Thinking in Waves Episode 24 ／ Part X · Breaking the assumptions
The one wave this series had never touched
Gravitational waves ──
radiation one order higher
Here what is waving is spacetime itself. And the measured quantity is a ratio of lengths, \(h=\Delta L/L\) ── dimensionless.
If electromagnetic radiation is "the second derivative of a dipole, squared", gravitational radiation is "the third derivative of a quadrupole, squared".
★ The ladder of orders from Episode 12 goes up one rung.
This series has kept saying "something is a wave". Only for gravitational waves is the thing that waves spacetime itself.
And the measured quantity is dimensionless from the start ── Episode 7's conclusion is here a definition.
01What we measure is dimensionless
| Event | \(h\) | \(\Delta L\) over a 4 km arm [m] | Comparison |
|---|---|---|---|
| GW150914 (2015) | \(10^{-21}\) | \(4\times10^{-18}\) | ★ 1/1000 of a proton |
| GW170817 (neutron stars) | \(10^{-22}\) | \(4\times10^{-19}\) | |
| Continuous waves (pulsars), bound | \(10^{-26}\) | \(4\times10^{-23}\) | |
| Ground motion (1 Hz) | \(10^{-9}\) | \(4\times10^{-6}\) | ★ 12 orders larger without isolation |
The arm stretched by \(4\times10^{-18}\) m ── a thousandth of a proton's diameter. Measured over 4 km. A relative precision of \(10^{-21}\).
02Why the quadrupole ── counting orders
| Multipole | Moment | Conservation law | Why it does not radiate |
|---|---|---|---|
| Monopole (mass) | \(\sum m\) | mass conservation | its time derivative vanishes |
| Dipole (centre of mass) | \(\sum m\,\mathbf{r}\) | momentum conservation | ★ its second derivative vanishes |
| ★ Quadrupole | \(\sum m\,r_ir_j\) | no conservation law | ★ radiation starts here |
The heart of §02 ── one rung up
In electromagnetism the dipole survives (charge is conserved, but the "centre of charge" is not).
In gravity the motion of the centre of mass is conserved, so even the dipole dies.
Set the radiation formulas side by side ──
Electromagnetic (Episode 12): \(P=\dfrac{1}{6\pi\varepsilon_0c^3}\bigl\langle(\ddot{\mathbf{p}})^2\bigr\rangle\) ★ second derivative of a dipole, squared
Gravitational: \(P=\dfrac{G}{5c^5}\bigl\langle(\dddot{Q})^2\bigr\rangle\) ★ third derivative of a quadrupole, squared
★★ One more order in space buys one more order in time.
03Why it is so weak ── the reference power
No mass, no length, no time ── built from \(G\) and \(c\) alone
| Event | Power [W] | Fraction of \(c^5/G\) | Comparison |
|---|---|---|---|
| GW150914 peak | \(3.6\times10^{49}\) | \(9.9\times10^{-4}\) | ★ the brightest event in the universe |
| The Sun | \(3.8\times10^{26}\) | \(1.1\times10^{-26}\) | |
| The whole Milky Way | \(5\times10^{36}\) | \(1.4\times10^{-16}\) | |
| Every star in the observable universe | \(\sim10^{49}\) | \(2.8\times10^{-4}\) | ★ the merger matches this |
| Planck power | \(3.6\times10^{52}\) | 1 | ★ the theoretical ceiling |
The heart of §03 ── the paradox of gravitational waves
For an instant GW150914 shone as brightly as every star in the universe. And that is still only a thousandth of \(c^5/G\).
★ Earth received a flux comparable to moonlight, and the arm moved by a thousandth of a proton.
The energy flowing is large; the coupling to matter is far too weak.
04The chirp ── time to merger sets the frequency
★ Only the "chirp mass" \(M_c\) enters
| Binary | \(m_1\) [M☉] | \(m_2\) [M☉] | \(M_c\) [M☉] | 35 Hz to merger [s] |
|---|---|---|---|---|
| GW150914 | 36 | 29 | 28.10 | 0.190 |
| GW170817 (neutron stars) | 1.46 | 1.27 | 1.185 | 37.19 |
| Equal masses 10+10 | 10 | 10 | 8.706 | 1.339 |
| Equal masses 100+100 | 100 | 100 | 87.06 | 0.029 |
GW150914 takes 0.19 s from 35 Hz to merger ── matching the observed 0.2 s. The neutron-star merger takes 37 s, which is why it could be tracked long enough to catch the gamma-ray burst.
| \(\tau\) to merger [s] | \(f\) [Hz] | \(f\cdot\tau^{3/8}\) |
|---|---|---|
| 10 | 7.92 | 18.776939 |
| 1 | 18.78 | 18.776939 |
| 0.1 | 44.53 | 18.776939 |
| 0.01 | 105.59 | 18.776939 |
§04 ── a different chirp from Episode 13's
Exactly \(-3/8\), not the \(-1\) of Episode 13's hyperbolic chirp ── so it is not log-periodic.
