Non-Uniqueness of the Lienard-Wiechert Potentials

Home , Lorenz gauge condition

a) Andrew T. Hyman 1, 1Institute for Intermediate Study, P.O. Box 197, Monroe, Connecticut, 06468, USA

The retarded Lienard-Wiechert (L-W) potentials of a classical point-charge have several important

properties. Five of those properties are these: conformity with the homogeneous wave equation in free

space, satisfaction of the Lorenz-Gauge condition in free space, inverse dependence upon distance,

retardation due to the speed of light, and incapability to be formulated as the gradient of a scalar. These

properties raise the question whether any other potentials could possibly satisfy the same five

enumerated properties, or instead whether the L-W potentials are unique in that respect. The answer is

that they are not unique. This non-uniqueness is proved by deriving a family of potentials that satisfy

all of these enumerated properties.

I. INTRODUCTION

In classical electrodynamics, the retarded Lienard-Wiechert (L-W) potentials of a point-charge have many interesting features. Here are five important ones: [A] they satisfy the homogeneous wave equation in free space, [B] they satisfy the Lorenz-Gauge condition (the spelling of “Lorenz” is discussed at Ref. 1), [C] they have a 1/r distance-dependence from a retarded point of the source, [D] they satisfy the retardation condition such that the distance from the retarded point of the source divided by the time-delay is the speed of light “c,” and [E] they are not merely the four-gradient of a scalar.

Various well-known electromagnetic potentials satisfy some — but not all — of these five properties.

For example, the potential of a magnetic point-dipole satisfies all of these properties, except for property [C]

a) Electronic mail: [email protected]. due to terms that have an inverse-square distance dependence. Up until now, the only known solution that satisfies all five of these properties has apparently been the solution of Lienard and Wiechert for an electric or magnetic monopole.

This raises an interesting question, at least mathematically: are the Lienard-Wiechert potentials unique in this regard, or not? I will demonstrate by counter-example that they are not unique. It is difficult to say exactly what physical significance this non-uniqueness may have. But, we can only benefit from learning more about the mathematical properties (including uniqueness or non-uniqueness) of physically important solutions like the L-W potentials.

Because I will prove non-uniqueness by counter-example, the end result of this article will be a new solution. This new solution happens to be very unusual, though perhaps not on a par with the “magical” solution that one gets from considering the flat-spacetime limit of a spinning and electrically charged black hole

(i.e. the Kerr-Newman solution with gravitational constant set to zero), as discussed in Ref. 2. According to the renowned compilation at Ref. 3 of exact solutions in General Relativity, solutions to the Einstein Field

Equations must be found before one can undertake a “mathematical and physical interpretation of the solutions thus obtained.” In the context of Special Relativity, things are often the other way around; because exact solutions are not so difficult to find, a particular physical problem will normally motivate the exact solution rather than vice versa. While the present calculations are carried out in the context of Special Relativity, nevertheless a new exact solution is obtained here without a clear understanding of its mathematical and physical interpretation, or even the charge distribution that gives rise to it. I will try to say a few words in the

Appendix about how to interpret this strange new solution, but a more complete interpretation is outside the scope of this article.

The perceptive reader will notice that the five properties listed above could be supplemented, for example by requiring that the potential satisfies the inhomogeneous wave equation for a delta-function source.

In other words, I have described a potential that satisfies all the right requirements except where the potential is

2 singular. However, it is worth keeping in mind that the source term in the usual inhomogenous wave equation is different depending upon whether the point-source has a dipole moment or not, so the source term in a sense depends upon the solution, rather than vice-versa. As we shall see, sticking with the five cardinal principles listed above leads to some intriguing results.

Immediately following this introduction, in Section II, the standard potentials and notation are briefly reviewed. Then, in Section III, the scalar wave equation is considered and solved in a new way that can be applied later to the vector wave equation. A potential is derived in Section IV that is different from the Lienard-

Wiechert potential, and yet satisfies all five properties of the L-W potentials described above. Finally, in the

Appendix, some interpretational issues are briefly discussed.

