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A nonzero covector $\ell$ is null when its squared dual Lorentzian inner product is zero: $g^{\alpha \beta }{\ell }_{\alpha }{\ell }_{\beta }=0$. Raising its index gives a null vector. Null covectors describe normals to characteristic light cones and are the test directions in the classical null condition for wave equations.

A quadratic derivative expression is a null form when its symbol vanishes on parallel null covectors. The Lorentz contraction $Q_0(f,g)=-f_t{g}_t+\nabla f\cdot \nabla g$ is the basic example. Its cancellation suppresses interactions of derivatives pointing along the same light ray. It is the algebraic structure behind the classical null condition for wave equations.

Small data global regularity for wave maps also holds in three spatial dimensions. For localized compatible data small in high weighted Sobolev norms, the key is the classical null condition for wave equations. Each derivative contraction is a null form for wave equations, vanishing for parallel null derivatives. The vector field method for wave equations exploits derivatives tangent to the light cone and weighted energy estimates to close the bootstrap argument; ordinary $(1+t{)}^{-1}$ decay alone is insufficient.

The vector field method commutes a wave equation with spacetime symmetry generators, estimates the resulting commuted wave energies, and converts weighted L2 norms to pointwise decay using a Klainerman-Sobolev inequality. The commutation vector fields for the wave equation include translations, rotations, Lorentz boost vector fields and the scaling vector field. It is useful for almost global existence for wave equations and for small-data global existence under the classical null condition for wave equations.