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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 $  \mathbf a $
 +
and $  \mathbf b $
 +
there corresponds a definite number, called the scalar product $  ( \mathbf a , \mathbf b ) $
 +
(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>
+
$$
 +
( \mathbf a , \mathbf b )  = ( \mathbf b , \mathbf a ) ;
 +
$$
  
 
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>
+
$$
 +
( \mathbf a , ( \mathbf b + \mathbf c ) )  = ( \mathbf a , \mathbf b ) + ( \mathbf a ,\
 +
\mathbf c ) ;
 +
$$
  
 
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>
+
$$
 +
( k \mathbf a , \mathbf b )  = k ( \mathbf a , \mathbf b ) ;
 +
$$
  
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
+
4) there exist $  n $
 +
vectors $  \mathbf a _ {i} $
 +
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>
+
$$
 +
( \mathbf a _ {c} , \mathbf a _ {c} )  > 0 ,\
 +
c \leq  l ; \  ( \mathbf a _ {d} , \mathbf a _ {d} )  < 0 , d > l ;
 +
$$
  
<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>
+
$$
 +
( \mathbf a _ {i} , \mathbf a _ {j} )  = 0 ,\  i \neq j .
 +
$$
  
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 number $  n $
 +
is called the dimension of the pseudo-Euclidean space, $  l $
 +
is called the index, the pair of numbers $  ( l , p ) $,  
 +
$  p = n - l $,  
 +
is called the signature. A pseudo-Euclidean space is denoted by $  E _ {( l , p ) }  $(
 +
or $  {}  ^ {l} E _ {n} $).  
 +
The space $  E _ {( 1 , 3 ) }  $
 +
is called the [[Minkowski space|Minkowski space]]. In any system of $  n $
 +
vectors $  \mathbf b _ {i} $
 +
in $  E _ {( l , p ) }  $
 +
for which $  ( \mathbf b _ {i} , \mathbf b _ {i} ) \neq 0 $
 +
and $  ( \mathbf b _ {i} , \mathbf b _ {j} ) = 0 $
 +
when $  i \neq j $,  
 +
the number of vectors $  \mathbf b _ {i} $
 +
for which $  ( \mathbf b _ {i} , \mathbf b _ {i} ) > 0 $
 +
is equal to $  l $
 +
and the number of vectors $  \mathbf b _ {i} $
 +
for which $  ( \mathbf b _ {i} , \mathbf b _ {i} ) < 0 $
 +
is equal to $  n - l $
 +
(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.
+
The modulus $  | \mathbf a | $
 +
of a vector $  \mathbf a $
 +
in a pseudo-Euclidean space can be defined as the non-negative root $  \sqrt {| ( \mathbf a , \mathbf a ) | } $.  
 +
The vectors that have scalar squares equal to 1 or $  - 1 $
 +
are called unit and pseudo-unit vectors, respectively. The vectors $  \mathbf x $
 +
for which $  ( \mathbf x , \mathbf x ) = 0 $
 +
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 three types of straight lines: Euclidean, having direction vector with positive scalar square $  ( ( \mathbf a , \mathbf a ) > 0 ) $,  
 +
pseudo-Euclidean $  ( ( \mathbf a , \mathbf a ) < 0 ) $
 +
and isotropic $  ( ( \mathbf a , \mathbf a ) = 0 ) $.  
 +
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.
+
In a pseudo-Euclidean space there are several types of planes: Euclidean planes $  E  ^ {2} $,  
 +
pseudo-Euclidean planes $  E _ {( 1 , 1 ) }  $
 +
and planes containing isotropic vectors, the so-called semi-Euclidean planes with signatures $  ( 0 , 1 ) $
 +
and $  ( 1 , 0 ) $
 +
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:
+
The distance between two points $  A ( x) $
 +
and $  B ( y) $
 +
is taken to be the modulus of the vector $  \overline{ {A B }}\; $
 +
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>
+
$$
 +
\overline{ {A B }}\; {}  ^ {2}  = \
 +
| \mathbf y - \mathbf x |  ^ {2}  = \
 +
| ( \mathbf y - \mathbf x , \mathbf y - \mathbf x ) | .
 +
$$
  
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:
+
A pseudo-Euclidean space is not a metric space, since the triangle inequality is not satisfied. If the vectors $  \mathbf a $
 +
and $  \mathbf b $
 +
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>
+
$$
 +
| \mathbf a + \mathbf b |  \geq  | \mathbf a | + | \mathbf b | .
 +
$$
  
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.
+
In a pseudo-Euclidean space there are three types of spheres: spheres with positive radius squared, $  ( \mathbf x , \mathbf x ) = \rho  ^ {2} $,  
 +
spheres with negative radius squared, $  ( \mathbf x , \mathbf x ) = - \rho  ^ {2} $,  
 +
and spheres of zero radius, $  ( \mathbf x , \mathbf x ) = 0 $,  
 +
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>
+
$$
 +
\mathbf x  ^  \prime  = \mathbf U \mathbf x + \mathbf a .
 +
$$
  
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.
+
The operator $  \mathbf U $
 +
satisfies the condition $  | \mathbf U \mathbf x | = | \mathbf x | $,  
 +
that is, it preserves distances between points. The motions of a pseudo-Euclidean space form a multiplicative group; it depends on $  n ( n + 1 ) / 2 $
 +
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" />.
+
Geometric transformations are called anti-motions when each vector $  \mathbf a $
 +
goes to a vector $  \mathbf a  ^  \prime  $
 +
for which $  ( \mathbf a , \mathbf a ) = - ( \mathbf a  ^  \prime  , \mathbf a  ^  \prime  ) $.
  
