Multipole Expansion of Gravitational Radiation

Every radiating system can be sorted by the angular shape of the field it throws off — the multipole expansion — and gravity is no exception, but the gravitational version carries a shock in its opening moves. Electromagnetism radiates copiously from an oscillating electric dipole: separate positive and negative charge, shake them, and out come waves. Gravity has only one sign of charge, mass-energy, and obeys conservation laws far stronger than charge conservation. Energy conservation freezes the mass monopole, momentum conservation the mass dipole, angular-momentum conservation the current dipole. The three lowest multipoles an electromagnetic analogy would hand you are struck out, one after another, before any wave escapes. Gravitational radiation cannot begin until the mass quadrupole — the $\ell=2$ stretching-and-squeezing of the source — and that single fact reshapes the whole subject. This article assembles the hierarchy from the ground up: from the linearized Einstein equations in Lorenz gauge, whose retarded solution integrates the source over its past light cone; expanding that integral in source size relative to wavelength (the slow-motion approximation); and showing that the expansion parameter is nothing but $v/c$ . So the multipole series is simultaneously a post-Newtonian series — each higher moment costs another factor $(v/c)^2$ in power. The mass quadrupole dominates almost every astrophysical source, while the current quadrupole and mass octupole supply the leading corrections that distinguish the harmonics of an eccentric or precessing binary. The organizing framework — the symmetric-trace-free decomposition due in modern form to Thorne — underlies gravitational-wave generation theory and every waveform template a detector uses.

Key takeaways

Radiation from a slow, weakly self-gravitating source expands in a hierarchy of mass multipoles $I_L$ and current multipoles $S_L$ — the spin-2 analogue of the electric/magnetic multipole expansion.
Energy conservation freezes the mass monopole ( $\dot M=0$ ), momentum the mass dipole ( $\ddot d^i=\dot P^i=0$ ), and angular momentum the current dipole ( $\dot J^i=0$ ): the lowest radiating moment is the mass quadrupole, $\ell=2$ .
The leading wave is the second time derivative of the mass quadrupole, $h_{ij}^{TT}=(2G/c^4 r)\,\Lambda_{ij,kl}\,\ddot I_{kl}(t-r/c)$ .
Each higher multipole carries an extra factor $v/c$ in amplitude and $(v/c)^2$ in luminosity — the multipole series is also the post-Newtonian series.
Mass moments ( $T^{00}$ , even parity) and current moments ( $T^{0i}$ , odd parity) radiate into orthogonal polarizations and do not interfere in the total power at a given order.
The absence of monopole and dipole emission is a structural prediction of GR that scalar and vector theories violate — which is why dipole-radiation searches are sharp tests of GR.
Key idea: energy, momentum, and angular-momentum conservation kill the mass monopole, mass dipole, and current dipole, so gravitational radiation begins at the mass quadrupole — and each higher multipole is suppressed by a further factor of $(v/c)^2$ in luminosity.

The core insight

Energy, momentum, and angular-momentum conservation kill the mass monopole, mass dipole, and current dipole, so gravitational radiation begins at the mass quadrupole — and each higher multipole is suppressed by an extra factor of $(v/c)^2$ in luminosity.

Where this lives today

The multipole hierarchy is the working language of every gravitational-wave observation. In a compact-binary inspiral the dominant signal is the mass quadrupole at twice the orbital frequency; the "higher modes" appear as extra harmonics scaling with $v/c$ , and detecting them sharpens the measured mass ratio and inclination. Modern waveform families used by LIGO–Virgo–KAGRA include these subdominant multipoles explicitly, and the unequal-mass merger GW190412 was the first with clearly measured higher-mode content. The absence of dipole radiation is itself a precision test: scalar-tensor theories predict dipole emission from the two bodies' differing gravitational binding, and its non-appearance in the Hulse–Taylor pulsar (whose orbital decay matches the quadrupole formula to better than a percent) and in mergers tightly bounds them. Continuous-wave searches hunt the mass quadrupole of a spinning neutron star with a tiny "mountain," turning non-detections into ellipticity limits of order $10^{-8}$ ; and the nonlinear "memory" effect — lower multipoles sourcing higher ones — predicts a permanent strain offset that future detectors aim to catch.

