---
title: The Quadrupole Formula
module: Gravitational Waves
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  The retarded solution of the linearized field equation gives the field of a
  moving source, and conservation of mass and momentum forbids monopole and
  dipole radiation, leaving the mass quadrupole as the leading emitter. The
  quadrupole formula fixes the strain and the radiated luminosity, and applied to
  a compact binary it predicts the inspiral chirp of rising frequency and
  amplitude. The Hulse-Taylor pulsar's orbital decay confirmed it to a fraction
  of a percent.
topics: [Gravitational Waves]
draft: false
sources:
  - book: Schutz
    ref: "A First Course in General Relativity, Ch. 9 — Gravitational Radiation"
  - book: Hartle
    ref: "Gravity, Ch. 23 — Gravitational Wave Emission"
---

The previous lesson solved the vacuum wave equation and read off the two
polarizations. Turning on the source, $\Box\bar h_{\mu\nu}=-16\pi G\,c^{-4}
T_{\mu\nu}$, gives the field a slowly moving mass distribution radiates. The
result is that the leading radiation comes from the second time derivative of the
mass quadrupole moment, the monopole and dipole being killed by conservation
laws. This lesson derives the quadrupole formula for the strain and the
luminosity, applies it to a compact binary to produce the inspiral chirp, and
compares it with the orbital decay of the Hulse-Taylor binary pulsar.

## The retarded field of a source

The sourced wave equation is solved by the retarded integral, the same Green's
function that governs the electromagnetic potential,

$$
\bar h_{\mu\nu}(t,\vec x)
= \frac{4G}{c^4}\int
\frac{T_{\mu\nu}\!\big(t - |\vec x - \vec x'|/c,\ \vec x'\big)}
{|\vec x - \vec x'|}\,\d^3 x'.
$$

The field at an event is the sum of source contributions on the past light cone,
each delayed by the light-travel time from $\vec x'$ to $\vec x$. Two
approximations reduce this to a usable form.

- **Far zone**: the observer is at distance $r$ much larger than the source size,
  so $|\vec x-\vec x'|\approx r$ in the denominator, which comes out of the
  integral.
- **Slow motion**: the source moves slowly compared with $c$, so its
  characteristic size is small compared with the wavelength, and the retardation
  can be evaluated at the single retarded time $t_r = t-r/c$.

With both, the spatial components are

$$
\bar h_{ij}(t,\vec x) \approx \frac{4G}{c^4 r}\int T_{ij}(t_r,\vec x')\,\d^3 x'.
$$

## The stress–energy identity

The integral of $T_{ij}$ is rewritten as a second time derivative of a mass
integral using conservation of the source. Local conservation
$\partial_\mu T^{\mu\nu}=0$ splits into
$\partial_t T^{00}+\partial_k T^{k0}=0$ and
$\partial_t T^{0i}+\partial_k T^{ki}=0$. Multiplying the first by $x^i x^j$ and
integrating twice by parts over a volume enclosing the source (so surface terms
vanish) gives the **tensor virial identity**,

$$
\int T^{ij}\,\d^3 x
= \tfrac{1}{2}\,\frac{\d^2}{\d t^2}\int T^{00}\,x^i x^j\,\d^3 x.
$$

In the slow-motion limit $T^{00}\approx\rho c^2$ is the rest-mass energy density.
Defining the **mass quadrupole moment**

$$
Q^{ij}(t) = \int \rho(t,\vec x')\,x'^i x'^j\,\d^3 x',
$$

the field of the source is

$$
\bar h_{ij}(t,\vec x) = \frac{2G}{c^4 r}\,\ddot Q^{ij}(t_r).
$$

The radiation is governed by the second time derivative of the mass quadrupole,
evaluated at the retarded time.

## Why there is no monopole or dipole radiation

Two lower multipoles are absent, each removed by a conservation law.

- **Monopole**: the analogue of the electric monopole is the total mass $M=\int
  \rho\,\d^3x$. Mass–energy is conserved, $\dot M=0$, so it cannot radiate. In
  electromagnetism the total charge is likewise conserved and monopole radiation
  is likewise absent.
- **Mass dipole**: the mass dipole is $\vec d=\int\rho\,\vec x\,\d^3x=M\vec
  x_{\text{cm}}$. Its first derivative is the total momentum, $\dot{\vec
  d}=\vec P$, and its second derivative $\ddot{\vec d}=\dot{\vec P}=0$ by
  momentum conservation for an isolated system. The dipole term in the radiation
  therefore vanishes identically.
- **Magnetic-type dipole**: the current analogue involves the total angular
  momentum $\vec J$, conserved for an isolated system, $\dot{\vec J}=0$, so it too
  does not radiate.

