dispersion of gravitational waves

Is there dispersion in gravitational waves?

No, and the reason for that is that the wave vector \(k_\mu\) is a null vector, \(g^{\mu\nu}k_\mu k_\nu=0\). That gives \((\omega/c)^2-k^2 = 0\) for the wave vector with components \(\omega/c\) and \(k^i\) in a locally Cartesian coordinate frame, such that the dispersion relation is linear, \(\omega = \pm ck\), with equal group and phase velocities \(\mathrm{d}\omega/\mathrm{d}k = \omega/k = c\).