View source for Pseudo-Euclidean space

In [[mathematics]] and [[theoretical physics]], a '''pseudo-Euclidean space''' of signature {{math|(''k'', ''n-k'')}} is a finite-[[dimension (mathematics)|dimensional]] [[real coordinate space|real {{math|''n''}}-space]] together with a non-[[degenerate form|degenerate]] [[quadratic form]] {{math|''q''}}. Such a quadratic form can, given a suitable choice of [[Basis (linear algebra)|basis]] {{math|(''e''<sub>1</sub>, …, ''e''<sub>''n''</sub>)}}, be applied to a vector {{math|1=''x'' = ''x''<sub>1</sub>''e''<sub>1</sub> + ⋯ + ''x''<sub>''n''</sub>''e''<sub>''n''</sub>}}, giving
<math display="block">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 {{math|''x''}}.<ref>{{citation |author=Élie Cartan |year=1981 |title= The Theory of Spinors |publisher=[[Dover Publications]] |isbn=0-486-64070-1 }}</ref>{{rp|3}}

For [[Euclidean space]]s, {{math|1=''k'' = ''n''}}, implying that the quadratic form is [[definite quadratic form|positive-definite]].<ref>Euclidean spaces are regarded as pseudo-Euclidean spaces – see for example {{citation|author1=Rafal Ablamowicz|author2=P. Lounesto|year=2013|title=Clifford Algebras and Spinor Structures|publisher=[[Springer Science+Business Media|Springer Science & Business Media]]|page=32}}.</ref> When {{math|0 < ''k'' < ''n''}}, then {{math|''q''}} is an [[isotropic quadratic form]]. Note that if {{math|1 ≤ ''i'' ≤ ''k'' < ''j'' ≤ ''n''}}, then {{math|1=''q''(''e''<sub>''i''</sub> + ''e''<sub>''j''</sub>) = 0}}, so that {{math|1=''e''<sub>''i''</sub> + ''e''<sub>''j''</sub>}} is a [[null vector]]. In a pseudo-Euclidean space with {{math|''k'' < ''n''}}, unlike in a Euclidean space, there exist vectors with [[negative number|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'''<ref>{{citation |author1=Rafal Ablamowicz| author2=P. Lounesto| year=2013| title=Clifford Algebras and Spinor Structures| publisher=[[Springer Science+Business Media|Springer Science & Business Media]]| page=32}} [https://books.google.com/books?id=pXvwCAAAQBAJ&pg=PA32]</ref> (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 geometry|affine structure]] is unchanged, and thus also the concepts [[line (geometry)|line]], [[plane (geometry)|plane]] and, generally, of an [[affine subspace]] ([[flat (geometry)|flat]]), as well as [[line segment]]s.

=== Positive, zero, and negative scalar squares ===
[[Image:DoubleCone.png|thumb|right|{{math|1=''n'' = 3}}, {{math|''k''}} is either 1 or 2 depending on the choice of [[sign (mathematics)|sign]] of {{math|''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-[[#Orthogonality|orthogonal]].
If the quadratic form is indefinite, a pseudo-Euclidean space has a [[linear cone]] of null vectors given by {{math|{{mset|{{hsp}}''x'' {{!}} ''q''(''x'') {{=}} 0{{hsp}}}}}}. When the pseudo-Euclidean space provides a model for [[spacetime]] (see [[#Examples|below]]), the null cone is called the [[light cone]] of the origin.

The null cone separates two [[open set]]s,<ref>The [[standard topology]] on {{math|'''R'''<sup>''n''</sup>}} is assumed.</ref> respectively for which {{math|''q''(''x'') > 0}} and {{math|''q''(''x'') < 0}}. If {{math|''k'' ≥ 2}}, then the set of vectors for which {{math|''q''(''x'') > 0}} is [[connected space|connected]]. If {{math|1=''k'' = 1}}, then it consists of two disjoint parts, one with {{math|''x''<sub>1</sub> > 0}} and another with {{math|''x''<sub>1</sub> < 0}}. Similarly, if {{math|''n'' − ''k'' ≥ 2}}, then the set of vectors for which {{math|''q''(''x'') < 0}} is connected. If {{math|1=''n'' − ''k'' = 1}}, then it consists of two disjoint parts, one with {{math|''x''<sub>''n''</sub> > 0}} and another with {{math|''x''<sub>''n''</sub> < 0}}.

