theory, while maintaining thepossibility of spinning solutions with quantized mass and charge.

PACS numbers: 03.50.De,04.40.Nr,04.20.-q,03.50.Kk

Contents

I. Introduction 1

II. Historical Review 2A. Development of Maxwell’s Equations 2B. Radiation, Self-Interaction, and Special

Relativity 3C. Issues with Self-Interaction 4D. Point Charges and Quantum Mechanics 5E. Gravity and Electromagnetism 7

III. Mathematical Review of Classical

Electrodynamics 7A. Notation 8B. Dynamics of a Charged Object 8C. Least Action Principle and Deriving

Maxwell’s Equations 11

IV. Completing the Stress-Energy Tensor 13A. Integrability 13B. One Possible Addition to the Stress-Energy

Tensor 14C. Principle of Least Action Revisited 15D. Least Action, Energy-Momentum

Conservation, and DiffeomorphismInvariance 19

V. Solutions to Equations of Motion 21A. Superluminal (Space-Like) Currents 21B. Spherical Solutions in Flat Space-Time 22

∗Electronic address: pflammer@mines.edu

C. Curved Space-Time and Mass/ChargeQuantization 22

VI. Summary 25

VII. Acknowledgments 26

References 26

I. INTRODUCTION

There is currently no completely suitable theory de-scribing the dynamics of charged bodies. Feynman, in hisfamous lectures, describes the situation as follows (see [1]Vol.2 Ch. 28):

You can appreciate that there is a failure ofall classical physics because of the quantum-mechanical effects. Classical mechanics isa mathematically consistent theory; it justdoesn’t agree with experience. It is inter-esting, though, that the classical theory ofelectromagnetism is an unsatisfactory theoryall by itself. There are difficulties associatedwith the ideas of Maxwell’s theory which arenot solved by and not directly associated withquantum mechanics. You may say, “Perhapsthere’s no use worrying about these difficul-ties. Since the quantum mechanics is goingto change the laws of electrodynamics, weshould wait to see what difficulties there areafter the modification.” However, when elec-tromagnetism is joined to quantum mechan-ics, the difficulties remain. So it will not be a

http://arxiv.org/abs/1802.02141v5mailto:pflammer@mines.edu

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waste of our time now to look at what thesedifficulties are.

The difficulties come in two flavors: (1) if a charged ob-ject is extended, i.e. it has some non-zero size, it has notbeen possible to develop self-consistent equations of mo-tion for that object; and (2) if we take the point-chargelimit (to resolve difficulty 1, or because we think physicalcharges are points), the energy of the charge becomes in-finite. Feynman continues by discussing examples in theliterature of researchers trying to resolve the infinite en-ergy of point charges, finally concluding that all attemptshave failed[1]:

We do not know how to make a consistenttheory-including the quantum mechanics-which does not produce an infinity for theself-energy of an electron, or any point charge.And at the same time, there is no satisfactorytheory that describes a non-point charge. It’san unsolved problem.

The infinite “self-energy” of a point charge is due toself-interaction: its constituent parts repel against theother parts; as one tries to pack charge into a ball, thesmaller the ball, the harder this is, and more work mustbe done to make it more compact. The result is the en-ergy required to form such a ball is inversely proportionalto the size of the ball; and point charges have infinite self-energy. This is in apparent contradiction to the physicalfact that a very compact electron exists.Another consequence of self-interaction is radiation: as

a charge is accelerated, its interaction with its own field(in addition to adding to its rest energy) can cause thecharge to recoil, as momentum is carried away by thefields in the form of radiation. Again, the classical reac-tion of a charge to radiation is intractable. J.D. Jacksonsummarizes this difficulty in his text as (See Sec. 16.1 of[2]):

...a completely satisfactory classical treat-ment of the reactive effects of radiation doesnot exist. The difficulties presented by thisproblem touch one of the most fundamen-tal aspects of physics, the nature of an ele-mentary particle. Although partial solutions,workable within limited areas, can be given,the basic problem remains unsolved.

As we will see in the next section, the root cause ofthese problems is our ignorance of what keeps an electron(or other compact charge) compact. As it turns out, ithas been very difficult to even postulate a self-consistenttheory of what could bind an electron (or other compactcharge) together; no such theory exists to date.In this paper, we attempt to address this fundamen-

tal question. This cannot be addressed by treating pointcharges, as they are compact by construction. Also, thiscannot be done in a quantum mechanical framework,because quantum mechanics presupposes point charges

(the idea of an extended object is foreign to the theory).Therefore, we investigate the classical electrodynamics ofextended bodies.

II. HISTORICAL REVIEW

We start by reviewing some of the history of the devel-opment of the theory of electrodynamics. The main pur-pose of this section is to explore the evolution of thought,evolving from Coulomb’s law to the theory of quantumelectrodynamics, focusing on the issue of self-interaction,the resulting self-inconsistency of electrodynamic theory,and the necessary introduction of point charges alongwith their pathologies.

A. Development of Maxwell’s Equations

With the invention of the Leyden Jar (a rudimentarycapacitor) in the middle of the 18th century, experimen-talists were able to repeatably apply charge to variousobjects, and determine how charged objects affect eachother. By 1785, Coulomb had established the mathemat-ical form of this electrostatic force[3], the law being verysimilar to Newton’s law of gravitation.

Near the turn of the 19th century, Alessandro Voltainvented the voltaic pile (battery). This enabled exper-imentalists to more reliably study electrical flow in cir-cuits. In the summer of 1820, Oersted discovered theamazing fact that magnetic needles were affected by elec-tric currents, linking what were before thought to be sep-arate phenomena, electricity and magnetism[4]. Withina few months, Biot and Savart successfully determinedthe mathematical behavior of the force between a cur-rent carrying wire and a magnetic pole[5–7]. By 1827,Ampere had also shown that solenoids of current carry-ing wire behaved similarly to bar magnets, and studiedthe magnetic force between two circuits[8].Also in the 1820s, Ohm successfully described that the

current in a conductor was proportional to the electromo-tive force and the conductance of the material[9]. Thiswas the primary “force law” (now called Ohm’s law) usedby physicists for electrodynamics until near the turn ofthe 20th century. In 1831, Faraday discovered that mov-ing a magnet near a wire circuit induced a current in thecircuit, discovering electromagnetic induction[10].

Various physicists worked to understand the interac-tion between magnets and currents for the next fewdecades[11, 12]. One theoretical achievement, which wasimportant to the development of electromagnetic theory,was the use of “potentials”. In 1857, Kirchhoff first wrotethe electric force as a combination of the gradient of ascalar potential (which had already been used for sometime in electrostatic problems) and the time derivative ofa newly introduced vector potential[13]. Kirchhoff alsoshowed, in that particular formulation, that the vector

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and scalar potential were related to one another (in mod-ern terminology, describing the particular gauge, whichhe was using).All of this work found some closure in the 1860s. In

1861 and 1862, Maxwell published “On physical lines offorce”[14] (where he added the necessary displacementcurrent1), and in 1865, he presented a complete frame-work of electromagnetism in “A Dynamical Theory of theElectromagnetic Field”[15]. This theory was extremelysuccessful at describing all of the electrical phenomenaknown at the time; he also calculated that electromag-netic waves propagate at a speed close to the speed oflight (which had recently been measured), thus identify-ing light as an electromagnetic wave.A key piece of this new theory was that important

dynamics took place in the space between electrified ob-jects. This was a major shift in thought: up to thattime, interactions were typically thought of as “actionsat a distance”. In Maxwell’s words:

These [old] theories assume, more or less ex-plicitly, the existence of substances the par-ticles of which have the property of actingon one another at a distance by attraction orrepulsion.[15]

Maxwell differentiated his new theory in this way:

The theory I propose may therefore be calleda theory of the Electromagnetic Field, be-cause it has to do with the space in theneighborhood of the electric or magneticbodies.[15]

This was the birth of physical field theories, where theoriginal concept of a “field” was that important dynamicsoccur (and propagate) throughout the space (or field)between interacting bodies.To motivate the fact that electromagnetic interactions

could propagate through “so-called vacuum”, Maxwellused the idea of disturbances propagating through anelastic medium, called the “luminiferous aether”. How-ever, although Maxwell used this idea of an underlyingelastic medium to develop the theory, he gave up on hy-pothesizing its exact character or role:

I have on a former occasion, attempted todescribe a particular kind of motion and aparticular kind of strain, so arranged as toaccount for the phenomena. In the presentpaper, I avoid any hypothesis of this kind;and in using words such as electric momen-tum and electric elasticity in reference tothe known phenomena of the induction ofcurrents and the polarization of dielectrics,

1 The displacement current is mathematically necessary to con-serve charge.

I wish merely to direct the mind of thereader to the mechanical phenomena whichwill assist him in understanding the electri-cal ones. All such phrases in the present pa-per are to be considered as illustrative, not asexplanatory.[15]

Immediately after the previous statement, however, hestresses the importance of the field:

In speaking of the Energy of the field, how-ever, I wish to be understood literally... Onthe old theories it resides in the electrifiedbodies, conducting circuits, and magnets...On our theory it resides in the electromag-netic field, in the space surrounding the elec-trified and magnetic bodies, as well as inthose bodies...[15]

The fact that the field could contain energy in its ownright allowed him to effectively describe how fields trans-port energy via radiation through a vacuum, and equatelight and heat with electromagnetic waves2.

