Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate indefinite quadratic form q. Such a quadratic form can, given a suitable choice of basis (e₁, ..., e_n), be applied to a vector x = x₁e₁ + ... + x_ne_n, giving (with 1 ≤ k < n)

$q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_{k+1}^2+\cdots + x_n^2\right)$ which is called the magnitude of the vector x.

For true Euclidean spaces, k = n, implying that the quadratic form is positive-definite rather than indefinite. Otherwise q is an isotropic quadratic form. Note that if i ≤ k and j > k, then q(e_i + e_j) = 0, so that e_i + e_j is a null vector. In a pseudo-Euclidean space, unlike in a Euclidean space, there exist vectors with negative magnitude.

As with the term Euclidean space, pseudo-Euclidean space may refer to either an affine space or a vector space (see point–vector distinction) over real numbers.

Geometry

The geometry of a pseudo-Euclidean space is consistent in spite of a breakdown of the some properties of Euclidean space; most notably that it is not a metric space as explained below. Though, its affine structure provides that concepts of line, plane and, generally, of an affine subspace (flat), can be used without modifications, as well as line segments.

Positive, zero, and negative magnitudes

n = 3, k is either 1 or 2 depending on the choice of sign of q

A null vector is a vector whose magnitude is zero. Unlike in a Euclidean space, it can be non-zero, in which case it is perpendicular to itself. Every pseudo-Euclidean space has a linear cone of null vectors given by {x : q(x) = 0 } . When the pseudo-Euclidean space provides a model for spacetime (see below), the null cone is called the light cone of the origin.

The null cone separates two open sets^[1] of positive-magnitude and negative-magnitude vectors. If k > 1, then the set of positive-magnitude vectors is connected. If k = 1, which means the quadratic form has the only x₁² square term with positive sign, then it consists of two disjoint parts, one with x₁ > 0 and another with x₁ < 0. Similar statements can be made for negative-magnitude vectors if k is replaced with n − k.

Distance

The magnitude q corresponds to the square of a vector (or its norm) in Euclidean case. To define the vector norm (and distance) in an invariant manner, one has to get square roots of magnitudes, which leads to possibly imaginary distances; see square root of negative numbers. But even for a triangle with positive magnitudes of all three sides (whose square roots are real and positive), the triangle inequality is not necessarily true.

That's why terms norm and distance are avoided in pseudo-Euclidean geometry, replaced with magnitude and interval respectively.

Though, for a curve whose tangent vectors all have the same sign of magnitude, the arc length is defined. It has important applications: see proper time, for example.

Rotations and spheres

The rotations group of such space is indefinite orthogonal group O(q), also denoted as O(k, n − k) without a reference to particular quadratic form.^[2] Such "rotations" preserve the form q and, hence, the magnitude of each vector whether is it 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 hyperboloid or unit 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:

$\langle x, y\rangle = \frac {1}{2}[q (x + y) - q(x) - q(y)] = \left(x_1y_1+\cdots + x_ky_k\right)-\left(x_{k+1}y_{k+1}+\cdots + x_ny_n\right).$

The quadratic form can be expressed in terms of the bilinear form: $\langle x, x \rangle = q(x)$.

When $\langle x, y \rangle = 0$, then x and y are orthogonal elements of the pseudo-Euclidean space. Some authors use the terms "inner product" or "dot product" for the bilinear form, but it does not define an inner product space and its properties do not match to dot product of Euclidean vectors, although these terms are seldom used to refer to this bilinear form.

The standard basis of the real n-space is orthogonal. There are no orthonormal bases in a pseudo-Euclidean space because there is no vector norm.

Subspaces and orthogonality

For a (positive-dimensional) subspace^[3] U of a pseudo-Euclidean space, when the magnitude form q is restricted to U, following three cases are possible:

1. q|_U is either positively or negatively definite. Then, U is essentially Euclidean (up to sign of q).
2. q|_U is indefinite, but non-degenerate. Then, U is itself pseudo-Euclidean. It is possible only if dim U ≥ 2; if dim U = 2, which means than U is a plane, then it is called a hyperbolic plane.
3. q|_U is degenerate.

One of most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their orthogonality. When two non-zero Euclidean vectors are perpendicular, they are certainly not collinear. Any Euclidean linear subspace intersects with its orthogonal complement only by the {0} subspace. But the definition from the previous subsection immediately implies that any vector ν of zero magnitude is perpendicular to itself. Hence, for the 1-subspace N = ⟨ ν ⟩ generated by such non-zero vector, its orthogonal complement N^⊥ will be a superspace of N.

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 magnitude form's non-degeneracy. It is just the condition

U ∩ U^⊥ = {0} or, equivalently, U + U^⊥ =all space^[4]

which can be broken if the subspace U contains a null direction.^[5] While subspaces form a distributive lattice, as in any vector space, they do not form a Boolean algebra with this ⊥ operation, as in inner product spaces.

