Page:The principle of relativity (1920).djvu/140

 equation, we obtain the energy-law for the motion of the material point and the expression

m (dt/dτ - 1) = m [1/[sqrt](1 - w^2) - 1] = m (1/2 |w_{1}^2  + 3/8 |w_{1}^4 + )

may be called the kinetic energy of the material point.

Since dt is always greater than dτ we may call the quotient (dt - dτ)/dτ as the "Gain" (vorgehen) of the time over the proper-time of the material point and the law can then be thus expressed;—The kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time.

The set of four equations (22) again shows the symmetry in (x, y, z, t), which is demanded by the relativity postulate; to the fourth equation however, a higher physical significance is to be attached, as we have already seen in the analogous case in electrodynamics. On the ground of this demand for symmetry, the triplet consisting of the first three equations are to be constructed after the model of the fourth; remembering this circumstance, we are justified in saying,—

"If the relativity-postulate be placed at the head of mechanics, then the whole set of laws of motion follows from the law of energy."

I cannot refrain from showing that no contradiction to the assumption on the relativity-postulate can be expected from the phenomena of gravitation.

If B^*(x^*, y^*, z^*, t^*) be a solid (fester) space-time point, then the region of all those space-time points B (x, y, z, t), for which

(23) (x - x^*)^2 + (y - y^*)^2 + (z - z^*)^2 = (t - t,^*)^2

t - t^* [>=] 0