Nullity of wave 4 - vector for grav. plane wave

[WannabeNewton]

How can one tell from $$\square ^{2}\bar{h^{\mu \nu }} = 0$$ that in the plane wave solution $$\bar{h^{\mu \nu }} = A^{\mu \nu }e^{ik_{\alpha }x^{\alpha }}$$ the wave 4 - vector is null. If you plug in the solution you just end up with the dispersion relation $$\omega ^{2} = \left | k \right |^{2}$$. Is it implied from this that it is null or am I just missing something obvious or is it not possible to deduce its nullity from the wave equation itself? Intuitively it would have to be null because the dispersion relation implies the waves travel at c but is this sufficient to conclude that $$k^{\mu }k_{\mu } = 0$$?

 

[Bill_K]

You've got it all right there, k^μk_μ = |k|² - ω²/c² = 0