Four-velocity

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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.

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.

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