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Why is the magnetic force similar to a Coriolis force?

It is pointed out that the underlying reason why the magnetic force is similar to a Coriolis force is that it is caused by Thomas rotations, induced by successions of non-collinear Lorentz boosts. The magnetic force may even be viewed as a kind of Coriolis force (making perhaps more acceptable the apparent non-existence of magnetic monopoles). We also show that under a change of inertial frames, Faraday lines of force Lorentz contract as if 'etched' in space, while 'Coriolis' terms get added on.


Introduction
What led me more or less directly to the special theory of relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field.
A. Einstein [1] This magnetic force has a strange directional character […] Magnetism is in reality a relativistic effect of electricity. R. P. Feynman [2a] As is well known, Lorentz covariance implies that a purely electric force in some inertial frame S 0 acquires a magnetic component in another inertial frame S [2][3][4][5][6][7][8][9][10][11][12]. This can be seen physically in various ways. For instance, an axially moving current-carrying wire exerts a radial force on a test charge at rest, because the density of conduction electrons, and that of positive charges, equal in the wire's rest frame, do not stay so in the moving wire due to different Lorentz contractions of the volumes they occupy; but the radial force on the test charge is the same in both frames (to order v 2 /c 2 ) [2b, 8a, 9a].
This pretty argument, however, does not work if the test charge moves transversally to the wire. Also, it does not reveal the underlying reason for the "strange directional character" of the magnetic force, perpendicular to the velocity, hence doing no work, similarly to a Coriolis force -of kinematic (rather than dynamic) origin.
We here point out that magnetic forces have their strange Coriolis-like character because they enact Thomas rotations, induced by successions of non-collinear boosts [10][11][12][13][14]. So if one views magnetism as a "relativistic effect of electricity" (Feynman), as was apparently also the * antoine.royer@umontreal.ca initial idea of Einstein (citations above), then magnetic forces are a kind of Coriolis force. More generally, any Newtonian force (i.e. depending on the positions, but not on the velocities of particles) in some inertial frame, is necessarily accompanied by a 'Coriolis' force in another inertial frame.
Let us first recall that, as was implicitly noted by Einstein in his founding 1905 article on special relativity [3], a succession of two non-collinear Lorentz boosts does not result in a pure boost, but in a boost times a rotation, usually called a Thomas (or Wigner) rotation [13,14].
Now, an electric field e 0 in inertial frame S 0 induces, on a charged particle, an infinitesimal boost during an infinitesimal time interval. In going from S 0 to another inertial frame S, this infinitesimal boost gets Lorentz transformed. Thereby non-collinear boosts get combined, whence an infinitesimal Thomas rotation. It results that an infinitesimal boost induced by e 0 in S 0 becomes in S an infinitesimal boost times an infinitesimal rotation. The latter rotates the velocity of the particle in S , and this rotation is attributed to a 'magnetic' field b. Yet what is acting is "nothing else but an electric field" [1].

