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 $t$:

${\vec {x}}=x^{i}(t)={\begin{bmatrix}x^{1}(t)\\x^{2}(t)\\x^{3}(t)\end{bmatrix}},$

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.

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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