Nonlinear Multipolar Waveforms

A nonlinear multipolar waveform is assembled only after three maps have been kept consistent: source moments to canonical moments, canonical moments to radiative moments, and radiative moments to strain or angular modes. The resulting field contains instantaneous and hereditary terms at declared post-Minkowskian and, when invoked, post-Newtonian orders. Its reliability depends as much on the truncation statement and scale cancellations as on the displayed coefficients.

Required background. Weak-Field Gravity supplies the perturbative metric and gauge projection, while Ordinary Differential Equations supplies the moment derivatives and hereditary integrals.

Helpful background. Einstein Equations supplies nonlinear coupling; Waves, Green Functions, and Geometric Optics supplies retarded propagation; and Relativistic Two-Body Dynamics can supply the source history. Detailed binary equations of motion and detector inference are not treated on this page.

Radiative moments to transverse strain

Work in radiative coordinates $(T,\boldsymbol X)$, with $R=\lvert\boldsymbol X\rvert$, propagation direction $N_i=X_i/R$, and retarded time $U=T-R$. Define $P_{ij}=\delta_{ij}-N_iN_j$ and

$$\mathcal P_{ijab} =P_{a(i}P_{j)b}-\frac12P_{ij}P_{ab}.$$

For the ordinary metric perturbation, rather than the oppositely signed gothic perturbation, the leading $1/R$ field is

$$\begin{aligned} h_{ij}^{\mathrm{TT}} =\frac{4G}{R}\mathcal P_{ijab} \sum_{\ell=2}^{\infty}\frac{1}{\ell!} \bigg[ &N_{L-2}U_{abL-2}\\ &-\frac{2\ell}{\ell+1} N_{cL-2}\epsilon_{cd(a}V_{b)dL-2} \bigg] +O(R^{-2}). \end{aligned}$$

$U_L(U)$ and $V_L(U)$ are the mass- and current-type radiative STF moments, $L=i_1\cdots i_\ell$, and $\epsilon_{123}=+1$. A string $L-2$ contains $\ell-2$ indices; it is empty for the quadrupole. This asymptotic definition and its exact normalization are given in Blanchet 2024, §2.4.1, eq. (178), p. 79 and in Thorne 1980, §§III–IV and Table II.

Choose transverse unit vectors $(\boldsymbol P,\boldsymbol Q)$ so that $(\boldsymbol N,\boldsymbol P,\boldsymbol Q)$ is oriented. Then

$$\begin{aligned} h_+&=\frac12(P_iP_j-Q_iQ_j)h_{ij}^{\mathrm{TT}},\\ h_\times&=\frac12(P_iQ_j+P_jQ_i)h_{ij}^{\mathrm{TT}}, \end{aligned}$$

and the complex strain convention is $h=h_+-ih_\times$. Rotating the polarization dyad changes the phase of $h$ but not the underlying tensor. A mode decomposition uses spin weight $-2$,

$$h(U,\theta,\phi) =\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} h_{\ell m}(U)\,{}_{-2}Y_{\ell m}(\theta,\phi),$$

with the harmonic normalization declared before numerical coefficients are compared.

Instantaneous and hereditary assembly

The radiative moments are nonlinear functionals, not merely differentiated source integrals. A useful organization is

$$\begin{aligned} U_L&=U_L^{\mathrm{inst}} +U_L^{\mathrm{tail}} +U_L^{\mathrm{tail^2}} +U_L^{\mathrm{memory}}+\cdots,\\ V_L&=V_L^{\mathrm{inst}} +V_L^{\mathrm{tail}} +V_L^{\mathrm{tail^2}}+\cdots. \end{aligned}$$

Instantaneous pieces depend on source or canonical moments evaluated at $U$. Tails and iterated tails integrate the earlier moment history through logarithmic kernels generated by monopole scattering. Nonlinear displacement memory integrates the radiated energy flux and contributes to mass-type radiative moments. These labels classify the time structure and coupling mechanism; they do not assign separate observable waveforms that can be truncated independently without an order count.

The construction pipeline is therefore:

1. compute source moments $I_L,J_L$ and the required gauge moments in one near-zone coordinate and regularization convention;
2. transform them to canonical $M_L,S_L$ through the nonlinear order needed;
3. form $U_L,V_L$, including every instantaneous and hereditary interaction entering that order;
4. substitute them into the TT field or into a consistently normalized spin-weighted mode basis.

No orbital dynamics is recomputed in these steps. The pipeline consumes the source history and exposes how it becomes radiation.

