Scales, Invariants, and Regimes of the Two-Body Problem

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The relativistic two-body problem is a family of regimes connected by overlaps, not a single perturbation series. This chapter supplies the coordinates of that theory space: dimensionless parameters, invariant observables, orbit classes, dynamical sectors, and timescale hierarchies. Its purpose is to let a reader choose an approximation and a validation target before beginning a long calculation.

Helpful background. Review The Einstein Equation for the common field law, Weak Metric Perturbations and Gauge Freedom for controlled expansions, Presymplectic Potentials and Currents for invariant generators, and Black-Hole Perturbation Theory for the fixed-background strong-field limit.

The regime-selection problem

A compact binary may be weakly bound but have comparable masses, strongly relativistic but have an extreme mass ratio, weakly deflected at arbitrary asymptotic speed, or slowly inspiralling close to a strong-field separatrix. These statements constrain different ratios. A method is selected only after the physical question, inputs, boundary conditions, and desired error are stated.

The recurring parameter set is

\mathcal P= (\nu,x,e,\boldsymbol\chi_1,\boldsymbol\chi_2, \Lambda_1,\Lambda_2, T_{\mathrm{orb}}/T_{\mathrm{rr}},\ldots).

Here $\nu$ measures mass symmetry, $x=(GM\Omega)^{2/3}$ labels quasi-circular orbital strength, $e$ must be tied to a particular eccentricity definition, $\boldsymbol\chi_A$ are dimensionless spins, $\Lambda_A$ characterize adiabatic quadrupolar response, and the timescale ratio controls averaging. An encounter replaces $x$ by asymptotic energy and a parameter such as $GM/b$ at fixed relative speed. At low speed, weak deflection requires the stronger condition $GM/(bv_\infty^2)\ll1$ ; at high energy the energy-dependent enhancement must also be checked.

The first route through the chapter therefore begins with Dimensionless Parameters of Compact Binaries. It explains why a small mass ratio does not imply a weak field, why weak binding does not imply slow radiation reaction in every region, and why internal resonances add a scale that an orbital compactness cannot capture.

Observables before representations

Coordinates and canonical variables make calculations possible, but comparisons require invariant relations. The second page, Gauge-Invariant Energetics and Frequencies, develops four reusable examples:

circular binding energy as a function of asymptotic orbital frequency;
radial-to-azimuthal frequency ratios and periastron advance;
redshift, spin-precession, and tidal invariants at small mass ratio;
scattering angle and radial action at fixed asymptotic energy and angular momentum.

This distinction prevents a common false disagreement. Two Hamiltonians related by a canonical transformation can have different potentials and coordinate radii while producing the same $E(\Omega)$ and $\chi(E,J)$ . Conversely, identical-looking potentials with different mass, spin, time, or boundary conventions need not describe the same physics.

The need for invariant cross-method data is central to modern two-body theory. PN dynamics and its invariant circular relations are organized in Blanchet 2024, §§1 and 3, while self-force observables and their use beyond the extreme-mass-ratio limit are reviewed in Barack and Pound 2019, §§6 and 8.

Orbit classes and dynamical sectors

Bound, Scattering, Capture, and Plunge Regimes classifies motion from asymptotic charges and radial turning-point structure. The key distinctions are global:

bound motion remains radially confined in the conservative limit;
scattering has incoming and outgoing asymptotic states;
conservative capture lacks an outgoing branch at fixed incoming data;
radiative capture occurs because losses bind a system that would scatter conservatively;
plunge begins when stable-orbit tracking fails near a separatrix.

None of these is defined by an instantaneous sign of $\dot r$ . Nor is plunge universally defined by crossing $r=6GM$ : that radius belongs to circular Schwarzschild geodesics, while spin, eccentricity, and finite mass ratio change the relevant separatrix.

