where is the dimension of the space, and
A very important pseudo-Euclidean space is the Minkowski space, for which and For true Euclidean spaces one has 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 in the space is defined as 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 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.
See also
References
- Szekeres, Peter A course in modern mathematical physics: groups, Hilbert space, and differential geometry. Cambridge University Press.
- Novikov, S. P.; Fomenko, A.T.; [translated from the Russian by M. Tsaplina] Basic elements of differential geometry and topology. Dordrecht; Boston: Kluwer Academic Publishers.
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Last updated on Monday June 23, 2008 at 19:23:21 PDT (GMT -0700)
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where is the dimension of the space, and
A very important pseudo-Euclidean space is the Minkowski space, for which and For true Euclidean spaces one has 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 in the space is defined as 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 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.
See also
References
- Szekeres, Peter A course in modern mathematical physics: groups, Hilbert space, and differential geometry. Cambridge University Press.
- Novikov, S. P.; Fomenko, A.T.; [translated from the Russian by M. Tsaplina] Basic elements of differential geometry and topology. Dordrecht; Boston: Kluwer Academic Publishers.
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