Dempster–Shafer theory

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 $t$:

$\overrightarrow{x}=x^i(t)=\left[\begin{array}{c}{x}^1(t)\\ x^2(t)\\ x^3(t)\end{array}\right],$

where the $x^i(t)$ denote the three spatial coordinates of the object at time t.

The components of the velocity $\overrightarrow{u}$ (tangent to the curve) at any point on the world line are

$\overrightarrow{u}=\left[\begin{array}{c}{u}^1\\ u^2\\ u^3\end{array}\right]=\frac{d\overrightarrow{x}}{dt}=\frac{d{x}^i}{dt}=\left[\begin{array}{c}\frac{d{x}^1}{dt}\\ \frac{d{x}^2}{dt}\\ \frac{d{x}^3}{dt}\end{array}\right].$

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.

$\mathbf{x}=x^{\mu }(\tau )=\left[\begin{array}{c}{x}^0(\tau )\\ x^1(\tau )\\ x^2(\tau )\\ x^3(\tau )\\ \end{array}\right]=\left[\begin{array}{c}ct\\ x^1(t)\\ x^2(t)\\ x^3(t)\\ \end{array}\right]$

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 $\overrightarrow{u}$:

$u=\left|\right|\overrightarrow{u}\left|\right|=\sqrt{(u^1{)}^2+(u^2{)}^2+(u^3{)}^2}$.

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:

$U^0=\frac{d{x}^0}{d\tau }=c\gamma$

Using the chain rule, for $\mu =i=$1, 2, 3, we have

$U^i=\frac{d{x}^i}{d\tau }=\frac{d{x}^i}{d{x}^0}\frac{d{x}^0}{d\tau }=\frac{d{x}^i}{d{x}^0}c\gamma =\frac{d{x}^i}{d(ct)}c\gamma =\frac{1}{c}\frac{d{x}^i}{dt}c\gamma =\gamma \frac{d{x}^i}{dt}=\gamma u^i$

where we have used the relationship

$u^i=\frac{d{x}^i}{dt}.$

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

$\mathbf{U}=\gamma (c,\overrightarrow{u})$

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 $\gamma \overrightarrow{u}=d\overrightarrow{x}/d\tau$ 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.

See also

• four-vector, four-acceleration, four-momentum, four-force.
• Special Relativity, Calculus, Derivative.
• Algebra of physical space
• Congruence (general relativity)

References

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