Radiative PM Scattering and Energy Loss

Retarded gravitational scattering has three coupled outputs: the two bodies’ final momenta, radiation at null infinity, and a permanent change in the asymptotic field. A conservative impulse cannot simply be supplemented by a positive energy loss after the calculation. Radiation reaction alters each body’s impulse, the center of mass recoils when radiation carries spatial momentum, and angular-momentum balance depends on a specified asymptotic frame. The reliable organizing principle is to compute all retained terms with causal boundary conditions and then enforce the full set of Poincaré-charge balance laws at the same PM order.

Required background. Events, Intervals, and Minkowski Geometry supplies four-momentum and angular-momentum tensors, and Weak Metric Perturbations and Gauge Freedom supplies the radiative weak field and its gauge dependence.

Helpful background. The Einstein Equation fixes nonlinear radiation sources; Presymplectic Potentials and Currents supplies asymptotic charges and their boundary terms; and Gravitational Radiation and Waveforms develops wave extraction, Bondi news, memory, and fluxes beyond the scattering-specific use made here.

Four-momentum and angular-momentum balance

Let $p_a^{-\mu}$ and $p_a^{+\mu}$ be the incoming and outgoing body momenta, and define $\Delta p_a^\mu=p_a^{+\mu}-p_a^{-\mu}$. If $P_{\rm rad}^\mu$ is the total four-momentum carried through future null infinity, translation invariance gives

$$\boxed{\; \Delta p_1^\mu+\Delta p_2^\mu+P_{\rm rad}^\mu=0 \;}$$

for a system with no other outgoing channel. In a physical retarded solution,

$$E_{\rm rad}\equiv P_{\rm rad}^0\geq0$$

in the incoming center-of-mass frame. The outgoing mechanical energy is lower by $E_{\rm rad}$, while $\boldsymbol P_{\rm rad}\neq0$ implies recoil of the two-body center of mass. The amplitude-based definition of $P_{\rm rad}^\mu$ and its exact balance with the impulses are derived in Kosower, Maybee, and O’Connell 2019, §§ 3.4–3.5 and §§ 4.4–4.5.

For spinless bodies, choose a common origin and write the mechanical angular momentum tensor asymptotically as

$$J_{\rm mech}^{\mu\nu} =\sum_{a=1}^2 2x_a^{[\mu}p_a^{\nu]} .$$

Then

$$J_{\rm mech}^{+\mu\nu} -J_{\rm mech}^{-\mu\nu} +J_{\rm rad}^{\mu\nu}=0.$$

A translation $x^\mu\mapsto x^\mu+a^\mu$ shifts angular momentum by $2a^{[\mu}P^{\nu]}$. The angular-momentum balance remains invariant because four-momentum balance shifts all terms consistently. At null infinity there is an additional supertranslation-frame issue: the zero-frequency field and the chosen early-time shear affect what is called radiated angular momentum. Static modes and the associated Poincaré form factors are analyzed in Manohar, Ridgway, and Shen 2022, eqs. (2)–(16).

PM order and causal boundary conditions

For two structureless nonspinning bodies, the conservative impulse begins at $O(G)$. The leading radiated four-momentum and the associated radiation-reaction contribution to the impulse begin at $O(G^3)$. This counting reflects the fact that a leading radiation amplitude is squared and integrated over on-shell phase space. Herrmann and collaborators compute both quantities to $O(G^3)$ while retaining the exact velocity dependence and verify their balance Herrmann et al. 2021, §§ 7.4–7.5, pp. 40–45.

At higher order it is unsafe to classify every radiation-mode contribution as dissipative. A retarded propagator may be decomposed into a time-symmetric part and a reactive part. The time-symmetric radiation sector produces conservative tails; the reactive part produces dissipation. In an in-in, or closed-time-path, formulation the causal propagator and the doubled fields preserve this distinction from the outset. A fourth-PM calculation includes linear, nonlinear, and hereditary radiation-reaction effects and provides the total impulse and radiated four-momentum in one causal framework Dlapa et al. 2023, main text around eqs. (2)–(8).

Thus a fixed-order workflow is:

1. expand the worldlines, field, and any lower-order iterations consistently through $G^n$;
2. impose retarded boundary conditions before evaluating the dissipative sector;
3. compute $\Delta p_a^\mu$, $P_{\rm rad}^\mu$, and, when needed, $J_{\rm rad}^{\mu\nu}$ in one asymptotic frame;
4. check each outgoing mass shell and both charge-balance equations; and
5. only then separate conservative, relative radiation-reaction, and recoil pieces.

The split in the last step can depend on representation, but the total retarded impulse and total radiated charges do not.

Radiation reaction as a response of the angle

At first order in radiation reaction, time symmetry gives a useful relation. Let $\chi_{\rm cons}(E,J)$ be the conservative scattering angle evaluated with incoming energy and angular momentum. Define the averages of the incoming and outgoing charges,

$$E_{\rm mid}=E-\frac12E_{\rm rad}, \qquad J_{\rm mid}=J-\frac12J_{\rm rad}.$$

These are asymptotic averages, not an assertion that mechanical energy or angular momentum in every gauge takes that value exactly at pericenter. The loss may be concentrated in a short passage; a slow variation over the whole orbit is unnecessary. Expanding the conservative angle at these average charges about the incoming data yields

$$\chi_{\rm ret} =\chi_{\rm cons} -\frac12 \left( E_{\rm rad}\frac{\partial\chi_{\rm cons}}{\partial E} +J_{\rm rad}\frac{\partial\chi_{\rm cons}}{\partial J} \right) +O(\text{loss}^2).$$

This is not an independent radiation model: it is a linear-response consequence of a time-antisymmetric force acting around a time-symmetric conservative orbit. It fails at quadratic order in radiation reaction and must be supplemented by recoil when $\boldsymbol P_{\rm rad}\neq0$. The derivation, including the distinction between relative and center-of-mass motion, appears in Bini, Damour, and Geralico 2021, § III.B–D and eqs. (3.23)–(3.43).

