Gravitational Waves, and How to Hear One · Elementary Particle Physics

Gravity’s charge is its own mass, and that one fact makes the radiation quadrupolar, feeble beyond any laboratory, and detectable only because an interferometer measures amplitude rather than power.

🎯 Why this matters

The wave is quadrupolar and the instrument is differential, and those are the same fact. A ruler that stretches with what it measures can only report a comparison, so the source’s shape and the detector’s shape were fixed by one conservation law.

Everything in this book so far has been the Standard Model, which describes three of the four interactions. This chapter is about the fourth.

General relativity is to gravity what Maxwell’s equations are to the Standard Model: a macroscopic approximation that works beautifully and is not the underlying theory. We know it is not, because the structure in the Universe — superclusters, galaxies, stars, us — grew from quantum fluctuations in the very early Universe, and a purely classical gravity would have left nothing behind to grow.

Building the quantum theory needs data. The Planck scale where quantum gravity should become obvious is $10^{19}$ GeV, fifteen orders of magnitude above the LHC, so the evidence will not come from an accelerator. It has started coming from the sky instead: light cannot reach us from before the Universe became transparent, but gravitational waves pass through everything, and on 11 February 2016 the LIGO and Virgo collaborations announced that one had been caught.

📐 Physics you need first — the four ideas of general relativity

The book assumes GR and gives you a paragraph. Four ideas carry everything in this chapter, and none of them needs tensor calculus.

1. The field is a geometry. In electromagnetism the field is a potential $A^\mu$ that lives on space-time. In GR the field is space-time: the metric tensor $g_{\mu\nu}$ , the thing that tells you how far apart two nearby events are. §1.2 gave you the flat version, $ds^2 = \eta_{\mu\nu}dx^\mu dx^\nu$ with $\eta$ = diag(−1,1,1,1); GR lets those numbers vary from place to place.

2. It has ten components, and it is dimensionless. $g_{\mu\nu}$ is a symmetric 4×4 array, so ten independent entries and ten field equations. And because it converts coordinate differences into lengths, it carries no units at all — where $A^\mu$ carries energy–momentum per unit charge. That is not cosmetic: it is why gravity has no natural coupling constant of the kind ch05–ch09 kept measuring.

3. The source is energy, not mass. Newton’s source was mass; GR’s is the energy–momentum tensor $T_{\mu\nu}$ . Anything carrying energy or momentum gravitates — including light, which is why gravitational lensing exists.

4. Free fall is not acceleration. A body with no force but gravity on it follows a geodesic , the straightest available path through curved space-time. It feels nothing. This is the fact that makes the whole experiment possible: an interferometer mirror hanging on a pendulum, at frequencies well above the pendulum’s resonance, is free along the beam direction, and so is a legitimate test particle.

The wave, and why it is quadrupolar

ds^2 = \htmlClass{t-g}{g_{\mu\nu}}\,dx^\mu dx^\nu, \qquad\qquad \htmlClass{t-g}{g_{\mu\nu}} = \htmlClass{t-0}{g^{(0)}_{\mu\nu}} + \htmlClass{t-h}{h_{\mu\nu}}

(11.1)

Bettini p. 482. The interval and the weak-field split — with ds² rather than the printed ds; see the erratum below.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — a ripple in the ruler, not something moving through space

Splitting the metric into a flat background plus a small correction is the weak-field approximation , and what it describes is a ripple in the ruler. Not something moving through space — a change in what “distance” means, propagating at the speed of light.

That sounds like it should be unmeasurable, and the reason it is not is that the change is anisotropic: at any instant the wave stretches one transverse direction and compresses the perpendicular one. Compare two perpendicular distances and the effect survives. Compare only one against a ruler that stretches with it, and it does not.