★ The reason is clear ── there is a special moment: the merger.
A scale-invariant world (Episode 20) has no special moment, which is why it gave \(1/t\).
── ★ Whether a special moment exists is what sets the chirp's exponent.
05The speed of gravitational waves ── is there dispersion?
§05 ── translated into Episode 6's language
Gravitational waves and light travel at the same speed to \(10^{-15}\). That swept away many models in which gravity is faster or slower than light.
And it bounds the graviton mass ── as a bound on the cutoff frequency in \(\omega^2=c^2k^2+(m_gc^2/\hbar)^2\).
| Particle | Mass bound [eV] | Range bound [m] | In familiar units |
|---|---|---|---|
| Photon (Episode 17) | \(10^{-18}\) | \(2.0\times10^{11}\) | 1.3 AU |
| Graviton | \(1.3\times10^{-23}\) | \(1.6\times10^{16}\) | 0.50 pc |
★ In length, the graviton's lower bound is \(7.9\times10^{4}\) times longer. Episode 6's "mass makes a cutoff" applies to gravity too ── and neither cutoff has been found.
06Polarisations at 45° ── spin sets the order
| Wave | Spin \(s\) | Angle between polarisations | Rank of the field |
|---|---|---|---|
| Sound (longitudinal) | 0 | ── | no polarisation |
| Electromagnetic | 1 | 90° | vector field; dipole radiation |
| ★ Gravitational | 2 | 45° | ★ rank-2 tensor; quadrupole radiation |
Conclusion of §06
In general, a spin-\(s\) field has polarisations orthogonal at \(90°/s\) and radiates at lowest order as a \(2^s\)-pole.
★ "Spin", "multipole order" and "polarisation angle"
are three ways of saying one thing.
07The detection limit ── joining up with Episode 22
LIGO's design sensitivity at 100 Hz ≈ \(4\times10^{-24}\)/√Hz ── ★ the same order
The heart of §07
LIGO already operates near the standard quantum limit. Which is why it has used squeezed light routinely since 2019 (Episode 22).
★ "The product is fixed but the split is yours" — implemented in a gravitational-wave observatory.
| Approach | What it changes | Content |
|---|---|---|
| Longer arms | raise \(L\) | \(h=\Delta x/L\). LISA uses 2.5 million km |
| Heavier mirrors | raise \(m\) | ★ SQL is \(\sqrt{\hbar\tau/m}\) |
| More laser power | lower shot noise | but radiation-pressure noise rises |
| ★ Squeezing | change the split | ★ 6 dB gives a factor of 2 (Episode 22) |
| Go to low frequency | lengthen \(\tau\) | ★ fighting seismic noise → go to space |
08The ladder of bands ── putting gravitational waves on Episode 3's window
| Method | Low [Hz] | High [Hz] | Period | Targets |
|---|---|---|---|---|
| Ground interferometers (LIGO) | 10 | 5000 | 0.1 s | stellar-mass black-hole mergers |
| Space interferometers (LISA) | \(10^{-4}\) | \(10^{-1}\) | 0.1 day | supermassive black holes |
| Pulsar timing | \(10^{-9}\) | \(10^{-7}\) | 32 yr | ★ the supermassive binary background |
| CMB (B modes) | \(10^{-18}\) | \(10^{-16}\) | 32 Gyr | ★ inflationary origin |
§08 ── Episode 3 applies unchanged
The bottom band has a period of 32 billion years ── longer than the universe. So it cannot be measured directly; one looks for its imprint in a polarisation pattern.
★ Episode 3's "\(\omega=0\) is out of reach" applies verbatim to gravitational waves.
And in 2023, in the pulsar-timing band (periods of years), evidence for a gravitational-wave background was reported ── a chorus of supermassive black-hole binaries. Verdict: several teams report consistent results, but the origin is not settled.
Exercises
- Why is the gravitational-wave frequency twice the orbital frequency?
