Capture, Scattering, and Strong-Field Encounters

Strong encounters connect asymptotic scattering data to unstable orbital structure, radiative capture, and possible merger. Their clean inputs are center-of-mass energy, angular momentum, impact parameter, asymptotic spins, and object response. Their central boundary is a separatrix: on one side an outer turning point produces scattering, while on the other the trajectory is captured. Near that boundary, zoom–whirl residence amplifies radiation and finite-size effects.

Helpful background. The Einstein Equation fixes the spacetime, Weak Metric Perturbations and Gauge Freedom supplies perturbative comparisons, Presymplectic Potentials and Currents supplies charges, and Computational Relativity supplies nonlinear benchmarks.

Enter strong-field encounter dynamics

For nonspinning asymptotic data, define

$$p_\infty =\frac{ \sqrt{ [E_{\mathrm{cm}}^2-(m_1+m_2)^2] [E_{\mathrm{cm}}^2-(m_1-m_2)^2] } }{2E_{\mathrm{cm}}}, \qquad b=\frac{J}{p_\infty}.$$

Near the parabolic limit $p_\infty\to0$, impact parameter diverges at fixed $J$, so energy and angular momentum are the regular labels. For spinning bodies the SSC, centroid, and asymptotic spin directions must also be declared.

A conservative capture threshold occurs when the radial potential has a double root. Invariantly, an outer turning point disappears. A dissipative encounter additionally changes energy and angular momentum:

$$E_{\mathrm{out}} =E_{\mathrm{in}}-\Delta E_{\mathrm{rad}}, \qquad J_{\mathrm{out}} =J_{\mathrm{in}}-\Delta J_{\mathrm{rad}}.$$

Thus an orbit can scatter conservatively but become bound after radiation, or cross directly into a plunge domain after losing angular momentum.

The six-page route

1. Relativistic Capture Thresholds defines the separatrix using double roots and derives exact Schwarzschild null and marginally bound benchmarks.
2. Zoom–Whirl Dynamics Near Separatrices derives logarithmic whirl counts and relates their slope to an unstable-orbit Lyapunov exponent.
3. Relativistic Hyperbolic and Parabolic Encounters organizes unbound and marginally bound motion and recovers Newtonian conic scattering.
4. Radiative Capture and Dynamical Binary Formation derives the capture inequality and the weak-field velocity scaling of the cross section.
5. Spinning and Finite-Size Strong Encounters compares point-particle, spin-multipole, response, perturbative, and nonlinear descriptions through the 2026-08-12 cutoff.
6. Analytic–Numerical Comparisons for Strong Scattering defines a reproducible environment, invariant alignment, double extrapolation, benchmarks, and pass criteria.

The route moves from conservative classification to critical dynamics, unbound kinematics, dissipative formation, body dependence, and finally a validation workflow.

Critical amplification

Near a conservative threshold $J_c(E)$, an encounter shadows an unstable orbit and its scattering angle grows as

$$\chi(E,J) \simeq -\frac{\Omega_{\phi,c}}{\lambda_t} \ln\left\lvert \frac{J-J_c}{J_0} \right\rvert+\chi_0.$$

Here $\Omega_{\phi,c}$ is the azimuthal frequency of the unstable circular orbit and $\lambda_t$ its radial Lyapunov exponent, both measured in the same asymptotically normalized time. The constants $J_0$ and $\chi_0$ set the nonuniversal scale and offset. The same long residence enhances radiation reaction, absorption, spin precession, and tides. A correction that is small relative to $J_c$ can still change the outcome if it is comparable with $\lvert J-J_c\rvert$. This is the chapter’s central validity rule:

$$\lvert\delta J_c^{\mathrm{omitted}}\rvert \ll\lvert J-J_c\rvert.$$

When it fails, scatter-versus-capture is unresolved at that approximation order. More digits in a coordinate trajectory cannot repair the model error.

The geodesic and comparable-mass relation between scattering and unstable orbits is discussed in Damour 2016, §§III–VIII and Levin and Perez-Giz 2008, §§II–IV.

Choosing an approximation

• PN expansion requires slow motion and is strongest for weak-field hyperbolic or nearly parabolic passages not too close to periapsis relativistic breakdown.
• PM expansion keeps arbitrary velocity but expands in $G$ and is naturally organized by scattering at modest deflection.
• Self-force resolves small-mass-ratio strong-field corrections.
• EOB maps and resums specified PN, PM, and self-force information and may be calibrated to numerical data.
• Numerical relativity supplies finite-amplitude strong-field dynamics but carries resolution, extraction, initial-data, and coverage errors.

These approaches overlap on invariant angles, radiated four-momentum, and critical data. Calibration points must be separated from validation points, and every numerical comparison must reconstruct incoming invariants after junk radiation.

Scope and outcomes

Possible outcomes include one-pass scattering, a radiatively bound eccentric binary, prompt plunge or merger, tidal disruption, and matter-producing collision. The chapter treats the approach to these outcomes and the invariant boundaries among analytic regimes. Finite-amplitude merger spacetimes belong to black-hole dynamics, while production simulations belong to computational relativity.

Check your understanding

Why can a small finite-size correction invalidate a near-critical point-particle classification even when the body-size expansion is globally small?

Solution

The outcome depends on the sign of $J-J_c$. A finite-size correction shifts $J_c$. If that shift is comparable with the small distance from the point-particle threshold, it can reverse the sign and change the predicted outcome. Control requires the omitted shift to be small compared with the distance to threshold, not merely with $J_c$ itself.

References

• Damour, Thibault. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94 (2016): 104015. DOI.
• Levin, Janna, and Gabe Perez-Giz. “A Periodic Table for Black Hole Orbits.” Physical Review D 77 (2008): 103005. DOI.

Further reading

• Pretorius, Frans, and Deepak Khurana. “Black Hole Mergers and Unstable Circular Orbits.” Classical and Quantum Gravity 24 (2007): S83–S108. DOI.