Pseudo-Euclidean space

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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 $x=(x_1, \dots, x_n),$ $n$ is the dimension of the space, and $1\le k < n.$

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.

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.

[edit] See also

Pseudo-Riemannian manifold

[edit] References

Szekeres, Peter (2004). A course in modern mathematical physics: groups, Hilbert space, and differential geometry. Cambridge University Press. ISBN 0521829607.

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.