Some Transformation Equations

by Richard Chace Tolman

Non-Newtonian Mechanics.– Some Transformation Equations.

By Richard C. Tolman, Ph.D., Assistant Professor of Chemistry at the University of California.^[1]

There have previously been presented some of the consequences^[2] of a system of mechanics which is based on the conservation laws of mass, energy, and momentum and the Einstein transformation equations for coordinate systems in relative motion. It will be shown in the present article, that the principles of this system of mechanics lead directly to a number of further transformation equations, and in particular to the same transformation equations for force as chosen by Planck^[3] to agree with electromagnetic considerations. An application of the equations to electromagnetic and gravitational problems will also be presented.

If we consider two systems of "space time" coordinates in relative motion in the X direction with the velocity $v$, the Einstein theory of relativity has led to the following equations for transforming the description of any kinematic phenomenon from the variables of system S to those of system S'.

$x'=\kappa (x-vt),$

(1)

$y'=y,$

(2)

$z'=z,$

(3)

$t'=\kappa \left(t-{\frac {v}{c^{2}}}x\right);$

(4)

where $c$ is the velocity of light and $\kappa$ is put equal to ${\frac {1}{\sqrt {1-v^{2}/c^{2}}}}$.

By the obvious differentiations and substitutions, Einstein has obtained the further equations:

${\frac {dt'}{dt}}=\kappa \left(1-{\frac {v{\dot {x}}}{c^{2}}}\right),$

(5)

${\dot {x}}'={\frac {{\dot {x}}-v}{1-{\frac {v{\dot {x}}}{c^{2}}}}},$

(6)

${\dot {y}}'={\frac {{\dot {y}}\kappa ^{-1}}{1-{\frac {v{\dot {x}}}{c^{2}}}}},$

(7)

${\dot {z}}'={\frac {{\dot {z}}\kappa ^{-1}}{1-{\frac {v{\dot {x}}}{c^{2}}}}},$

(8)

where for simplicity we have put

${\frac {dx}{dt}}={\dot {x}},\ {\frac {dx'}{dt'}}={\dot {x}}',$

If, for an observer in system S, a point is moving with the velocity $\left({\dot {x}},{\dot {y}},{\dot {z}}\right)$ its velocity $\left({\dot {x}}',{\dot {y}}',{\dot {z}}'\right)$, as seen by an observer in system S', is given by equations (6), (7), and (8). It is interesting to note that if to one observer a particle appears to have a constant velocity, that is not to be acted on by any force, it appears so to any other observer who is in uniform motion.

By further differentiation and simplification it is possible to obtain from equations (6), (7), and (8) three new equations for transforming measurements of acceleration from system S to S', viz. :-

${\ddot {x}}'=\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)^{-3}\kappa ^{-3}{\ddot {x}},$

(9)

${\ddot {y}}'=\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)^{-2}\kappa ^{-2}{\ddot {y}}+{\dot {y}}{\frac {v}{c^{2}}}\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)^{-3}\kappa ^{-2}{\ddot {x}},$

(10)

${\ddot {z}}'=\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)^{-2}\kappa ^{-2}{\ddot {z}}+{\dot {z}}{\frac {v}{c^{2}}}\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)^{-3}\kappa ^{-2}{\ddot {x}},$

(11)

In contrast to the relation holding for the case of uniform velocity, it may be pointed out in connexion with the above equations that, if a point has a uniform acceleration $\left({\ddot {x}},{\ddot {y}},{\ddot {z}}\right)$ with respect to an observer in system S, it will not in general have a uniform acceleration $\left({\ddot {x}}',{\ddot {y}}',{\ddot {z}}'\right)$ in another system S', since the acceleration in system S' depends not only on the constant acceleration but also on the velocity in system S which is necessarily varying.

We may next obtain transformation equations for a useful function of the velocity, namely, ${\frac {1}{\sqrt {1-q^{2}/c^{2}}}}$, where we have placed $q^{2}={\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}$. By substitution of equations (6), (7), and (8) and simplification we obtain

${\frac {1}{\sqrt {1-q^{2}/c^{2}}}}={\frac {\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)\kappa }{\sqrt {1-q^{2}/c^{2}}}}$

(12)

It has been shown in an earlier article^[4] that the principles of non-Newtonian mechanics lead to the equation $m={\frac {m_{0}}{\sqrt {1-q^{2}/c^{2}}}}$ for the mass of a moving body, where $m_{0}$ is the mass of the body at rest and $q$ is its velocity. By substitution of equation (12) we may obtain the following equation for transforming measurements of mass from one system of coordinates to the other:

$m'=\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)\kappa m,$

(13)

where $m$ is the mass of the body and $x$ the X component of its velocity as measured in system S and $m'$ its mass as measured in system S'.