What a controlled truncation must state

“The waveform is known to a given order” is incomplete unless several independent expansions are identified:

Expansion	Required statement
Post-Minkowskian	Highest retained power of $G$ in each multipole interaction
Post-Newtonian, if used	Highest relative power of $v$ or $x\sim v^2$ in phase and amplitude
Angular multipoles	Largest retained $\ell$ and any symmetry-based omissions
Asymptotic radius	Whether only $1/R$ strain is kept or finite-radius terms are included
Hereditary history	Past-stationarity or convergence assumption and included kernel classes
Frames and scales	Harmonic, canonical, and radiative coordinates; $b_0$, $r_0$, and their cancellation

The weakest retained layer controls the claim. For example, a high-PN source quadrupole combined with a lower-order canonical-to-radiative map is not a waveform of the higher order. Similarly, retaining many spherical-harmonic modes does not repair a missing tail contribution within a low-$\ell$ mode.

At any fixed slow-motion order only finitely many multipole ranks contribute, because higher ranks carry additional powers of source size relative to wavelength. Outside a slow-motion regime, one may still use the PM and multipolar organization, but that PN suppression argument cannot be invoked.

Independent waveform checks

The same radiative moments should reproduce strain and flux. With the polarization normalization above, the luminosity at future null infinity is

$$\mathcal F =\frac{R^2}{16\pi G} \int d\Omega\, \left[ (\partial_U h_+)^2+(\partial_U h_\times)^2 \right].$$

At leading mass-quadrupole order,

$$\mathcal F =\frac{G}{5}I_{ij}^{(3)}I_{ij}^{(3)},$$

at the leading instantaneous radiative order. A cycle average is optional when reporting the mean luminosity of a periodic source; it is not needed to define the Bondi flux. Deriving this expression from the angular integral of the same $h_{ij}^{\mathrm{TT}}$ checks the TT normalization, the factor $4G/R$, the STF angular integral, and the derivative order simultaneously.

Parity supplies another check. Mass-type rank-$\ell$ radiation has electric parity $(-1)^\ell$, whereas current-type radiation has magnetic parity $(-1)^{\ell+1}$. A coupling assigned to the wrong family violates the angular selection rules before any detailed coefficient is inspected.

Finally, arbitrary scales must cancel. Changing $b_0$ can be absorbed into the radiative retarded-time origin. Dependence on the finite-part scale $r_0$ cancels between source and propagation pieces at a consistent order. The final strain can depend on physical masses, frequencies, distances, and a declared frame, but not on an unphysical analytic-continuation scale.

Stationary and weak-field limits

For a source stationary throughout its relevant past, $M_L$ and $S_L$ are constant. Then $U_L=M_L^{(\ell)}=0$ and $V_L=S_L^{(\ell)}=0$ at leading order for $\ell\ge2$; the differentiated tail kernels and flux also vanish. The source may retain nonzero Coulombic multipoles. A source that settles down after a burst is different: tails can persist from its earlier history, and memory can leave a constant strain offset even after the news vanishes.

In the linear weak-field limit, discard nonlinear dictionaries and hereditary integrals. Then $M_L=I_L$, $S_L=J_L$, $U_L=I_L^{(\ell)}$, and $V_L=J_L^{(\ell)}$. For $\ell=2$ with no current moment,

$$h_{ij}^{\mathrm{TT}} =\frac{2G}{R}\mathcal P_{ijab}I_{ab}^{(2)}.$$

Any nonlinear implementation must recover this quadrupole formula before more elaborate terms are trusted.

Common pitfalls

Mixing gothic and ordinary metric signs. A formula imported for $\sqrt{-g}g^{ij}-\eta^{ij}$ changes sign when translated to the leading ordinary spatial metric perturbation. The field definition must accompany the coefficient.

Combining unmatched orders. Adding a tail or memory term without the source-moment correction of the same order can leave spurious scale dependence and a false accuracy label.

Treating mode content as frame invariant. The tensor field is geometrical, but individual $h_{\ell m}$ values depend on the angular frame, polarization dyad, and harmonic convention. Comparisons require the complete frame dictionary.

Exercise

Let a source quadrupole be $I_{ij}=Q_{ij}\cos(\Omega U)$, with constant STF $Q_{ij}$. Determine the scaling of the leading strain, leading luminosity, and first tail correction. Check the stationary limit and dimensions.

Solution

Two derivatives give $I_{ij}^{(2)}=-\Omega^2Q_{ij}\cos(\Omega U)$, so the strain scales as $h\sim G\Omega^2Q/R$. A third derivative enters the quadrupole luminosity, giving $\mathcal F\sim G\Omega^6Q^2$ after cycle averaging. Since $Q$ has dimensions mass times length squared, both expressions have the required dimensions: $h$ is dimensionless and $\mathcal F$ is energy per unit time. The leading monopole-scattering tail is smaller than the strain by $GM\Omega$ and contains its logarithmic phase. Sending $\Omega\to0$ removes strain, flux, and tail while leaving the stationary source quadrupole itself nonzero.

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.