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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a pseudo-Euclidean space of signature is a finite- dimensional real -space together with a non- degenerate
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
. Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x) = \left(x_1^2 + \dots + x_k^2\right) - \left( x_^2 + \dots + x_n^2\right) which is called the ''scalar square'' of the vector . For
Euclidean space 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 ...
s, , implying that the quadratic form is positive-definite. When , then is an isotropic quadratic form. Note that if , then , so that is a null vector. In a pseudo-Euclidean space with , unlike in a Euclidean space, there exist vectors with negative scalar square. As with the term ''Euclidean space'', the term ''pseudo-Euclidean space'' may be used to refer to an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
or 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 ...
depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space (see point–vector distinction).


Geometry

The geometry of a pseudo-Euclidean space is consistent despite some properties of Euclidean space not applying, most notably that it is not a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
as explained below. The affine structure is unchanged, and thus also the concepts line, plane and, generally, of an
affine subspace In mathematics, an affine space is a geometry, geometric structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance (mathematics), distance ...
( flat), as well as
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
s.


Positive, zero, and negative scalar squares

A null vector is a vector for which the quadratic form is zero. Unlike in a Euclidean space, such a vector can be non-zero, in which case it is self-
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
. If the quadratic form is indefinite, a pseudo-Euclidean space has a linear cone of null vectors given by . When the pseudo-Euclidean space provides a model for
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 ...
(see below), the null cone is called the light cone of the origin. The null cone separates two
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s, respectively for which and . If , then the set of vectors for which is connected. If , then it consists of two disjoint parts, one with and another with . Similarly, if , then the set of vectors for which is connected. If , then it consists of two disjoint parts, one with and another with .


Interval

The quadratic form corresponds to the square of a vector in the Euclidean case. To define the vector norm (and distance) in an invariant manner, one has to get square roots of scalar squares, which leads to possibly imaginary distances; see square root of negative numbers. But even for a
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
with positive scalar squares of all three sides (whose square roots are real and positive), the triangle inequality does not hold in general. Hence terms ''norm'' and ''distance'' are avoided in pseudo-Euclidean geometry, which may be replaced with ''scalar square'' and ''interval'' respectively. Though, for a curve whose tangent vectors all have scalar squares of the same sign, the
arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
is defined. It has important applications: see proper time, for example.


Rotations and spheres

The rotations group of such space is the
indefinite orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimension (vector space), dimensional real number, real vector space that leave invariant a nondegenerate form, nondegenerate, symmetric bi ...
, also denoted as without a reference to particular quadratic form. Such "rotations" preserve the form and, hence, the scalar square of each vector including whether it is positive, zero, or negative. Whereas Euclidean space has a unit sphere, pseudo-Euclidean space has the hypersurfaces and . Such a hypersurface, called a quasi-sphere, is preserved by the appropriate indefinite orthogonal group.


Symmetric bilinear form

The quadratic form gives rise to a symmetric bilinear form defined as follows: : \langle x, y\rangle = \tfrac12 (x + y) - q(x) - q(y)= \left(x_1 y_1 + \ldots + x_k y_k\right) - \left(x_y_ + \ldots + x_n y_n\right). The quadratic form can be expressed in terms of the bilinear form: . When , then and are
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
vectors of the pseudo-Euclidean space. This bilinear form is often referred to as the scalar product, and sometimes as "inner product" or "dot product", but it does not define an inner product space and it does not have the properties of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
of Euclidean vectors. If and are orthogonal and , then is hyperbolic-orthogonal to . The
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For exampl ...
of the real -space is
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
. There are no ortho''normal'' bases in a pseudo-Euclidean space for which the bilinear form is indefinite, because it cannot be used to define a vector norm.


Subspaces and orthogonality

For a (positive-dimensional) subspace of a pseudo-Euclidean space, when the quadratic form is restricted to , following three cases are possible: # is either positive or negative definite. Then, is essentially Euclidean (up to the sign of ). # is indefinite, but non-degenerate. Then, is itself pseudo-Euclidean. It is possible only if ; if , which means than is a plane, then it is called a hyperbolic plane. # is degenerate. One of the most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their orthogonality. When two non-zero Euclidean vectors are orthogonal, they are not
collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
. The intersections of any Euclidean
linear subspace In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...
with its orthogonal complement is the subspace. But the definition from the previous subsection immediately implies that any vector of zero scalar square is orthogonal to itself. Hence, the isotropic line generated by a null vector ν is a subset of its orthogonal complement . The formal definition of the orthogonal complement of a vector subspace in a pseudo-Euclidean space gives a perfectly well-defined result, which satisfies the equality due to the quadratic form's non-degeneracy. It is just the condition : or, equivalently, all space, which can be broken if the subspace contains a null direction. While subspaces form a lattice, as in any vector space, this operation is not an orthocomplementation, in contrast to inner product spaces. For a subspace composed ''entirely'' of null vectors (which means that the scalar square , restricted to , equals to ), always holds: : or, equivalently, . Such a subspace can have up to
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 coo ...
s. For a (positive) Euclidean -subspace its orthogonal complement is a -dimensional negative "Euclidean" subspace, and vice versa. Generally, for a -dimensional subspace consisting of positive and negative dimensions (see Sylvester's law of inertia for clarification), its orthogonal "complement" has positive and negative dimensions, while the rest ones are degenerate and form the intersection.


