Pseudo-Euclidean space
In
(e1, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, givingFor Euclidean spaces, k = n, implying that the quadratic form is positive-definite.[2] When 0 < k < n, q is an isotropic quadratic form, otherwise it is anisotropic. Note that if 1 ≤ i ≤ k < j ≤ n, then q(ei + ej) = 0, so that ei + ej is a null vector. In a pseudo-Euclidean space with k < n, unlike in a Euclidean space, there exist vectors with negative scalar square.
As with the term Euclidean space, the term pseudo-Euclidean space may be used to refer to an
Geometry
The geometry of a pseudo-Euclidean space is consistent despite some properties of Euclidean space not applying, most notably that it is not a
Positive, zero, and negative scalar squares
A null vector is a vector for which the quadratic form is zero. Unlike in a Euclidean space, such a vector can be non-zero, in which case it is self-orthogonal. If the quadratic form is indefinite, a pseudo-Euclidean space has a
The null cone separates two open sets,[4] respectively for which q(x) > 0 and q(x) < 0. If k ≥ 2, then the set of vectors for which q(x) > 0 is connected. If k = 1, then it consists of two disjoint parts, one with x1 > 0 and another with x1 < 0. Similarly, if n − k ≥ 2, then the set of vectors for which q(x) < 0 is connected. If n − k = 1, then it consists of two disjoint parts, one with xn > 0 and another with xn < 0.
Interval
The quadratic form q corresponds to the square of a vector in the Euclidean case. To define the
Hence terms norm and distance are avoided in pseudo-Euclidean geometry, which may be replaced with scalar square and interval respectively.
Though, for a curve whose tangent vectors all have scalar squares of the same sign, the arc length is defined. It has important applications: see proper time, for example.
Rotations and spheres
The rotations group of such space is the indefinite orthogonal group O(q), also denoted as O(k, n − k) without a reference to particular quadratic form.[5] Such "rotations" preserve the form q and, hence, the scalar square of each vector including whether it is positive, zero, or negative.
Whereas Euclidean space has a unit sphere, pseudo-Euclidean space has the hypersurfaces { x | q(x) = 1 } and { x | q(x) = −1 }. Such a hypersurface, called a quasi-sphere, is preserved by the appropriate indefinite orthogonal group.
Symmetric bilinear form
The quadratic form q gives rise to a symmetric bilinear form defined as follows:
The quadratic form can be expressed in terms of the bilinear form: q(x) = ⟨x, x⟩.
When ⟨x, y⟩ = 0, then x and y are orthogonal vectors of the pseudo-Euclidean space.
This bilinear form is often referred to as the
If x and y are orthogonal and q(x)q(y) < 0, then x is
The
Subspaces and orthogonality
For a (positive-dimensional) subspace
- q|U is either positive or negative definite. Then, U is essentially Euclidean (up to the sign of q).
- q|U is indefinite, but non-degenerate. Then, U is itself pseudo-Euclidean. It is possible only if hyperbolic plane.
- q|U is degenerate.
One of the most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their
The formal definition of the orthogonal complement of a vector subspace in a pseudo-Euclidean space gives a perfectly well-defined result, which satisfies the equality dim U + dim U⊥ = n due to the quadratic form's non-degeneracy. It is just the condition
- U ∩ U⊥ = {0} or, equivalently, U + U⊥ = all space,
which can be broken if the subspace U contains a null direction.
For a subspace N composed entirely of null vectors (which means that the scalar square q, restricted to N, equals to 0), always holds:
- N ⊂ N⊥ or, equivalently, N ∩ N⊥ = N.
Such a subspace can have up to min(k, n − k) dimensions.[8]
For a (positive) Euclidean k-subspace its orthogonal complement is a (n − k)-dimensional negative "Euclidean" subspace, and vice versa. Generally, for a (d+ + d− + d0)-dimensional subspace U consisting of d+ positive and d− negative dimensions (see Sylvester's law of inertia for clarification), its orthogonal "complement" U⊥ has (k − d+ − d0) positive and (n − k − d− − d0) negative dimensions, while the rest d0 ones are degenerate and form the U ∩ U⊥ intersection.