For a subspace N composed entirely of null vectors (which means that the magnitude q, restricted to N, equals to 0), always holds:

N ⊂ N^⊥ or, equivalently, N ∩ N^⊥ = N.

Such subspaces can have up to min(k, n − k) dimensions.^[6]

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

$q(x) + q(y) = \frac{1}{2}(q(x+y) + q(x-y)).$

Using the square of the sum identity, for an arbitrary triangle one can express the magnitude of the third side from magnitudes of two sides and their bilinear form product:

$q(x+y) = q(x) + q(y) + 2\langle x,y \rangle.$

This demonstrates that, for perpendicular vectors, a pseudo-Euclidean analog of the Pythagorean theorem holds:

$\langle x,y \rangle = 0 \Rightarrow q(x) + q(y) = q(x+y).$

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 positive-magnitude vectors:

$|\langle x, y\rangle| \ge \sqrt{q(x)q(y)}\,,$

which permits to define hyperbolic angle, an analog of angle between these vectors through inverse hyperbolic cosine:

$\operatorname{arcosh}\frac{|\langle x, y\rangle|}{\sqrt{q(x)q(y)}}\,.$ ^[7]

It corresponds to the distance on a (n − 1)-dimensional hyperbolic space. This is known as rapidity in the context of theory of relativity discussed below. Unlike Euclidean angle, it takes values from [0, +∞) and equals to 0 for antiparallel vectors.

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, a pseudo-Euclidean space possesses geometric algebra. Unlike properties above, where replacement of q to −q changed numbers but not geometry, the sign reversal of the magnitude form actually alters Cℓ, so for example Cℓ_1,2(R) and Cℓ_2,1(R) are not isomorphic.

Just like over any vector space, there are pseudo-Euclidean tensors. Like with a Euclidean structure, there are raising and lowering indices operators but, unlike the case with Euclidean tensors, there is no bases where these operations do not change values of components. If there is a vector v^β, the corresponding covariant vector is:

$v_\alpha = q_{\alpha\beta} v^\beta\,,$

and with the standard-form

$q_{\alpha\beta} = \begin{pmatrix}I_{k\times k} & 0 \\ 0 & -I_{(n-k)\times (n-k)}\end{pmatrix}$

the first k components of v_α are numerically the same as ones of v^β, but the rest n − k have opposite signs.

The correspondence between contravariant and covariant tensors makes a tensor calculus on pseudo-Riemannian manifolds analogous to one on Riemannian manifolds.

Examples

A very important pseudo-Euclidean space is Minkowski space, which is the mathematical setting in which Albert Einstein's theory of special relativity is conveniently formulated. For Minkowski space, n = 4 and k = 3^[8] so that

$q(x) = x_1^2+ x_2^2 + x_3^2-x_4^2,$

The geometry associated with this pseudo-metric was investigated by Poincaré. 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 + y j consisting of split-complex numbers, equipped with the quadratic form

$\lVert z \rVert = z z^* = z^* z = x^2 - y^2.$

This is the simplest case of a pseudo-Euclidean space (n = 2, k = 1) and the only one where the null cone dissects the space to four open sets. The group SO⁺(1, 1) consists of so named hyperbolic rotations.

See also

• Hyperbolic equation
• Hyperboloid model
• Paravector

Footnotes

1. The standard topology on Rⁿ is assumed.
2. What is the "rotations group" depends on exact definition of a rotation. "O" groups contain improper rotations. Transforms which preserves 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-magnitude parts separately, is a (connected) analog of Euclidean rotations group SO(n). Indeed, all these groups are Lie groups of dimension n(n − 1)/2 .
3. A linear subspace is assumed, but same conclusions are true for an affine flat with the only complication that the magnitude form is always defined on vectors, not points.
4. Violation of this equality makes the term "orthogonal complement" itself an oxymoron.
5. Actually, U ∩ U^⊥ is not zero if and only if the magnitude form q restricted to U is degenerate.
6. Thomas E. Cecil (1992) Lie Sphere Geometry, page 24, Universitext Springer ISBN 0-387-97747-3
7. Note that cos(i⋅arcosh s) = s, so for s > 0 these can be understood as imaginary angles.
8. 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.

References

• Werner Greub (1963) Linear Algebra, 2nd edition, §12.4 Pseudo-Euclidean Spaces, pp. 237–49, Springer-Verlag.
• Walter Noll (1964) "Euclidean geometry and Minkowskian chronometry", American Mathematical Monthly 71:129–44.
• 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.
• Poincaré, Science and Hypothesis 1906 referred to in the book B.A. Rosenfeld, A History of Non-Euclidean Geometry Springer 1988 (English translation) p. 266.
• Szekeres, Peter (2004). A course in modern mathematical physics: groups, Hilbert space, and differential geometry. Cambridge University Press. ISBN 0-521-82960-7.
• Shafarevich, I. R.; A. O. Remizov (2012). Linear Algebra and Geometry. Springer. ISBN 978-3-642-30993-9.

External links

• D.D. Sokolov (originator), Pseudo-Euclidean space, Encyclopedia of Mathematics