By differentiation of equation (13) and simplification we may obtain the following transformation equation for the rate at which the mass of a body is changing owing to change in velocity :

${\dot {m}}'={\dot {m}}-{\frac {mv}{c^{2}}}{\ddot {x}}\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)^{-1}$

(14)

${\dot {x}}$ and ${\ddot {x}}$ are the X components of the velocity and acceleration of the body in question as measured in system S.

We are now in a position to obtain transformation equations for the force acting on a particle. The force acting on a body may be best defined as equal to the rate of increase of momentum^[5], i. e. by the equation

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

or

${\begin{array}{l}{\mathsf {F}}_{x}=m{\ddot {x}}+{\dot {m}}{\dot {x}},\\{\mathsf {F}}_{y}=m{\ddot {y}}+{\dot {m}}{\dot {y}},\\{\mathsf {F}}_{z}=m{\ddot {z}}+{\dot {m}}{\dot {z}}.\end{array}}$

By the substitution of the previous equations presented in this article we obtain:

${\mathsf {F}}_{x'}={\frac {{\mathsf {F}}_{x}-{\dot {m}}v}{1-{\frac {v{\dot {x}}}{c^{2}}}}}={\mathsf {F}}_{x}-{\frac {v{\dot {y}}}{c^{2}-v{\dot {x}}}}{\mathsf {F}}_{y}-{\frac {v{\dot {z}}}{c^{2}-v{\dot {x}}}}{\mathsf {F}}_{z}$

(15)

${\mathsf {F}}_{y'}={\frac {\kappa ^{-1}}{1-{\frac {v{\dot {x}}}{c^{2}}}}}{\mathsf {F}}_{y},$

(16)

${\mathsf {F}}_{z'}={\frac {\kappa ^{-1}}{1-{\frac {v{\dot {x}}}{c^{2}}}}}{\mathsf {F}}_{z},$

(17)

which are the desired transformation equations of force. These equations, which have here been derived from the principles of non-Newtonian mechanics, are those which were chosen by Planck to agree with electromagnetic considerations^[6].

Field around a Moving Charge.

As an application of these transformation equations, we may calculate the force with which a point charge in uniform motion acts on any other point charge, merely assuming Coulomb’s inverse square law for the force exerted by a stationary charge^[7].

Consider a set of coordinates S(x, y, z, t), and let there be a charge $\epsilon$ in uniform motion along the X axis with the velocity $v$. We desire to know the force acting at the time $t$ on any other charge $\epsilon _{1}$, which has any desired coordinates $x,y,$ and $z$ and any desired velocity ${\dot {x}},{\dot {y}},{\dot {z}}$.

Assume a system of coordinates $S'(x',y',z',t')$ moving with the same velocity as the charge $\epsilon$ which is situated at the origin. To an observer moving with the system S', the charge always appears at rest an to be surrounded by a pure electrostatic field. Hence in system S' the force with which $\epsilon$ acts on $\epsilon _{1}$, will be in accord with Coulomb’s law.

${\mathsf {F}}'={\frac {\epsilon \epsilon _{1}{\mathsf {r}}'}{r^{3}}},$

or

${\mathsf {F}}_{x}={\frac {\epsilon \epsilon _{1}x'}{\left(x'^{2}+y'^{2}+z'^{2}\right)^{3/2}}},$

(18)

${\mathsf {F}}_{y}={\frac {\epsilon \epsilon _{1}y'}{\left(x'^{2}+y'^{2}+z'^{2}\right)^{3/2}}},$

(19)

${\mathsf {F}}_{z}={\frac {\epsilon \epsilon _{1}z'}{\left(x'^{2}+y'^{2}+z'^{2}\right)^{3/2}}},$

(20)

where $x',y',$ and $z'$ are the coordinates of charge $\epsilon _{1}$, at the time $t'$. For simplicity let us consider the force at the time $t'=0$, then from transformation equations (1)-(3) we shall have

$x'=\kappa ^{-1}x,\ y'=y,\ z'=z.$

Substituting into (18), (19), and (20) and also making use of the transformation equations of force (15), (16), and (17), we obtain the following equations for the force acting on $\epsilon _{1}$, as it appears to an observer in system S.

