Four-velocity

# Four-velocity

Discussion

Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, in particular in special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, 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. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...

(vector in four-dimensional spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

) that replaces classical velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

(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
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

. In relativity theory events
Event (relativity)
In physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference.Strictly speaking, the...

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 relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

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...

. 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 $x^i\left(t\right),\; i \in \\left\{1,2,3\\right\}$ as a function of (absolute) time t: $\vec\left\{x\right\} = x^i\left(t\right) = \begin\left\{bmatrix\right\} x^1\left(t\right) \\ x^2\left(t\right) \\ x^3\left(t\right) \\ \end\left\{bmatrix\right\}$ where the $x^i\left(t\right)$ denote the three spatial positions of the object at time t. The components of the classical velocity $\left\{\vec\left\{u\right\}\right\}$ at a point p (tangent to the curve) are $\left\{\vec\left\{u\right\}\right\} = \left(u^1,u^2,u^3\right) = \left\{\mathrm\left\{d\right\} \vec\left\{x\right\} \over \mathrm\left\{d\right\}t\right\} = \left\{\mathrm\left\{d\right\}x^i \over \mathrm\left\{d\right\}t\right\} = \left\left(\frac\left\{\mathrm\left\{d\right\}x^1\right\}\left\{\mathrm\left\{d\right\}t\right\}\;,\frac\left\{\mathrm\left\{d\right\}x^2\right\}\left\{\mathrm\left\{d\right\}t\right\}\;,\frac\left\{\mathrm\left\{d\right\}x^3\right\}\left\{\mathrm\left\{d\right\}t\right\}\right\right)$ where the derivatives are taken at the point p. So they are the difference in two nearby positions $\mathrm\left\{d\right\}x^a$ divided by the time interval $\mathrm\left\{d\right\}t$.

## Theory of relativity

In Einstein's theory of relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions $x^\left\{\mu\right\}\left(\tau\right),\; \mu \in \\left\{0,1,2,3\\right\}$ (where $x^\left\{0\right\}$ denotes the time coordinate multiplied by c), each function depending on one parameter $\tau$, called its proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

. $\mathbf\left\{x\right\} = x^\left\{\mu\right\}\left(\tau\right) = \begin\left\{bmatrix\right\} x^0\left(\tau\right)\\ x^1\left(\tau\right) \\ x^2\left(\tau\right) \\ x^3\left(\tau\right) \\ \end\left\{bmatrix\right\}$

### Time dilation

From time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

, we know that$t = \gamma \tau \,$ where $\gamma$ is the Lorentz factor
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

, which is defined as: $\gamma = \frac\left\{1\right\}\left\{\sqrt\left\{1-\frac\left\{u^2\right\}\left\{c^2\right\}\right\}\right\}$ and u is the Euclidean norm of the classical velocity vector $\vec\left\{u\right\}$: $u = || \ \vec\left\{u\right\} \ || = \sqrt\left\{ \left(u^1\right)^2 + \left(u^2\right)^2 + \left(u^3\right)^2\right\}$.

### 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...

. The four velocity of world line $\mathbf\left\{x\right\}\left(\tau\right)$ is defined as: $\mathbf\left\{U\right\} = \frac\left\{\mathrm\left\{d\right\}\mathbf\left\{x\right\}\right\}\left\{\mathrm\left\{d\right\} \tau\right\}$ where $\tau \,$ is the proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

.

### 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: $U^0 = \frac\left\{\mathrm\left\{d\right\}x^0\right\}\left\{\mathrm\left\{d\right\}\tau\;\right\} = c \gamma$ Using the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

, for $\mu = i =$1, 2, 3, we have $U^i = \frac\left\{\mathrm\left\{d\right\}x^i\right\}\left\{\mathrm\left\{d\right\}\tau\right\} = \frac\left\{\mathrm\left\{d\right\}x^i\right\}\left\{\mathrm\left\{d\right\}x^0\right\} \frac\left\{\mathrm\left\{d\right\}x^0\right\}\left\{\mathrm\left\{d\right\}\tau\right\} = \frac\left\{\mathrm\left\{d\right\}x^i\right\}\left\{\mathrm\left\{d\right\}x^0\right\} c\gamma = \frac\left\{\mathrm\left\{d\right\}x^i\right\}\left\{\mathrm\left\{d\right\}\left(ct\right)\right\} c\gamma = \left\{1 \over c\right\} \frac\left\{\mathrm\left\{d\right\}x^i\right\}\left\{\mathrm\left\{d\right\}t\right\} c\gamma = \gamma \frac\left\{\mathrm\left\{d\right\}x^i\right\}\left\{\mathrm\left\{d\right\}t\right\} = \gamma u^i$ where we have used the relationship $u^i = \left\{dx^i \over dt \right\}$ from classical mechanics. Thus, we find for the four-velocity $U$: $U = \gamma \left\left( c, \vec\left\{u\right\} \right\right)$ 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
In relativity, proper-velocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observer-measured...

$\gamma \vec\left\{u\right\} = d\vec\left\{x\right\}/d\tau$ i.e. the rate at which distance is covered in the reference map-frame per unit proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

elapsed on clocks traveling with the object.

## Interpretation

For a rest frame, of course, $\gamma = 1$ and $\vec\left\{u\right\} = 0$, hence $U = \left(c,0,0,0\right) \,$ thus justifying the statement about traveling in the time direction. In every frame of reference, in both special and general relativity, we have $U_\mu U^\mu = -c^2 \,$ if the signature of the metric is (-1,1,1,1). Otherwise, if the signature is taken to be (1,-1,-1,-1), we have $U_\mu U^\mu = +c^2 \,$ However, in both cases: $|| \mathbf\left\{U\right\} || = \sqrt\left\{ | U_\mu U^\mu | \right\} = c$ In other words, the norm or magnitude of the four-velocity of a rest-massive object is always exactly equal to the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. Thus all rest-massive objects can be thought of as moving through spacetime at the speed of light. This provides a way of understanding time-dilation. As an object like a rocket accelerates from our perspective, it moves not only faster through space, but also faster through time in order to keep the magnitude of the four-velocity constant. As the speed through time {{math|dt/dτ}} increases, the rate of aging {{math|dτ/dt}} decreases. Thus to an observer, a clock on the rocket moves slower, as do the clocks in any reference frame that is not comoving with them. The four-velocity defined above via the rest frame of an object does not exist for rest-massless objects such as light, since light does not have a Lorentz inertial rest frame. However, the four-velocity can also be thought of as the tangent vector to an object's spacetime path. With this definition, the 4-velocity of light has zero spacetime "length". Formally, this is equivalent to a photon's path being a null geodesic. Heuristically, a photon always travels "equally" in coordinate space and time, its path being the diagonal in a spacetime diagram in which a particular Lorentz inertial coordinate system has been chosen. The two pictures are equivalent, since the diagonal is a null geodesic.

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• 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. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...

, 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:whereandand \gamma_u is the Lorentz factor for the speed u...

, 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.
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Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, Calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, 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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

.
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Algebra of physical space
In physics, the algebra of physical space is the use of the Clifford or geometric algebra Cℓ3 of the three-dimensional Euclidean space as a model for -dimensional space-time, representing a point in space-time via a paravector .The Clifford algebra Cℓ3 has a faithful representation, generated by...

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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...

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