Derivation of Fifth Fundamental Equation

by Richard Chace Tolman

XXXV. Note on the Derivation from the Principle of Relativity of the Fifth Fundamental Equation of the Maxwell-Lorentz Theory.

By Richard C. Tolman, Ph. D., Instructor in Physical Chemistry at the University of Michigan.^[1]

If we consider two systems of "space time coordinates” S and S' in relative motion in the X direction with the velocity v, any kinematic phenomenon which occurs may be described in terms of the variables x, y, z and t belonging to the system S or z', y', z' and t' belonging to the system S'. The Einstein theory of relativity has led to the following equations for transforming the description of a kinematic phenomenon from one set of coordinates to the other^[2].

$t'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left(t-{\frac {v}{c^{2}}}x\right)$

(1)

$x'={\frac {1}{\sqrt {1-\beta ^{2}}}}(x-vt)$

(2)

$y'=y\,$

(3)

$z'=z\,$

(4)

(where $c$ is the velocity of light and $\beta$ is substituted for the fraction ${\tfrac {v}{c}}$).

The content of these equations may be expressed in words, by saying that an observer in the moving system S' (S having been arbitrarily taken as at rest) uses a metre stick which, although the same length as a stationary metre stick when held perpendicular to the line of relative motion of the two systems, is shortened in the ratio of ${\sqrt {1-\beta ^{2}}}:1$ when held parallel to OX, that clocks in the moving system beat elf seconds which are longer than those of stationary clocks in the ratio $1:{\sqrt {1-\beta ^{2}}}$, and that a clock in the moving system which is $x'$ units to the rear of the one at the centre of coordinates is set ahead by $x'{\tfrac {v}{c^{2}}}$ seconds, although the two clocks appear synchronous to the moving observer. A simple non-analytical derivation of these relations has been given in another place^[3].

Let us now take the Maxwell-Hertz equations for the electromagnetic field

$\mathrm {curl} \ {\mathsf {H}}=4\pi \rho {\mathsf {u}}+{\frac {1}{c}}{\frac {\partial {\mathsf {E}}}{\partial t}}$

(5)

$\mathrm {curl} \ {\mathsf {E}}=-{\frac {1}{c}}{\frac {\partial {\mathsf {H}}}{\partial t}}$

(6)

$\mathrm {div} .\ {\mathsf {E}}=4\pi \rho$

(7)

$\mathrm {div} .\ {\mathsf {H}}=0.$

(8)

If these equations are true descriptions of electromagnetic phenomena, it is evident then by the first postulate of relativity that similar equations

$\mathrm {curl} \ {\mathsf {H}}'=4\pi \rho '{\mathsf {u}}'+{\frac {1}{c}}{\frac {\partial {\mathsf {E}}'}{\partial t'}}$

(9)

$\mathrm {curl} \ {\mathsf {E}}'=-{\frac {1}{c}}{\frac {\partial {\mathsf {H'}}}{\partial t'}}$

(10)

$\mathrm {div} .\ {\mathsf {E}}'=4\pi \rho '$

(11)

$\mathrm {div} .\ {\mathsf {H}}'=0.$

(12)

must hold when the phenomena are described by an observer moving with the system S'.

It has been shown by Einstein that the following equations, together with the kinematic relations (1-4) already given are necessary and sufficient for transforming equations (5, 6, 7 and 8) into (9, 10, 11 and 12)^[4].

${\mathsf {E}}_{x}'={\mathsf {E}}_{x}$

(13)

${\mathsf {E}}_{y}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {E}}_{y}-{\frac {v}{c}}{\mathsf {H}}_{z}\right)$

(14)

${\mathsf {E}}_{z}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {E}}_{z}+{\frac {v}{c}}{\mathsf {H}}_{y}\right)$

(15)

${\mathsf {H}}_{x}'={\mathsf {H}}_{x}$

(16)

${\mathsf {H}}_{y}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {H}}_{y}+{\frac {v}{c}}{\mathsf {E}}_{z}\right)$

(17)

${\mathsf {H}}_{z}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {H}}_{z}-{\frac {v}{c}}{\mathsf {E}}_{y}\right)$

(18)

Thus at a given point in space, we may distinguish between the electric vector E as measured by a stationary observer and the vector E as measured in units of his own system by an observer who is moving past the stationary system with the velocity $v$ in the X direction. If $\epsilon$E is the force acting on a small stationary test charge of magnitude $\epsilon$, then $\epsilon$E' will be the force acting on the same test charge or electron when it is moving through the point in question with the velocity $v$, the force $\epsilon$E' being measured in units of the moving system^[5].

