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Concept# Four-velocity

Summary

In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.
Physical 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, so that its speed is necessarily less than the speed of light, the world line 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 observer, with respect to the observer's time.
The value of the magnitude of an object's four-velocity, i.e. the quantity obtained by applying the metric tensor g to the four-velocity U, that is 2 = U ⋅ U = gμνUνUμ, is always equal to ±c2, where 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 U0 = 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.
The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functions xi(t) of time t, where i is an index which takes values 1, 2, 3.
The three coordinates form the 3d position vector, written as a column vector
The components of the velocity (tangent to the curve) at any point on the world line are
Each component is simply written
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 xμ(τ), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates.

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