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

where x = (x₁, ..., x_n), n is the dimension of the space, and 1 ≤ k < n.

A very important pseudo-Euclidean space is Minkowski space, which is the mathematical setting in which Albert Einstein's theory of special relativity is most conveniently formulated. For Minkowski space, 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

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

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.

[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.