II. REVIEW OF STANDARD POTENTIALS AND NOTATION

This section can be skimmed or skipped by those who feel comfortable with their background knowledge. It will be simplest to work in units such that the speed of light equals unity (c=1). Except as otherwise noted, I will use standard notations and conventions, as described at Ref. 4 for example. The standard potentials of Lienard and Wiechert are these:

qz& A = λ λ ω & (1) R z ω

λ Greek indices go from zero to three, and repeated indices are summed. The quantity z is the retarded position of the source, and “q” is the associated charge which is constant (units are selected so that no universal constant

λ needs to be used). The quantity x represents the field point. The distance vector is defined like this:

Rα ≡ x α − z α(p) (2)

This distance vector satisfies the retardation condition:

α R R α = 0. (3)

3 λ One consequence of equations (2) and (3) is that, at any particular point-event x , there will be a particular

λ value of the retarded parameter of motion “p,” so that “p” is a function of x . Overdots denote differentiation with respect to the parameter of motion (“p”), and one can easily see that equation (1) is parameter-independent as discussed at Ref. 5, so it does not matter whether the parameter of motion is retarded proper time, or retarded coordinate time, or something else. As usual, indices are raised and lowered by summation with the Minkowski tensor:

 1- 0 0 0   µν  0 1 0 0 η = ηµν = (4)  0 0 1 0    0 0 0 1

The d’Alembertian of the potential (1) vanishes in free space (i.e. in vacuo):

λ α A , , α = 0 (5)

Likewise, equation (1) also satisfies the Lorenz Gauge condition:

α A , α = 0 (6)

The fields can be obtained from these potentials by partial differentiation:

Fλβ = Aλ,β − Aβ ,λ (7)

It is often desireable to differentiate retarded quantities, either to directly confirm that the potentials

4 satisfy equations (5) and (6), or to obtain retarded fields, or for other reasons. The usual covariant technique for differentiating retarded fields is elegant, and I will now describe it briefly, and then use it to directly confirm that equation (1) satisfies (6). To start with, differentiate the retardation condition (3):

α Rβ− R z& α p, β = 0. (8)

Then solve (8) for the gradient of the parameter of motion:

ω −1 p, β = Rβ(R z& ω) . (9)

This will be a very handy formula. Next, differentiate (1):

&& zq λ p,β zq& ω ω A = − λ []R z&& p + z& − p z& z& (10) λ,β ω & ω & 2 ω ,β β ,β ω R zω (R zω )

Inserting (9) into (10), we get:

&& zq λ Rβ zq& ω ω ω A = − λ R z&& R + z& R z& − R z& z& (11) λ,β ω 2 ω 3 []ω β β ω β ω (R z&ω ) (R z&ω )

It is then trivial to contract indices in (11) to confirm the Lorenz Gauge Condition (6). The wave equation (5) could be confirmed in the same way. Alternatively, one can confirm (5) by recognizing that each component of

(1) is of the following form:

f ( p) φ = (12) ω R z&ω

5 Here “f(p)” is an arbitrary retarded function. Differentiation of equation (12), using the method described above, yields the scalar wave equation:

,α φ ,α = 0 (13)

Since each component of (1) is of the form (12), therefore it follows that each component of (1) satisfies (13), and so the vector wave equation (5) is confirmed.

Notice that we could easily modify equation (1) without violating (5) and (6):

zq&  G( p)  A = λ +   λ Rω z& Rω z& (14) ω  ω  ,λ

However, adding the four-gradient of a scalar in this manner would contribute nothing to the retarded fields.

