# Pseudo-Euclidean space

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 $x ^ {j} = i y ^ {j}$, $l < j \leq n$.

#### 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)