\htmlClass{t-q}{Q_{ij}} = \int \mu(\mathbf{r})\left(3x_ix_j - \delta_{ij}r^2\right)dV \qquad\qquad h_{ij} = \frac{2G}{3c^4}\,\frac{1}{\htmlClass{t-d}{D}}\,\htmlClass{t-dd}{\frac{d^2Q_{ij}}{dt^2}}

(11.2, 11.3)

Bettini pp. 482–483. The mass quadrupole, and the wave it radiates.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — why the quadrupole, and why gravity is so quiet

The quadrupole formula above says the leading radiating multipole is the second one, and the reason is a conservation law. Expand any radiating source in multipoles. Electromagnetism starts at the dipole, because the monopole — the total charge — is conserved and cannot change. Gravity has to start one order later, because two things are conserved: the monopole (total mass) and, uniquely, the dipole (total momentum).

There is no gravitational analogue of an antenna, in which you slosh charge from one end to the other. To slosh mass from one end to the other you would need to push against something, and that something is part of the system.

The cost is enormous. Quadrupole radiation is suppressed relative to dipole by roughly $(v/c)^2$ , and on top of that $G/c^4 \approx 10^{-45}$ in SI. That is why the only detectable sources are two black holes at a third of the speed of light, and why nothing you can build in a laboratory will ever radiate a measurable gravitational wave.

⚙️ Engineer’s bridge — amplitude detection, and why it changed everything

Two distinctions from your own instruments do most of the work in this chapter.

Multipole expansion = a series whose leading terms are killed by conservation laws. You have met this as the multipole expansion of an antenna pattern. The new part is that gravity’s charge is its mass, so momentum conservation removes the dipole term that electromagnetism relies on.

Amplitude detector, not square-law. A photodiode is a square-law device: it absorbs energy, so its output goes as the square of the field, and the signal from a distant source falls as $1/D^2$ . An interferometer is not — it measures a phase, which is linear in the field, so its signal falls as $1/D$ .

That is the difference between a power meter and a lock-in amplifier, and here it is worth a factor of a thousand: since the sensitive volume goes as $D^3$ ,

\text{sensitivity} \times 10 \;\Longrightarrow\; \text{observable volume} \times 1000

Thirty years of incremental noise reduction were rational because each factor of ten in strain sensitivity bought a factor of a thousand in the rate of events. LIGO’s first decade found nothing; the upgrade that found GW150914 was a factor of ten.

Where it breaks: an amplitude detector needs the wave’s period to be long compared with the measurement, so the test masses can follow the metric adiabatically. At 100 Hz that is comfortable. It is also why these detectors have a band: below about 10 Hz seismic noise wins, above a few kHz the photon shot noise does.

The strain, and the shape of the detector

Strain is the observable: $\Delta L/L = h$ , about $10^{-21}$ . Over a 1 m ruler that is $10^{-21}$ m. The book’s Fig. 11.1 shows four free masses on a circle; the widget below does the same thing with twelve, and adds the part that matters for building a detector.

What a gravitational wave does to space — and why the detector is an L

arm along ψ
arm at ψ + 90°
the difference — what the fringe measures

phase of the wave = 0.00 periods · the ring is a circle again twice per period, at ¼ and ¾detector orientation ψ = 0° — full response

antenna response

1.000

strain h

1e-21

arm length

4 km

peak differential ΔL

4.0e-18 m

Drag the orientation slider. At ψ = 0 the detector responds fully to the + polarisation and is exactly blind to ×; at 45° the two swap. One interferometer measures one polarisation, which is why LIGO built two and why Virgo joining mattered — and it is why the response is written cos 2ψ and sin 2ψ rather than cos ψ and sin ψ: a 90° rotation of an L maps it onto itself.

The two arms always move in opposite directions, so the difference is twice either one — which is the whole reason a Michelson is the matched detector for a quadrupolar wave. Watch the blue and red curves: they are mirror images, and the dark curve is their difference.

Bettini Fig. 11.1, extended. The dashed circle is where the masses would sit with no wave. The orientation slider is the point: an interferometer at ψ = 0 responds fully to the + polarization and is exactly blind to ×, and at 45° the two swap. That is why LIGO built two observatories and why Virgo joining in 2017 mattered — and why the response goes as cos 2ψ rather than cos ψ.

💡 What this really says — a Michelson is not a convenient choice — it is the matched detector

Three things the still figure cannot show.

The two arms move oppositely. One stretches exactly when the other compresses, so their difference is twice either one. A Michelson interferometer is not a convenient choice for measuring a gravitational wave — it is the matched detector for a quadrupolar field.

