Relativistic Capture Thresholds

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A conservative encounter with a radial reduction scatters if its incoming branch has an accessible outer turning point and plunges if none exists outside the horizon. The boundary is a separatrix where the radial potential develops a double root. The potential depends on coordinates, but the existence and multiplicity of this root survive smooth invertible changes of radial coordinate. Asymptotic energy, angular momentum, spin data, and body response determine the threshold; a coordinate “capture radius” does not.

Helpful background. The Einstein Equation fixes the spacetime, Weak Metric Perturbations and Gauge Freedom supplies perturbative comparisons, Presymplectic Potentials and Currents supplies invariant charges, Computational Relativity supplies nonlinear benchmarks, Relativistic Two-Body Dynamics supplies scattering invariants, and Black-Hole Geometry and Mechanics supplies horizon and geodesic structure.

The invariant separatrix criterion

For equatorial Schwarzschild geodesics of unit rest mass, define specific energy $\mathcal E$ and specific angular momentum $\ell$ . With $u=GM/r$ ,

\left(\frac{dr}{d\tau}\right)^2 =\mathcal E^2 -\left(1-2u\right) \left(1+\frac{\ell^2}{r^2}\right) \equiv R(r;\mathcal E,\ell).

An incoming orbit scatters when $R=0$ has an accessible outer root. At the critical orbit,

R(r_c;\mathcal E,\ell_c)=0, \qquad \partial_rR(r_c;\mathcal E,\ell_c)=0.

Solving gives the unstable circular orbit associated with that incoming energy. On one side, the trajectory turns around; on the other, it crosses the horizon. The double-root statement survives radial coordinate changes even though the number assigned to $r_c$ does not.

This criterion can be solved explicitly. Put $\rho=r/(GM)$ and $j=\ell/(GM)\geq0$ , choosing the azimuthal orientation so that the angular momentum is nonnegative. Differentiating $V=(1-2/\rho)(1+j^2/\rho^2)$ at fixed $j$ gives

j_c^2=\frac{\rho_c^2}{\rho_c-3}, \qquad \mathcal E^2=\frac{(\rho_c-2)^2}{\rho_c(\rho_c-3)}.

For an incoming massive particle, $\mathcal E\geq1$ , the relevant root is

\rho_c= \frac{8}{4-3\mathcal E^2+\mathcal E\sqrt{9\mathcal E^2-8}}, \qquad 3\lt\rho_c\leq4.

The expression has been rationalized to remain regular at $\mathcal E=1$ . The second derivative $V''(\rho_c)=2(\rho_c-6)/[\rho_c^3(\rho_c-3)]$ is negative on this branch: the orbit is a maximum of the effective potential, hence unstable. An incoming particle with $j\gt j_c$ encounters a barrier and turns back; $j\lt j_c$ permits passage toward the horizon. This is a test-particle, nonspinning result with radiation reaction neglected, not a formula for the capture threshold of two comparable masses.

For a massive particle arriving from infinity with speed $v_\infty$ ,

\mathcal E=\gamma_\infty, \qquad \ell=\gamma_\infty v_\infty b,

where $b$ is the impact parameter defined in the asymptotic inertial frame. The critical impact parameter is therefore

b_c(\mathcal E) =\frac{\ell_c(\mathcal E)} {\sqrt{\mathcal E^2-1}}.

In a comparable-mass problem, replace specific test-body quantities by total center-of-mass energy and angular momentum, and state the mass normalization used in $b=J/p_\infty$ .

Null and marginally bound benchmarks

For a Schwarzschild null geodesic,

\left(\frac{dr}{d\lambda}\right)^2 =E^2-\frac{L^2}{r^2} \left(1-\frac{2GM}{r}\right).

The double root occurs at

r_c=3GM, \qquad b_c=\frac{L}{E}=3\sqrt3\,GM.

This photon-sphere result is an exact benchmark for the ultrarelativistic test-mass limit.

