null vector
{{Short description|Vector on which a quadratic form is zero}}
{{about|zeros of a quadratic form|the zero element in a vector space|Zero vector|null vectors in Minkowski space|Minkowski space#Causal structure}}
In mathematics, given a vector space X with an associated quadratic form q, written {{nowrap|(X, q)}}, a null vector or isotropic vector is a non-zero element x of X for which {{nowrap|1=q(x) = 0}}.
In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.
A quadratic space {{nowrap|(X, q)}} which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces,Emil Artin (1957) Geometric Algebra, [https://archive.org/details/geometricalgebra033556mbp/page/n129/mode/2up?view=theater&q=isotropic isotropic] and anisotropic space for a quadratic space without null vectors.
A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, {{nowrap|1=X = A + B}}, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres:
The null cone is also the union of the isotropic lines through the origin.
Split algebras
A composition algebra with a null vector is a split algebra.Arthur A. Sagle & Ralph E. Walde (1973) Introduction to Lie Groups and Lie Algebras, page 197, Academic Press
In a composition algebra (A, +, ×, *), the quadratic form is q(x) = x x*. When x is a null vector then there is no multiplicative inverse for x, and since x ≠ 0, A is not a division algebra.
In the Cayley–Dickson construction, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the complex number field as the foundation of this doubling construction due to L. E. Dickson (1919). In particular, these algebras have two imaginary units, which commute so their product, when squared, yields +1:
: Then
: so 1 + hi is a null vector.
The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light tracking into and out of 0 ∈ A, suggest spacetime topology.
Examples
The light-like vectors of Minkowski space are null vectors.
The four linearly independent biquaternions {{nowrap|1=l = 1 + hi}}, {{nowrap|1=n = 1 + hj}}, {{nowrap|1=m = 1 + hk}}, and {{nowrap|1=m∗ = 1 – hk}} are null vectors and {{nowrap|{ l, n, m, m∗ }{{void}}}} can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.Patrick Dolan (1968) [http://projecteuclid.org/euclid.cmp/1103840725 A Singularity-free solution of the Maxwell-Einstein Equations], Communications in Mathematical Physics 9(2):161–8, especially 166, link from Project Euclid
In the Verma module of a Lie algebra there are null vectors.
References
{{Reflist}}
- {{cite book |first=B. A. |last=Dubrovin |first2=A. T. |last2=Fomenko |authorlink2=Anatoly Fomenko |first3=S. P. |last3=Novikov |authorlink3=Sergei Novikov (mathematician) |year=1984 |title=Modern Geometry: Methods and Applications |translator-first=Robert G. |translator-last=Burns |page=[https://archive.org/details/moderngeometryme000dubr/page/50 50] |publisher=Springer |isbn=0-387-90872-2 |url=https://archive.org/details/moderngeometryme000dubr |url-access=registration }}
- {{cite book |first=Ronald |last=Shaw |year=1982 |title=Linear Algebra and Group Representations |volume=1 |page=151 |publisher=Academic Press |isbn=0-12-639201-3 |url=https://books.google.com/books?id=C6DgAAAAMAAJ }}
- {{cite book | last = Neville | first = E. H. (Eric Harold) | author-link =Eric Harold Neville | title =Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions | publisher =Cambridge University Press | date = 1922 |page=[https://archive.org/details/prolegomenatoana00nevi/page/204 204]| url =https://archive.org/details/prolegomenatoana00nevi }}