=== Interval ===
The quadratic form {{math|''q''}} corresponds to the square of a vector in the Euclidean case. To define the [[vector norm]] (and distance) in an [[invariant (mathematics)|invariant]] manner, one has to get [[square root]]s of scalar squares, which leads to possibly [[imaginary number|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 vector]]s 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 ===
[[Image:Hyperboloid1.png|thumb|right]]
The [[rotation (mathematics)|rotations]] [[group (mathematics)|group]] of such space is the [[indefinite orthogonal group]] {{math|O(''q'')}}, also denoted as {{math|O(''k'', ''n'' − ''k'')}} without a reference to particular quadratic form.<ref>What is the "rotations group" depends on exact definition of a rotation. "O" groups contain [[improper rotation]]s. Transforms that preserve [[orientation (geometry)|orientation]] form the group {{math|SO(''q'')}}, or {{math|SO(''k'', ''n'' − ''k'')}}, but it also is not [[connected space|connected]] if both {{math|''k''}} and {{math|''n'' − ''k''}} are positive. The group {{math|SO<sup>+</sup>(''q'')}}, which preserves orientation on positive and negative scalar square parts separately, is a (connected) analog of Euclidean rotations group {{math|SO(''n'')}}. Indeed, all these groups are [[Lie group]]s of dimension {{math|{{sfrac|1|2}}''n''(''n'' − 1)}}.</ref> Such "rotations" preserve the form {{math|''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 [[hypersurface]]s {{math|{{mset|{{hsp}}''x'' {{!}} ''q''(''x'') {{=}} 1{{hsp}}}}}} and {{math|{{mset|{{hsp}}''x'' {{!}} ''q''(''x'') {{=}} −1{{hsp}}}}}}. Such a hypersurface, called a [[quasi-sphere]], is preserved by the appropriate indefinite orthogonal group.

=== Symmetric bilinear form ===
The quadratic form {{math|''q''}} gives rise to a [[symmetric bilinear form]] defined as follows:
: <math display=block>\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: {{math|1=''q''(''x'') = {{langle}}''x'', ''x''{{rangle}}}}.

When {{math|1={{langle}}''x'', ''y''{{rangle}} = 0}}, then {{math|''x''}} and {{math|''y''}} are [[orthogonality|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 {{math|''x''}} and {{math|''y''}} are orthogonal and {{math|''q''(''x'')''q''(''y'') < 0}}, then {{math|''x''}} is [[hyperbolic-orthogonal]] to {{math|''y''}}.

{{anchor|basis}}The [[standard basis]] of the real {{math|''n''}}-space is [[orthogonal basis|orthogonal]]. There are no ortho''normal'' 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<ref>A [[linear subspace]] is assumed, but same conclusions are true for an affine [[flat (geometry)|flat]] with the only complication that the quadratic form is always defined on vectors, not points.</ref> {{math|''U''}} of a pseudo-Euclidean space, when the quadratic form {{math|''q''}} is [[Function restriction|restricted]] to {{math|''U''}}, following three cases are possible:
# {{math|''q''{{!}}<sub>''U''</sub>}} is either [[definite quadratic form|positive or negative definite]]. Then, {{math|''U''}} is essentially [[Euclidean space|Euclidean]] (up to the sign of {{math|''q''}}).
# {{math|''q''{{!}}<sub>''U''</sub>}} is indefinite, but non-degenerate. Then, {{math|''U''}} is itself pseudo-Euclidean. It is possible only if {{math|[[dimension (vector space)|dim]] ''U'' ≥ 2}}; if {{math|1=dim ''U'' = 2}}, which means than {{math|''U''}} is a [[plane (geometry)|plane]], then it is called a [[hyperbolic plane (quadratic forms)|hyperbolic plane]].
# {{math|''q''{{!}}<sub>''U''</sub>}} is degenerate.

{{anchor|Orthogonality}}
One of the most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their [[orthogonality]]. When two non-zero [[Euclidean vector]]s are orthogonal, they are not [[collinear]]. The intersections of any Euclidean [[linear subspace]] with its [[orthogonal complement]] is the [[zero vector space|{{math|{0}{{null}}}} subspace]]. But the definition from the previous subsection immediately implies that any vector {{math|'''ν'''}} of zero scalar square is orthogonal to itself. Hence, the [[isotropic line]] {{math|1=''N'' = [[linear span|{{langle}}'''ν'''{{rangle}}]]}} generated by a [[null vector]] '''ν''' is a subset of its orthogonal complement {{math|''N''<sup>⊥</sup>}}.

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 {{math|1=dim ''U'' + dim ''U''<sup>⊥</sup> = ''n''}} due to the quadratic form's non-degeneracy. It is just the condition
: {{math|1=''U'' ∩ ''U''<sup>⊥</sup> = {{mset|0}}}} or, equivalently, {{math|1=''U'' + ''U''<sup>⊥</sup> =}} all space,
which can be broken if the subspace {{math|''U''}} contains a null direction.<ref>Actually, {{math|''U'' ∩ ''U''<sup>⊥</sup>}} is not zero only if the quadratic form {{math|''q''}} restricted to {{math|''U''}} is degenerate.<!-- can somebody find a source with this simple consequence of the law of inertia? --></ref> While subspaces [[linear subspace#Lattice of subspaces|form a lattice]], as in any vector space, this {{math|⊥}} operation is not an [[orthocomplementation]], in contrast to [[inner product space]]s.

