{"history":[],"article":{"id":"7dbd8330-b294-4910-b162-21a5777da709","title":"Pseudo-Euclidean space","slug":"pseudo-euclidean-space","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) and [theoretical physics](https://bliptext.com/articles/theoretical-physics), a **pseudo-Euclidean space** of signature\n(k, n-k)\nis a finite-[dimensional](https://bliptext.com/articles/dimension-mathematics) [[real coordinate space|real\nn\n-space]] together with a non-[degenerate](https://bliptext.com/articles/degenerate-form) [quadratic form](https://bliptext.com/articles/quadratic-form)\nq\n. Such a quadratic form can, given a suitable choice of [basis](https://bliptext.com/articles/basis-linear-algebra)\n(e1, …, en)\n, be applied to a vector\nx = x1e1 + ⋯ + xnen\n, giving\nwhich is called the scalar square of the vector\nx\n.\nFor [Euclidean spaces](https://bliptext.com/articles/euclidean-space),\nk = n\n, implying that the quadratic form is [positive-definite](https://bliptext.com/articles/definite-quadratic-form).\nWhen\n0 < k < n\n, then\nq\nis an [isotropic quadratic form](https://bliptext.com/articles/isotropic-quadratic-form).\nNote that if\n1 ≤ i ≤ k < j ≤ n\n, then\nq(ei + ej) = 0\n, so that\nei + ej\nis a [null vector](https://bliptext.com/articles/null-vector).\nIn a pseudo-Euclidean space with\nk < n\n, unlike in a Euclidean space, there exist vectors with [negative](https://bliptext.com/articles/negative-number) scalar square.\nAs with the term Euclidean space, the term pseudo-Euclidean space may be used to refer to an [affine space](https://bliptext.com/articles/affine-space) or a [vector space](https://bliptext.com/articles/vector-space) depending on the author, with the latter alternatively being referred to as a **pseudo-Euclidean [vector space](https://bliptext.com/articles/vector-space)** (see [point–vector distinction](https://bliptext.com/articles/point-vector-distinction)).\n\n# Geometry\nThe 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](https://bliptext.com/articles/metric-space) as explained below.\nThe [affine structure](https://bliptext.com/articles/affine-geometry) is unchanged, and thus also the concepts [line](https://bliptext.com/articles/line-geometry), [plane](https://bliptext.com/articles/plane-geometry) and, generally, of an [affine subspace](https://bliptext.com/articles/affine-subspace) ([flat](https://bliptext.com/articles/flat-geometry)), as well as [line](https://bliptext.com/articles/line-geometry) segments.\n\n## Positive, zero, and negative scalar squares\n[[Image:DoubleCone.png|thumb|right|\nn = 3\n,\nk\nis either 1 or 2 depending on the choice of [sign](https://bliptext.com/articles/sign-mathematics) of\nq\n]] A [null vector](https://bliptext.com/articles/null-vector) is a vector for which the quadratic form is zero.\nUnlike in a Euclidean space, such a vector can be non-zero, in which case it is self-[orthogonal](https://bliptext.com/articles/).\nIf the quadratic form is indefinite, a pseudo-Euclidean space has a [linear cone](https://bliptext.com/articles/linear-cone) of null vectors given by\n. When the pseudo-Euclidean space provides a model for [spacetime](https://bliptext.com/articles/spacetime) (see [below](https://bliptext.com/articles/)), the null cone is called the [light cone](https://bliptext.com/articles/light-cone) of the origin.\nThe null cone separates two [open sets](https://bliptext.com/articles/open-set), respectively for which\nq(x) > 0\nand\nq(x) < 0\n. If\nk ≥ 2\n, then the set of vectors for which\nq(x) > 0\nis [connected](https://bliptext.com/articles/connected-space).\nIf\nk = 1\n, then it consists of two disjoint parts, one with\nx1 > 0\nand another with\nx1 < 0\n. Similarly, if\nn − k ≥ 2\n, then the set of vectors for which\nq(x) < 0\nis connected.\nIf\nn − k = 1\n, then it consists of two disjoint parts, one with\nxn > 0\nand another with\nxn < 0\n.\n\n## Interval\nThe quadratic form\nq\ncorresponds to the square of a vector in the Euclidean case.\nTo define the [vector norm](https://bliptext.com/articles/vector-norm) (and distance) in an [invariant](https://bliptext.com/articles/invariant-mathematics) manner, one has to get [square roots](https://bliptext.com/articles/square-root) of scalar squares, which leads to possibly [imaginary](https://bliptext.com/articles/imaginary-number) distances; see [square root of negative numbers](https://bliptext.com/articles/square-root-of-negative-numbers).\nBut even for a [triangle](https://bliptext.com/articles/triangle) with positive scalar squares of all three sides (whose square roots are real and positive), the [triangle](https://bliptext.com/articles/triangle) inequality does not hold in general.\nHence terms norm and distance are avoided in pseudo-Euclidean geometry, which may be replaced with scalar square and interval respectively.\nThough, for a [curve](https://bliptext.com/articles/curve) whose [tangent vectors](https://bliptext.com/articles/tangent-vector) all have scalar squares of the same sign, the [arc length](https://bliptext.com/articles/arc-length) is defined.\nIt has important applications: see [proper time](https://bliptext.com/articles/proper-time), for example.\n\n## Rotations and spheres\nThe [rotations](https://bliptext.com/articles/rotation-mathematics) [group](https://bliptext.com/articles/group-mathematics) of such space is the indefinite orthogonal [group](https://bliptext.com/articles/group-mathematics)\nO(q)\n, also denoted as\nO(k, n − k)\nwithout a reference to particular quadratic form.\nSuch \"rotations\" preserve the form\nq\nand, hence, the scalar square of each vector including whether it is positive, zero, or negative.\nWhereas Euclidean space has a [unit sphere](https://bliptext.com/articles/unit-sphere), pseudo-Euclidean space has the [hypersurfaces](https://bliptext.com/articles/hypersurface)\nand\n. Such a hypersurface, called a [quasi-sphere](https://bliptext.com/articles/quasi-sphere), is preserved by the appropriate indefinite orthogonal group.\n# [pseudo-Euclidean space | upload.wikimedia.org/wikipedia/commons/4/46/Hyperboloid1.png]\n\n## Symmetric bilinear form\nThe quadratic form\nq\ngives rise to a [symmetric bilinear form](https://bliptext.com/articles/symmetric-bilinear-form) defined as follows:\nThe quadratic form can be expressed in terms of the bilinear form:\nq(x) = ⟨x, x⟩\n.\nWhen\n⟨x, y⟩ = 0\n, then\nx\nand\ny\nare [orthogonal](https://bliptext.com/articles/orthogonality) vectors of the pseudo-Euclidean space.\nThis bilinear form is often referred to as the [scalar product](https://bliptext.com/articles/scalar-product), and sometimes as \"inner product\" or \"[dot product](https://bliptext.com/articles/dot-product)\", but it does not define an [inner product space](https://bliptext.com/articles/inner-product-space) and it does not have the properties of the [dot product](https://bliptext.com/articles/dot-product) of Euclidean vectors.\nIf\nx\nand\ny\nare orthogonal and\nq(x)q(y) < 0\n, then\nx\nis [hyperbolic-orthogonal](https://bliptext.com/articles/hyperbolic-orthogonal) to\ny\n.\nThe [standard basis](https://bliptext.com/articles/standard-basis) of the real\nn\n-space is [orthogonal](https://bliptext.com/articles/orthogonal-basis).\nThere 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](https://bliptext.com/articles/vector-norm).\n\n## Subspaces and orthogonality\nFor a (positive-dimensional) subspace\nU\nof a pseudo-Euclidean space, when the quadratic form\nq\nis [restricted](https://bliptext.com/articles/function-restriction) to\nU\n, following three cases are possible:\nq\nis either [positive or negative definite](https://bliptext.com/articles/definite-quadratic-form).\nThen,\nU\nis essentially [Euclidean](https://bliptext.com/articles/euclidean-space) (up to the sign of\nq\n).\nq\nis indefinite, but non-degenerate.