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Four-velocity
 
 
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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, in particular in special relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
 and general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
, the four-velocity of an object is a four-vector
Four-vector

In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations....
 (vector in four-dimensional spacetime
Spacetime

In physics, spacetime is any mathematical model that combines space and Time in physics into a single continuum . Spacetime is usually interpreted with space being Three-dimensional space and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions....
) that replaces classical velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
 (a three-dimensional vector). It is chosen in such a way that the velocity of light is a constant as measured in every inertial reference 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
Proper time

In theory of relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events....
 of the object.






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Encyclopedia


In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, in particular in special relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
 and general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
, the four-velocity of an object is a four-vector
Four-vector

In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations....
 (vector in four-dimensional spacetime
Spacetime

In physics, spacetime is any mathematical model that combines space and Time in physics into a single continuum . Spacetime is usually interpreted with space being Three-dimensional space and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions....
) that replaces classical velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
 (a three-dimensional vector). It is chosen in such a way that the velocity of light is a constant as measured in every inertial reference 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
Proper time

In theory of relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events....
 of the object. This curve is called its world line
World line

In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime.The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein perception straight paths are recalculated to...
. 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.

Classical mechanics


In classical mechanics the path of an object in three-dimensional space is determined by three coordinate functions as a function of (absolute) time t:

where the 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

where the derivatives are taken at the point p. So they are the difference in two nearby positions divided by the time interval .

Theory of relativity


In Einstein's theory of relativity
Theory of relativity

File:spacetime curvature.pngThe theory of relativity, or simply relativity, generally refers specifically to two theories of Albert Einstein: special relativity and general relativity....
, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions (where denotes the time coordinate multiplied by c), each function depending on one parameter , called its proper time
Proper time

In theory of relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events....
.

Time dilation


From time dilation
Time dilation

Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock....
, we know that

where is the Lorentz factor
Lorentz transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....
, which is defined as:

and u is the Euclidean norm
Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
 of the classical velocity vector :

.

Definition of the four-velocity


The four-velocity is the tangent four-vector of a world line
World line

In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime.The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein perception straight paths are recalculated to...
. The four velocity of world line is defined as:

where

is the proper time
Proper time

In theory of relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events....
.

Components of the four-velocity

The relationship between the time t and the coordinate time is given by

Taking the derivative with respect to the proper time , we find the velocity component for μ = 0:

Using the chain rule
Chain rule

In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...
, for 1, 2, 3, we have

where we have used the relationship

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

In terms of the yardsticks (and synchronized clocks) associated with a particular slice of flat spacetime, the three spacelike components of 4-velocity define a traveling object's proper velocity
Proper velocity

Proper-velocity, the distance traveled per unit time elapsed on the clocks of a traveling object, equals velocity at low speeds. At any speed it equals momentum per unit mass, and it therefore has no upper limit....
  i.e. the rate at which distance is covered in the reference map-frame per unit proper time
Proper time

In theory of relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events....
 elapsed on clocks traveling with the object.

Interpretation


For a rest frame, of course, and , hence thus justifying the statement about traveling in the time direction.

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

and therefore

In other words, the norm or magnitude of the four-velocity is always exactly equal to the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
.

See also


  • four-vector
    Four-vector

    In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations....
    , four-acceleration
    Four-acceleration

    In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particle's proper time:where...
    , four-momentum
    Four-momentum

    In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime....
    , four-force
    Four-force

    In the special theory of relativity four-force is a four-vector that replaces the classical force; the four-force is the four-vector defined as the change in four-momentum over the particle's own Proper Time:...
    .
  • Special Relativity
    Special relativity

    Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
    , Calculus
    Calculus

    Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
    , Derivative
    Derivative

    In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
    .
  • Algebra of physical space
    Algebra of physical space

    In physics, the algebra of physical space is the Clifford algebra or geometric algebra of the three-dimensional Euclidean space, with emphasis in its paravector structure....
  • Congruence (general relativity)
    Congruence (general relativity)

    In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime....