Frequency of gravitational waves

We show that there are physically relevant situations where gravitational waves do not inherit the frequency spectrum of their source but its wavenumber spectrum.

Figure 1

The quantity ${sin}^2((k-{\omega }_s)\Delta )/(k-{\omega }_s{)}^2$, $\Delta =(t_{\mathrm{fin}}-t_{\mathrm{in}})/2$, is shown as a function of $k-{\omega }_s$ for the values of $\Delta =$ 1 (black, solid) 0.3 (red dotted) and 10 (blue, dashed). For clarity the red (dotted) and black (solid) curves are displaced.

Figure 2

The past light cone (on which gravitons propagate) of a detector at $x=0$ observing for a given time interval sweeps a portion of space in the temporally confined source (solid, red line). Therefore, the wave number of the source is registered as the frequency of the emitted GW observed at $x$. On the other hand, the detector observes the time evolution of a spatially confined source (dashed, blue line), and therefore sees its frequency.