Four-velocity

In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector, a vector in four-dimensional spacetime.^{[nb 1]} It is the relativistic counterpart of velocity, which is a three-dimensional vector in space.

Events constitute the mathematical description of 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 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 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 magnitude of an object's four-velocity (the quantity obtained by applying the metric tensor to the four-velocity 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 four-velocity is parallel to the direction of the time coordinate. 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.^{[nb 2]}

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(\tau) \\ x^2(\tau) \\ x^3(\tau) \\ \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

Taking the derivative of the other 3 components to proper time we get:

U^i = \frac{dx^i}{d\tau} = \frac{dx^i}{dt} \frac{dt}{d\tau} = \frac{dx^i}{dt} \gamma = \gamma u^i

where we have used the relationships

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

and

t = \gamma \tau \,

which gives

\frac{dt}{d\tau} = \gamma

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

\mathbf{U} = \gamma \begin{bmatrix} c\\ \vec{u} \\ \end{bmatrix} .

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.

Remarks

↑ Technically, the 4-vector should be though 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 4-vector can be thought of as residing in spacetime itself.
↑ The set of 4-vectors is a subset of the tangent space (which is a vector space) at an event. The label 4-vector 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. ISBN 0-19-853952-5.