= Classical null condition for wave equations
{title2=$A^{\alpha\beta}\ell_\alpha\ell_\beta=0\quad\text{for null }\ell$}
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.
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