The point of view that "magnetism is in reality a relativistic effect of electricity" [2a]
is not the most economical for the mind. It is usually more efficient to think of electric and magnetic fields as equally 'real' components of a 2-tensor. On the other hand, it is often useful to look at mathematical equations from different perspectives, and to have different physical pictures and interpretations, even if this has no consequences in practice.
Observe, however, that different physical interpretations may lead to different physical expectations. Thus, if one views magnetism as a 'fictive' relativistic 'Coriolis' effect, then one does not expect magnetic (i.e., 'Coriolis') monopoles to exist -as is apparently the case (for now). 1 In the 'equally real' point of view, on the contrary, they would be most welcome: Classically, magnetic charges make Maxwell's equations totally symmetric [15a]; while in quantum mechanics, which requires magnetic potentials (incompatible, classically, with magnetic charges), the mere existence of a single magnetic monopole would imply (hence explain) the quantization of electric charges [16][17][18].
Of course, the view that magnetism is a 'fictive' relativistic 'Coriolis' effect only makes magnetic monopoles seem less likely, but in no way forbids them. Indeed, quantum mechanics often preys on 'fictive' classical entities and gives them 'reality'. For instance, magnetic potentials, long considered 'fictive', 2 appear quite 'real' in the Bohm-Aharonov effect [2d].
The electromagnetic field: If the magnetic force is a relativistic 'Coriolis' effect, so should the magnetic field b (to which it is convenient to ascribe this force). Indeed, let e 0 be a Coulomb field in inertial frame S 0 , obeying the electrostatic equations ( t t ) t 0 e 0 = 0 (static), 0 e 0 = 0 (inverse square), Then Lorentz transforming from S 0 to S yields Maxwell's equations wherein b is again due to Thomas rotations. Thus Faraday's magnetic induction law (d) (a changing magnetic field creates an electric field), the other facet of magnetism alongside the magnetic force, emerges. Interestingly, Ampere's law (a), including Maxwell's correction t e (a changing electric field creates a magnetic field), is also generated by a Galilean transformation, but with no physical effect, for no magnetic induction, nor a magnetic force, are produced thereby.
Electric and magnetic fields, e and b, boost and rotate the velocities of charged particles (making these fields detectable in the first place); that is, they Lorentz transform these velocities, hence act as Lorentz generators. So by analysing how an infinitesimal Lorentz transformation, inside some inertial frame S, itself transforms in going to S , one can deduce, and understand kinematically, the way e and b Lorentz transform, namely [2][3][4][5][6][7][8][9][10][11][12]: Here, e || and e are the components parallel and perpendicular, respectively, to the velocity v of frame S relative to S ; c is the velocity of light, and v = (1 v 2 / c 2 ) 1 2 . We will see that  19,20], is often presented as an isolated and rather startling curiosity ("an extraordinary coincidence" [9b]). 4 Here we see on the contrary that everything is exactly as should be: Lorentz contraction of space with everything in it, plus '(anti)Coriolis' terms due to rotations.
The propagation of electromagnetic waves has been described as "a kind of a dance -one making the other" [2e] between electric and magnetic fields inducing and sustaining one another.
In the 'Coriolis' point of view, this pas de deux is between a 'real' electric field and its 'fictive' magnetic 'Coriolis' by-product. Such a symbiosis between 'real' and 'fictive' entities has precedents: For instance, in the endless precession-nutation of a frictionless spinning top, 'real' gravitational and 'fictive' Coriolis forces alternatively reanimate one another.
Let us now briefly recall the history of Thomas rotations, and discuss notation.
Thomas rotations: Two inertial frames S and S are said to be parallel if their Cartesian frames are oriented such that if an observer in S sees the velocity of S (relative to himself) as v, then his friend in S sees the velocity of S as v . Since this still allows arbitrary rotations of the Cartesian frames about v as axis, one must ask moreover that vectors perpendicular to v have identical coordinates in the two frames. 5 Let then S be parallel to S and have velocity v in S , which we denote by S || v S (read right to left). Let in turn S || u S , so that S || u S || v S . 3 As another example, the circular magnetic lines around a current carrying electrically neutral wire, at rest, appear as ellipses to an observer moving transversally to the wire. 4 See also [8b]. Heaviside himself found it "rather remarkable" that the field stays radial [15d].
If we denote by u v the velocity of S in S , then it turns out that v u is not equal to u v if u and v are not collinear. In his founding 1905 article, Einstein evaluated u v , and its magnitude, noting [3]: "It is worthy of remark that u and v enter into the expression for the resultant speed in a symmetrical manner".
That is, v u and u v have equal magnitudes, hence differ by a rotation. This means that observers in S and in S would disagree by this rotation about their relative velocity, implying that one frame is rotated relative to the other (parallelism of inertial frames is not transitive).
Thus, a boost by velocity v, followed by a boost by u, results in a boost by u v , times a rotation. This was emphasized by Silberstein [10] in 1914. Yet these rotations are usually called Thomas rotations (especially in the infinitesimal case), or Wigner rotations (in the finite case) (after Thomas, who showed in 1926 one of their subtle effects in atomic physics [13], and Wigner, who gave in 1939 a famous analysis of the Lorentz-Poincaré group [14]).
About notation: To analyse the interplay of Lorentz boosts and spatial rotations, tensor notation is not practical. We will use instead vector-matrix notation in a (3-space,1-time) block form. This notation was (implicitly) used by Silberstein [10] in his treatment of Thomas rotations, 6 though once done with these rotations, he prefered quaternions [12,21] for providing "just enough 'union' to express the relativistic standpoint, and yet enough distinction not to amalgamate time and space entirely." We believe that the block matrix notation achieves the same end in a more intuitive way. 7 We will add to that notation spatial projectors, both to expedite calculations and to highlight the physical and geometrical meanings of quantities. This notation will allow us to make otherwise messy computations relatively simple and transparent. 6 Silberstein displayed separately space and time components, using 3-vector notation, which amounts to using block matrices (Cayley's matrices were little familiar to physicists at the time). 7

Vector-matrix notation
Bold letters will denote 3 3 matrices A, or column 3-vectors a. The transpose a T is a row 3-vector, so that a T b = a b , and the 'dyad' ab T is a 3 3 matrix. Four-dimensional spacetime vectors and matrices will be written in (3-space, 1-time) block form as follows: In this way, manipulations reduce to those of 2-dimensional vector-matrix algebra, except that the elements in these matrices do not commute.