Show answer
Because it is a quadrupole. The quadrupole moment \(\sum m\,r_ir_j\) returns to its original configuration after half an orbit (the two stars have swapped, and if their masses are equal they are indistinguishable).
So the quadrupole repeats at half the orbital period, doubling the frequency.
★ In general \(2^s\)-pole radiation runs at \(s\) times the orbital frequency ── spin sets the multiplier. - Explain "moonlight-level energy arrived and nothing happened" in Episode 6's language.
Show answer
Because the coupling is weak. A large flux does nothing if matter cannot absorb it.
Light couples to charge with strength \(\alpha=1/137\). Gravity couples to mass with strength \(Gm^2/(\hbar c)\) ── for two protons, \(5.9\times10^{-39}\).
★ Thirty-six orders of magnitude. So the same flux passes almost entirely through.
★ And "not absorbed" is also a virtue ── nothing attenuates it, so the moment of a black-hole merger is visible from anywhere in the universe. - What does it mean that the chirp exponent is \(-3/8\) here and \(-1\) in Episode 13?
Show answer
Whether there is a special moment.
Episodes 13 and 20 gave \(1/t\) in a scale-invariant world where the origin of time carries no meaning (\(t\to\mu t\) leaves the form unchanged).
A binary merger has a special moment \(t_c\), and \(\tau=t_c-t\) is meaningful. That breaks scale invariance and shifts the exponent away from \(-1\).
★ The \(-3/8\) comes from the quadrupole formula's \(f^{11/3}\) ── an instance of Episode 14's "dimensions and dynamics set the exponent".
Summary Radiation one order higher
In gravitational waves, what waves is spacetime itself. The measured quantity \(h=\Delta L/L\) is dimensionless from the start ── Episode 7's conclusion has become a definition. GW150914's \(h=10^{-21}\) means a 4 km arm moved by a thousandth of a proton.
The monopole dies by mass conservation and the dipole by momentum conservation, so radiation starts at the quadrupole. Side by side ── electromagnetic is the second derivative of a dipole squared; gravitational is the third derivative of a quadrupole squared. ★ One order higher.
The reference power is \(c^5/G=3.6\times10^{52}\) W (built from \(G\) and \(c\) alone). GW150914's peak is a thousandth of it ── for an instant, as bright as every star in the universe. The flux at Earth was 3 mW/m², the order of moonlight, and the arm moved a thousandth of a proton. Large energy, far too weak a coupling.
The chirp goes as \(f\propto\tau^{-3/8}\) (verified numerically). It differs from Episode 13's \(1/t\) because there is a special moment: the merger. GW150914 takes 0.19 s from 35 Hz; the neutron stars take 37 s.
GW170817 gives \(|\Delta v|/c<4\times10^{-16}\). The polarisations sit at 45° because spin 2 means a rank-2 tensor ── spin, multipole order and polarisation angle are three names for one thing. And LIGO's sensitivity is already near the standard quantum limit, which is where Episode 22's squeezed light goes to work.
★ If electromagnetic radiation is the second derivative of a dipole, gravitational radiation is the third derivative of a quadrupole.
And what we measure is a ratio of lengths ── "only dimensionless is physics" is here a definition.
What do the strong interaction and the elementary particles look like as waves? ★ Quarks are confined not in a box but on a string, and that string has a ladder of modes ── Regge trajectories.
And the origin of mass (Episode 18) is completed here.
This document is Episode 24 of the series "Thinking in Waves". The computation is in kenshou/juryoku.js. On the numbers: the strains, peak powers and masses are representative published LIGO-Virgo values, not analysed here. The "35 Hz to merger" times from the chirp mass are point-particle Newtonian (0 PN) calculations; real waveforms include higher-order corrections and spin ── 0.19 s is the value within that approximation. The 3 mW/m² flux is an instantaneous value for \(h=10^{-21}\) and \(f=100\) Hz, and the real peak lasts only tens of milliseconds (time-averaged it is far smaller). Moonlight is compared against a representative irradiance of about \(10^{-3}\) W/m². The graviton mass bound is a representative LIGO-Virgo value from a parameterised dispersion relation and is model-dependent. The \(10^{49}\) W for "every star in the observable universe" is a very rough estimate. The 2023 pulsar-timing results come from several collaborations, but whether they originate in supermassive binaries is not settled. The SQL in §07 is the same free-mass formula as Episode 22; LIGO's real noise budget is considerably more complicated. ── To print, use your browser's Print → Save as PDF.