In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.
Contents
- Velocity
- Theory of relativity
- Time dilation
- Definition of the four velocity
- Components of the four velocity
- References
Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its world line. If the object is massive, so that its speed is less than the speed of light, the world line 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 observer, with respect to the observer's time.
The value of the magnitude of an object's four-velocity, i.e. the quantity obtained by applying the metric tensor g to the four-velocity u, that is || u ||2 = u ⋅ u = gμνuνuμ, is always equal to ±c2, where c is the speed of light. Whether the plus or minus sign applies depends on the choice of metric signature. For an object at rest its four-velocity is parallel to the direction of the time coordinate with u0 = c. 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.
Velocity
The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functions xi(t) of time t, where i is an index which takes values 1, 2, 3.
The three coordinates form the 3d position vector, written as a column vector
The components of the velocity
Each component is simply written
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 xμ(τ), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates. The zeroth component is defined as the time coordinate multiplied by c,
Each function depends on one parameter τ called its proper time. As a column vector,
Time dilation
From time dilation, the differentials in coordinate time t and proper time τ are related by
where the Lorentz factor,
is a function of the Euclidean norm u of the 3d velocity vector
Definition of the four-velocity
The four-velocity is the tangent four-vector of a timelike world line. The four-velocity
where
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 x0 is defined to be related to coordinate time by
Taking the derivative of this with respect to the proper time τ, we find the Uμ velocity component for μ = 0:
and for the other 3 components to proper time we get the Uμ velocity component for μ = 1, 2, 3:
where we have used the chain rule and the relationships
Thus, we find for the four-velocity
Written in standard 4-vector notation this is:
where
In terms of the synchronized clocks and rulers associated with a particular slice of flat spacetime, the three spacelike components of four-velocity define a traveling object's proper velocity
Unlike most other 4-vectors, the 4-Velocity has only 3 independent components
When certain Lorentz scalars are multiplied by the 4-Velocity, one then gets new physical 4-Vectors that have 4 independent components. For example:
4-Momentum:Effectively, the