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)

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)