The spectrum of radiation at low frequencies

For both electromagnetic and gravitational radiation the spectrum and polarization at low frequency can be related to the time dependence of the potentials at asymptotically early and late times. A low-frequency analysis of the spectrum in two cases is presented. For compact sources of gravitational radiation the quadrupole formula is generalized to strong-field fast-motion sources, and the low-frequency spectrum is shown to be proportional to the fourth power of frequency, with a proportionality constant that depends only on the initial and final quadrupole moments. For electromagnetic or gravitational radiation sources involving incoming or outgoing particles interacting via inverse-square-of-distance forces, the spectrum is nonanalytic at zero frequency. A technique for computing a low-frequency approximation is given and applied to simple examples.