In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the line element or length element can be informally thought of as a line segment associated with an
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
displacement vector in a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
. The length of the line element, which may be thought of as a differential
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
, is a function of 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 allow ...
and is denoted by ''
''.
Line elements are used in
physics
Physics is the natural science that studies matter, its 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 which ...
, especially in theories of
gravitation
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
(most notably
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
) where
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
is modelled as a curved
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
with an appropriate
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 allow ...
.
General formulation
Definition of the line element and arclength
The
coordinate-independent definition of the square of the line element ''ds'' in an ''n''-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al
Riemannian or
Pseudo Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
(in physics usually a
Lorentzian manifold) is the "square of the length" of an infinitesimal displacement
(in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length:
where ''g'' is 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 allow ...
, · denotes
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, and ''d''q an
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
displacement on the (pseudo) Riemannian manifold. By parametrizing a curve
, we can define the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
of the curve length of the curve between
, and
as the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
:
:
To compute a sensible length of curves in pseudo Riemannian manifolds, it is best to assume that the infinitesimal displacements have the same sign everywhere. E.g. in physics the square of a line element along a timeline curve would (in the
signature convention) be negative and the negative square root of the square of the line element along the curve would measure the proper time passing for an observer moving along the curve.
From this point of view, the metric also defines in addition to line element the
surface and
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV ...
s etc.
Identification of the square of the line element with the metric tensor
Since
is arbitrary "square of the arc length"
completely defines the metric, it is therefore usually best to consider the expression for
as a definition of the metric tensor itself, written in a suggestive but non tensorial notation:
:
This identification of the square of arc length
with the metric is even more easy to see in ''n''-dimensional general
curvilinear coordinates , where it is written as a symmetric rank 2 tensor coinciding with the metric tensor:
:
.
Here the
indices ''i'' and ''j'' take values 1, 2, 3, ..., ''n'' and
Einstein summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
is used. Common examples of (pseudo) Riemannian spaces include
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
(no inclusion of
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
coordinates), and indeed
four-dimensional
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called '' dimensions'' ...
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
.
Line elements in Euclidean space
Following are examples of how the line elements are found from the metric.
Cartesian coordinates
The simplest line element is in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
- in which case the metric is just the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
:
:
(here ''i, j'' = 1, 2, 3 for space) or in
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
form (''i'' denotes row, ''j'' denotes column):
:
The general curvilinear coordinates reduce to Cartesian coordinates:
:
so
:
Orthogonal curvilinear coordinates
For all
orthogonal coordinates the metric is given by:
:
where
:
for ''i'' = 1, 2, 3 are
scale factors, so the square of the line element is:
:
Some examples of line elements in these coordinates are below.
:
General curvilinear coordinates
Given an arbitrary basis of a space of dimension
, the metric is defined as the inner product of the basis vectors.
Where
and the inner product is with respect to the ambient space (usually its
)
In a coordinate basis
The coordinate basis is a special type of basis that is regularly used in differential geometry.
Line elements in 4d spacetime
Minkowskian spacetime
The
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
is:
:
where one sign or the other is chosen, both conventions are used. This applies only for
flat spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
. The coordinates are given by the
4-position
In special relativity, a four-vector (or 4-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 vector space considered as a ...
:
:
so the line element is:
:
Schwarzschild coordinates
In
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coor ...
coordinates are
, being the general metric of the form:
:
(note the similitudes with the metric in 3D spherical polar coordinates).
so the line element is:
:
General spacetime
The coordinate-independent definition of the square of the line element d''s'' in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
is:
[Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ]
:
In terms of coordinates:
:
where for this case the indices α and β run over 0, 1, 2, 3 for spacetime.
This is the
spacetime interval - the measure of separation between two arbitrarily close
events in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. In
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
it is invariant 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 velo ...
s. In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
it is invariant under arbitrary
invertible differentiable coordinate transformations.
See also
*
Covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notat ...
*
First fundamental form
*
List of integration and measure theory topics
{{TOCright
This is a list of integration and measure theory topics, by Wikipedia page.
Intuitive foundations
*Length
*Area
*Volume
*Probability
*Moving average
Riemann integral
*Riemann sum
*Riemann–Stieltjes integral
*Bounded variation
* Jorda ...
*
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 allow ...
*
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
*
Raising and lowering indices In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Mat ...
References
{{reflist
Affine geometry
Riemannian geometry
Special relativity
General relativity
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