Parallelogram law and Pythagorean theorem

The parallelogram law takes the form : q(x) + q(y) = \tfrac12(q(x + y) + q(x - y)). Using the square of the sum identity, for an arbitrary triangle one can express the scalar square of the third side from scalar squares of two sides and their bilinear form product: : q(x + y) = q(x) + q(y) + 2\langle x, y \rangle. This demonstrates that, for orthogonal vectors, a pseudo-Euclidean analog of the Pythagorean theorem holds: : \langle x, y \rangle = 0 \Rightarrow q(x) + q(y) = q(x + y).


Angle

Generally, absolute value of the bilinear form on two vectors may be greater than , equal to it, or less. This causes similar problems with definition of
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
(see ) as appeared above for distances. If (only one positive term in ), then for vectors of positive scalar square: , \langle x, y\rangle, \ge \sqrt\,, which permits definition of the hyperbolic angle, an analog of angle between these vectors through inverse hyperbolic cosine: \operatorname\frac\,. It corresponds to the distance on a -dimensional hyperbolic space. This is known as
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 ...
in the context of theory of relativity discussed below. Unlike Euclidean angle, it takes values from and equals to 0 for antiparallel vectors. There is no reasonable definition of the angle between a null vector and another vector (either null or non-null).


Algebra and tensor calculus

Like Euclidean spaces, every pseudo-Euclidean vector space generates a Clifford algebra. Unlike properties above, where replacement of to changed numbers but not
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the sign reversal of the quadratic form results in a distinct Clifford algebra, so for example and are not isomorphic. Just like over any vector space, there are pseudo-Euclidean
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s. Like with a Euclidean structure, there are raising and lowering indices operators but, unlike the case with Euclidean tensors, there is no bases where these operations do not change values of components. If there is a vector , the corresponding covariant vector is: : v_\alpha = q_ v^\beta\,, and with the standard-form : q_ = \begin I_ & 0 \\ 0 & -I_ \end the first components of are numerically the same as ones of , but the rest have opposite signs. The correspondence between contravariant and covariant tensors makes a tensor calculus on pseudo-Riemannian manifolds a generalization of one on Riemannian manifolds.


Examples

A very important pseudo-Euclidean space is Minkowski space, which is the mathematical setting in which the theory of
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 ...
is formulated. For Minkowski space, and so that : q(x) = x_1^2 + x_2^2 + x_3^2 - x_4^2, The geometry associated with this pseudo-metric was investigated by Poincaré.B. A. Rosenfeld (1988) ''A History of Non-Euclidean Geometry'', page 266, Studies in the history of mathematics and the physical sciences #12, Springer Its rotation group is the Lorentz group. The Poincaré group includes also translations and plays the same role as Euclidean groups of ordinary Euclidean spaces. Another pseudo-Euclidean space is the plane consisting of
split-complex number In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...
s, equipped with the quadratic form : \lVert z \rVert = z z^* = z^* z = x^2 - y^2. This is the simplest case of an indefinite pseudo-Euclidean space (, ) and the only one where the null cone dissects the remaining space into ''four'' open sets. The group consists of so named hyperbolic rotations.


See also

* Pseudo-Riemannian manifold * Hyperbolic equation * Hyperboloid model * Paravector


Footnotes


References

* * Werner Greub (1963) ''Linear Algebra'', 2nd edition, §12.4 Pseudo-Euclidean Spaces, pp. 237–49, Springer-Verlag. * Walter Noll (1964) "Euclidean geometry and Minkowskian chronometry",
American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
71:129–44. * * * {{cite book , last = Shafarevich , first = I. R. , author-link = Igor Shafarevich , author2 = A. O. Remizov , title = Linear Algebra and Geometry , publisher =
Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...
, year = 2012 , url = https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9 , isbn = 978-3-642-30993-9


External links

* D.D. Sokolov (originator)
Pseudo-Euclidean space
Encyclopedia of Mathematics Lorentzian manifolds