In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, in particular in
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
and
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, a four-velocity is a
four-vector
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vect ...
in four-dimensional
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
[Technically, the four-vector should be thought of as residing in the ]tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
of a point in spacetime, spacetime itself being modeled as a smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
. This distinction is significant in general relativity. that represents the relativistic counterpart of
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
, which is a
three-dimensional
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
in space.
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
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from c ...
. If the object has
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
, so that its speed is necessarily less than the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
, the world line may be
parametrized by the
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
of the object. The four-velocity is the rate of change of
four-position
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vect ...
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
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of an object's four-velocity, i.e. the quantity obtained by applying the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
to the four-velocity , that is , is always equal to , where is the speed of light. Whether the plus or minus sign applies depends on the choice of
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and z ...
. For an object at rest its four-velocity is parallel to the direction of the time coordinate with . 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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
.
[The set of four-velocities is a subset of the tangent space (which ''is'' a vector space) at an event. The label ''four-vector'' stems from the behavior under ]Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
s, namely under which particular representation they transform.
Velocity
The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functions of time , where is an
index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
which takes values 1, 2, 3.
The three coordinates form the 3d
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
, written as a
column vector
In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some , c ...
The components of the velocity
(tangent to the curve) at any point on the world line are
Each component is simply written
Theory of relativity
In Einstein's
theory of relativity
The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical ph ...
, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions , where 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 ,
Each function depends on one parameter ''τ'' called its
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
. As a column vector,
Time dilation
From
time dilation
Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unsp ...
, the
differentials in
coordinate time and
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
are related by
where the
Lorentz factor
The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
,
is a function of the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
of the 3d velocity vector
Definition of the four-velocity
The four-velocity is the tangent four-vector of a
timelike
In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold.
Lorentzian manifolds can be classified according to the types of causal structures they admit (''ca ...
world line
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from c ...
.
The four-velocity
at any point of world line
is defined as:
where
is the
four-position
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vect ...
and
is the
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
.
The four-velocity defined here using the proper time of an object does not exist for world lines for massless objects such as photons travelling at the speed of light; nor is it defined for
tachyon
A tachyon () or tachyonic particle is a hypothetical particle that always travels Faster-than-light, faster than light. Physicists posit that faster-than-light particles cannot exist because they are inconsistent with the known Scientific law#L ...
ic world lines, where the tangent vector is
spacelike
In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold.
Lorentzian manifolds can be classified according to the types of causal structures they admit (''ca ...
.
Components of the four-velocity
The relationship between the time and the coordinate time is defined by
Taking the derivative of this with respect to the proper time , we find the velocity component for :
and for the other 3 components to proper time we get the velocity component for :
where we have used the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
and the relationships
Thus, we find for the four-velocity
Written in standard four-vector notation this is:
where
is the temporal component and
is the spatial component.
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 i.e. the rate at which distance is covered in the reference map frame per unit
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
elapsed on clocks traveling with the object.
Unlike most other four-vectors, the four-velocity has only 3 independent components
instead of 4. The
factor is a function of the three-dimensional velocity
.
When certain Lorentz scalars are multiplied by the four-velocity, one then gets new physical four-vectors that have 4 independent components.
For example:
*
Four-momentum
In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum i ...
:
where
is the
rest mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
*
Four-current density:
where
is the
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
Effectively, the
factor combines with the Lorentz scalar term to make the 4th independent component
and
Magnitude
Using the differential of the four-position in the rest frame, the magnitude of the four-velocity can be obtained by the
Minkowski metric
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model ...
with signature :
in short, the magnitude of the four-velocity for any object is always a fixed constant:
In a moving frame, the same norm is:
so that:
which reduces to the definition of the Lorentz factor.
See also
*
Four-acceleration
*
Four-momentum
In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum i ...
*
Four-force
In the special theory of relativity, four-force is a four-vector that replaces the classical force.
In special relativity
The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper t ...
*
Four-gradient
*
Algebra of physical space
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
*
Congruence (general relativity)
*
Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
*
Rapidity
In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velo ...
Remarks
References
*
* {{cite book, author=Rindler, Wolfgang, title=Introduction to Special Relativity (2nd), location=Oxford, publisher=Oxford University Press, year=1991, isbn=0-19-853952-5, url-access=registration, url=https://archive.org/details/introductiontosp0000rind
Four-vectors