\nThen,\nU\nis itself pseudo-Euclidean.\nIt is possible only if\ndim U ≥ 2\ndim U = 2\n, which means than\nU\nis a [plane](https://bliptext.com/articles/plane-geometry), then it is called a hyperbolic [plane](https://bliptext.com/articles/plane-geometry).\nq\nis degenerate.\nOne of the most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their [orthogonality](https://bliptext.com/articles/orthogonality).\nWhen two non-zero [Euclidean vectors](https://bliptext.com/articles/euclidean-vector) are orthogonal, they are not [collinear](https://bliptext.com/articles/collinear).\nThe intersections of any Euclidean [linear subspace](https://bliptext.com/articles/linear-subspace) with its [orthogonal complement](https://bliptext.com/articles/orthogonal-complement) is the [[zero vector space|\n{0}\nsubspace]].\nBut the definition from the previous subsection immediately implies that any vector\nν\nof zero scalar square is orthogonal to itself.\nHence, the [isotropic line](https://bliptext.com/articles/isotropic-line)\nN = ⟨ν⟩\ngenerated by a [null vector](https://bliptext.com/articles/null-vector) **ν** is a subset of its orthogonal complement\nN⊥\n.\nThe 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\ndim U + dim U⊥ = n\ndue to the quadratic form's non-degeneracy.\nIt is just the condition\nU ∩ U⊥ =\nor, equivalently,\nU + U⊥ =\nall space,\nwhich can be broken if the subspace\nU\ncontains a null direction.\nWhile subspaces [form a lattice](https://bliptext.com/articles/linear-subspace), as in any vector space, this\n⊥\noperation is not an [orthocomplementation](https://bliptext.com/articles/orthocomplementation), in contrast to [inner product spaces](https://bliptext.com/articles/inner-product-space).\nFor a subspace\nN\ncomposed entirely of null vectors (which means that the scalar square\nq\n, restricted to\nN\n, equals to\n0\n), always holds:\nN ⊂ N⊥\nor, equivalently,\nN ∩ N⊥ = N\n.\nSuch a subspace can have up to\nmin(k, n − k)\n[dimensions](https://bliptext.com/articles/dimension-vector-space).\nFor a (positive) Euclidean\nk\n-subspace its orthogonal complement is a\n(n − k)\n-dimensional negative \"Euclidean\" subspace, and vice versa.\nGenerally, for a\n(d+ + d− + d0)\n-dimensional subspace\nU\nconsisting of\nd+\npositive and\nd−\nnegative dimensions (see [Sylvester's law of inertia](https://bliptext.com/articles/sylvester-s-law-of-inertia) for clarification), its orthogonal \"complement\"\nU⊥\nhas\n(k − d+ − d0)\npositive and\n(n − k − d− − d0)\nnegative dimensions, while the rest\nd0\nones are degenerate and form the\nU ∩ U⊥\nintersection.\n\n## Parallelogram law and Pythagorean theorem\nThe [parallelogram law](https://bliptext.com/articles/parallelogram-law) takes the form\nUsing the [square of the sum](https://bliptext.com/articles/binomial-theorem) 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:\nThis demonstrates that, for orthogonal vectors, a pseudo-Euclidean analog of the [Pythagorean theorem](https://bliptext.com/articles/pythagorean-theorem) holds:\n\n## Angle\nGenerally, absolute value\nof the bilinear form on two vectors may be greater than\n√\n, equal to it, or less.\nThis causes similar problems with definition of [angle](https://bliptext.com/articles/angle) (see ) as [appeared above](https://bliptext.com/articles/) for distances.\nIf\nk = 1\n(only one positive term in\nq\n), then for vectors of positive scalar square:\nwhich permits definition of the [hyperbolic angle](https://bliptext.com/articles/hyperbolic-angle), an analog of angle between these vectors through [inverse hyperbolic cosine](https://bliptext.com/articles/inverse-hyperbolic-function):\nIt corresponds to the distance on a\n(n − 1)\n-dimensional [hyperbolic space](https://bliptext.com/articles/hyperbolic-space).\nThis is known as [rapidity](https://bliptext.com/articles/rapidity) in the context of theory of relativity discussed [below](https://bliptext.com/articles/).