Difference between revisions of "Pseudo-Euclidean space"
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− | + | A real [[Affine space|affine space]] in which to any vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756802.png" /> there corresponds a definite number, called the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756803.png" /> (cf. also [[Inner product|Inner product]]), satisfying | |
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− | A real [[Affine space|affine space]] in which to any vectors | ||
− | and | ||
− | there corresponds a definite number, called the scalar product | ||
− | cf. also [[Inner product|Inner product]]), satisfying | ||
1) the scalar product is commutative: | 1) the scalar product is commutative: | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756804.png" /></td> </tr></table> | |
− | |||
− | |||
2) the scalar product is distributive with respect to vector addition: | 2) the scalar product is distributive with respect to vector addition: | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756805.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
3) a scalar factor can be taken out of the scalar product: | 3) a scalar factor can be taken out of the scalar product: | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756806.png" /></td> </tr></table> | |
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− | |||
− | 4) there exist | + | 4) there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756807.png" /> vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756808.png" /> such that |
− | vectors | ||
− | such that | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p0756809.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568010.png" /></td> </tr></table> | |
− | |||
− | |||
− | The number | + | The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568011.png" /> is called the dimension of the pseudo-Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568012.png" /> is called the index, the pair of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568014.png" />, is called the signature. A pseudo-Euclidean space is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568015.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568016.png" />). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568017.png" /> is called the [[Minkowski space|Minkowski space]]. In any system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568018.png" /> vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568020.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568022.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568023.png" />, the number of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568024.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568025.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568026.png" /> and the number of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568027.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568028.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568029.png" /> (the law of inertia for a quadratic form). |
− | is called the dimension of the pseudo-Euclidean space, | ||
− | is called the index, the pair of numbers | ||
− | |||
− | is called the signature. A pseudo-Euclidean space is denoted by | ||
− | or | ||
− | The space | ||
− | is called the [[Minkowski space|Minkowski space]]. In any system of | ||
− | vectors | ||
− | in | ||
− | for which | ||
− | and | ||
− | when | ||
− | the number of vectors | ||
− | for which | ||
− | is equal to | ||
− | and the number of vectors | ||
− | for which | ||
− | is equal to | ||
− | the law of inertia for a quadratic form). | ||
− | The modulus | + | The modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568030.png" /> of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568031.png" /> in a pseudo-Euclidean space can be defined as the non-negative root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568032.png" />. The vectors that have scalar squares equal to 1 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568033.png" /> are called unit and pseudo-unit vectors, respectively. The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568034.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568035.png" /> have zero modulus and are called isotropic vectors. The directions of the isotropic vectors are the isotropic directions. |
− | of a vector | ||
− | in a pseudo-Euclidean space can be defined as the non-negative root | ||
− | The vectors that have scalar squares equal to 1 or | ||
− | are called unit and pseudo-unit vectors, respectively. The vectors | ||
− | for which | ||
− | have zero modulus and are called isotropic vectors. The directions of the isotropic vectors are the isotropic directions. | ||
− | In a pseudo-Euclidean space there are three types of straight lines: Euclidean, having direction vector with positive scalar square | + | In a pseudo-Euclidean space there are three types of straight lines: Euclidean, having direction vector with positive scalar square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568036.png" />, pseudo-Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568037.png" /> and isotropic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568038.png" />. The union of all the isotropic straight lines passing through a certain point is called the isotropic cone. |
− | pseudo-Euclidean | ||
− | and isotropic | ||
− | The union of all the isotropic straight lines passing through a certain point is called the isotropic cone. | ||
− | In a pseudo-Euclidean space there are several types of planes: Euclidean planes | + | In a pseudo-Euclidean space there are several types of planes: Euclidean planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568039.png" />, pseudo-Euclidean planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568040.png" /> and planes containing isotropic vectors, the so-called semi-Euclidean planes with signatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568042.png" /> and deficiency 1 (see [[Semi-Euclidean space|Semi-Euclidean space]]) and isotropic planes, all vectors of which are isotropic. |
− | pseudo-Euclidean planes | ||
− | and planes containing isotropic vectors, the so-called semi-Euclidean planes with signatures | ||
− | and | ||
− | and deficiency 1 (see [[Semi-Euclidean space|Semi-Euclidean space]]) and isotropic planes, all vectors of which are isotropic. | ||
− | The distance between two points | + | The distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568044.png" /> is taken to be the modulus of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568045.png" /> and is computed from: |
− | and | ||
− | is taken to be the modulus of the vector | ||
− | and is computed from: | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568046.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | |||
− | A pseudo-Euclidean space is not a metric space, since the triangle inequality is not satisfied. If the vectors | + | A pseudo-Euclidean space is not a metric space, since the triangle inequality is not satisfied. If the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568048.png" /> belong to a Euclidean plane (or to a pseudo-Euclidean plane of index 0), then they satisfy the triangle inequality, but if they belong to a pseudo-Euclidean plane of index 1, then they satisfy the so-called inverse triangle inequality: |
− | and | ||
