In physics, in particular in special relativity and general relativity, a fourvelocity is a fourvector in fourdimensional spacetime that represents the relativistic counterpart of velocity, which is a threedimensional vector in space.^{[nb 1]}
Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical fourdimensional spacetime. The history of an object traces a curve in spacetime, called its world line. If the object is massive, so that its speed is less than the speed of light, the world line may be parametrized by the proper time of the object. The fourvelocity is the rate of change of fourposition with respect to the proper time along the curve. The velocity, in contrast, is the rate of change of the position in (threedimensional) space of the object, as seen by an observer, with respect to the observer's time.
The magnitude of an object's fourvelocity (the quantity obtained by applying the metric tensor to the fourvelocity and itself) is always equal to the square of c, the speed of light. For an object at rest (with respect to the coordinate system) its fourvelocity is parallel to the direction of the time coordinate. A fourvelocity is thus the normalized futuredirected timelike tangent vector to a world line, and is a contravariant vector. Though it is a vector, addition of two fourvelocities does not yield a fourvelocity: the space of fourvelocities is not itself a vector space.^{[nb 2]}
Contents

Velocity 1

Theory of relativity 2

Time dilation 2.1

Definition of the fourvelocity 2.2

Components of the fourvelocity 2.3

See also 3

Remarks 4

References 5
Velocity
The path of an object in threedimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functions x^{i}(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

\vec{x}(t) = \begin{bmatrix} x^1(t) \\ x^2(t) \\ x^3(t) \end{bmatrix} \,.
The components of the velocity {\vec{u}} (tangent to the curve) at any point on the world line are

{\vec{u}} = \begin{bmatrix}u^1 \\ u^2 \\ u^3\end{bmatrix} = {d \vec{x} \over dt}= \begin{bmatrix}\tfrac{dx^1}{dt} \\ \tfrac{dx^2}{dt} \\ \tfrac{dx^3}{dt}\end{bmatrix}.
Each component is simply written

u^i = {dx^i \over dt}
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 x^{μ}(τ), where 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 c,

x^{0} = ct\,,
Each function depends on one parameter τ called its proper time. As a column vector,

\mathbf{x} = \begin{bmatrix} x^0(\tau)\\ x^1(\tau) \\ x^2(\tau) \\ x^3(\tau) \\ \end{bmatrix}\,.
Time dilation
From time dilation, the differentials in coordinate time t and proper time τ are related by

dt = \gamma(u) d\tau
where the Lorentz factor,

\gamma(u) = \frac{1}{\sqrt{1\frac{u^2}{c^2}}}\,,
is a function of the Euclidean norm u of the 3d velocity vector \vec{u}:

u =  \ \vec{u} \  = \sqrt{ (u^1)^2 + (u^2)^2 + (u^3)^2} \,.
Definition of the fourvelocity
The fourvelocity is the tangent fourvector of a timelike world line. The fourvelocity at any point of world line \mathbf{X}(\tau) is defined as:

\mathbf{U} = \frac{d\mathbf{X}}{d \tau}
where \mathbf{X} is the fourposition and \tau is the proper time.^{[1]}
The fourvelocity 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 tachyonic world lines, where the tangent vector is spacelike.
Components of the fourvelocity
The relationship between the time t and the coordinate time x^{0} is defined to be related to coordinate time by

x^0 = ct .
Taking the derivative of this with respect to the proper time τ, we find the U^{μ} velocity component for μ = 0:

U^0 = \frac{dx^0}{d\tau} = \frac{d(ct)}{d\tau}= c\frac{dt}{d\tau} = c \gamma(u)
and for the other 3 components to proper time we get the U^{μ} velocity component for μ = 1, 2, 3:

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
where we have used the chain rule and the relationships

u^i = {dx^i \over dt } \,,\quad \frac{dt}{d\tau} = \gamma (u)
Thus, we find for the fourvelocity \mathbf{U}:

\mathbf{U} = \gamma \begin{bmatrix} c\\ \vec{u} \\ \end{bmatrix} .
Written in standard 4vector notation this is:

\mathbf{U} = \gamma (c,\vec{u}) = (\gamma c,\gamma \vec{u})
where \gamma c is the temporal component and \gamma \vec{u} 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 fourvelocity define a traveling object's proper velocity \gamma \vec{u} = d\vec{x}/d\tau 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 4vectors, the 4Velocity has only 3 independent components u_x, u_y, u_z instead of 4. The \gamma factor is a function of the 3velocity \vec{u}.
When certain Lorentz scalars are multiplied by the 4Velocity, one then gets new physical 4Vectors that have 4 independent components. For example:

4Momentum: \mathbf{P} = m_o\mathbf{U} = \gamma m_o(c,\vec{u}) = m(c,\vec{u}) = (mc,m\vec{u}) = (mc,\vec{p})= (\frac{E}{c},\vec{p}), where m_o is the invariant rest mass

4CurrentDensity: \mathbf{J} = \rho_o\mathbf{U} = \gamma \rho_o(c,\vec{u}) = \rho(c,\vec{u}) = (\rho c,\rho\vec{u}) = (\rho c,\vec{j}), where \rho_o is the invariant rest charge density
Effectively, the \gamma factor combines with the Lorentz scalar rest term to make the 4th independent component

m=\gamma m_o and \rho=\gamma \rho_o
See also

^ Technically, the 4vector should be thought of as residing in the tangent space of a point in spacetime, spacetime itself being modeled as a smooth manifold. This technical distinction is important in general relativity, but in special relativity, where spacetime is modeled as a vector space, the 4vector can be thought of as residing in spacetime itself.

^ The set of 4vectors is a subset of the tangent space (which is a vector space) at an event. The label 4vector stems from the behavior under Lorentz transformations, namely under which particular representation they transform.
References

Einstein, Albert; translated by Robert W. Lawson (1920). Relativity: The Special and General Theory. New York: Original: Henry Holt, 1920; Reprinted: Prometheus Books, 1995.

Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press.
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