ABSTRACT (original article): 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 of the CCB mass shell was obtained via the Einstein field equations. At the time of coalescence , the resulting surface energy depends on the reduced mass , the symmetric mass ratio , 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 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, with 1:1 ratios spanning from 0.828 to 0.997. Three events exhibit ratios in the range 0.721–0.779, 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. CITATION (original article): MacKay, N. M. (2026). Comparing a compact-binary mass-shell model with select observed gravitational waves. Classical and Quantum Gravity, 43, 115015. https://doi.org/10.1088/1361-6382/ae72e4

Introduction

Observable gravitational waves (GWs) are sourced by coalescing compact binaries (CCBs), whose structure is quadrupolar at basic leading order. Rather than producing trigonometric waves quintessential to classical simple harmonic oscillators, the waveforms that describe emitted and incoming GWs are reflected in the complicated dynamics of CCBs. Current models that attempt to capture the dynamics (well enough to become part of GW analysis) is the state-of-the-art effective one-body (EOB) model. However, the model heavily relies on calibration with data and numerical relativity. We introduce an “easier” model that in spirit reinterprets EOB -- which essentially depicts the CCB as its reduced mass spiraling down a funnel -- as a rotating and contracting hollow shell.

Modeling the Compact-binary mass-shell

Compact binaries are made of two black holes, two neutron stars, or a mixed pair between one black hole and one neutron star. As a representative case, let’s consider two black holes.
To draw an ideal compact-binary, we basically make a dumbbell where the bar is a dashed line, showing the separation between the two.
In[]:=
f1=Graphics3D[{(*Twoblackspheres*)Directive[Opacity[1],Black],Sphere[{5,0,0},1.18],(*Firstblacksphere*)​​(*Equatorialplaneasadashedline*)Directive[Dashed,Thickness[0.02],Black],Table[Line[{{-5,0,0},{5,0,0}}],{n,-5,5,1}],​​Directive[Opacity[1],Black],Sphere[{-5,0,0},1.18](*Secondblacksphere*)​​},Boxed->False]​​​​
Out[]=
The mass-shell model is basically encapsulates the two binary masses such that it draws a Gaussian sphere between the surfaces of each mass. The shell diameter in the main text is S=L-(r1+r2)+4GM, so it is expected for the shell’s diameter to be larger than the separation. We model this with respect to the ideal compact-binary, which we place inside the shell.
In[]:=
f2=Graphics3D[{(*Largerspherewithlowopacity*)Directive[Opacity[0.08],Blue],Sphere[{0,0,0},6.19],(*Twoblackspheres*)Directive[Opacity[0.75],Gray],Sphere[{5,0,0},1.18],(*Firstblacksphere*)​​(*Equatorialplaneasadashedline*)Directive[Dashed,Thickness[0.02],Black],Table[Line[{{-5,0,0},{5,0,0}}],{n,-5,5,1}],​​Directive[Opacity[0.75],Gray],Sphere[{-5,0,0},1.18](*Secondblacksphere*)​​},Boxed->False]​​
Out[]=
To view them side-by-side, as seen in the paper,
TableForm[{{f1,f2}}]
Out[]//TableForm=

