---
title: Gravitational Waves from Inspiraling Binaries
draft: false
module: Binaries and Gravitational Waves
moduleNumber: 9
lessonNumber: 3
order: 903
summary: >
  A time-varying mass quadrupole radiates gravitational waves, ripples in spacetime
  that stretch and squeeze a ring of free masses along two polarizations. The
  radiated power drains a binary's orbital energy, shrinking the orbit and sweeping
  the wave frequency upward in a chirp whose rate fixes the chirp mass. Laser
  interferometers with kilometre arms measure the resulting strain of order ten to
  the minus twenty-one. The first detection, GW150914, matched a template for two
  merging black holes near thirty solar masses each.
topics: [Binaries and Gravitational Waves]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 17 §17.6 — Gravitational Radiation"
  - book: Maoz
    ref: "Ch. 4 — Stellar Death and Remnants"
---

General relativity predicts that accelerating masses radiate gravitational waves,
propagating distortions of spacetime that travel at the speed of light. The waves are
extraordinarily weak, because the coupling constant of gravity to matter is
$G/c^4 \sim 10^{-45}\ \mathrm{s^2\,kg^{-1}\,m^{-1}}$, and only the most violent
astrophysical events, the merger of two compact objects, produce a signal detectable
on Earth. This lesson derives the quadrupole formula and the reason gravitational
radiation is so faint, describes the two polarizations and their effect on free
masses, follows the orbital decay that drives an inspiral into a rising chirp, and
extracts the chirp mass from the frequency sweep. It closes with the interferometric
detection principle and the first observed event, GW150914.

## Linearized gravity and the quadrupole formula

Far from any source, spacetime is nearly flat and the metric is a small perturbation
on the Minkowski background,

$$
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},
\qquad |h_{\mu\nu}| \ll 1.
$$

Linearizing the Einstein field equations in $h_{\mu\nu}$ and choosing the harmonic
(Lorenz) gauge reduces them to a wave equation with the stress-energy tensor as
source,

$$
\Box\, \bar h_{\mu\nu} = -\frac{16\pi G}{c^4}\, T_{\mu\nu},
$$

where $\bar h_{\mu\nu}$ is the trace-reversed perturbation. In vacuum the right side
vanishes and $h_{\mu\nu}$ propagates as a wave at speed $c$. The lowest radiating
multipole is not the monopole or dipole. Mass conservation forbids monopole
radiation, and momentum conservation forbids mass-dipole radiation, because the
mass dipole's second derivative is $\ddot{\!\sum m_i \vec r_i} = \dot{\vec P} = 0$
for an isolated system. Gravitational radiation therefore begins at the **quadrupole**
order.

> **Theorem (Quadrupole formula).** The gravitational-wave strain at distance $r$
> from a slowly moving, weak-field source is
> $$
> h_{ij} = \frac{2G}{c^4 r}\,\ddot{I}_{ij}\!\left(t - \tfrac{r}{c}\right),
> $$
> where $I_{ij} = \int \rho\,\bigl(x_i x_j - \tfrac{1}{3}\delta_{ij} r^2\bigr)\d V$
> is the reduced (traceless) mass quadrupole moment. The radiated luminosity is
> $$
> L_{\mathrm{GW}} = \frac{G}{5 c^5}\,
>   \bigl\langle \dddot{I}_{ij}\,\dddot{I}_{ij} \bigr\rangle .
> $$

The prefactor $G/c^5$ carries the units of inverse power and has the numerical value
$1/L_0$ with $L_0 = c^5/G \approx 3.6 \times 10^{52}\ \mathrm{W}$, an enormous
luminosity scale. Gravitational-wave luminosities are large only when a substantial
fraction of a compact object's rest energy is set into quadrupolar motion at the
dynamical frequency, which happens only in the final orbits of a compact-binary
merger.

## Two polarizations and their effect on free masses

A gravitational wave propagating along $z$ has two independent polarization states,
transverse and traceless. In the transverse-traceless gauge the perturbation is

$$
h_{ij} =
\begin{pmatrix} h_+ & h_\times & 0 \\ h_\times & -h_+ & 0 \\ 0 & 0 & 0 \end{pmatrix}
\cos\bigl[\omega(t - z/c)\bigr].
$$

The wave does not move test masses through space; it changes the proper distances
between them. Consider a ring of freely floating masses in the plane transverse to the
propagation direction. The **plus** polarization stretches the ring along one axis
while squeezing it along the perpendicular axis, then reverses each half-cycle. The
**cross** polarization does the same along axes rotated by $45^\circ$. A displacement
$L$ between two free masses oscillates by

$$
\frac{\delta L}{L} = \frac{1}{2}\, h,
$$

so the wave amplitude $h$ is a dimensionless strain, the fractional change in length.