Spatial projectors:
We denote by u the projector onto the 3-vector u, and by u that onto planes perpendicular to u, that is (u is the length of vector u): These satisfy u 2 = u , u u 2 = u u and u u u u = u u u u = 0 , so that, for scalars a, b : Components parallel and perpendicular to u, namely a || u a and a u a , satisfy: Infinitesimal rotations: Let be an infinitesimal scalar. To rotate a 3-vector u by angle b about axis b, one adds to u a perpendicular vector b u . So a 3 3 infinitesimal rotation matrix R b may be defined by R b u = u b u , for any u, that is: where the antisymmetric matrix J b can be read off from The standard identities (a b) c = c (b a) = b(a c) a(b c) = (ba T ab T )c imply that

Coriolis forces
We wish here to underline some aspects of Coriolis forces. Let a Cartesian frame S rotate at a uniform angular frequency relative to an inertial frame S 0 . Then a trajectory x(t ) in S becomes x 0 (t) = R(t)x(t) in S 0 , where the rotation matrix R(t ) satisfies R = J R (overdots indicate time derivatives). Hence, or letting R = 1 at the instant considered [22]: The velocity dependent term 2 x is the Coriolis acceleration. The factor 2 stems from two different effects: (i) rotation of the velocity ˙ x , and (ii) changes in the tangential velocity v tan = x due to changes in x. The centripetal acceleration ( x) comes from the rotation of v tan . If now a Newtonian force f 0 acts on a particle of mass m in S 0 , so that m x 0 = f 0 , then its equation of motion inside S is, by (11): where f cent and f cor are 'fictive forces'. For instance, if you sit on a frictionless turntable S , then you follow a curved path relative to the turntable rotating under you, and you can view this curving relative to S as caused by these 'forces'. If, however, there is friction, and you manage to crawl along a straight path x(t ) on the turntable, then you do feel these 'fictive' forces (here better called 'inertial'). How can you detect that you are on a rotating frame? By varying, at the same position, your velocity ˙ x relative to S , for the sideways Coriolis force will then vary.
Otherwise, all forces seem 'real' (i.e., to depend only on position). 8 You do not feel Coriolis forces just on a merry-go-round. Actually, the most common manifestation of Coriolis forces is the gyroscopic effect in spinning objects like tops and wheels.
This is usually discussed in terms of angular momentum and torque, which is more efficient mathematically [22]. But as emphasized by Feynman [23], the gyroscopic effect is really due to 8 You cannot tell fictive from real forces if your velocity x at x on the turntable is always the same: so if you were told that it turns at rate , then you would call m ( x) 2m x fictive, and all the rest 'real'. But by varying x , you would find that only = yields a 'real' force independent of x .
Coriolis forces felt by the individual particles constituting the spinning object. 9 In some cases, Coriolis and 'real' forces can sustain one another in an endless oscillation, as in the precessionnutation of a frictionless spinning top (see section 12).

Galilean relativity
To better understand how Lorentz transformations interact with electric and magnetic fields, it will help to first examine which aspects result from simple Galilean relativity, where our intuition is more at ease.

Galilean transformations: A general Galilean transformation may be written as a boost
x x + vt = B v x by velocity v, times a spatial rotation R : Let be an infinitesimal, and consider, using (8), an infinitesimal Galilean transformation because electric and magnetic fields act in this way on the velocities of charged particles). To first order in : Let now S have velocity v relative to S and be parallel to it, so that x = x + vt . Then in S : where (14)(c) acts on x alone, and not on v . We shall call J b v an 'anti-Coriolis' term. 10 9 Consider a spinning vertical bicycle wheel: Particles at its front end have large downwards velocities, so that inclining the wheel to the right forces these particles to acquire leftwards velocity components, whence Coriolis forces to the right. Particles at the back end of the wheel have upwards velocities, hence exert Coriolis forces to the left. Whence a torque turning the wheel to the right. 10 Put otherwise, a rotation R b about the origin of S (e.g., the center of a flying frisbee), moving with uniform velocity v relative to S (the ground), is equivalent to the same rotation about the origin of S (yourself say), but with an added translation b vt undoing the rotation of vt .

