Inphysics, in particular inspecial relativity andgeneral relativity, afour-velocity is afour-vector in four-dimensionalspacetime^{[nb 1]} that represents the relativistic counterpart ofvelocity, which is athree-dimensionalvector in space.

Physicalevents 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 itsworld line. If the object hasmass, so that its speed is necessarily less than thespeed of light, the world line may beparametrized by theproper time of the object. The four-velocity is the rate of change offour-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 themagnitude of an object's four-velocity, i.e. the quantity obtained by applying themetric tensorg to the four-velocityU, that is‖U‖² =U ⋅U =g_μνU^νU^μ, is always equal to±c², wherec is the speed of light. Whether the plus or minus sign applies depends on the choice ofmetric signature. For an object at rest its four-velocity is parallel to the direction of the time coordinate withU⁰ =c. A four-velocity is thus the normalized future-directed timelike tangent vector to a world line, and is acontravariant 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 avector space.^{[nb 2]}

The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functionsxⁱ(t) of timet, wherei is anindex which takes values 1, 2, 3.

The three coordinates form the 3dposition vector, written as acolumn vector$${\displaystyle \overrightarrow{x}(t)=\left[\begin{array}{c}{x}^1(t)\\ x^2(t)\\ x^3(t)\end{array}\right].}$$

The components of the velocity${\displaystyle \overrightarrow{u}}$ (tangent to the curve) at any point on the world line are

$${\displaystyle \overrightarrow{u}=\left[\begin{array}{c}{u}^1\\ u^2\\ u^3\end{array}\right]=\frac{d\overrightarrow{x}}{dt}=\left[\begin{array}{c}\frac{d{x}^1}{dt}\\ \frac{d{x}^2}{dt}\\ \frac{d{x}^3}{dt}\end{array}\right].}$$

Each component is simply written$${\displaystyle u^i=\frac{d{x}^i}{dt}}$$

In Einstein'stheory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functionsx^μ(τ), whereμ 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 byc,$${\displaystyle x^0=ct,}$$

Each function depends on one parameterτ called itsproper time. As a column vector,$${\displaystyle \mathbf{x}=\left[\begin{array}{c}{x}^0(\tau )\\ x^1(\tau )\\ x^2(\tau )\\ x^3(\tau )\end{array}\right].}$$

Fromtime dilation, thedifferentials incoordinate timet andproper timeτ are related by$${\displaystyle dt=\gamma (u)d\tau }$$where theLorentz factor,$${\displaystyle \gamma (u)=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}},}$$is a function of theEuclidean normu of the 3d velocity vector${\displaystyle \overrightarrow{u}}$:$${\displaystyle u=\Vert \overrightarrow{u}\Vert =\sqrt{{\left(u^1\right)}^2+{\left(u^2\right)}^2+{\left(u^3\right)}^2}.}$$

The four-velocity is the tangent four-vector of atimelikeworld line.The four-velocity${\displaystyle \mathbf{U}}$ at any point of world line${\displaystyle \mathbf{X}(\tau )}$ is defined as:$${\displaystyle \mathbf{U}=\frac{d\mathbf{X}}{d\tau }}$$where${\displaystyle \mathbf{X}}$ is thefour-position and${\displaystyle \tau }$ is theproper time.^[1]

The four-velocity defined here using the proper time of an object does not exist for world lines for massless objects such as photons travelling at the speed of light; nor is it defined fortachyonic world lines, where the tangent vector isspacelike.

The relationship between the timet and the coordinate timex⁰ is defined by$${\displaystyle x^0=ct.}$$

Taking the derivative of this with respect to the proper timeτ, we find theU^μ velocity component forμ = 0:$${\displaystyle U^0=\frac{d{x}^0}{d\tau }=\frac{d(ct)}{d\tau }=c\frac{dt}{d\tau }=c\gamma (u)}$$

and for the other 3 components to proper time we get theU^μ velocity component forμ = 1, 2, 3:$${\displaystyle U^i=\frac{d{x}^i}{d\tau }=\frac{d{x}^i}{dt}\frac{dt}{d\tau }=\frac{d{x}^i}{dt}\gamma (u)=\gamma (u)u^i}$$where we have used thechain rule and the relationships$${\displaystyle u^i=\frac{d{x}^i}{dt},\frac{dt}{d\tau }=\gamma (u)}$$

