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

50 $$\tau$$ 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

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

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

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

We therefore see that

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,