Quasi-Equilibrium Binary Initial Data

Binary initial data are called quasi-equilibrium when their freely specified fields and inner conditions approximate a slowly inspiraling configuration on one slice. Constraint satisfaction does not determine the orbital angular velocity, radial approach, spins, or residual gravitational radiation. Those parameters must be calibrated through global relations and short evolutions, with numerical error kept below the targeted eccentricity. This page develops that iterative computational workflow without claiming that exact helical symmetry exists in an asymptotically flat radiating spacetime.

Required background. Hypersurfaces and Foliations supplies the intrinsic and extrinsic slice geometry and horizon projections.

Helpful background. Formulations of General Relativity develops conformal thin-sandwich systems, Ordinary Differential Equations supplies nonlinear solution and fitting methods, and Relativistic Matter and Compact Objects supplies equilibrium and compact-object charge concepts.

Approximate symmetry as numerical data

With $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$, approximate helical symmetry in a corotating frame motivates setting the conformal-metric time derivative and often $\partial_tK$ to zero. An extended conformal thin-sandwich solve then determines the conformal factor, lapse, and shift after free data and boundary conditions are chosen. On a black-hole excision surface, the normal shift holds the surface approximately stationary while its tangential part controls spin.

The asymptotic shift contains orbital and possibly radial motion,

$$\boldsymbol\beta\longrightarrow \boldsymbol\Omega_0\times\mathbf r+\dot a_0\mathbf r$$

in a chosen asymptotic frame. Neither $\Omega_0$ nor $\dot a_0$ follows from the constraints. A virial-like comparison between ADM and Komar masses can help choose $\Omega_0$, but it is a diagnostic tied to stationarity assumptions, not a theorem that eccentricity vanishes. Pfeiffer, Cook, and Teukolsky 2002, §§II–V compares direct decomposition choices and their global quantities.

Close the loop with short evolutions

Solve the data, evolve for a few orbits, and fit a separation or frequency diagnostic to a secular trend plus an oscillation,

$$\dot\Omega(t)=A_0+A_1t+B\cos(\omega_et+\phi).$$

Use $B$ and $\phi$ to update $\Omega_0$ and $\dot a_0$, regenerate the constraints, and repeat. The exact update depends on the fitted observable and orbital model, so the implementation must record the formula rather than merely state that eccentricity was reduced. Pfeiffer et al. 2007, §§2–4, Table 1 and Figs. 2–6 demonstrates this iterative strategy and the sensitivity to initial radial velocity.

Coordinate separation is gauge dependent. Fit more than one indicator—proper separation where meaningful, orbital frequency, waveform frequency after causal delay—and require consistent oscillation phase. Exclude startup gauge transients using a predeclared window, then vary the window. A fit that changes substantially is not a stable eccentricity estimate.

A Newtonian estimate of the oscillation amplitude

For a nearly circular Kepler orbit, write $r(t)=a[1-e\cos(nt)]$ to first order in eccentricity $e\ll1$. Conservation of specific angular momentum gives $\Omega(t)=n[1+2e\cos(nt)]+O(e^2)$ and therefore $\dot\Omega=-2en^2\sin(nt)+O(e^2)$. A fitted amplitude $B$ in $\dot\Omega$ thus suggests

$$e_\Omega\simeq\frac{|B|}{2\Omega_0\omega_e}.$$

This is a leading-order interpretation of the fit, not a relativistic eccentricity invariant or a universal update rule. It assumes that the secular inspiral changes slowly over a radial cycle; precession, spin-dependent oscillations, and gauge motion require a more complete model. For example, $\Omega_0=\omega_e=0.02/m$ and $|B|=8\times10^{-7}/m^2$ give $e_\Omega\simeq10^{-3}$, where $m=G(M_1+M_2)$ is a time or length in $c=1$ units. Both sides are dimensionless after this conversion. Check the phase as well as the amplitude before deciding whether to change tangential or radial initial motion.

Benchmark and error hierarchy

Use an equal-mass, nonspinning binary at stated target masses and separation. For each iteration record the free-data prescription, domain maps, inner conditions, $\Omega_0$, $\dot a_0$, ADM energy and angular momentum, Komar diagnostic, horizon masses and spins, independent constraint norms, and fitted eccentricity. Run at least three elliptic resolutions and two outer radii before interpreting small changes in orbital parameters.

The initial-data error should be below the eccentricity target, and the evolution phase error over the fit interval should be smaller than the oscillatory signal. Otherwise the iteration calibrates truncation error. Repeat the final short evolution at a finer resolution and altered outer boundary. Compare junk-radiation energy and component charges before and after the burst; low eccentricity does not imply low junk radiation.

For neutron stars, surface treatment and fluid equilibrium add errors; for spinning or precessing black holes, quasi-equilibrium spin conditions and orbital diagnostics become more model dependent. The method’s validity is restricted to systems evolving slowly relative to an orbit and to the declared conformal and symmetry assumptions.

Environment and failure modes

Pin the decomposition, free-data files, horizon boundary conditions, target charges, domain resolution, elliptic tolerances, evolution formulation and gauge, mesh hierarchy, fit observable, fit interval, update formula, and code revision. A rerun interface could be iterate-binary-data --config equal-mass.toml --output run/, emitting every initial-data solve, evolution series, fit covariance, updates, constraints, and environment metadata. This specifies the complete experiment without implying that an executable is present.

Reject an iteration when the fitted oscillation is not resolved, parameter updates vary with fit window, constraints or charges drift with resolution, or the outer boundary contaminates the fitting interval. A small ADM–Komar difference can coexist with eccentric motion, and a small coordinate-separation oscillation can result from gauge. Stopping because successive parameters barely change is another failure mode if the physical eccentricity has plateaued above tolerance.

Diagnostic

After three iterations, coordinate separation shows no visible oscillation, but waveform frequency and proper separation show a common oscillation with the same phase. The ADM and Komar masses agree to $10^{-5}$. Are the data low-eccentricity?

Solution

No. The coordinate separation is gauge dependent, while two less-correlated diagnostics identify residual eccentricity. ADM–Komar agreement tests an approximate stationarity relation but does not remove radial orbital error. Update $\Omega_0$ and $\dot a_0$ using the resolved oscillation, repeat the solve and short evolution, and demonstrate stability under resolution and fit-window changes.

References

• Pfeiffer, H. P., Cook, G. B., and Teukolsky, S. A. (2002). “Comparing initial-data sets for binary black holes.” Physical Review D 66, 024047. arXiv.
• Pfeiffer, H. P., Kidder, L. E., Scheel, M. A., and Teukolsky, S. A. (2007). “Reducing orbital eccentricity in binary black hole simulations.” Classical and Quantum Gravity 24, S59–S81. arXiv.