The physical picture

The perturbation reaching a distant observer integrates the source's stress-energy over the retarded past, and because the source is small compared to the wavelength, we Taylor-expand the retardation across it. Each term is a moment of successively higher angular order — monopole, dipole, quadrupole — as in electrostatics, but now every moment is a tensor and the field is spin-2. The first figure is the ladder of moments, with conservation laws crossing out the rungs that cannot radiate.

Figure

The radiation ladder: mass moments (from energy density T⁰⁰, even parity) left, current moments (from momentum density T⁰ⁱ, odd parity) right. Conservation freezes the three lowest — monopole M, dipole dⁱ, current dipole Jⁱ. Selection rule Ṁ = 0, d̈ⁱ = Ṗⁱ = 0, J̇ⁱ = 0 means radiation begins at the mass quadrupole ℓ = 2

Why the quadrupole and not the dipole? The electromagnetic dipole $\sum_a q_a \mathbf{x}_a$ has two signs of charge, so its second derivative need not vanish — shake the charges and it radiates. Gravity's counterpart is the mass dipole $\int \rho\,\mathbf{x}\,d^3x$ , just the total mass times the center-of-mass position. Its second derivative is the rate of change of momentum, zero for an isolated system: internal forces cannot accelerate the center of mass, so gravity cannot radiate at dipole order. The lowest surviving motion is a change of shape at fixed center of mass — a prolate body becoming oblate and back — precisely the quadrupole, matching the helicity-2 graviton whose lowest content is $\ell=2$ . Each higher moment carries one more power of $\mathbf{x}'$ inside the retardation, hence one more factor of (source size)/(wavelength), which the virial theorem makes equal to $v/c$ for a bound system. The multipole expansion is a velocity expansion in disguise: the current quadrupole and mass octupole cost one factor $v/c$ in amplitude, hence $(v/c)^2$ in power. The second figure makes that suppression quantitative at a tenth the speed of light.

Figure

Luminosity per multipole order for v/c = 0.1, on a log scale. The mass quadrupole ℓ = 2 sets the reference; the current quadrupole and mass octupole enter at relative (v/c)² = 10⁻², about 1 percent; the next order at (v/c)⁴ = 10⁻⁴. Since mass ℓ-poles scale as (v/c)^2(ℓ−2) and current ℓ-poles as (v/c)^2(ℓ−1) relative to the mass quadrupole, each new multipole costs another (v/c)² and the multipole series is also the post-Newtonian series

Finally, mass and current moments carry opposite parity, like electric and magnetic multipoles: a mass moment is built from the energy density $T^{00}$ (even), a current moment from the momentum density $T^{0i}$ (odd). They excite orthogonal polarizations and never interfere in the total energy — the luminosity is a sum of squares of individual moments.

The core equations

The starting point is the linearized field equation with a source. In Lorenz gauge the trace-reversed perturbation obeys a flat-space wave equation driven by the stress-energy tensor, so the whole apparatus of retarded Green's functions carries over from electromagnetism.

\Box\,\bar h_{\mu\nu} = -\frac{16\pi G}{c^4}\,T_{\mu\nu} \qquad (1)

where: $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ — trace-reversed metric perturbation; $\Box=-c^{-2}\partial_t^2+\nabla^2$ — flat d'Alembertian; $T_{\mu\nu}$ — source stress-energy tensor; $G,c$ — Newton's constant and the speed of light. Reading: $T_{\mu\nu}$ sources the perturbation just as the four-current sources the EM potential, with coupling $16\pi G/c^4$ . The retarded solution integrates the source over the past light cone — the field at $(t,\mathbf{x})$ is built from what the source did at the earlier time $t-|\mathbf{x}-\mathbf{x}'|/c$ .

