The Speed of Gravity, and the Mass of the Graviton

A 1.74-second head start over 26 megaparsecs, and a chirp that must arrive in the shape it left. Two of the tightest bounds in physics, and neither of them needed an accelerator.

🎯 Why this matters

The graviton bound is five orders of magnitude tighter than the photon’s, and gravity’s weakness has nothing to do with it. The limit comes from the shape of a waveform rather than the strength of a force, so being barely detectable costs it nothing.

§11.2 established that gravitational waves exist. This section gets two pieces of fundamental physics out of them — and both come from comparisons that were impossible before 2015, because there was nothing to compare.

The first is the speed of gravity, measured against the speed of light over 26 megaparsecs. The second is an upper bound on the graviton mass, from the fact that a massive graviton would make the wave dispersive and smear the chirp.

17 August 2017

🔬 Experiment card — GW170817, and the birth of multi-messenger astronomy

Apparatus

The two Advanced LIGO interferometers and Advanced Virgo, plus — and this is the point — the Fermi Gamma-ray Space Telescope and INTEGRAL in orbit, and then within hours a worldwide campaign of optical, infrared, X-ray, radio and microwave observatories.

What is measured

A gravitational-wave signal several minutes long, quite unlike the black-hole mergers seen before, and a short gamma-ray burst from the same patch of sky. Two numbers matter: the time delay between them, and the distance to the source — the latter obtained twice, once from the gravitational wave alone and once from the host galaxy’s redshift.

The result

GRB 170817A arrived $1.74 \pm 0.05$ s after the gravitational-wave merger. The probability of that coincidence in time and sky position happening by chance is $5.0\times10^{-8}$ . Follow-up found a kilonova in NGC 4993 — a transient about a thousand times brighter than a classical nova, powered by the decay of freshly made heavy nuclei.

What it proved

Three things at once. That neutron star mergers make short gamma-ray bursts. That they are where much of the heavy-element inventory of the Universe comes from. And — the only part this chapter pursues — that gravity and light travel at the same speed to about one part in $10^{15}$ .

It is also the founding event of multi-messenger astronomy : one source, observed through gravitational waves and through the whole electromagnetic spectrum, with each messenger supplying what the other cannot.

Bettini Fig. 11.4, redrawn as the one comparison that matters. The book stacks four panels — two Fermi energy bands, INTEGRAL, and the gravitational-wave time–frequency map; here they are two, on one time axis. The frequency track is computed from the chirp equation below for GW170817’s measured chirp mass of 1.188 solar masses. The whole result of §11.3 is the length of the red bar.

The chirp equation, and what it measures

\frac{d\Omega}{dt} = \frac{96}{\htmlClass{t-k}{5}} \left(\frac{G\,\htmlClass{t-m}{M_c}}{c^3}\right)^{5/3}\htmlClass{t-o}{\Omega^{11/3}}, \qquad\qquad \htmlClass{t-m}{M_c} = \frac{(m_1m_2)^{3/5}}{(m_1+m_2)^{1/5}}

(11.4)

Bettini p. 486, with the coefficient corrected from the printed 96/3 — see the erratum below. This is the Peters result for a circular binary.

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 — the chirp rate is a property of the source, with no distance in it

The chirp rate alone gives you the masses. Measure the frequency and how fast it is rising and you have the chirp mass — with no assumption beyond general relativity being right. Nothing has to be calibrated and nothing about the source has to be known in advance.

Note what is not in the equation: the distance. The rate of chirping is a property of the source alone. That is what makes the next step possible.

🪜 How one event measures its own distance

Step 1 of 5 — Measure Ω and dΩ/dt

Why you may do this: Both are read straight off the waveform. The frequency is where the signal is, and its rate of change is how fast the signal sweeps — and §11.2 showed nine cycles is enough to fit both.

Bettini pp. 486–488. The chain that makes a compact binary a standard siren — a distance measurement with no calibration ladder behind it.

🔢 Worked example — which coefficient in (11.4) is right

Two facts about GW150914 are quoted more often than any others: its chirp mass is 28.6 solar masses, and the part of the signal above 35 Hz lasts about 0.2 s. Either one tests the equation.