Thus, we find for the four-velocity${\displaystyle \mathbf{U}}$:$${\displaystyle \mathbf{U}=\gamma \left[\begin{array}{c}c\\ \overrightarrow{u}\end{array}\right].}$$

Written in standard four-vector notation this is:$${\displaystyle \mathbf{U}=\gamma \left(c,\overrightarrow{u}\right)=\left(\gamma c,\gamma \overrightarrow{u}\right)}$$where${\displaystyle \gamma c}$ is the temporal component and${\displaystyle \gamma \overrightarrow{u}}$ 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 four-velocity define a traveling object'sproper velocity${\displaystyle \gamma \overrightarrow{u}=d\overrightarrow{x}/d\tau }$ i.e. the rate at which distance is covered in the reference map frame per unitproper time elapsed on clocks traveling with the object.

Unlike most other four-vectors, the four-velocity has only 3 independent components${\displaystyle u_x,u_y,u_z}$ instead of 4. The${\displaystyle \gamma }$ factor is a function of the three-dimensional velocity${\displaystyle \overrightarrow{u}}$.

When certain Lorentz scalars are multiplied by the four-velocity, one then gets new physical four-vectors that have 4 independent components.

For example:

• Four-momentum:$${\displaystyle \mathbf{P}=m_o\mathbf{U}=\gamma m_o\left(c,\overrightarrow{u}\right)=m\left(c,\overrightarrow{u}\right)=\left(mc,m\overrightarrow{u}\right)=\left(mc,\overrightarrow{p}\right)=\left(\frac{E}{c},\overrightarrow{p}\right),}$$ where${\displaystyle m_o}$ is therest mass
• Four-current density:$${\displaystyle \mathbf{J}={\rho }_o\mathbf{U}=\gamma {\rho }_o\left(c,\overrightarrow{u}\right)=\rho \left(c,\overrightarrow{u}\right)=\left(\rho c,\rho \overrightarrow{u}\right)=\left(\rho c,\overrightarrow{j}\right),}$$ where${\displaystyle {\rho }_o}$ is thecharge density

Effectively, the${\displaystyle \gamma }$ factor combines with the Lorentz scalar term to make the 4th independent component$${\displaystyle m=\gamma m_o}$$ and$${\displaystyle \rho =\gamma {\rho }_o.}$$

Using the differential of the four-position in the rest frame, the magnitude of the four-velocity can be obtained by theMinkowski metric with signature(−, +, +, +):$${\displaystyle {\Vert \mathbf{U}\Vert }^2={\eta }_{\mu \nu }{U}^{\mu }{U}^{\nu }={\eta }_{\mu \nu }\frac{d{X}^{\mu }}{d\tau }\frac{d{X}^{\nu }}{d\tau }=-c^2,}$$in short, the magnitude of the four-velocity for any object is always a fixed constant:$${\displaystyle {\Vert \mathbf{U}\Vert }^2=-c^2}$$

In a moving frame, the same norm is:$${\displaystyle {\Vert \mathbf{U}\Vert }^2={\gamma (u)}^2\left(-c^2+\overrightarrow{u}\cdot \overrightarrow{u}\right),}$$so that:$${\displaystyle -c^2={\gamma (u)}^2\left(-c^2+\overrightarrow{u}\cdot \overrightarrow{u}\right),}$$

which reduces to the definition of the Lorentz factor.

• Four-acceleration
• Four-momentum
• Four-force
• Four-gradient
• Algebra of physical space
• Congruence (general relativity)
• Hyperboloid model
• Rapidity

• Einstein, Albert (1920).Relativity: The Special and General Theory. Translated by Robert W. Lawson. New York: Original: Henry Holt, 1920; Reprinted: Prometheus Books, 1995.
• Rindler, Wolfgang (1991).Introduction to Special Relativity (2nd). Oxford: Oxford University Press.ISBN 0-19-853952-5.

• Four-vectors

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• Short description matches Wikidata