\bar h_{\mu\nu}(t,\mathbf{x}) = \frac{4G}{c^4}\int \frac{T_{\mu\nu}\!\left(t-\tfrac{|\mathbf{x}-\mathbf{x}'|}{c},\,\mathbf{x}'\right)}{|\mathbf{x}-\mathbf{x}'|}\,d^3x' \qquad (2)

where: the argument $t-|\mathbf{x}-\mathbf{x}'|/c$ is the retarded time; $\mathbf{x}$ is the field point and $\mathbf{x}'$ ranges over the source. Reading: the retarded time enforces causal, outgoing-wave conditions. For a source small compared to the wavelength, expand the retardation, $|\mathbf{x}-\mathbf{x}'|\approx r-\hat n\cdot\mathbf{x}'$ ; Taylor-expanding in $\hat n\cdot\mathbf{x}'/c$ gives one term per power of the source coordinate — the multipole series.

\bar h_{ij}(t,\mathbf{x}) \simeq \frac{4G}{c^4 r}\sum_{n=0}^{\infty}\frac{1}{n!}\,\frac{\hat n_{k_1}\cdots\hat n_{k_n}}{c^n}\,\frac{\partial^n}{\partial t^n}\!\int T_{ij}\,x'_{k_1}\cdots x'_{k_n}\,d^3x' \qquad (3)

where: $r=|\mathbf{x}|$ — far-zone distance; $\hat n_k$ — unit vector toward the observer; $n$ — order of the retardation expansion. Reading: each power of $\mathbf{x}'$ raises the moment by one, and $\hat n\cdot\mathbf{x}'/c\sim v/c$ makes successive terms small. The $n=0$ term contains the volume integral of the spatial stress $T^{ij}$ , which the tensor virial identity $\tfrac{d^2}{dt^2}\int T^{00}x^ix^j\,d^3x=2c^2\int T^{ij}\,d^3x$ (derived below from conservation) trades for the second time derivative of the mass second moment. That moment, made trace-free, is the mass quadrupole — the central object of the whole theory.

I^{ij}(t)=\int \rho\left(x^i x^j-\tfrac13\delta^{ij}\,r^2\right)d^3x \qquad (4)

where: $\rho=T^{00}/c^2$ — mass density; $\delta^{ij}$ — Kronecker delta; the subtraction removes the trace so $I^{ij}$ carries only the genuine $\ell=2$ shape, i.e. how far the mass departs from spherical symmetry. Combining the leading term with the virial identity and projecting onto the two physical (transverse-traceless) polarizations gives the quadrupole formula in field form.

h_{ij}^{TT}(t,\mathbf{x}) = \frac{2G}{c^4 r}\,\Lambda_{ij,kl}(\hat n)\,\ddot I_{kl}\!\left(t-\frac{r}{c}\right) \qquad (5)

where: $\Lambda_{ij,kl}=P_{ik}P_{jl}-\tfrac12 P_{ij}P_{kl}$ with $P_{ij}=\delta_{ij}-\hat n_i\hat n_j$ — the TT projector along $\hat n$ ; $\ddot I_{kl}$ — second time derivative of the mass quadrupole at retarded time. Reading: only the acceleration of the source's shape radiates, and only its TT part is physical. One order higher in $v/c$ come the current moments, from the momentum density: the current dipole is the (conserved, non-radiating) angular momentum; the current quadrupole is the leading radiating current moment.

S^{ij}=\int \epsilon_{kl(i}\,x^{\,j)} x^k\,T^{0l}\,d^3x, \qquad J^i=\int \epsilon^{ijk}x^j\,T^{0k}\,d^3x \qquad (6)

where: $S^{ij}$ — current quadrupole; $J^i$ — current dipole = total angular momentum; $\epsilon_{ijk}$ — Levi-Civita symbol; parentheses denote symmetrization. Reading: current moments are the gravitational analogues of magnetic multipoles, sourced by mass flow not density. Squaring the field and integrating over the sphere gives the total luminosity as a sum over multipoles, the mass quadrupole leading and each successive term down by $(v/c)^2$ .

L = \frac{G}{5c^5}\big\langle\dddot I_{ij}\,\dddot I_{ij}\big\rangle + \frac{G}{c^7}\left[\frac{1}{189}\big\langle I^{(4)}_{ijk}\,I^{(4)}_{ijk}\big\rangle + \frac{16}{45}\big\langle\dddot S_{ij}\,\dddot S_{ij}\big\rangle\right]+\cdots \qquad (7)

where: $L$ — total radiated power; $I_{ij},I_{ijk}$ — mass quadrupole and octupole; $S_{ij}$ — current quadrupole; $I^{(4)}$ — four time derivatives; $\langle\cdot\rangle$ — average over several wavelengths. Reading: the coefficients $\tfrac15,\tfrac{1}{189},\tfrac{16}{45}$ are Thorne's standard values; the sum-of-squares structure confirms mass and current moments do not interfere.

