Comparing a Compact-Binary Mass-Shell Model with Select Observed Gravitational Waves

In a recent work, coalescing compact binaries (CCBs) were modeled as a rotating and contracting compact mass shell, providing an alternative effective representation to the state-of-the-art effective-one-body approach. Using a variational methodology, the Laplace-Beltrami formulation of the Ricci tensor was applied to a Kerr metric Ansatz, and the corresponding energy density $T_{00}$ of the CCB mass shell was obtained via the Einstein field equations. At the time of coalescence $t_C$, the resulting surface energy depends on the reduced mass $μ$, the symmetric mass ratio $α$, and the normalized orbital spin velocity of the CCB. In this work, we evaluate the radiated energy predicted by this variational approach for 45 select gravitational wave (GW) events from the O1–O4 runs, and compare these values with those inferred from observational catalogs, either directly or via the total-minus-remnant mass difference. For 38 of the 45 events analyzed, the predicted radiated energies agree with observationally inferred values within the reported uncertainties, with 1:1 ratios spanning from $0.828$ to $0.997$ (mean $0.942$, median $0.955$). Three events exhibit ratios in the range $0.721\sim0.779$, one event yields a ratio of $0.466$, and for the remaining events the radiated energy is either unconstrained or inaccessible due to undocumented total-minus-remnant mass differences. These results indicate that the analytical approximation captures, for the most part, the leading-order energy scaling of compact binary mergers, while also suggesting clear avenues for further systematic improvement, including incorporating post-Newtonian corrections due to e.g. a non-zero eccentricity or combined tidal deformability.

💡 Research Summary

This paper presents a comparative analysis between a novel analytical model for coalescing compact binaries (CCBs) and observational data from selected gravitational wave (GW) events. The authors propose an alternative to the computationally intensive effective-one-body (EOB) framework by modeling a CCB as an effective, rotating and contracting hollow mass shell. This model reinterprets the binary as a single object with total mass M and reduced mass μ distributed over a shell whose radius shrinks until it reaches the total mass Schwarzschild radius (2GM) at coalescence time t_C.

The core methodological innovation lies in deriving the energy radiated in GWs at coalescence. Instead of solving the Einstein Field Equations (EFEs) from a prescribed energy-momentum tensor, the authors employ a variational approach inspired by quantum mechanics. They start with a known metric Ansatz—the Kerr metric, suitable for a rotating object—and apply the Laplace-Beltrami differential operator formulation to compute the Ricci tensor. By inserting this into the EFEs, they effectively “reverse-engineer” the corresponding energy-momentum tensor component T_00, which gives the surface energy density. Focusing on the coalescence instant and using the Kretschmann scalar as a curvature surrogate, they derive a closed-form expression for the radiated energy: E(t_C) ≃ (π/6) α μ