 
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>
+
$$
 +
g _ {ij}  =
 +
\left \|
 +
\begin{array}{cc}
 +
\left .
 +
\begin{array}{ccccc}
 +
1  & {}    & {}    & {}  & {} \\
 +
{}  & \cdot & {}    & {}  & {} \\
 +
{}  & {}    & \cdot & {}  & {} \\
 +
{}  & {}    & {}    & \cdot& {} \\
 +
{}  & {}    & {}    & {}  & 1  \\
 +
\end{array}
 +
\right \}  l & 0 \\
 +
0 &
 +
\left .  
 +
\begin{array}{ccccc}
 +
-1  & {}    & {}    & {}  & {} \\
 +
{}  & \cdot & {}    & {}  & {} \\
 +
{}  & {}    & \cdot & {}  & {} \\
 +
{}  & {}    & {}    & \cdot& {} \\
 +
{}  & {}    & {}    & {}  & -1 \\
 +
\end{array}
 +
\right\} p
 +
\end{array}
 +
\right \| .
 +
$$
  
 
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 <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.
+
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 $  E _ {( 1 , n - 1 ) }  $
 
+
a sphere with positive radius squared is an $  ( n - 1 ) $-
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
+
dimensional space isometric to the hyperbolic space.
 
 
<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 <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" />.
+
The pseudo-Euclidean space $  E _ {( l , p ) }  $(
 
+
$  l + p = n $)
====References====
+
and the Euclidean space  $ ^ {n} $
<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>
+
can be considered as subspaces of a complex space with form  $  d s ^ {2} = \sum_{i=1}^ {n} d z _ {i} ^ {2} $.  
 +
If  $  x  ^ {j} $
 +
are coordinates in the pseudo-Euclidean space$ y ^ {j} $
 +
are those of the real Euclidean space and  $ z ^ {j} $
 +
those of the complex Euclidean space, then the equations of the subspaces have the form
  
 +
$$
 +
x  ^ {j}  =  \mathop{\rm Re}  z  ^ {j} ,\ \
 +
0 < j \leq  l ; \ \
 +
x  ^ {j}  =  \mathop{\rm Im}  z  ^ {j} ,\  y  ^ {j}  =  \mathop{\rm Re}  z  ^ {j} ,\  l <
 +
j \leq  n .
 +
$$
  
 +
The metric of the pseudo-Euclidean space can be formally obtained from the metric of the Euclidean space by the substitution  $  x  ^ {j} = i y  ^ {j} $,
 +
$  l < j \leq  n $.
  
 
====Comments====
 
====Comments====
Line 68: Line 185:
  
 
====References====
 
====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>
+
<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>
 +
<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>

Latest revision as of 08:54, 13 January 2024


A real affine space in which to any vectors $ \mathbf a $ and $ \mathbf b $ there corresponds a definite number, called the scalar product $ ( \mathbf a , \mathbf b ) $ (cf. also Inner product), satisfying

1) the scalar product is commutative:

$$ ( \mathbf a , \mathbf b ) = ( \mathbf b , \mathbf a ) ; $$

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

$$ ( \mathbf a , ( \mathbf b + \mathbf c ) ) = ( \mathbf a , \mathbf b ) + ( \mathbf a ,\ \mathbf c ) ; $$

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

$$ ( k \mathbf a , \mathbf b ) = k ( \mathbf a , \mathbf b ) ; $$

4) there exist $ n $ vectors $ \mathbf a _ {i} $ such that

$$ ( \mathbf a _ {c} , \mathbf a _ {c} ) > 0 ,\ c \leq l ; \ ( \mathbf a _ {d} , \mathbf a _ {d} ) < 0 , d > l ; $$

$$ ( \mathbf a _ {i} , \mathbf a _ {j} ) = 0 ,\ i \neq j . $$

The number $ n $ is called the dimension of the pseudo-Euclidean space, $ l $ is called the index, the pair of numbers $ ( l , p ) $, $ p = n - l $, is called the signature. A pseudo-Euclidean space is denoted by $ E _ {( l , p ) } $( or $ {} ^ {l} E _ {n} $). The space $ E _ {( 1 , 3 ) } $ is called the Minkowski space. In any system of $ n $ vectors $ \mathbf b _ {i} $ in $ E _ {( l , p ) } $ for which $ ( \mathbf b _ {i} , \mathbf b _ {i} ) \neq 0 $ and $ ( \mathbf b _ {i} , \mathbf b _ {j} ) = 0 $ when $ i \neq j $, the number of vectors $ \mathbf b _ {i} $ for which $ ( \mathbf b _ {i} , \mathbf b _ {i} ) > 0 $ is equal to $ l $ and the number of vectors $ \mathbf b _ {i} $ for which $ ( \mathbf b _ {i} , \mathbf b _ {i} ) < 0 $ is equal to $ n - l $ (the law of inertia for a quadratic form).