Derivation AWhy there is no monopole or dipole radiation

The goal is to prove the selection rule that governs the entire subject: the three lowest moments an electromagnetic analogy would suggest — mass monopole, mass dipole, current dipole — are each frozen in time by a conservation law and so cannot source outgoing waves. We differentiate each moment and use the linearized conservation law $\partial_\mu T^{\mu\nu}=0$ (forced by the linearized Einstein equations) to show the relevant derivative vanishes for any bounded, isolated system.

Step 1. The mass monopole is $M=\tfrac1{c^2}\int T^{00}\,d^3x$ . Differentiate once and use $\partial_0 T^{00}=-\partial_k T^{0k}$ , so $\dot M=-\tfrac1{c^2}\int\partial_k T^{0k}\,d^3x=0$ because the integral of a divergence is a surface term that vanishes for a bounded source. Why: a constant monopole gives the static Newtonian/Schwarzschild field, not radiation — mass-energy conservation as a no-radiation theorem.

Step 2. The mass dipole is $d^{\,i}=\tfrac1{c^2}\int T^{00}x^i\,d^3x$ . Integrating by parts with the same conservation law, its first derivative is $\dot d^{\,i}=\tfrac1{c}\int T^{0i}\,d^3x=P^i/c$ , the total momentum. So radiation would require the second derivative $\ddot d^{\,i}=\dot P^i/c$ . Why: this identifies the would-be dipole source with the acceleration of the center of mass.

Step 3. Apply momentum conservation: from $\partial_0 T^{0i}=-\partial_k T^{ki}$ , $\dot P^i=-\int\partial_k T^{ki}\,d^3x=0$ . Hence $\ddot d^{\,i}=0$ and there is no mass-dipole radiation; equivalently, work in the center-of-momentum frame. Why: internal forces cannot shift the center of mass — exactly momentum conservation.

Step 4. The current dipole $J^i=\int\epsilon^{ijk}x^j T^{0k}\,d^3x$ is the total angular momentum. For an isolated system, $\dot J^i=0$ , so it too is frozen and cannot radiate. Why: the magnetic-type $\ell=1$ moment is conserved, the rotational analogue of momentum conservation.

Step 5. The lowest moment whose relevant derivative can be nonzero is thus the mass quadrupole at $\ell=2$ , the current quadrupole one order higher in $v/c$ . The selection rule is structural — it follows from conservation laws alone and fails only in theories whose radiation is not tied to a conserved rank-2 source. Why: this is why the quadrupole formula heads gravitational-wave theory.

Go deeper · Derivation B · 1 minFrom the retarded integral to the quadrupole field

Derivation BFrom the retarded integral to the quadrupole field

The goal is to reduce the leading ( $n=0$ ) term of the expansion — the unfamiliar volume integral of the spatial stress $T^{ij}$ — to the clean, computable second time derivative of the mass quadrupole via the tensor virial identity. This turns the formal series (3) into the working quadrupole formula (5) and shows why only the second derivative of the source's shape radiates.

Step 1. Keep the $n=0$ term of (3): $\bar h_{ij}\approx\dfrac{4G}{c^4 r}\int T_{ij}\!\left(t-\tfrac{r}{c},\mathbf{x}'\right)d^3x'$ , the spatial stress over the source at retarded time. Why: at lowest order in $v/c$ the retardation across the source is negligible, so it all contributes at the single time $t-r/c$ .

Step 2. Establish the virial identity. Two uses of conservation give $\partial_0^2 T^{00}=\partial_k\partial_l T^{kl}$ ; multiply by $x^i x^j$ and integrate by parts twice, with boundary terms vanishing and $\partial_k\partial_l(x^i x^j)=2\delta^{i}_{k}\delta^{j}_{l}$ symmetrized, leaving $\int\partial_0^2 T^{00}x^i x^j\,d^3x=2\int T^{ij}\,d^3x$ . Why: this turns the opaque stress integral into the acceleration of a mass moment set by the density.