\nUnlike Euclidean angle, it takes values from and equals to 0 for [antiparallel vectors](https://bliptext.com/articles/antiparallel-vectors).\nThere is no reasonable definition of the angle between a null vector and another vector (either null or non-null).\n# [pseudo-Euclidean space | upload.wikimedia.org/wikipedia/commons/d/db/Minkowski///lightcone///lorentztransform.svg]\n\n# Algebra and tensor calculus\nLike Euclidean spaces, every pseudo-Euclidean vector space generates a [Clifford algebra](https://bliptext.com/articles/clifford-algebra).\nUnlike properties above, where replacement of\nq\nto\n−q\nchanged numbers but not [geometry](https://bliptext.com/articles/geometry), the sign reversal of the quadratic form results in a distinct Clifford algebra, so for example\nCl1,2(R)\nand\nCl2,1(R)\nare not isomorphic.\nJust like over any vector space, there are pseudo-Euclidean [tensors](https://bliptext.com/articles/tensor).\nLike with a Euclidean structure, there are [raising and lowering indices](https://bliptext.com/articles/raising-and-lowering-indices) operators but, unlike the case with [Euclidean tensors](https://bliptext.com/articles/euclidean-tensor), there is [no bases where these operations do not change values of components](https://bliptext.com/articles/).\nIf there is a vector\nv\n, the corresponding [covariant vector](https://bliptext.com/articles/covariant-vector) is:\nand with the standard-form\nthe first\nk\ncomponents of\nvα\nare numerically the same as ones of\nv\n, but the rest\nn − k\nhave [opposite signs](https://bliptext.com/articles/additive-inverse).\nThe correspondence between contravariant and covariant tensors makes a [tensor calculus](https://bliptext.com/articles/tensor-calculus) on [pseudo-Riemannian manifolds](https://bliptext.com/articles/pseudo-riemannian-manifold) a generalization of one on Riemannian manifolds.\n\n# Examples\nA very important pseudo-Euclidean space is [Minkowski space](https://bliptext.com/articles/minkowski-space), which is the mathematical setting in which the theory of [special relativity](https://bliptext.com/articles/special-relativity) is formulated.\nFor Minkowski space,\nn = 4\nand\nk = 3\nso that\nThe geometry associated with this pseudo-metric was investigated by [Poincaré](https://bliptext.com/articles/henri-poincar).\nIts rotation group is the [Lorentz group](https://bliptext.com/articles/lorentz-group).\nThe [Poincaré group](https://bliptext.com/articles/poincar-group) includes also [translations](https://bliptext.com/articles/translation-geometry) and plays the same role as [Euclidean groups](https://bliptext.com/articles/euclidean-group) of ordinary Euclidean spaces.\nAnother pseudo-Euclidean space is the [plane](https://bliptext.com/articles/plane-mathematics)\nz = x + yj\nconsisting of [split-complex numbers](https://bliptext.com/articles/split-complex-number), equipped with the quadratic form\nThis is the simplest case of an indefinite pseudo-Euclidean space (\nn = 2\n,\nk = 1\n) and the only one where the null cone dissects the remaining space into four open sets. The group\nSO+(1, 1)\nconsists of so named [hyperbolic rotations](https://bliptext.com/articles/hyperbolic-rotation).\n\n# Footnotes","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T09:32:06.807Z","updated_at":"2025-01-26T09:32:06.807Z","search_vector":"'euclidean':3 'pseudo':2 'pseudo-euclidean':1 'space':4","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) and [theoretical physics](https://bliptext.com/articles/theoretical-physics), a **pseudo-Euclidean space** of signature\n(k, n-k)\nis a finite-[dimensional](https://bliptext.com/articles/dimension-mathematics) [[real coordinate space|real\nn\n-space]] together with a non-[degenerate](https://bliptext.com/articles/degenerate-form) [quadratic form](https://bliptext.com/articles/quadratic-form)\nq\n. Such a quadratic form can, given a suitable choice of [basis](https://bliptext.com/articles/basis-linear-algebra)\n(e1, …, en)\n, be applied to a vector\nx = x1e1 + ⋯ + xnen\n, giving\nwhich is called the scalar square of the vector\nx\n.