For a subspace {{math|''N''}} composed ''entirely'' of null vectors (which means that the scalar square {{math|''q''}}, restricted to {{math|''N''}}, equals to {{math|0}}), always holds:
: {{math|1=''N'' ⊂ ''N''<sup>⊥</sup>}} or, equivalently, {{math|1=''N'' ∩ ''N''<sup>⊥</sup> = ''N''}}.

Such a subspace can have up to {{math|min(''k'', ''n'' − ''k'')}} [[dimension (vector space)|dimension]]s.<ref>Thomas E. Cecil (1992) ''Lie Sphere Geometry'', page 24, Universitext Springer {{isbn|0-387-97747-3}}</ref>

For a (positive) Euclidean {{math|''k''}}-subspace its orthogonal complement is a {{math|(''n'' − ''k'')}}-dimensional negative "Euclidean" subspace, and vice versa.
Generally, for a {{math|(''d''<sub>+</sub> + ''d''<sub>−</sub> + ''d''<sub>0</sub>)}}-dimensional subspace {{math|''U''}} consisting of {{math|''d''<sub>+</sub>}} positive and {{math|''d''<sub>−</sub>}} negative dimensions (see [[Sylvester's law of inertia]] for clarification), its orthogonal "complement" {{math|''U''<sup>⊥</sup>}} has {{math|(''k'' − ''d''<sub>+</sub> − ''d''<sub>0</sub>)}} positive and {{math|(''n'' − ''k'' − ''d''<sub>−</sub> − ''d''<sub>0</sub>)}} negative dimensions,<!-- it is an easily accessible fact which follows from dim(''U'' ∩ ''U''<sup>⊥</sup>) = ''d''<sub>0</sub>, which itself is a direct consequence of the definition of ''U''<sup>⊥</sup> and Sylvester's law, but it would be better to find a source. --> while the rest {{math|''d''<sub>0</sub>}} ones are degenerate and form the {{math|''U'' ∩ ''U''<sup>⊥</sup>}} intersection.

=== Parallelogram law and Pythagorean theorem ===
The [[parallelogram law]] takes the form
: <math>q(x) + q(y) = \tfrac12(q(x + y) + q(x - y)).</math>

Using the [[binomial theorem#Examples|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>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>\langle x, y \rangle = 0 \Rightarrow q(x) + q(y) = q(x + y).</math>

=== Angle ===
[[Image:Minkowski lightcone lorentztransform.svg|thumb|right]]
Generally, absolute value {{math|{{abs|⟨''x'', ''y''⟩}}}} of the bilinear form on two vectors may be greater than {{math|{{sqrt|{{abs|''q''(''x'')''q''(''y'')}}}}}}, equal to it, or less. This causes similar problems with definition of [[angle]] (see {{section link|Dot product|Geometric definition}}) as [[#Distance|appeared above]] for distances.

If {{math|1=''k'' = 1}} (only one positive term in {{math|''q''}}), then for vectors of positive scalar square:
<math display="block">|\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 function|inverse hyperbolic cosine]]:<ref>Note that {{math|1=[[cosine|cos]](''i'' arcosh ''s'') = ''s''}}, so for {{math|''s'' > 0}} these can be understood as imaginary angles.</ref>
<math display="block">\operatorname{arcosh}\frac{|\langle x, y\rangle|}{\sqrt{q(x)q(y)}}\,.</math>

It corresponds to the distance on a {{math|(''n'' − 1)}}-dimensional [[hyperbolic space]]. This is known as [[rapidity]] in the context of theory of relativity discussed [[#Examples|below]]. Unlike Euclidean angle, it takes values from {{closed-open|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, every pseudo-Euclidean vector space generates a [[Clifford algebra]]. Unlike properties above, where replacement of {{math|''q''}} to {{math|−''q''}} changed numbers but not [[geometry]], the sign reversal of the quadratic form results in a distinct Clifford algebra, so for example {{math|Cl<sub>1,2</sub>('''R''')}} and {{math|Cl<sub>2,1</sub>('''R''')}} are not isomorphic.