Step 3. Rearrange: $\displaystyle\int T^{ij}\,d^3x=\frac{1}{2c^2}\frac{d^2}{dt^2}\int T^{00}x^i x^j\,d^3x=\frac12\ddot M_{ij}$ , with $M_{ij}=\tfrac1{c^2}\int T^{00}x^i x^j\,d^3x$ . Why: the factor $\tfrac12$ and the second derivative are the whole content of "the quadrupole radiates through its acceleration."

Step 4. Substitute: $\bar h_{ij}=\dfrac{2G}{c^4 r}\,\ddot M_{ij}(t-r/c)$ , then project with $\Lambda_{ij,kl}$ ; since it removes any trace, $M_{ij}$ and its trace-free $I_{ij}$ give the same field, $h_{ij}^{TT}=\dfrac{2G}{c^4 r}\Lambda_{ij,kl}\,\ddot I_{kl}$ . Why: the TT projection strips the gauge parts, leaving the two physical polarizations. *Note: the TT projector $\Lambda_{ij,kl}=P_{ik}P_{jl}-\tfrac12 P_{ij}P_{kl}$ with $P_{ij}=\delta_{ij}-\hat n_i\hat n_j$ is idempotent, transverse, and traceless.*

Go deeper · Derivation C · 1 minThe multipole series as a post-Newtonian velocity expansion

Derivation CThe multipole series as a post-Newtonian velocity expansion

The goal is to show that the multipole index and the post-Newtonian order are the same counting: advancing one step up the ladder — mass quadrupole to current quadrupole or mass octupole — costs one power of $v/c$ in amplitude and $(v/c)^2$ in luminosity. This is what makes the quadrupole formula overwhelmingly dominant for slow sources and turns the expansion into a controlled approximation.

Step 1. In (3), the $n$ -th term carries $(\hat n\cdot\mathbf{x}'/c)^n$ times $\partial_t^n$ on the source. With $\mathbf{x}'\sim L$ and $\partial_t\sim\omega$ , it scales as $(\omega L/c)^n$ . Why: the whole series reduces to powers of the single number $\omega L/c$ .

Step 2. Identify $\omega L/c$ with velocity: for a bound system $v\sim\omega L$ , so $\omega L/c\sim v/c$ , and each order multiplies the amplitude by $v/c$ . Why: the virial theorem ties frequency, size, and velocity, so "small compared to a wavelength" is the same as "moving slowly."

Step 3. Square for power: luminosity is the square of a field derivative, so $v/c$ in amplitude becomes $(v/c)^2$ in power. Counting from the mass quadrupole, a mass $\ell$ -pole enters at $L\sim(v/c)^{2(\ell-2)}L_{\text{quad}}$ and a current $\ell$ -pole one power of $v/c$ higher, $L\sim(v/c)^{2(\ell-1)}L_{\text{quad}}$ — which is why the mass octupole ( $\ell=3$ ) and the current quadrupole ( $\ell=2$ ) both enter together at relative $(v/c)^2$ . Why: the quantitative content of the second figure — for $v/c=0.1$ , about one percent.

Step 4. Consequence: for a slow source the series converges fast and the mass quadrupole dominates; only precision modeling of fast, asymmetric sources near merger needs the higher moments. Why: it explains both the quadrupole formula's success and where it must be corrected. *Note: the relative $1/c^2$ between successive lines of Eq. (7) is exactly this argument's $(v/c)^2$ .*

