We have already seen that the relativistic expressions for momentum, mass, and energy are quite different from those applying in Newtonian theory, reducing to the latter only in the limit v/c 0. For an isolated system, the laws of movement then take the relativistic form
(21–1)
To these we must add the law of conservation of mass. In Newton’s theory this was implicit in the assumption that m
i, the mass of the ith body is a constant, or that dmi/dt=0
[Eq. (18–3)]. In Einstein’s theory, however, the mass of a given body can vary. However, mass and energy are equivalent, according to the relationship E=mc2. Therefore, the conservation of the total mass and the conservation of the total energy, of an isolated system are essentially the same law, in the sense that one follows from the other.
The conservation law dM/dt=0, or, alternatively, dE/dt=0, now replaces the nonrelativistic law dm
i/dt=0, as well as the nonrelativistic expression for the conservation
of the total energy of the system.
We shall now generalize these laws to a nonisolated system, i.e., to a system on which a net force F is acting. To simplify the problem, let us consider a system consisting of a single body, with velocity v. Then, we tentatively propose as the proper relativistic laws
(21–2)
(21–3)
These have the same form as do the corresponding expressions in the Newtonian theory. However, their physical meaning is different, because p and E are now defined by the relativistic expression p=mv, E=mc2, with, , rather than by the Newtonian expression.
It is evident, however, that to obtain equations of motion which remain invariant in form (i.e., which constitute the same relationships) in every Lorentz frame, it is not enough to give p and E a proper relativistic definition. It is also necessary to define the
Charged Particles in an Electromagnetic Field 79
force F in such a way that it will express the same kind of relationship, independent of the speed of the reference frame. Now, this cannot actually be done until we have some more specific expressions for the force, such as that due to an electromagnetic field, to gravitation, or to other forces (e.g., those arising in nuclear interactions). In this work we shall in fact discuss only the electromagnetic forces, showing in detail that they do lead to invariant relationships for the equations of motion. It may be stated, however, that all forces with properties that are known can be expressed in such a way as to lead to similarly invariant equations of motion, but the proof of the statement is beyond the scope of the present work.1
The force on a body of charge q under an electric field and a magnetic field is
(21–4)
Noting that , we obtain the well-known Lorentz equations of motion for such a body:
(21–5)
(21–6)
For our purposes these can more conveniently be expressed in differential form with ,
(21–7)
(21–8)
1 There are further forces (notably the forces between atomic nuclei), which are so poorly
understood as yet that little can be said about them in this regard. However, there is at present no reason to suppose that they lead to equations of motion that are not invariant under a Lorentz transformation.
80 The Special Theory of Relativity
when dx is the vector for the distance moved by the body in the time interval dt.
The above laws were first observed to hold in frames of reference which are such that the velocity v of the electron is small compared with c. However, we are now investigating the conditions under which these laws will hold, independent of the speed of the frame of reference. In other words, if (21–7) and (21–8) hold in some frame A, we wish to find out how the quantities and , as observed in another frame B, must be related to and in order that the equations in frame B will have the same form, when expressed in terms of the new variables.
(21–9)
(21–10)
We now express dp and dE! in terms of dp and dE by the Lorentz transformations (20–7) and (20–8) and express dx and dt! in terms of dx and dt by the similar transformation (15–12). In doing this we take the differentials of the corresponding equations, noting that V and are constants. We obtain [with
(21–11)
(21–12)
Substitution of (21–7) and (21–8) for dE and dp yields
Charged Particles in an Electromagnetic Field 81
(21–14)
Equations (21–11) and (21–12) together yield [with ]
(21–15)
Now, the above equation must be true for arbitrary particle velocity v=dx/dt. Hence it must hold independent of dx and dt. The reader will readily verify that this is possible only if the coefficients of dx and dt are separately zero, or if
(21–16)
It will now be convenient to express the field quantities and , in terms of components , ; , which are parallel to V and , ; , , which are perpendicular to V. From it follows that .
Since and , it follows [using
and ] that
(21–17)
By going through a similar procedure with Eqs. (21–11) and (21–13) the reader can verify that we obtain the corresponding equations:
82 The Special Theory of Relativity
(21–19)
The equations for and can be combined into the set
(21–20)
(21–21)
The above equations define the transformation laws for and that will lead to the same equations of motion [(21–7) to (21–10)] for a charged particle, independent of the speed of the frame of reference.
It should be noted that the transformation relationships (21–20) and (21–21) can also be shown to lead to an invariant form for Maxwell’s equations. (To do this is beyond the scope of the present work, but for a further discussion on this point see C.C.Moller, The Theory of Relativity, and W.Panofsky and M.Phillips, Classical Electricity and Magnetism.) Therefore, what has been achieved is the demonstration that the laws of electrodynamics (Maxwell’s equations) and the laws of motion of a charged particle in an electromagnetic field can both be expressed in an invariant form (i.e., as the same set of relationships in all frames of reference connected by Lorentz transformations).
Finally, it should be noted that the transformation laws for and give the kind of results that would be expected from a consideration of the laws of electrodynamics. Thus Faraday’s law of induction implies that a wire passing through a magnetic field with velocity V will have emf induced in it proportional to V and the field but perpendicular to both. Equations (21–20) and (21–21) express what is essentially the same conclusion. Thus, if in frame A we have and , then on going to the frame B, moving at speed V in which the wire is at rest, we will have . The emf (as experienced in the frame on which the wire is at rest) will then be proportional to
Similarly, if in frame A we have and , then (21–18) and (21–19) imply that in a frame moving at a velocity V relative to A, there will appear a magnetic field . This can be shown to lead to results equivalent to Maxwell’s “displacement current,” , which implies that an object passing through a static electric field will, in the frame in which it is at rest, experience a corresponding magnetic field (which could be shown up, for example, if the object were a magnetic dipole tending to orient itself in relationship to this “induced”