Editing Four-velocity (section)

== Theory of relativity ==

In Einstein's [[theory of relativity]], the path of an object moving relative to a particular frame of reference is defined by four coordinate functions {{math|''x<sup>μ</sup>''(''τ'')}}, where {{mvar|μ}} is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates. The zeroth component is defined as the time coordinate multiplied by {{math|''c''}},
<math display="block">x^0 = ct\,,</math>  

Each function depends on one parameter ''τ'' called its [[proper time]]. As a column vector,
<math display="block">
\mathbf{x} = \begin{bmatrix}
  x^0(\tau) \\ x^1(\tau) \\ x^2(\tau) \\ x^3(\tau) \\
\end{bmatrix}\,.
</math>

=== Time dilation ===

From [[time dilation]], the [[differential of a function|differential]]s in [[coordinate time]] {{mvar|t}} and [[proper time]] {{mvar|τ}} are related by
<math display="block">dt = \gamma(u) d\tau</math>
where the [[Lorentz factor]],
<math display="block">\gamma(u) = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}\,,</math>
is a function of the [[Norm (mathematics)#Euclidean norm|Euclidean norm]] {{mvar|u}} of the 3d velocity vector {{nowrap|<math>\vec{u}</math>:}}
<math display="block">u =  \left\|\ \vec{u}\ \right\| = \sqrt{ \left(u^1\right)^2 + \left(u^2\right)^2 + \left(u^3\right)^2} \,.</math>

=== Definition of the four-velocity ===

The four-velocity is the tangent four-vector of a [[timelike curve|timelike]] [[world line]].
The four-velocity <math>\mathbf{U}</math> at any point of world line <math>\mathbf{X}(\tau)</math> is defined as:
<math display="block">\mathbf{U} = \frac{d\mathbf{X}}{d \tau}</math>
where <math>\mathbf{X}</math> is the [[four-position]] and <math>\tau</math> is the [[proper time]].<ref>{{cite book | last1 = McComb | first1 = W. D. | title=Dynamics and relativity | date=1999 | publisher=Oxford University Press | location = Oxford [etc.] | isbn=0-19-850112-9 | pages=230}}</ref>

The four-velocity defined here using the proper time of an object does not exist for world lines for massless objects such as photons travelling at the speed of light; nor is it defined for [[tachyon]]ic world lines, where the tangent vector is [[spacelike]].

=== 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>

=== Magnitude ===
Using the differential of the four-position in the rest frame, the magnitude of the four-velocity can be obtained by the [[Minkowski metric]] with signature {{math|(−, +, +, +)}}:
<math display="block">\left\|\mathbf{U}\right\|^2 = \eta_{\mu\nu} U^\mu U^\nu = \eta_{\mu\nu} \frac{dX^\mu}{d\tau} \frac{dX^\nu}{d\tau} = - c^2 \,,</math>
in short, the magnitude of the four-velocity for any object is always a fixed constant:
<math display="block">\left\|\mathbf{U}\right\|^2 = - c^2 </math>

In a moving frame, the same norm is:
<math display="block">\left\|\mathbf{U}\right\|^2 = {\gamma(u)}^2 \left( - c^2 + \vec{u} \cdot \vec{u} \right) ,</math>
so that:
<math display="block"> - c^2 = {\gamma(u)}^2 \left( - c^2 + \vec{u} \cdot \vec{u} \right) ,</math>

which reduces to the definition of the Lorentz factor.