Relativistic Binary Equilibrium and Initial Data

Late-inspiral neutron-star data are built as a quasiequilibrium snapshot: the orbital period is short compared with radiation reaction, so the geometry and fluid are approximately invariant along a helical direction. Realistic old neutron stars are nearly irrotational rather than tidally synchronized, because viscosity is too weak to lock the spin during inspiral. The construction must solve Einstein’s constraints and the stellar surface simultaneously; placing two isolated stars on an orbit is not constraint-consistent initial data.

Helpful background. Stress–Energy and Conservation supplies fluid first integrals; the Einstein equations supply the constraints; Relativistic Two-Body Dynamics supplies orbital invariants; Relativistic Matter and Compact Objects supplies stellar equilibria; Gravitational Radiation and Waveforms supplies radiation reaction; and Computational Relativity supplies elliptic solvers and evolution checks.

Approximate helical symmetry

Let $t^\mu$ be an asymptotic time direction and $\phi^\mu$ an orbital rotation generator. Quasiequilibrium assumes approximate invariance along

$$k^\mu=t^\mu+\Omega\phi^\mu, \qquad {\cal L}_k g_{\mu\nu}\simeq0,$$

with orbital angular frequency $\Omega$. This cannot be an exact global symmetry of an asymptotically flat radiating binary: exact helical symmetry would require standing radiation. It is controlled when the radiation-reaction timescale is much longer than an orbit and the stars are outside disruption or merger.

In a 3+1 split with spatial metric $\gamma_{ij}$ and extrinsic curvature $K_{ij}$, the initial slice must satisfy

$${\cal H}\equiv {}^{(3)}R+K^2-K_{ij}K^{ij}-16\pi G E=0,$$ $${\cal M}^i\equiv D_j(K^{ij}-\gamma^{ij}K)-8\pi G S^i=0 .$$

A conformal thin-sandwich or related decomposition turns these into coupled elliptic equations for conformal geometry, lapse, and shift after free data and gauge conditions are chosen. Conformal flatness is a common approximation but not a consequence of helical symmetry; its bias belongs in the error budget. Baiotti and Rezzolla review binary-neutron-star initial data and evolutions in Baiotti and Rezzolla 2017, §III for evolution variables and §IV for initial data.

Irrotational and spinning fluid states

Relativistic circulation uses the specific enthalpy $h=(e+P)/n_b$ and canonical momentum $h u_\mu$. An isentropic irrotational flow has vanishing canonical vorticity,

$$\nabla_{[\mu}(h u_{\nu]})=0, \qquad h u_\mu=\nabla_\mu\Psi$$

locally for a velocity potential $\Psi$. Irrotational does not mean that the fluid is at rest in an inertial frame; it means zero circulation in this canonical sense. Helical stationarity of the potential yields a Bernoulli-type first integral, while baryon conservation supplies an elliptic equation for $\Psi$ with a no-through-flow condition at the stellar surface.

Corotating data instead take the fluid four-velocity parallel to $k^\mu$ inside each star. They are useful as a limiting construction but generally overestimate tidal locking. Controlled spinning data add a rotational component and must define spin or circulation quasi-locally; coordinate angular velocity alone is gauge-dependent. Gourgoulhon and collaborators formulate and test synchronized and irrotational sequences in Gourgoulhon et al. 2001, §§II–IV and Appendices A–B.

The continuum inputs are the component baryon masses or central enthalpies, equation of state and table version, desired circulation or spin, orbital separation or frequency, conformal and gauge choices, outer and compact-object boundary conditions, stellar-surface representation, and elliptic tolerances. A dimension check uses $GM\Omega$ and binding energy divided by total mass as dimensionless sequence variables.

Surface, orbit, and eccentricity

The fluid boundary is part of the solution: pressure vanishes at a surface whose shape is tidally deformed. Enthalpy per baryon $h$ generally tends to a nonzero surface value $h_s$; it is the log-enthalpy $\ln(h/h_s)$ that vanishes there. Spectral domains fitted to that surface can converge rapidly, but an unresolved cusp near mass shedding causes Gibbs oscillations and loss of a solution branch. Finite-difference methods must prevent atmosphere and interpolation errors from displacing the surface.