Equation of motion:
Let a particle of mass m have position x(t ) and velocity u = x = dx dt in inertial frame S. Let also t be an infinitesimal time interval. Writing u(t + t) = u(t) + u t to first order in t , we will use Newton's law of motion m u = f in the form Now, although magnetism is a Lorentzian correction of order c 2 , we can still imagine, for the sake of discussion, a magnetic force acting in a Galilean world ( c ). So let f = q(e b u) in (16), where q is the charge on the particle, and e, b are electric and magnetic fields in S .
Using (8), we get, to first order in q t m [compare (14)]: namely an infinitesimal Galilean transformation on u(t) : During t , the electric field e boosts the velocity u by e , and the magnetic field b rotates it by R b . Thus, e, b act here as Galilean generators. So the way they transform, namely Eqs (15), is just kinematics. Note that by (15), and since the particle's velocity in i.e., the particle feels the same force in both frames, f = f , as is evident.
Observe that if e = 0 in S, but b 0 , then one gets e = J b v in S , that is: magnetic forces in a Galilean world necessarily imply 'anti-Coriolis' electric forces. However, electric forces do not imply magnetic forces (since b = b ), so that electric forces, that is, Newtonian forces, can exist solitarily in a Galilean world (unlike in our Lorentzian world).
Field equations: We next examine how the electrostatic field equations transform under a Galilean transformation. Coulomb's inverse square force law, and the superposition principle, imply that a static charge density 0 , in inertial frame S 0 , creates an electric field e 0 satisfying the electrostatic equations (1) (Coulomb's law in differential form): It is perfectly legitimate to submit these equations (which also describe Newtonian gravity) to a Galilean transformation. So let S 0 have velocity v 0 in S, so that x = x 0 + v 0 t . Then e(x,t) = e 0 (x v 0 t) , and we get in S : where e = e 0 and = 0 . Noting next 11 that (v 0 )e 0 = v 0 ( e 0 ) (v 0 e 0 ) and v 0 ( e 0 ) = v 0 0 = j , we obtain (the reason for the tilde on b will appear in section 9): (20) have the advantage over (19) that they do not refer explicitly to v 0 . So if there are charges of various velocities v i in S, then one can invoke the superposition principle and put (if e i is due to charge i in its rest frame S i ): The fields e, b can now vary arbitrarily in both space and time. If frame S has velocity v in S , then e (x ,t) = e(x vt,t) , so that t e = e (v )e . Using e = b j , and again (v e) (and e = e 0 ), we get: Observe that (20) induction) with Galilean relativity which motivated special relativity. 11 We use here the identities . 12 Note however that b v 0 e , and t e = (v 0 )e in (19),

Special relativity, Lorentz transformations
Lorentz transformations X X = X preserve space-time intervals 13 where g is the metric. Thus, X T gX = X T gX , whence the requirement: In view of (25)(c), with every contravariant vector Y is associated a covariant vector ˜ Y = gY , and vice versa. For instance, the contravariant vector associated with in (27) is Let inertial frames S and S be related by X = X . Then a transformation T inside S Thus, in contrast to (27) Note, however, in view (25)(b), that T Tg 1 T = T T is a contravariant tensor.
Explicit form of Lorentz transformations: We will denote, using the projectors (5): x , we see that K v compresses transverse components by v 1 1, hence shortens and rotates 3-vectors. Now, any matrix satisfying (25) can uniquely be written as a product = RB v of a spatial rotation R and of a boost B v [7,10]: whence the familiar relations Time-dilation and length contraction: Applying (32) to intervals X = ( x, t ) gives us Let us recall then the two most famous relativistic effects: Two events separated by a time interval t at the same location in S, so that x = 0 in (34), are separated in S by By the inverse X = B v X of (32), two events distant by x but simultaneous in S , so that (6): exhibiting the Lorentz-Fitzgerald contraction of longitudinal lengths. Using dx = v 1 dx || + dx (if dt = 0) and (7), we find that surface elements ds = dx dy transform as (the longitudinal dimension of the surface, i.e., the component ds of its normal, contracts).
The 4-velocity: Consider a particle of position x(t ) and velocity u(t) = dx dt in inertial frame S: since it stays fixed ( x 0 = 0 ) in its rest frame S 0 , its 'proper time' t 0 = t u [by (35)] is an invariant. It follows that the 4-velocity U dX d transforms like X. Thus:

