transversal waves

The transversality of electromagnetic waves in vacuum is ensured by the field equation \(\partial_\mu F^{\mu\nu} = 0\), which, for a harmonic ansatz \(\exp(\pm\mathrm{i}k_\mu x^\mu)\) yields \(F^{\mu\nu}k_\mu = 0\), i.e. for $\nu = 0$ the orthognality of the electric field on the direction of propagation, \(k_iE^i = 0\). Why is the magnetic field likewise orthogonal, \(k_iB^i = 0\)? Could you apply these ideas to gravitational waves as well?