Mass-Shell Waveforms

GW waveforms gives us information of the source from whence they came. The structure and profile of the waveforms themselves encode the complicated dynamics of inspiral, where the wave amplitude and frequency both change with the binary separation and its orbital frequency. In the mass-shell model, the concurrent rotation and contracting of the shell’s surface scale impacts the radiation it emits.
◼
  • Variable Frequency and Separation
    • In[]:=
    Omega=Ω[t];​​Radius=(S[t]/2);​​hplus=(2G/D)*D[μ*Radius^2*Cos[Omega*t]^2,{t,2}]//Simplify//TrigReduce​​hcross=(2G/D)*D[μ*Radius^2*Cos[Omega*t]*Sin[Omega*t],{t,2}]//Simplify//TrigReduce
    Out[]=
    1
    2D
    (-2GμCos[2tΩ[t]]
    2
    S[t]
    2
    Ω[t]
    -4GμS[t]Sin[2tΩ[t]]Ω[t]
    ′
    S
    [t]+Gμ
    2
    ′
    S
    [t]
    +GμCos[2tΩ[t]]
    2
    ′
    S
    [t]
    -2Gμ
    2
    S[t]
    Sin[2tΩ[t]]
    ′
    Ω
    [t]-4GtμCos[2tΩ[t]]
    2
    S[t]
    Ω[t]
    ′
    Ω
    [t]-4GtμS[t]Sin[2tΩ[t]]
    ′
    S
    [t]
    ′
    Ω
    [t]-2G
    2
    t
    μCos[2tΩ[t]]
    2
    S[t]
    2
    ′
    Ω
    [t]
    +GμS[t]
    ′′
    S
    [t]+GμCos[2tΩ[t]]S[t]
    ′′
    S
    [t]-Gtμ
    2
    S[t]
    Sin[2tΩ[t]]
    ′′
    Ω
    [t])
    Out[]=
    1
    2D
    (-2Gμ
    2
    S[t]
    Sin[2tΩ[t]]
    2
    Ω[t]
    +4GμCos[2tΩ[t]]S[t]Ω[t]
    ′
    S
    [t]+GμSin[2tΩ[t]]
    2
    ′
    S
    [t]
    +2GμCos[2tΩ[t]]
    2
    S[t]
    ′
    Ω
    [t]-4Gtμ
    2
    S[t]
    Sin[2tΩ[t]]Ω[t]
    ′
    Ω
    [t]+4GtμCos[2tΩ[t]]S[t]
    ′
    S
    [t]
    ′
    Ω
    [t]-2G
    2
    t
    μ
    2
    S[t]
    Sin[2tΩ[t]]
    2
    ′
    Ω
    [t]
    +GμS[t]Sin[2tΩ[t]]
    ′′
    S
    [t]+GtμCos[2tΩ[t]]
    2
    S[t]
    ′′
    Ω
    [t])
    ◼
  • Fixed Frequency and Separation (for comparison)
    • In[]:=
    Omega2=Ω;​​Radius2=(S/2);​​hplus2=(2G/D)*D[μ*Radius2^2*Cos[Omega2*t]^2,{t,2}]//Simplify//TrigReduce​​hcross2=(2G/D)*D[μ*Radius2^2*Cos[Omega2*t]*Sin[Omega2*t],{t,2}]//Simplify//TrigReduce
    Out[]=
    -
    G
    2
    S
    μ
    2
    Ω
    Cos[2tΩ]
    D
    Out[]=
    -
    G
    2
    S
    μ
    2
    Ω
    Sin[2tΩ]
    D
    The “fixed” case recovers the simple harmonic-like GWs, but with a factor of 4 missing due to the choice of quadrupolar geometry. One can also introduce off-circular or other forms of orbital trajectory in this stage by replacing sinusoidal waves with e.g. elliptical functions, in practice.
    ​
    The mass-shell geometry of the model allows one to link the change in binary separation (end-to-end shell diameter) with the orbital frequency (effective axial rotation) via the conservation of angular momentum:
    ◼
  • Conserved Angular Momentum of a mass-shell (c.f. https://iopscience.iop.org/article/10.1088/1361-6382/ae2560)
    • In[]:=
    J[t_]:=μ*Radius^2*Omega​​D[J[t],t]==0//Simplify(*ConservationLaw*)
    Out[]=
    μS[t](2Ω[t]
    ′
    S
    [t]+S[t]
    ′
    Ω
    [t])0
    In[]:=
    Solve[2Ω[t]
    ′
    S
    [t]+S[t]
    ′
    Ω
    [t]==0,S'[t]]
    Out[]=
    
    ′
    S
    [t]-
    S[t]
    ′
    Ω
    [t]
    2Ω[t]
    
    In[]:=
    Solve[D[2Ω[t]
    ′
    S
    [t]+S[t]
    ′
    Ω
    [t],t]==0,S''[t]](*Forsecond-orderterms*)
    Out[]=
    
    ′′
    S
    [t]
    -3
    ′
    S
    [t]
    ′
    Ω
    [t]-S[t]
    ′′
    Ω
    [t]
    2Ω[t]
    