Electromagnetic radiation is dominated by the electric dipole because positive
and negative charges can separate and oscillate. Mass has one sign only, so the
mass dipole is rigidly tied to the centre of mass and the first radiating
multipole is the quadrupole. This is why gravitational waves are intrinsically
weak: the leading term is one multipole order higher than in electromagnetism.

$$
% caption: Conservation laws remove the two lowest multipoles: constant total
% mass kills the monopole, constant total momentum kills the mass dipole, and
% constant angular momentum kills the current dipole, leaving the quadrupole as
% the first radiating moment.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \y/\lab/\why in {2.4/monopole/mass constant, 1.4/mass dipole/momentum constant, 0.4/current dipole/spin constant} {
    \draw[black, thick] (0,\y-0.25) rectangle (2.6,\y+0.25);
    \node[black, anchor=center] at (1.3,\y) {\lab};
    \draw[black] (0.2,\y+0.28) -- (2.4,\y-0.28);
    \node[black, anchor=west] at (2.8,\y) {\why};
  }
  \draw[acc, thick, fill=acc!10] (0,-0.85) rectangle (2.6,-0.35);
  \node[acc, anchor=center] at (1.3,-0.6) {quadrupole};
  \node[acc, anchor=west] at (2.8,-0.6) {radiates};
\end{tikzpicture}
$$

## The luminosity

The energy carried away per unit time follows from the effective stress–energy of
the wave, averaged over a wavelength. Using the traceless **reduced quadrupole**

$$
I^{ij} = \int \rho\left(x^i x^j - \tfrac{1}{3}\delta^{ij}\,r'^2\right)\d^3 x'
= Q^{ij} - \tfrac{1}{3}\delta^{ij}\,Q^k{}_k,
$$

the total radiated power is the **Einstein quadrupole formula**,

$$
L_{\text{GW}} = \frac{G}{5c^5}\left\langle \dddot I_{ij}\,\dddot I^{ij}\right\rangle,
$$

where angle brackets denote an average over several periods. The prefactor
$G/c^5$ carries the units: $c^5/G\approx 3.6\times10^{52}\ \text{W}$ is the natural
luminosity scale of general relativity, so any realistic source with a
dimensionless $\dddot I$ factor far below unity radiates a tiny fraction of it,
yet a source approaching that scale would be a merger of compact objects moving
near $c$.

## The compact binary

The cleanest application is two masses $m_1$ and $m_2$ in a circular orbit. Work
in the centre-of-mass frame, where the problem reduces to a single body of
reduced mass $\mu=m_1 m_2/M$ at separation $a$, with $M=m_1+m_2$. Place the orbit
in the $x$–$y$ plane with angular frequency $\omega$,

$$
x'(t) = a\cos\omega t, \qquad y'(t) = a\sin\omega t.
$$

The quadrupole components, using double-angle identities, are

$$
Q_{xx} = \tfrac{1}{2}\mu a^2(1+\cos 2\omega t),\quad
Q_{yy} = \tfrac{1}{2}\mu a^2(1-\cos 2\omega t),\quad
Q_{xy} = \tfrac{1}{2}\mu a^2\sin 2\omega t.
$$

Every component oscillates at $2\omega$: the gravitational-wave frequency is twice
the orbital frequency, because the mass distribution returns to itself after half
an orbit. Differentiating three times and inserting into the luminosity formula
gives

$$
L_{\text{GW}} = \frac{32}{5}\,\frac{G}{c^5}\,\mu^2 a^4 \omega^6.
$$

Eliminating $\omega$ with Kepler's third law $\omega^2=GM/a^3$ expresses the
luminosity in terms of the masses and separation,

$$
L_{\text{GW}} = \frac{32}{5}\,\frac{G^4}{c^5}\,
\frac{m_1^2 m_2^2\,(m_1+m_2)}{a^5}.
$$

The steep $a^{-5}$ dependence is decisive: as the orbit shrinks the luminosity
climbs sharply, driving a runaway.