B. Radiation, Self-Interaction, and SpecialRelativity

With the connection of electromagnetism and light, itbecame clear that currents, which change in time, gener-ate electromagnetic waves, i.e. radiation. The radiatedenergy due to a varying electrical current was calculatedby Fitzgerald in 1883[18], and a general vectorial lawfor the flow of electromagnetic energy and its conserva-tion was derived in 1884 by Poynting[19]. Experimentalgeneration and measurement of electromagnetic radia-tion at lower-than-optical frequencies was achieved byHertz in 1887 using oscillating electrical circuits[4, 20].Poincaré immediately noted that such radiation mustcause damping within the oscillator due to the energyit carries away[21].Also in 1887, the concept of the aether was discounted

by the experiment of Michelson and Morley3. With theaether gone (or at least going), the “field” was no longera description of “space in the neighborhood”. The elec-tromagnetic field necessarily took on a life of its own; the

2 Later in the 1860s, Lorenz and Riemann alternatively describedthe interactions between currents and charged objects as re-tarded integrals of the charge and current rather than focusingon the dynamics of the field[16, 17]. This point of view had someadvantages; in particular, it didn’t motivate the existence of theaether. However, in the late 19th and early 20th centuries, elec-tromagnetic theory predominantly grew out of Maxwell’s theory,and the contributions of Lorenz and Riemann were somewhatforgotten until later[4]

3 This showed that the speed of light was independent of direction;very unlikely if it is a disturbance of an underlying medium thatthe earth was likely moving through.

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field (or radiation, or the energy-momentum it carries,etc.) was its own substance.

Up until the late 19th century, experimentally and the-oretically, continuous charge and current densities (in cir-cuits) were the primary focus of study: Maxwell’s equa-tions were used to calculate the field, and Ohm’s law wasused to calculate how fields caused the current to evolve.However, in the 1880s, many researchers turned theirattention to calculating the fields of discrete, compactcharges, including Heaviside[22], who is often creditedwith writing Maxwell’s equations in their more modernform.

In 1892, Lorentz published “Maxwell’s electromagnetictheory and its application to moving bodies”; in this pa-per, he wrote down the force from an external electro-magnetic field on a charged particle (point charge), nowcalled the Lorentz force; he also formalized the gaugeinvariance of electromagnetism[23]. Lorentz also notedthat, in general, one must account for the electromag-netic force of a discrete charge on itself. In his 1892 pa-per, Lorentz evaluated this self-force and calculated theequations of motion for a “relativistically rigid” spheri-cal shell of charge (where the sphere maintains its shapein its proper frame)4. This was done in the limit of thesphere being small, so higher order terms in the size of thesphere could be ignored. It was found that the self-forcecontains a term, with magnitude inversely proportionalto the size of the sphere, which can effectively be addedto its inertial mass.

Additionally, a term appeared in the force equation,which is independent of the size. This force came tobe known as the “radiation reaction” or “field reaction”,although it seems Lorentz initially did not connect thisreaction to radiation; Planck appears to be the first todo so in 1897[25]. Also in 1897, J.J. Thomson discoveredthe existence of the electron, which fueled further studyof small, discrete charges.

Lorentz initially calculated this self-field reaction inthe low-velocity limit (or, if you like, in the proper frameof the charged body), but by the early 1900s, Abraham(and then Lorentz) had extended this theory to arbitraryvelocities[26, 27]. Also, during this time, the hypothesisthat the electron mass was due entirely to the electromag-netic self-interaction gained some favor (Abraham explic-itly assumed it was the only contributor to the electronmass5).

In 1905, Einstein published “On the Electrodynamics

4 Lorentz called this model a deformable sphere, because he no-ticed (before Einstein’s theory of relativity) in a moving frame,the electron would contract, but this model is now called rela-tivistically rigid[24].

5 It appears this was done, at least in part, because at the time,it was thought (before Einstein’s Special Relativity) that anyother mass would not transform between reference frames in thesame way as electromagnetic mass; see [24] for a discussion ofthis history.

of moving bodies”, where he introduced his concept ofspecial relativity[28]. In a paper later that year, he pro-posed that the inertial mass of a body was directly pro-portional to its energy content[29]. With this, one couldcalculate the mass due to the energy stored in the elec-tromagnetic field for a charged object. It’s interestingto note that, although they preceded special relativity,the equations of Lorentz and Abraham exhibited manyspecial-relativistic effects (e.g. the fact that the speed ofthe charge can only asymptotically approach the speedof light).

C. Issues with Self-Interaction

All of these developments gave some hope that a fullysuccessful model of the electron was within reach. How-ever, there were serious issues with the model. In 1904,Abraham derived a power equation of motion for the rigidmodel of an electron. Unfortunately, the power equationwas not consistent with the force equation derived ear-lier: the scalar product of the velocity and the force doesnot equal the power. Also, in the context of relativity,the power and the force do not form a 4-vector.

There is an issue with the inertial mass as well, whichone calculates from the Lorentz-Abraham equations: fora spherical shell, it is 4/3 times the mass that one obtainsfrom the energy stored in the electrostatic fields (the self-energy). This was not noticed originally by Lorentz orAbraham as their theory preceded special relativity, butin the second edition of Abraham’s book, Abraham men-tions this discrepancy[24, 30].

The equations of motion also violate causality. If aforce is instantaneously “turned on” and one excludesrunaway solutions, pre-acceleration solutions exist (thecharge accelerates before the force is turned on)[24].

In 1906, Poincaré pointed out the source of most ofthese problems: in order for a stable charged object to ex-ist, there must be non-electromagnetic forces, which bindthe electron together (keeping it from exploding due toits self-electric field): “Therefore it is indeed necessary toassume that in addition to electromagnetic forces, thereare other forces or bonds”[24, 31]. He came to the conclu-sion that while this other binding force integrated to zeroover the object, the integrated power from the bindingforce was not zero, and exactly canceled the discrepancybetween the force and power equations. However, thisdid not resolve the “4/3 problem”. In order to correctthat, one must include some “bare mass” of the charge6,which was set to zero by early authors.

6 The bare mass of a charge is what its mass would be if it had noelectromagnetic field. One cannot set this to zero for arbitrarygeometries of charge. If one sets the geometry, the bare massmust take a specific value in order to be consistent with the self-energy[24, 32]

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The problems associated with the radiation reaction,the 4/3 problem, pre-acceleration, etc., continue to re-ceive some attention in the literature. See the follow-ing references for examples from the 21st century[33–44].Ref. [45] has a concise historical overview of the problem.A full history and detailed treatment of the spherical

shell, with a description of the cause of these paradoxesmay be found in a comprehensive monograph by ArthurYaghjian[24]. In particular it’s worth noting, the pre-acceleration issue can be traced to the fact that “turningon” a force creates a non-analytic point in the force as afunction of time, which invalidates the derivation of theequations of motion. If the force is analytic as a functionof time, no pre-acceleration appears in the point-chargelimit[24]. The remainder of the problems are due to twomissing items in electromagnetic theory: the bare massof electromagnetic charge and a suitable binding force tobind a compact charge together.It is an interesting fact of relativity and electrodynam-

ics, that one has less freedom in choosing the problemsone can treat than in general classical mechanics, such asinstantaneously “turning on” a force (although concep-tually possible, it’s difficult to think a real relativisticallyvalid force can turn on instantly). Most strikingly, onecannot consistently consider the dynamics of a chargedobject without appropriately balancing the forces on itsconstituent parts and knowing its internal mass density.The idea of considering a blob of charge, without con-sidering what it is made of and what binds it together,yields inconsistent equations of motion.One may attempt to model a certain structure, like the

rigid sphere, and after the fact, add in what the bindingforce must have been, and state what it’s bare mass musthave been. However, this inherently violates causality,since the binding force is required to react across theentire object instantaneously.Thus, in order to really create a self-consistent dy-

namical theory for extended charged bodies, one mustknow two things a priori: the local binding force den-sity that creates stability, and the bare mass density ofthe charge. There was some effort in the early 20th cen-tury to this end. In the 1910s, an idea originated by Miegenerated some hope, albeit short-lived[46, 47]. Einsteincommented on these developments in 1919:

Great pains have been taken to elaborate atheory which will account for the equilibriumof the electricity constituting the electron.G. Mie, in particular, has devoted deep re-searches to this question. His theory, whichhas found considerable support among theo-retical physicists, is based mainly on the in-troduction into the energy-tensor of supple-mentary terms depending on the componentsof the electro-dynamic potential, in additionto the energy terms of the Maxwell-Lorentztheory. These new terms, which in outsidespace are unimportant, are nevertheless effec-tive in the interior of the electrons in main-

taining equilibrium against the electric forcesof repulsion. In spite of the beauty of theformal structure of this theory, as erected byMie, Hilbert, and Weyl, its physical resultshave hitherto been unsatisfactory. On theone hand the multiplicity of possibilities isdiscouraging, and on the other hand those ad-ditional terms have not as yet allowed them-selves to be framed in such a simple form thatthe solution could be satisfactory[48].

In the same paper, Einstein proposed gravity as a possi-ble binding force for the electron by modifying his fieldequations; this admitted stable solutions, but could notexplain charge quantization, causing him to abandon thatline of thought[48].None of these studies came to result in any suitable

theory, and eventually support for this direction waned.To quote Weyl from the early 1920s,

Meanwhile I have quite abandoned thesehopes, raised by Mie’s theory; I do not believethat the problem of matter is to be solved bya mere field theory[47].

