Editing Four-velocity (section)

=== Components of the four-velocity ===

The relationship between the time {{mvar|t}} and the coordinate time {{math|''x''<sup>0</sup>}} is defined by
<math display="block">x^0 = ct .</math>

Taking the derivative of this with respect to the proper time {{mvar|τ}}, we find the {{math|''U<sup>μ</sup>''}} velocity component for {{math|1=''μ'' = 0}}:
<math display="block">U^0 = \frac{dx^0}{d\tau} = \frac{d(ct)}{d\tau} = c\frac{dt}{d\tau} = c \gamma(u)</math>

and for the other 3 components to proper time we get the {{math|''U<sup>μ</sup>''}} velocity component for {{math|1=''μ'' = 1, 2, 3}}:
<math display="block">U^i = \frac{dx^i}{d\tau} = 
  \frac{dx^i}{dt} \frac{dt}{d\tau} = 
  \frac{dx^i}{dt} \gamma(u) =
  \gamma(u) u^i
</math>
where we have used the [[chain rule]] and the relationships
<math display="block">u^i = {dx^i \over dt } \,,\quad \frac{dt}{d\tau} = \gamma (u)</math>

Thus, we find for the four-velocity {{nowrap|<math>\mathbf{U}</math>:}}
<math display="block">\mathbf{U} = \gamma \begin{bmatrix} c \\ \vec{u} \\ \end{bmatrix}.</math>

Written in standard four-vector notation this is:
<math display="block">\mathbf{U} = \gamma \left(c, \vec{u}\right) = \left(\gamma c, \gamma \vec{u}\right)</math>
where <math>\gamma c</math> is the temporal component and <math>\gamma \vec{u}</math> is the spatial component.

In terms of the synchronized clocks and rulers associated with a particular slice of flat spacetime, the three spacelike components of four-velocity define a traveling object's [[proper velocity]] <math>\gamma \vec{u} = d\vec{x} / d\tau</math> i.e. the rate at which distance is covered in the reference map frame per unit [[proper time]] elapsed on clocks traveling with the object.

Unlike most other four-vectors, the four-velocity has only 3 independent components <math>u_x, u_y, u_z</math> instead of 4.  The <math>\gamma</math> factor is a function of the three-dimensional velocity <math>\vec{u}</math>.

When certain Lorentz scalars are multiplied by the four-velocity, one then gets new physical four-vectors that have 4 independent components.

For example:
* [[Four-momentum]]: <math display="block">\mathbf{P} = m_o\mathbf{U} = \gamma m_o\left(c, \vec{u}\right) = m\left(c, \vec{u}\right) = \left(mc, m\vec{u}\right) = \left(mc, \vec{p}\right) = \left(\frac{E}{c},\vec{p}\right),</math> where <math>m_o</math> is the [[Rest Mass|rest mass]]
* [[Four-current|Four-current density]]: <math display="block">\mathbf{J} = \rho_o\mathbf{U} = \gamma \rho_o\left(c, \vec{u}\right) = \rho\left(c, \vec{u}\right) = \left(\rho c, \rho\vec{u}\right) = \left(\rho c, \vec{j}\right) ,</math> where <math>\rho_o</math> is the [[charge density]]

Effectively, the <math>\gamma</math> factor combines with the Lorentz scalar term to make the 4th independent component
<math display="block">m = \gamma m_o</math> and <math display="block">\rho = \gamma \rho_o.</math>