Infinitesimal Lorentz transformations
Let be an infinitesimal. Since e = 1 to first order in , and R b = 1 J b by (8), an infinitesimal Lorentz transformation has the general form where G e,b is a Lorentz generator. If inertial frames S and S are related by X = X , then by (29) a transformation 1+ G e,b inside S becomes 1+ G e,b 1 inside S . Since this is also a Lorentz transformation, we must have with transformed vectors e ,b . If = B v is a pure boost, one gets (see Appendix B.1): [identical to (3)]. Note that the term v b v in e survives in the Galilean limit c (since v 1 ), and is just the 'anti-Coriolis' term J b v in (15) (the role of K v 1 will be seen later on). We note also that transforming a pure boost B e with B v yields Here, If one defines a 'dual' G e,b G b, e c 2 , then one finds that [7]:

Relativistic equation of motion
All our special relativistic considerations up to now were purely kinematic. We now turn to dynamics. We will start with a particle at rest in inertial frame S 0 , and let a Newtonian force (depending only on position) act on it. Since the particle has zero velocity, Newton's law applies at time t 0 = 0 . Lorentz transforming to another inertial frame will then yield the relativistic equation of motion. We will find that due to the Thomas rotation in (42), a velocity-rotating force arises, so that Newtonian forces imply Coriolis-like forces. The reciprocal is also true, as in the Galilean case (15). Hence, in Lorentzian relativity, any (rest-mass preserving) force necessarily has both an 'electric' and a 'magnetic' component [7]. This being so, we may as well assume that the Newtonian force acting in S 0 is an electric force.
So consider a particle of mass m and electric charge q, at rest in S 0 at time t 0 = (its proper time). Its velocity is u 0 ( ) = dx 0 d = 0 , and its 4-velocity (38) is U 0 ( ) = 0 1 . Let a Newtonian force f 0 = qe 0 act on the particle, where e 0 is an electric field in S 0 . Newton's law (which applies at zero velocity) implies, by (16), that an infinitesimal time later, u 0 ( + ) = (f 0 m) = e 0 , where q m . Hence, to first order in : (where e 0 T c 2 could be added in since it multiplies 0). Thus, during , the 4-velocity gets boosted by B e 0 . Now if the particle (hence S 0 ) had, at its proper time , velocity u( ) in inertial frame S, then its 4-velocity in S at time + will be, by (42):  (12). There is no factor 2 in qb u because the Thomas rotation here acts only on velocities, not on positions, so that no tangential velocity comes into play (hence there is no 'centrifugal' force either). 14 We started with a particle at rest to be able to apply Newton's law, and therefrom derive the relativistic equation of motion. But once the latter is known, we may start with an inertial frame S wherein b = 0 , but the particle can have any velocity u. The equation of motion in S is then U( + ) = B e U( ) , by (46). If S has velocity v in S , then similarly to (46): boosts induced by e i (due to q i ) inside the rest frame S i of q i , wherein b i = 0. 14 Note that a particle at rest in S 0 would not feel a field b 0 0 , so that one could also write (45) as U 0 ( + ) = R b 0 B e 0 U 0 ( ) , with any b 0 . This would modify e and b in S, but not the total f in (48). So for a single particle of velocity u, it is largely arbitrary how the force f acting on it is broken up into f = e b u . However, by observing particles of various velocities u, one would find that a unique e, b makes (48) valid for all u. This is analogous to the turntable case, see footnote 8.

