Page:The Meaning of Relativity - Albert Einstein (1922).djvu/62

$\tau$ the proper time of the material particle. As opposed to $dl,d\tau$ is therefore an invariant, and is practically equivalent to $dl$ for motions whose velocity is small compared to that of light. Hence we see that

$u_{\sigma }={\frac {dx_{\sigma }}{d\tau }}$

(39)

has, just as the $dx_{\nu }$, the character of a vector; we shall designate $(u_{\sigma })$ as the four-dimensional vector (in brief, 4-vector) of velocity. Its components satisfy, by (38), the condition

$\sum u_{\sigma }^{2}=-1.$

(40)

We see that this 4-vector, whose components in the ordinary notation are

${\frac {\mathbf {q} _{x}}{\sqrt {1-q^{2}}}},{\frac {\mathbf {q} _{y}}{\sqrt {1-q^{2}}}},{\frac {\mathbf {q} _{z}}{\sqrt {1-q^{2}}}},{\frac {\mathbf {i} }{\sqrt {1-q^{2}}}}$

(41)

is the only 4-vector which can be formed from the velocity components of the material particle which are defined in three dimensions by

$\mathbf {q} _{x}={\frac {dx}{dl}},\mathbf {q} _{y}={\frac {dy}{dl}},\mathbf {q} _{z}={\frac {dz}{dl}}.$

We therefore see that

$\left(m{\frac {dx_{\mu }}{d\tau }}\right)$

(42)

must be that 4-vector which is to be equated to the 4-vector of momentum and energy whose existence we have proved above. By equating the components, we obtain, in three-dimensional notation,