$$
% caption: The radiated luminosity of a circular binary rises as the inverse
% fifth power of the separation and as the square of each mass, so emission is
% negligible at wide separation and diverges as the bodies approach contact.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[anchor=north] {separation};
  \draw[->, black] (0,0) -- (0,3.4) node[anchor=east] {luminosity};
  % L ~ a^-5: large near small a (left), tends to zero at large a (right)
  \draw[acc, very thick] (0.55,3.2)
    .. controls (0.9,1.4) and (1.4,0.55) .. (2.4,0.3)
    .. controls (4.0,0.08) and (5.6,0.04) .. (7.2,0.03);
  \node[acc, anchor=west] at (2.4,1.5) {rises as inverse fifth power};
  \node[black, anchor=south west] at (0.15,2.7) {near contact};
  \node[black, anchor=south] at (5.8,0.1) {wide orbit};
\end{tikzpicture}
$$

$$
% caption: The orbiting binary carries a rotating mass quadrupole; because the
% configuration repeats every half orbit, the emitted wave oscillates at twice
% the orbital frequency, and the luminosity rises as the inverse fifth power of
% the separation.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, densely dotted] (0,0) circle (1.3);
  \fill (1.3,0) circle (0.16);
  \fill[black] (-1.3,0) circle (0.11);
  \draw[->] (1.3,0.3) arc (90:150:1.3 and 0.9);
  \node[anchor=west] at (1.5,0) {m1};
  \node[black, anchor=east] at (-1.5,0) {m2};
  \node[black, anchor=south] at (0,1.35) {separation a};
  \fill[black] (0,0) circle (0.05);
  \node[black, anchor=north] at (0,-0.1) {center of mass};
  % emitted double-frequency wave to the side
  \draw[acc, thick] (2.6,0) sin (3.0,0.4) cos (3.4,0) sin (3.8,-0.4) cos (4.2,0) sin (4.6,0.4) cos (5.0,0);
  \node[acc, anchor=south] at (3.8,0.45) {wave at twice orbit rate};
\end{tikzpicture}
$$

## The inspiral chirp

The orbit is not closed: radiated energy comes from the orbital energy, which for
a bound circular orbit is $E = -Gm_1 m_2/(2a)$. Energy balance
$\dot E = -L_{\text{GW}}$ gives

$$
\frac{Gm_1 m_2}{2a^2}\,\dot a = -\frac{32}{5}\frac{G^4}{c^5}
\frac{m_1^2 m_2^2 M}{a^5}
\quad\Longrightarrow\quad
\dot a = -\frac{64}{5}\frac{G^3}{c^5}\frac{m_1 m_2 M}{a^3}.
$$

The separation shrinks, $\omega$ rises, the luminosity rises, and the shrinkage
accelerates: an **inspiral**. The gravitational-wave frequency $f=\omega/\pi$ and
amplitude both sweep upward toward merger, a **chirp**. Introducing the **chirp
mass**

$$
\mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1+m_2)^{1/5}} = \mu^{3/5}M^{2/5},
$$

the frequency evolution takes a form depending on the masses only through
$\mathcal M$,

$$
\dot f = \frac{96}{5}\,\pi^{8/3}
\left(\frac{G\mathcal M}{c^3}\right)^{5/3} f^{11/3}.
$$

Measuring $f(t)$ and $\dot f(t)$ from the waveform therefore delivers the chirp
mass directly, the single best-determined parameter of an inspiral. The
$f^{11/3}$ dependence makes the sweep slow at low frequency and explosively fast
near merger, the audible-analogue rise that named the chirp.

$$
% caption: As the binary loses energy the separation shrinks, the frequency and
% amplitude climb, and the waveform sweeps upward into the merger; the rate of
% frequency increase fixes the chirp mass.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[anchor=north] {time};
  \draw[->, black] (0,-1.5) -- (0,1.5) node[anchor=east] {strain};
  % chirp: increasing frequency and amplitude via explicit sine segments
  \draw[acc, thick]
    (0.3,0) sin (0.9,0.25) cos (1.5,0) sin (2.1,-0.25) cos (2.7,0)
    sin (3.15,0.4) cos (3.6,0) sin (4.05,-0.4) cos (4.5,0)
    sin (4.85,0.6) cos (5.2,0) sin (5.55,-0.6) cos (5.9,0)
    sin (6.15,0.9) cos (6.4,0) sin (6.65,-0.9) cos (6.9,0)
    sin (7.1,1.25) cos (7.3,0);
  \node[acc, anchor=west] at (7.4,0.9) {merger};
  \node[black, anchor=north] at (1.5,-0.35) {inspiral};
\end{tikzpicture}
$$

