Editing Four-velocity (section)

{{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>