It will be useful to now briefly consider the generalization of (1) for the case of a magnetic point dipole, which is our best classical model for the fields of a classical electron. Accordingly, a term can be added to equation (1):

λ λβ λ zq &  KS ( p)  A = +   (15) ω & ω & R zω  R zω  ,β

λβ The function S is antisymmetric. Unlike the added term in (14), the added term in (15) affects the retarded fields. The potential (15) clearly still satisfies the Lorenz Gauge condition, on account of the equality of mixed partials. Likewise, the potential (15) clearly still satisfies the wave equation in free space, given that any component of the term inside the brackets has the same form as equation (12). We know empirically that the

λβ retarded antisymmetric function S ( p) has certain properties:

6 αβ αβ S z& β = 0 and S Sαβ = 1. (16)

Incidentally, notice that the constant “K” can be used along with the charge “q” and the mass “m” of the source- particle to form a dimensionless constant N=q 3[2mK] −1 . Modern science has yet to provide any explanation for the measured value of the dimensionaless constant N, and “all good theoretical physicists put this number up on their wall and worry about it,” as Feynman 6 put it.

It is often convenient to decompose the spin tensor as follows:

λβ µνλβ λ β β λ S ( p) = ε Uµ ( p)z&ν + W ( p)z& −W ( p)z& (17)

Instead of using the antisymmetric tensor function “S” as in equation (15), we instead employ the two vector functions “U” and “W.” The constant Levi-Civita tensor is defined like this:

+1 if αβµν is even permutatio n αβµν ε ≡ { −1 if αβµν is odd permutatio n (18) if 0 if any two indices are the same

Without loss of generality, we can assume that both “U” and “W” are null vectors. Notice that when (17) is inserted into (15), it becomes apparent that neither the “U” vector nor the “W” vector depends upon what parameter of motion is employed (whereas the tensor “S” does depend upon what parameter of motion is employed). The empirical limitations upon “U” and “W” can be written as follows:

µ W =0 (19)

7 ν & &λ & U zν = − − z zλ (20)

Equation (19) signifies no electric dipole moment, though experimenters continue to search for one, according to Ref. 7. The square root in (20) maintains the parameter-independence.

It is useful to keep in mind that equation (7), which gives the fields in terms of the potentials, can be generalized to include two different potentials:

α,τ Fλβ = Aλ,β − Aβ,λ + ελβατ P (21)

In the literature, this is often called the “Cabibbo-Ferrari” relation, though Shanmugadhasan 8 proposed it earlier.

By equating the “P” four-vector to the Lienard-Wiechert potentials (1), we thus obtain the fields of a particle having a magnetic charge (i.e. a magnetic monopole). More generally, and putting aside empirical limitations ~ ~ upon the four arbitrary null vectors (U ,W ,U ,W ), we have these two formulae for the two potentials in

(21):

λ  µνλβ & λ & β β & λ  λ zq & ε U µ ( p)zν + W ( p)z − W ( p)z A = + K   (22) R ω z&  R ω z&  ω  ω  ,β

and

~ λ  µνλβ ~ & ~ λ & β ~ β & λ  λ q z& ε U µ ( p)zν + W ( p)z − W ( p)z P = + K   (23) Rω z&  R ω z&  ω  ω  ,β

8 Here q~ represents magnetic charge, whereas q represents electric charge. Without any loss of generality, and in view of the following mathematical identity that is a consequence of the definition (18) of the Levi-Civita ~ ~ tensor per Ref. 9, we can absorb the vector U into the vector W and thus set U equal to zero.

αβµν β µ ν β µ ν β µ ν -ε εαλτσ = η λη τη σ + η ση λη τ + η τη ση λ

β µ ν β µ ν β µ ν - η λη ση τ - η ση τη λ - η τη λη σ. (24)

The “dual” electromagnetic field strength tensor is defined like this:

1 ατ F ≡ ε F (25) λβ 2 λβατ

So, according to (21) and (24), we have this: F α,τ λβ = Pλ,β − Pβ,λ +ελβατ A (26) ~ ~ Again without loss of generality, we can absorb the vector W into the vector U and thus set W equal to zero. Equations (22) and (23) simplify:

λ  µνλβ & λ & β β & λ  λ zq & ε U µ ( p)zν + W ( p)z − W ( p)z A = + K   (27) R ω z&  R ω z&  ω  ω  ,β and

q~z& λ P λ = (28) ω & R zω

9 This section has summarized the standard way of constructing retarded fields using retarded potentials of point-sources. If the two arbitrary null vectors (U ,W ) in equation (27) vanish, then the resulting potential has a 1/r dependence. As we shall see, that resulting potential can be generalized without violating the five principles described above in the Introduction.

