Page:TolmanEquations.djvu/7

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

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

1. These equations would accord with the electromagnetic theory of gravitation proposed by D. L. Webster, Proc. Amer. Acad. xlvii. p. 561 (1912).