Four-velocity
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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
As an introduction, note that in classical mechanics a path of an object in three-dimensional space is determined by three coordinate functions <math>x^i(t),\; i=1,2,3<math> as a function of (absolute) time t, where the <math>x^i(t)<math> denote the three spatial positions of the object at time t. The components of the classical velocity <math>{\mathbf u}<math> at a point p (tangent to the curve) are
- <math>{\mathbf u} = (u^1,u^2,u^3) =
\left(\frac{dx^1}{dt}\;,\frac{dx^2}{dt}\;,\frac{dx^3}{dt}\right)<math> where the derivatives are taken at the point p. So they are the difference in two nearby positions <math>dx^a<math> divided by the time interval <math>dt<math>.
In relativity theory a path of an object is defined by four coordinate functions <math>x^a(\tau),\; a=0,1,2,3<math> (where <math>x^{0}<math> denotes the time coordinate multiplied by c), each function depending on one parameter <math>\tau<math>, called its proper time. The components of the four-velocity at a point p (and tangent to the curve) are defined as:
where the derivatives are taken at the point p.
In special relativity the relation between the proper time <math>\tau<math> and the coordinate time <math>x^0<math> is given by
- <math> \frac{dx^0}{d\tau\;} = c \gamma <math>
where <math>\gamma<math> is the so-called Lorentz factor defined as:
- <math> \gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} <math>
with <math>u<math> the absolute value of the velocity <math>u^2= (u^1)^2 + (u^2)^2 + (u^3)^2<math>.
This formula, where written x0 = ct is known as Time dilation and written as:
- <math> \frac{d t}{d\tau\;} = \gamma <math>
in that context.
Using the chain rule
- <math> \frac{dx^i}{d\tau} =
\frac{dx^i}{dx^0} \frac{dx^0}{d\tau} = \frac{dx^i}{dx^0} \gamma = v^i \gamma<math>
where we have used that <math>dx^i/dx^0<math> is the spatial velocity <math>u^i<math>, we find for the four-velocity <math>U^a<math>:
- <math>U^a = \gamma \left( c, \mathbf{u} \right) <math>
See also
- four-vector, four-acceleration, four-momentum, four-force.
- Special Relativity, Calculus, Derivative.