\nFor [Euclidean spaces](https://bliptext.com/articles/euclidean-space),\nk = n\n, implying that the quadratic form is [positive-definite](https://bliptext.com/articles/definite-quadratic-form).\nWhen\n0 0\nand\nq(x) 0\nis [connected](https://bliptext.com/articles/connected-space).\nIf\nk = 1\n, then it consists of two disjoint parts, one with\nx1 > 0\nand another with\nx1 0\nand another with\nxn < 0\n.\n\n## Interval\nThe quadratic form\nq\ncorresponds to the square of a vector in the Euclidean case.\nTo define the [vector norm](https://bliptext.com/articles/vector-norm) (and distance) in an [invariant](https://bliptext.com/articles/invariant-mathematics) manner, one has to get [square roots](https://bliptext.com/articles/square-root) of scalar squares, which leads to possibly [imaginary](https://bliptext.com/articles/imaginary-number) distances; see [square root of negative numbers](https://bliptext.com/articles/square-root-of-negative-numbers).\nBut even for a [triangle](https://bliptext.com/articles/triangle) with positive scalar squares of all three sides (whose square roots are real and positive), the [triangle](https://bliptext.com/articles/triangle) inequality does not hold in general.\nHence terms norm and distance are avoided in pseudo-Euclidean geometry, which may be replaced with scalar square and interval respectively.\nThough, for a [curve](https://bliptext.com/articles/curve) whose [tangent vectors](https://bliptext.com/articles/tangent-vector) all have scalar squares of the same sign, the [arc length](https://bliptext.com/articles/arc-length) is defined.\nIt has important applications: see [proper time](https://bliptext.com/articles/proper-time), for example.\n\n## Rotations and spheres\nThe [rotations](https://bliptext.com/articles/rotation-mathematics) [group](https://bliptext.com/articles/group-mathematics) of such space is the indefinite orthogonal [group](https://bliptext.com/articles/group-mathematics)\nO(q)\n, also denoted as\nO(k, n − k)\nwithout a reference to particular quadratic form.\nSuch \"rotations\" preserve the form\nq\nand, hence, the scalar square of each vector including whether it is positive, zero, or negative.\nWhereas Euclidean space has a [unit sphere](https://bliptext.com/articles/unit-sphere), pseudo-Euclidean space has the [hypersurfaces](https://bliptext.com/articles/hypersurface)\nand\n. Such a hypersurface, called a [quasi-sphere](https://bliptext.com/articles/quasi-sphere), is preserved by the appropriate indefinite orthogonal group.\n# [pseudo-Euclidean space | upload.wikimedia.org/wikipedia/commons/4/46/Hyperboloid1.png]\n\n## Symmetric bilinear form\nThe quadratic form\nq\ngives rise to a [symmetric bilinear form](https://bliptext.com/articles/symmetric-bilinear-form) defined as follows:\nThe quadratic form can be expressed in terms of the bilinear form:\nq(x) = ⟨x, x⟩\n.\nWhen\n⟨x, y⟩ = 0\n, then\nx\nand\ny\nare [orthogonal](https://bliptext.com/articles/orthogonality) vectors of the pseudo-Euclidean space.\nThis bilinear form is often referred to as the [scalar product](https://bliptext.com/articles/scalar-product), and sometimes as \"inner product\" or \"[dot product](https://bliptext.com/articles/dot-product)\", but it does not define an [inner product space](https://bliptext.com/articles/inner-product-space) and it does not have the properties of the [dot product](https://bliptext.com/articles/dot-product) of Euclidean vectors.\nIf\nx\nand\ny\nare orthogonal and\nq(x)q(y) < 0\n, then\nx\nis [hyperbolic-orthogonal](https://bliptext.com/articles/hyperbolic-orthogonal) to\ny\n.\nThe [standard basis](https://bliptext.com/articles/standard-basis) of the real\nn\n-space is [orthogonal](https://bliptext.com/articles/orthogonal-basis).