Null vector

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A null cone where [math]\displaystyle{ q(x,y,z) = x^2 + y^2 - z^2 . }[/math]

In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero 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 does there exist a nonzero null vector.

A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space.

A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal 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:

[math]\displaystyle{ \bigcup_{r\ge0} \{x = a + b : q(a) = -q(b) = r, a \in A, b \in B \}. }[/math]

The null cone is also the union of the isotropic lines through the origin.

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]

A composition algebra splits when it has a null vector; otherwise it is a division algebra.

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