The ring returns to a circle twice per period, at ¼ and ¾. It is not “squashed and then less squashed”; it passes through undeformed.

There are two polarizations, 45° apart rather than 90°. For light the two polarizations are perpendicular; here they are at 45°, because the field is a tensor of rank 2 rather than a vector. That is the same factor of two that makes the graviton spin 2, and it is visible in the widget as the $2\psi$ in the response.

🔢 Worked example — what 10⁻²¹ costs you

The strain, in engineering units

import numpy as np
G, c, Msun = 6.67430e-11, 2.99792458e8, 1.98892e30

print("=== what a strain of 1e-21 means ===")
h = 1.0e-21
for L, lab in ((1.0, 'a 1 m ruler'), (4.0e3, "LIGO's 4 km arm"), (2.0e6, 'the 2000 km folded path')):
    print(f"  {lab:24s} L = {L:8.3g} m -> dL = {h*L:8.2e} m")
rp = 8.4e-16
print(f"\n  a proton radius is {rp:.1e} m, so LIGO's 4e-18 m is 1 part in {rp/(h*4e3):.0f} of one")
print(f"  the Fabry-Perot fold is {2.0e6/4.0e3:.0f} bounces of a 4 km arm")
print(f"  amplitude detection: signal ~ 1/D, so x10 in sensitivity is x{10**3} in volume")

print("\n=== how long does an inspiral spend in the band? ===")
Mc = 30*Msun                               # near enough to GW150914's 28.6
A = (5/256)*(G*Mc/c**3)**(-5/3)*np.pi**(-8/3)
tau = lambda f: A*f**(-8/3)                # time from frequency f to merger
for f in (35., 100., 250.):
    print(f"  from {f:5.1f} Hz to merger: {tau(f)*1e3:7.1f} ms")
span = tau(35.) - tau(250.)
t = np.linspace(0, span, 200001)           # count the cycles in the sweep
f_of_t = (A/(tau(35.) - t))**0.375
print(f"\n  the 35 -> 250 Hz sweep the book quotes lasts {span*1e3:.0f} ms")
print(f"  and contains {np.trapezoid(f_of_t, t):.0f} wave cycles")

prints

=== what a strain of 1e-21 means ===
a 1 m ruler              L =        1 m -> dL = 1.00e-21 m
LIGO's 4 km arm          L =    4e+03 m -> dL = 4.00e-18 m
the 2000 km folded path  L =    2e+06 m -> dL = 2.00e-15 m

a proton radius is 8.4e-16 m, so LIGO's 4e-18 m is 1 part in 210 of one
the Fabry-Perot fold is 500 bounces of a 4 km arm
amplitude detection: signal ~ 1/D, so x10 in sensitivity is x1000 in volume

=== how long does an inspiral spend in the band? ===
from  35.0 Hz to merger:   170.4 ms
from 100.0 Hz to merger:    10.4 ms
from 250.0 Hz to merger:     0.9 ms

the 35 -> 250 Hz sweep the book quotes lasts 169 ms
and contains 9 wave cycles

Four attometres over four kilometres. That is one part in 210 of a proton radius, and it is measured by a machine sitting on a planet whose crust moves by microns. Folding the path 500 times inside the Fabry–Pérot cavities buys back three orders of magnitude, and the rest is noise engineering.

The second block is the other side of the problem and the reason it is soluble: the whole event lasts 169 milliseconds and contains about nine wave cycles. Nine cycles of a known waveform is enough to matched-filter out of noise a hundred times larger — and the shape is known, because general relativity predicts it.

Erratum — a strain cannot have units

p. 484 says “the oscillation amplitude due to a GW is typically $\Delta L/L \approx 10^{-18}$ m”. The fraction bar should not be there.

Two independent refutations, both from the page before. $\Delta L/L$ is a ratio of two lengths and therefore dimensionless, so it cannot carry a metre. And §11.1 has already given it: $h = \Delta L/L \approx 10^{-21}$ .

