Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real $n$ -space together with a non-degenerate quadratic form $q$ . Such a quadratic form can, given a suitable choice of basis $(e 1, \dots, e n)$ , be applied to a vector $x = x 1 e 1 + \dots + x n e n$ , giving [math]\displaystyle{ q(x) = \left(x_1^2 + \dots + x_k^2\right) - \left( x_{k+1}^2 + \dots + x_n^2\right) }[/math] which is called the scalar square of the vector $x$ .^[1]^:3

For 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 $1 \leq i \leq k < j \leq n$ , then $q (e i + e j) = 0$ , so that $e i + e j$ 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 affine space or a vector space depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space^[3] (see point–vector distinction).

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 metric space as explained below. The affine structure is unchanged, and thus also the concepts line, plane and, generally, of an affine subspace (flat), as well as line segments.

Positive, zero, and negative scalar squares

n = 3

,

k

is either 1 or 2 depending on the choice of sign of

q

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 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,^[4] respectively for which $q (x) > 0$ and $q (x) < 0$ . If $k \geq 2$ , then the set of vectors for which $q (x) > 0$ is connected. If $k = 1$ , then it consists of two disjoint parts, one with $x 1 > 0$ and another with $x 1 < 0$ . Similarly, if $n - k \geq 2$ , 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 $x n > 0$ and another with $x n < 0$ .

Interval

The quadratic form $q$ corresponds to the square of a vector in the Euclidean case. To define the vector norm (and distance) in an invariant manner, one has to get square roots of scalar squares, which leads to possibly imaginary distances; see square root of negative numbers. But even for a triangle with positive scalar squares of all three sides (whose square roots are real and positive), the triangle inequality does not hold in general.

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:

[math]\displaystyle{ \langle x, y\rangle = \tfrac12[q(x + y) - q(x) - q(y)] = \left(x_1 y_1 + \ldots + x_k y_k\right) - \left(x_{k+1}y_{k+1} + \ldots + x_n y_n\right). }[/math]

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 scalar product, and sometimes as "inner product" or "dot product", but it does not define an inner product space and it does not have the properties of the dot product of Euclidean vectors.

If $x$ and $y$ are orthogonal and $q (x) q (y) < 0$ , then $x$ is hyperbolic-orthogonal to $y$ .

The standard basis of the real $n$ -space is orthogonal. There are no orthonormal bases in a pseudo-Euclidean space for which the bilinear form is indefinite, because it cannot be used to define a vector norm.

Subspaces and orthogonality

For a (positive-dimensional) subspace^[6] $U$ of a pseudo-Euclidean space, when the quadratic form $q$ is restricted to $U$ , following three cases are possible:

$q | U$ 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 $dim U \geq 2$ ; if $dim U = 2$ , which means than $U$ is a plane, then it is called a hyperbolic plane.
$q | U$ is degenerate.

One of the most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their orthogonality. When two non-zero Euclidean vectors are orthogonal, they are not collinear. The intersections of any Euclidean linear subspace with its orthogonal complement is the ${0}$ subspace. But the definition from the previous subsection immediately implies that any vector $ν$ of zero scalar square is orthogonal to itself. Hence, the isotropic line $N = ⟨ ν ⟩$ generated by a null vector ν is a subset of its orthogonal complement $N ⊥$ .

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

U \cap U ⊥ = {0}

or, equivalently,

U + U ⊥ =

all space,

which can be broken if the subspace $U$ contains a null direction.^[7] While subspaces form a lattice, as in any vector space, this $⊥$ operation is not an orthocomplementation, in contrast to inner product spaces.

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 \subset N ⊥

or, equivalently,

N \cap 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 $(d + + d - + d 0)$ -dimensional subspace $U$ consisting of $d +$ positive and $d -$ negative dimensions (see Sylvester's law of inertia for clarification), its orthogonal "complement" $U ⊥$ has $(k - d + - d 0)$ positive and $(n - k - d - - d 0)$ negative dimensions, while the rest $d 0$ ones are degenerate and form the $U \cap U ⊥$ intersection.