Parallelogram law and Pythagorean theorem
The parallelogram law takes the form
Using the square of the sum identity, for an arbitrary triangle one can express the scalar square of the third side from scalar squares of two sides and their bilinear form product:
This demonstrates that, for orthogonal vectors, a pseudo-Euclidean analog of the Pythagorean theorem holds:
Angle
Generally, absolute value |⟨x, y⟩| of the bilinear form on two vectors may be greater than √|q(x)q(y)|, equal to it, or less. This causes similar problems with definition of angle (see Dot product § Geometric definition) as appeared above for distances.
If k = 1 (only one positive term in q), then for vectors of positive scalar square:
which permits definition of the
It corresponds to the distance on a (n − 1)-dimensional
There is no reasonable definition of the angle between a null vector and another vector (either null or non-null).
Algebra and tensor calculus
Like Euclidean spaces, every pseudo-Euclidean vector space generates a Clifford algebra. Unlike properties above, where replacement of q to −q changed numbers but not geometry, the sign reversal of the quadratic form results in a distinct Clifford algebra, so for example Cl1,2(R) and Cl2,1(R) are not isomorphic.
Just like over any vector space, there are pseudo-Euclidean
and with the standard-form
the first k components of vα are numerically the same as ones of vβ, but the rest n − k have opposite signs.
The correspondence between contravariant and covariant tensors makes a tensor calculus on pseudo-Riemannian manifolds a generalization of one on Riemannian manifolds.
Examples
A very important pseudo-Euclidean space is Minkowski space, which is the mathematical setting in which the theory of special relativity is formulated. For Minkowski space, n = 4 and k = 3[10] so that
The geometry associated with this pseudo-metric was investigated by Poincaré.[11][12] Its rotation group is the Lorentz group. The Poincaré group includes also translations and plays the same role as Euclidean groups of ordinary Euclidean spaces.
Another pseudo-Euclidean space is the plane z = x + yj consisting of split-complex numbers, equipped with the quadratic form
This is the simplest case of an indefinite pseudo-Euclidean space (n = 2, k = 1) and the only one where the null cone dissects the remaining space into four open sets. The group SO+(1, 1) consists of so named
See also
- Pseudo-Riemannian manifold
- Hyperbolic equation
- Hyperboloid model
- Paravector
Footnotes
- ISBN 0-486-64070-1
- ^ Euclidean spaces are regarded as pseudo-Euclidean spaces – see for example Rafal Ablamowicz; P. Lounesto (2013), Clifford Algebras and Spinor Structures, Springer Science & Business Media, p. 32.
- ^ Rafal Ablamowicz; P. Lounesto (2013), Clifford Algebras and Spinor Structures, Springer Science & Business Media, p. 32 [1]
- standard topologyon Rn is assumed.
- ^ What is the "rotations group" depends on exact definition of a rotation. "O" groups contain improper rotations. Transforms that preserve orientation form the group SO(q), or SO(k, n − k), but it also is not connected if both k and n − k are positive. The group SO+(q), which preserves orientation on positive and negative scalar square parts separately, is a (connected) analog of Euclidean rotations group SO(n). Indeed, all these groups are Lie groups of dimension 1/2n(n − 1).
- ^ A linear subspace is assumed, but same conclusions are true for an affine flat with the only complication that the quadratic form is always defined on vectors, not points.
- ^ Actually, U ∩ U⊥ is not zero only if the quadratic form q restricted to U is degenerate.
- ISBN 0-387-97747-3
- cos(i arcosh s) = s, so for s > 0 these can be understood as imaginary angles.
- ^ Another well-established representation uses k = 1 and coordinate indices starting from 0 (thence q(x) = x02 − x12 − x22 − x32), but they are equivalent up to sign of q. See Sign convention § Metric signature.
- Rendiconti del Circolo Matematico di Palermo
- ISBN 0-387-96458-4
References
- MR 0631850
- Werner Greub (1963) Linear Algebra, 2nd edition, §12.4 Pseudo-Euclidean Spaces, pp. 237–49, Springer-Verlag.
- American Mathematical Monthly71:129–44.
- 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 0-7923-1009-8.)
{{cite book}}
: CS1 maint: multiple names: authors list (link - Szekeres, Peter (2004). A course in modern mathematical physics: groups, Hilbert space, and differential geometry. ISBN 0-521-82960-7.
- ISBN 978-3-642-30993-9.
External links
- D.D. Sokolov (originator), Pseudo-Euclidean space, Encyclopedia of Mathematics