The Merger Spacetime

A binary black-hole merger is one spacetime, not a sequence of coordinate snapshots in which two objects simply touch. Different questions select different events: the event horizon has a global causal history; marginally outer trapped surfaces depend on the foliation; waveform peaks depend on the asymptotic frame and extraction procedure; and coordinate trajectories depend on gauge. A controlled account therefore names the diagnostic before naming a “merger time.”

Helpful background. The distinctions among horizon types build on Black-Hole Geometry and Mechanics; the late-time linear description belongs to Black-Hole Perturbation Theory; evolution variables and gauges come from Formulations of General Relativity; validation principles are developed in Computational Relativity; and the inspiral-to-plunge interface is treated in Relativistic Two-Body Dynamics.

Four clocks for one spacetime

Let $(\mathcal M,g_{ab})$ be an asymptotically flat vacuum spacetime evolved from binary-black-hole initial data. In a numerical foliation $\{\Sigma_t\}$, find the first slice containing a closed marginally outer trapped surface $S$ enclosing both individual holes, with $\theta_{(\ell)}=0$ and the inward expansion negative. This supplies a common-MOTS time $t_{\rm MOTS}$; the individual holes already have marginal surfaces, so searching for any MOTS would answer a different question. That number changes if the same spacetime is sliced differently. It is nevertheless useful because it marks when a particular Cauchy data set first contains a common marginal surface and initiates a quasi-local horizon history.

The event horizon $\mathcal H^+=\partial J^-(\mathscr I^+)$ is invariant once the conformal completion and future evolution are fixed. Its generators generally join along a crease set before any chosen slice first displays a common apparent horizon. Locating it requires future information, so it is a retrospective causal diagnostic, not an online signal available during evolution. Pook-Kolb and collaborators show explicitly how the common marginal surfaces and their stability structure differ from this global object in a head-on merger Pook-Kolb et al. 2021, §§ II–VI.

At future null infinity one may define a radiative time, for example the maximum of the Bondi news norm or strain amplitude in a fixed Bondi frame. A strong-field curvature diagnostic can define another time, such as the onset of decreasing horizon shear or approximate Kerr multipole relations. Neither is guaranteed to coincide with $t_{\rm MOTS}$. Finally, a coordinate time at which puncture tracks meet has no invariant status and should not be used without a demonstrated relation to a geometric observable.

Continuum problem and numerical evidence

The continuum problem is the vacuum Einstein initial-value problem with asymptotically flat constraint-satisfying data, a declared mass ratio, spin vectors, eccentricity estimate, and no incoming radiation condition to the accuracy intended. A numerical realization additionally chooses formulation, gauge, outer boundary treatment, mesh or spectral truncation, excision or punctures, and horizon and waveform diagnostics. The physical outputs are invariant or asymptotically defined quantities: marginal-surface geometry and stability eigenvalues, Bondi fluxes, quasi-local mass and spin, curvature scalars, and constraint residuals—not coordinate separation alone.

A useful benchmark is a symmetric, nonspinning head-on merger for which reflection symmetry, vanishing recoil, area increase, and a well-resolved sequence of marginal surfaces give discriminating checks. A quasi-circular case then tests radiation extraction and remnant budgets. The broad numerical-relativity workflow and its validation standards are summarized by Centrella and collaborators Centrella et al. 2010, arXiv PDF §§ IV–V, pp. 8–27. One important failure mode is a local elliptic horizon finder locking onto only one branch because its convergence depends on the initial guess; branch-completeness tests therefore require continuation or multiple seeds Thornburg 2007, §§ 3.1.7, 3.2.5, and 3.3.1. Other failure modes include gauge transients masquerading as merger features and finite-radius waveform extraction shifting the inferred radiative clock.

Convergence must be shown in the diagnostic being interpreted. At least three resolutions—or an equivalently discriminating spectral-tolerance sequence—should demonstrate decreasing constraint norms and stable values of common-surface formation time after converting to a specified asymptotic time. The error budget separates truncation error, finite-radius or extraction error, initial-data junk radiation, horizon-finder tolerance, and gauge alignment. A coordinate trajectory may look converged while an inferred causal ordering remains gauge contaminated.

This page reports published continuum formulations and validation requirements; it does not claim a local rerun. Reproducibility would require simulation identifiers, initial-data parameters, formulation and gauge, resolution hierarchy, extraction method, horizon-finder settings, code revision, and machine-readable diagnostics. Failure modes include missed inner MOTS branches, boundary reflections, under-resolved near-horizon gradients, loss of constraint convergence, waveform time-shift ambiguity, and comparing quantities in inequivalent asymptotic frames.

A diagnostic hierarchy

The safest interpretation is hierarchical. First establish that all quoted surfaces and waveforms arise from a convergent solution of the same continuum problem. Next state whether an event horizon, a MOTS world tube, or an asymptotic radiation feature is being used. Then state the foliation or Bondi-frame dependence. Only afterward compare times or claim a transition.

For example, “the common apparent horizon forms before the waveform peak” is meaningful only after the slicing, retarded-time construction, and propagation offset are specified. It does not imply that the event horizon forms at either time. Conversely, a waveform peak need not locate a sharp local event in the strong-field region: radiation reaching $\mathscr I^+$ integrates source dynamics and propagation. The physically robust statement is a relation between well-defined diagnostics with an uncertainty, not a universal timestamp.

Check your understanding

Suppose two evolutions of the same physical initial data use different lapse conditions. Their first common-MOTS coordinate times differ by $8GM/c^3$, where $M$ is the binary’s total mass, but after waveform alignment their Bondi news and remnant budgets agree within error. What has changed, and what has not?

Solution

The coordinate labels of the slices and therefore the reported value of $t_{\rm MOTS}$ have changed. The existence and detailed foliation of the marginal tube may also be sampled differently. Agreement of Bondi radiation and closed remnant budgets indicates that the asymptotic physical solution agrees within uncertainty. One cannot infer a physical delay of $8GM/c^3$, nor identify either common-MOTS time with event-horizon formation. The comparison should report the slicing-dependent MOTS event separately from the aligned asymptotic observables.

References

• Centrella, Joan, John G. Baker, Bernard J. Kelly, and James R. van Meter. “Black-Hole Binaries, Gravitational Waves, and Numerical Relativity.” Reviews of Modern Physics 82 (2010): 3069–3119. doi:10.1103/RevModPhys.82.3069.
• Pook-Kolb, Daniel, Ofek Birnholtz, José Luis Jaramillo, Badri Krishnan, and Erik Schnetter. “Horizons in a Binary Black Hole Merger I: Geometry and Area Increase.” Physical Review D 103 (2021): 026005. doi:10.1103/PhysRevD.103.026005.
• Thornburg, Jonathan. “Event and Apparent Horizon Finders for 3+1 Numerical Relativity.” Living Reviews in Relativity 10 (2007): 3. doi:10.12942/lrr-2007-3.