 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.