D. Point Charges and Quantum Mechanics

Without knowing the bare mass density and bindingforce for a charged object, one cannot solve for what sta-ble objects should exist. But as mentioned in the lastsection, one can assume a structure, and add in what thebinding force and bare mass should have been after thefact, unfortunately violating causality since the force isrequired to react instantaneously across the object (tomaintain the pre-ordained shape). But if the object is apoint, no time is required for signals to cross the objectand causality is restored7. Also, this removes all internaldegrees of freedom, so conservation of total momentumdetermines the motion completely.One cannot directly calculate the energy or mass of an

object, because we do not know the bare mass. Whatwe can calculate, from the electromagnetic energy of thecharge, diverges in the point-charge limit[22]. But themass of the electron is a measurable quantity; ratherthan calculate it, one may simply use its measured value.This process of replacing an infinite calculated value witha measured value is often called “renormalization”. Inthe context of classical dynamics of an electron, Diracis credited with writing down the “renormalized” classi-cal equations of motion of a charged particle in 1938[49],where he developed these equations in a manifestly co-variant method8. These renormalized equations of mo-

7 The external force still must be analytic as a function of time, oryou will still have the pre-acceleration issues discussed earlier[24].

8 Note Von Laue had already written down the covariant radiationreaction much earlier[50].

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tion are often called the Lorentz-Abraham-Dirac equa-tions of motion.

In any case, in the early 1900s, atomic structure wasforcing physicists to rethink their perception of reality.With the discovery of the atomic nucleus around 1910,the idea that electrons orbit the nucleus (in the sameway as planets orbit the sun) took root. However, anysimple classical model of the electron (such as Lorentz’ssphere of charge) cannot produce stable orbits aroundan atomic nucleus, precisely due to the damping effectof the radiation reaction: an electron in orbital motionwill radiate energy away and its orbit will decay. If oneignores the radiation reaction, then classically one findsa continuum of possible orbits, which is also not what ismeasured: discrete, stable energy levels are observed foratomic orbits.

Due to these difficulties, in the 1910s and 1920s, newways of thinking about these physical systems emerged,which were more successful at describing atomic phe-nomena: quantum mechanics. In the “old quantum me-chanics” (sometimes called the Bohr model, or Bohr-Sommerfeld model), there was not much departure fromclassical thought. The electron was assumed to exist asa point (or at least very small) charge; classical orbitswere then solved for the electron, and integrals of gen-eralized momenta along the orbits were required to beinteger multiples of the Planck constant, which yieldedthe correct energy levels (see Sommerfeld’s 1921 book onthe subject[51]). Note however, the classical equations ofmotion, which were used, ignored the radiation reaction,and the topic of the stability or self-interaction of parti-cles was avoided altogether. In any case, this was verysuccessful at predicting energy levels for simple systems,such as Hydrogen.

In the last half of the 1920s, the more modern quan-tum mechanics took shape. A new “wave mechanics”approach was developed where classical equations of mo-tion, such as the classical Hamiltonian (again withoutany self-interaction/radiation reaction), are taken, and“quantized” (dynamic variables become operators on awave function, which describes the state of the system)to develop equations such as the Schrodinger equation,which was published in 1926[52]. At the same time, adifferent formulation, “matrix mechanics” was developedby Heisenberg, Born, and Jordan[53], which was shownto be equivalent to the wave mechanics approach.

Initially, all of this was done for low, non-relativisticvelocities. However, in 1930, Dirac developed the rel-ativistic generalization of Schrodinger’s equation forelectrons[54]. With the success of the Dirac equationin predicting energy levels in simple atoms (includingthe interaction with the electron spin), attention turnedto describing self-interaction/radiative corrections in theframework of quantum mechanics. This was done bystarting with a non-interacting solution, and “perturb-ing” it by adding in successive interaction terms (thesedays, “Feynman diagrams” are used to do bookkeepingon what terms are needed)[55]. However, any attempt to

add in certain self-interaction terms resulted in infinity(similar to the classical case). To illustrate some of thefrustration of the time, in 1945, Feynman and Wheelerpublished “Interaction with the absorber as the mecha-nism of radiation”, where they proposed that electronsdo not interact with themselves at all[56] (see [1] Vol. 2,Ch. 28 for more discussion and other examples of effortsto remove this infinity). But by 1949, Schwinger andTomonaga developed methods, which circumvent the is-sue of infinite self-interaction, while accurately predictingthe Lamb shift and the anomalous magnetic moment ofthe electron. Infinite self-interaction terms are absorbedinto quantities, such as mass and charge, and the exper-imentally measured values are used in place of the infi-nite calculated ones[55, 57]. As mentioned above, thisprocess of dealing with infinite calculated values is calledrenormalization9. The perturbative process of adding inappropriate interaction terms, in conjunction with renor-malization is what is now called quantum electrodynam-ics.

The “standard model”, built on these principles, isextremely successful at predicting quantities outside ofthose which require renormalization. While the renor-malization program allows physicists to do useful cal-culations, the lack of ability to calculate the masses ofparticles is less than ideal. In 1979, Dirac, speaking ofrenormalization, said

It’s just a stop-gap procedure. There must besome fundamental change in our ideas, prob-ably a change just as fundamental as the pas-sage from Bohr’s orbit theory to quantum me-chanics. When you get a number turning outto be infinite which ought to be finite, youshould admit that there is something wrongwith your equations, and not hope that youcan get a good theory just by doctoring upthat number.[58]

Feynman, who shared a Nobel prize, in part for devel-oping the renormalization program, also was skeptical inhis later years. In his 1986 book, he wrote

The shell game that we play to find n [baremass] and j [bare charge] is technically called“renormalization.” But no matter how cleverthe word, it is what I would call a dippyprocess! Having to resort to such hocus-pocus has prevented us from proving thatthe theory of quantum electrodynamics ismathematically self-consistent. It’s surpris-ing that the theory still hasn’t been provedself-consistent one way or the other by now;

9 Often in practice, one assumes a bare mass or bare charge thatare also infinity in just the way needed to cancel the infiniteself-interaction and result in the measured value.

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I suspect that renormalization is not mathe-matically legitimate.[59]10

In addition to failing to predict quantities such as massand charge of particles, renormalization is somewhat atodds with general relativity: general relativity is non-renormalizable (one cannot play the same game and getany useful predictions from calculations). This troublingfact is a motivator in the study of string theory, whereparticles are stretched into strings: different excitationsof strings are the different particles of nature, with finiteself-energies. Unfortunately, to this date, despite signifi-cant effort, string theory has yet to demonstrate itself asa suitable theory that can predict experimental results.

E. Gravity and Electromagnetism

Speaking of general relativity, we skipped over somedetails on historical attempts to integrate the theory ofgravitation and electromagnetism. Einstein introducedhis theory of gravitation, general relativity, in 1915[60].However, since gravity is most important at astronomi-cal scales, for planets, stars, black holes, etc., which arenot likely to carry significant excess charge, the vast ma-jority of theoretical and computational studies in generalrelativity consider uncharged situations[61].There have been various efforts throughout the 20th

century to “unify” gravity and electromagnetism, whereresearchers have attempted to describe electromagnetismin the context of a generalized theory of the geometry ofspace-time (see [62] for a review). However, more discus-sion of this type of unification of gravity with electromag-netism does not contribute to our purpose here: for theentirety of this paper we take a conventional “dualisticview”, where matter is treated separate from geometry;it is the source of geometry’s curvature.Concerning the presence of electromagnetic charge in

conventional general relativity: The metric for the spaceoutside of a charged spherical object was published in1916 and 1918 by Reissner[63] and Nordstrom[64]. Studyof the interior of charged objects was not attempted untilmore recently than other history outlined here, startingmainly in the latter half of the 20th century. For exam-ple, charged polytropic stars have been studied[65], aswell as charged situations with various other equationsof state and space-times[66–68]. For a fairly comprehen-sive discussion and characterization scheme of sphericallycharged solutions in general relativity, see Ref. [61].Because the electromagnetic stress-energy tensor has a

non-zero divergence in the presence of charge, one can-not use it as the sole source in Einstein’s field equations:

10 It is interesting that some, who were so integral to developingquantum electrodynamics into its current state, had such opin-ions; it may be the only time someone has described a goodportion of their Nobel Prize winning topic as “dippy”.

Einstein’s tensor has a zero divergence due to the Bianchiidentity, and cannot be equated to a tensor with non-zerodivergence. Therefore, treating situations with electro-magnetic charge in general relativity is even more dif-ficult than in special relativity: without some additionto the electromagnetic stress-energy tensor, one cannotstart to solve the simplest problem.

Also, one cannot introduce point particles to supple-ment the stress-energy tensor: their infinite energy den-sity creates singularities in space-time. Therefore, in theliterature where electric charge is studied in general rel-ativity, the electromagnetic stress-energy tensor is aug-mented typically using a fluid, where the density of thefluid is proportional to the density of its constituent par-ticles, i.e. a continuum approximation. The addition ofthe fluid results in 6 dynamic degrees of freedom at eachpoint in space-time (3 in the fluid and 3 in the electro-magnetic current). This makes the field equations un-derdetermined (there are only 3 dynamical equations ofmotion at each point in space-time), and the charge dis-tribution must be set as a model parameter, rather thansolved for by the dynamics[61]. Interestingly, this makesfinding many solutions easier, since one has free param-eters to tune11.

