Bound-Orbit Information from Scattering Data

Scattering data can determine important bound-orbit invariants, but the continuation is conditional rather than automatic. For a local conservative two-body Hamiltonian, the hyperbolic radial action analytically continues across the energy threshold and reconstructs periastron advance, orbital frequencies, and related invariants up to canonical equivalence. Hereditary tails and dissipation change the problem: generic bound motion samples its history repeatedly, whereas a hyperbolic encounter supplies a different history and different analytic branches. Claims on this page were checked through 2026-08-12; later calculations, corrections, and independent reproductions lie outside this evidence cutoff.

Required background. Events, Intervals, and Minkowski Geometry supplies the asymptotic invariants to be continued, and Weak Metric Perturbations and Gauge Freedom supplies the PM expansion and its domain.

Helpful background. The Einstein Equation fixes the dynamics; Presymplectic Potentials and Currents supplies radial actions and canonical invariance; and Gravitational Radiation and Waveforms supplies tails, fluxes, and memory that limit a purely conservative continuation.

Local conservative continuation

Assume nonspinning, planar dynamics described by a time-independent local Hamiltonian, with a radial momentum $p_r(r;E,J)$ analytic in the parameters away from turning-point collisions. For scattering, the squared asymptotic momentum satisfies $p_\infty^2\gt 0$ and there is one physical positive turning point. For bound motion, analytic continuation below the two-body threshold gives $p_\infty^2\lt 0$ and two positive turning points.

Define the unnormalized bound radial action

$$I_r^{\rm bound}(E,J) =\oint p_r(r;E,J)\,dr .$$

Hamilton–Jacobi theory gives the azimuth accumulated in one radial period,

$$\Phi(E,J) =-\frac{\partial I_r^{\rm bound}}{\partial J}, \qquad \Delta\Phi=\Phi-2\pi .$$

The continuation can be expressed directly in terms of the oriented scattering angle:

$$\boxed{\; \Delta\Phi(E,J) =\chi(E,J)+\chi(E,-J) \;}$$

after continuing $E$ onto the bound sheet. The $J\to-J$ operation reaches the second turning-point branch; it is not the statement that the physical magnitude of angular momentum is negative. The derivation from turning-point continuation and its all-PM check appear in Kälin and Porto 2020b, §§ 2–4.

If

$$\chi(E,J) =\sum_{n\geq1}\frac{\chi_n(E)}{J^n},$$

then

$$\Delta\Phi(E,J) =2\sum_{k\geq1}\frac{\chi_{2k}(E)}{J^{2k}} .$$

The leading odd $1/J$ deflection produces no Newtonian periastron advance; the first even term produces the relativistic advance. Integrating the continued angle fixes the bound action only up to a function $C(E)$. Periastron advance is insensitive to this function, but the radial period $T_r=\partial_E I_r^{\rm bound}$ is not. Frequencies therefore require its normalization from the full radial action or additional boundary data. The complete reconstruction is developed in Kälin and Porto 2020a, §§ 2–5.

The Newtonian example makes the missing information concrete. Let $M=m_1+m_2$, $\mu=m_1m_2/M$, and $\mathcal E=E-M\lt0$ be the binding energy. The unnormalized Kepler action is

$$I_{r,\rm N}^{\rm bound} =2\pi\left(\frac{GM\mu^2}{\sqrt{-2\mu\mathcal E}}-J\right).$$

Its $J$ derivative gives $\Phi=2\pi$ for every ellipse, while its energy derivative gives $T_r=2\pi\sqrt{a^3/(GM)}$, with $\mathcal E=-GM\mu/(2a)$. A vanishing periastron advance alone cannot determine that energy-dependent period.

A Schwarzschild check

In the weak-binding test-body limit, the even 2PM term approaches

$$\chi_{\rm 2PM}^{\rm test} =\frac{3\pi G^2M^2}{J_{\rm sp}^2} +O(E_{\rm bind}),$$

where $J_{\rm sp}=J/\mu$ is the test body’s specific angular momentum. Since this term is even under $J_{\rm sp}\to-J_{\rm sp}$, the dictionary gives

$$\Delta\Phi =\frac{6\pi G^2M^2}{J_{\rm sp}^2} +O(E_{\rm bind}).$$

Using the Newtonian relation $J_{\rm sp}^2=GMa(1-e^2)$,

$$\Delta\Phi =\frac{6\pi GM}{a(1-e^2)},$$

the leading Schwarzschild periastron advance. This check tests the branch operation, the factor of two, and the distinction between total and specific angular momentum. It does not prove that the continuation works for nonlocal or dissipative dynamics.

