The paper computes how an electrically charged pair of black holes radiates both light-like (electromagnetic) and gravity waves. The authors extend an earlier calculation of the binary’s motion to include the radiation seen far away, working to second post-Newtonian (2PN) order. The post-Newtonian expansion is an approximation that expands general relativity in powers of the objects’ speed divided by the speed of light, so 2PN means keeping terms up to order (v/c)^4 beyond leading order.

The study focuses on non-spinning black holes that move slowly and follow nearly circular, slowly shrinking orbits. At this level of approximation the authors produce a 2PN expression for the total energy flux carried to infinity. Compared with a neutral binary, charge introduces a new leading electromagnetic contribution of dipole type. This dipole electromagnetic emission appears one PN order earlier than the leading quadrupole gravitational flux. In addition, the electromagnetic field changes the stress-energy that sources the metric and so induces extra, charge-dependent corrections in the gravitational radiation itself.

To arrive at these results the authors extend two complementary theoretical tools to the Einstein–Maxwell system. One is the multipolar post-Minkowskian (MPM) wave-generation formalism, which relates the binary’s near-zone multipole structure to the radiative field seen at null infinity and accounts for nonlinear propagation effects such as tails and memory. The other is an effective field theory (EFT) approach adapted for charged systems. The paper reports exact agreement between the parts computed independently with the two methods. The authors derive the radiative symmetric-trace-free multipole moments of both the vector (electromagnetic) and tensor (gravitational) fields at 2PN order and compute the corresponding spherical-harmonic waveform modes.