Recently, some attempts at modeling a “charged fluid”appear in the literature, where the electromagneticcharge is stuck on the fluid (to remove the extra degreesof freedom): the energy density of the fluid is tied in anad-hoc way to the density of the charge. For instance,this has been done (in a spherically static case) by addinga perfect fluid stress-energy tensor to the electromagneticstress-energy tensor and setting the energy density of thefluid to be proportional to the charge density squared[69].To obtain stable solutions, negative pressure is required(since the charge self-repels), and the equation of state(the relationship between the energy density, ǫ, and pres-sure, P , of the fluid) is set to P = −ǫ[70–78]. This equa-tion of state has been called the “false vacuum,” “degen-erate vacuum” and “vacuum fluid” among other names.All of these attempts center around special cases (e.g.static situations with spherical symmetry), rather thantreating the general problem, which again is unsolved.

III. MATHEMATICAL REVIEW OFCLASSICAL ELECTRODYNAMICS

Having reviewed some of the history of the develop-ment of electrodynamics, let us now review the currentstate of the associated mathematics. The electromag-netic field generated by a charge distribution is calculatedvia Maxwell’s equations, and is without pathology. We

11 As Ivanov writes, “The presence of charge serves as a safetyvalve, which absorbs much of the fine tuning, necessary in theuncharged case.”[61]

8

refer the reader to [2] for review. However, as mentionedin the history, due to our lack of knowledge of the baremass and binding force for fundamental charged objects,a full treatment of the dynamics of a charged object, in-cluding self-interaction, is intractable. We now lay downthe mathematics of why this is so.

A. Notation

For the remainder of the paper, the following nota-tion will be used (unless otherwise noted). Capital itali-cized variables with Greek superscripts or subscripts, i.e.Fµν or F

µν , are tensors defined in 4-space; Greek indicesvary from 0 to 3, with the 0th element being the timecomponent, and 1-3 being space components. Lowercaseitalicized variables with Greek superscripts, i.e. jµ, aretensor densities. Bold italic variables, i.e. F , are dif-ferential forms (totally antisymmetric covariant tensors).Bold non-italic variables, i.e. B, are spatial vectors, anditalicized variables with Latin indices, i.e. Bi, are alsospatial vectors, with indices varying from 1 to 3. Re-peated indices in a product, unless otherwise noted, areimplicitly summed over.

We assume a space-time characterized by coordinatesxµ = (t, xi), with a metric, gµν with signature (-+++).The determinant of the metric is written as g (with noindices). The totally antisymmetric (Levi-Civita) tensor

is written as ηαβγδ =√

|g|ǫαβγδ, where ǫαβγδ has compo-nents±1, 0. We may also write ηαβγδ = 1√

|g|ǫαβγδ, where

ǫαβγδ = −ǫαβγδ (more generally, ǫαβγδ = (−1)neǫαβγδ,where ne is the number of negative eigenvalues of gµν).∇ is the spatial gradient operator (operating on spatialvectors),∇µ is the covariant derivative (operating on ten-sors), d is the exterior derivative (operating on differen-tial forms), ∗ is the Hodge star operator (operating ondifferential forms), δ ≡ ∗d∗ is the codifferential opera-tor (operating on differential forms)12, and ∂µ or ∂i isthe partial derivative with respect to the coordinate ofthe subscript. Relativistic (geometrized) units are usedthroughout.

The covariant representations of kinematic variablesare:

rµ = (t, ri)vµ = (γ, γvi)

(1)

where r = ri is the position of a point (i.e. the center ofmass of an object) at time t, both with units of length;v = vi is the unitless fraction of the velocity of a pointto the speed of light, c (or equivalently, c = 1); γ is the

Lorentz factor, γ = 1/√1− v2.

12 For background on differential forms and the operations de-scribed here, see Refs.[79, 80]

The antisymmetric part of a tensor may be writ-ten using square brackets in the indices as A[µBν] ≡12 (AµBν − BνAµ); likewise, parentheses in indices rep-resent the symmetric part of a tensor. Square bracketsaround two operators signifies the commutator, for in-stance, [∇µ,∇ν ] ≡ ∇µ∇ν − ∇ν∇µ. Square brackets orparentheses elsewhere have no special meaning.

The covariant representations of the electromagneticvariables are:

Aµ = (φ,Ai)Fµν ≡ 2∇[µAν] (or F ≡ dA)

Fµν =

0 E1 E2 E3

−E1 0 B3 −B2−E2 −B3 0 B1−E3 B2 −B1 0

Jµ = (ρ, J i) ≡ ∇νFµν ,

(2)

where Aµ is the electromagnetic potential (made of thescalar and vector potentials, φ and A = Ai), which areunitless; Fµν is the electromagnetic field tensor (made upof the electromagnetic fields, E = Ei and B = Bi), withunits of 1/distance; Jµ is the electromagnetic currentdensity (made up of charge and current density, ρ andJ = J i), with units of 1/distance2.

B. Dynamics of a Charged Object

In this section, we develop the center-of-mass dynamicsof a discrete charged object in an electromagnetic field.First, we will write down local momentum conservationin a general form, and integrate it to arrive at the center-of-mass equations of motion. For simplicity, in this sec-tion flat space-time will be assumed.

Consider a compact, stable distribution of charge (asurface can be drawn around the distribution, whichcompletely contains the charge). Internally, the distri-bution has local charge density ρ and current density J(which can vary with time and position within the ob-ject; we make no constraints on those at present). Sta-bility requires some non-electromagnetic binding forcedensity[31], which we will call fb. From Maxwell’s equa-tions, the electromagnetic field loses momentum densityat a rate given by the negative of the Lorentz force den-sity, fem ≡ ρE+ J×B13.Now, consider the charge density contained in a small

spatial volume dV . It could inherently carry some mo-mentum density in its own right: call this dpbare. Notethis is not related to the momentum in the electromag-netic field; this would be the momentum related to the

13 This makes no assumption on the structure of the charge; this ex-pression of the local loss of momentum from the electromagneticfield can be derived directly from Maxwell’s equations assumingJµ is defined as ∇νFµν [2].

9

mass of the charge density if it were stripped of its elec-tromagnetic field (thus, in the literature this is called the“bare mass”[24]).For completeness, allow for some other non-

electromagnetic, external force density fext, which actsdirectly on the charge in some way. Then, in order to beconserved, the momentum leaving the electromagneticfield plus the momentum supplied by the binding andexternal force must completely be absorbed by the mo-mentum of the charge in dV :

(ρE+J×B+ fb+ fext)dV =∂

∂t(dpbare)+∇·dpbare. (3)

Integrating over the extent of the charge gives∫

(ρE+ J×B+ fb + fext)dV =dpbaredt

, (4)

where pbare =∫

dpbare is the integrated (total) momen-tum of the charge (again, not including the contributionfrom its field).The binding force is the non-electromagnetic self-force

from one portion of the object on another, which cre-ates stability. While we don’t know what it is, we maysay something about its integral without any knowledgeof its local form. If it is not associated with any (non-electromagnetic) radiation, then the integral of the bind-ing self-force should be zero by Newton’s third law (sinceno momentum is carried away). However, if there is othermass in the object (which is not due directly to the pres-ence of charge), which is bound to the charge by fb, thenas the charge is accelerated, this other mass, gets draggedalong with the charge. Assuming this other bound massdoes not interact externally, i.e. with fext

14, we can writethe integral of the binding force as[32]:

∫

fbdV = −dpother

dt. (5)

To put it another way, pother is the integrated momentumof any extra mass associated with the source of fb. Notethat it is possible to conceive of an fb that acts directlyfrom charge to charge, in which case pother would be zero.Now separate the electromagnetic field into a self-field

(Eself and Bself) due to the object itself, and an externalfield (Eext and Bext) due to other charges outside theobject. Assuming the distribution is sufficiently smallcompared to the variation of the external electromag-netic field, we can immediately integrate terms with theexternal field, and momentum conservation becomes:

qEext + qv ×Bext +∫

ρEself + J×BselfdV + Fext= d

dt(pbare + pother),

(6)

14 This would be true for some short-range binding force like thestrong or weak interaction; distant particles don’t interact viathose forces.

where q =∫

ρdV and v = 1q

∫

JdV ; if the charge is suf-

ficiently stable (i.e. rigid), then v represents the center-of-mass motion of our compact object. Fext =

∫

fextdVis the non-electromagnetic external force.The integral of the self-field over the distribution re-

sults in the “field reaction”, i.e. the rate of change of themomentum of the self-electromagnetic field due to thedistribution’s motion[24]. The field reaction will resultin a term, which looks like − d

dt(γmfieldv), where mfield

represents the contribution of the field-energy to the in-ertial mass of the object[24].This field energy (and hence mfield) is given by[2]

mfield =

∫ (

1

2E2self,rest +

1

2B2self,rest

)

dV, (7)

where the fields are evaluated when the charge is isolatedand at rest, and the integral is over all space; this integralis inversely proportional to the size of the object (this iseasy to show, for example, for a sphere of charge)[2, 24,81]. Therefore, this self-energy, and its associated inertialmass, approaches infinity as one take the point chargelimit, leading to the “infinite self-energy” problem. If weremove this inertial-mass contribution, we are left withthe “radiation reaction”15[24, 82–84], called thus as itrepresents momentum carried away as radiation from theobject (causing the object to recoil).With the assumption that the integrated momenta are

proportional to γv (an assumption of sufficient rigidity),we may also now define masses for the different momentaas pbare = γmbarev and pother = γmotherv

16. Replacingthe momenta, extracting the contribution to the inertialmass from the self-field integrals, and rearranging Eq. 6,we obtain

qEext + qv ×Bext + Fext= d

dt(γmv) + (radiation reaction),

(8)

where m = mbare + mfield + mother is the total inertialmass (what one would measure as the inertial mass inthe laboratory); The radiation reaction,

(radiation reaction)= −

∫

ρEself + J×BselfdV − ddt (γmfieldv),(9)

again, is what is left from∫

ρEself +J×BselfdV after re-moving the portion which contributes to the inertial massof the object. Note that the radiation reaction stays fi-nite as the size of the charge approaches zero: the partwhich approaches infinity can all be rolled into the mass.