Integrate (11.4) both ways

import numpy as np
G, c, Msun, Mpc = 6.67430e-11, 2.99792458e8, 1.98892e30, 3.086e22

def sweep(K, Mc, f1, f2):            # seconds for the GW frequency to go f1 -> f2
    B = (G*Mc/c**3)**(5/3)
    return 3*((np.pi*f1)**(-8/3) - (np.pi*f2)**(-8/3))/(8*K*B)

print("=== which coefficient in (11.4) describes the real events? ===")
print("  LIGO: GW150914 has chirp mass 28.6 Msun and lasts about 0.2 s above 35 Hz")
for n in (5, 3):
    K, dt = 96/n, sweep(96/n, 28.6*Msun, 35., 250.)
    Mc = (3*(np.pi*35.)**(-8/3)*(1-(35/250)**(8/3))/(8*K*0.2))**0.6*c**3/G/Msun
    print(f"   96/{n}:  sweep = {dt*1e3:5.0f} ms   and a 0.2 s sweep implies "
          f"{Mc:5.1f} Msun")
print("   96/5 reproduces both numbers; 96/3 misses both by about 35 %\n")

print("=== and the same equation on GW170817, a neutron-star binary ===")
Mc = 1.188*Msun                       # LIGO's chirp mass for GW170817
for f0 in (30., 20.):
    print(f"   from {f0:.0f} Hz to merger: {sweep(96/5, Mc, f0, 2000.):6.1f} s")
print("   the book calls it 'several minutes long' — and from 20 Hz it is")

prints

=== which coefficient in (11.4) describes the real events? ===
LIGO: GW150914 has chirp mass 28.6 Msun and lasts about 0.2 s above 35 Hz
 96/5:  sweep =   184 ms   and a 0.2 s sweep implies  27.2 Msun
 96/3:  sweep =   110 ms   and a 0.2 s sweep implies  20.0 Msun
 96/5 reproduces both numbers; 96/3 misses both by about 35 %

=== and the same equation on GW170817, a neutron-star binary ===
 from 30 Hz to merger:   55.9 s
 from 20 Hz to merger:  164.7 s
 the book calls it 'several minutes long' — and from 20 Hz it is

The second block is worth as much as the first. The same equation, with a chirp mass 24 times smaller, predicts that GW170817 should be 165 seconds long from 20 Hz — and the book opens §11.3 by calling it “a signal several minutes long, completely different from all the previous ones”. A black-hole merger lasts a fifth of a second and a neutron-star merger lasts minutes, and the reason is one factor in one equation.

Erratum — (11.4)‘s coefficient is 96/5, not 96/3

The book prints $\dfrac{d\Omega}{dt} = \dfrac{96}{3}\left(\dfrac{GM_c}{c^3}\right)^{5/3}\Omega^{11/3}$ .

96/5 is the Peters result for a circular binary, and the snippet above is the test: with 96/5 the equation says GW150914’s 35 → 250 Hz sweep should last 184 ms (observed: about 200) and that a 0.2 s sweep implies a chirp mass of 27.2 solar masses (measured: 28.6). With 96/3 the same two numbers come out 110 ms and 20.0 — both low by about 35 %.

Nothing in the chapter depends on it, because the book never evaluates the equation. But it is the only equation in §11.3 a reader could use, and it is the one that turns a waveform into a mass.

Two distances, and the speed of gravity

The source’s distance was obtained twice, by routes that share nothing.

Erratum — 42.9 Mpc is the distance, not the Hubble constant

p. 488 says the electromagnetic distance “was evaluated from the ratio of its Hubble flow velocity … measured as 3017 ± 166 km s⁻¹, and the Hubble constant, 42.9 ± 3.2 Mpc”.

A Hubble constant is a velocity per distance, in km s⁻¹ Mpc⁻¹; 42.9 Mpc is a distance. And it is not the divisor at all — it is the quotient, the number the division produces, which the same sentence then compares with the gravitational-wave distance of 40⁺⁸₋₁₄ Mpc. Those are the same quantity, which is what “in perfect agreement” means.

The actual divisor is $3017 / 42.9 = 70.3\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ — which is the Hubble constant, and a familiar number.

-3\times10^{-15} \;\le\; \frac{\htmlClass{t-v}{v_{\rm GW} - v_{\rm EM}}}{v_{\rm EM}} \;\le\; \htmlClass{t-u}{+6.5\times10^{-16}}

(11.5)

Bettini p. 488, with the upper limit recomputed from the book's own recipe — it prints +1.7×10⁻¹⁶; see the erratum below. LIGO published +7×10⁻¹⁶.

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 — and why the two ends are not equally hard

The two limits look symmetric and are not.

The upper one is a measurement. It says: even if gravity were faster, it cannot have been faster by more than the observed head start allows.

The lower one is a model. It says: the burst cannot have been emitted more than 10 seconds after the merger, so gravity cannot have been slower than that by more than the same arithmetic allows. The 10 s comes from astrophysics — how long a newly merged remnant plausibly takes to launch a jet — and not from the data.

The conclusion is nonetheless overwhelming, and it settles something Poincaré asserted in 1905 with no evidence at all: gravity does not act instantaneously, and it propagates at the speed of light.