For a marginally bound massive orbit, $\mathcal E=1$ . Solving the two conditions yields

r_c=4GM, \qquad \ell_c=4GM.

The impact parameter diverges as $v_\infty\to0$ because $b=\ell/(\gamma v_\infty)$ ; angular momentum remains the smooth invariant for the parabolic limit. These two benchmarks detect confusion between null, marginally bound, and circular stability thresholds.

The relation between unstable geodesics, scattering, and EOB descriptions is developed in Damour 2016, §§III–VIII.

From a test body to two comparable masses

For two finite masses, the conservative scattering angle $\chi(E,J)$ is a clean organizing observable. A threshold appears as the boundary of the domain in which the radial action has a real outer turning point; near it, $\chi$ grows without bound in an ideal conservative model. PN, PM, and EOB descriptions may assign different radial coordinates and effective potentials, but they must predict the same threshold in the common invariants after all mappings are applied.

A perturbative critical value for the reduced angular momentum $j_c=J_c/(GM\mu)$ has the structure

j_c(\widehat E,\nu,\boldsymbol\chi_A,C_i) =j_c^{(0)}(\widehat E) +\nu j_c^{(1)}(\widehat E) +j_c^{\mathrm{spin}} +j_c^{\mathrm{finite\ size}} +\cdots.

Here $\widehat E$ is a specified dimensionless energy variable, held fixed as $\nu\to0$ , and the spin and finite-size terms may themselves depend on $\nu$ . The reduced normalization makes the test-mass limit finite. If $\lvert j-j_c^{(0)}\rvert$ is comparable with an omitted term, the classification itself is uncertain: quoting a precise scatter/capture label would exceed the approximation.

The conservative threshold also differs from prompt merger in a dissipative numerical spacetime. A captured pair may form a long-lived eccentric bound orbit, while a horizon-crossing effective trajectory may require a separate merger completion. Each outcome definition must be stated before datasets are combined.

Dissipation and body dependence

With radiation, capture is no longer determined by a conservative double root alone. Let incoming total energy in a fixed asymptotic inertial frame be $E_{\mathrm{in}}$ . Radiation to infinity changes the total energy of the remaining bodies according to

E_{\mathrm{out}} =E_{\mathrm{in}} -E_{\mathrm{rad}}^\infty.

Energy absorbed by horizons or internal degrees of freedom remains in those bodies; it must not be subtracted a second time from this total. In a nonrelativistic orbital-energy budget, by contrast, $\Delta E_{\mathrm{orb}}=-E_{\mathrm{rad}}^\infty-E_{\mathrm{abs}}$ , with the absorbed energy added to the bodies’ internal energies or masses. For spinning holes its signed value can be negative through superradiance. At relativistic energies one must also transform to the outgoing center-of-mass frame and use the updated body masses. A trajectory that would scatter conservatively can become bound or plunge after losing enough energy and angular momentum.

Spin changes the critical potential and asymptotic spin orientation. Finite-size response adds tidal excitation, absorption, and possible disruption. Near criticality, the residence time grows, so parametrically small forces may accumulate. A finite-order point-particle prediction must therefore be checked against its own enhancement factor rather than extended to the threshold automatically.

For numerical comparisons, specify the incoming ADM energy and angular momentum after junk radiation has left, extract outgoing momenta and radiation at controlled radius, and propagate their covariance into the critical parameter. Fitting a coordinate initial separation or puncture momentum without reconstructing the invariants does not define a universal threshold.

Exercises

Derive the Schwarzschild null critical impact parameter.

Solution

The null effective potential is $V=L^2(1-2GM/r)/r^2$ . Setting $dV/dr=0$ gives $r_c=3GM$ . At the double root, $E^2=V(r_c)=L^2/(27G^2M^2)$ , hence $b_c=L/E=3\sqrt3\,GM$ .

Why is angular momentum preferable to impact parameter in the parabolic limit?