15 Many authors include the contribution to the inertial mass inwhat they call the “radiation reaction”[24]

16 Any required assumption of rigidity will necessarily be violatedover short time-scales as changes in external forces propagateacross the object; however, this assumption of rigidity is requiredto produce well-determined equations of motion, as will be dis-cussed in more detail later.

10

Assuming thatmbare cancels this infinity to give the mea-sured m constitutes renormalization.

Eq. 8 are the equations of motion for the center-of-massdynamics of a sufficiently stable charge, and is the forcelaw found in text books for a charged particle, or in pa-pers discussing radiation reaction. We tried to keep thisas general as possible: we didn’t assume much about thestructure of the charge, only that it is small compared tovariations of the external field and stable enough that vis well-defined (the different momenta are proportional toit). If one ignores the radiation reaction, these equationsof motion are readily solvable, and effectively describemany experiments17.

However, including the radiation reaction is much moredifficult. The radiation reaction term depends heavilyon the charge distribution. Therefore, in order to solvedynamical problems for the center-of-mass motion of adiscrete charged body including the radiation reaction,one must use a process summarized as follows (this is theprocess used in all examples in the literature of which theauthor is aware):

1. Assume an internal distribution of charge for allinvolved charged bodies (such as rigid spheres, orpoint charges). This enables solving for the radia-tion reaction as a function of v and its time deriva-tives. It also implicitly sets the binding force ev-erywhere in the bodies.

2. Assume the mass of each charged body (a measuredvalue if you’re treating a real charge), m; this isnecessary because of the lack of knowledge of howto calculate mbare and mother.

3. With the radiation reaction known as a functionof v and its derivatives, Eq. 8 provides a well-determined system of equations for the center-of-mass dynamics of each body: given initial condi-tions, Eq. 8 may be solved.

As stated before, this process necessarily violates causal-ity on the time scale of light crossing the object.

Also, if individual bodies are too close to each other,our assumption that Eext, Bext are constant over thecharge fails. Also, you will not be able to effectivelysolve for the radiation reaction before solving the dy-namics: if the radiation fields significantly overlap, theassociated radiated momentum/power does not obey su-perposition (the fields add, but their momentum/powerdo not). Therefore, this methodology is only effective forbodies that do not interact too closely.

17 The radiation reaction is negligible for many situations and maybe ignored without too much effect. See [2], Ch. 16 for a discus-sion of when radiation reaction becomes important for variousexperiments.

Also, while this process may be used to solve for thecenter-of-mass dynamics of charged bodies under cer-tain circumstances (while unfortunately violating causal-ity on short time scales), solving for the internal dy-namics of a charge distribution (which is equivalent tosolving systems where charged objects interact closely)is completely intractable without knowledge of the bind-ing force18.

Without this, mathematically, the only way to producewell-posed equations of motion, which do not manifestlyviolate causality, is to take the point charge (particle)limit, which also results in no internal degrees of freedom.Then, for particles, which do not interact too closely,Eq. 8 becomes the Lorentz-Abraham-Dirac equations ofmotion. Even in the case of closely interacting particles(where the radiated energy/momentum cannot be prede-termined), in principle, if one is careful enough, one couldtrack all the electromagnetic momentum/power emittedor absorbed through a small surface surrounding each in-teracting particle, and use that to calculate the changein momentum of each particle from one time to a slightlylater time (assuming you know/measure each particle’smass). Therefore, in the point charge limit, one can cre-ate a well-posed, causally correct problem, which can besolved.

Without knowing fb and dpbare, taking the pointcharge limit appears to be the only way of doing this ina self-consistent, causally correct way. The cost, mathe-matically, of taking the point charge limit, is that mfieldis infinite, creating the need for renormalization (setmbare = −∞ so m is the measured finite value). If onewants to develop equations of motion for extended ob-jects, one must use the local force law of Eq. 3 with apriori knowledge of the local form of both fb and dpbare.

All of these difficulties with developing equations ofmotion for charged objects may be summarized conciselyand covariantly as follows. Conservation of momentum(and energy) density is written by setting the divergenceof the total stress-energy tensor (call it T µν) to zero[79]:

∇µT µν = 0. (10)

The electromagnetic stress-energy tensor (the contribu-tion to the stress-energy tensor from the electromagnetic

18 With some assumptions on the binding force, one can make someprogress in developing internal dynamics. For instance, one maydevelop equations of motion by assuming a spherical charge iscomprised of spherical shells, which are tied together by somelinear restoring force. This gives enough information about theinternal binding force, that with some other assumptions on themotion, one can solve for the center-of-mass motion and the in-duced dipole moment of such a structure; this has been donein [32]. However, this still violates causality due to requiringthat the spherical shell components remain spherical; any for-mulation that assumes any structure cannot produce a causallycorrect theory.

11

field),

T µνEM = gαβFµαF νβ − 1

4gµνFαβF

αβ , (11)

has divergence[79]

∇µT µνEM = JµFµν , (12)

which is manifestly non-zero in the presence of electro-magnetic charge. Without some addition to T µν , energy-momentum cannot be conserved. This is the source of allthe problems/paradoxes associated with developing clas-sical electrodynamics of extended bodies[2, 24, 45, 85].Some other contribution to the total stress-energy tensoris necessary to allow the total divergence to be zero, butno reasonable addition has been found, outside of includ-ing point charges, with their associated infinite masses[2].If a reasonable non-particle addition were to be included,all of the paradoxes and problems with electromagnetismwould be resolved. In the language of this section, the di-vergence of an appropriate addition to the stress-energytensor would supply expressions for fb and dpbare, whichwould allow one to solve the local equations of motionEq. 3.

C. Least Action Principle and Deriving Maxwell’sEquations

To use the principle of least action to develop a fieldtheory, one defines a scalar “action integral”, S, whichis the integral over our entire space-time manifold of ascalar density, the Lagrangian density L, which can de-pend on various tensor fields, Ti

µ1,µ2...ν1,ν2..., and their

derivatives (i here labels the different fields, not a coor-dinate), as

S =

∫

L(Tiµ1,µ2...ν1,ν2...,∇αTiµ1,µ2...ν1,ν2..., ...)d4x.(13)

The derivatives are often, but not always limited to firstorder19. One then requires variations of this action to bezero against smooth, arbitrary infinitesimal variations ofthe different fields (signified by an operator δ),

δS =

∫

∂L∂Ti

µ1,µ2...ν1,ν2...

δTiµ1,µ2...

ν1,ν2...d4x = 0, (14)

with the understanding that integration by parts is usedto convert variations of ∇αT µ1,µ2...i ν1,ν2... to variations ofT µ1,µ2...i ν1,ν2... and a total divergence, which can be con-

verted to a surface integral and assumed zero (as varia-tions are assumed to be zero on the bounds of integra-tion). Since each field can vary independently, and the

19 See [86] for some discussion.

variations are allowed to deform in any way in space-time, the following must be true for each of the fields atevery point in space-time:

∂L∂Ti

µ1,µ2...ν1,ν2...

= 0. (15)

This produces the same number of equations as there aredegrees of freedom in all the fields (e.g. 6 equations ateach point in space-time for an antisymmetric 2-tensor).To develop electromagnetism from a Lagrangian den-

sity, one starts with (in our units, using the sign conven-tion of Jackson)[2]

LEM = −1

4

√

|g|FαβFαβ . (16)

If the “field” is considered to be Aβ , then

∂LEM∂Aβ

=√

|g|∇βFαβ , (17)

and the equations of motion for the field are ∇βFαβ = 0,which means there is no electromagnetic charge. Eq. 16is the “free-space Lagrangian”. To add charge, an extraterm is necessary. To do this, consider an independentvector field Jβ (not yet identified as electromagnetic cur-rent). Adding the typical “interaction Lagrangian”, Lint,makes the total Lagrangian[2]

L = LEM + LintLint = −

√

|g|JβAβ .(18)

Since Jβ is assumed independent of Aβ ,

∂Lint∂Aβ

= −√

|g|Jβ∂L∂Aβ

=√

|g|(

∇βFαβ − Jβ)

= 0,(19)

the last equation being the inhomogeneous Maxwellequations with Jβ as the electromagnetic current, andwe have recovered the correct inhomogeneous Maxwell’sequations. One can’t just vary Jβ independent of Aµ,since its variation would lead to Aβ = 0, which meansthere is no field. As before, we unfortunately need tomake some assumption about the structure of what iscarrying the current. To the author’s knowledge, the onlyoption in the literature that yields reasonable equationsof motion is the “particle hypothesis”[87]

Jβ ≡ (ρ,J) ≡∑

i

qiδ3(x− ri) (1,vi) , (20)

where Jβ is comprised of a flow of particles: qi, ri, andvi are the charge, position, and velocity of the i

th pointcharge (here i is summed over discrete charges, not di-mensions), and δ3 is the 3-dimensional Dirac delta func-tion, not an arbitrary variation.Using this particle hypothesis, and adding the particle

Lagrangian density[88] to the action, we have the total

12

action:

S =∫

LEM + Lint + Lparticled4xLparticle =

√

|g|∑

i12mivi,αv

αi δ

3(x− ri)= −

√

|g|∑i miγi δ3(x− ri),

(21)

where mi is the mass of the ith particle, vαi is its 4-

velocity, and γi is its Lorentz factor. In the second twoterms of the action, we can immediately integrate overthe spatial coordinates using the Dirac delta functions,

S =∫

LEMd4x+∑

i

∫

−qiφ+ qivi ·A− miγi dt, (22)where in the second integral, all fields are implicitly eval-uated at ri. The first integral does not depend on ri, andthe variation of each ri yields the Lorentz force equationfor each particle[2, 88]:

ddt

(γimivi) = qE+ qvi ×B. (23)However, these are not the correct equations of motion:they are missing the radiation reaction. It isn’t surprisingthat the radiation reaction is left out. It is well knownthat the principle of least action depends on the systembeing conservative. In 1900, when Joseph Larmor usedthe principle of least action to obtain both Maxwell’sequations and the Lorentz force[89], at the beginning ofhis treatment, he states

If the individual molecules are to be perma-nent, the system...must be conservative; sothat the Principle of Least Action supplies afoundation certainly wide enough...