Memory and the angular-momentum subtlety

Gravitational memory is the difference between the early and late asymptotic shear. It probes the zero-frequency limit of the waveform and is not identical to the positive energy flux, which is quadratic in the Bondi news. Memory can therefore influence soft and angular-momentum bookkeeping without constituting an additional packet of radiated energy.

This distinction resolves an apparent PM-order puzzle. In an “intrinsic” Bondi frame, static or zero-frequency pieces can make an angular-momentum change appear at $O(G^2)$, even though radiated energy and the canonical radiated angular momentum derived from propagating amplitudes begin at $O(G^3)$. The relation between intrinsic and canonical frames, and why the radiation-reaction response can remain correct, is explained in Veneziano and Vilkovisky 2022, §§ 2–4. A page that quotes $J_{\rm rad}$ without its origin and Bondi frame is therefore incomplete.

Checks and limitations

Four checks are discriminating. Momentum balance must hold component by component, not only for energy. Each $p_a^+$ must satisfy $(p_a^+)^2=-m_a^2$. The radiated energy must be nonnegative for a classical outgoing solution, although individual decomposed terms need not be. Finally, the low-velocity expansion must reproduce the known bremsstrahlung hierarchy and the soft limit must agree with the zero-frequency waveform.

The treatment is valid for weak-field scattering with freely separated asymptotic bodies, perturbatively small loss, and negligible absorption or finite-size excitation unless those channels are included explicitly. It breaks down near capture, when radiation changes the orbit by order unity, or when a truncated PM series is used near a strong-field separatrix. It also does not supply source-independent waveform propagation or detector response; those belong to the gravitational-radiation volume.

Exercises

Derive the linear-loss correction

Assume the retarded orbit has midpoint invariants $E_{\rm mid}=E-E_{\rm rad}/2$ and $J_{\rm mid}=J-J_{\rm rad}/2$. Derive the correction to $\chi_{\rm cons}(E,J)$ to first order in the losses.

Solution

Taylor expansion gives

$$\chi_{\rm cons}(E_{\rm mid},J_{\rm mid}) =\chi_{\rm cons}(E,J) -\frac{E_{\rm rad}}{2}\frac{\partial\chi_{\rm cons}}{\partial E} -\frac{J_{\rm rad}}{2}\frac{\partial\chi_{\rm cons}}{\partial J} +O(\text{loss}^2).$$

Time symmetry of the unperturbed orbit identifies this midpoint evaluation with the retarded angle through linear order in a time-antisymmetric radiation-reaction force. The result does not include quadratic reaction or the boost to a recoiling outgoing center-of-mass frame.

Diagnose an incomplete impulse

A calculation reports $\Delta p_1^\mu+\Delta p_2^\mu=(-\mathcal E,\boldsymbol 0)$ with $\mathcal E\gt 0$ for a retarded encounter but reports no radiation. What is missing?

Solution

The mechanical system has lost energy $\mathcal E$. Four-momentum balance requires

$$P_{\rm rad}^\mu=(\mathcal E,\boldsymbol 0)$$

in this symmetric example. If the masses or trajectory are not symmetric, a spatial recoil component may also be required. Declaring the two-body impulse “conservative” cannot repair the mismatch: either a radiative channel is missing or the calculated impulse is inconsistent.

References

• Bini, Donato, Thibault Damour, and Andrea Geralico. “Radiative Contributions to Gravitational Scattering.” Physical Review D 104, 084031 (2021). DOI. Open PDF.
• Dlapa, Christoph, Gregor Kälin, Zhengwen Liu, Jakob Neef, and Rafael A. Porto. “Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order.” Physical Review Letters 130, 101401 (2023). DOI. Open PDF.
• Herrmann, Enrico, Julio Parra-Martinez, Michael S. Ruf, and Mao Zeng. “Radiative Classical Gravitational Observables at $\mathcal O(G^3)$ from Scattering Amplitudes.” Journal of High Energy Physics 2021, 148 (2021). DOI. Open PDF.
• Kosower, David A., Ben Maybee, and Donal O’Connell. “Amplitudes, Observables, and Classical Scattering.” Journal of High Energy Physics 2019, 137 (2019). DOI. Open PDF.
• Manohar, Aneesh V., Alexander K. Ridgway, and Chia-Hsien Shen. “Radiated Angular Momentum and Dissipative Effects in Classical Scattering.” Physical Review Letters 129, 121601 (2022). DOI. Open PDF.
• Veneziano, Gabriele, and Gregory A. Vilkovisky. “Angular Momentum Loss in Gravitational Scattering, Radiation Reaction, and the Bondi Gauge Ambiguity.” Physics Letters B 834, 137419 (2022). DOI. Open PDF.