Conservative, Dissipative, and Hereditary Dynamics makes an independent three-way distinction. Conservative dynamics is time symmetric but can be nonlocal in time. Dissipative dynamics is selected by retarded boundary conditions and appears in asymptotic fluxes and local radiation reaction. Hereditary terms depend on the orbit over an interval of time: retarded terms sample the past, while the time-symmetric conservative tail samples both past and future. A “nonlocal term” is therefore not automatically a loss term.

Together, these pages separate orbit geometry from dynamical mechanism. Scattering can be treated conservatively or radiatively; a bound inspiral combines conservative orbital structure with dissipative secular drift; hereditary tails affect both.

Timescale route

Readers concerned with long evolutions should continue to Adiabatic, Post-Adiabatic, and Resonant Timescales. Its central scaling is

T_{\mathrm{orb}}\ll T_{\mathrm{rr}},

which permits fast phases to be averaged while actions evolve on a slow time. Write $\varepsilon=T_{\mathrm{orb}}/T_{\mathrm{rr}}\ll1$ for the dimensionless slow-evolution parameter. The leading adiabatic phase is often $O(\varepsilon^{-1})$ , so a formally small local correction can accumulate an order-unity phase. Post-adiabatic accuracy therefore requires a bookkeeping of conservative, oscillatory dissipative, and higher-order averaged terms.

At a resonance,

\mathbf k\cdot\boldsymbol\Omega\approx0,

one Fourier mode stops averaging away. For an isolated simple crossing with nonzero detuning sweep and a nonvanishing forcing harmonic, the layer lasts $O(\varepsilon^{-1/2})$ orbital times and produces an $O(\varepsilon^{1/2})$ change in normalized actions. Tangential crossings, vanishing harmonics, and resonance trapping need separate analysis. Near a separatrix, a period itself degenerates and a different inner scaling is required. These are different breakdown mechanisms and should not be grouped under a vague label such as “nonadiabatic effects.”

Cross-method route

Researchers comparing calculations can go directly to Map of Approximation Overlaps. The page treats the major intersections:

PN with self-force in the weak-field, small-mass-ratio corner;
PN with PM through the low-velocity expansion of scattering data;
amplitudes with Hamiltonians through impulse, scattering angle, and radial action;
self-force with EOB through strong-field small- $\nu$ invariant functions;
analytic models with numerical relativity through withheld invariant benchmarks.

An overlap test specifies both small parameters, the quantities held fixed, the common observable, and the combined truncation. It also checks whether limits commute. The PM expansion at fixed Lorentz factor, for example, need not remain uniform if the high-energy limit enhances nominally higher-order terms.

This route leads naturally to the later Cross-Regime Matching, Calibration, and Consistency chapter, where dictionaries, calibration discipline, and uncertainty propagation are developed in detail.

What this chapter does not replace

This chapter does not derive high-order PN or PM dynamics, regularize a self-force, construct an EOB Hamiltonian, or evolve a numerical spacetime. It provides the definitions needed to judge those calculations. Detailed binary insertion of black-hole response belongs later in this volume, while the response itself belongs to Black-Hole Perturbation Theory. Source-independent radiation theory belongs to Gravitational Radiation and Waveforms, and production simulations belong to Computational Relativity.

After completing an appropriate route, a reader should be able to state:

the invariant physical inputs;
every small expansion parameter and what is held fixed;
the orbit and boundary-condition class;
the common observable used for validation;
the expected breakdown surface and independent limit check.

Review the chapter

1. Regime diagnosis. A binary has $\nu=10^{-5}$ , $x=0.18$ , and $T_{\mathrm{orb}}/T_{\mathrm{rr}}=10^{-4}$ . Which approximations are directly supported, and which is not guaranteed?

Solution

The small $\nu$ supports a small-mass-ratio expansion. The small timescale ratio supports adiabatic evolution away from resonances and separatrices. The value $x=0.18$ is not parametrically small, so low-order PN accuracy is not guaranteed by power counting; strong-field self-force or a validated resummed model may be preferable.

2. Transfer task. Two groups disagree on a coordinate capture radius but agree on the critical angular momentum $J_\ast(E)$ and the radiated four-momentum. What should be compared next?