− | belong to a Euclidean plane (or to a pseudo-Euclidean plane of index 0), then they satisfy the triangle inequality, but if they belong to a pseudo-Euclidean plane of index 1, then they satisfy the so-called inverse triangle inequality: | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568049.png" /></td> </tr></table> | |
− | |||
− | |||
− | In a pseudo-Euclidean space there are three types of spheres: spheres with positive radius squared, | + | In a pseudo-Euclidean space there are three types of spheres: spheres with positive radius squared, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568050.png" />, spheres with negative radius squared, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568051.png" />, and spheres of zero radius, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568052.png" />, which are just the isotropic cones. |
− | spheres with negative radius squared, | ||
− | and spheres of zero radius, | ||
− | which are just the isotropic cones. | ||
The motions of a pseudo-Euclidean space are affine transformations (cf. [[Affine transformation|Affine transformation]]) and can be written in the form | The motions of a pseudo-Euclidean space are affine transformations (cf. [[Affine transformation|Affine transformation]]) and can be written in the form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568053.png" /></td> </tr></table> | |
− | |||
− | |||
− | The operator | + | The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568054.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568055.png" />, that is, it preserves distances between points. The motions of a pseudo-Euclidean space form a multiplicative group; it depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568056.png" /> independent parameters. The motions of a pseudo-Euclidean space are called motions of the first or second kind if they are affine transformations of the corresponding kind. |
− | satisfies the condition | ||
− | that is, it preserves distances between points. The motions of a pseudo-Euclidean space form a multiplicative group; it depends on | ||
− | independent parameters. The motions of a pseudo-Euclidean space are called motions of the first or second kind if they are affine transformations of the corresponding kind. | ||
− | Geometric transformations are called anti-motions when each vector | + | Geometric transformations are called anti-motions when each vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568057.png" /> goes to a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568058.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568059.png" />. |
− | goes to a vector | ||
− | for which | ||
The basic operations of vector and tensor algebra can be introduced into a pseudo-Euclidean space. The basic differential-geometric concepts are constructed in accordance with the rules of the geometry of [[Pseudo-Riemannian space|pseudo-Riemannian space]]. The [[Metric tensor|metric tensor]] of a pseudo-Euclidean space has the form (in a Galilean coordinate system) | The basic operations of vector and tensor algebra can be introduced into a pseudo-Euclidean space. The basic differential-geometric concepts are constructed in accordance with the rules of the geometry of [[Pseudo-Riemannian space|pseudo-Riemannian space]]. The [[Metric tensor|metric tensor]] of a pseudo-Euclidean space has the form (in a Galilean coordinate system) | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568060.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | |||
A pseudo-Euclidean space is flat, that is, its [[Riemann tensor|Riemann tensor]] is zero. If the Riemann tensor of a pseudo-Riemannian space is identically zero, then it is a locally pseudo-Euclidean space. | A pseudo-Euclidean space is flat, that is, its [[Riemann tensor|Riemann tensor]] is zero. If the Riemann tensor of a pseudo-Riemannian space is identically zero, then it is a locally pseudo-Euclidean space. | ||
− | Subsets of a pseudo-Euclidean space can carry various metrics: A positive- or negative-definite Riemannian metric, a pseudo-Riemannian metric or a degenerate metric (see [[Indefinite metric|Indefinite metric]]). For example, the spheres of a pseudo-Euclidean space carry a (generally speaking, indefinite) metric of constant curvature. In | + | Subsets of a pseudo-Euclidean space can carry various metrics: A positive- or negative-definite Riemannian metric, a pseudo-Riemannian metric or a degenerate metric (see [[Indefinite metric|Indefinite metric]]). For example, the spheres of a pseudo-Euclidean space carry a (generally speaking, indefinite) metric of constant curvature. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568061.png" /> a sphere with positive radius squared is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568062.png" />-dimensional space isometric to the hyperbolic space. |
− | a sphere with positive radius squared is an | ||
− | dimensional space isometric to the hyperbolic space. | ||
− | The pseudo-Euclidean space | + | The pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568063.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568064.png" />) and the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568065.png" /> can be considered as subspaces of a complex space with form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568066.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568067.png" /> are coordinates in the pseudo-Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568068.png" /> are those of the real Euclidean space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568069.png" /> those of the complex Euclidean space, then the equations of the subspaces have the form |
− | |||
− | and the Euclidean space | ||
− | can be considered as subspaces of a complex space with form | ||
− | If | ||
− | are coordinates in the pseudo-Euclidean space, | ||
− | are those of the real Euclidean space and | ||
− | those of the complex Euclidean space, then the equations of the subspaces have the form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568070.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | The metric of the pseudo-Euclidean space can be formally obtained from the metric of the Euclidean space by the substitution | + | The metric of the pseudo-Euclidean space can be formally obtained from the metric of the Euclidean space by the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075680/p07568072.png" />. |
− | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1962) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1962) (Translated from Russian)</TD></TR></table> | ||
+ | |||
+ | |||
====Comments==== | ====Comments==== |
Revision as of 14:53, 7 June 2020
A real affine space in which to any vectors and
there corresponds a definite number, called the scalar product
(cf. also Inner product), satisfying
1) the scalar product is commutative:
![]() |
2) the scalar product is distributive with respect to vector addition:
![]() |
3) a scalar factor can be taken out of the scalar product:
![]() |
4) there exist vectors
such that
![]() |
![]() |
The number is called the dimension of the pseudo-Euclidean space,
is called the index, the pair of numbers
,
, is called the signature. A pseudo-Euclidean space is denoted by
(or
). The space
is called the Minkowski space. In any system of
vectors
in
for which
and
when
, the number of vectors
for which
is equal to
and the number of vectors
for which
is equal to
(the law of inertia for a quadratic form).