Erratum — (11.5)‘s upper limit should be +6.5×10⁻¹⁶

The book gives the recipe on the same page — attribute the whole 1.74 s delay to the gravitational wave arriving early, over the conservatively small distance $D = 26$ Mpc — and then prints $+1.7\times10^{-16}$ .

Both ends of (11.5), and the Hubble line

import numpy as np
c, Mpc, hP, hbarc = 2.99792458e8, 3.086e22, 4.135667696e-15, 1.9732698e-7

print("=== (11.5): both bounds, from the book's own recipe ===")
D = 26*Mpc
for dt, lab in ((1.74, 'the whole 1.74 s delay is the GW arriving early'),
                (1.74-10, 'the burst was emitted 10 s after the merger')):
    print(f"  {lab:46s} dv/v = {c*dt/D:+.2e}")
print(f"  the book prints -3e-15 <= dv/v <= +1.7e-16\n")

print("=== the Hubble constant, from the same paragraph ===")
print(f"  3017 km/s divided by 'the Hubble constant, 42.9 Mpc' = {3017/42.9:.1f} km/s/Mpc")
print(f"  which IS the Hubble constant — so 42.9 Mpc is the distance, not the divisor\n")

print("=== a massive graviton smears the chirp ===")
D2 = 400*Mpc                                    # a typical binary black hole
for m in (1.73e-23, 1.0e-22):
    d = lambda f: (D2/c)*(m/(hP*f))**2/2        # extra delay vs a massless graviton
    lag = d(35.) - d(250.)
    print(f"  m_g = {m:.2e} eV:  35 Hz lands {lag*1e3:6.3f} ms behind 250 Hz"
          f"  = {lag*35:.3f} of a cycle")
print("\n=== the two bounds, as ranges rather than masses ===")
for m, lab in ((1.73e-23, 'graviton'), (1.0e-18, 'photon  ')):
    print(f"  {lab}: m < {m:.2e} eV  ->  Compton wavelength > {hbarc/m:.2e} m"
          f" = {hbarc/m/9.461e15:8.2g} light years")
print(f"  the graviton bound is {1e-18/1.73e-23:.0f} times tighter in mass")

prints

=== (11.5): both bounds, from the book's own recipe ===
the whole 1.74 s delay is the GW arriving early dv/v = +6.50e-16
the burst was emitted 10 s after the merger    dv/v = -3.09e-15
the book prints -3e-15 <= dv/v <= +1.7e-16

=== the Hubble constant, from the same paragraph ===
3017 km/s divided by 'the Hubble constant, 42.9 Mpc' = 70.3 km/s/Mpc
which IS the Hubble constant — so 42.9 Mpc is the distance, not the divisor

=== a massive graviton smears the chirp ===
m_g = 1.73e-23 eV:  35 Hz lands  0.288 ms behind 250 Hz  = 0.010 of a cycle
m_g = 1.00e-22 eV:  35 Hz lands  9.633 ms behind 250 Hz  = 0.337 of a cycle

=== the two bounds, as ranges rather than masses ===
graviton: m < 1.73e-23 eV  ->  Compton wavelength > 1.14e+16 m =      1.2 light years
photon  : m < 1.00e-18 eV  ->  Compton wavelength > 1.97e+11 m =  2.1e-05 light years
the graviton bound is 57803 times tighter in mass

The lower bound comes out at $-3.09\times10^{-15}$ , exactly the $-3\times10^{-15}$ printed. The upper one comes out at $+6.5\times10^{-16}$ — four times what the book gives, and consistent with the $+7\times10^{-16}$ LIGO published. Note that “1.7” is the leading pair of digits of the 1.74 s on the same page.

The mass of the graviton

If gravity is a quantum gauge theory like the other three, it has a mediator. The classical field is a tensor, so the graviton would have spin 2; the interaction has infinite range, so its mass should be exactly zero. The question is how well that can be checked.

\htmlClass{t-d}{E^2 = p^2c^2 + m_g^2c^4} \qquad\Longrightarrow\qquad \htmlClass{t-v}{v = \frac{pc^2}{E}} = c\sqrt{1 - \frac{m_g^2c^4}{E^2}}

(11.6, 11.7)

Bettini p. 489. A massive graviton gets an ordinary relativistic dispersion relation — and with it, a propagation speed that depends on frequency.

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 comparison of the signal against itself, so no emission time is needed

A massless wave has every frequency travelling at exactly $c$ , so a chirp emitted 400 Mpc away arrives with its cycles in the order they were sent. Give the graviton a mass and the low-frequency components fall behind — the chirp stretches, and stops matching the template.

That is a far more sensitive test than measuring an arrival time, because it is a comparison of the signal against itself. The bound does not depend on knowing when the source went off.