We are more particularly interested, however, in the vector F which determines the force $\epsilon$F that nets on the moving charge but which is measured in "stationary units," thus determining the equations of motion of the test charge $\epsilon$ with respect to stationary coordinates. Since, however, it is possible to obtain relations between the units of force used by stationary and moving observers, a method is presented of calculating F from the values of E' already given by the transformation equations (13-18). As a matter of fact the expression for F which can thus be obtained is identical with the fifth fundamental equation of the Maxwell-Lorentz theory.

Relation between the Units of Force used in Moving and Stationary Systems.

Consider a body having; the mass $m_{0}$ when at rest and moving with the same velocity $v$ as a system of coordinates S'. Evidently its acceleration with respect to those coordinates is determined by Newton's laws of motion, and its acceleration with respect to stationary coordinates can be found by making the proper substitutions, giving us

${\mathsf {F}}_{x}'=m_{0}{\dot {u}}_{x}'=m_{0}{\frac {{\dot {u}}_{x}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}$

(19)

${\mathsf {F}}_{y}'=m_{0}{\dot {u}}_{y}'=m_{0}{\frac {{\dot {u}}_{y}}{\left(1-\beta ^{2}\right)}}$

(20)

${\mathsf {F}}_{z}'=m_{0}{\dot {u}}_{z}'=m_{0}{\frac {{\dot {u}}_{z}}{\left(1-\beta ^{2}\right)}}$

(21)

The substitutions

${\dot {u}}_{x}'={\frac {{\dot {u}}_{x}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}},\ {\dot {u}}_{y}'={\frac {{\dot {u}}_{y}}{\left(1-\beta ^{2}\right)}}\ \mathrm {and} \ {\dot {u}}_{z}'=m_{0}{\frac {{\dot {u}}_{z}}{\left(1-\beta ^{2}\right)}}$

are an obvious consequence of the relations between the units of length and time used in the two systems. For example, if a body has an acceleration in the Y direction, of magnitude ${\dot {u}}_{y}$when measured in the system S, evidently its acceleration ${\dot {u}}_{y}'$ as measured in the system S' will be greater because the units of time used in that system are "lengthened" in the ratio $1:{\sqrt {1-\beta ^{2}}}$. Remembering that the units of length in the Y direction are the same in both systems, and noticing the time enters to the second power in the expression for acceleration, the relation ${\dot {u}}_{y}'={\tfrac {{\dot {u}}_{y}}{\left(1-\beta ^{2}\right)}}$ is evident. The other relations may be obtained in a similar way.

If now we define force as the increase in momentum per second we shall have, as has already been pointed out by Lewis^[6],

${\mathsf {F}}={\frac {d}{dt}}(mu)=m{\frac {du}{dt}}+u{\frac {dm}{dt}},$

where a possible change in mass as well as a change in velocity is allowed for. It has, moreover, been shown by Professor Lewis and the writer^[7], that the two postulates of relativity, themselves, combined simply with the principle of the conservation of momentum are sufficient for a proof that the mass of a body is increased when set in motion in the ratio $1:{\sqrt {1-\beta ^{2}}}$, so that in general the mass of moving body $m={\tfrac {m_{0}}{\left(1-\beta ^{2}\right)}}$. Substituting in the equation above, we have

${\mathsf {F}}={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\frac {du}{dt}}+u{\frac {d}{dt}}{\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},$

or in the case where $u_{x}=v,\ u_{y}=u_{z}=0$ :-

${\begin{array}{l}{\mathsf {F}}_{x}={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\frac {du_{x}}{dt}}+u_{x}{\frac {d}{dt}}{\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}{\dot {u}}_{x},\\\\{\mathsf {F}}_{y}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {1}{2}}}}{\dot {u}}_{y},\\\\{\mathsf {F}}_{z}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {1}{2}}}}{\dot {u}}_{z},\end{array}}$

and by the further substitution of equations (19-20-21) we obtain

${\mathsf {F}}_{x}={\mathsf {F}}_{x}'$

(22)

${\mathsf {F}}_{y}=\left(1-\beta ^{2}\right){\mathsf {F}}_{y}'$

(23)

${\mathsf {F}}_{z}=\left(1-\beta ^{2}\right)F_{z}'$

(24)

which are the desired relations connecting measurements of force in the two systems.

The Fifth Equation.

