Null vector

For the additive identity of a vector space, see zero element.

File:Conformalsphere.pdf In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is an element x of X for which 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 there exists a nonzero null vector. Where such a vector exists, (X, q) is called a pseudo-Euclidean space.

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:

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

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.

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.