The quantity that is about $10^{-18}$ m is $\Delta L$ itself — the block above computes $4\times10^{-18}$ m for a 4 km arm at $h = 10^{-21}$ , which is where the number comes from. Strain and displacement are the two things it is most important not to confuse in this chapter, because the first is a property of the wave and the second is a property of your detector.

Erratum — three small slips in §11.1 that will make a careful reader stop

p. 482, the interval. The book writes $ds = g_{\mu\nu}dx^\mu dx^\nu$ . The right-hand side is quadratic in the coordinate differences, so the left must be $ds^2$ — as it is in the flat-space version the reader already knows.

p. 482, the flat metric. It is given as $g^{(0)}_{ij} = \delta_{ik}$ : the left-hand side carries the indices $i,j$ and the right carries $i,k$ . It should be $\delta_{ij}$ .

p. 483, Fig. 11.1’s caption. Panel (c) is captioned “maximum displacement in perpendicular direction after ¼ period”, but the text on the same page says “and vice versa, after half a period, as in Fig. 11.1(c)”. Half is right: with $h(t) \propto \cos\omega t$ the deformation reverses at $t = T/2$ , and at $T/4$ the ring is an undeformed circle. Drag the phase slider above to $0.25$ and then to $0.5$ and the point makes itself.

The instrument

🔬 Experiment card — LIGO, Virgo and GW150914

Apparatus

Three laser Michelson interferometers. LIGO has two with 4 km arms, at Hanford (Washington) and Livingston (Louisiana), 3002 km apart. Virgo has one with 3 km arms at Cascina near Pisa. In each arm a Fabry–Pérot cavity bounces the light hundreds of times, folding 4 km into an effective 2000 km — roughly half a typical gravitational wavelength. The mirrors hang from multi-stage pendulums that suppress ground motion by a factor of $10^{12}$ , and above the ~1 Hz pendulum resonance they are effectively in free fall along the beam.

What is measured

The differential arm length. A passing wave stretches one arm and shortens the other, so the two beams return to the beam splitter with a phase difference and the intensity at the photodetector swings between its maximum and its minimum. Two widely separated observatories are required — for coincidence against local disturbances, and because the arrival-time difference locates the source on the sky.

The result

“On September 14, 2015 at 09:50:45 UTC the two detectors … simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of $1.0\times10^{-21}$ . It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole.”

What it proved

That gravitational waves exist and can be detected — a century after Einstein predicted them and Poincaré argued they had to exist. It was also the first direct observation of a binary black hole, and of a black hole merger. LIGO ran from 2002 to 2010 and saw nothing; that decade was spent learning how to gain the factor of ten in sensitivity that made 2015 possible.

🛠️ Fig. 11.2 — the Michelson interferometer, folded

Click a numbered marker for what that piece does.

Bettini Fig. 11.2. The beam splitter divides the laser into two perpendicular arms; each arm is a Fabry–Pérot cavity between an input and an end mirror. Click a callout.

📏 Where the numbers in the instrument come from

Why 4 km and not 40. The response grows with arm length only until the light spends about half a wave period in the arm; beyond that the wave reverses while the light is still travelling and the gain stops. At 100 Hz, half a period of light travel is 1500 km of optical path — which is why the folded length is 2000 km and the physical length is 4 km. The cavity buys the length that civil engineering cannot.

Why 10¹² of seismic isolation. Ground motion at 10 Hz is of order $10^{-9}$ m; the signal is $4\times10^{-18}$ m. That is nine orders of magnitude, and the pendulum chain provides twelve.

Why above 1 Hz. A pendulum follows its support below resonance and ignores it above. The suspension resonance is put at about 1 Hz precisely so that the whole detection band, 10 Hz to a few kHz, sits in the free-fall regime.

