Page:TolmanEquations.djvu/2

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