> **Worked example.** For two neutron stars of $1.4\,M_\odot$ each at separation
> $a$, the reduced mass is $\mu=0.7\,M_\odot$ and $M=2.8\,M_\odot$. When the
> orbit reaches $a=100\ \text{km}$, Kepler gives $\omega=\sqrt{GM/a^3}$. With
> $GM_\odot/c^2=1.48\ \text{km}$, $GM/c^2=4.1\ \text{km}$, so
> $\omega=c\sqrt{(GM/c^2)/a^3}=c\sqrt{4.1/10^6}\ \text{km}^{-1}\approx
> 6.1\times10^2\ \text{s}^{-1}$, an orbital frequency near $97\ \text{Hz}$ and a
> gravitational-wave frequency near $195\ \text{Hz}$, in the band of ground-based
> detectors. The luminosity at this separation, from
> $L=\tfrac{32}{5}(G^4/c^5)m_1^2m_2^2M/a^5$, is of order $2\times10^{45}\
> \text{W}$, comparable to the electromagnetic output of a large galaxy, and it
> climbs steeply as $a$ falls toward the tens of kilometres of contact.

## The Hulse-Taylor binary pulsar

The first evidence came not from a detector but from timing a pulsar. PSR
B1913+16 is a neutron star in an $7.75$-hour eccentric orbit with a companion
neutron star; the pulsar's radio pulses act as a precise clock, and their arrival
times map the orbit. General relativity predicts the orbit loses energy to
gravitational radiation, so the orbital period should shrink at a computed rate.
The quadrupole formula, generalized to the eccentric orbit, predicts a period
derivative

$$
\dot P_b \approx -2.40\times10^{-12},
$$

a decrease of about $76.5$ microseconds per year. The measured cumulative shift
of the time of periastron follows a parabola that matches the general-relativity
prediction to better than $0.2\%$ over decades of timing, the first confirmation
that gravitational waves carry energy exactly as the quadrupole formula says.

$$
% caption: The cumulative shift in the time of periastron of the Hulse-Taylor
% pulsar accumulates as a parabola in time; the data points track the general
% relativity prediction from the quadrupole formula, confirming orbital energy
% loss to gravitational radiation.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.3) -- (7.2,0.3) node[anchor=north] {year};
  \draw[->, black] (0,0.3) -- (0,-3.2) node[anchor=east, pos=0.9] {shift};
  % parabola opening downward (cumulative negative shift)
  \draw[acc, very thick] (0,0.3)
    .. controls (2.4,0.0) and (4.0,-1.1) .. (5.0,-2.0)
    .. controls (5.7,-2.65) and (6.1,-3.0) .. (6.4,-3.1);
  \node[acc, anchor=west] at (5.2,-1.2) {GR prediction};
  % data markers along the curve
  \foreach \x/\y in {0.6/0.24, 1.6/0.05, 2.6/-0.28, 3.6/-0.78, 4.6/-1.55, 5.6/-2.5} {
    \fill[black] (\x,\y) circle (0.06);
  }
  \node[black, anchor=south] at (2.2,0.55) {timing data};
\end{tikzpicture}
$$

The luminosity scaling makes the physical picture concrete: the power radiated
grows as the fifth inverse power of separation and the square of the reduced
mass, so the last seconds of a compact-object inspiral outshine, in
gravitational-wave luminosity, the electromagnetic output of every star in the
observable universe combined. Detecting that final burst directly is the subject
of the next lesson.

[^schutz-quad]: **Schutz**, _A First Course in General Relativity_, Ch. 9 — the retarded solution, the tensor virial identity, the quadrupole formula for the field and luminosity, and the exclusion of monopole and dipole radiation.

[^hartle-quad]: **Hartle**, _Gravity_, Ch. 23 — gravitational-wave emission, the compact-binary inspiral and chirp, and the Hulse-Taylor binary pulsar as the first confirmation of the quadrupole formula.