Null form for wave equations ($Q_0(f,g)=-f_t{g}_t+\nabla f\cdot \nabla g$)

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.

For a quadratic derivative source $A^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u$, the classical null condition is $A^{\alpha \beta }{\ell }_{\alpha }{\ell }_{\beta }=0$ whenever $-{\ell }_0^2+|{\ell }_{\mathrm{sp}}{|}^2=0$. It removes the leading interaction of parallel lightlike derivatives. In three dimensions this structure gives global smooth solutions for sufficiently small, localized data through the vector field method for wave equations. This PDE condition is distinct from the null condition asserting that a single vector is lightlike.