A Mass-Shell Model of Compact Binary Coalescence

Abstract

The final pulse of gravitational wave (GW) emission is released at the peak of the chirp rise before compact binary merger. LIGO detections since GW150914 reveal a correlation between the radiated energy $E_\mathrm{rad}$ and the ad hoc scaling of one-tenth of the chirp mass $\mathcal{M}$, which begs to ask if this is physically grounded. Motivated by current effective one-body models, this work models compact binary coalescence (CBC) as a rotating, compact mass shell that is contracting towards the total mass horizon. Using a variational methodology, the Laplace-Beltrami formulation for the Ricci tensor is applied to a Kerr metric Ansatz, retrieving the energy density $T_{00}$ of the CB mass shell via the Einstein field equations. At the time of coalescence $t_C$, the corresponding surface energy ultimately depends on the reduced mass $\mu$ of the CB, the symmetric mass ratio $\alpha$, and the CB's normalized orbital spin velocity. In other words, this surface energy is the anticipated energy radiated as GWs, which is not one-tenth of the chirp mass systematically. Under simple assumptions, the anticipated energy for GW150914 -- a representative example -- is $2.08 M_\odot c^2$ using documented center values. Under a more rigorous analysis in comparison, the anticipated energy for GW150914 is $3.27M_\odot c^2$. This is compared with the GWTC recorded value of $3.1^{+0.4}_{-0.4}M_\odot c^2$ for GW150914, with the latter analysis providing a closer approximation to the actual value. This study also includes the derivation of gravitational waveforms from the CB mass shell model, which depend on dynamic frequencies and decreasing CB separations.