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

: $q(x)\; =\; left(x\_1^2+cdots\; +\; x\_k^2\; ight)-left(x\_\{k+1\}^2+cdots\; +\; x\_n^2\; ight)$

where $x=(x\_1,\; dots,\; x\_n),$ $n$ is the dimension of the space, and

A very important pseudo-Euclidean space is the Minkowski space, for which $n=4$ and $k=3.$ For true Euclidean spaces one has $k=n,$ so the quadratic form is positive-definite, rather than indefinite.

Another pseudo-Euclidean space is the plane "z" = "x" + "y" j consisting of split-complex numbers, equipped with the quadratic form: $lVert\; z\; Vert\; =\; z\; z^*\; =\; z^*\; z\; =\; x^2\; -\; y^2.$

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

: $langle\; x,\; y\; angle\; =\; left(x\_1y\_1+cdots\; +\; x\_ky\_k\; ight)-left(x\_\{k+1\}y\_\{k+1\}+cdots\; +\; x\_ny\_n\; ight).$

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

An interesting property of pseudo-Euclidean space is that it has not only a unit sphere {"x" : "q(x)" = 1 }, but also a counter-sphere {"x" : "q(x)" = − 1}. The sets are actually generalized hyperboloids; the term "sphere" is for consistency with the Euclidean space terminology.

ee also

* Pseudo-Riemannian manifold

References

*cite book
last = Szekeres
first = Peter
title = A course in modern mathematical physics: groups, Hilbert space, and differential geometry
publisher = Cambridge University Press
date = 2004
pages =
isbn = 0521829607

*cite book
last = Novikov
first = S. P.
coauthors = Fomenko, A.T.; [translated from the Russian by M. Tsaplina]
title = Basic elements of differential geometry and topology
publisher = Dordrecht; Boston: Kluwer Academic Publishers
date = 1990
pages =
isbn = 0792310098