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Appendix A: Relativity and Electromagnetism In this appendix we examine and, relativistically, modify Newtonian particle mechanics; thus, it would be natural to look with the same intention at Maxwell’s electrodynamics, at first in a vacuum. That theory, however, turns out to be already “special-relativistic.” Unlike Newtonian mechanics, classical electrodynamics is consistent with special relativity. Maxwell’s equations and Lorentz’s force law can be applied legitimately in any inertial system (e.g., system S). The only two assumptions we need to make about the electromagnetic force are that it is pure force—in other words, rest mass is preserving—and that it acts on particles in proportion to point charge q, which they carry. Beyond that only “simplicity” and some analogies with Newton’s gravitational theory will guide us, which what this appendix is all about. A.1 Introduction With corrected Newtonian mechanics, we are now in a position to develop a complete and consistent formulation of relativistic electrodynamics. We will not be changing the rules of electrodynamics at all, however; rather, we will be expressing these rules in a notation that exposes and illustrates their relativistic character. As we said the at the beginning of this appendix, the intention is to look at Maxwell’s electrodynamics and its basic laws, as they have been summarized by the four Maxwell’s equations, plus Lorentz’s force law , are for an invariant condition under Lorentz transformations from one inertial frame to another. As also mentioned before, by inertial frame with only two assumptions: (1) the force induced from electromagnetics (EM) will be a pure force; and (2) the rest mass of moving particles persevere, and that it acts on particles in proportion to point charge q, which the particle carries. Indeed, it was the problem of finding a transformation that leaves © Springer International Publishing AG, part of Springer Nature 2019 493 B. Zohuri, Scalar Wave Driven Energy Applications, https://doi.org/10.1007/978-3-319-91023-9 494 Appendix A: Relativity and Electromagnetism Maxwell’s equations invariant, which led Lorentz to the discovery of the equations now associated with his name. Nevertheless, even though relativity did not modify Maxwell’s theory in a vacuum, it added immeasurably to our understanding of it and also gave us a new means for working in it. On the other hand, the theory of electromagnetism in moving media, originally because of Minkowski, is purely a relativistic development [1]. Maxwell’s theory, within its old Galilean framework, seems quite far-fetched. Galileo was the first one to observe that a brick on a table, to which is applied a shoving force, soon comes to a stop when a certain time elapses after the initial shove because of friction between the two surfaces (i.e., brick and table). He also con- cluded, however, that the more you polish the surface of a brick, or a table, the farther the brick travels. Moreover, if the table itself is moving, the brick comes to rest with respect to the table, not the floor across which the table is moving or standing. Galileo concluded that the brick stops because of the friction that may exist between the brick and the table, as we said. If you can eliminate all existing friction by some means of polishing the surfaces to perfect smoothness, then the brick would keep moving forever. Newton, of course, incorporated this discovery into his first law of motion into the classical mechanics of motion [2]. In the real world and in practice, some friction always exists; however, this does not bring an object to an “absolute” state of rest, but merely to rest with respect to whatever it is rubbing against. Within relativity, on the other hand, it is one of the two or three simplest possible theories of a field force such as Yukawa-Klein- Gordon’s scalar meson field theory and Nordström’s attempt at a special-relativistic theory of gravity. In fact, rather than verifying that Maxwell’s equations are Lorentz invariant,we will use a synthetic approach that will parallel what we know about classical mechanics and highlight once more the “human-made” aspect of physical laws. We will construct a relativistic field theory that is consistent with just a very few of the basic facts of electromagnetism, and we intend to find that it is Maxwell’s. Thus, the main purpose of this appendix is to provide a deeper understanding of the structure of electrodynamics laws that had seemed arbitrary and unrelated before taking on a kind of coherence and