Conservative, Dissipative, and Hereditary Dynamics

Relativistic two-body dynamics is not exhausted by an instantaneous conservative Hamiltonian. Retarded propagation carries energy and angular momentum away, horizons and matter absorb radiation, and waves scattered by the spacetime generate dependence on the system’s past. A useful decomposition separates conservative, dissipative, and hereditary structures while keeping in view that the split depends on boundary conditions and perturbative organization.

Required background. The Einstein Equation supplies the nonlinear field dynamics, and Weak Metric Perturbations and Gauge Freedom supplies the distinction between a perturbative field and invariant response.

Helpful background. Presymplectic Potentials and Currents explains conservative generators and fluxes, while Black-Hole Perturbation Theory develops retarded response and horizon boundary conditions on a fixed background.

Time reversal and boundary conditions

For positions $\mathbf x_A$, momenta $\mathbf p_A$, and spins $\mathbf S_A$, time reversal acts schematically as

$$t\mapsto-t, \qquad \mathbf x_A\mapsto\mathbf x_A, \qquad \mathbf p_A\mapsto-\mathbf p_A, \qquad \mathbf S_A\mapsto-\mathbf S_A.$$

A conservative Hamiltonian is invariant under this transformation when external time-odd structures are absent. Its flow preserves phase-space volume and a conserved energy. A retarded solution is not time-reversal invariant: reversing it produces an advanced solution with radiation arriving from infinity or emerging from a future horizon.

For a linear field with retarded and advanced Green functions, define

$$G_{\mathrm{sym}}=\frac12(G_{\mathrm{ret}}+G_{\mathrm{adv}}), \qquad G_{\mathrm{rad}}=\frac12(G_{\mathrm{ret}}-G_{\mathrm{adv}}).$$

The time-symmetric part is a natural source of conservative dynamics; the time-antisymmetric part encodes radiative response. This split is exact only after the operator, gauge, state, asymptotics, and source history have been fixed. In nonlinear gravity, different perturbative rearrangements can move local terms between sectors while leaving invariant observables unchanged.

Conservative dynamics

An instantaneous conservative model with a time-independent Hamiltonian has equations

$$\dot q^i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q^i}, \qquad \frac{dH}{dt}=0.$$

At high PN order, however, the conservative action itself can be nonlocal in time. A schematic tail term is

$$S_{\mathrm{tail}} \sim G^2M \int dt\, I_{ij}^{(3)}(t) \operatorname{Pf}\!\int\frac{dt'}{\lvert t-t'\rvert} I_{ij}^{(3)}(t'),$$

where $I_{ij}$ is a source quadrupole, superscript $(3)$ denotes three time derivatives, and $\operatorname{Pf}$ indicates a regulated finite part. The kernel is time symmetric, so nonlocality does not imply dissipation. The 4PN conservative tail and its Hamiltonian treatment are reviewed in Schäfer and Jaranowski 2024, §§6.2–6.4.

This distinction matters: “conservative” refers to time-reversal and energy-balance structure, not to temporal locality. Likewise, a local-looking force can contain dissipative information after auxiliary radiation degrees of freedom have been integrated out with retarded boundary conditions.

Dissipation and balance laws

Let $E_{\mathrm{orb}}$ and $J_{\mathrm{orb}}$ denote appropriately defined orbital quantities. For an isolated binary with retarded conditions,

$$\frac{dE_{\mathrm{orb}}}{dt} =-\mathcal F_E^\infty-\mathcal F_E^{\mathrm H} -\frac{dE_{\mathrm{Schott}}}{dt},$$ $$\frac{dJ_{\mathrm{orb}}}{dt} =-\mathcal F_J^\infty-\mathcal F_J^{\mathrm H} -\frac{dJ_{\mathrm{Schott}}}{dt}.$$

The fluxes through null infinity and horizons are invariant after their generators and normalizations are fixed. Horizon fluxes transfer orbital energy and angular momentum into body masses and spins; they are not an additional loss from the total system when those body contributions are included. The Schott terms represent reversible near-zone storage and depend on the definition of orbital energy. Therefore a local radiation-reaction force need not satisfy “mechanical loss equals far-zone flux” instant by instant. After averaging a periodic adiabatic orbit, total derivatives often vanish, leaving the familiar balance law.

