Hereditary Tails and Nonlocal Dynamics

Gravitational waves do not propagate only on the light cone of a curved spacetime. They scatter from the long-range curvature generated by the system’s total mass and return to influence the source. This tail makes the radiative field and orbital dynamics depend on the source’s history, produces logarithms of frequency, and couples near and radiation zones through a cancellation of arbitrary scales.

Helpful background. The Einstein Equation fixes nonlinear wave propagation, Weak Metric Perturbations and Gauge Freedom supplies radiative variables, Presymplectic Potentials and Currents supplies conserved fluxes, Partial Differential Equations and Green Functions supplies inside-the-light-cone propagation, Gravitational Radiation and Waveforms supplies radiative multipoles, and Relativistic Two-Body Dynamics supplies the near-zone source.

The origin of a tail

At leading nonlinear order, an outgoing quadrupolar wave scatters from the monopolar curvature $GM/r$. The retarded solution contains a hereditary integral schematically

$$U_{ij}(t) =I_{ij}^{(2)}(t) +2GM \int_0^\infty d\tau\, I_{ij}^{(4)}(t-\tau) \left[ \ln\left(\frac{\tau}{2b}\right)+c \right] +\cdots.$$

$U_{ij}$ is a radiative quadrupole, $I_{ij}$ a source multipole, and $b$ an intermediate time scale tied to radiative coordinates or matching. The coefficient and constant $c$ depend on the exact multipole convention. The integral reaches over the entire past, so no finite collection of instantaneous derivatives reproduces it for arbitrary motion.

The tail is causal in the retarded waveform: only $t-\tau$, which is earlier than the observation time $t$, occurs. Its appearance and multipolar matching are developed in Blanchet 2024, §2.

This effect differs from ordinary propagation delay. A direct signal samples the source at one retarded time selected by the null cone. A tail samples a continuum of earlier times because curvature scatters radiation along a continuum of longer routes. In a Green-function decomposition it belongs to the smooth inside-the-light-cone term, rather than the singular direct term.

The monopole–quadrupole interaction gives the standard leading binary tail, but higher multipoles and repeated scattering create further hereditary structures. Nonlinear memory is hereditary too, yet it has a different mechanism: memory accumulates radiated flux, while a tail backscatters a wave from background curvature. Their time parity, kernels, and coefficients must not be interchanged.

Conservative and dissipative kernels

At the level of two-body dynamics, a tail contribution can be written schematically as

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

The symmetric kernel $1/\lvert t-t'\rvert$ defines the conservative, time-symmetric part after a specified Hadamard finite part $\operatorname{Pf}_{T}$. In the source convention $T=2s/c$, where $s$ is the near-zone matching length; with $c=1$, $T=2s$, and absorbing the factor of two changes only a scheme-dependent constant. The positive coefficient and this two-sided, single-history normalization follow from Damour, Jaranowski, and Schäfer 2014, § IV, p. 5, Eqs. (4.2)–(4.5). The retarded prescription adds a time-antisymmetric piece responsible for dissipation. A formula without its prescription cannot be labeled conservative or radiative merely from the logarithm.

The finite part has a concrete meaning. For a smooth function $f$ with suitable decay, define

$$\operatorname{Pf}_T\int_{-\infty}^{\infty} \frac{f(t')}{|t-t'|}\,dt' =\int_0^\infty\frac{d\tau}{\tau} \left[f(t+\tau)+f(t-\tau)-2f(t)\Theta(T-\tau)\right].$$

Near $\tau=0$ the odd Taylor terms cancel and the bracket is $O(\tau^2)$, so the subtraction removes the logarithmic divergence. For a harmonic $f(t)=e^{-i\omega t}$ with $\omega\ne0$, interpret the large-time integral by an Abel regulator and obtain

$$\operatorname{Pf}_T\int_{-\infty}^{\infty} \frac{e^{-i\omega t'}}{|t-t'|}\,dt' =-2\left[\gamma_{\mathrm E}+\ln(|\omega|T)\right]e^{-i\omega t},$$

where $\gamma_{\mathrm E}$ is Euler’s constant. This directly displays the real, even logarithm and supplies an exact test for a history-integral code. Differentiating either representation with respect to $\ln T$ gives $-2f(t)$, fixing the local term needed for scale cancellation.

A causal doubled-history calculation is a different functional. For the $e^{-i\omega t}$ convention its renormalized tail term is

$$W_{\mathrm{tail}} =-\frac{2G^2M}{5} \int\frac{d\omega}{2\pi}\, \omega^6 I_-^{ij}(-\omega)I_{+\,ij}(\omega) \left[ \ln\frac{\omega^2}{\mu^2} -i\pi\operatorname{sgn}\omega \right].$$

The complete outer-minus and bracket convention matters when checking the imaginary sign. The real even logarithm contributes to time-symmetric dynamics; the imaginary odd term supplies dissipative phase and flux. The causal frequency- and past-time kernels are derived in Galley et al. 2016, § 3.1, pp. 7–8, Eqs. (3.1)–(3.6), with the conservative/dissipative split in Galley et al. 2016, § 3.2, pp. 8–9, Eqs. (3.7)–(3.9). Additive constants depend on the finite-part convention, but this parity split does not.