Limiting cases

Case	Result	Meaning
Slow-motion / compact source	$\lambda\gg L,\quad v/c\ll1$	Retardation across the source is small; the series converges fast and the mass quadrupole dominates — the quadrupole-formula regime.
Leading order (mass quadrupole)	$L=\dfrac{G}{5c^5}\langle\dddot I_{ij}\dddot I_{ij}\rangle$	The lowest radiating term; $I_{ij}$ is the trace-free mass quadrupole and every other moment is suppressed by powers of $v/c$ .
Forbidden lowest orders	$\dot M=0,\ \ddot d^i=\dot P^i=0,\ \dot J^i=0$	Monopole, mass dipole, and current dipole cannot radiate — their derivatives vanish by conservation of energy, momentum, angular momentum.
Higher multipoles	$L^{\text{mass}}_\ell\sim(v/c)^{2(\ell-2)}L_{\text{quad}}$ , $L^{\text{curr}}_\ell\sim(v/c)^{2(\ell-1)}L_{\text{quad}}$	Mass octupole and current quadrupole enter at relative $(v/c)^2$ ; the series terminates effectively at modest order for slow sources.
Spherical symmetry	$h^{TT}_{ij}=0$ exactly	Only a static monopole; by Birkhoff's theorem the exterior is Schwarzschild regardless of radial pulsation — no radiation.

Worked examples

1 · How feeble a laboratory source is (fully worked)

A steel rod of mass $M=500\ \mathrm{kg}$ , length $\ell=20\ \mathrm{m}$ is spun end-over-end about its center at $\omega=30\ \mathrm{rad/s}$ (tip speed $300\ \mathrm{m/s}$ , near steel's tensile limit). What gravitational-wave luminosity does it emit?

Step 1. With the rod along $(\cos\omega t,\sin\omega t,0)$ and $I=M\ell^2/12$ , the mass second moment oscillates as $M_{xx}=\tfrac12 I\cos2\omega t$ , $M_{yy}=-\tfrac12 I\cos2\omega t$ , $M_{xy}=\tfrac12 I\sin2\omega t$ — emission at $2\omega$ , the quadrupole hallmark, with zero oscillating trace so $M_{ij}=I_{ij}$ .

Step 2. Each $d/dt$ brings down $2\omega$ , so $\dddot I_{ij}\sim 8I\omega^3$ per component; squaring and summing the independent components gives $\langle\dddot I_{ij}\dddot I_{ij}\rangle=32\,I^2\omega^6$ .

Step 3. Insert $L=\dfrac{G}{5c^5}\langle\dddot I_{ij}\dddot I_{ij}\rangle$ , $I=M\ell^2/12$ :

L=\frac{32\,G\,I^2\omega^6}{5c^5}=\frac{2}{45}\,\frac{G}{c^5}\,M^2\ell^4\omega^6.

Step 4. With $M^2=2.5\times10^5$ , $\ell^4=1.6\times10^5$ , $\omega^6=7.29\times10^8$ , $G/c^5=2.75\times10^{-53}$ SI,

L=\frac{2}{45}\times2.75\times10^{-53}\times(2.5\times10^5)(1.6\times10^5)(7.29\times10^8)\approx\mathbf{3.6\times10^{-35}\ W}.

Even a rod spun to destruction radiates $\sim10^{-35}$ W. Gravity's weak coupling $G/c^5$ means no laboratory source is ever visible; only relativistic astrophysical masses radiate detectably.

2 · A neutron-star binary as a half-billion suns

Two neutron stars of $m_1=m_2=1.4\,M_\odot$ ( $2.78\times10^{30}\ \mathrm{kg}$ ) orbit in a circle of separation $a=10^7\ \mathrm{m}$ . Estimate the luminosity and orbital velocity, and check the quadrupole formula applies.

Step 1. For a circular binary, formula (7) reduces (Peters–Mathews) to

L=\frac{32}{5}\,\frac{G^4}{c^5}\,\frac{(m_1 m_2)^2(m_1+m_2)}{a^5},

using $\langle\dddot I_{ij}\dddot I_{ij}\rangle=32\,\mu^2 a^4\Omega^6$ , $\Omega^2=G(m_1+m_2)/a^3$ .

Step 2. With $m_1=m_2=2.78\times10^{30}\ \mathrm{kg}$ and $a=10^7\ \mathrm{m}$ ,

L=\frac{32}{5}\times\frac{(6.674\times10^{-11})^4}{(2.998\times10^8)^5}\times\frac{(2.78\times10^{30})^4\,(5.57\times10^{30})}{(10^7)^5}\approx\mathbf{1.8\times10^{35}\ W}.