At leading order for a quasi-circular nonspinning binary,

$$E_{\mathrm N}(x)=-\frac12\mu x, \qquad \mathcal F_{\mathrm N}(x)=\frac{32}{5G}\nu^2x^5.$$

Using $dE/dt=-\mathcal F$ gives

$$\dot x=\frac{64}{5GM}\nu x^5.$$

Dimensions provide an independent check: with $c=1$, $G\mathcal F$ is dimensionless and $GM\dot x$ is dimensionless. The positive sign correctly describes inspiral toward larger frequency. PN radiation reaction, flux balance, and nonlinear propagation effects are developed in Blanchet 2024, §§2.5, 3.3, and 3.4.

Hereditary dependence

An instantaneous term depends on dynamical variables and finitely many derivatives at the current time. A hereditary term depends on an interval of the past:

$$\mathcal H(t) =\int_{-\infty}^{t}dt'\, K(t-t')\,\mathcal S(t').$$

In gravity, tails arise because radiation scatters from background curvature generated by the total mass. Memory accumulates from sustained radiation. Both are hereditary, but they have different kernels and observational signatures. A tail can contribute to conservative and dissipative sectors; memory is not simply another local radiation-reaction force.

For a periodic source with frequency $\Omega$, a tail integral can be evaluated mode by mode and produces logarithms and phase shifts. That reduction to an instantaneous function of orbital elements assumes a sufficiently stationary history. During capture, resonance, or rapid transition, replacing the integral by a circular steady-state value can be wrong.

Curved-spacetime self-force gives another hereditary structure. The regular field on the worldline contains a tail integral over prior motion because the Green function has support inside the null cone. The split into singular and regular fields is not the same as the conservative–dissipative split: each regular-field sector can have both time-symmetric and time-antisymmetric pieces Barack and Pound 2019, §§2.3 and 3.5.

What the decomposition can and cannot say

The decomposition is useful when it is tied to a question:

• Conservative dynamics determines orbit structure, invariant energies, precession, and scattering in a time-symmetric problem.
• Dissipative dynamics determines secular losses, capture by radiation, and inspiral through retarded response.
• Hereditary terms identify history dependence but do not by themselves decide whether the effect is conservative or dissipative.

The split is not always gauge invariant term by term. A coordinate transformation can change a local radiation-reaction force and its Schott contribution while preserving averaged flux balance and accumulated phase. A canonical transformation can change the appearance of a nonlocal Hamiltonian. Comparisons should therefore use invariant energies, frequencies, scattering data, asymptotic fluxes, or waveform phases with matched conventions.

Exercises

Use the Newtonian circular energy and leading flux above to derive $\dot x$ and the adiabatic ratio $T_{\mathrm{orb}}/T_{\mathrm{rr}}$, up to an overall choice of $2\pi$ in the orbital time.

Solution

Since $dE_{\mathrm N}/dx=-\mu/2$,

$$-\frac{\mu}{2}\dot x =-\frac{32}{5G}\nu^2x^5,$$

and $\mu=\nu M$ gives

$$\dot x=\frac{64}{5GM}\nu x^5.$$

Taking $T_{\mathrm{orb}}\sim\Omega^{-1}=GMx^{-3/2}$ and $T_{\mathrm{rr}}\sim x/\dot x$ gives

$$\frac{T_{\mathrm{orb}}}{T_{\mathrm{rr}}} \sim\frac{64}{5}\nu x^{5/2}.$$

Thus slow inspiral is controlled by $\nu x^{5/2}\ll1$ at leading order, not by $x$ alone.

References

• Barack, Leor, and Adam Pound. “Self-Force and Radiation Reaction in General Relativity.” Reports on Progress in Physics 82 (2019): 016904. DOI.
• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI.
• Schäfer, Gerhard, and Piotr Jaranowski. “Hamiltonian Formulation of General Relativity and Post-Newtonian Dynamics of Compact Binaries.” Living Reviews in Relativity 27 (2024): 2. DOI.