Four-velocity and four-momentum definitions

An Universe line is a curve α = α(τ) such us the tangent vector α'(τ) is temporal future and his norm (of Minkowski) is1. The parameter τ is called Own-Time.
A Material particle is an Universe line, α with a postive number m, called a mass.

Note that if the particle material is not in relative motion, its universe line is

α(τ) = (t(τ), x(τ), y(τ), z(τ)), then

||α'(τ)|| = 1 Because α' is temporal

-1 = -(dt/dτ)2 + (dx/dτ)2 + (dy/dτ)2 + (dz/dτ)2, therefore

2 = dt2 - dx2 - dy2 - dz2

Thus, if there are no infinitesimal changes in space and α' is future, we have dτ = dt

In nonrelativistic mechanics, we have the speed and linear momentum, in spacial relativity there are define its analogues in following forms

Four-velocity is the vector U = α'.

Four-momentum is the vector P = mU = mα'.

If a particle travels at constant speed v = (v1, v2, v3), t is holds

(x, y, z) = vt. Then the universe line of this particle must be

α(τ) = (t(τ), vt(τ))

Then, again

||α'(τ)|| = 1 => -1 = -(dt/dτ)2 + (dx/dτ)2 + (dy/dτ)2 + (dz/dτ)2

and therefore

dt/dτ = (1-v2)-1/2
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