# Four-velocity

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).

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.

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 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.

## Velocity

The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three coordinate functions $x^i(t),\; i \in \{1,2,3\}$ of time $t$:

$\vec{x} = x^i(t) = \begin{bmatrix} x^1(t) \\ x^2(t) \\ x^3(t) \end{bmatrix} ,$

where the $x^i(t)$ denote the three spatial coordinates of the object at time t.

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

${\vec{u}} = \begin{bmatrix}u^1 \\ u^2 \\ u^3\end{bmatrix} = {d \vec{x} \over dt} = {dx^i \over dt} = \begin{bmatrix}\tfrac{dx^1}{dt} \\ \tfrac{dx^2}{dt} \\ \tfrac{dx^3}{dt}\end{bmatrix}.$

## 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 $x^{\mu}(\tau),\; \mu \in \{0,1,2,3\}$ (where $x^{0}$ denotes the time coordinate multiplied by c), each function depending on one parameter $\tau$, called its proper time.

$\mathbf{x} = x^{\mu}(\tau) = \begin{bmatrix} x^0(\tau)\\ x^1(\tau) \\ x^2(\tau) \\ x^3(\tau) \\ \end{bmatrix} = \begin{bmatrix} ct \\ x^1(t) \\ x^2(t) \\ x^3(t) \\ \end{bmatrix}$

### Time dilation

From time dilation, we know that

$t = \gamma \tau \,$

where $\gamma$ is the Lorentz factor, which is defined as:

$\gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$

and u is the Euclidean norm of the velocity vector $\vec{u}$:

$u = || \ \vec{u} \ || = \sqrt{ (u^1)^2 + (u^2)^2 + (u^3)^2}$.

### Definition of the four-velocity

The four-velocity is the tangent four-vector of a world line. The four-velocity 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 four-position and $\tau$ is the proper time.

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 tachyonic world lines, where the tangent vector is spacelike.

### Components of the four-velocity

The relationship between the time t and the coordinate time $x^0$ is given by

$x^0 = ct = c \gamma \tau \,$

Taking the derivative with respect to the proper time $\tau \,$, we find the $U^\mu \,$ velocity component for μ = 0:

$U^0 = \frac{dx^0}{d\tau} = c \gamma$

Using the chain rule, for $\mu = i =$1, 2, 3, we have

$U^i = \frac{dx^i}{d\tau} = \frac{dx^i}{dx^0} \frac{dx^0}{d\tau} = \frac{dx^i}{dx^0} c\gamma = \frac{dx^i}{d(ct)} c\gamma = {1 \over c} \frac{dx^i}{dt} c\gamma = \gamma \frac{dx^i}{dt} = \gamma u^i$

where we have used the relationship

$u^i = {dx^i \over dt } .$

Thus, we find for the four-velocity $\mathbf{U}$:

$\mathbf{U} = \gamma \left( c, \vec{u} \right)$

In terms of the yardsticks (and synchronized clocks) associated with a particular slice of flat spacetime, the three spacelike components of four-velocity 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.