inevitability when approached from the point of view of relativity. To begin with we will show why there needs to be such a thing as magnetism, given electrostatics and relativity, and how, in particular, we can calculate the magnetic force between a current-carrying wire and a moving charge without ever invoking the laws of magnetism. Using Fig. A.1, suppose you had a string of positive charges moving along to the right at speed υ. We will assume the charges are close enough together so that we can regard them as a continuous line charge λ. Superimposed on this positive string is a negative one, Àλ, proceeding to the left at the same speed υ. We then have a net current to the right of magnitude: I ¼ 2λυ ðA:1Þ Appendix A: Relativity and Electromagnetism 495 a –––() –λ v – –– v + + + ()+λ +++ s u q b ––– (λ – v –) – – – v + + ()λ+ + + + s q Fig. A.1 Illustration of moving particle q Meanwhile, distance s away there is a point charge q traveling to the right at speed u < υ, as shown in Fig. A.1a. Because, the two line charges cancel each other, there is no electrical force on charge q in this system S. We now examine the same situation from another system’s point of view— namely, system S, which moves to the right with speed u, as illustrated in Fig. A.1b. In this frame of reference charge particle q is at rest. According to Einstein’s theory and his velocity addition rule, however, the velocities of the positive and negative lines are now defined by the following equation: υ Ç u υÆ ¼ ðA:2Þ 1 Ç υu=c2 Because υÀ has a greater value than υ+, the Lorentz contraction of the space between negative charges is more severe than that between positive charges; in this frame, therefore, the wire carries net negative charges [2]. In fact, λÆ ¼ÆðÞγÆ λ0 ðA:3Þ where 496 Appendix A: Relativity and Electromagnetism 1 γÆ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA:4Þ 2 2 1 À υÆ=c and λ0 is the charge density of the positive line in its own rest system. Still, that is not the same as λ; indeed, in system S they already are moving at speed υ, thus: λ ¼ γλ ð : Þ 0 A 5 It takes some algebra to put γÆ into a simple form as: 2 Ç υ γ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic u Æ ÀÁ 1 À2 2 2 1 À ðÞυ Ç u 2 1 Ç υu=c2 ðÞc2 Ç uυ À c2ðÞυ Ç u c2 ð : Þ () A 6 2 Ç υ Ç υ= 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic u ¼ γ p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu c ðÞÀc2 À υ2 ðÞc2 À u2 1 À u2=c2 Naturally, then, the net line change in frame system S is going to be: ÀÁÀ2λuυ λtotal ¼ λþ þ λÀ ¼ λ0 γþ À γÀ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA:7Þ c2 1 À u2=c2 At the conclusion, as the result of unequal Lorentz contraction of the positive and negative lines, we can see that a electrically neutral current-carrying wire in one inertial system will be charged in another. Now a line charge λtotal sets up an electric field as: λ E ¼ total ðA:8Þ 2πε0s But if there is a force on charge particle q in system S, there must be one in system S as well; in fact, we can calculate it by using the transformation rules for forces. Because charge q is at rest in system S and force F is perpendicular to u, the force in system S is given by a set of equations as follows: 8 < ~ 1~ F⊥ ¼ F⊥ γ ðA:9Þ : Fk ¼ Fk This equation is valid under the condition that particle q is instantaneously at rest in system S, as we mentioned in the preceding as part of the two conditions and assumptions. Equation A.9 is the component of vector force ~F parallel to the motion of system S and is unchanged, whereas components perpendicular are divided by γ [2]. Note that perhaps it may occur to us that we could avoid the bad transformation behavior of force ~F by introducing a “proper” force, analogous to proper velocity, Appendix A: Relativity and Electromagnetism 497 that would be the derivative of momentum with respect to proper time; it can be written as: dpμ Kμ ðA:10Þ dτ Equation A.10 is a presentation of what is called Minkowski force, and it is plainly a 4-vector because pμ is a 4-vector and proper time is invariant. The spatial components of Kμ are related to the “ordinary” force by: dt d ~p K~ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~F ðA:11Þ dτ 1 À u2=c2 while the zeroth component is provided as: dp0 1 dE K0 ¼ ¼ ðA:12Þ dτ c dτ Remember that γ and β pertain to the motion of system S with respect to system S, and they are constant values and that ~u is the velocity of the particle q with respect to S. See Chap. 4 of this book, section “Electrodynamics and Relativity,”for some further details.

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