The field equations
The preceding derivation of the relativistic equation of motion, and the emergence of the magnetic force, did not depend on the form of the Newtonian force field e 0 (x 0 ,t 0 ) in S 0 . We will now see that if e 0 (x 0 ) is a static Coulomb field in S 0 , then Lorentz transforming to another inertial frame S yields Maxwell's equations, so that Faraday's magnetic induction, the other facet of magnetism besides the Lorentz force, also emerges. where the arrows on the differential operators signify that they act towards the left, so that To make the calculations more transparent, it is convenient here to introduce the antisymmetric contravariant tensor F = Gg 1 and its dual, see (44): Then, in view of (27), the electrostatic eqs (50) can be written as  (44): Since Eqs (55) do not refer explicitly to v 0 , one can set, similarly to (22): Note that Eqs (20) can be obtained directly [bypassing (19)] 15  Observe, nonetheless, that since b v and v e both appear in the Galilean expressionseven if in an incoherent way -so must they, in a coherent way, in the Lorentzian case, since transforming G e,b , F e,b , or F e,b , with B v , necessarily produces the same e , b . So in a sense, all aspects of magnetism are 'rooted' in Galilean kinematics (see also Footnote 12). The field e is often represented by Faraday lines of force, of density (number of lines piercing a unit transverse area) proportional to e. If b = 0 in (41), then by (37)  As a famous special case, discovered by Heaviside in 1888 [19,20], the electric lines of force of a point charge of uniform velocity u are just the isotropic radial lines of the charge at rest, but compressed by u 1 along u . This has generally been considered surprising.

Lines of force
Suppose you were to draw on a piece of paper the field lines for a charge at rest, and then set the picture to travelling with the speed v. Then, of course, the whole picture would be compressed by the Lorentz contraction […] The miracle of it is that the picture you would see as the page flies by would still represent the field lines of the point charge.

Feynman [2f]
On the contrary, we see here that everything is exactly as should be, since Lorentz contraction applies to space with everything in it.
Let us write Eq (47) as follows [noting that (b u) u = 0 ]: Or equivalently, since = t u , and defining an infinitesimal velocity w f m : Let us denote here relativistic velocity addition by v u = B v u ( signifies 'linear fractional map', see Appendix C): while the notation v u conveys the notion of 'addition', B v u conveys that of 'transformation'. We may then write the spatial parts of (46) and (62) as: (46)] is just Newton's law in the instantaneous rest frame of the particle. The equation u(t + t) = B w u(t) [eq (62)] (which one would use for a step by step numerical integration of the motion) suggests that a particle of velocity u may be viewed as a 'quasi-particle' of 'rest mass' m u = u m 0 : an impulse f t gives to that "particle-ofvelocity u" object an infinitesimal velocity w , relative to which the 'bare' particle maintains its velocity u (so that the new velocity of the particle in the laboratory is B w u = w u ). Thereby it is possible to treat a bunch of particles of various velocities as an 'object' of 'rest mass' equal to the sum of the individual relativistic masses [6]. This possibility, obvious in the Galilean case, has of course been crucial to the development of physics.
Note that Newton's equation of motion (17) may also be written in the forms (62)-(63), but with Galilean boosts B v (instead of Lorentzian boosts B v ), that is: That B u w = B w u = u + w is obvious [recall that here f = f = f 0 , see after eq (17)]. By contrast, relativistic 'velocity addition' u v = B u v is non-commutative, u v v u , for the two sides differ by a Thomas rotation. It follows that B u w 0 can equal B w u only if f and f 0 differ by a transverse term (i.e., a magnetic force) enacting the Thomas rotation.

