Black-Hole Mergers and Common Horizons

A binary merger joins two strong-field regions into one trapped region and then a relaxing remnant, but no single surface or timestamp captures the whole transition. This chapter organizes the merger by geometry: causal horizons, marginal surfaces, quasi-local fluxes, curvature and multipoles, and radiation at null infinity. Its purpose is to make those diagnostics agree where they should and remain distinct where their definitions demand it.

Helpful background. Readers will benefit from Black-Hole Geometry and Mechanics, Black-Hole Perturbation Theory, Formulations of General Relativity, and Computational Relativity. The orbital history immediately before merger is developed in Relativistic Two-Body Dynamics.

One merger, several geometrical questions

The event horizon is a global null boundary whose location depends on the full future spacetime. A marginally outer trapped surface is quasi-local but tied to a Cauchy slice. Bondi news and radiative charges live at future null infinity in a declared asymptotic frame. Coordinate tracks are gauge variables. The central discipline of the chapter is therefore to attach every claimed merger event to one of these objects.

The first common MOTS is especially useful, but it normally appears with an inner companion rather than replacing the two individual surfaces. The principal eigenvalue of the MOTS stability operator distinguishes the outer and inner branches. Multiple branches, their later interactions, and a switch in the outermost surface explain apparent-horizon “jumps” without demanding a discontinuous spacetime. The detailed head-on calculations of Pook-Kolb and collaborators make this solution-space structure concrete Pook-Kolb et al. 2021, §§ II–VI.

Two routes through the chapter

The geometric route begins with The Merger Spacetime, which separates causal, marginal-surface, waveform, and coordinate clocks. It then studies the Formation of a Common Marginal Surface and the fold-like creation of stable and unstable branches. Horizon Topology, Branching, and Jumps states the stable-MOTS topology theorem with its energy, regularity, dimension, and outermostness hypotheses.

The quantitative route begins with Nonlinear Merger and Ringdown Onset, where onset is defined through stable predictive accuracy or invariant geometric decay rather than a universal timestamp. Mass, Spin, Area, and Charge Budgets then closes asymptotic and quasi-local conservation laws with numerical uncertainty. Finally, Mass Ratio, Spin, Eccentricity, and Precession maps where parameter trends are smooth and where capture thresholds or changing encounter number create qualitatively new regimes.

These are complementary routes. The first explains what the surfaces mean; the second asks whether a computed spacetime quantitatively supports the interpretation. A researcher interested in ringdown should still read the first page before comparing start times. A reader interested in horizon topology should still consult the budget page before treating a numerical branch diagram as validated physics.

Recurring checks

Three checks recur throughout. First, definitions must remain compatible: null normals, extrinsic curvature, spin, retarded time, and asymptotic frame are declared before use. Second, convergence must be demonstrated in the observable being interpreted—not just in a coordinate trajectory. Third, independent channels should close. ADM initial charges, Bondi fluxes, and quasi-local remnant measurements must agree within an error budget; horizon multipoles and waveform models should approach a consistent remnant mass and spin Centrella et al. 2010, §§ IV–V and VIII.

The chapter does not teach production evolution algorithms, construct waveform likelihoods, or replace the spectral theory of quasinormal modes. Numerical formulation belongs to Computational Relativity, waveform modeling to Gravitational Radiation and Waveforms, and mode response to Black-Hole Perturbation Theory. Here those ingredients are evidence for the finite-amplitude spacetime transition.

What the chapter establishes

After the geometric route, a reader can distinguish event-horizon merger from common-MOTS formation, explain the paired common surfaces in the resolved head-on family, state the exact stability and energy hypotheses under which a horizon cross-section in a four-dimensional spacetime is spherical—including the exceptional flat case—and recognize branch switching rather than call every area jump a discontinuity Galloway and Schoen 2006, Theorem 2.1 and concluding remark 1, pp. 2–5. After the quantitative route, the reader can design a controlled ringdown-onset test, close mass and angular-momentum budgets, and identify extrapolations beyond a simulation catalog.

The limitations matter as much as the results. MOTS histories depend on foliation; event horizons are teleological; ringdown onset depends on observable, norm, and tolerance; and empirical parameter fits are only as broad as their calibration. None of those qualifications makes the diagnostics arbitrary. They specify which comparisons remain physically meaningful.

The geometrical and radiative closure developed here supplies the initial data for the next question: what remnant was actually formed? Continue to Recoil, Remnants, and Final-State Formation for kick velocities, final mass and spin, Kerrness diagnostics, and the scope of present final-state theorems.

Chapter review

An evolution reports, in order, a common-MOTS coordinate time, a peak of $|h_{22}|$, and a late plateau in fitted Kerr parameters. Another gauge changes the first time but not the aligned waveform or the plateau. Which event is invariant, and what evidence is still missing before claiming a controlled linear regime?

Solution

The numerical value of the first common-MOTS coordinate time is slicing dependent, although the existence and intrinsic properties of the located surface are meaningful within that foliation. The aligned asymptotic waveform and stable late remnant parameters are more robust, but neither alone supplies a universal onset. One still needs resolution and extraction errors, start-time and mode-set stability, an independently measured remnant, and preferably a horizon or curvature diagnostic. The event horizon is a separate global object and cannot be dated from the listed online diagnostics alone.

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.
• Galloway, Gregory J., and Richard Schoen. “A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions.” Communications in Mathematical Physics 266 (2006): 571–576. doi:10.1007/s00220-006-0019-z.
• 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.