four-velocity

In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector (vector in four-dimensional spacetime) that replaces classical velocity (a three-dimensional vector). It is chosen in such a way that the velocity of light is a constant as measured in every inertial refererence frame. In relativity theory events are described in time and space, together forming four-dimensional spacetime. The history of an object traces a curve in spacetime, parametrized by a curve parameter, the proper time of the object. This curve is called its world line. The four-velocity is the rate of change of both time and space coordinates with respect to the proper time of the object. The four-velocity is a tangent vector to the world line. For comparison: in classical mechanics events are described by its (three-dimensional) position at each moment in time. The path of an object is a curve in three-dimensional space, parametrized by the time. The classical velocity is the rate of change of the space coordinates of the object with respect to the time. The classical velocity of an object is a tangent vector to its path. The length of the four-velocity (in the sense of the metric used in special relativity) is always equal to c (it is a normalized vector). For an object at rest (with respect to the coordinate system) its four-velocity points in the direction of the time coordinate.

Four-velocity in special relativity

It will be shown here that the four-velocity

{\mathbf v}

in special relativity in components is given by

{\mathbf v} = \gamma \left( c, v^x,  v^y, v^z \right)

Here c is the velocity of light and

v^x,  v^y, v^z

are the components of the spatial velocity in the three spatial directions x, y and z, in the following also denoted by

v^1, v^2, v^3

\gamma

is the so called Lorentz factor

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

with

v

the absolute value of the velocity

v^2= (v^x)^2 + (v^y)^2 + v^z)^2

. As an introduction, note that in classical mechanics a path of an object in three-dimensional space is determined by three coordinate functions

x^i(t),\; i=1,2,3

as a function of (absolute) time t, where the

x^i(t)

denote the three spatial positions of the object at time t. The components of the classical velocity

{\mathbf v}

at a point p (tangent to the curve) are

{\mathbf v} = (v^1,v^2,v^3) =

\left(\frac{dx^1}{dt}\;,\frac{dx^2}{dt}\;,\frac{dx^3}{dt}\right) where the derivatives are taken at the point p. So they are the difference in two nearby positions

dx^i

divided by the time interval

dt

. In relativity theory a path of an object is defined by four coordinate functions

x^a(\tau),\; a=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. The components of the four-velocity at a point p (and tangent to the curve) are defined as

{\mathbf v} = (v^0,v^1,v^2,v^3) =  \left(\frac{dx^0}{d\tau} \;,\frac{dx^1}{d\tau}\;,  \frac{dx^2}{d\tau} \;, \frac{dx^3}{d\tau}\right)

definition of four-velocity
where the derivatives are taken at the point p. In special relativity the relation between the proper time

\tau

and the coordinate time

x^0

is given by

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

where

\gamma

is the Lorentz factor defined above. This formula, where written x⁰ = ct is known as Time dilation and written as

\frac{d t}{d\tau\;} = \gamma

in that context. Using the chain rule

\frac{dx^i}{d\tau} =

\frac{dx^i}{dx^0} \frac{dx^0}{d\tau} = \frac{dx^i}{dx^0} \gamma = v^i \gamma where we have used that

dx^i/dx^0

is the spatial velocity

v^i

, we find for the four-velocity

{\mathbf v}

in special relativity the four components

{\mathbf v} = \gamma  \left( c, v_x,  v_y, v_z \right)

four-velocity in special relativity
Here c is the velocity of light and

v_x,  v_y , v_z

are the components of the spatial velocity in the three spatial directions.

\gamma

is the Lorentz factor.

Four-velocity

Four-velocity in special relativity

See also