The Intermediate Binary Black Hole (IBBH) problem - computing the gravitational waveform from the last ~10 orbits of binary-black-hole inspiral, between the failure of post-Newtonian methods and the reach of then-standard numerical relativity - was posed by Brady, Creighton, and Thorne in 1998 together with a specific solution program: evolve Einstein's equations in coordinates co-rotating with the binary, with lapse and shift fixed on each slice by variational 'minimal strain' elliptic equations that lock the coordinates onto the spacetime's almost-Killing vector, the ADM system parabolized by discarding small second time derivatives, matched-asymptotic initial data, and Cauchy-characteristic-type boundaries. The authors themselves listed what remained unproven: attractor behavior of the gauge, well-posedness of the elliptic boundary-value problems, fidelity of the parabolized evolution, the effect of the almost-Killing vector becoming spacelike beyond the speed-of-light surface, the matched initial data, and the practical machinery. The subsequent literature shows: the elliptic gauge equations were largely settled (strong ellipticity, existence, and uniqueness conditions) by 2000, matched initial data were constructed for nonspinning holes, and the goal-level IBBH problem was solved in 2005-2006 by generalized-harmonic and moving-puncture methods that do not use co-rotation. What remains open is the dynamical core of the source question: whether the coupled Einstein-minimal-strain co-rotating evolution is well posed, stable, and attracting, the size of the parabolization error, and the rigorous treatment of the superluminal gauge region - now a question of mathematical viability and historical interest rather than practical capability.

Background

Gravitational waves from the inspiral and merger of binary black holes (BBHs) are prime sources for interferometric detectors such as LIGO and VIRGO, and optimal searches require accurate waveform models. In the late 1990s the last ∼10\sim 10 orbits before merger (∼100\sim 100 radians of gravitational-wave phase) formed a computational gap: the post-Newtonian (PN) expansion (an approximation in powers of the orbital speed vv) was estimated to fail at v≃0.3v\simeq 0.3, while numerical relativity in standard, asymptotically inertial coordinates could not afford the ≳1200\gtrsim 1200 dynamical timescales (each of order the total mass MM) needed to evolve the binary across the gap, since the holes sweep across the grid on the short orbital timescale 1/Ω1/\Omega. Brady, Creighton, and Thorne named this the Intermediate Binary Black Hole (IBBH) problem and proposed a concrete program to solve it.

Numerical relativity decomposes spacetime into spatial slices of constant time, writing the metric as ds2=−α2dt2+γij(dxi+βidt)(dxj+βjdt)ds^2=-\alpha^2dt^2+\gamma_{ij}(dx^i+\beta^idt)(dx^j+\beta^jdt). The freely specifiable lapse α\alpha and shift βi\beta^i fix the coordinate system (the gauge), while the three-metric γij\gamma_{ij} and the extrinsic curvature KijK_{ij} (which measures how γij\gamma_{ij} changes from slice to slice) evolve via the ADM (Arnowitt-Deser-Misner) equations, the standard evolution equations for this 3+1 split. Smarr and York had earlier proposed 'minimal distortion'/'minimal strain' conditions: elliptic equations for the shift that minimize the size of γ˙ij\dot\gamma_{ij}, partly so that coordinates track the symmetry of a stationary spacetime - its Killing vector, the vector field whose flow leaves the metric unchanged. The IBBH spacetime has no exact Killing vector but an 'almost Killing vector' ξ\xi with Lξg∼λ/τ∗\mathcal{L}_\xi g\sim\lambda/\tau_* (Lξg\mathcal{L}_\xi g denotes the Lie derivative of the metric along ξ\xi, the amount by which ξ\xi fails to preserve the metric), where λ∼M\lambda\sim M is the curvature length scale and τ∗\tau_* the slow radiation-reaction timescale on which the orbit shrinks.

The Brady-Creighton-Thorne program: (i) on each slice, determine α\alpha and βi\beta^i from a variational principle minimizing ∫γ˙ijγ˙klγikγjld3x\int \dot\gamma_{ij}\dot\gamma_{kl}\gamma^{ik}\gamma^{jl}d^3x (the 'minimal strain equations', their Eqs. (5)) or a spacetime-integral variant (their Eq. (11)); solving these elliptic equations should lock ∂t\partial_t onto the almost Killing vector up to fractional errors λ/τ∗\lambda/\tau_*, so all metric coefficients evolve on the slow timescale τ∗\tau_*. (ii) Parabolize the ADM system: in co-rotating coordinates, terms with two time derivatives are smaller than single-derivative terms by the factor 1/(Ωτ∗)1/(\Omega\tau_*), so they can be discarded or back-differenced and the resulting parabolic system integrated implicitly with steps of order τ∗\tau_*, 'freezing out' spurious short-wavelength numerical modes that would otherwise force Courant-limited steps (steps kept small by a numerical stability condition on the ratio of time step to grid spacing). (iii) Build initial data just before PN failure by matched asymptotic expansions (joining the PN exterior metric onto the metrics of two tidally distorted Kerr black holes and transforming to co-rotating coordinates). (iv) Treat inner and outer boundaries by Cauchy-characteristic matching (matching the interior Cauchy evolution to an exterior evolution along outgoing null hypersurfaces) together with two-surface analogs of the gauge equations.

