Page:The Meaning of Relativity - Albert Einstein (1922).djvu/61

Rh Since the four-dimensional element of volume is an invariant, and $$(K_1,K_2,K_3,K_4)$$ forms a 4-vector, the four-dimensional integral extended over the shaded portion transforms as a 4-vector, as does also the integral between the limits $$l_1$$ and $$l_2$$, because the portion of the region which is not shaded contributes nothing to the integral. It follows, therefore, that $$\Delta I_x,\Delta I_y,\Delta I_z,i\Delta E$$ form a 4-vector. Since the quantities themselves transform in the same way as their increments, it follows that the aggregate of the four quantities

has itself the properties of a vector; these quantities are referred to an instantaneous condition of the body (e.g. at the time $$l = l_1$$).

This 4-vector may also be expressed in terms of the mass $$m$$, and the velocity of the body, considered as a material particle. To form this expression, we note first, that

is an invariant which refers to an infinitely short portion of the four-dimensional line which represents the motion of the material particle. The physical significance of the invariant $$d\tau$$ may easily be given. If the time axis is chosen in such a way that it has the direction of the line differential which we are considering, or, in other words, if we reduce the material particle to rest, we shall then have $$d\tau = dl$$; this will therefore be measured by the light-seconds clock which is at the same place, and at rest relatively to the material particle. We therefore call 4