$$
% caption: The plus and cross polarizations act on a ring of free test masses: plus
% stretches and squeezes along the horizontal and vertical axes over one cycle, and
% cross does the same along axes rotated by forty-five degrees.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% --- plus polarization: three phases ---
\node[black, anchor=south] at (2.0,3.7) {plus polarization};
\draw[black] (0.7,2.6) circle (0.5);
\draw[acc, very thick] (2.0,2.6) ellipse (0.7 and 0.35);
\draw[black] (3.3,2.6) circle (0.5);
\node[black, anchor=north] at (0.7,2.0) {phase 0};
\node[acc, anchor=north] at (2.0,2.0) {stretch x};
\node[black, anchor=north] at (3.3,2.0) {reverse};
% --- cross polarization ---
\node[black, anchor=south] at (6.4,3.7) {cross polarization};
\draw[black] (5.1,2.6) circle (0.5);
\draw[acc, very thick, rotate around={45:(6.4,2.6)}] (6.4,2.6) ellipse (0.7 and 0.35);
\draw[black] (7.7,2.6) circle (0.5);
\node[black, anchor=north] at (5.1,2.0) {phase 0};
\node[acc, anchor=north] at (6.4,1.75) {stretch diagonal};
\node[black, anchor=north] at (7.7,2.0) {reverse};
\end{tikzpicture}
$$

## Orbital decay and the inspiral chirp

A binary of masses $m_1$ and $m_2$ in a circular orbit of separation $a$ has a mass
quadrupole that rotates at the orbital frequency, and the quadrupole formula gives its
gravitational-wave luminosity. Evaluating $\langle \dddot{I}_{ij}\dddot{I}_{ij}
\rangle$ for the orbit yields

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

This power is drawn from the orbital energy $E = -G m_1 m_2 / 2a$. Setting
$\dot E = -L_{\mathrm{GW}}$ and solving for the shrinking separation gives

$$
\frac{\d a}{\d t} = -\frac{64}{5}\,\frac{G^3\, m_1 m_2 (m_1 + m_2)}{c^5\, a^3}.
$$

The orbit decays slowly at first and then catastrophically as $a$ shrinks, since the
rate diverges as $a^{-3}$. Integrating from an initial separation $a_0$ to coalescence
gives a finite inspiral time

$$
t_{\mathrm{inspiral}} = \frac{5}{256}\,\frac{c^5\, a_0^4}{G^3\, m_1 m_2 (m_1 + m_2)}.
$$

As $a$ falls, the orbital frequency rises (Kepler's law, $\omega^2 = GM/a^3$), and so
does the gravitational-wave frequency, which is twice the orbital frequency because
the quadrupole returns to its configuration twice per orbit. The amplitude grows in
concert. The signal is a **chirp**: an oscillation that sweeps upward in both
frequency and amplitude until merger.

$$
% caption: The inspiral-merger-ringdown waveform: a chirp of rising frequency and
% amplitude during the inspiral, a peak at merger, and a damped ringdown as the
% remnant settles to a stationary black hole.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0.4,2.2) -- (8.4,2.2) node[right, black!70] {time};
\draw[->, black] (0.4,0.7) -- (0.4,3.7) node[above, black!70] {strain};
% chirp: increasing frequency + amplitude, then damped ringdown
\draw[acc, very thick]
 (0.6,2.2)
 .. controls (0.9,2.55) and (1.2,2.55) .. (1.5,2.2)
 .. controls (1.8,1.85) and (2.1,1.85) .. (2.4,2.2)
 .. controls (2.65,2.65) and (2.95,2.65) .. (3.2,2.2)
 .. controls (3.42,1.75) and (3.72,1.75) .. (3.95,2.2)
 .. controls (4.13,2.8) and (4.4,2.8) .. (4.6,2.2)
 .. controls (4.76,1.55) and (5.02,1.55) .. (5.2,2.2)
 .. controls (5.36,3.2) and (5.66,3.2) .. (5.85,2.2)
 .. controls (6.0,1.5) and (6.25,1.5) .. (6.4,2.2)
 .. controls (6.55,2.7) and (6.78,2.7) .. (6.9,2.2)
 .. controls (7.02,1.85) and (7.22,1.85) .. (7.32,2.2)
 .. controls (7.42,2.4) and (7.58,2.4) .. (7.68,2.2);
\node[acc, anchor=south] at (2.3,2.9) {inspiral};
\node[black, anchor=south] at (5.5,3.2) {merger};
\node[black, anchor=south] at (7.3,2.5) {ringdown};
\end{tikzpicture}
$$

## The chirp mass

The rate at which the frequency sweeps depends on a single combination of the two
masses. Rewriting the decay in terms of the gravitational-wave frequency
$f = \omega_{\mathrm{GW}}/2\pi$ gives

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

where the mass dependence collapses entirely into the **chirp mass**.