Step 3. Against the Sun's $L_\odot=3.8\times10^{26}\ \mathrm{W}$ , the ratio is $L/L_\odot\approx\mathbf{4.6\times10^{8}}$ — nearly half a billion suns in gravitational waves from two city-sized stars.

Step 4. Check the approximation: $\Omega=\sqrt{G(m_1+m_2)/a^3}=0.61\ \mathrm{rad/s}$ gives $v=\Omega a\approx6.1\times10^6\ \mathrm{m/s}$ , so $v/c\approx0.02\ll1$ . The slow-motion expansion holds and the quadrupole dominates, corrections entering at $\sim(v/c)^2\approx4\times10^{-4}$ . For this equal-mass, non-spinning system that correction is not a higher multipole at all: the mass octupole and the current quadrupole both carry a factor $(m_1-m_2)$ and vanish identically, so the leading $(v/c)^2$ correction is the first post-Newtonian correction to the mass quadrupole itself. A $v/c=0.02$ neutron-star binary shines at $\sim5\times10^{8}\,L_\odot$ , essentially all in the quadrupole.

Physics of the portrait

The multipole expansion spans a century: Einstein wrote the quadrupole formula in 1918, and six decades later Kip Thorne recast the whole hierarchy in the symmetric-trace-free language modelers now use.

Origins · 1 minKip Thorne and how this was discovered

Kip Thorne

born 1940, American theoretical physicist, Nobel laureate 2017 · the modern multipole formalism of gravitational-wave generation

Kip Thorne turned gravitational radiation from an elegant weak-field prediction into a working quantitative science. Einstein had derived the quadrupole formula in 1918, but for half a century the higher multipoles lived in a thicket of inequivalent conventions, and it was unclear how to define the mass and current moments of a strongly self-gravitating source unambiguously. Thorne's 1980 review "Multipole expansions of gravitational radiation" (Rev. Mod. Phys. 52, 299) settled it: he organized the field into symmetric-trace-free mass moments $I_L$ and current moments $S_L$ , gave the luminosity as a clean sum over $\ell$ with the coefficients $\tfrac15,\tfrac{1}{189},\tfrac{16}{45},\dots$ , and made the multipole order coincide with the post-Newtonian order. That framework — refined with Blanchet, Damour, and others — is the direct ancestor of the analytic waveform templates LIGO and Virgo match against. Thorne also co-founded LIGO with Weiss and Drever and shared the 2017 Nobel Prize for the first detection of the waves whose multipole structure he had spent his career codifying — the rare theorist whose formalism, instrument, and detection all came to fruition in one lifetime.

History

1916

— Einstein predicts gravitational waves from the linearized field equations

1918

— Einstein derives the quadrupole formula for the radiated power

1974

— Hulse and Taylor discover the binary pulsar PSR B1913+16

1980

— Thorne casts the multipole expansion in symmetric-trace-free form

2015

— LIGO detects GW150914, the first observed binary black hole merger

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §36.1–36.10 · Thorne, "Multipole expansions of gravitational radiation," Rev. Mod. Phys. 52, 299 (1980) · Wald, General Relativity (1984), §4.4 · Maggiore, Gravitational Waves Vol. 1 (2008), Ch. 3 · Carroll, Spacetime and Geometry (2004), §7.5 · Schutz, A First Course in General Relativity, 2nd ed. (2009), Ch. 9. The TT projector is $\Lambda_{ij,kl}=P_{ik}P_{jl}-\tfrac12 P_{ij}P_{kl}$ , $P_{ij}=\delta_{ij}-\hat n_i\hat n_j$ ; the luminosity coefficients $\tfrac15,\tfrac{1}{189},\tfrac{16}{45}$ in Eq. (7) are Thorne's standard values. The worked numbers are: the spinning rod gives $L=(2/45)(G/c^5)M^2\ell^4\omega^6\approx3.6\times10^{-35}\ \mathrm{W}$ ; the $1.4$ – $1.4\,M_\odot$ circular binary at $a=10^7\ \mathrm{m}$ gives $L\approx1.8\times10^{35}\ \mathrm{W}\approx4.6\times10^{8}\,L_\odot$ , $v/c\approx0.02$ .