\nThere 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](https://bliptext.com/articles/vector-norm).\n\n## Subspaces and orthogonality\nFor a (positive-dimensional) subspace\nU\nof a pseudo-Euclidean space, when the quadratic form\nq\nis [restricted](https://bliptext.com/articles/function-restriction) to\nU\n, following three cases are possible:\nq\nis either [positive or negative definite](https://bliptext.com/articles/definite-quadratic-form).\nThen,\nU\nis essentially [Euclidean](https://bliptext.com/articles/euclidean-space) (up to the sign of\nq\n).\nq\nis indefinite, but non-degenerate.\nThen,\nU\nis itself pseudo-Euclidean.\nIt is possible only if\ndim U ≥ 2\ndim U = 2\n, which means than\nU\nis a [plane](https://bliptext.com/articles/plane-geometry), then it is called a hyperbolic [plane](https://bliptext.com/articles/plane-geometry).\nq\nis degenerate.\nOne of the most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their [orthogonality](https://bliptext.com/articles/orthogonality).\nWhen two non-zero [Euclidean vectors](https://bliptext.com/articles/euclidean-vector) are orthogonal, they are not [collinear](https://bliptext.com/articles/collinear).\nThe intersections of any Euclidean [linear subspace](https://bliptext.com/articles/linear-subspace) with its [orthogonal complement](https://bliptext.com/articles/orthogonal-complement) is the [[zero vector space|\n{0}\nsubspace]].\nBut the definition from the previous subsection immediately implies that any vector\nν\nof zero scalar square is orthogonal to itself.\nHence, the [isotropic line](https://bliptext.com/articles/isotropic-line)\nN = ⟨ν⟩\ngenerated by a [null vector](https://bliptext.com/articles/null-vector) **ν** is a subset of its orthogonal complement\nN⊥\n.\nThe 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\ndim U + dim U⊥ = n\ndue to the quadratic form's non-degeneracy.\nIt is just the condition\nU ∩ U⊥ =\nor, equivalently,\nU + U⊥ =\nall space,\nwhich can be broken if the subspace\nU\ncontains a null direction.\nWhile subspaces [form a lattice](https://bliptext.com/articles/linear-subspace), as in any vector space, this\n⊥\noperation is not an [orthocomplementation](https://bliptext.com/articles/orthocomplementation), in contrast to [inner product spaces](https://bliptext.com/articles/inner-product-space).\nFor a subspace\nN\ncomposed entirely of null vectors (which means that the scalar square\nq\n, restricted to\nN\n, equals to\n0\n), always holds:\nN ⊂ N⊥\nor, equivalently,\nN ∩ N⊥ = N\n.\nSuch a subspace can have up to\nmin(k, n − k)\n[dimensions](https://bliptext.com/articles/dimension-vector-space).\nFor a (positive) Euclidean\nk\n-subspace its orthogonal complement is a\n(n − k)\n-dimensional negative \"Euclidean\" subspace, and vice versa.\nGenerally, for a\n(d+ + d− + d0)\n-dimensional subspace\nU\nconsisting of\nd+\npositive and\nd−\nnegative dimensions (see [Sylvester's law of inertia](https://bliptext.com/articles/sylvester-s-law-of-inertia) for clarification), its orthogonal \"complement\"\nU⊥\nhas\n(k − d+ − d0)\npositive and\n(n − k − d− − d0)\nnegative dimensions, while the rest\nd0\nones are degenerate and form the\nU ∩ U⊥\nintersection.\n\n## Parallelogram law and Pythagorean theorem\nThe [parallelogram law](https://bliptext.com/articles/parallelogram-law) takes the form\nUsing the [square of the sum](https://bliptext.com/articles/binomial-theorem) 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:\nThis demonstrates that, for orthogonal vectors, a pseudo-Euclidean analog of the [Pythagorean theorem](https://bliptext.com/articles/pythagorean-theorem) holds:\n\n## Angle\nGenerally, absolute value\nof the bilinear form on two vectors may be greater than\n√\n, equal to it, or less.\nThis causes similar problems with definition of [angle](https://bliptext.com/articles/angle) (see ) as [appeared above](https://bliptext.com/articles/) for distances.