> **Definition (Chirp mass).** The combination of component masses that governs the
> leading-order inspiral,
> $$
> \mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}.
> $$
> The gravitational-wave frequency sweep $\dot f$ depends on the two masses only
> through $\mathcal{M}$, so measuring $f$ and $\dot f$ during the inspiral determines
> the chirp mass directly, with no assumption about distance or inclination.

The chirp mass is the best-measured parameter of any inspiral, because it controls the
phase evolution that a matched-filter analysis tracks over hundreds or thousands of
cycles. The individual masses, the spins, and the distance enter at higher order and
are extracted from the full waveform including the merger and ringdown. Measuring
$\dot f$ and $f$ gives $\mathcal{M}$; combined with the observed amplitude $h$, which
scales as $\mathcal{M}^{5/3} f^{2/3}/r$, the analysis also yields the luminosity
distance $r$, making a compact binary a self-calibrating **standard siren**, the tool
developed in the [next
lesson](/astrophysics-cosmology/binaries-and-gravitational-waves/multimessenger-astronomy-and-gamma-ray-bursts).

## Interferometric detection

A strain of $h \sim 10^{-21}$ changes a $4\ \mathrm{km}$ interferometer arm by
$\delta L \sim 10^{-18}\ \mathrm{m}$, a thousandth of a proton radius. A Michelson
interferometer measures exactly this differential arm-length change. Laser light is
split and sent down two perpendicular arms, reflected, and recombined; a passing wave
lengthens one arm and shortens the other, shifting the interference fringe at the
output. LIGO, Virgo, and KAGRA fold the light through **Fabry-Pérot cavities** in each
arm, so the light traverses the arm hundreds of times and multiplies the accumulated
phase shift, and add power recycling to build up the circulating laser power.

$$
% caption: A Michelson interferometer with Fabry-Perot arm cavities: the beam
% splitter divides the laser between two perpendicular arms whose length difference,
% modulated by a passing wave, is read out at the photodetector.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% laser
\node[black, anchor=east] at (0.5,2.0) {laser};
\draw[very thick, ->] (0.6,2.0) -- (2.6,2.0);
% beam splitter
\draw[black, very thick] (2.75,1.75) -- (3.25,2.25);
\node[black, anchor=north] at (3.0,1.6) {splitter};
% arm x (horizontal)
\draw[very thick] (3.25,2.0) -- (6.6,2.0);
\draw[black!70, very thick] (6.6,1.65) -- (6.6,2.35);
\node[black, anchor=west] at (6.75,2.0) {mirror};
% arm y (vertical)
\draw[very thick] (3.0,2.25) -- (3.0,4.4);
\draw[black!70, very thick] (2.65,4.4) -- (3.35,4.4);
\node[black, anchor=south] at (3.0,4.5) {mirror};
% cavity markers
\draw[black, densely dashed] (4.6,1.65) -- (4.6,2.35);
\node[black, anchor=north] at (4.9,1.7) {arm cavity};
% photodetector
\draw[very thick, ->] (3.0,1.75) -- (3.0,0.7);
\draw[black, very thick] (2.7,0.7) rectangle (3.3,0.45);
\node[black, anchor=north] at (3.0,0.4) {photodetector};
\end{tikzpicture}
$$

The overwhelming challenge is noise. Seismic vibration dominates at low frequency,
thermal motion of the mirror suspensions in the middle band, and photon shot noise at
high frequency. The mirrors hang as pendulums to isolate them from ground motion, and
the detectors are sensitive between roughly $10$ and $1000\ \mathrm{Hz}$, the band in
which stellar-mass compact binaries chirp through merger. A real signal is confirmed by
coincident detection in widely separated instruments, which also triangulates the sky
position from the arrival-time differences.

## GW150914

On 14 September 2015 the two LIGO detectors recorded a chirp sweeping from $35$ to
$250\ \mathrm{Hz}$ over about $0.2\ \mathrm{s}$, matching the template for two inspiraling
black holes. The measured chirp mass and full waveform fit gave component masses of
about $36\ M_\odot$ and $29\ M_\odot$ merging into a final black hole of about
$62\ M_\odot$.

