Null vector

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For the additive identity of a vector space, see zero element.
A null cone where n = 3

In mathematics, a null vector, or isotropic vector, is an element x of a real vector space X with an associated quadratic form q, i.e. a pseudo-Euclidean space (X, q), for which q(x) = 0.

A pseudo-Euclidean space may be decomposed (non-uniquely) into subspaces A and B, 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:

$\bigcup_{r>0} \{x = a + b : q(a) = - q(b) = r \}.$

In the theory of bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter there exists a nonzero null vector. In this context, a null vector is termed an isotropic vector.

Examples

The light-like vectors of Minkowski space are null vectors.

The four linearly independent biquaternions l = 1 + hi, n = 1 + hj, m = 1 + hk, and m* = 1 – hk are null vectors and { l, n, m, m* } 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.[1]

In the Verma module of a Lie algebra there are null vectors.

References

• B.A. Dubrovin, A. T. Fomenko, S.P. Novikov (1984) Modern Geometry — Methods and Applications, Robert G. Burns translator, page 50, Springer ISBN 0-387-90872-2.
• Ronald Shaw (1982) Linear Algebra and Group Representations, v. 1, p. 151, Academic Press ISBN 0-12-639201-3.