Dimensions provide a check. In $c=1$ units, $[I]=\text{mass}\,\text{length}^2$ and $[I^{(3)}]=\text{mass}/\text{length}$. Including $G^2M\,dt\,dt'/\lvert t-t'\rvert$ leaves the action dimensionless in $\hbar=1$ bookkeeping, or with dimensions of action when units are restored. More practically, the kernel must contain a dimensionless logarithm such as $\ln(\tau/s)$; $\ln\tau$ alone is incomplete.

Scale cancellation and circular logarithms

The finite-part scale $s$ is not physical. A local near-zone contribution has the opposite scale dependence:

$$H_{\mathrm{near}} \supset A(I^{(3)})^2\ln\frac{r}{s}, \qquad H_{\mathrm{tail}} \supset A(I^{(3)})^2\ln(\Omega s)+\cdots.$$

Their sum contains

$$\ln(\Omega r),$$

or an equivalent invariant combination after the circular relation between $r$ and $\Omega$ is applied. Differentiating the complete observable gives

$$\frac{d}{d\ln s} \left( H_{\mathrm{near}}+H_{\mathrm{tail}} \right)=0.$$

This cancellation is a decisive check that near and radiation momentum regions have not been double counted or omitted. The remaining circular logarithm may be expressed in $x=(GM\Omega)^{2/3}$, typically as $\ln x$ plus a convention-dependent constant. Comparing constants before translating the radiative time and multipole normalization is unsafe.

The cancellation can also be phrased as classical renormalization-group invariance. Changing the separation scale alters a local composite-source coefficient and the explicit tail kernel by opposite amounts. Evolving the coefficient between orbital and radiation scales can resum leading logarithms. The running coefficient is not observable; the complete scale-independent energy, phase, or flux is. In the EFT notation $\mu$ is an inverse-length scale, so translating to the finite-part time scale uses $\mu\sim T^{-1}$ up to scheme constants. The far-zone ultraviolet/near-zone infrared cancellation and the resulting running are derived in Galley et al. 2016, § 3.3, pp. 9–12, Eqs. (3.10)–(3.25).

The EFT treatment of tails, logarithms, and radiation reaction is reviewed in Porto 2016, §§7.4–7.7, pp. 55–68.

Domain and numerical treatment

For a periodic source, Fourier transformation diagonalizes the kernel: history integrals become functions of $\ln\lvert\omega\rvert$ plus a retarded imaginary part. For an inspiral, one may use a slow-evolution expansion only when orbital parameters change little over the kernel’s dominant support. Near a resonance or sudden encounter, direct history integration or matched asymptotics is required.

Three representations are useful. A time-domain convolution keeps causality and arbitrary source history manifest. A frequency-domain kernel is efficient for quasiperiodic motion and exposes logarithmic parity. A local slow-time expansion is cheapest for adiabatic evolution, but it must carry a remainder and cannot cross a nonanalytic resonance uniformly. They agree only where the needed transforms and expansions exist.

A numerical implementation must control the remote-past model, singular behavior near $t'=t$, and cancellation of $s$. It should reproduce a harmonic test source analytically, verify time symmetry of the conservative kernel, reverse the dissipative sign under retarded-to-advanced exchange, and show that changing $s$ moves local and nonlocal terms separately but not their sum.

The PN tail expansion breaks when the source changes on the curvature scattering scale without a small parameter, or when strong-field merger dynamics replaces the perturbative source multipoles. The existence of tails is general; the displayed coefficients and truncation are not.

A clean validation benchmark uses one sinusoidal STF quadrupole switched on smoothly in the remote past. The analytic Fourier response fixes the logarithm, imaginary sign, and scale dependence. The numerical history integral should approach it as the start time is moved earlier and the near-coincidence quadrature is refined. A second benchmark uses a slow chirp and identifies where the frozen-frequency approximation loses its predicted order.

Exercises

Show the scale cancellation in the schematic circular Hamiltonian.

Solution

Adding the logarithms gives

$$\ln\frac{r}{s}+\ln(\Omega s) =\ln(\Omega r).$$

Differentiation with respect to $\ln s$ gives $-1+1=0$. If either momentum region is omitted, an arbitrary scale remains in the observable.

Why is the symmetric tail action conservative even though its value depends on both past and future times?

Solution

Its kernel is unchanged under $t\leftrightarrow t'$ and under time reversal with the appropriate parity of the multipoles. Variation produces a time-symmetric force. It is a nonlocal representation of the conservative part, not a causal initial-value prescription; the physical retarded problem adds the antisymmetric dissipative component.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI.
• 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 (2014): 064058. DOI.
• Galley, Chad R., Adam K. Leibovich, Rafael A. Porto, and Andreas Ross. “The Tail Effect in Gravitational Radiation-Reaction: Time Non-Locality and Renormalization Group Evolution.” Physical Review D 93 (2016): 124010. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.