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Non-Uniqueness of the Lienard-Wiechert Potentials

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Non-Uniqueness of the Lienard-Wiechert Potentials Non-Uniqueness of the Lienard-Wiechert Potentials 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.
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