The modulus of a vector
in a pseudo-Euclidean space can be defined as the non-negative root
. The vectors that have scalar squares equal to 1 or
are called unit and pseudo-unit vectors, respectively. The vectors
for which
have zero modulus and are called isotropic vectors. The directions of the isotropic vectors are the isotropic directions.
In a pseudo-Euclidean space there are three types of straight lines: Euclidean, having direction vector with positive scalar square , pseudo-Euclidean
and isotropic
. The union of all the isotropic straight lines passing through a certain point is called the isotropic cone.
In a pseudo-Euclidean space there are several types of planes: Euclidean planes , pseudo-Euclidean planes
and planes containing isotropic vectors, the so-called semi-Euclidean planes with signatures
and
and deficiency 1 (see Semi-Euclidean space) and isotropic planes, all vectors of which are isotropic.
The distance between two points and
is taken to be the modulus of the vector
and is computed from:
![]() |
A pseudo-Euclidean space is not a metric space, since the triangle inequality is not satisfied. If the vectors and
belong to a Euclidean plane (or to a pseudo-Euclidean plane of index 0), then they satisfy the triangle inequality, but if they belong to a pseudo-Euclidean plane of index 1, then they satisfy the so-called inverse triangle inequality:
![]() |
In a pseudo-Euclidean space there are three types of spheres: spheres with positive radius squared, , spheres with negative radius squared,
, and spheres of zero radius,
, which are just the isotropic cones.
The motions of a pseudo-Euclidean space are affine transformations (cf. Affine transformation) and can be written in the form
![]() |
The operator satisfies the condition
, that is, it preserves distances between points. The motions of a pseudo-Euclidean space form a multiplicative group; it depends on
independent parameters. The motions of a pseudo-Euclidean space are called motions of the first or second kind if they are affine transformations of the corresponding kind.
Geometric transformations are called anti-motions when each vector goes to a vector
for which
.
The basic operations of vector and tensor algebra can be introduced into a pseudo-Euclidean space. The basic differential-geometric concepts are constructed in accordance with the rules of the geometry of pseudo-Riemannian space. The metric tensor of a pseudo-Euclidean space has the form (in a Galilean coordinate system)
![]() |
A pseudo-Euclidean space is flat, that is, its Riemann tensor is zero. If the Riemann tensor of a pseudo-Riemannian space is identically zero, then it is a locally pseudo-Euclidean space.
Subsets of a pseudo-Euclidean space can carry various metrics: A positive- or negative-definite Riemannian metric, a pseudo-Riemannian metric or a degenerate metric (see Indefinite metric). For example, the spheres of a pseudo-Euclidean space carry a (generally speaking, indefinite) metric of constant curvature. In a sphere with positive radius squared is an
-dimensional space isometric to the hyperbolic space.
The pseudo-Euclidean space (
) and the Euclidean space
can be considered as subspaces of a complex space with form
. If
are coordinates in the pseudo-Euclidean space,
are those of the real Euclidean space and
those of the complex Euclidean space, then the equations of the subspaces have the form
![]() |
The metric of the pseudo-Euclidean space can be formally obtained from the metric of the Euclidean space by the substitution ,
.
References
[1] | N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian) |
[2] | B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian) |
[3] | L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1962) (Translated from Russian) |
Comments
The concept of a pseudo-Euclidean space was generalized by E. Witt in 1937, see [a1]–[a2].
References
[a1] | E. Witt, "Theorie der quadratischen Formen in beliebigen Körpern" J. Reine Angew. Math. , 176 (1937) pp. 31–44 |
[a2] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) |
[a3] | S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973) |
[a4] | C.W. Misner, K.S. Thorne, J.A. Wheeler, "Gravitation" , Freeman (1973) |
[a5] | B. O'Neill, "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press (1983) |
Pseudo-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-Euclidean_space&oldid=49380