It is with great pleasure that we present today the first second-order (in the mass-ratio) self-force result. From the monopole piece of the second-order metric perturbation, and specializing to circular orbits in Schwarzschild spacetime, we compute the binding energy of the binary. We have compared the results of our calculation with the result from the first law of binary mechanics (FLBM) and find great agreement — see the figure below.

In comparing against the FLBM we find a slight discrepancy in the very strong field. This discrepancy is not unexpected as the FLBM only applies to a conservative spacetime (no inspiral) whereas we include the effects of dissipation in our calculation.

This work builds upon almost 7 years of effort since the foundational second-order papers. It’s been a long road with 10+ papers already out and I can think of at least another 5 in preparation.

This first concrete calculation really marks the end of the beginning. As we compute higher modes we will get the binding angular momentum, the radiated flux, gauge invariants, local force, and then the first complete post-adiabatic waveform. Watch this space!