Tails and Tail-of-Tail Terms

A gravitational-wave tail is a causal record of earlier emission scattered by the source’s long-range monopolar curvature. The radiative moment at retarded time $U$ therefore contains an integral over canonical moments at $U-\tau$ for every $\tau\gt0$. A second scattering produces a tail-of-tail, with squared logarithms and a new regularization scale. These effects are nonlinear propagation corrections, not changes to the underlying source orbit.

Required background. Weak-Field Gravity supplies the wave field and perturbative order, while Ordinary Differential Equations supplies the causal convolution and Fourier-transform techniques.

Helpful background. Einstein Equations supplies the nonlinear interactions; Waves, Green Functions, and Geometric Optics explains retarded Green functions and scattering; and Relativistic Two-Body Dynamics may provide a concrete moment history. Detailed binary equations of motion and detector inference are not treated here.

Monopole scattering and the tail integral

Let $M$ be the conserved mass monopole and $M_L(U)$ a canonical mass-type STF moment of rank $\ell\ge2$. The leading asymptotic field is linear in $M_L^{(\ell)}$. At the next relevant post-Minkowskian order, the nonlinear interaction $M\times M_L$ acts as an extended source for the outgoing wave. Retarded integration through the $1/r$ monopolar field produces support inside the flat-spacetime light cone and a logarithmic kernel.

With radiative retarded time $U=T-R$, the leading mass-type tail contribution is

$$\begin{aligned} U_L^{\mathrm{tail}}(U) ={}&2GM\int_0^\infty d\tau\, M_L^{(\ell+2)}(U-\tau) \left[ \ln\left(\frac{\tau}{2b_0}\right)+\kappa_\ell \right], \end{aligned}$$

where $b_0$ is a length defining the logarithmic transformation from harmonic to radiative time and

$$\kappa_\ell =\frac{2\ell^2+5\ell+4}{\ell(\ell+1)(\ell+2)} +H_{\ell-2}, \qquad H_p=\sum_{k=1}^{p}\frac1k, \qquad H_0=0.$$

Thus $U_L=M_L^{(\ell)}+U_L^{\mathrm{tail}}+\cdots$, but the dots do not start only at $G^2$: other multipole interactions, including nonlinear memory, can already contribute at order $G$ to a radiative moment. The displayed integral isolates the quadratic $M\times M_L$ sector. Since the metric waveform has an additional overall factor of $G$, this tail is quadratic in $G$ in the metric itself. For $\ell=2$, the constant is $\kappa_2=11/12$.

The current-type moment $V_L$ has the same causal structure with a different constant in the kernel. These formulas, their coordinate translation, and the conditions on the remote past are derived in Blanchet 2024, §2.4.2, pp. 82–86.

Every factor passes a dimension check. $GM$, $\tau$, and $b_0$ are lengths; the logarithm is dimensionless; and $GM\,d\tau\,M_L^{(\ell+2)}$ has the same dimension as $M_L^{(\ell)}$. The integral is causal because $U-\tau\le U$. If the source was stationary sufficiently far in the past, the differentiated moment vanishes there and the integral converges. More general past histories are allowed only after their convergence has been checked rather than assumed.

Why the contribution is hereditary

An instantaneous term depends on a finite number of moments and derivatives evaluated at $U$. A hereditary term is a functional of an interval of history:

$$\mathcal T_L[U] =\int_0^\infty d\tau\,K_\ell(\tau) M_L^{(\ell+2)}(U-\tau).$$

Two sources with identical values of $M_L$ and all derivatives at one time can have different $\mathcal T_L[U]$ if their earlier histories differ. The effect is therefore not reproducible by merely adding a higher local time derivative.

The word “tail” here has a precise mechanism: back-scattering from the long-range mass monopole. It should not be confused with a detector impulse response, a late-time black-hole branch-cut tail, or the nonlinear memory integral. Those phenomena can also be nonlocal in time, but their kernels and physical origins differ.

For a gravitationally bound, virialized slow-motion source, $GM/a\sim v^2$ and $\omega\sim v/a$, so $GM\omega\sim v^3$. The tail is therefore a relative 1.5PN correction in that source regime. A prescribed weak source need not obey the virial relation; its more general small parameter is $GM\omega$. The MPM construction itself did not assume $v\ll1$.

