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| In [[mathematics]] and [[theoretical physics]], a '''pseudo-Euclidean space''' is a finite-[[dimension (mathematics)|dimensional]] [[real coordinate space|real {{mvar|n}}-space]] together with a non-[[degenerate form|degenerate]] [[definite bilinear form|indefinite]] [[quadratic form]], called the '''magnitude''' of a [[vector space|vector]]. Such a quadratic form can, after a suitable change of coordinates, be written as
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| : <math>q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_{k+1}^2+\cdots + x_n^2\right)</math> | |
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| where {{math|1=''x'' = (''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>)}}, {{mvar|n}} is the dimension of the space, and {{math|1=1 ≤ ''k'' < ''n''}}. For true [[Euclidean space]]s, {{math|1=''k'' = ''n''}}, implying that the quadratic form is positive-definite rather than indefinite. Otherwise {{mvar|q}} is an [[isotropic quadratic form]]. In a pseudo-Euclidean space, unlike in a Euclidean space, there exist non-[[additive identity|zero]] vectors with [[zero]] magnitude, and also vectors with [[negative number|negative]] magnitude.
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| 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 number]]s.
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| == Geometry ==
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| 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 geometry|affine structure]] provides that concepts of [[line (geometry)|line]], [[plane (geometry)|plane]] and, generally, of an [[affine subspace]] ([[flat (geometry)|flat]]), can be used without modifications, as well as [[line segment]]s.
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| ===Positive, zero, and negative magnitudes===
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| [[Image:DoubleCone.png|thumb|right|{{math|1=''n'' = 3}}, {{mvar|k}} is either 1 or 2 depending on the choice of [[sign (mathematics)|sign]] of {{mvar|q}}]]
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| 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 [[#Orthogonality|perpendicular to itself]].
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| Every pseudo-Euclidean space has a [[linear cone]] of null vectors given by {{math|1={''x'' : ''q''(''x'') = 0 } }}. When the pseudo-Euclidean space provides a model for [[spacetime]] (see [[#Examples|below]]), the null cone is called the [[light cone]] of the origin.
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| The null cone separates two [[open set]]s<ref>The [[standard topology]] on {{math|'''R'''<sup>''n''</sup>}} is assumed.</ref> of positive-magnitude and negative-magnitude vectors. If {{math|1=''k'' > 1}}, then the set of positive-magnitude vectors is [[connected space|connected]]. If {{math|1=''k'' = 1}}, which means the quadratic form has the only {{math|''x''<sub>1</sub><sup>2</sup>}} square term with positive sign, then it consists of two disjoint parts, one with {{math|''x''<sub>1</sub> > 0}} and another with {{math|''x''<sub>1</sub> < 0}}. Similar statements can be made for negative-magnitude vectors if {{mvar|k}} is replaced with {{math|''n'' − ''k''}}.
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| ===Distance===
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| The magnitude {{mvar|q}} corresponds to the [[square (algebra)|square]] of a vector (or its [[normed vector space|norm]]) in Euclidean case. To define the [[vector norm]] (and distance) in an [[invariant (mathematics)|invariant]] manner, one has to get [[square root]]s of magnitudes, which leads to possibly [[imaginary number|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.
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| That's why terms ''norm'' and ''distance'' are avoided in pseudo-Euclidean geometry, replaced with ''magnitude'' and ''interval'' respectively.
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| Though, for a [[curve]] whose [[tangent vector]]s all have the same sign of magnitude, the [[arc length]] is defined. It has important applications: see [[proper time]], for example.
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| ===Rotations and spheres===
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| [[Image:Hyperboloid1.png|thumb|right]]
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| The [[rotation (mathematics)|rotations]] [[group (mathematics)|group]] of such space is [[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 which preserves [[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 {{mvar|k}} and {{math|''n'' − ''k''}} are positive. The group {{math|SO<sup>+</sup>(''q'')}}, which preserves orientation on positive- and negative-magnitude parts separately, is a (connected) analog of Euclidean rotations group {{math|SO(''n'')}}. Indeed, all these groups are [[Lie group]]s of dimension {{math|''n''(''n'' − 1)/2 }}.</ref> Such "rotations" preserve the form {{mvar|q}} and, hence, the magnitude of each vector whether is it positive, zero, or negative.
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| Whereas Euclidean space has a [[unit sphere]], pseudo-Euclidean space has the [[hypersurface]]s {{math|1={''x'' : ''q''(''x'') = 1 } }} and {{math|1={''x'' : ''q''(''x'') = −1 } }}. Such a hypersurface called a [[hyperboloid]] or [[Hyperboloid#Relation to the sphere|unit quasi-sphere]] is preserved by the appropriate indefinite orthogonal group.
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| ===Symmetric bilinear form===
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| The quadratic form {{mvar|q}} gives rise to a [[symmetric bilinear form]] defined as follows:
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| : <math>\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).</math>
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| The quadratic form can be expressed in terms of the bilinear form: <math> \langle x, x \rangle = q(x)</math>.