What the literature establishes

The following comparison summarizes the evidence available at the cutoff date.

Construction	Supported result	Essential limitation
Local conservative radial action	Periastron advance, radial action, frequencies, and redshift-type invariants can be reconstructed order by order in the PM expansion.	Requires a suitable local conservative Hamiltonian, controlled analytic branches, and no unresolved turning-point singularity.
Radiative boundary-to-bound relations	Integrated energy and angular-momentum losses admit parity-based hyperbolic-to-elliptic relations in a perturbative treatment.	A flux map is not the same as reconstructing a dissipative trajectory or waveform phase.
Conservative tail sector	Local-in-time pieces and universal logarithms continue between unbound and bound motion.	Generic nonlogarithmic, nonlocal pieces obtained at large angular momentum do not determine quasi-circular or mildly eccentric motion.
High-order PM scattering	Explicit tail calculations let one subtract hereditary pieces and isolate a candidate local component.	At 5PM, memory-like contributions and analytic singularities remain incompletely resolved at the cutoff.

The third row is not merely a conservative warning. The authors who developed the radiative extension show that the full nonlocal map holds only in the large-eccentricity, large-$J$ limit; local terms and characteristic binding-energy logarithms are more universal, while other nonlocal terms fail to reproduce generic bound orbits Cho, Kälin, and Porto 2022, §§ 4.3–4.5 and § 6.

At fourth PM order, an amplitude-derived hyperbolic radial action includes a tail contribution and yields an effectively local Hamiltonian valid on hyperbolic orbits. Its authors explicitly identify conversion to the bound problem in the presence of conservative radiation as a further step, rather than treating the hyperbolic Hamiltonian as automatically universal Bern et al. 2022, discussion after eqs. (5)–(8).

Competing positions at the cutoff

One position is constructive: once the complete conservative scattering action is known and its nonlocal tail part is separated, the remaining local part is universal and can be continued through the boundary-to-bound dictionary. This program has passed nontrivial PM–PN overlap checks and has reconstructed known invariants. From this perspective, tails are a calculable complication rather than a fundamental obstruction.

A more restrictive position says that scattering data determine generic bound invariants only after additional assumptions about locality, branch choice, and hereditary completion. Two actions may agree on the large-eccentricity hyperbolic history yet differ for a periodic history, so a naive continuation of the full scattering answer is underdetermined. Dissipative loss is still more restrictive: a conservative scattering angle cannot reconstruct accumulated inspiral phase without fluxes and a causal evolution prescription.

The 2026 tail calculation sharpens both positions. It derives nonlocal-in-time 5PM tail contributions through tenth self-force order and thereby supplies data needed to isolate a local component. It also states that naive continuation of complete scattering results fails for generic elliptic motion, that nonlogarithmic bound nonlocal coefficients are not yet known in a PM representation, and that memory-like terms leave the full 5PM conservative value unsettled Dlapa et al. 2026, §§ I, VII, and VIII. In particular, a proposed treatment of a spurious $\gamma=3$ singularity is not yet a unique, independently established prescription. The credible positions therefore differ mainly in claimed scope, not in the established local dictionary.

Analytic branches, tails, and dissipation

Three failure mechanisms must be separated.

Branch failure. Continuing $p_\infty=\sqrt{\lambda}/(2E)$ below threshold makes it imaginary. The turning points, logarithms, and square roots must be continued together on a specified sheet. Continuing isolated closed-form terms can create imaginary pieces that cancel only in the full invariant.

History failure. A tail action contains integrals over two times. Evaluating it on one hyperbolic passage and then replacing the energy by a bound value does not reproduce the infinite sequence of returns in an elliptic orbit. Localized or logarithmic pieces can be universal even when the finite nonlocal remainder is orbit dependent.

Dissipative failure. Retarded evolution changes $E$ and $J$ from cycle to cycle. Analytic continuation of a conservative angle cannot determine the arrow of time, emitted waveform, recoil, or accumulated phase. Those require the radiated charges on Radiative PM Scattering and Energy Loss and a causal evolution law.

The continuation also loses uniform validity near a separatrix, where turning points merge and the radial period diverges. A finite PM truncation at large impact parameter cannot be assumed accurate there.