The authors themselves flagged the program's load-bearing uncertainties: they had not shown that the desired gauge solution is an attractor of the coupled evolution; the well-posedness of the elliptic boundary-value problems was assumed without proof; the fidelity of the parabolized evolution to the true Einstein evolution was unquantified; the almost-Killing vector becomes spacelike beyond the 'speed-of-light surface' (where the co-rotating coordinates rotate faster than light), with unknown consequences for the nonlinear evolution; the matched initial data were only sketched; and the practical machinery (per-slice elliptic solvers, three-dimensional characteristic matching, outer-boundary reflections) remained to be built.

The subsequent history frames the question at issue. Within two years the elliptic gauge equations were analyzed rigorously (existence, uniqueness, and degeneracy conditions; uniqueness was shown to hold exactly when a certain conformal-Killing-type Dirichlet kernel is empty), a derivative-loss concern was recorded for the coupled evolution (the minimal-strain lapse is a quotient of contractions of KijK_{ij}, so the evolved KijK_{ij} loses two derivatives per step), and negative numerical evidence appeared in spherical toy models. The matched initial-data ingredient was realized for nonspinning holes. Then, in 2005-2006, the goal-level IBBH problem was solved by different techniques: Pretorius evolved a BBH through plunge, merger, and ringdown using generalized harmonic coordinates (coordinates satisfying a wave equation, making the evolution manifestly hyperbolic), and the moving-puncture method (representing each hole by a puncture in the conformal metric that moves through the grid) achieved the same without excision or a co-rotating shift. Numerical-relativity waveforms now routinely cover the late inspiral, so the practical motivation for the co-rotating program has disappeared. What was never settled is the dynamical stability of the scheme itself - the question at issue here.

Problem Statement

As posed in the source, the question is whether the Brady-Creighton-Thorne computational program for the IBBH regime can be carried out and trusted: an evolution of Einstein's equations in coordinates co-rotating with the binary, with lapse and shift determined on each slice by the variational minimal-strain equations (Eqs. (5)) or the spacetime-integral variant (Eq. (11)) together with the proposed inner and outer boundary conditions, the ADM system parabolized by discarding or back-differencing the second time derivatives (justified by the small factor 1/(Ωτ∗)1/(\Omega\tau_*)), matched-asymptotic initial data, and Cauchy-characteristic-type boundaries, yielding waveforms accurate enough for detector data analysis. The subsequent literature resolved the elliptic well-posedness ingredient in substantial part, constructed the matched initial data for nonspinning holes, and solved the goal-level IBBH problem by other methods, so the surviving open core of the source question is the following. For the coupled system consisting of the full nonlinear Einstein evolution together with the minimal-strain (or Eq. (11)) gauge equations and the source's proposed boundary conditions: (a) prove or disprove that the evolution is well posed and dynamically stable, with the co-rotating gauge solution acting as an attractor for arbitrary reasonable initial choices of the lapse and shift - or exhibit a definite obstruction; (b) quantify the error introduced by the parabolized evolution relative to the true Einstein evolution, to the accuracy needed for gravitational-wave phase prediction over the last ∼10\sim 10 orbits (in the cited source, the remaining inspiral waveform is ≳100\gtrsim 100 radians of phase, and an error of more than 2 radians seriously affects LIGO/VIRGO data analysis); and (c) analyze rigorously the effect on the nonlinear evolution of the almost-Killing vector becoming spacelike beyond the speed-of-light surface. These three components jointly answer the single question of whether the symmetry-seeking co-rotating scheme is mathematically viable as a numerical-relativity method; a decisive affirmative of all three, or a definitive obstruction to any one of them, closes the question. The practical urgency is gone - IBBH waveforms are now computed by generalized-harmonic and moving-puncture techniques - so this is a question about the mathematical viability and historical soundness of the specific scheme, not about current capability.

References

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