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

[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. 
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