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| When <math>\langle x, y \rangle = 0 </math>, then {{mvar|x}} and {{mvar|y}} are [[orthogonality|orthogonal]] elements of the pseudo-Euclidean space.
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| 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.
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| {{anchor|basis}}The [[standard basis]] of the real {{mvar|n}}-space is [[orthogonal basis|orthogonal]]. There are no ortho''normal'' bases in a pseudo-Euclidean space because there is no [[vector norm]].
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| ===Subspaces and orthogonality===
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| {{unreferenced section|date=June 2013}}
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| 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 magnitude form is always defined on vectors, not points.</ref> {{mvar|U}} of a pseudo-Euclidean space, when the magnitude form {{mvar|q}} is [[Function restriction|restricted]] to {{mvar|U}}, following three cases are possible:
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| # {{math|''q''{{!}}<sub>''U''</sub>}} is either [[definite form|positively or negatively definite]]. Then, {{mvar|U}} is essentially [[Euclidean space|Euclidean]] (up to sign of {{mvar|q}}).
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| # {{math|''q''{{!}}<sub>''U''</sub>}} is indefinite, but non-degenerate. Then, {{mvar|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 {{mvar|U}} is a [[plane (geometry)|plane]], then it is called a [[hyperbolic plane (quadratic forms)|hyperbolic plane]].
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| # {{math|''q''{{!}}<sub>''U''</sub>}} is degenerate.
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| {{anchor|Orthogonality}}
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| One of most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their [[orthogonality]]. When two non-zero [[Euclidean vector]]s are perpendicular, they are certainly not [[collinear]]. Any Euclidean [[linear subspace]] intersects with its [[orthogonal complement]] only by the [[zero vector space|{0} subspace]]. But the definition from the previous subsection immediately implies that any vector {{math|'''ν'''}} of zero magnitude is perpendicular to itself. Hence, for the 1-subspace {{math|1=''N'' = [[linear span|{{langle}} '''ν''' {{rangle}}]]}} generated by such non-[[additive identity|zero]] vector, its orthogonal complement {{math|''N''<sup>⊥</sup>}} will be a [[superset|superspace]] of {{mvar|N}}.
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| 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 magnitude form's non-degeneracy. It is just the condition
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| : {{math|1=''U'' ∩ ''U''<sup>⊥</sup> = {0}}} or, equivalently, {{math|1=''U'' + ''U''<sup>⊥</sup> = }}all space<ref>Violation of this equality makes the term "orthogonal [[complementary subspaces|complement]]" itself an [[oxymoron]].</ref>
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| which can be broken if the subspace {{mvar|U}} contains a null direction.<ref>Actually, {{math|''U'' ∩ ''U''<sup>⊥</sup>}} is not zero if and only if the magnitude form {{mvar|q}} restricted to {{mvar|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 distributive lattice]], as in any vector space, they do not form a [[Boolean algebra (structure)|Boolean algebra]] with this ⊥ operation, as in [[inner product space]]s.
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| For a subspace {{mvar|N}} composed ''entirely'' of null vectors (which means that the magnitude {{mvar|q}}, restricted to {{mvar|N}}, equals to 0), always holds:
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| : {{math|1=''N'' ⊂ ''N''<sup>⊥</sup>}} or, equivalently, {{math|1=''N'' ∩ ''N''<sup>⊥</sup> = ''N''}}. | |
| Such subspaces can have up to {{math|min(''k'', ''n'' − ''k'')}} [[dimension (vector space)|dimensions]].
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| For a (positive) Euclidean {{mvar|k}}-subspace its orthogonal complement is a {{math|(''n'' − ''k'')}}-dimensional negative "Euclidean" subspace, and vice versa.
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| Generally, for a {{math|(''d''<sub>+</sub> + ''d''<sub>−</sub> + ''d''<sub>0</sub>)}}-dimensional subspace {{mvar|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.
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| ===Parallelogram law and Pythagorean theorem=== | |
| The [[parallelogram law]] takes the form
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| :<math>q(x) + q(y) = \frac{1}{2}(q(x+y) + q(x-y)).</math>
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| Using the [[binomial theorem#Examples|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:
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| :<math>q(x+y) = q(x) + q(y) + 2\langle x,y \rangle.</math>
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| This demonstrates that, for perpendicular vectors, a pseudo-Euclidean analog of the [[Pythagorean theorem]] holds:
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| :<math>\langle x,y \rangle = 0 \Rightarrow q(x) + q(y) = q(x+y).</math>
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| ===Angle===
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| [[Image:Minkowski lightcone lorentztransform.svg|thumb|right]]
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| Generally, absolute value {{math|{{!}}{{langle}}''x'', ''y''{{rangle}}{{!}}}} of the bilinear form on two vectors may be greater than {{math|{{sqrt|{{!}} ''q''(''x'') ''q''(''y'') {{!}}}}}}, equal to it, or less. This causes similar problems with definition of [[angle]] (see [[dot product#Geometric definition]]) as [[#Distance|appeared above]] for distances.