\nIf\nk = 1\n(only one positive term in\nq\n), then for vectors of positive scalar square:\nwhich permits definition of the [hyperbolic angle](https://bliptext.com/articles/hyperbolic-angle), an analog of angle between these vectors through [inverse hyperbolic cosine](https://bliptext.com/articles/inverse-hyperbolic-function):\nIt corresponds to the distance on a\n(n − 1)\n-dimensional [hyperbolic space](https://bliptext.com/articles/hyperbolic-space).\nThis is known as [rapidity](https://bliptext.com/articles/rapidity) in the context of theory of relativity discussed [below](https://bliptext.com/articles/).\nUnlike Euclidean angle, it takes values from and equals to 0 for [antiparallel vectors](https://bliptext.com/articles/antiparallel-vectors).\nThere is no reasonable definition of the angle between a null vector and another vector (either null or non-null).\n# [pseudo-Euclidean space | upload.wikimedia.org/wikipedia/commons/d/db/Minkowski///lightcone///lorentztransform.svg]\n\n# Algebra and tensor calculus\nLike Euclidean spaces, every pseudo-Euclidean vector space generates a [Clifford algebra](https://bliptext.com/articles/clifford-algebra).\nUnlike properties above, where replacement of\nq\nto\n−q\nchanged numbers but not [geometry](https://bliptext.com/articles/geometry), the sign reversal of the quadratic form results in a distinct Clifford algebra, so for example\nCl1,2(R)\nand\nCl2,1(R)\nare not isomorphic.\nJust like over any vector space, there are pseudo-Euclidean [tensors](https://bliptext.com/articles/tensor).\nLike with a Euclidean structure, there are [raising and lowering indices](https://bliptext.com/articles/raising-and-lowering-indices) operators but, unlike the case with [Euclidean tensors](https://bliptext.com/articles/euclidean-tensor), there is [no bases where these operations do not change values of components](https://bliptext.com/articles/).\nIf there is a vector\nv\n, the corresponding [covariant vector](https://bliptext.com/articles/covariant-vector) is:\nand with the standard-form\nthe first\nk\ncomponents of\nvα\nare numerically the same as ones of\nv\n, but the rest\nn − k\nhave [opposite signs](https://bliptext.com/articles/additive-inverse).\nThe correspondence between contravariant and covariant tensors makes a [tensor calculus](https://bliptext.com/articles/tensor-calculus) on [pseudo-Riemannian manifolds](https://bliptext.com/articles/pseudo-riemannian-manifold) a generalization of one on Riemannian manifolds.\n\n# Examples\nA very important pseudo-Euclidean space is [Minkowski space](https://bliptext.com/articles/minkowski-space), which is the mathematical setting in which the theory of [special relativity](https://bliptext.com/articles/special-relativity) is formulated.\nFor Minkowski space,\nn = 4\nand\nk = 3\nso that\nThe geometry associated with this pseudo-metric was investigated by [Poincaré](https://bliptext.com/articles/henri-poincar).\nIts rotation group is the [Lorentz group](https://bliptext.com/articles/lorentz-group).\nThe [Poincaré group](https://bliptext.com/articles/poincar-group) includes also [translations](https://bliptext.com/articles/translation-geometry) and plays the same role as [Euclidean groups](https://bliptext.com/articles/euclidean-group) of ordinary Euclidean spaces.\nAnother pseudo-Euclidean space is the [plane](https://bliptext.com/articles/plane-mathematics)\nz = x + yj\nconsisting of [split-complex numbers](https://bliptext.com/articles/split-complex-number), equipped with the quadratic form\nThis is the simplest case of an indefinite pseudo-Euclidean space (\nn = 2\n,\nk = 1\n) and the only one where the null cone dissects the remaining space into four open sets. The group\nSO+(1, 1)\nconsists of so named [hyperbolic rotations](https://bliptext.com/articles/hyperbolic-rotation).\n\n# Footnotes","revisionCount":0},"hasMore":false,"nextCursor":null}

History