A Mass-Shell Model of Compact Binary Coalescence

TL;DR

The paper presents a physically grounded semi-analytical model of compact binary coalescence (CBC) by treating the binary as a rotating, contracting mass shell with mass measure $\mu$, inspired by effective one-body ideas. A variational-Laplace-Beltrami approach applied to a Kerr metric Ansatz yields the energy density $T_{00}$ and a coalescence surface energy $E(t_C)$ that depends on the reduced mass $\mu$, symmetric mass ratio $\alpha$, and the normalized spin velocity $\beta$, with simple estimates $E(t_C) \approx \frac{\pi}{6}\alpha\mu(1-5\beta_C^2)$ and refined curvature-based corrections capable of reproducing GW150914 energetics after a scaling factor. The model also provides TT gravitational-wave waveforms driven by a shrinking shell geometry and evolving orbital frequency, and discusses extensions to eccentric orbits, ringdown, spins, charges, and environmental effects, as well as potential tests with the Einstein Telescope. While initial predictions align with observed energetics within order-of-magnitude, calibrations of curvature contributions (e.g., via a reduced Kretschmann scalar with a scaling $\lambda$) are needed, and future work should incorporate PN corrections, tidal effects, and formation-channel dependencies. Overall, the mass-shell framework offers a physically interpretable, semi-analytical bridge between traditional EOB/NR waveform modeling and observational energetics, with clear paths for refinement and broader applicability.

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 $μ$ of the CB, the symmetric mass ratio $α$, 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.