Close Hyperbolic and Parabolic Encounters

An initially unbound or marginally bound compact-object encounter can scatter, become radiatively captured, or form a common horizon promptly. Near the boundary between escape and merger, the trajectory may execute several zoom–whirl cycles and radiate a large fraction of its excess orbital energy. The outcome is controlled by asymptotic energy, angular momentum, spins, and finite-size physics—not by a coordinate impact parameter alone.

Helpful background. Black-Hole Geometry and Mechanics supplies unstable circular orbits; Black-Hole Perturbation Theory supplies radiative loss in limiting regimes; Formulations of General Relativity supplies initial data; Relativistic Two-Body Dynamics supplies scattering invariants; Computational Relativity supplies numerical evolution and validation.

Outcome variables at infinity

For asymptotically flat data, characterize the incoming state by total center-of-mass energy $E_{\rm ADM}$, total angular momentum $J_{\rm ADM}$, mass ratio, and spin vectors. In a weak-field incoming region one may define

$$b=\frac{J_{\rm orb}}{p_\infty}, \qquad \gamma_\infty=\frac{E_1E_2+\mathbf p_\infty^2}{m_1m_2},$$

Here $E_i=\sqrt{m_i^2+p_\infty^2}$ and $c=1$. The parameter $b$ is meaningful only after the asymptotic frame and spin–orbital split are declared. Useful outputs include the scattering angle in an aligned asymptotic frame, Bondi energy–momentum flux, final horizon charges, and the existence of a common MOTS in a specified foliation. Common-MOTS time and appearance are slicing dependent; they are not the global event-horizon definition of capture.

For Newtonian conics, positive orbital energy gives a hyperbola and zero energy a parabola. Dissipation can change that classification. For a weakly bound or unbound encounter with small net recoil and two distinguishable bodies after the first passage, define excess orbital energy relative to the sum of their rest masses. To leading order in this bookkeeping,

$$\Delta E_{\rm out}^{\rm orb} \simeq\Delta E_{\rm in}^{\rm orb} -E_{\rm rad}^{\infty}-\Delta m_1-\Delta m_2.$$

Absorption raises the bodies’ rest masses by $\Delta m_i$ in the positive-absorption case; that energy has left the orbital sector but remains in the total system. Thus an initially unbound pair can be captured when the combined radiative and absorptive orbital losses exceed its incoming excess energy. Do not subtract absorption a second time from the total ADM-minus-radiation balance. With appreciable recoil, first calculate the invariant mass of the remaining system from its four-momentum, rather than treating energy in the original frame as its center-of-mass energy. Spins, residual tides, and surviving matter require additional terms or definitions. At still smaller angular momentum or pericenter, a common horizon can form promptly.

Zoom–whirl near the threshold

The geodesic limit provides the organizing benchmark. Near an unstable circular orbit, a trajectory approaches along a “zoom,” completes $N$ nearly circular “whirls,” and escapes or plunges. For a geodesic one-parameter family whose initial radial displacement from the critical orbit is proportional to $p-p_*$, linearization in Boyer–Lindquist time gives

$$N\simeq-\frac{\Omega_{\rm orb}}{2\pi\lambda} \log|p-p_*|+\text{constant},$$

where $p$ is a one-parameter initial-data label, $p_*$ its threshold value, and $\lambda$ the orbit’s Lyapunov exponent. Comparable-mass dynamics modifies the coefficients and the background interpretation, but logarithmic sensitivity and repeated whirls provide a useful diagnostic.

The coefficient $\Omega_{\rm orb}/(2\pi\lambda)$ is therefore a geodesic result in a specified time normalization, not a universal comparable-mass exponent. Equal-mass nonspinning simulations found analogous logarithmic behavior for one tuned data family, with a fitted exponent and finite-resolution uncertainty Pretorius and Khurana 2007, Eq. (9), Figs. 1–2, and §IV.A, pp. 3–6.

The black-hole scattering problem and its relation to large-eccentricity simulations are reviewed in Pretorius 2007, §II.A.2, p. 7, §IV.D, pp. 26–28, and §V.B, pp. 31–32. Strong-field simulations find threshold behavior analogous to unstable circular motion, but no coordinate track should be used as the sole evidence.

Continuum computation and benchmark ladder

The continuum problem is the vacuum Einstein equation, or Einstein–matter system for extended bodies, with constraint-satisfying asymptotically flat initial data. A reproducible calculation records formulation, gauge, initial-data solver, masses, spins, boosts, separation, boundary conditions, grid or spectral representation, horizon finder, and radiation extraction.

Use a benchmark ladder:

1. large impact parameter: compare scattering angle and radiation with post-Minkowskian or perturbative results;
2. extreme mass ratio: compare the threshold and whirl frequency with Kerr geodesics plus self-force corrections;
3. quasi-circular limit: recover established merger budgets;
4. finite resolution: close ADM, horizon, and radiative energy–angular-momentum balances.

At least three effective resolutions or an equivalent tolerance sequence should converge the critical impact parameter, scattering angle, common-horizon time, and radiated flux. Threshold uncertainty must be separated from discretization error: near $p_*$, tiny initial-data changes produce physically large differences.

Failure modes and reproducibility boundary

A principal failure mode is gauge-dependent classification from puncture tracks. Other systematic errors include junk radiation changing $E_{\rm in}$, finite initial separation, outer-boundary reflection, under-resolved horizons during close passage, ambiguous asymptotic-frame alignment, and failure to search for a common marginal surface. Near threshold, a simulation that ends before the final zoom cannot classify escape versus capture.

No simulation was rerun for this page. Published outcomes are accepted only within their documented data families and convergence tests. A later rerun should start from clean recorded initial data, reproduce a weak-field or geodesic benchmark, regenerate invariant fluxes and horizon diagnostics, and retain machine-readable convergence data.

Check the outcome

An encounter has incoming excess orbital energy $\Delta E_\infty=0.015M$, radiates $0.012M$ to infinity, and deposits $0.006M$ through horizons without immediate common-horizon formation. Is the outgoing orbit unbound?

Solution

The orbital sector loses $0.018M\gt\Delta E_\infty$. Its post-encounter energy is negative by $0.003M$, so the pair is captured by radiation and absorption even though no common horizon formed during first passage. It may return and merge later. The calculation assumes negligible recoil, that the absorbed energy raises the individual rest masses as in the leading balance above, and a budget closing within less than $0.003M$ uncertainty. Only the $0.012M$ radiated to infinity is lost from the total system energy.

References

• Pretorius, Frans, “Binary Black Hole Coalescence”, in Physics of Relativistic Objects in Compact Binaries, Springer, pp. 305–369, 2009. Open PDF.
• Pretorius, Frans, and Deepak Khurana, “Black Hole Mergers and Unstable Circular Orbits”, Classical and Quantum Gravity 24, S83–S108, 2007. Open PDF.

Further reading

• Gold, Roman, and Bernd Brügmann, “Eccentric Black Hole Mergers and Zoom–Whirl Behavior from Elliptical Inspirals to Hyperbolic Encounters”, Physical Review D 88, 064051, 2013. Open PDF.