Null vector

A null cone where n = 3

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.^[1]

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

References

1. Patrick Dolan (1968) A Singularity-free solution of the Maxwell-Einstein Equations, Communications in Mathematical Physics 9(2):161–8, especially 166, link from Project Euclid

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