$$
% caption: The GW150914 detection: the measured strain tracks the relativistic
% template through the rising inspiral to the merger peak, with residual scatter
% consistent with instrumental noise.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0.4,2.0) -- (8.4,2.0) node[right, black!70] {time};
\draw[->, black] (0.4,0.6) -- (0.4,3.5) node[above, black!70] {strain};
% template (smooth accent)
\draw[acc, very thick]
 (0.6,2.0)
 .. controls (0.95,2.3) and (1.3,2.3) .. (1.65,2.0)
 .. controls (1.95,1.72) and (2.28,1.72) .. (2.6,2.0)
 .. controls (2.86,2.4) and (3.18,2.4) .. (3.45,2.0)
 .. controls (3.68,1.6) and (3.98,1.6) .. (4.22,2.0)
 .. controls (4.42,2.55) and (4.72,2.55) .. (4.95,2.0)
 .. controls (5.12,1.4) and (5.42,1.4) .. (5.62,2.0)
 .. controls (5.78,2.8) and (6.06,2.8) .. (6.24,2.0)
 .. controls (6.4,1.55) and (6.62,1.55) .. (6.78,2.0)
 .. controls (6.9,2.3) and (7.08,2.3) .. (7.2,2.0);
\node[acc, anchor=south west] at (0.7,2.55) {template};
% measured data (dotted, following template with jitter)
\draw[black, densely dotted, thick]
 (0.6,2.05)
 .. controls (0.95,2.28) and (1.3,2.33) .. (1.65,1.97)
 .. controls (1.95,1.78) and (2.28,1.68) .. (2.6,2.04)
 .. controls (2.86,2.36) and (3.18,2.45) .. (3.45,1.96)
 .. controls (3.68,1.64) and (3.98,1.55) .. (4.22,2.05)
 .. controls (4.42,2.5) and (4.72,2.62) .. (4.95,1.95)
 .. controls (5.12,1.45) and (5.42,1.34) .. (5.62,2.06)
 .. controls (5.78,2.72) and (6.06,2.86) .. (6.24,1.96)
 .. controls (6.4,1.6) and (6.62,1.5) .. (6.78,2.03)
 .. controls (6.9,2.26) and (7.08,2.34) .. (7.2,1.98);
\node[black, anchor=north west] at (4.5,1.42) {measured};
\end{tikzpicture}
$$

> **Worked example.** Energy accounting for GW150914. The initial total mass is
> $36 + 29 = 65\ M_\odot$; the remnant is $62\ M_\odot$. The difference,
> $\approx 3\ M_\odot$, was radiated as gravitational waves in a few tenths of a
> second. Converting, $3\,M_\odot c^2 \approx 5 \times 10^{47}\ \mathrm{J}$ over
> $\sim 0.2\ \mathrm{s}$ gives a peak luminosity of order
> $L \sim 3 \times 10^{49}\ \mathrm{W} \sim 10^{-3}\,(c^5/G)$, briefly exceeding the
> combined electromagnetic output of every star in the observable universe. The event
> occurred at a luminosity distance of about $410\ \mathrm{Mpc}$, and the peak strain
> at Earth was $h \approx 10^{-21}$.

GW150914 confirmed three predictions at once: gravitational waves exist and travel as
the quadrupole formula requires, stellar-mass black holes exist and form binaries that
merge within a Hubble time, and the full inspiral-merger-ringdown waveform matches the
numerical solution of the Einstein equations. It opened gravitational-wave astronomy as
an observational field.

## Summary

Gravitational radiation begins at quadrupole order, with strain $h = (2G/c^4 r)
\ddot{I}_{ij}$ and luminosity $L_{\mathrm{GW}} = (G/5c^5)\langle
\dddot{I}_{ij}\dddot{I}_{ij}\rangle$; the tiny factor $G/c^5$ makes the waves weak. The
plus and cross polarizations stretch and squeeze a ring of free masses along axes
$45^\circ$ apart, producing a fractional strain $\delta L/L = h/2$. A binary radiates
orbital energy, so the separation shrinks as $\dot a \propto -a^{-3}$ and the
gravitational-wave frequency sweeps upward in a chirp whose rate $\dot f \propto
\mathcal{M}^{5/3} f^{11/3}$ fixes the chirp mass $\mathcal{M} = (m_1 m_2)^{3/5}/
(m_1+m_2)^{1/5}$. Laser interferometers with Fabry-Pérot arms measure the $h \sim
10^{-21}$ strain, and GW150914 recorded two merging black holes of $\sim 36$ and
$29\ M_\odot$ radiating $\sim 3\,M_\odot c^2$, launching gravitational-wave astronomy.

[^co-gw]: Carroll & Ostlie, Ch. 17 §17.6 — gravitational radiation, the quadrupole formula, orbital decay, and interferometric detection.
[^maoz-gw]: Maoz, Ch. 4 — compact-binary inspiral, the chirp, and the detection of gravitational waves.
[^ligo]: LIGO Scientific and Virgo Collaborations, "Observation of Gravitational Waves from a Binary Black Hole Merger," and the GWTC event catalog — https://www.ligo.org.