Conclusion
That Lorentz covariance implies magnetic forces, alongside electric forces, has been argued in various ways. One way (recalled in section 1) is to consider the radial force exerted by a current-carrying wire on a test charge moving parallel to it [2,8,9]. This argument, however, does not work if the test charge moves transversally to the wire. Also, it gives no clue as to why the magnetic force has its strange, Coriolis-like, velocity-rotating character.
Now, due to the spatial anisotropy (expressed by K v ) forced into Lorentz boosts by spacetime interval invariance, products of non-collinear Lorentz boosts produce Thomas rotations (intimately tied to the non-commutativity of velocity 'addition'). It is noteworthy that Silberstein [10] ended his discussion of Thomas rotations with the remark: ; another example (perhaps) is the magnetic field associated with the spin of a 'point' particle with (apparently) no internal structure, like an electron. 16 Realizing that magnetism is a kind of 'Coriolis' effect changes nothing, except maybe to make more palatable the fact that magnetic monopoles have not shown up (yet) -but in no way forbids them [16][17][18]9]. Heaviside, for one, liked to write Maxwell's equations in a symmetric 'duplex form', including magnetic charges and currents, thereby "throwing all potentials over- Moreover, the mere existence of a single magnetic monopole would explain the quantisation of electric charge [16][17][18].
The mere existence of one pole of strength g would require all electric charges to be quantized in units of c / 2g […]. The quantization of electricity is one of the most fundamental and striking features of atomic physics, and there seems to be no explanation for it apart from the theory of poles. This provides some grounds for believing in the existence of these poles. Dirac 1948 [17] But "none has been found" (Feynman -see Footnote 1). 16 Feynman, in his Lectures, seemed to visualize electrons as tiny charged gyroscopes [2e].
Electric and magnetic fields e and b act as Lorentz generators on the velocities of charged particles. So by considering how an infinitesimal Lorentz transformation inside a given inertial frame transforms to another inertial frame, one can understands kinematically the way e and b Lorentz transform: as should be, electric and magnetic lines of force Lorentz contract as if 'etched' in space, while '(anti)Coriolis' terms due to rotations get added on.
The contraction of the radial electric lines of force of a moving point charge provides a nice way of understanding electromagnetic radiation (due to J. J. Thomson [25]): If the charge suddenly accelerates to a new velocity, then transverse kinks are created in these lines of force, since the contraction is different before and after the acceleration. As these kinks propagate outwards, their separations grow like the radial distance r, hence their density decreases like r 1 . In contrast, the density of radial lines (hence the radial electric field) decreases like r 2 .
The kinks ~r 1 constitute the radiation field: We have a little piece of field which is travelling through space all by itself. The fields have "taken off"; they are propagating freely through space, no longer connected in any way with the source. The caterpillar has turned into a butterfly! Feynman [2f] Thus, a maximum speed c for signals implies Lorentz covariance; implying in turn that electric fields produce (due to Thomas rotations) 'Coriolis' magnetic fields, such that these two fields, when changing in time, induce one another, enabling them to co-propagate at speed c.
How can this bundle of electric and magnetic fields maintain itself? The answer is: by the combined effects of the Faraday law e = t b , and the new term of Maxwell, c 2 b = t e . They cannot help maintaining themselves. Suppose the magnetic field were to disappear. There would be a changing magnetic field which would produce an electric field. If this electric field tries to go away, the changing electric field would create a magnetic field back again […] They must go on forever. They maintain themselves in a kind of a dance -one making the other. Feynman [2f] Recall from Eq (20) [15g]) is also a 'dance' balancing gravitational and 'fictive' inertial forces [26]. 18 We noted that the relativistic equation of motion allows to treat a bunch of particles of varied velocities as an 'object' of rest mass equal to the sum of the individual relativistic masses.
Indeed, physics would be in a sorry state had it not been possible to treat agregates of elementary particles (whatever that means) as 'elementary' objects. As Bohm stressed [6]: The property possessed by bulk matter -being capable alternatively of analysis into parts or treatment as a single whole -is a general feature of the world. This feature must therefore be implied by any proposed set of laws of mechanics. David Bohm [6] This demand in fact suffices [6] to deduce the relativistic equation of motion (48). As often in physics, 'simplicity' is here an unreasonably effective guide: One has no other guide for guessing the right laws than the ideal of simplicity. Max Born [5] This is not always the case in science: Elegance and simplicity are, in biology, dangerous guides. Biologists must constantly remind themselves that what they see was not designed but evolved. F. H. Crick [27] Appendix A Connection with tensor notation Block matrix notation allows one to keep track simultaneously of Lorentz covariance and separate space and time aspects. We here relate (with two-way arrows ) matrix equations with the corresponding tensor equations.
Vectors X X μ transform as X = X X μ = μ X , where μ satisfies (25)(a): Treating the Lorentz transformation matrix μ as a 'tensor', we can write 1T = g g 1 μ = g μ g [28], so that The electromagnetic tensor has the Lorentz-generator form G G μ , and the antisymmetric form F Gg 1 F μ = G μ g . These transform as In explicit 4 4 matrix notation: Eqs (53) are G μ = j μ and G μ = 0 , where G is the dual electromagnetic tensor, obtained by doing e b , c 2 b e , or F μ = μ F , where μ is the totally antisymmetric tensor. This last tensor relation makes obvious the duality (44).

Appendix B Explicit computations
B.1 Derivation of Eqs (41): It is a bit simpler to work with F = Gg 1 than with G . Using (10), we first compute: where we used ve T K v K v ev T  (since trab T = a b and tr1 = 3 ), whence the invariants (43). We also find that If y is an invertible square matrix (e.g., a non-zero scalar), then since T = xy 1 : Applying (73) to dx = udt and dX = B v dX , where dX = (dx, dt) , so that u plays the role of T in x = Ty , and B v that of M , we get: