View source for Pseudo-Euclidean space

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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:

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

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

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

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

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|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|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|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|pseudo-Riemannian space]]. The [[Metric tensor|metric tensor]] of a pseudo-Euclidean space has the form (in a Galilean coordinate system)

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

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.

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 ) $-
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 $.

====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>