View source for Pseudo-Euclidean space

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

1) the scalar product is commutative:

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2) the scalar product is distributive with respect to vector addition:

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3) a scalar factor can be taken out of the scalar product:

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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

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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).

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.

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.

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.

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:

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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:

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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.

The motions of a pseudo-Euclidean space are affine transformations (cf. [[Affine transformation|Affine transformation]]) and can be written in the form

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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.

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" />.

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)

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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 <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.

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

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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====
<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====
The concept of a pseudo-Euclidean space was generalized by E. Witt in 1937, see [[#References|[a1]]]–[[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Witt,   "Theorie der quadratischen Formen in beliebigen Körpern"  ''J. Reine Angew. Math.'' , '''176'''  (1937)  pp. 31–44</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Dieudonné,   "La géométrie des groups classiques" , Springer  (1955)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.W. Hawking,   G.F.R. Ellis,   "The large scale structure of space-time" , Cambridge Univ. Press  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C.W. Misner,   K.S. Thorne,   J.A. Wheeler,   "Gravitation" , Freeman  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. O'Neill,   "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press  (1983)</TD></TR></table>