Frequency-domain phase and amplitude

Take a positive-frequency canonical moment $M_L(U)=A_L e^{-i\omega U}$ with $\omega\gt0$, and regulate the remote past by $e^{-\varepsilon\tau}$. The required logarithmic transform is

$$\lim_{\varepsilon\to0^+} \int_0^\infty d\tau\, e^{(i\omega-\varepsilon)\tau} \ln\left(\frac{\tau}{\tau_0}\right) =-\frac{\pi}{2\omega} -\frac{i}{\omega} \left[ \gamma_{\mathrm E}+\ln(\omega\tau_0) \right].$$

The real $-\pi/(2\omega)$ part changes the amplitude after multiplication by the two additional time derivatives, while the imaginary logarithm shifts the phase. Including the constant $\kappa_\ell$ gives the explicit ratio

$$\frac{U_L^{\mathrm{tail}}}{M_L^{(\ell)}} =GM\omega\left\{\pi+2i\left[\gamma_{\mathrm E}+\ln(2\omega b_0)-\kappa_\ell\right]\right\}.$$

The positive real term and the sign of the phase are independent checks on the retarded convention. The ratio scales as $GM\omega$, consistent with the bound-source 1.5PN count. An eternal sinusoid is understood here as the Abel-regulated frequency-domain limit, not as a past-stationary source.

This calculation also fixes the sign translation. With $e^{-i\omega U}$, the delayed value is $e^{-i\omega(U-\tau)}=e^{-i\omega U}e^{+i\omega\tau}$. Using the opposite Fourier convention without conjugating the kernel would reverse the tail phase.

Tail-of-tail structure and scale cancellation

At the next scattering order, the cubic interaction $M\times M\times M_L$ produces a tail-of-tail. Its mass-type structure is schematically

$$\begin{aligned} U_L^{\mathrm{tail^2}}(U) =2G^2M^2\int_0^\infty d\tau\, M_L^{(\ell+3)}(U-\tau) \bigg[ &\ln^2\left(\frac{\tau}{2b_0}\right) +a_\ell\ln\left(\frac{\tau}{2b_0}\right)\\ &+b_\ell\ln\left(\frac{\tau}{2r_0}\right) +c_\ell \bigg], \end{aligned}$$

where $a_\ell,b_\ell,c_\ell$ are fixed rational or harmonic-number combinations and $r_0$ is the finite-part scale used in the MPM iteration. The squared logarithm produces a relative $(GM\omega)^2\sim v^6$, or 3PN, contribution.

$b_0$ is a radiative-coordinate scale: changing it corresponds to a retarded-time shift and cannot change an invariant waveform. $r_0$ is a regularization scale. It cancels only after the canonical moment is expressed consistently in terms of source moments, which carry the compensating near-zone logarithm. Blanchet 2024, §2.4.2 and §§2.4.5–2.4.6, pp. 82–86 and 95–103 traces both cancellations. A residual $b_0$ or $r_0$ in the completed observable signals that terms from different approximation orders have been combined.

Limiting and normalization checks

Three checks are especially sharp.

1. Setting $M=0$ removes the long-range scattering potential and every tail term.
2. A moment stationary throughout its relevant past has $M_L^{(\ell+2)}=0$, so no tail is produced even when $M\ne0$. Becoming stationary only recently does not remove the earlier history from the integral.
3. The leading nonhereditary multipole field must reduce to the standard radiative STF normalization Thorne 1980, §§III–IV and Table II before any tail correction is added.

The first check separates propagation from source generation; the second separates Coulombic multipoles from radiation; the third detects a misleading tail coefficient caused by a mismatched definition of the underlying radiative moment.

Exercise

For $M_L=A_L e^{-i\omega U}$, use the logarithmic transform above to show the scaling and causal phase structure of the leading tail. You need not simplify the rank-dependent constant $\kappa_\ell$.

Solution

$M_L^{(\ell+2)}(U-\tau)=(-i\omega)^{\ell+2}A_Le^{-i\omega U}e^{i\omega\tau}$. The integral is therefore the displayed logarithmic transform plus $\kappa_\ell\int_0^\infty e^{(i\omega-\varepsilon)\tau}d\tau$, whose regulated limit is $i\kappa_\ell/\omega$. Relative to $M_L^{(\ell)}$, the two extra derivatives contribute $-\omega^2$, while the integral contributes $1/\omega$. The correction is thus proportional to $GM\omega$. Its real part contains the $\pi$ amplitude term, and its imaginary part contains $\ln(2\omega b_0)+\gamma_{\mathrm E}-\kappa_\ell$, producing a phase shift. Because the integrand uses only $U-\tau$ with $\tau\ge0$, the construction is retarded. Taking either $M\to0$ or $\omega\to0$ makes the tail vanish.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024) 4. DOI. Open PDF.
• Thorne, Kip S. “Multipole Expansions of Gravitational Radiation.” Reviews of Modern Physics 52 (1980) 299–339. DOI.