Orbital force balance can be imposed by requiring vanishing enthalpy gradient at each stellar center, and the center of mass can be adjusted to suppress linear momentum. The ADM mass, baryon mass, and total angular momentum are useful global diagnostics along a sequence; explicit expressions for a conformally flat construction appear in Baumgarte et al. 1998, §IV.A, Eqs. (59)–(66). The Newtonian limit also permits the virial check $2T+W+3\int p\,dV=0$, where $T$ is total kinetic energy and $W$ gravitational potential energy; Gourgoulhon et al. 2001, §V.C.1, Fig. 7 tests its convergence. In relativistic constructions that supply a consistent approximate time symmetry, an ADM–Komar mass comparison serves a related purpose; it is not an exact identity for an outgoing radiating binary.

Quasiequilibrium does not determine the small inspiral radial velocity exactly. Evolving nominally circular data commonly reveals residual eccentricity through oscillations in separation or orbital frequency. Iterative eccentricity reduction adjusts tangential and radial velocities using a short evolution. The measured eccentricity and fitting window must be reported rather than hidden inside the word “circular”.

Benchmark and convergence tests

At large separation, each component should recover an isolated TOV star with the same baryon mass and equation of state, while binding energy tends toward zero. Along a sequence, Hamiltonian and momentum constraint norms, ADM mass, angular momentum, orbital frequency, stellar surface, circulation, and ADM–Komar difference should remain controlled. A published dimensionless equal-mass benchmark supplies an independent reference.

A rerunnable computation must pin code commit, compiler and libraries, floating-point precision, EOS file and checksum, domain map or grid, decomposition and gauge choices, nonlinear and linear tolerances, outer boundary, spin prescription, and deterministic settings. Convergence studies refine spectral basis or grid, domain boundaries, outer radius, surface fit, and solver tolerance. Report both global norms and local surface errors; constraint convergence alone does not show that the intended masses and spins were obtained.

Each failure mode needs a diagnostic: surface Gibbs error, lost branches near mass shedding, outer-boundary contamination, inconsistent spin definitions, conformal-flatness bias, failure of the velocity-potential boundary condition, and residual eccentricity. An elliptic residual much smaller than the physical truncation error can give false confidence.

No initial-data solver was run for this page. A reproducible companion would provide one documented command that regenerates machine-readable sequence and refinement tables from a pinned software environment and fails when constraint, virial, surface, mass, circulation, benchmark, or convergence tolerances are missed.

Check a constraint convergence rate

For grid spacings $h$ and $h/2$, an $L^2$ Hamiltonian-constraint norm decreases from $8.0\times10^{-5}$ to $2.0\times10^{-5}$. Assuming $\lVert{\cal H}\rVert_2=C h^p$, estimate $p$.

Solution

The ratio is

$$\frac{\lVert{\cal H}\rVert_2(h)} {\lVert{\cal H}\rVert_2(h/2)} =4=2^p,$$

so $p=2$. This is consistent with second-order convergence over those two resolutions. It is not sufficient validation: a third resolution should test whether the rate persists, and masses, circulation, surface location, virial error, and evolution eccentricity need independent convergence checks.

References

• Baiotti, Luca, and Luciano Rezzolla. “Binary Neutron Star Mergers: A Review of Einstein’s Richest Laboratory.” Reports on Progress in Physics 80 (2017): 096901. arXiv.
• Baumgarte, Thomas W., Gregory B. Cook, Mark A. Scheel, Stuart L. Shapiro, and Saul A. Teukolsky. “General Relativistic Models of Binary Neutron Stars in Quasiequilibrium.” Physical Review D 57 (1998): 7299–7311. DOI.
• Gourgoulhon, Eric, Philippe Grandclément, Keisuke Taniguchi, Jérôme-A. Marck, and Silvano Bonazzola. “Quasiequilibrium Sequences of Synchronized and Irrotational Binary Neutron Stars in General Relativity. I. Method and Tests.” Physical Review D 63 (2001): 064029. DOI.