📏 Notation — this chapter reuses four letters that already mean something else

Gravitational-wave notation was fixed long before particle physics settled its own, and the collisions are unlucky. None is ambiguous in context; all four will trip you if you are skimming.

here	elsewhere on this site
$h$ , $h_{\mu\nu}$ — the metric perturbation, and the strain	the Planck constant in every other chapter, and the reduced Hubble constant in §10.6. All three appear within four pages of each other
$\Omega$ — the orbital angular frequency of the binary	the cosmological density parameter $\Omega_m$ , $\Omega_\Lambda$ in §10.6, and the $\Omega^-$ baryon of §4.2
$Q_{ij}$ — the mass quadrupole moment	an electric charge in ch05, the energy release $Q_{\beta\beta}$ in §10.7, and the momentum transfer $Q^2$ in ch06
$g_{\mu\nu}$ — the metric tensor, dimensionless	the SU(2) coupling $g$ of §9.3, and the gluon

The one worth watching is $\Omega$ : §10.6 is four pages earlier in the book and uses it for a density, and this chapter uses it for a frequency in the equation that does all the work.

What is making the waves

Every source detected so far is a compact binary inspiral , and it passes through three phases — inspiral, merger and ringdown .

🪜 A compact binary, from inspiral to ringdown

Step 1 of 6 — Two compact objects in orbit

Why you may do this: The quadrupole moment of a binary changes as it orbits, so it radiates. Nothing else in the sky both has a large enough quadrupole and changes it fast enough — which is why the sources are all black holes and neutron stars rather than ordinary stars.

The orbital frequency Ω is set by the separation and the two masses, so measuring Ω over time measures the system.

Bettini p. 485. Every source detected so far is a compact binary — two black holes, two neutron stars, or one of each.

h(t) for a 30 M☉ chirp mass

Bettini Fig. 11.3, computed rather than traced: the amplitude goes as f^{2/3} and the frequency follows the chirp equation (11.4), which §11.3 derives the chirp mass from. Nine cycles, 169 ms, and the frequency doubling in the last two — that is what LIGO recorded on 14 September 2015, and it is why the event is called a chirp.

💡 What this really says — the waveform is its own clock, which makes one binary a standard siren

Look at where the cycles are. The signal spends most of its 169 ms near 35 Hz and crosses 100 to 250 Hz in the last ten milliseconds. The waveform is its own clock: the rate at which the frequency rises encodes the masses, and nothing else in the signal is needed to get them.

That is why §11.3 can turn one event into a distance measurement with no calibration ladder at all. GR supplies the intrinsic luminosity from the frequency and its rate of change; comparing with the observed amplitude gives $D$ . A single binary is a standard siren.

🔑 If you remember only three things

Momentum conservation is what removes the dipole. Electromagnetism radiates from a sloshing charge; gravity cannot, because moving mass one way means pushing on something inside the system.
Ten times the sensitivity is a thousand times the volume. An interferometer reads a phase rather than a power, so its reach falls as 1/D — which is what made three decades of noise reduction rational.
A century passed between the prediction and the detection. Einstein published in 1916 and LIGO recorded in 2015, and nothing in between was a failure of theory — only of sensitivity.

Where this goes next

You now have the wave, the reason it is quadrupolar, the strain it produces and the machine that measures it. What you do not yet have is any physics out of it beyond “they exist”.

§11.3 extracts two fundamental results from a single event. On 17 August 2017 a neutron star merger, GW170817, arrived 1.74 s before a gamma ray burst from the same place, after travelling 26 Mpc — which bounds the difference between the speed of gravity and the speed of light at the $10^{-15}$ level, confirming an assumption Poincaré made in 1905. And because a massive graviton would make gravitational waves dispersive — smearing the chirp you have just seen — the measured waveforms bound the graviton mass more tightly than any experiment bounds the photon’s.

✅ Check yourself — gravitational waves and their detection

0/6 answered · 0 correct

1.Electromagnetic radiation starts at the dipole; gravitational radiation starts at the quadrupole. Why the difference?
2.The book notes that improving the strain sensitivity by a factor of 10 makes the observable volume 1000 times larger. Where does that come from?
3.In the widget, set the polarisation to + and drag the detector orientation from 0° to 45°. What happens, and what does it imply?
4.Which of these follow from the numbers in the worked example? (Select all that apply.)
5.Why is a hanging mirror a legitimate 'free' test mass, when it is obviously attached to the Earth?
6.The signal from a compact binary is a chirp — the frequency sweeps upwards and the amplitude grows. Why does the frequency accelerate rather than settle?