What could change the conclusions

This assessment should be revisited if any of the following occurs:

• a general derivation controls nonlocal tails, memory, and all analytic branches for generic eccentricity rather than only large $J$;
• a counterexample shows that two admissible conservative completions have identical scattering data but different bound invariants;
• the full 5PM conservative scattering angle, including memory-like regions, is fixed without the spurious $\gamma=3$ ambiguity and independently reproduced; or
• a causal boundary-to-bound construction predicts generic bound fluxes and phases and is verified against an independent PN, self-force, or numerical calculation.

Until such a trigger is met, the safe statement is precise: local conservative dynamics and universal logarithmic tail information map systematically; generic hereditary remainders and dissipative evolution require additional input.

Exercises

Select the terms that control periastron advance

Let

$$\chi(E,J) =\frac{a_1(E)}{J} +\frac{a_2(E)}{J^2} +\frac{a_3(E)}{J^3} +\frac{a_4(E)}{J^4} +O(J^{-5}).$$

Use the boundary-to-bound relation to find $\Delta\Phi$ through $J^{-4}$.

Solution

Under $J\to-J$, odd inverse powers change sign and even inverse powers do not. Therefore

$$\Delta\Phi =\chi(E,J)+\chi(E,-J) =\frac{2a_2(E)}{J^2} +\frac{2a_4(E)}{J^4} +O(J^{-6}),$$

after all coefficients and branches are analytically continued to the bound-energy sheet. The cancellation of $a_1$ explains the absence of Newtonian periastron advance.

Identify where a naive continuation fails

Consider the time-symmetric quadrupolar tail action

$$S_{\rm tail} =\frac{G^2M}{5}\int dt\,I_{ij}^{(3)}(t) \operatorname{Pf}_{T_0}\int_{-\infty}^{\infty} \frac{I_{ij}^{(3)}(t')}{\lvert t-t'\rvert}\,dt'.$$

Here $I_{ij}$ is the Newtonian symmetric trace-free mass quadrupole and $\operatorname{Pf}_{T_0}$ regularizes the coincidence singularity with time scale $T_0$. This is the $c=1$ form of Damour, Jaranowski, and Schäfer 2014, § IV, eq. (4.4), PDF. Why can evaluating it on a hyperbolic orbit and continuing $E$ fail for a bound orbit?

Solution

The integrand depends on pairs of times, not only on instantaneous $E$, $J$, and $r$. A hyperbolic orbit has one passage and asymptotically inertial ends. A bound orbit revisits the source indefinitely, adding correlations between different radial periods. Analytic continuation of the energy changes the instantaneous trajectory but does not automatically change the integration domain and retarded or time-symmetric boundary prescription into the periodic one. Local pieces and universal logarithms may still match; the finite nonlocal remainder requires a separate bound-history calculation or a proven general continuation theorem.

References

• Bern, Zvi, Julio Parra-Martinez, Radu Roiban, Michael S. Ruf, Chia-Hsien Shen, Mikhail P. Solon, and Mao Zeng. “Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at $\mathcal O(G^4)$.” Physical Review Letters 128, 161103 (2022). DOI. Open PDF.
• Cho, Gihyuk, Gregor Kälin, and Rafael A. Porto. “From Boundary Data to Bound States III: Radiative Effects.” Journal of High Energy Physics 2022, 154 (2022), with erratum Journal of High Energy Physics 2022, 2 (2022). DOI. Open PDF.
• Damour, Thibault, Piotr Jaranowski, and Gerhard Schäfer. “Nonlocal-in-Time Action for the Fourth Post-Newtonian Conservative Dynamics of Two-Body Systems.” Physical Review D 89, 064058 (2014). DOI. Open PDF.
• Dlapa, Christoph, Gregor Kälin, Zhengwen Liu, and Rafael A. Porto. “Nonlocal-in-Time Tail Effects in Gravitational Scattering to Fifth Post-Minkowskian and Tenth Self-Force Orders.” Physical Review D, accepted 3 June 2026. DOI. Open PDF.
• Kälin, Gregor, and Rafael A. Porto. “From Boundary Data to Bound States II: Scattering Angle to Dynamical Invariants (with Twist).” Journal of High Energy Physics 2020, 120 (2020). DOI. Open PDF.
• Kälin, Gregor, and Rafael A. Porto. “From Boundary Data to Bound States.” Journal of High Energy Physics 2020, 72 (2020). DOI. Open PDF.