Editing Four-velocity

{{Short description|Analogue of velocity in four-dimensional spacetime}}
In [[physics]], in particular in [[special relativity]] and [[general relativity]], a '''four-velocity''' is a [[four-vector]] in four-dimensional [[spacetime]]<ref group=nb>Technically, the four-vector should be thought of as residing in the [[tangent space]] of a point in spacetime, spacetime itself being modeled as a [[smooth manifold]]. This distinction is significant in general relativity.</ref> that represents the relativistic counterpart of [[velocity]], which is a [[Three-dimensional space|three-dimensional]] [[Vector (mathematics and physics)|vector]] in space.

Physical [[Event (relativity)|events]] correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its [[world line]]. If the object has [[Mass in special relativity|mass]], so that its speed is necessarily less than the [[speed of light]], the world line may be [[Parametrization (geometry)|parametrized]] by the [[proper time]] of the object. The four-velocity is the rate of change of [[four-position]] with respect to the proper time along the curve. The velocity, in contrast, is the rate of change of the position in (three-dimensional) space of the object, as seen by an observer, with respect to the observer's time.

The value of the [[Magnitude_(mathematics)#Pseudo-Euclidean_space|magnitude]] of an object's four-velocity, i.e. the quantity obtained by applying the [[Metric tensor (general_relativity)|metric tensor]] {{math|''g''}} to the four-velocity {{math|'''U'''}}, that is {{math|1={{norm|'''U'''}}<sup>2</sup> = '''U''' ⋅ '''U''' = ''g''<sub>''μν''</sub>''U''<sup>''ν''</sup>''U''<sup>''μ''</sup>}}, is always equal to {{math|±''c''<sup>2</sup>}}, where {{mvar|c}} is the speed of light. Whether the plus or minus sign applies depends on the choice of [[metric signature]]. For an object at rest its four-velocity is parallel to the direction of the time coordinate with {{math|1=''U''<sup>0</sup> = ''c''}}. A four-velocity is thus the normalized future-directed timelike tangent vector to a world line, and is a [[contravariant vector]]. Though it is a vector, addition of two four-velocities does not yield a four-velocity: the space of four-velocities is not itself a [[vector space]].<ref group=nb>The set of four-velocities is a subset of the tangent space (which ''is'' a vector space) at an event. The label ''four-vector'' stems from the behavior under [[Lorentz transformation]]s, namely under which particular [[Representation theory of the Lorentz group|representation]] they transform.</ref>

== Velocity ==

The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functions {{math|''x<sup>i</sup>''(''t'')}} of time {{mvar|t}}, where {{mvar|i}} is an [[index notation|index]] which takes values 1, 2, 3.

The three coordinates form the 3d [[position vector]], written as a [[column vector]]
<math display="block">\vec{x}(t) = \begin{bmatrix} x^1(t) \\[0.7ex] x^2(t) \\[0.7ex] x^3(t) \end{bmatrix} \, .</math>

The components of the velocity <math>\vec{u}</math> (tangent to the curve) at any point on the world line are

<math display="block">\vec{u} = \begin{bmatrix} u^1 \\ u^2 \\ u^3 \end{bmatrix} = \frac{d \vec{x}}{dt} =
\begin{bmatrix} \tfrac{dx^1}{dt} \\ \tfrac{dx^2}{dt} \\ \tfrac{dx^3}{dt} \end{bmatrix}.</math>

Each component is simply written
<math display="block">u^i = \frac{dx^i}{dt}</math>

== 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.

==See also==
{{Portal|Physics}}

* [[Four-acceleration]]
* [[Four-momentum]]
* [[Four-force]]
* [[Four-gradient]]
* [[Algebra of physical space]]
* [[Congruence (general relativity)]]
* [[Hyperboloid model]]
* [[Rapidity]]

==Remarks==
{{Reflist|group=nb}}

== References ==
* {{cite book | author = Einstein, Albert | translator =  Robert W. Lawson | title = Relativity: The Special and General Theory | location = New York | publisher = Original: Henry Holt, 1920; Reprinted: Prometheus Books, 1995 | year = 1920 }}
* {{cite book| author=Rindler, Wolfgang| title=Introduction to Special Relativity (2nd)| location=Oxford| publisher=Oxford University Press| year=1991| isbn=0-19-853952-5| url-access=registration| url=https://archive.org/details/introductiontosp0000rind}}

[[Category:Four-vectors]]