Source, Canonical, and Radiative Moments

The phrase “the quadrupole moment” hides three different constructions. Source moments are integrals or finite-part functionals of the near-zone fields; canonical moments parametrize a particularly economical exterior solution; radiative moments are the coefficients of the asymptotic waveform in radiative coordinates. They coincide only at the appropriate leading order. Keeping the three families distinct is what makes nonlinear corrections, coordinate scales, and hereditary propagation intelligible.

Required background. Weak-Field Gravity supplies the metric and gauge expansion, while Ordinary Differential Equations supplies the time derivatives and causal integrals used to relate the moment families.

Helpful background. Einstein Equations supplies the nonlinear vacuum equations; Waves, Green Functions, and Geometric Optics explains retarded solutions; and Relativistic Two-Body Dynamics can provide concrete trajectories from which source moments are evaluated. This page does not derive detailed binary equations of motion or perform detector inference.

Three moment families and their comparison axes

Let $L=i_1\cdots i_\ell$ denote an STF multi-index and let $u=t-r$ be harmonic retarded time. In the standard multipolar post-Minkowskian construction, the relevant families differ along four axes:

Family	Defined from	Natural coordinates	Primary role
Source moments $I_L,J_L$ with gauge moments $W_L,X_L,Y_L,Z_L$	Near-zone matter and gravitational fields, with specified regularization	Harmonic source coordinates	Connect the exterior field to a physical source model
Canonical moments $M_L,S_L$	Two-function parametrization of the exterior vacuum solution	Canonical harmonic or related exterior coordinates	Organize nonlinear multipole interactions efficiently
Radiative moments $U_L,V_L$	Coefficients of the leading $1/R$ transverse waveform	Radiative coordinates with retarded time $U=T-R$	Build strain, modes, and fluxes at future null infinity

The mass-type families are $I_L,M_L,U_L$; the current-type families are $J_L,S_L,V_L$. Source and canonical moments have dimensions of mass times length$^\ell$ in units $c=1$. Radiative moments already include the time derivatives required by radiation, so $U_L$ and $V_L$ have different dimensions from their undifferentiated ancestors. The terminology labels stages in a map, not competing names for one tensor.

Blanchet 2024, §§2.1.4 and 2.4.5–2.4.6, pp. 46–53 and 95–103 gives the two nonlinear dictionaries and makes their coordinate assumptions explicit. Thorne’s radiative multipoles provide the asymptotic end of the construction Thorne 1980, §§III–IV and Table II.

From a general source description to canonical moments

At linear order, the most general stationary-in-the-past exterior solution in harmonic gauge contains two physical STF families and four gauge families. Use the gothic perturbation $\mathfrak h^{\alpha\beta}=\sqrt{-g}\,g^{\alpha\beta}-\eta^{\alpha\beta}$; at linear order it is minus the raised trace reversal of the ordinary metric perturbation. Its linear coefficient can be written schematically as

$$\mathfrak h^{\alpha\beta}_{(1)} =\mathfrak h^{\alpha\beta}_{\mathrm{can}}[I_L,J_L] +\partial^\alpha\varphi^\beta +\partial^\beta\varphi^\alpha -\eta^{\alpha\beta}\partial_\mu\varphi^\mu,$$

where $\varphi^\alpha$ is built from $W_L,X_L,Y_L,Z_L$. The mass monopole and mass and current dipoles obey conservation laws; time-dependent radiative content begins at $\ell=2$.

When the nonlinear field is iterated, the gauge part of the linear solution participates in products with the physical part. A nonlinear coordinate transformation can again put the exterior solution in canonical form, but its two moments are then nonlinear functionals of all six source families:

$$\begin{aligned} M_L&=I_L+\Delta M_L[I,J,W,X,Y,Z],\\ S_L&=J_L+\Delta S_L[I,J,W,X,Y,Z]. \end{aligned}$$

The corrections $\Delta M_L,\Delta S_L$ vanish at leading linear order. Their detailed post-Newtonian onset depends on the moment and source class; treating them as identically zero beyond leading order would erase real gauge-moment contributions. Conversely, the presence of $W_L,\ldots,Z_L$ does not add physical radiative polarizations. Their role is to complete the general harmonic-coordinate representation before the canonical reorganization.