Returning now to the consideration of an electron which is moving with the same velocity as the system S', we see that the transformation equations (13-14-15) together with the above equations lead to the relation

${\begin{array}{l}{\mathsf {F}}_{x}={\mathsf {F}}_{x}'={\mathsf {E}}_{x}'={\mathsf {E}}_{x}\\\\{\mathsf {F}}_{y}=\left(1-\beta ^{2}\right){\mathsf {F}}_{y}'={\sqrt {1-\beta ^{2}}}{\mathsf {E}}_{y}'=\left({\mathsf {E}}_{y}-{\frac {v}{c}}{\mathsf {H}}_{z}\right)\\\\{\mathsf {F}}_{z}=\left(1-\beta ^{2}\right){\mathsf {F}}_{z}'={\sqrt {1-\beta ^{2}}}{\mathsf {E}}_{z}'=\left({\mathsf {E}}_{z}-{\frac {v}{c}}{\mathsf {H}}_{y}\right),\end{array}}$

which is the desired equation:

${\mathsf {F}}={\mathsf {E}}+{\frac {1}{c}}{\mathsf {v}}\times {\mathsf {H}}$

This result agrees with that obtained by Einstein in his second treatment (Jahrbuch der Radioaktivität, iv. p. 411, 1907), where instead of defining force as equal to mass times acceleration, he defined it by the equations

${\mathsf {F}}_{x}={\frac {d}{dt}}{\frac {m_{0}u_{x}}{\sqrt {1-\beta ^{2}}}},\ {\mathsf {F}}_{y}={\frac {d}{dt}}{\frac {m_{0}u_{y}}{\sqrt {1-\beta ^{2}}}},\ {\mathsf {F}}_{z}={\frac {d}{dt}}{\frac {m_{0}u_{z}}{\sqrt {1-\beta ^{2}}}},$

which agree with our definition of force as equal to the rate of increase of momentum.

The special purpose of this note is to make clear that the fifth fundamental equation of electromagnetic theory may be derived from the four field equations and the principle of relativity without making any arbitrary convention as to the mass of a moving body. It is quite unnecessary to place the transverse mass of a moving body equal to ${\tfrac {m_{0}}{\left(1-\beta ^{2}\right)}}$ and the longitudinal mass equal ${\tfrac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}$. The simple relation for the mass of a moving bod $m={\tfrac {m_{0}}{\left(1-\beta ^{2}\right)}}$, which was derived directly from the principle of relativity by Lewis and Tolman (loc. cit.) and from ideas of light pressure by Lewis (loc. cit.) is sufficient.

The fact that the fifth equation can be derived by combining the principle of relativity with the four field equations is one of the chief pieces of evidence which support the theory of relativity.

University of Michigan,

Ann Arbor, Mich.,

November 11, 1910.

---

1. Communicated by the Author.
2. Einstein, Arm. d. Physik, xvii. p. 891 (1905); Jahrbuch der Radioaktivität, iv. p. 411 (1907).
3. Lewis and Tolman, Proc. Amer. Acad. xliv. p. 711 (1909); Phil. Mag. xviii. p. 510 (1909).
4. For the purposes of this transformation, it is necessary to use not only the simple kinematic relations (1-4), but also the following relations which can be directly derived from them:-

   $u_{x'}={\frac {u_{x}-v}{1-{\frac {u_{x}v}{c^{2}}}}},\ u_{y'}={\frac {u_{y}{\sqrt {1-\beta ^{2}}}}{1-{\frac {u_{x}v}{c^{2}}}}},\ u_{z'}={\frac {u_{z}{\sqrt {1-\beta ^{2}}}}{1-{\frac {u_{x}v}{c^{2}}}}},$ $\rho '={\frac {\partial {\mathsf {E}}_{x}'}{\partial x'}}+{\frac {\partial {\mathsf {E}}_{y}'}{\partial y'}}+{\frac {\partial {\mathsf {E}}_{z}'}{\partial z'}}={\frac {\left(1-{\frac {u_{x}v}{c^{2}}}\right)}{\left(1-\beta ^{2}\right)}}\rho .$

5. It should be noticed that according to the first postulate of relativity, if the charge of a stationary electron, for example a hydrogen ion, is $\epsilon$, then when the electron is in motion it must still appear to have the charge $\epsilon$ to an observer who is moving along with it, otherwise the possibility would be presented of distinguishing between relative and absolute motion. This justifies us in taking $\epsilon$E' as the force acting on the moving electron and measured in the moving system.
6. Lewis, Phil. Mag. xvi. p. 705 (1908).
7. Lewis and Tolman, loc. cit.