Pseudo-Euclidean space

A pseudo-Euclidean space is a finite-dimensional real vector space together with a non-degenerate indefinite quadratic form. Such a quadratic form can, after a change of coordinates, be written as

$q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_{k+1}^2+\cdots + x_n^2\right) \,$

where x = (x₁, ..., x_n), n is the dimension of the space, and 1 ≤ k < n. For true Euclidean spaces one has k = n, so the quadratic form is positive-definite, rather than indefinite.

A very important pseudo-Euclidean space is Minkowski space, which is the mathematical setting in which Albert Einstein's theory of special relativity is conveniently formulated. For Minkowski space, n = 4 and k = 3 so that

$q(x) = x_1^2+ x_2^2 + x_3^2-x_4^2,$

The geometry associated with this pseudo-metric was investigated by Poincaré who showed its consistency in spite of a total breakdown of the usual properties of Euclidean space. For example a straight line may be perpendicular to itself.

Another pseudo-Euclidean space is the plane z = x + y j consisting of split-complex numbers, equipped with the quadratic form

$\lVert z \rVert = z z^* = z^* z = x^2 - y^2. \,$

The magnitude of a vector x in the space is defined as q(x). In a pseudo-Euclidean space, unlike in a Euclidean space, there exist non-zero vectors with zero magnitude, and also vectors with negative magnitude.

Associated with the quadratic form q is the pseudo-Euclidean inner product

$\langle x, y\rangle = \left(x_1y_1+\cdots + x_ky_k\right)-\left(x_{k+1}y_{k+1}+\cdots + x_ny_n\right). \,$

This bilinear form is symmetric, but not positive-definite, so it is not a true inner product.

Whereas Euclidean space has a unit sphere, pseudo-Euclidean space has the hypersurfaces {x : q(x) = 1 } and {x : q(x) = −1}. Such a hypersurface, called a hyperboloid or unit quasi-sphere, is preserved by the appropriate indefinite orthogonal group.

Every pseudo-Euclidean space has a linear cone given by {x : q(x) = 0 }. When the pseudo-Euclidean space provides a model for spacetime, the linear cone is called the light cone of the origin.

See also

• Pseudo-Riemannian manifold

References

• Walter Noll (1964) "Euclidean geometry and Minkowskian chronometry", American Mathematical Monthly 71:129–44.
• Novikov, S. P.; Fomenko, A.T.; [translated from the Russian by M. Tsaplina] (1990). Basic elements of differential geometry and topology. Dordrecht; Boston: Kluwer Academic Publishers. ISBN 0792310098. 
• Poincaré, Science and Hypothesis 1906 referred to in the book B.A. Rosenfeld, A History of Non-Euclidean Geometry Springer 1988 (English translation) p.266.
• Szekeres, Peter (2004). A course in modern mathematical physics: groups, Hilbert space, and differential geometry. Cambridge University Press. ISBN 0521829607.