## What determines the frequency of gravitational waves?

The frequency of gravitational waves, like the frequency of electromagnetic waves, is determined by the frequency of the oscillations in the source.

For example, with the appropriate choice of gauge, the gravitational radiation far from a non-relativistic source localized near the origin is described to leading order by the well-known

(

For example, with the appropriate choice of gauge, the gravitational radiation far from a non-relativistic source localized near the origin is described to leading order by the well-known

**quadrupole formula:**\[h_{ij}(\mathbf{x}, t) \approx \frac{2G}{c^4 r} \ddot{Q}_{ij}(t_r)\] where \(Q\) is the energy density quadrupole moment, and \(t_r\) is the retarded time, as in electrodynamics. The quantity \(h_{ij}\) is a perturbation to the background flat metric \(\eta_{ij}\) and represents the field of gravitational radiation. So it is easy to see that if the source oscillates with a frequency \(f\) in such a way that its quadrupole moment has nonzero second derivative, then it will emit gravitational radiation with frequency \(f\).(

*Note*: Compare this with the formula from electrodynamics, \(\mathbf{A} \approx \frac{\mu_0}{4\pi r} \dot{\mathbf{p}}(t_r)\).)