With the understanding that charged particles inherentlycan lose a significant amount of energy due to radiation,this argument doesn’t hold for a charged particle in gen-eral, and calls into question the ability to use the princi-ple of least action for electromagnetism.There have been efforts to contrive a Lagrangian for

charged objects, which directly includes radiation re-action. For instance, researchers have developed La-grangians for such dissipative systems by combining itwith a time-reversed copy, doubling the phase space,but producing something where energy is conserved[90].In any case, one does not obtain anything like the La-grangians used in the standard model, and such La-grangians are dependent on the geometry of the charge,so we’d be back to pre-defining the geometry and violat-ing causality.This lack of self-interaction/radiation in particle the-

ories, which are developed using the principle of leastaction, is apparent in both classical and quantum me-chanics. In quantum electrodynamics, radiative/self-interaction effects are absent until they are added in (af-ter the fact) via perturbation theory, using the constructof virtual particles and virtual photons20. All of this to

20 The interaction with virtual particles is indistinguishable fromself-interaction. See Sec. 3 from [55].

say that the principle of least action does not appear tobe able to help us develop equations of motion for pointcharges in a non-perturbative way.However, for non-point charges, the situation is less

grim. In fact, self-interaction for extended charged ob-jects, where the charge/current density is bounded, doesnot need to be accounted for at all at the fundamentallevel. If the current density, Jµ, is bounded, then in asmall volume dV , the magnitude of the self-field is pro-portional to dV , and the charge enclosed is proportionalto dV , so the self-force is proportional to dV 2. Whereasthe force from a finite external field (the field generatedby charge/current outside of dV ) is proportional to dV .Therefore, in the limit of dV → 0, the self-interaction isnegligible compared to the interaction with the externalfield.That is why with bounded Jµ, conservation of energy

locally is given by Eq. 3, without any explicit representa-tion of the self-interaction. The local equations of motionare conservative; it is not until an integral is performedover a finite charge that the radiation reaction appears,as in Eq. 8. The world has yet to produce any Lagrangianthat can produce Maxwell’s equations and sensible equa-tions of motion outside of Eq. 23, and we are again stuckwith point charges and their self-interaction problems.However, one does not need a Lagrangian to reproduce

Maxwell’s equations. With the definition (written as atensor equation or in the language of differential forms)

Fµν ≡ 2∇[µAν]F ≡ dA, (24)

the homogeneous Maxwell’s equations

∇αFβγ +∇βFγα +∇γFαβ = 0dF = ddA = 0

(25)

are identically true (since dd of any form is zero[79]).This is well-known; the inhomogeneous Maxwell’s equa-tions are what Lint was meant to produce (we needed Lintand then to get reasonable equations of motion, neededto introduce particles, Lparticle). But this is also not nec-essary. Purely, from the antisymmetry of Fµν , a currentdefined as

Jµ ≡ ∇νFµνJ ≡ ∗d ∗ F (26)

is guaranteed to be conserved identically[79]:

∇µJµ = ∇µ∇νFµν = 0∗d ∗ J = − ∗ ddF = 0. (27)

Without making any assumptions or constraints on Aµ,and using the two definitions Eqs. 24, 26, we have allof Maxwell’s equations; we obtain the inhomogeneousMaxwell’s equations by simply defining it to be true (aslong as we don’t define Jµ in any other way, or contra-dict it, we’re free to define it as in Eq. 26). Or if one

13

likes, we never introduce Jµ at all; in any case, it’s justshort-hand for ∇νFµν .It’s worth noting that this isn’t anything new. In al-

most any introductory electromagnetism text[2], or gen-eral relativity text for that matter[79], electromagneticcurrent is defined by Eq. 26; historically, before the dis-covery of the fundamental particles, this was the onlydefinition of Jµ. In fact, this definition is exactly how Jµ

is experimentally measured (some measurement of thefield is used in combination with this definition to deter-mine the charge and current that produced the field). Nomatter what the internal dynamics of a charged “parti-cle” is, its integral will result in Eq. 8 as long as it is smalland stable enough, and that is all that has been measuredexperimentally. So from an experimental standpoint, wehave complete freedom to explore our options.This doesn’t help us to determine how the principle

of least action can be used to develop useful equationsof motion for electromagnetic charge, but it does allowus to continue without belaboring how we might ever re-cover Maxwell’s equations without imposing the actionof Eq. 21; we will just need to make sure that whateverformulation we choose, we don’t get equations that con-tradict Maxwell’s equations and our definitions.For the remainder of the paper, we will removeLint and

Lparticle from the Lagrangian and their associated stress-energy tensor. We use as a starting point only LEM andits associated stress-energy tensor, T µνEM.

IV. COMPLETING THE STRESS-ENERGYTENSOR

We’ll return to the principle of least action later; sincethe non-zero divergence of the electromagnetic stress-energy tensor is central to the inconsistencies plaguingelectromagnetic theory of extended bodies, let us takesome time to investigate possible ways to “complete” theelectromagnetic stress-energy tensor directly, by findingadditions that can produce energy-momentum conserva-tion equations that are self-consistent and solvable. Tosummarize our problem from an energetic standpoint,electrodynamic theory suffers from the fact that the elec-tromagnetic stress-energy tensor, Eq. 11, in the presenceof electric charge, has manifestly non-zero divergence:from a purely mathematical perspective, one obtains un-solvable problems without an appropriate addition. Theonly suitable addition that has been found is that ofcharged point particles. This point-charge assumptionproduces solvable equations in flat space-time, but at acost of infinite self-energies. In flat space-time, these maybe dealt with via a renormalization procedure, but un-fortunately preclude integrating with general relativity.This method also requires perturbative methods to in-clude remaining non-infinite self-interaction terms (ra-diative corrections).In this section, we ask the question: Is there a purely

electromagnetic solution? More explicitly, there are 3

degrees of freedom per point in space-time in the elec-tromagnetic field: 4 in Aα minus 1 since you have onedegree of gauge freedom (or if you like, 4 in Jα mi-nus 1 since it is identically conserved). There are 4energy-momentum conservation equations, but often en-ergy conservation can be related to momentum conserva-tion, which would yield 3 equations of motion per pointin space-time. Without adding any more degrees of free-dom per point in space-time, what can we add to T µνEM toproduce self-consistent equations of motion?

A. Integrability

Since we only have 3 degrees of freedom per point inspace-time in the electromagnetic field, we need to ensurethat the energy-momentum conservation equations alsoonly result in 3 independent equations of motion, so asnot to overdetermine its evolution. This hopefully willallow us to formulate a dynamic system of equations that,given initial conditions, can be solved.If our resulting conservation of energy-momentum sat-

isfies a scalar identity, leaving us with only 3 independentequations of motion, which with conservation of charge,uniquely evolves the electromagnetic current, that wouldsuffice. The electromagnetic field energy-momentum pro-vides us with just such an identity. As mentioned before,∇µT µνEM is:

∇µT µνEM = JµFµν . (28)

While this divergence is non-zero, it has the propertythat it is identically orthogonal to Jν :

Jν∇µT µνEM = JνJµFµν = 0, (29)

since JνJµ is symmetric and Fµν is anti-symmetric. This

can be interpreted physically in the following way: in alocal frame where J = 0, the power delivered from/to

the electromagnetic field (∇µT µ0EM) is identically 0. Thisis obvious from the expression JµF

µ0 = −J · E.This is analogous to a similar condition for particles.

For a particle with 4-momentum, pµ, the rate of changeof pµ in the direction of some time-like vector field tν istν∇νpµ. This measure of the rate of change and pµ areidentically perpendicular, if the mass of the particle isconstant,

pµtν∇νpµ =

1

2tν∇ν (pµpµ) =

1

2tν∇ν

(

−m2)

= 0. (30)

This can be seen as being from the fact that in a framewhere pi = 0 (the instantaneous center-of-mass frame),the energy is a minimum compared to frames boostedout of the center-of-mass frame (a non-moving particlehas less energy than a moving one). For an acceleratedparticle as it passes through its center-of-mass frame, thepower delivered must be zero due to this minimum.

