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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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They contact me Emilia. One of the very very best issues in the world for him is to collect badges but he is struggling to discover time for it. Minnesota is exactly where he's been living for many years. She is a librarian but she's usually needed her personal business.
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