Erratum — pc²/E is the group velocity, not the phase velocity

Eq. (11.7) is introduced as “the GW phase velocity $v = pc^2/E$ ”. For $E^2 = p^2c^2 + m^2c^4$ :

v_{\rm phase} = \frac{\omega}{k} = \frac{E}{p} > c, \qquad v_{\rm group} = \frac{d\omega}{dk} = \frac{dE}{dp} = \frac{pc^2}{E} < c

The printed expression is the group velocity. It is also the right one for the argument that follows: what carries the signal, and what makes low-frequency components arrive late, is the group velocity. The phase velocity of a massive wave exceeds $c$ and transmits nothing — a fact worth stating, because a reader who notices $v > c$ in the other formula and does not know which is which will think something is badly wrong.

m_g = 1.73×10⁻²³ eV, the bound
m_g = 1×10⁻²² eV, already excluded

How a massive graviton would distort the chirp, over a typical binary black hole distance of 400 Mpc. The delay goes as 1/f², so the early low-frequency cycles fall behind the late high-frequency ones. At the limiting mass the 35 Hz components land 0.29 ms behind the 250 Hz ones — one hundredth of a cycle. That is the phase precision the bound rests on, and it is why the graviton limit is so much tighter than the photon's.

Bettini Eqs. (11.8) and (11.9). The same statement, in mass and in range.

Mediator↕	mass limit↕	range exceeds↕	measured by↕
graviton		1.1×10¹⁶ m ≈ 1.2 ly
photon	1×10⁻¹⁸ eV		a variety of laboratory and solar-system tests

A mass bound on a force mediator is equivalent to a bound on the range of the force, through the Compton wavelength ħc/mc².

⚙️ Engineer’s bridge — a massive graviton is a dispersive channel

Everything in §11.4 is one idea you already own: a channel whose group delay depends on frequency smears a wideband pulse.

Give the graviton a mass and space becomes exactly that channel, with $v_{\rm group}(f) = c\sqrt{1-(m c^2/hf)^2}$ . Low frequencies travel slower, so they arrive late, so a chirp that left the source in one shape arrives in another.

Three consequences fall straight out of the engineering picture:

The distortion accumulates with path length, so the best bounds come from the most distant events, not the loudest.
It goes as $1/f^2$ , so the low-frequency end of the sweep carries almost all the information — the same reason a dispersive fibre hurts the long-wavelength edge of a pulse most.
You need a known pulse shape, and general relativity supplies one. This is matched filtering used as a metrology tool rather than a detection one: the template is the reference waveform, and the bound is how far the received signal is allowed to drift from it.

Where it breaks: an optical fibre’s dispersion can be measured and calibrated out by sending a known pulse through a known length. Here neither the length nor the emitted shape is independently known — both come from the same fit — so the bound is a statement about self-consistency of the waveform, not a subtraction.

🔑 If you remember only three things

A distance measured twice by unrelated routes is one you can trust. The waveform gave 40 Mpc and the host galaxy’s redshift gave 43, and neither borrowed the other’s calibration.
The chirp mass is the only mass a waveform reports directly. Two binaries that share it evolve identically, so the individual masses have to be teased out of higher-order corrections.
The coincidence is what licenses the comparison. A burst in the right patch of sky 1.74 s after a merger has a chance probability of $5\times10^{-8}$ , and without that the two arrivals are unrelated events.

Where this goes next

Chapter 11 is a short excursion outside the Standard Model, and it ends with the Model’s largest hole exactly where it started: gravity is not in it, there is no quantum theory of it, and the Planck scale is fifteen orders of magnitude out of reach.

What has changed since 2015 is that there is now data. Gravitational waves propagate at the speed of light to a part in $10^{15}$ , the graviton — if it exists — is lighter than $10^{-23}$ eV, and a single binary can measure its own distance. None of that existed when this book’s first edition was written.

The Epilogue takes the open questions in turn — gravity first, then dark matter, dark energy, supersymmetry, strong CP, the matter–antimatter asymmetry, and the structural problems of the Standard Model itself.

✅ Check yourself — the speed of gravity and the graviton mass

0/6 answered · 0 correct

1.A compact binary is called a 'standard siren'. What does that mean, and what does it assume?
2.The book gives (11.4) with a coefficient of 96/3. How can you tell from published facts about GW150914 that it should be 96/5?
3.The two ends of the speed-of-gravity bound in (11.5) are not equally secure. Why not?
4.How does a bound on the graviton mass come out of waveforms, with no arrival-time measurement at all? (Select all that apply.)
5.Eq. (11.7) gives v = pc²/E and calls it the phase velocity. What is wrong, and why does it matter?
6.The graviton mass bound is 1.73×10⁻²³ eV and the photon's is 10⁻¹⁸ eV. What is the more informative way to compare them?