Just like over any vector space, there are pseudo-Euclidean [[tensor]]s. Like with a Euclidean structure, there are [[raising and lowering indices]] operators but, unlike the case with [[Euclidean tensor]]s, there is [[#basis|no bases where these operations do not change values of components]]. If there is a vector {{math|''v''{{i sup|''β''}}}}, the corresponding [[covariant vector]] is:
: <math>v_\alpha = q_{\alpha\beta} v^\beta\,,</math><!-- here "q" is the bilinear form; possibly another letter should be used -->
and with the standard-form
: <math>q_{\alpha\beta} = \begin{pmatrix}
  I_{k\times k} &                     0 \\
              0 & -I_{(n-k)\times(n-k)}
\end{pmatrix}</math>
the first {{math|''k''}} components of {{math|''v''<sub>''α''</sub>}} are numerically the same as ones of {{math|''v''{{i sup|''β''}}}}, but the rest {{math|''n'' − ''k''}} have [[additive inverse|opposite signs]].

The correspondence between contravariant and covariant tensors makes a [[tensor calculus]] on [[pseudo-Riemannian manifold]]s 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, {{math|1=''n'' = 4}} and {{math|1=''k'' = 3}}<ref>Another well-established representation uses {{math|1=''k'' = 1}} and coordinate indices starting from {{math|0}} (thence {{math|1=''q''(''x'') = ''x''<sub>0</sub><sup>2</sup> − ''x''<sub>1</sub><sup>2</sup> − ''x''<sub>2</sub><sup>2</sup> − ''x''<sub>3</sub><sup>2</sup>}}), but they are equivalent [[additive inverse|up to sign]] of {{math|''q''}}. See {{section link|Sign convention|Metric signature}}.</ref> so that
: <math>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 [[Henri Poincaré|Poincaré]].<ref>H. Poincaré (1906) [[s:Translation:On the Dynamics of the Electron (July)|On the Dynamics of the Electron]], [[Rendiconti del Circolo Matematico di Palermo]]</ref><ref>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}}</ref> Its rotation group is the [[Lorentz group]]. The [[Poincaré group]] includes also [[translation (geometry)|translations]] and plays the same role as [[Euclidean group]]s of ordinary Euclidean spaces.

Another pseudo-Euclidean space is the [[Plane (mathematics)|plane]] {{math|1=''z'' = ''x'' + ''yj''}} consisting of [[split-complex number]]s, equipped with the quadratic form
: <math>\lVert z \rVert = z z^* = z^* z = x^2 - y^2.</math>

This is the simplest case of an indefinite pseudo-Euclidean space ({{math|1=''n'' = 2}}, {{math|1=''k'' = 1}}) and the only one where the null cone dissects the remaining space into ''four'' open sets. The group {{math|SO<sup>+</sup>(1, 1)}} consists of so named [[hyperbolic rotation]]s.

== See also ==
* [[Pseudo-Riemannian manifold]]
* [[Hyperbolic equation]]
* [[Hyperboloid model]]
* [[Paravector]]

== Footnotes ==
{{reflist}}

== References ==
* {{citation | last1=Cartan | first1=Élie | author-link=Élie Cartan|title=The Theory of Spinors |page=3| orig-year=1938 | url=https://books.google.com/books?isbn=0486640701 | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-64070-9 | mr=631850 | year=1981}}
* Werner Greub (1963) ''Linear Algebra'', 2nd edition, §12.4 Pseudo-Euclidean Spaces, pp.&nbsp;237–49, Springer-Verlag.
* [[Walter Noll]] (1964) "Euclidean geometry and Minkowskian chronometry", [[American Mathematical Monthly]] 71:129–44.
* {{cite book
 | last1      = Novikov
 | first1     = S. P.
 | last2      = Fomenko
 | first2     = A.T.; [translated from the Russian by M. Tsaplina]
 | title      = Basic elements of differential geometry and topology
 | publisher  = Dordrecht; Boston: Kluwer Academic Publishers
 | year       = 1990
 | isbn       = 0-7923-1009-8
}}
* {{cite book
 | last       = Szekeres
 | first      = Peter
 | title      = A course in modern mathematical physics: groups, Hilbert space, and differential geometry
 | publisher  = [[Cambridge University Press]]
 | year       = 2004
 | isbn       = 0-521-82960-7
}}
* {{cite book
 | last = Shafarevich
 | first = I. R. 
 | author-link = Igor Shafarevich
 | author2 = A. O. Remizov
 | title = Linear Algebra and Geometry
 | publisher = [[Springer Science+Business Media|Springer]]
 | year = 2012
 | url = https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9
 | isbn = 978-3-642-30993-9
}}

== External links ==
* D.D. Sokolov (originator), [http://www.encyclopediaofmath.org/index.php/Pseudo-Euclidean_space Pseudo-Euclidean space], [[Encyclopedia of Mathematics]]

[[Category:Lorentzian manifolds]]