${\mathsf {F}}_{x}={\frac {\epsilon \epsilon _{1}\kappa ^{-1}}{\left(\kappa {}^{-2}x^{2}+y{}^{2}+z{}^{2}\right)^{3/2}}}\left[x+{\frac {v}{c^{2}}}\kappa ^{2}\left(y{\dot {y}}+z{\dot {z}}\right)\right],$

(21)

${\mathsf {F}}_{y}={\frac {\epsilon \epsilon _{1}\kappa \left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)y}{\left(\kappa {}^{-2}x^{2}+y{}^{2}+z{}^{2}\right)^{3/2}}},$

(22)

${\mathsf {F}}_{z}={\frac {\epsilon \epsilon _{1}\kappa \left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)z}{\left(\kappa {}^{-2}x^{2}+y{}^{2}+z{}^{2}\right)^{3/2}}},$

(23)

These equations give the force acting on $\epsilon _{1}$ at the time $t$. From equation (4) we have $t={\frac {v}{c^{2}}}x$ since $t'=0$. At this time, the charge $\epsilon$ which is moving with the uniform velocity $v$ along the X axis will evidently have the position

$x_{\epsilon }={\frac {v^{2}}{c^{2}}}x,\ y_{\epsilon }=0,\ z_{\epsilon }=0.$

For convenience we may now refer our results to a system of coordinates whose origin coincides with the position of the charge $\epsilon$ at the instant under consideration. If X, Y, and Z are the coordinates of $\epsilon _{1}$ with respect to this new system, we evidently have the relations

${\mathsf {X}}=x-{\frac {v^{2}}{c^{2}}}x=\kappa ^{-2}x,\ {\mathsf {Y}}=y,\ {\mathsf {Z}}=z,\ {\mathsf {\dot {X}}}={\dot {x}},\ {\mathsf {\dot {Y}}}={\dot {y}},\ {\mathsf {\dot {Z}}}={\dot {z}}.$

Substituting into (21), (22), and (23) we may obtain:—

${\mathsf {F}}_{x}={\frac {\epsilon \epsilon _{1}}{x^{3}}}\left(1-\beta ^{2}\right)\left\{{\mathsf {X}}+{\frac {v}{c^{2}}}\left({\mathsf {Y{\dot {Y}}}}+{\mathsf {Z{\dot {Z}}}}\right)\right\},$

(24)

${\mathsf {F}}_{y}={\frac {\epsilon \epsilon _{1}}{x^{3}}}\left(1-\beta ^{2}\right)\left(1-{\frac {v{\mathsf {\dot {X}}}}{c^{2}}}\right){\mathsf {Y}},$

(25)

${\mathsf {F}}_{z}={\frac {\epsilon \epsilon _{1}}{x^{3}}}\left(1-\beta ^{2}\right)\left(1-{\frac {v{\mathsf {\dot {X}}}}{c^{2}}}\right){\mathsf {Z}},$

(26)

where for simplicity we have placed $\beta ={\tfrac {v}{c}}$, and

$s={\sqrt {{\mathsf {X}}^{2}+\left(1-\beta ^{2}\right)\left({\mathsf {Y}}^{2}+{\mathsf {Z}}^{2}\right)}}.$

These same equations could also be obtained by substituting the well-known formula for the strength of the electric and magnetic field around a moving point charge into the fifth fundamental equation of the Maxwell-Lorentz theory ${\mathsf {F}}={\mathsf {E}}+1/c\ {\mathsf {v}}\times {\mathsf {H}}$. It is interesting to see that they can be obtained so directly, merely from Coulomb's law.

If we consider the particular case that the charge $\epsilon _{1}$ is stationary (i. e. ${\mathsf {\dot {X}}}={\mathsf {\dot {Y}}}={\mathsf {Z}}=0$) and equal to unity, equations (24), (25) and (26) should give us the strength of the electric field produced by the moving point charge $\epsilon$, and in fact they do reduce as expected to the known expression

${\mathsf {F}}={\mathsf {E}}={\frac {\epsilon }{s^{3}}}\left(1-\beta ^{2}\right){\mathsf {r}}$

where

${\mathsf {r}}={\mathsf {Xi}}+{\mathsf {Yj}}+{\mathsf {Zk}},$

Gravitational Field.

This method of obtaining from Coulomb’s law the expected expression for the force exerted by a moving electric charge is of special interest, since it suggests the possibility of obtaining from Newton’s law an expression for the gravitational force exerted by a moving mass.