Builds on: Linearized Einstein Equations · Lorenz (Harmonic) Gauge · Metric Perturbation and Trace-Reversed Field · Transverse-Traceless Gauge · Gravitational Wave Equation. Leads to: Quadrupole Formula · Gravitational-Wave Luminosity and Energy Flux · Plus and Cross Polarizations · Helicity and Spin-2 Nature of Gravitons · Isaacson Stress-Energy of Gravitational Waves.

Ask the tutor: Ask me to derive the Peters–Mathews orbital decay from the quadrupole luminosity, or to show how the current quadrupole enters one post-Newtonian order above the mass quadrupole.

Why does electromagnetism radiate at dipole order but gravity does not?

Because the conservation laws differ. In electromagnetism only charge is conserved ( $\dot Q=0$ ), which freezes the monopole but leaves the electric dipole $\sum_a q_a\mathbf{x}_a$ free to oscillate — two opposite charges shaken against each other radiate. In gravity the analogue is the mass dipole, whose first derivative is the total momentum and whose second is $\dot P^i=0$ by momentum conservation. So energy conservation removes the monopole and momentum conservation the dipole — one order higher than in electromagnetism. The current dipole (angular momentum) is conserved too, leaving the mass quadrupole as the lowest survivor.

What does it mean that the multipole expansion is also a post-Newtonian expansion?

Each step up the ladder inserts one more factor of (source size)/(wavelength) inside the retardation, which the virial theorem makes equal to $v/c$ . So climbing one multipole order and advancing one power of $v/c$ are the same operation: the mass octupole and current quadrupole are "1PN" corrections, down by $(v/c)^2$ in power — which is why for $v/c\sim0.1$ the quadrupole formula is already good to about one percent. The identification is about where each NEW multipole first enters: a fixed multipole, the mass quadrupole included, also receives its own post-Newtonian corrections at $(v/c)^2$ , $(v/c)^4$ , and so on, so "multipole order = PN order" counts the leading appearance of each moment, not the whole content of a given PN order.

Why does only the second time derivative of the quadrupole appear, not the quadrupole itself?

The retarded solution with the tensor virial identity gives $\bar h_{ij}\propto\ddot M_{ij}$ — the field is fixed by the acceleration of the mass distribution, not its instantaneous shape. A static or uniformly changing quadrupole gives near-zone tidal fields but no outgoing waves, just as a static charge gives a Coulomb field but no radiation. The radiated power involves one more derivative, $\langle\dddot I_{ij}\dddot I_{ij}\rangle$ , since luminosity is the flux of $\dot h^2$ — hence three dots.

What is the difference between mass multipoles and current multipoles?

Mass multipoles are built from the energy density $T^{00}$ and describe how mass is distributed — the even-parity, "electric-type" moments. Current multipoles come from the momentum density $T^{0i}$ and describe how mass is flowing — the odd-parity, "magnetic-type" moments. The current dipole is the (conserved, non-radiating) total angular momentum; the current quadrupole is the leading radiating current moment. Having opposite parity, the two families add as separate squares in the total power and never interfere.

Could a spherically symmetric but violently pulsating star emit gravitational waves?

No — one of the cleanest illustrations of the selection rule. A spherically symmetric distribution has only a monopole, and for an isolated system that monopole (the total mass-energy) is conserved and time-independent. By Birkhoff's theorem, the exterior of any spherically symmetric source is exactly Schwarzschild, fixed by that mass alone, however violently the star pulsates radially. With no time-varying quadrupole or higher moment there is nothing to radiate — a star must become non-spherical to emit at all.

How do these higher multipoles actually show up in real detections?

In a compact binary the dominant harmonic is at twice the orbital frequency — the mass quadrupole. Higher multipoles add further harmonics (three times the orbital frequency from the octupole and current quadrupole) whose amplitudes grow as the binary speeds toward merger. For equal-mass, non-spinning systems these higher modes nearly cancel; for unequal-mass or precessing systems they are appreciable, and GW190412 (a roughly 3-to-1 mass-ratio merger) was the first event with clearly measured higher-mode content. Bounding the modes also tests GR, since alternative theories predict extra (e.g. dipole) emission that GR forbids.