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Finite-Amplitude Tidal Dynamics and Strong-Field Encounters

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Strong encounters connect two controlled limits—a weak external tide and a widely separated scattering orbit—to capture, disruption, and merger, where neither limit alone controls the spacetime. The bridge must be built from dimensionless regime parameters, quasi-local horizon fluxes, asymptotic charges, and converged nonlinear calculations. Coordinate tracks and visual deformation can guide intuition, but they are not the shared observables.

Helpful background. Black-Hole Geometry and Mechanics supplies horizons and capture scales; Black-Hole Perturbation Theory supplies weak-response limits; Formulations of General Relativity supplies nonlinear variables; Relativistic Two-Body Dynamics supplies orbital approximations.

From response to outcome

Section titled “From response to outcome”

Nonlinear Tidal Deformation separates tidal amplitude, induced displacement, and driving-frequency parameters. It also distinguishes a material body’s deformation from a black hole’s quasi-local horizon geometry.

Close Hyperbolic and Parabolic Encounters classifies scattering, radiative capture, prompt merger, and zoom–whirl behavior using asymptotic energy and angular momentum. Horizon Absorption at Finite Amplitude replaces a low-order heating formula with exact dynamical-horizon balance on smooth branches.

Mode Coupling and Nonlinear Resonance derives frequency and angular selection rules and explains why combination tones need not be resonant. Tidal Disruption and the Capture Boundary compares the supplied material self-gravity scale with relativistic capture. Matching Perturbative and Full Nonlinear Dynamics defines a common invariant and uncertainty-controlled overlap test.

The scale hierarchy

Section titled “The scale hierarchy”

Let body 1 have mass M1M_1, radius R1R_1, and characteristic internal-mode frequency ω∗\omega_*, while body 2 has mass M2M_2 and instantaneous separation rr; Ωdrive\Omega_{\rm drive} is the tidal driving frequency. In a black-hole–material-body encounter define q=MBH/M∗q=M_{\rm BH}/M_* and material compactness C∗=GM∗/(c2R∗)\mathcal C_*=GM_*/(c^2R_*). For coupled modes, Δω\Delta\omega is the frequency mismatch and γ\gamma the daughter-mode damping rate. Four ratios recur:

ϵtide∼M2M1(R1r)3,ϵad=Ωdriveω∗,\epsilon_{\rm tide} \sim\frac{M_2}{M_1} \left(\frac{R_1}{r}\right)^3, \qquad \epsilon_{\rm ad} =\frac{\Omega_{\rm drive}}{\omega_*}, rtGMBH/c2∼1C∗q−2/3,∣Δω∣γ.\frac{r_t}{GM_{\rm BH}/c^2} \sim\frac{1}{\mathcal C_*}q^{-2/3}, \qquad \frac{|\Delta\omega|}{\gamma}.

They answer different questions: Is the tide weak? Is it slow? Does disruption begin outside the black-hole scale? Is a nonlinear coupling coherent for longer than its damping time? A single label such as “strong field” does not replace this map.

Shared observables

Section titled “Shared observables”

Across analytic and nonlinear descriptions, prefer:

  • ADM or Bondi energy and angular momentum in a declared asymptotic frame;
  • scattering angle χ(E,J)\chi(E,J);
  • invariant frequency–energy relations;
  • horizon area, spin, shear flux, and intrinsic multipoles;
  • material rest mass outside the horizon with a declared unboundness criterion;
  • asymptotic radiation after frame alignment.

The exact horizon balance gives a local absorption measure Ashtekar and Krishnan 2004, §§3.2–3.3 and 4.2, pp. 16–36. Scattering and strong merger calculations require numerical validation and asymptotic budgets Centrella et al. 2010, §§IV–VI, pp. 8–33, with strong-encounter diagnostics summarized in Pretorius 2007, §II.A.2, p. 7, and §IV.D, pp. 26–28. For disruption by a material body, the corresponding regime and numerical systematics are reviewed in Kyutoku, Shibata, and Taniguchi 2021, §§2.4 and 3.3–3.4.

Testing a numerical encounter

Section titled “Testing a numerical encounter”

For each computational outcome, state continuum equations, initial-data family, physical parameters, symmetry, boundary conditions, resolution or tolerance sequence, diagnostic definitions, and uncertainties. Benchmark against geodesics, perturbation theory, equilibrium stars, or weak-field scattering as appropriate. Monitor constraints and close energy–angular-momentum budgets.

Important failure modes include coordinate-track classification, junk radiation, outer-boundary reflection, unresolved horizons or stellar surfaces, atmosphere effects, ambiguous asymptotic frames, and calibration on validation data. No simulation was rerun in this chapter.

Chapter diagnostic

Section titled “Chapter diagnostic”

An extended body has ϵtide=0.03\epsilon_{\rm tide}=0.03 but ϵad=1.1\epsilon_{\rm ad}=1.1. A nonlinear simulation and a static-tide model disagree by 20%. Does the disagreement demonstrate nonlinear-amplitude response?

Solution

No. The tidal amplitude is small enough that amplitude nonlinearity may remain subleading, while ϵad=1.1\epsilon_{\rm ad}=1.1 shows that the static or adiabatic approximation has failed. The first comparison should use frequency-dependent linear response. Only residual disagreement that scales nonlinearly with amplitude, survives invariant matching, and exceeds numerical and truncation uncertainties supports nonlinear-amplitude response.

Scope boundary

Section titled “Scope boundary”

This chapter takes material equations of state and transport as inputs and develops finite-amplitude spacetime mechanisms, invariant distinctions between outcomes, and tests for comparing approximations. Computational resources explain numerical implementation; the treatments of perturbative Love numbers and waveform construction develop the corresponding linear response and radiation calculations. Once a common marginally trapped surface forms, continue to Black-Hole Mergers and Common Horizons for common-horizon formation, merger geometry, and the transition to ringdown.

References

Section titled “References”