III. NEW SOLUTION OF THE SCALAR WAVE EQUATION

We have seen that the scalar wave equation (13), and its standard 1/r retarded solution (12), are useful in the context of studying the retarded solutions of the vector wave equation (5). Therefore, in order to construct a potential that satisfies the five principles discussed in the Introduction, it seems reasonable to start by seeing whether we can construct new 1/r solutions of the scalar wave equation. We might posit the following form of solution:

 λ λ & σβρτ & &  1 R U R U ε zσU β RρUτ φ = • F λ , λ , , p (29) ω &  β & β λ  R zω  R zβ R U β R U λ 

A solution like this would have a 1/r distance-dependence from a retarded point of the source, but plugging this form of solution into the wave equation creates a big mess. Through a process of trial and error, it turns out to be much simpler if we choose a different set of independent variables that are constructed out of the four parenthesized variables in equation (29), while assuming that “U” is a null vector function of the retarded parameter “p.” So, instead of the ansatz (29), let our supposition be this:

1 φ = • F v,θ,w, p (30) ω () R z&ω

10  λ & &ψ  λ & &ψ  − 2/1 v 1 β & R U λ z Uψ ψ 2R zλ z Uψ e cos θ ≡ z& U β − z&ψ z& −  (31) & σ &  RβU  RβU  U Uσ  β  β 

σβρτ & &  λ & &ψ − /1 2 v − ε zσU β RρUτ ψ 2R zλ z Uψ e sin θ ≡ z&ψ z& −  (32) λ & σ &  RβU  R Uλ U Uσ  β 

 λ & & µ  − /1 2 w µ 2R zλ z U µ & & (33) e ≡ zµ z − β   R U β 

µ U µU = 0 (34)

Except for equation (34), there is no essential difference between these last five equations and equation (29).

While (30) thru (33) at first appear much more complicated than (29), the results are much simpler.

The variables defined in (31) and (32) are not independent of each other. This becomes apparent from & & & considering the rest-frame where instantaneously z1 = z2 = z3 =U1 = U 2 = 0 . By squaring both sides of

(31) and (32), and adding the results together, I find this simple result:

2v e =1 (35)

11 Consequently, of course, “ υ” must equal zero. Plugging (30) into the wave equation (13), I find that the wave equation will surprisingly be satisfied if the function “F” satisfies the following simple partial differential equation:

∂2 F ∂ 2 F 0 = + (36) ∂w2 ∂θ 2

This is the Laplace Equation in two dimensions, and is readily solved:

w w w w F()θ,w, p = H (e cos θ + ie sin θ, p) + J (e cos θ − ie sin θ, p). (37)

So, we can write our new solution of the scalar wave equation as follows, where the overbar denotes complex conjugation:

V (Ψ, p) +W(Ψ, p) φ = Re   ω & (38)  R zω 

−1 λ σ σβρτ & λ σ  R U& U z& ε z& U R U 2R z& U z&  &β & λ σ σ β ρ τ λ σ & &λ Ψ ≡ z Uβ − β −i λ  β − zλ z  (39)  R Uβ R Uλ  R Uβ 

Keep in mind that the operation of complex conjugation is not analytic (see Ref. 10 for example), and that is why the two functons in the numerator of (38) are not combined into one single function.

12 I have not been able to find that anyone has ever considered this solution before, although the literature about classical electrodynamics is vast and may hold surprises. The notion of a complex retarded scalar potential has been discussed at Ref. 11, in connection with time-harmonic sources and the zeroth component of a four-vector potential. Here, however, we have a source in arbitrary rather than harmonic motion.