From canonical moments to radiative moments

The asymptotic observer uses radiative coordinates $(T,\boldsymbol X)$ for which $U=T-R$ approaches a null coordinate without the large logarithms present in harmonic radius. The ordinary metric perturbation has the leading transverse–traceless form

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

where $\mathcal P_{ijab}(\boldsymbol N)$ is the TT projector, $\epsilon_{123}=+1$, and $(\boldsymbol N,\boldsymbol P,\boldsymbol Q)$ is an oriented wave-frame triad. A superscript $(p)$ on a moment below means $p$ derivatives with respect to its time argument. This equation defines $U_L,V_L$ operationally after the radiative frame and polarization orientation have been fixed.

At leading order in the gravitational coupling,

$$U_L=M_L^{(\ell)}+O(G), \qquad V_L=S_L^{(\ell)}+O(G).$$

The first nonlinear corrections include both scattering tails proportional to the mass monopole $M$ and nonlinear memory, each at order $G$ in the radiative moments; tail-of-tail interactions enter at higher post-Minkowskian order. Their post-Newtonian orders need not coincide. Thus “radiative” does not mean “instantaneous.” The value at $U$ can contain integrals over the earlier history of canonical moments.

For the mass quadrupole, combining both leading maps recovers the familiar result

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

This quadrupole limit is the quickest normalization check on the full dictionary.

Coordinates, scales, and observable invariance

Two auxiliary lengths commonly appear. A scale $b_0$ labels the logarithmic transformation between harmonic and radiative retarded times; changing it shifts the origin of $U$ and reshuffles a tail phase. A finite-part scale $r_0$ regularizes singular multipolar integrals. At sufficiently high order, $r_0$ appears both in the exterior iteration and in the source moments. A consistently assembled waveform must be independent of $r_0$ and describe the same physical signal when $b_0$ and the radiative time coordinate are transformed together.

These checks are more discriminating than inspecting one intermediate formula. Varying $b_0$ while holding the numerical coordinate $U$ fixed generally changes $U_L(U)$; that is expected because the same coordinate label then refers to a different physical cut. The waveform must agree after the corresponding retarded-time shift. By contrast, residual $r_0$ dependence in the fully assembled waveform signals that source and propagation pieces have not been combined at a consistent order.

The stationary limit supplies a second check. For moments constant throughout the relevant past, all derivatives with $\ell\ge2$ vanish; the spacetime may retain Coulombic multipoles, but $U_L=V_L=0$ at leading radiative order and the tail integrals vanish. A source that only recently settled down can still have a tail from its earlier evolution, and accumulated memory can leave a constant strain offset after the news vanishes.

Common pitfalls

Substituting a raw source moment into strain. The mass quadrupole enters through two retarded-time derivatives, and a rank-$\ell$ mass moment enters through $\ell$ derivatives at leading order. Omitting these derivatives fails both dimensional analysis and the stationary limit.

Calling canonical moments observable. Canonical moments are an efficient intermediate parametrization. The asymptotic radiative moments, together with a declared Bondi or wave frame, are the quantities directly tied to the waveform.

Dropping arbitrary scales term by term. Individual source, canonical, or tail contributions may contain $b_0$ or $r_0$. Scale independence is required of the consistently assembled observable, not of every intermediate expression.

Exercise

Suppose a weak source has only a mass quadrupole

$$I_{ij}(t)=Q\,e_{ij}\cos(\Omega t), \qquad e_{ii}=0,$$

with constant STF polarization tensor $e_{ij}$. Find the leading radiative quadrupole and strain, and use the result to diagnose the stationary limit.

Solution

At leading linear order $M_{ij}=I_{ij}$, while the radiative definition gives $U_{ij}=M_{ij}^{(2)}=-\Omega^2Qe_{ij}\cos(\Omega U)$. The current moment vanishes. Therefore $h_{ij}^{\mathrm{TT}}=-(2G\Omega^2Q/R)\mathcal P_{ijab}e_{ab}\cos(\Omega U)$. Since $Q$ has dimensions mass times length squared, $G\Omega^2Q/R$ is dimensionless. Letting $\Omega\to0$ leaves a constant source quadrupole but sends both $U_{ij}$ and the waveform to zero. This distinguishes a stationary exterior deformation from radiation and verifies that the source moment cannot be inserted into strain without its required derivatives.

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.