Information About Four-velocity

	CATEGORIES ABOUT FOUR-VELOCITY
	minkowski spacetime
	relativity

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(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 their (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. This observation, though trivial (as we will see from the formulas below), has a rather nice interpretation: in spacetime, an object is always in motion (at the speed of light!); it's just that in a rest frame, this motion is all in the time direction. CLASSICAL MECHANICS In classical mechanics the path of an object in three-dimensional space is determined by three coordinate functions $x^i(t),\backslash ;\; i=1,2,3$ as a function of (absolute) time ''t'': :$$ ec{x} = x^i(t) = \begin{bmatrix} x^1(t) \ x^2(t) \ x^3(t) \ \end{bmatrix} where the $x^i(t)$ denote the three spatial positions of the object at time ''t''. The components of the classical velocity at a point ''p'' (tangent to the curve) are : \left(rac{\mathrm{d}x^1}{\mathrm{d}t}\;,rac{\mathrm{d}x^2}{\mathrm{d}t}\;,rac{\mathrm{d}x^3}{\mathrm{d}t} ight) where the derivatives are taken at the point ''p''. So they are the difference in two nearby positions $\backslash mathrm\{d\}x^a$ divided by the time interval $\backslash mathrm\{d\}t$. THEORY OF RELATIVITY In Einstein's Theory Of Relativity , the path of an object moving relative to a particular frame of reference is defined by four coordinate functions $x^\{\backslash mu\}(\; au),\backslash ;\; \backslash mu\; =0,1,2,3$ (where $x^\{0\}$ denotes the time coordinate multiplied by ''c''), each function depending on one parameter $au$, called its Proper Time . :$$ \mathbf{x} = x^{\mu}( au) = \begin{bmatrix} x^0( au)\ x^1( au) \ x^2( au) \ x^3( au) \ \end{bmatrix} = \begin{bmatrix} ct \ x^1(t) \ x^2(t) \ x^3(t) \ \end{bmatrix} Time dilation From Time Dilation , we know that :$t\; =\; \backslash gamma\; au\; \backslash ,$ where $\backslash gamma$ is the Lorentz Factor , which is defined as: : and ''u'' is the Euclidean Norm of the classical velocity vector :

where

:$au\; \backslash ,$ is the Proper Time .


Components of the four-velocity

The relationship between the improper time ''t'' and the coordinate time $x^0$ is given by

:$x^0\; =\; ct\; =\; c\; \backslash gamma\; au\; \backslash ,$

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

:

Using the Chain Rule , for $\backslash mu\; =\; i\; =$1, 2, 3, we have

:
rac{\mathrm{d}x^i}{\mathrm{d}x^0} rac{\mathrm{d}x^0}{\mathrm{d} au} =
rac{\mathrm{d}x^i}{\mathrm{d}x^0} c\gamma = rac{\mathrm{d}x^i}{\mathrm{d}(ct)} c\gamma =
{1 \over c} rac{\mathrm{d}x^i}{\mathrm{d}t} c\gamma = \gamma rac{\mathrm{d}x^i}{\mathrm{d}t} = \gamma u^i

where we have used the relationship

:$u^i\; =\; \{dx^i\; \backslash over\; dt\; \}$

from classical mechanics. Thus, we find for the four-velocity $U$:

:


INTERPRETATION


For a rest frame, of course, $\backslash gamma\; =\; 1$ and , hence $U\; =\; (c,0,0,0)\; \backslash ,$ thus justifying the statement about traveling in the time direction.

In every frame of reference, in both special and general relativity, we have

:$U\_\backslash mu\; U^\backslash mu\; =\; -c^2\; \backslash ,$

and therefore