The accuracy of the solution given by (38) and (39) can be confirmed by direct differentiation, which recovers the scalar wave equation (13). That differentiation can be tediously done by hand, or more humanely done by using a computer program like Mathematica . Equation (39) is written in parameter-independent form, so the parameter of motion could be retarded proper time, or retarded coordinate time, or something else. The retarded four-vector “U” is null, as is the distance vector “R.” If the four-vector “U” is static then “ Ψ” vanishes, in which case this solution reduces to the well-known solution (12). As we shall see, equations (38) and (39) can be used to construct solutions of the vector wave equation that satisfy the five cardinal principles described in the introductory section above.

IV. SATISFYING THE VECTOR WAVE EQUATION AND LORENZ-GAUGE CONDITION

Based upon the findings in the last section, we might posit the following form of solution of the vector wave equation (5):

µ  1 µ µ & µ µβλτ &  A = Re ()U F + z& F +U F + ε z& U U F (40)  λ & 0 1 2 β λ τ 3  R zλ 

Fn =Vn(Ψ, p)+Wn(Ψ, p) (41)

Each term on the right-hand-side of (40) is of the form (38), so equation (40) clearly satisfies the vector wave equation (5). We still need to ensure that the “F” functions are chosen so as to satisfy the Lorenz Gauge condition (6). A solution of the form (40) could possibly satisfy all five of the principles described in the

13 introductory section above. However, just as equation (29) turned out to be suboptimal, so too equation (40) turns out to be less useful than the following form:

  ε µβλτ z& U& U Ω  µ 1  & µ & µ β λ τ 2  A = Re  λ  2z Ω 3 + U Ω1 + ψ   2R z&λ  z& Uψ 

 µ 1 & σ & λ & β − r ()U U Ω + z& z& Ω + z& U Ω (42) λ & &σ σ 0 λ 3 β 1  2R zλ z U σ 

Ωn =Vn (Ψ, p)+Wn (Ψ, p) (43)

If we insert (42) into (6), then it turns out the Lorenz Gauge condition will be satisfied provided that the omega functions satisfy this requirement:

σβρτ & λ α λ & µ Ω ε z& U R U  Ω 2R z& z& U  Ω  R U z& U  Ω = 2  σ β ρ τ  + 3  λ α − z&λ z&  − 1 z&µU& − λ µ  o & σ & α & σ & µ λ & σ & µ τ U Uσ  R Uα  U Uσ  R Uµ  U Uσ  R Uτ 

(44)

All four of the omega functions must satisfy (43), while also satisfying (44), and this algebraically implies several relations among the “V” and “W” functions that are used in (43). The following identity will be useful:

−1  2Rλ z& Uψ z&  & σ & & &λ λ ψ ΨΨ =U Uσ zλ z − β  (45)  R Uβ 

So, equation (44) can be rewritten like this:

14 −1  Ψ − Ψ Ψ + Ψ Ωo = Ω3 + Ω2 • −Ω1 •  (46) ΨΨ  2i 2 

We have to ensure that Ωo will have the form required by (43). That is done as follows:

2V3(Ψ, p) V (Ψ, p) = B(p)+iV (Ψ, p)+ . (47) 1 2 Ψ

2W (Ψ, p) W(Ψ, p) = H(p)−iW (Ψ, p)+ 3 . 1 2 Ψ (48)

Putting (44) into (42) gives this:

  µ  ε ωβλτ z& U& U Ω  µ 1 µ U Rω ω & ω β λ τ 2 A = Re  η ω − 2z& Ω +U Ω +  λ &  β  3 1 &σ  2R zλ  R Uβ  z Uσ 

(49)

Inserting into (49) the omega formulae (43), (47) and (48), we have this result:

  µ   µ 1 µ U R ω ω A = Re  • η ω − Γ  (50) λ &  β   R z λ  R U β  

15 ωβλτ & & ω ω ω ω ε zβUλUτ Γ ≡U& f (p)+ z& (ΨH + ΨJ )+U& (iH −iJ + H + J )+ (H + J ). 1 1 2 2 1 1 &σ 2 2 z Uσ

(51)

H n ≡ H n (Ψ, p) and J n ≡ J n (Ψ, p). (52)

While this is admittedly a complicated formula for the potentials, its basic principles (listed in the Introduction) are simple. The “H” and “J” functions are arbitrary analytic complex functions. The complex quantity “ Ψ” was defined at (39), and the function f(p) is an arbitrary real function of the retarded parameter of motion. The four- vector “U” is an arbitrary real function of the retarded parameter of motion, except that it is null. Equation (50) is in parameter-independent form, so it does not matter whether the retarded parameter of motion “p” is the retarded proper time, the retarded coordinate time, or some other parameter. It can be readily seen that this potential (50) vanishes if the four-vector “U” is static, in which case the solution collapses to the usual form given in Section II.

