Relativistic Hyperbolic and Parabolic Encounters

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Relativistic unbound encounters are organized by asymptotic center-of-mass energy and angular momentum, not by coordinate eccentricity alone. Positive Newtonian orbital energy becomes $\mathcal E\gt 1$ for a test body, the marginally bound parabolic limit is $\mathcal E=1$ , and a hyperbolic scattering orbit additionally requires an outer turning point. Strong gravity can remove that turning point and produce capture at the same asymptotic energy.

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

Asymptotic invariant data

For two masses $m_1,m_2$ with total center-of-mass energy $E_{\mathrm{cm}}$ , define the relative momentum magnitude by the Källén function,

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}}}.

The invariant impact parameter is

b=\frac{J}{p_\infty},

with $J$ the asymptotic orbital angular momentum when spins are absent. For spinning bodies, total $J$ and asymptotic spin directions must be supplied, and the orbital part depends on the centroid convention.

Hyperbolic motion has $p_\infty\gt 0$ . The parabolic limit is $p_\infty\to0$ at fixed finite $J$ , so $b$ diverges. This makes $(E_{\mathrm{cm}},J)$ a smoother parameter pair than $(v_\infty,b)$ near marginal binding.

The Newtonian derivation and conic geometry belong to Parabolic and Hyperbolic Encounters. Here the focus is how relativistic turning points, radiation, and capture alter that baseline.

Test-body radial classification

For a Schwarzschild test body of unit rest mass,

\dot r^2 =\mathcal E^2 -\left(1-\frac{2GM}{r}\right) \left(1+\frac{\ell^2}{r^2}\right).

Here the dot means $d/d\tau$ , where $\tau$ is the particle’s proper time, and $\ell$ is its specific angular momentum, with dimensions of length.

The asymptotic limits are:

$\mathcal E\gt 1$ : unbound energy; the orbit scatters only if a turning point exists outside the horizon;
$\mathcal E=1$ : marginally bound or parabolic energy;
$\mathcal E\lt 1$ : bound energy, subject to radial turning-point structure.

Thus energy determines asymptotic binding, while energy plus angular momentum determine whether an unbound trajectory scatters or is captured. The critical boundary is a double root of the radial potential.

For $\mathcal E=1$ , the turning-point equation reduces to

0 =\frac{2GM}{r} -\frac{\ell^2}{r^2} +\frac{2GM\ell^2}{r^3}.

The double root gives $r_c=4GM$ and $\ell_c=4GM$ . Parabolic trajectories with larger angular momentum turn outside the unstable orbit; those with smaller angular momentum plunge.

Newtonian weak-field limit

At low asymptotic speed and large impact parameter, define Newtonian specific energy $\varepsilon=v_\infty^2/2$ and specific angular momentum $h=bv_\infty$ . The conic eccentricity is

e =\sqrt{ 1+\frac{2\varepsilon h^2}{(GM)^2} } =\sqrt{ 1+\frac{b^2v_\infty^4}{(GM)^2} }.

The Newtonian deflection satisfies

\tan\frac{\chi_{\mathrm N}}{2} =\frac{GM}{bv_\infty^2}.

Taking $\varepsilon\to0^+$ at fixed periapsis sends $e\to1$ smoothly and recovers a parabola. Relativistic formulas must reproduce these limits before their strong-field terms are trusted.

At fixed $v_\infty$ , decreasing $b$ first increases the deflection, then enters a zoom–whirl regime, and finally crosses the capture separatrix. A finite PM expansion at small deflection cannot be extrapolated through all three regions without resummation or comparison.

The invariant scattering-angle and boundary-to-bound dictionaries are developed in Kälin and Porto 2020, §§2–6, pp. 6–34.

Radiation and encounter outcomes

During a close passage, radiation changes the outgoing invariants:

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

where $E$ and $J$ are total charges of the remaining system in the same asymptotic frame and the displayed losses are carried to infinity. Horizon absorption transfers orbital energy and angular momentum into the bodies’ masses and spins; it is included separately when evolving orbital quantities and is not an additional loss of total system energy. Possible outcomes include one-pass scattering, radiative formation of a bound eccentric binary, direct capture, or disruption for material bodies. “Parabolic encounter” should therefore refer to the incoming energy, not assume a conservative outgoing orbit.

Approximation choice depends on more than energy class. PN methods require slow motion, PM methods expand in $G$ at fixed velocity and favor modest deflection, self-force requires a small mass ratio, and EOB resums specified inputs. Near threshold, compare a shared invariant and propagate the enhanced sensitivity to omitted orders.

A cross-method check fixes $E_{\mathrm{cm}}$ and $J$ , computes the weak-deflection angle in both PN-expanded and PM-expanded form, and takes the low-velocity and low- $G$ limits in both orders. Their common coefficient must agree. A mismatch often exposes a mass, momentum, or impact-parameter normalization error rather than new physics.

Exercises

Show that the Newtonian parabolic limit is smooth at fixed angular momentum.

Solution

The eccentricity is $e=\sqrt{1+2\varepsilon h^2/(GM)^2}$ . At fixed $h$ , taking $\varepsilon\to0^+$ gives $e\to1$ continuously. The impact parameter $b=h/\sqrt{2\varepsilon}$ diverges, which is why $h$ or $J$ is the regular label.

Why does $\mathcal E\gt 1$ not guarantee scattering?