Let us assume, in accordance with Newton’s law, that a stationary mass $m_{1}$ will act on any other mass $m_{2}$ with the force $F=-km_{1}m_{2}{\frac {\mathsf {r}}{r^{3}}}$, where $m_{1}$ and $m_{2}$ are the masses which the particles would have if they were at rest, isolated, and at the absolute zero of temperature, and r the radius vector from $m_{1}$ to $m_{2}$. The determination of the force exerted by a mass in uniform motion may now be carried out in exactly the same manner as for the force exerted by a moving charge. In fact in analogy to equations (24), (25), and (26), we may write–

${\mathsf {F}}_{x}=-k{\frac {m_{1}m_{2}}{s^{3}}}\left(1-\beta ^{2}\right)\left\{{\mathsf {X}}+{\frac {v}{c^{2}}}\left({\mathsf {Y{\dot {Y}}+Z{\dot {Z}}}}\right)\right\},$

(27)

${\mathsf {F}}_{y}=-k{\frac {m_{1}m_{2}}{s^{3}}}\left(1-\beta ^{2}\right)\left(1-{\frac {v{\mathsf {\dot {X}}}}{c^{2}}}\right){\mathsf {Y}},$

(28)

${\mathsf {F}}_{x}=-k{\frac {m_{1}m_{2}}{s^{3}}}\left(1-\beta ^{2}\right)\left(1-{\frac {v{\mathsf {\dot {X}}}}{c^{2}}}\right){\mathsf {Z}}.$

(29)

These are the components of the force with which a particle of "stationary" mass $m_{1}$, in uniform motion in the X direction with the velocity $v$, acts on another particle of "stationary" mass $m_{2}$. Taking $m_{1}$ as the centre of coordinates, $m_{2}$ has the coordinates X, Y, and Z and the velocity $\left({\mathsf {{\dot {X}},{\dot {Y}},{\dot {Z}}}}\right)$. $k$ is the constant of gravitation, $\beta$ is placed equal to ${\tfrac {v}{c}}$, and $s$ has been substituted for ${\sqrt {{\mathsf {X}}^{2}+\left(1-\beta ^{2}\right)\left({\mathsf {Y}}^{2}+{\mathsf {Z}}^{2}\right)}}$^[8].

It may be noted that the particle $m_{1}$ must be in uniform motion, although the particle $m_{2}$ may have any motion, its instantaneous velocity being $\left({\mathsf {{\dot {X}},{\dot {Y}},{\dot {Z}}}}\right)$. It is unfortunate that the method does not also permit a determination of the force which an accelerated particle exerts. For cases, however, where the acceleration is slow enough to be neglected, it would be of great interest to see if these equations lead to expressions for the orbits of heavenly bodies differing appreciably from those hitherto in use.

Summary.

In this article it has been shown that the Einstein transformation equations, and the other principles of non-Newtonian mechanics, lead to a number of further transformation equations for acceleration, mass, rate of change of mass and force. The transformation equations of force are identical with those chosen by Planck. Two applications of the transformation equations have been given. By combining them with Coulomb’s law, the expected equations have been derived for the force with which an electric charge in uniform motion acts on any other charge, and by combining them with Newton’s law a new expression has been derived for the gravitational force with which a particle in uniform motion acts on another particle.

July 19, 1912.

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1. Communicated by the Author.
2. Lewis and Tolman, Phil. Mag. xviii. p. 510 (1909), Online; Tolman, Phil. Mag. xxi. p. 296 (1911), xii. p. 458 (1911), xxiii. p. 375 (1912).
3. Planck, Ann. d. Physik, xxvi. p. 1 (1908).
4. Phil. Mag. xxiii. p. 375 (1912).
5. See Phil. Mag. xxii. p. 458 (1911).
6. In an article by Lewis and Tolman (loc. cit.) an attempt was made to deduce the transformation equation of force, which was unsuccessful, owing to the authors' assumption that the turning moment around a right-angled lever in uniform irrotational motion should be zero. This error and the interesting fact that in general, if we accept the relativity theory, the actual presence of a turning moment is necessary to produce a pure translatory motion in an elastically stressed body was pointed out by Laue, Verh. d. Deutsch. Phys. Ges. xiii. p. 513 (1911). For the particular case that the body on which the force is acting is stationary with respect to one of the systems, the transformation equations of force were correctly derived by the present author, Phil. Mag. xxi. p. 296 (1911).
7. In its simplest form, Coulomb’s law merely states the force acting between two stationary charges. It should be noted that our derivation assumes the same law for the force with which a stationary charge acts on a moving charge.
8. These equations would accord with the electromagnetic theory of gravitation proposed by D. L. Webster, Proc. Amer. Acad. xlvii. p. 561 (1912).