The modulus $ | \mathbf a | $ of a vector $ \mathbf a $ in a pseudo-Euclidean space can be defined as the non-negative root $ \sqrt {| ( \mathbf a , \mathbf a ) | } $. The vectors that have scalar squares equal to 1 or $ - 1 $ are called unit and pseudo-unit vectors, respectively. The vectors $ \mathbf x $ for which $ ( \mathbf x , \mathbf x ) = 0 $ 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 $ ( ( \mathbf a , \mathbf a ) > 0 ) $, pseudo-Euclidean $ ( ( \mathbf a , \mathbf a ) < 0 ) $ and isotropic $ ( ( \mathbf a , \mathbf a ) = 0 ) $. 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 $ E ^ {2} $, pseudo-Euclidean planes $ E _ {( 1 , 1 ) } $ and planes containing isotropic vectors, the so-called semi-Euclidean planes with signatures $ ( 0 , 1 ) $ and $ ( 1 , 0 ) $ and deficiency 1 (see Semi-Euclidean space) and isotropic planes, all vectors of which are isotropic.

The distance between two points $ A ( x) $ and $ B ( y) $ is taken to be the modulus of the vector $ \overline{ {A B }}\; $ and is computed from:

$$ \overline{ {A B }}\; {} ^ {2} = \ | \mathbf y - \mathbf x | ^ {2} = \ | ( \mathbf y - \mathbf x , \mathbf y - \mathbf x ) | . $$

A pseudo-Euclidean space is not a metric space, since the triangle inequality is not satisfied. If the vectors $ \mathbf a $ and $ \mathbf b $ 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:

$$ | \mathbf a + \mathbf b | \geq | \mathbf a | + | \mathbf b | . $$

In a pseudo-Euclidean space there are three types of spheres: spheres with positive radius squared, $ ( \mathbf x , \mathbf x ) = \rho ^ {2} $, spheres with negative radius squared, $ ( \mathbf x , \mathbf x ) = - \rho ^ {2} $, and spheres of zero radius, $ ( \mathbf x , \mathbf x ) = 0 $, 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

$$ \mathbf x ^ \prime = \mathbf U \mathbf x + \mathbf a . $$

The operator $ \mathbf U $ satisfies the condition $ | \mathbf U \mathbf x | = | \mathbf x | $, that is, it preserves distances between points. The motions of a pseudo-Euclidean space form a multiplicative group; it depends on $ n ( n + 1 ) / 2 $ 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 $ \mathbf a $ goes to a vector $ \mathbf a ^ \prime $ for which $ ( \mathbf a , \mathbf a ) = - ( \mathbf a ^ \prime , \mathbf a ^ \prime ) $.

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)

$$ g _ {ij} = \left \| \begin{array}{cc} \left . \begin{array}{ccccc} 1 & {} & {} & {} & {} \\ {} & \cdot & {} & {} & {} \\ {} & {} & \cdot & {} & {} \\ {} & {} & {} & \cdot& {} \\ {} & {} & {} & {} & 1 \\ \end{array} \right \} l & 0 \\ 0 & \left . \begin{array}{ccccc} -1 & {} & {} & {} & {} \\ {} & \cdot & {} & {} & {} \\ {} & {} & \cdot & {} & {} \\ {} & {} & {} & \cdot& {} \\ {} & {} & {} & {} & -1 \\ \end{array} \right\} p \end{array} \right \| . $$

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 $ E _ {( 1 , n - 1 ) } $ a sphere with positive radius squared is an $ ( n - 1 ) $- dimensional space isometric to the hyperbolic space.

The pseudo-Euclidean space $ E _ {( l , p ) } $( $ l + p = n $) and the Euclidean space $ E ^ {n} $ can be considered as subspaces of a complex space with form $ d s ^ {2} = \sum_{i=1}^ {n} d z _ {i} ^ {2} $. If $ x ^ {j} $ are coordinates in the pseudo-Euclidean space, $ y ^ {j} $ are those of the real Euclidean space and $ z ^ {j} $ those of the complex Euclidean space, then the equations of the subspaces have the form

$$ x ^ {j} = \mathop{\rm Re} z ^ {j} ,\ \ 0 < j \leq l ; \ \ x ^ {j} = \mathop{\rm Im} z ^ {j} ,\ y ^ {j} = \mathop{\rm Re} z ^ {j} ,\ l < j \leq n . $$

The metric of the pseudo-Euclidean space can be formally obtained from the metric of the Euclidean space by the substitution $ x ^ {j} = i y ^ {j} $, $ l < j \leq n $.

Comments

The concept of a pseudo-Euclidean space was generalized by E. Witt in 1937, see [a1][a2].

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)
[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)
How to Cite This Entry:
Pseudo-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-Euclidean_space&oldid=16529
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article