    In[]:=
    hplus/.S''[t]->
    -3
    ′
    S
    [t]
    ′
    Ω
    [t]-S[t]
    ′′
    Ω
    [t]
    2Ω[t]
    /.S'[t]->-
    S[t]
    ′
    Ω
    [t]
    2Ω[t]
    //FullSimplify​​hcross/.S''[t]->
    -3
    ′
    S
    [t]
    ′
    Ω
    [t]-S[t]
    ′′
    Ω
    [t]
    2Ω[t]
    /.S'[t]->-
    S[t]
    ′
    Ω
    [t]
    2Ω[t]
    //FullSimplify
    Out[]=
    -
    1
    4D
    2
    Ω[t]
    Gμ
    2
    S[t]
    (-2
    2
    ′
    Ω
    [t]
    +Ω[t]
    ′′
    Ω
    [t]+2tSin[2tΩ[t]]Ω[t](-2
    2
    ′
    Ω
    [t]
    +Ω[t]
    ′′
    Ω
    [t])+Cos[2tΩ[t]](4
    4
    Ω[t]
    +8t
    3
    Ω[t]
    ′
    Ω
    [t]-2
    2
    ′
    Ω
    [t]
    +4
    2
    t
    2
    Ω[t]
    2
    ′
    Ω
    [t]
    +Ω[t]
    ′′
    Ω
    [t]))
    Out[]=
    1
    4D
    2
    Ω[t]
    Gμ
    2
    S[t]
    (2tCos[2tΩ[t]]Ω[t](-2
    2
    ′
    Ω
    [t]
    +Ω[t]
    ′′
    Ω
    [t])-Sin[2tΩ[t]](4
    4
    Ω[t]
    +8t
    3
    Ω[t]
    ′
    Ω
    [t]-2
    2
    ′
    Ω
    [t]
    +4
    2
    t
    2
    Ω[t]
    2
    ′
    Ω
    [t]
    +Ω[t]
    ′′
    Ω
    [t]))

    The Approximated Energy

    While it is conventional to extract the energy density (and energy flux density) of GWs via
    2
    h
    +
    +
    2
    h
    x
    , a more heuristic approach was employed where the relevant quantity was obtained through a variational treatment of the EFEs. In short, 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:=μ/M, and the normalized orbital spin velocity of the CCB.
    ◼
  • Kerr metric elements
    • In[]:=
    rs=2M;​​Sigma=r^2+a^2Cos[x]^2;​​Delta=r^2-rs*r+a^2;​​​​g00=-(1-rs*r/Sigma);​​g11=Sigma/Delta;​​g22=Sigma;​​g33=(r^2+a^2+rs*r*a^2Sin[x]^2/Sigma)Sin[x]^2;​​g03=-(2rs*r*a*Sin[x]^2/Sigma)/2;​​g30=g03;
    ◼
  • Laplace-Beltrami formalism with Kerr Ansatz
    • One can, in practice, obtain a Laplace-Beltrami-formulated Ricci scalar via the trace (https://iopscience.iop.org/article/10.1088/1361-6382/ae2560):
    Because the mass-shell model gives a spherical geometry to the binary, the inclination angles become effective polar angles. Thus, the polar sector must be integrated over the whole range θ ∈ [0, π].
    In the Laplace-Beltrami formalism, the Ricci scalar diverges after integration over all polar angles. Instead, a surrogate in terms of the Kretschmann scalar is used.

    Model vs. Observation

    ◼
  • GW150914 (sourced by binary black holes)
    • This is compared to the actual, observed energy of 3.1 ± 0.4 Msol.
    ◼
  • GW170817 (sourced by binary neutron stars)
    • This is compared to an unconstrained energy ≥ 0.04 Msol.

    Conclusion

    In this notebook, we review the derivation steps and computation of the mass-shell waveforms (generally) and the approximated energy via the Laplace-Beltrami formulation of the Ricci tensor. One is encouraged to explore Ansätze for e.g. mass-shell frequency, to see how the waveforms change and become specifically defined, and use Ec[m1, m2, f] for other observed GW events (see https://gwosc.org/eventapi/html/GWTC/).

    CITATIONS (Cite when using this notebook for scientific work)

    ◼
  • Noah M MacKay 2026 Class. Quantum Grav. 43 115015
    • ◼
  • Noah M MacKay 2025 Class. Quantum Grav. 42 245003
    • ◼
  • Gravitational Wave Open Science Catalog (https://gwosc.org/eventapi/html/GWTC/)

    • CITE THIS NOTEBOOK

    Comparing a compact-binary mass-shell model with select observed gravitational waves​
    by Noah MacKay​
    Wolfram Community, STAFF PICKS, June 26, 2026
    ​https://community.wolfram.com/groups/-/m/t/3739638