14

Whatever 4-force is applied to particles must have thesame property to produce consistent energy and momen-tum evolution equations: it must be identically perpen-dicular to pµ (this is equivalent to requiring that thepower delivered by a force, F, on a particle is F · v,where v is the velocity of the particle). For the case ofcharged particles, the electromagnetic current, Jµ, andthe momentum, pµ, coincide up to a constant, and thetwo identities, Eqs. 29 and 30, are consistent.In our search for some non-particle matter to absorb

the energy-momentum lost by the electromagnetic field inthe presence of Jµ, let us use this same identity, and seeif there are any appropriate additions that can identicallysatisfy Eq. 29. Qualitatively, this means the energy den-sity contained in our addition must be at an extremumin frames where J = 0, such that power density deliveredin that frame will be 0.Mathematically, if we write the complete stress-energy

tensor T µν, comprised of T µνEM and some addition, Tµνadd,

T µν ≡ T µνEM + Tµνadd, (31)

we seek additions such that:

Jν∇µT µνadd = 0. (32)

This will ensure energy density conservation is automat-ically conserved if momentum density is conserved (justas for particles the power is determined by the force asF · v), leaving us with 3 independent equations to deter-mine our 3 degrees of freedom21.

B. One Possible Addition to the Stress-EnergyTensor

Using the guidance from the last section, we will re-quire the divergence of T µνadd to be orthogonal to J

µ identi-cally, and the most obvious choices are terms that explic-itly include Jµ. There are only two quadratic, symmetric2-tensors, which involve the current:

JµJν , gµνJαJα, (33)

which suggests the following addition,

T µνadd = agµνJαJ

α + bJµJν . (34)

where a and b are constants. Taking the divergence ofT µνadd yields

∇µT µνadd = agµν∇µ(JαJα) + b∇µ(JµJν)= 2agµνJα∇µJα + b(∇µJµJν + Jµ∇µJν)= 2aJµ∇νJµ + bJµ∇µJν ,

(35)

21 For space-like currents, this identity can make one of the momen-tum equations redundant rather than the energy equation as wewill see shortly. In any case, there are 3 independent equations.

and taking a = − 12b, we have

∇µT µνadd = bJµ (∇µJν −∇νJµ) , (36)

which is perpendicular to Jν , and Eq. 36 satisfies Eq. 32(the energy density of T µνadd is an extremum in a framewhere J=0). Therefore, one possible addition to the elec-tromagnetic stress-energy tensor is

T µνJ ≡ kJ(

JµJν − 12gµνJαJ

α

)

, (37)

where kJ is a constant with units of distance squared.The form of T µνJ is similar to T

µνEM. The components

explicitly in flat space-time are

T 00J =kJ2

(

ρ2 + J2)

T 0kJ = kJρJk

T ikJ = kJJiJk + kJ2 g

ik(ρ2 − J2),(38)

where gik in flat space-time is the Kronecker delta func-tion. The 00 component looks like what one might guessfor the energy stored in a charge density (it has some-thing that looks like a rest term, ρ2, and a kinetic term,J2; with kJ > 0, the energy density is a minimum in theframe where J = 0). The 0k components also look likewhat one might guess for the momentum carried by acurrent.Taking the divergence of T µν yields the equations of

motion:

∇µTµν = ∇µTEM,µν +∇µTJ,µν= Jµ

(

Fµν + 2kJ∂[µJν])

= 0.(39)

Note that we have replaced some covariant derivativeswith partial derivatives, since (in the absence of torsion),the antisymmetric derivatives coincide.Let’s explore them in flat space-time, since it will re-

veal some interesting properties of our new stress-energytensor. Taking into account charge conservation, theyare:

∇µT µν = 0 =(

−J · E+ kJ(

1cJ · ∂J

∂t+ J · ∇ρ

)

−(ρE+ J×B) + kJ(

1cρ∂J

∂t+ ρ∇ρ− J× (∇× J)

)

)

.

(40)The time component (power equation) is redundant byEq. 32 as long a ρ is non-zero (dot the momentum equa-tion with J and divide that by ρ), which is guaranteedfor time-like currents22. Thus, for time-like currents allthe information contained in Eq. 40 may be written as

ρE+ J×B = kJ(

1

cρ∂J

∂t+ ρ∇ρ− J× (∇× J)

)

. (41)

22 For space-like currents where ρ is zero but J is not, the powerequation provides one equation, and the momentum equation isguaranteed to be perpendicular to J leaving only two indepen-dent equations there.

15

This is a well-defined, local force law on the current den-sity, which given initial conditions, can be evolved. Hadwe chosen any other relationship between a and b, wewould have found inconsistent momentum and energyequations.Eq. 41 is reminiscent of the equations of motion of a

fluid. Using the identity J×(∇×J) = 12∇(J2)−(J ·∇)J,and with some algebra, the force law becomes

kJcρ∂J

∂t+kJ(J·∇)J = −∇

(

kJ2

(

ρ2 − J2)

)

+ρE+J×B.(42)

This is a Navier-Stokes-like equation, where the left handside represents the total change in momentum of thefluid. The right hand side has a pressure-like term, withpressure P = kJ2 (ρ

2 − J2), and a body force from theelectromagnetic field.In terms of our general local conservation of momen-

tum equation, Eq. 3, we can make the associations

∂∂t(dpbare) +∇ · dpbare = kJc ρ∂J∂t + kJ (J · ∇)J

fb = −∇(

kJ2

(

ρ2 − J2))

.(43)

Eq. 42 can be made to look exactly like the Navier-Stokes equation by making the replacement J = ρu(again, if ρ is non-zero):

kJcρ2

(

∂u

∂t+ (u · ∇)u

)

= −∇P+ρ2(∇·u)u+ρE+J×B.(44)

This is now the Navier-Stokes equation[91] with no vis-cosity for a fluid of mass density kJ2 ρ

2, velocity u, pres-

sure P = kJ2 (ρ2 − J2), and a body force, ρ2(∇ · u)u +

ρE+ J×B23.Covariantly, a perfect (non-viscous) fluid has a stress-

energy tensor given by

T µνpf = (ǫ+ P )uµuν + Pgµν , (45)

where ǫ is the energy density, P is the pressure, anduµ is the fluid 4-velocity, which satisfies uµu

µ = −1[2,92]. Writing the electromagnetic 4-current as Jµ =√−JαJαuµ =

√

ρ2 − J2uµ (assuming time-like cur-rents), T µνJ takes on the form of the stress-energy tensorof a perfect fluid with the following equation of state,

ǫ = P = −kJ2JµJ

µ =kJ2

(

ρ2 − J2)

. (46)

To review, we’ve established conservation of momen-tum equations that can, for the first time in a self-consistent way to the author’s knowledge, describe the

23 Note the “mass” conservation law for this fluid is slightly differentthan for a typical fluid. Using conservation of charge, one finds1c

∂(ρ2)∂t

+ 2ρ2∇ ·u+ u · ∇(ρ2) = 0; the factor of 2 on the secondterm is not found in the typical conservation of mass equationassociated with the Navier-Stokes equation.

evolution of an extended charged object. In this theory,kJ has the role of a fundamental physical constant, likethe gravitational constant, G, or the speed of light, c.Given general relativity and electromagnetism, we couldalways reduce variables to some unit of distance (usingappropriate factors of G and c), like in the geometricunits we use in this paper. In quantum mechanics, we cando away with all units using ~ (or equivalently the Plancklength) to set a fundamental length scale. Similarly, inthe equations of motion, Eq. 39, kJ sets the length scalefor any dynamical problem. One could rewrite all theequations in completely dimensionless form by modify-ing the ∇µ operator to be unitless using kJ , and addingappropriate powers of kJ to all variables to also makethem unitless.Because T µνJ results in equations of motion so similar

to the Navier-Stokes equation, this may allow use of wellestablished methods to solve the equations of motion (forinstance, to search for stable solutions). The connectionto a relativistic perfect fluid should also make availablevarious existing methods for solving these equations inthe context of general relativity.While it may be self-consistent, it may not be rep-

resentative of the physical world. We will inspect theproperties and solutions of this theory later. For now,let us see how we can arrive at this theory using varia-tional methods, which will be informative on how otherextended body theories may be developed.

