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

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

$I_{x},I_{y},I_{z},\mathbf {i} E$

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

$-ds^{2}=d\tau ^{2}=-(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})-dx_{4}^{2}=dl^{2}(1-q^{2})$

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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 call4