Consider what happens if we make the following substitution in equations (50) and (39) where g(p) is arbitrary:

U α = g( p) •(U α )'

Remove the primes, and equations of the same form are recovered. We could exploit this freedom without loss of generality, for example by letting the null vector “U” satisfy (20).

Equations (50) and (51) describe a potential that satisfies all five properties listed above in the introductory section. This can be confirmed by direct differentiation, by hand or by using a computer program like Mathematica . Consequently, the Lienard-Wiechert potentials are not unique in that regard, Q.E.D.

16 ACKNOWLEDGMENTS

I am very grateful to Professor David Griffiths of Reed College for helpful and detailed comments.

Thanks also to Professor José Luis López-Bonilla of the National Polytechnic Institute, for several useful communications. And Professor Richard Beals of Yale University kindly gave me many pointers regarding the mathematical aspects of this article.

APPENDIX

The potential (50) is unusual, and is not easily interpreted. That applies to both the physical interpretation as well as to the mathematical interpretation, and both are largely beyond the scope of the present article. These interpretational issues are not important for demonstrating the non-uniqueness of the L-W potentials, but curiosity now compels a few tentative interpretational remarks.

The potential (30) is singular not just at the retarded point of the source, but also at any point-event satisfying the following condition:

β R U β = .0 (A1)

The four-vector “U” is defined as a null function of retarded proper time (analogous to a spin vector), and “R” is the distance vector connecting the field point to the retarded position. Because both of these four-vectors are null, we can rewrite (A1) in three-dimensional form like this:

r r R •U = .1 RU (A2) r r So, the potential will be singular at any point-event such that R points in the direction of U . This looks like a classical tail (or “string”) extending from the retarded point of the source.

17 One way to look at (50) is as a classical electron model. This is not to suggest that the electron is really classical, or that quantum mechanics is wrong. Rather, it is merely to recognize that there is more than one classical electron model that can be quantized. Much effort has been devoted over the years to studying alternative classical electron models in hopes of removing self-energy problems in quantum electrodynamics, as

Feynman 12 described in his famous lectures. Perhaps the arbitrary function in equation (50) can be reduced by imposing some further requirements upon the potential, such as finiteness of self-energy or some conditions related to the flux of radiated self-energy.

It is well-known that we can calculate the flux of energy-momentum and angular momentum from a

Lienard-Wiechert particle, and it should be possible to do the same if we add the new potentials (50) to the L-W potentials. This sort of flux calculation has typically been done by visualizing a spherical surface surrounding the retarded position of the point-source, the surface being defined as follows where “Q” is a constant and the retardation condition (3) holds true:

2 β & Q + R zβ = .0 (A3)

In the rest-frame, this simply represents a sphere of radius Q 2. If we are interested in radiation, then we take “Q” to be very large, but if we are interested in the flux very near to the particle then we take “Q” to be very small, and the latter technique is described in detail at Ref. 13.

In the present situation, the potential (50) blows up not just at a point but also along a tail extending through the imagined spherical surface (A3). It might therefore be advisable to use a different surface of integration either instead of (A3) or in addition to (A3). For example, the following surface might be especially useful to calculate the flux very near to the tail:

2 β Q + R U β = .0 (A4)

18 In the rest-frame, this simplifies:

r r 2 RU − R •U = Q . (A5)

We can rotate coordinates in the rest-frame so that U x = U y = 0 and U z = U . Accordingly, (A5) simplifies some more:

2 R − Z = Q /U . (A6)

2 This is the formula for a paraboloid. The quantity Q /U is technically called the “semi-latus-rectum” of the paraboloid, per Ref. 14. This surface encloses the singular tail of the new potentials (50).

