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| In [[physics]], in particular in [[special relativity]] and [[general relativity]], a '''four-velocity''' is a [[four-vector]] (vector in four-dimensional [[spacetime]]) that replaces [[velocity]] (a three-dimensional vector).
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| [[Event (relativity)|Events]] are described in time and space, together forming four-dimensional spacetime. The history of an object traces a curve in spacetime, called its [[world line]], which may be 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 [[inertial]] observer, with respect to the observer's time.
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| 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]].
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| The ''magnitude'' of an object's four-velocity is always equal to ''c'', the [[speed of light]]. For an object at rest (with respect to the coordinate system) its four-velocity points in the direction of the time coordinate.
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| == Velocity ==
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| The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three coordinate functions <math>x^i(t),\; i \in \{1,2,3\}</math> of time <math>t</math>:
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| :<math>\vec{x} = x^i(t) = \begin{bmatrix} x^1(t) \\ x^2(t) \\ x^3(t) \end{bmatrix} ,</math>
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| where the <math>x^i(t)</math> denote the three spatial coordinates of the object at time ''t''.
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| The components of the velocity <math>{\vec{u}}</math> (tangent to the curve) at any point on the world line are
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| :<math>{\vec{u}} = \begin{bmatrix}u^1 \\ u^2 \\ u^3\end{bmatrix} = {d \vec{x} \over dt} = {dx^i \over dt} =
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| \begin{bmatrix}\tfrac{dx^1}{dt} \\ \tfrac{dx^2}{dt} \\ \tfrac{dx^3}{dt}\end{bmatrix}.</math>
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| == Theory of relativity ==
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| 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^{\mu}(\tau),\; \mu \in \{0,1,2,3\}</math> (where <math>x^{0}</math> denotes the time coordinate multiplied by ''c''), each function depending on one parameter <math>\tau</math>, called its [[proper time]].
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| :<math>
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| \mathbf{x} = x^{\mu}(\tau) =
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| \begin{bmatrix}
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| x^0(\tau)\\ x^1(\tau) \\ x^2(\tau) \\ x^3(\tau) \\
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| \end{bmatrix}
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| = \begin{bmatrix}
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| ct \\ x^1(t) \\ x^2(t) \\ x^3(t) \\
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| \end{bmatrix}
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| </math> | |
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| === Time dilation ===
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| From [[time dilation]], we know that
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| :<math>t = \gamma \tau \, </math>
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| where <math>\gamma</math> is the [[Lorentz transformation|Lorentz factor]], which is defined as:
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| :<math> \gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} </math>
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| and ''u'' is the [[Norm (mathematics)#Euclidean norm|Euclidean norm]] of the velocity vector <math>\vec{u}</math>:
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| :<math>u = || \ \vec{u} \ || = \sqrt{ (u^1)^2 + (u^2)^2 + (u^3)^2} </math>. | |
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| === Definition of the four-velocity ===
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| The four-velocity is the tangent four-vector of a [[world line]].
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| The four-velocity at any point of world line <math>\mathbf{x}(\tau)</math> is defined as:
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| :<math>\mathbf{U} = \frac{d\mathbf{x}}{d \tau} </math> | |
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| where <math>\mathbf{x}</math> is the [[four-position]] and <math>\tau</math> is the [[proper time]].
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| The four-velocity defined here using the proper time of an object does not exist for world lines for 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]].
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| === Components of the four-velocity ===
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| The relationship between the time ''t'' and the coordinate time <math>x^0</math> is given by
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| :<math> x^0 = ct = c \gamma \tau \, </math>
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| Taking the derivative with respect to the proper time <math> \tau \, </math>, we find the <math>U^\mu \,</math> velocity component for μ = 0:
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| :<math>U^0 = \frac{dx^0}{d\tau} = c \gamma </math>
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| Using the [[chain rule]], for <math>\mu = i = </math>1, 2, 3, we have
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| :<math>U^i = \frac{dx^i}{d\tau} =
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| \frac{dx^i}{dx^0} \frac{dx^0}{d\tau} =
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| \frac{dx^i}{dx^0} c\gamma = \frac{dx^i}{d(ct)} c\gamma =
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| {1 \over c} \frac{dx^i}{dt} c\gamma = \gamma \frac{dx^i}{dt} = \gamma u^i </math>
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| where we have used the relationship
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| :<math> u^i = {dx^i \over dt } .</math>
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| Thus, we find for the four-velocity <math>\mathbf{U}</math>:
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| :<math>\mathbf{U} = \gamma \left( c, \vec{u} \right) </math>
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| In terms of the yardsticks (and synchronized clocks) associated
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| with a particular slice of flat spacetime, the three spacelike
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| components of four-velocity define a traveling object's
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| [[proper velocity]] <math>\gamma \vec{u} = d\vec{x}/d\tau</math> i.e.
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| the rate at which distance is covered in the reference map frame
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| per unit [[proper time]] elapsed on clocks traveling with the object.
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| ==See also==
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| * [[four-vector]], [[four-acceleration]], [[four-momentum]], [[four-force]].
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| * [[Special Relativity]], [[Calculus]], [[Derivative]].
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| * [[Algebra of physical space]]
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| * [[Congruence (general relativity)]]
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| == References ==
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| {{Reflist}}
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| * {{cite book | author = Einstein, Albert; translated by Robert W. Lawson | title = Relativity: The Special and General Theory | location = New York | publisher = Original: Henry Holt, 1920; Reprinted: Prometheus Books, 1995 | year = 1920 }}
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| * {{cite book | author = Rindler, Wolfgang | title=Introduction to Special Relativity (2nd)| location= Oxford | publisher=Oxford University Press | year=1991 | isbn=0-19-853952-5}}
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| [[Category:Minkowski spacetime]]
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| [[Category:Theory of relativity]]
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