Parallelogram law and Pythagorean theorem

The parallelogram law takes the form

[math]\displaystyle{ q(x) + q(y) = \tfrac12(q(x + y) + q(x - y)). }[/math]

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:

[math]\displaystyle{ q(x + y) = q(x) + q(y) + 2\langle x, y \rangle. }[/math]

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

[math]\displaystyle{ \langle x, y \rangle = 0 \Rightarrow q(x) + q(y) = q(x + y). }[/math]

Angle

Generally, absolute value $| ⟨ x, y ⟩ |$ of the bilinear form on two vectors may be greater than $\sqrt | q (x) q (y) |$ , 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: [math]\displaystyle{ |\langle x, y\rangle| \ge \sqrt{q(x)q(y)}\,, }[/math]

which permits definition of the hyperbolic angle, an analog of angle between these vectors through inverse hyperbolic cosine:^[9] [math]\displaystyle{ \operatorname{arcosh}\frac{|\langle x, y\rangle|}{\sqrt{q(x)q(y)}}\,. }[/math]

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 $[0, +\infty)$ 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, 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 $Cl 1,2 (R)$ 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:

[math]\displaystyle{ v_\alpha = q_{\alpha\beta} v^\beta\,, }[/math]

and with the standard-form

[math]\displaystyle{ q_{\alpha\beta} = \begin{pmatrix} I_{k\times k} & 0 \\ 0 & -I_{(n-k)\times(n-k)} \end{pmatrix} }[/math]

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 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

[math]\displaystyle{ q(x) = x_1^2 + x_2^2 + x_3^2 - x_4^2, }[/math]

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

[math]\displaystyle{ \lVert z \rVert = z z^* = z^* z = x^2 - y^2. }[/math]

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 hyperbolic rotations.

Footnotes

↑ Élie Cartan (1981), The Theory of Spinors, Dover Publications, 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]
↑ The standard topology on $R n$ 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 $1 / 2 n (n - 1)$ .
↑ 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, $U \cap U ⊥$ is not zero only if the quadratic form $q$ restricted to $U$ is degenerate.
↑ Thomas E. Cecil (1992) Lie Sphere Geometry, page 24, Universitext Springer ISBN 0-387-97747-3
↑ Note that $cos (i arcosh s) = s$ , 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 $q (x) = x 02 - x 12 - x 22 - x 32$ ), but they are equivalent up to sign of $q$ . See Sign convention § Metric signature.
↑ H. Poincaré (1906) On the Dynamics of the Electron, Rendiconti del Circolo Matematico di Palermo
↑ B. A. Rosenfeld (1988) A History of Non-Euclidean Geometry, page 266, Studies in the history of mathematics and the physical sciences #12, Springer ISBN:0-387-96458-4

References

Cartan, Élie (1981), The Theory of Spinors, New York: Dover Publications, p. 3, ISBN 978-0-486-64070-9, https://books.google.com/books?isbn=0486640701
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.
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. https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9.

External links

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

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[1] Élie Cartan (1981), The Theory of Spinors, Dover Publications, ISBN 0-486-64070-1

[2] 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 .

[3] Rafal Ablamowicz; P. Lounesto (2013), Clifford Algebras and Spinor Structures, Springer Science & Business Media, p. 32 [1]

[4] The standard topology on $R n$ is assumed.

[5] 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 $1 / 2 n (n - 1)$ .

[6] 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.

[7] Actually, $U \cap U ⊥$ is not zero only if the quadratic form $q$ restricted to $U$ is degenerate.

[8] Thomas E. Cecil (1992) Lie Sphere Geometry, page 24, Universitext Springer ISBN 0-387-97747-3

[9] Note that $cos (i arcosh s) = s$ , so for $s > 0$ these can be understood as imaginary angles.

[10] Another well-established representation uses $k = 1$ and coordinate indices starting from $0$ (thence $q (x) = x 02 - x 12 - x 22 - x 32$ ), but they are equivalent up to sign of $q$ . See Sign convention § Metric signature.

[11] H. Poincaré (1906) On the Dynamics of the Electron, Rendiconti del Circolo Matematico di Palermo

[12] B. A. Rosenfeld (1988) A History of Non-Euclidean Geometry, page 266, Studies in the history of mathematics and the physical sciences #12, Springer ISBN:0-387-96458-4

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