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| If {{math|1=''k'' = 1}} (only one positive term in {{mvar|q}}), then for positive-magnitude vectors:
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| : <math>|\langle x, y\rangle| \ge \sqrt{q(x)q(y)}\,,</math>
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| which permits to define [[hyperbolic angle]], an analog of angle between these vectors through [[inverse hyperbolic function|inverse hyperbolic cosine]]:
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| : <math>\operatorname{arcosh}\frac{|\langle x, y\rangle|}{\sqrt{q(x)q(y)}}\,.</math> <ref>Note that {{math|1=[[cosine|cos]](''i''⋅arcosh ''s'') = ''s''}}, so for {{math|''s'' > 0}} these can be understood as imaginary angles.</ref>
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| 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 (mathematics)|antiparallel]] vectors.
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| There is no reasonable definition of the angle between a null vector and another vector (either null or non-null).
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| == Algebra and tensor calculus ==
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| Like Euclidean spaces, a pseudo-Euclidean space possesses [[geometric algebra]]. Unlike properties above, where replacement of {{mvar|q}} to {{math|−''q''}} changed numbers but not [[geometry]], the sign reversal of the magnitude form actually ''alters'' [[Cℓ]], so for example {{math|Cℓ<sub>1,2</sub>('''R''')}} and {{math|Cℓ<sub>2,1</sub>('''R''')}} are not isomorphic.
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| Just like over any vector space, there are pseudo-Euclidean [[tensor]]s. Like with an 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 {{mvar|v<sup>β</sup>}}, the corresponding [[covariant vector]] is:
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| : <math>v_\alpha = q_{\alpha\beta} v^\beta\,,</math><!-- here “q” is the bilinear form; possibly another letter should be used -->
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| and with the standard-form
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| : <math>q_{\alpha\beta} =
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| \begin{pmatrix}I_{k\times k} & 0 \\ 0 & -I_{(n-k)\times
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| (n-k)}\end{pmatrix}
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| </math>
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| the first {{mvar|k}} components of {{mvar|v<sub>α</sub>}} are numerically the same as ones of {{mvar|v<sup>β</sup>}}, but the rest {{math|''n'' − ''k''}} have [[additive inverse|opposite signs]].
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| The correspondence between contravariant and covariant tensors makes a [[tensor calculus]] on [[pseudo-Riemannian manifold]]s analogous to one on Riemannian manifolds.
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| == Examples ==
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| 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, {{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 {{mvar|q}}. See [[sign convention#Metric signature]].</ref> so that
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| : <math>q(x) = x_1^2+ x_2^2 + x_3^2-x_4^2, </math>
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| The geometry associated with this pseudo-metric was investigated by [[Henri Poincaré|Poincaré]]. 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.
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| Another pseudo-Euclidean space is the [[two-dimensional space|plane]] {{math|1=''z'' = ''x'' + ''y j''}} consisting of [[split-complex number]]s, equipped with the quadratic form
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| : <math>\lVert z \rVert = z z^* = z^* z = x^2 - y^2.</math>
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| This is the simplest case of a pseudo-Euclidean space ({{math|1=''n'' = 2}}, {{math|1=''k'' = 1}}) and the only one where the null cone dissects the space to ''four'' open sets. The group {{math|SO<sup>+</sup>(1, 1)}} consists of so named [[hyperbolic rotation]]s.
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| ==See also==
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| * [[Hyperbolic equation]]
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| * [[Hyperboloid model]]
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| * [[Paravector]]
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| ==Footnotes==
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| {{reflist}}
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| ==References==
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| *Werner Greub (1963) ''Linear Algebra'', 2nd edition, §12.4 Pseudo-Euclidean Spaces, pp. 237–49, Springer-Verlag.
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| *[[Walter Noll]] (1964) "Euclidean geometry and Minkowskian chronometry", [[American Mathematical Monthly]] 71:129–44.
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| *{{cite book
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| | last = Novikov
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| | first = S. P.
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| | coauthors = Fomenko, A.T.; [translated from the Russian by M. Tsaplina]
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| | title = Basic elements of differential geometry and topology
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| | publisher = Dordrecht; Boston: Kluwer Academic Publishers
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| | year = 1990
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| | pages =
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| | isbn = 0-7923-1009-8
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| }}
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| *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.
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| *{{cite book
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| | last = Szekeres
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| | first = Peter
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| | title = A course in modern mathematical physics: groups, Hilbert space, and differential geometry
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| | publisher = Cambridge University Press
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| | year = 2004
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| | pages =
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| | isbn = 0-521-82960-7
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| }}
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| * {{cite book
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| | last = Shafarevich
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| | first = I. R.
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| | authorlink = Igor Shafarevich
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| | coauthors = A. O. Remizov
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| | title = Linear Algebra and Geometry
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | year = 2012
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| | url = http://www.springer.com/mathematics/algebra/book/978-3-642-30993-9
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| | isbn = 978-3-642-30993-9}}
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| == External links ==
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| * ''Pseudo-Euclidean space''. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php/Pseudo-Euclidean_space
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| [[Category:Lorentzian manifolds]]
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