C. Principle of Least Action Revisited

Having found a self-consistent stress-energy tensor. Weturn our attention to whether a suitable Lagrangian ex-ists that will generate the stress-energy tensor of Eq. 37.In general relativity, one can associate a Lagrangian withits contribution to the stress-energy tensor as[93]

δLδgµν

= 12√

|g|T µν . (47)

The connection of our addition to a perfect fluid makesthis possible, in the case of time-like currents. In [94],the Lagrangian density is derived for a barotropic fluid(a fluid whose pressure/energy are only functions of therest mass density). They show that given a conservationlaw (conservation of matter in [94])∇µ(ρmuµ) = 0, whereuµ is the time-like 4-velocity of the fluid and ρm is therest mass density, one can relate the variation of ρm tothe variation of the metric as[94, 95]

δρm =1

2(gµν + uµuν)δg

µν . (48)

As in Sec. IVB, we cast the electromagnetic current asJµ =

√−JαJαuµ, and conservation of charge takes the

form ∇µ(ρmuµ) = 0 with ρm =√−JαJα. Our pressure

and energy are then P = ǫ = 12kJρ2m. In [94], they show

that if the pressure can be written purely as a functionof ρm, the Lagrangian density that produces the perfect

16

fluid stress-energy tensor is −√

|g|ǫ, and the energy den-sity must obey

ǫ = Cρm + ρm

∫

P

ρmdρm, (49)

where C is an arbitrary integration constant. In our case,P and ǫ are pure functions of ρm, and if we set C = 0, ourenergy is indeed given by Eq. 49. Since we satisfy all therequirements of [94], we can say, for time-like currentsthe Lagrangian density, which produces T µνJ , is

LJ =1

2

√

|g|kJJαJα. (50)

However, note there is an implicit assumption here ofwhat is held constant during the metric variation to ar-rive at the result of [94]; charge conservation was explic-itly maintained during metric variation. How can we dothat more directly in terms of electromagnetic variables?The simplest way is to to hold the current density,

jµ ≡√

|g|Jµ, (51)

constant during metric variation. This is because chargeconservation,

∇µJµ =1

√

|g|∂µ

(

√

|g|Jµ)

= 0, (52)

can be written as ∂µjµ = 0, completely independent of

the metric (even in curved space-time). If jµ is held con-stant during metric variation, then charge is guaranteedto be conserved independent of the metric24.It is simple to write LJ as a function of jµ and the

metric:

LJ =1

2√

|g|kJgµαj

µjα. (53)

Varying gµν holding jµ constant gives25

(

δLJδgµν

)

jµ=const= 12

√

|g|kJ(

JµJν − 12gµνJαJα)

= 12√

|g|T µνJ ,(54)

which is the correct relationship between a Lagrangianand its stress-energy tensor. Therefore, using charge con-servation as an argument to hold jµ constant, we obtainthe same result as [94] (but without any time-like currentassumption): our Lagrangian density is given by Eq. 50

24 This isn’t to say charge conservation would be violated if wehold anything else constant. By the definition of Jµ, one canshow Eq. 52 still holds during metric conservation, for instance,if you hold Aµ constant. This is just to say, if Jµ were just ageneral 4-vector, this would ensure its conservation during metricvariation.

25 δ√

|g| = 12

√

|g|gµνδgµν , see [79]

(or Eq. 53), which produces the correct stress-energy ten-sor.This Lagrangian has been studied before[96–99] in

what has come to be called “Bopp-Podolsky” electrody-namics. However, in all the cases the author is awareof, these studies maintain Lint and Lparticle in the La-grangian, and are interested in how these terms mod-ify the point-charge theory. We have done somethingdifferent; we have removed point charges from the the-ory altogether, to obtain a point-charge-free theory thatpotentially could yield small, stable, charged (but non-point-charge) solutions.It’s also worth noting that holding jµ constant is a

departure from what is typically done in developing elec-tromagnetism from a Lagrangian, as in Sec. III C orBopp-Podolsky electrodynamics, where Aµ is held con-stant during metric variation, and then independentlyvaried in its own right to obtain Maxwell’s inhomoge-neous equations. What are the consequences of using jµ

rather than Aµ as independent from the metric? Possiblymost importantly, what does this do to the electromag-netic stress-energy tensor? If that has changed form, ouroriginal goal of trying find an addition that is consistentwith it is moot.To address this question, it’s useful to introduce the

electromagnetic field tensor density,

fµν ≡√

|g|Fµν , (55)

and using the relation jµ = ∂νfµν (even in curved space-

time), the variation of fµν and jµ are simply related(removing a total divergence) by

Bµδjµ = ∇[µBν]δfµν , (56)

for any one form Bµ. Eq. 56 does not involve the metricat all (treating fµν or jµ as independent results in thesame Einstein’s equations with the same stress-energytensor). Now let us address the electromagnetic field La-grangian:

LEM = − 14√

|g|FαβFαβ= − 1

4√

|g|gµαgνβf

µνfαβ . (57)

The conventional variation, considering Aµ as the inde-pendent EM field, is[2]

δLEM = 12√

|g|T µνEMδgµν − JµδAµ. (58)

Calculating the variation using fµν as independent re-sults in

δLEM = − 12√−gT µνEMδgµν − 12Fµνδfµν . (59)

From this, it’s clear that using jµ as independent resultsin

δLEM = − 12√−gT µνEMδgµν −Aµδjµ. (60)

Whether we use jµ or Aµ as independent, we still getthe same, standard electromagnetic stress-energy tensor

17

from LEM, although with opposite sign. The sign is oflittle importance, however, since we can change its signin total the Lagrangian, or other signs, such as of kJ ,to compensate as necessary (the overall sign of the La-grangian does not effect the equations of motion).Varying LJ gives

δLJ =1

2

√−gT µνJ δgµν + kJJµδjµ (61)

=1

2

√−gT µνJ δgµν + kJ∇[µJν]δfµν (62)

or treating Aµ as independent,

δLJ = 12√−gT µνJ,Aδgµν +

√

|g|kJ∇ν∇[µJν]δAµ (63)

T µνJ,A ≡ TµνJ + kJ

(

∇[αJβ]Fαβgµν−2∇[µJα]F νβgαβ − 2∇[νJα]Fµβgαβ

)

.(64)

The stress-energy tensor of Eq 64 can be found inRef. [98] where it was studied in the context of Bopp-Podolsky electrodynamics. It is more complicated thanT µνJ , but its divergence is fairly simple,

∇νT µνJ,A = 2kJFµν∇α∇[νJα], (65)

which does not satisfy our original guiding condition ofJµ∇νT µνadd = 0. However, the divergence of the completestress-energy tensor is

∇ν(

T µνEM + TµνJ,A

)

= Fµν(

−Jν + 2kJ∇α∇[νJα])

,

(66)which is identically perpendicular to the conserved cur-rent in parentheses, and hence also has the correct num-ber of independent equations to specify Aµ (or Jµ).At this point, it’s useful to define a total “matter”

Lagrangian density (everything except the Lagrangiandensity for gravity, which we’ll add later). The matterLagrangian with our current terms is

Lmatter =√

|g|(

14kEMFµνF

µν + 12kJJµJµ)

. (67)

Note kEM is +1 if we’re holding jµ constant during vari-

ation, and −1 if we’re holding Aµ constant during varia-tion to obtain the conventional sign of T µνEM when varyingthe metric.The field equations for Aµ are

1√|g|

∂Lmatter∂Aµ

= kEMJµ + 2kJ∇α∇[µJα] = 0. (68)

One thing worth noting is that, for any Lagrangian, con-servation of energy-momentum is guaranteed trivially ifthe field equations for all the fields outside of the metricare satisfied, as we’ll discuss more in the next section.This is obvious from our current example: if Eq. 68 issatisfied, then Eq. 66 is trivially satisfied. Thus, fromthe standpoint of the principle of least action, the fieldequations fully determine the dynamics, without any helpfrom conservation of energy-momentum.

With this understanding, consider how the field equa-tions differ if we use jµ versus Aµ as independent. Thefield equations, with jµ independent, are

∂Lmatter∂jµ

= kEMAµ + kJJµ = 0, (69)

or plugging in the definition for Jµ,

kEMAµ + 2kJ∇ν∇[µAν] = 0. (70)

Eq. 70 and Eq. 68 are of the exact same form, one in termsof Aµ, and the other in terms of Jµ. Both field equationsreduce to the Proca equation in flat space-time (as longas one uses the Lorenz gauge for Aµ).It’s also clear, in flat space-time, that any solution to

Eq. 70 is a solution to Eq. 68. More generally, removingtotal divergences, for any two-form Bµν ,

Bµνδfµν

=√

|g|(

12BαβF

αβgµν − 2BµαF νβgαβ)

δgµν+ 2

√

|g|∇νBµνδAµ,(71)

and for any one form Bµ,

Bµδjµ

=√

|g|(

12∇[αBβ]Fαβgµν − 2∇[µBα]F νβgαβ

)

δgµν+ 2

√

|g|∇ν(

∇[µBν])

δAµ.(72)

So the electromagnetic field equations, treating Aµ asindependent, can be written as

2∇ν∇[µBν] = 0Bµ =

∂Lmatter∂jµ

.(73)

Thus, for any Lagrangian, in flat space-time, any solutionto the field equations using jµ as independent is also asolution to the field equations using Aµ as independent.However, the stress-energy tensors differ, and gµν

(gravity) will differ. Since the sizes of fundamental parti-cles are smaller than can currently be detected, it’s pos-sible space-time curvature can be important to their in-tradynamics, or even close enough interactions: as oneapproaches a small enough particle, curvature eventuallymust be important under the theory of general relativity.Typically space-time curvature is ignored in the studyof fundamental particles, due to difficulties in integrat-ing with quantum mechanics, on the grounds that theyare light or general relativity may not hold at all for suchlength scales, or possibly for simplicity. But since there isno obvious alternative to general relativity, and the factthat, if compact enough, curvature will definitely play arole in in internal dynamics, it could be important. We’venow found two extended body theories that allow us toinclude space-time curvature from the outset.With this in mind, consider what other scalars can

be added to the Lagrangian. Outside of LEM and LJ ,there are three other, independent quadratic scalars thatdepend on Fµν and its first derivatives, two of which

18

are linear in ηαβµν , and thus change sign under a paritytransformation (their inclusion would lead to a “parityviolating” theory)26:

L∗F ≡ 14√

|g|ηαβµνFαβFµνLdF ≡ 14

√

|g|∇αFµν∇αFµνLd∗F ≡ 14

√

|g|ηµνσρ∇αFµν∇αF σρ.(74)

Using dd(δA) = 0 (the homogeneous Maxwell’s equa-tions), the variation of L∗F can be rewritten as[97]

δL∗F = −2√

|g|∇ν(

ηµναβAν(

∂[αδAβ]