Since (50) is not linear in “U,” it is possible that a single source could have a second associated “U” vector. In this connection, it seems worth mentioning that a tensor form of the Dirac Wave Equation has been described by Asenjo et al. involving a further four-vector beyond the spin four-vector.

Besides allowing a second associated “U” vector, another interesting exercise might be to suppose that motion is at the speed of light along a curved path (instead of slower than the speed of light), and that the null four-vector “U” equals the four-velocity which would be a null vector. The existence of massless electrically charged particles is, after all, consistent with field theory, at least from a classical viewpoint according to Ref.

16. While these exercises are interesting, they are beyond the scope of the present article.

By the way, equation (38) easily yields a further potential that satisfies the wave equation and the

Lorenz-Gauge condition, and this further solution has inverse-square terms just like the dipole potential (15):

µλ µλ µ V (Ψ, p)+W (Ψ, p) A = Re   (A7) ω &  R zω ,λ

19 Here, the “V” and “W” tensors are antisymmetric.

REFERENCES

1J.D. Jackson and L.B. Okun, “Historical Roots of Gauge Invariance,” Rev. Mod. Phys., 73 , 663-680 (2001). See also: Robert Nevels and Chang-Seok Shin, “Lorenz, Lorentz, and the Gauge,” IEEE Antennas and Propagation Mag., 43 , 70-72 (2001).

2Donald Lynden-Bell, “Electromagnetic Magic: The Relativistically Rotating Disk,” Phys. Rev. D, 70 , 105017 (2004).

3Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt, Exact Solutions of Einstein’s Field Equations (Cambridge U. Press, Cambridge, 2003), p. 1.

4Andrew Hyman, “Relativistic Charged-Particle Motion in a Constant Field According to the Lorentz Force Law,” Am. J. Phys ., 65 , 195-198 (1997).

5Andrew Hyman, “A Parameter-Independent Formulation of the Retarded Electromagnetic Fields,” Am. J. Phys. , 67 , 643-645 (1999).

6Richard Feynman, QED: The Strange Theory of Light and Matter (Princeton U. Press, Princeton NJ, 1985), p. 129.

7Norval Fortson, Patrick Sandars, and Stephen Barr, “The Search for a Permanent Electric Dipole Moment,” Phys. Today, 56 (6), 33-39 (2003).

8S. Shanmugadhasan, “The Dynamical Theory of Magnetic Monopoles,” Can. J. Phys ., 30 , 218-225 (1952). See also: David Fryberger, “On Generalized Electromagnetism and Dirac Algebra,” Found.Phys ., 19 , 125-159 (1989).

9Robert C. Wrede, Introduction to Vector and Tensor Analysis (Wiley, New York, 1963), p. 311.

10 Richard Saeks, Generalized Networks (Holt, Rinehart and Winston, New York, 1972), p. 120.

11 Zoya Popovi ć and Branko Popovi ć, Introductory Electromagnetics , 377 (Prentice Hall, Saddle River NJ, 2000).

12 Richard Feynman, The Feynman Lectures on Physics , II- 28-5 (Addison-Wesley, Reading MA, 1964).

13 H.J. Bhabha and H.C. Corben, “General Classical Theory of Spinning Particles in a Maxwell Field,” Proc. Roy. Soc. London, 178 , 273-314 (1941).

14 Andrew Hyman, “A Simple Cartesian Treatment of Planetary Motion,” Eur. J. Phys ., 14 , 145-147 (1993).

15 Felipe A. Asenjo, Victor Muñoz, J. Alejandro Valdivia, and Swadesh M. Mahajan, “A Hydrodynamical Model for Relativistic Spin Quantum Plasmas,” Phys. Plasmas, 18 , 012107 (2011).

20 16 P.O. Kazinski and A.A. Sharapov, “Radiation Reaction for a Massless Charged Particle,” Classical and Quantum Grav ., 20 , 2715-2725 (2003).