Gravitational Waves: Hearing Spacetime Ring

Here is a number to sit with before any physics arrives. On 14 September 2015, two black holes finished a four-billion-year dance by colliding 1.3 billion light-years from Earth, and the only thing that reached us was a change in the length of a 4 km steel-and-glass instrument of about $4 \times 10^{-18}$ metres. That is roughly a thousandth the diameter of a single proton. We measured it. We measured a fifth of a second of spacetime ringing, from an event so distant the light leaving it has been travelling since before complex life existed on this planet, and the whole signal moved a mirror by less than the width of the atomic nucleus you are made of.

The claim worth resisting is the one your intuition reaches for first: that a wave this faint must be a wave passing through space, like sound through air or a swell across the ocean. It isn't. There is no medium. The wave is the space, flexing. When the signal arrived, the ground under the detector got momentarily longer in one direction and shorter at right angles to it, not because anything pushed the atoms apart but because the geometry that defines "how far apart" briefly changed. The detector didn't hear a sound. It felt the distance between two points stop being constant. Everything below is an attempt to make that sentence ordinary.

What Einstein predicted and then doubted

In 1916, a year after he finished general relativity, Einstein worked out a consequence he found almost embarrassing. If gravity is the curvature of spacetime, and if a mass that accelerates changes the curvature around it, then that change should not appear everywhere at once. It should propagate outward at the speed of light, a ripple in the geometry itself. He had predicted gravitational waves. Then he spent the next twenty years unsure whether they were real or an artefact of the coordinates, even submitting a 1936 paper titled Do Gravitational Waves Exist? that argued, wrongly, that they didn't.

The reason for his doubt is the reason they're so hard to detect. Start from the metric, the object that tells you the distance between nearby points in spacetime. Far from any strong source, it's almost flat, and you can write it as flat space plus a tiny wrinkle.

$$g_{ab} = \eta_{ab} + h_{ab}, \quad |h_{ab}| \ll 1$$

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Here $\eta_{ab}$ is the flat background of special relativity, and $h_{ab}$ is the gravitational wave: a small, dimensionless distortion riding on top. The number that matters is the size of $h$, called the strain. It's a fractional length change, $\Delta L / L$: a strain of $10^{-21}$ means a 1-metre ruler changes length by $10^{-21}$ metres as the wave passes. Feed equation (1) into Einstein's field equations and, after discarding everything quadratic in the small quantity $h$, what survives is a wave equation. The wrinkle propagates at $c$. That much Einstein had by 1916.

Why gravity radiates from the quadrupole

Now to the part that makes gravitational waves a thousand-billion-billion times subtler than light. An accelerating electric charge radiates electromagnetic waves, and the dominant term is the dipole: separate a positive and a negative charge, shake them, and they broadcast. Gravity has no negative mass, so it has no dipole radiation at all. There is a deeper reason too. The would-be gravitational dipole is just the system's total momentum, and momentum is conserved, so its time-derivative vanishes and there's nothing to radiate.

That kills the leading term. The first source that survives is the next one up: the quadrupole, a measure not of where the mass is but of how lopsided its distribution is, and crucially how that lopsidedness is accelerating. A perfectly spherical pulsing star radiates nothing, however violently it pulses. You need asymmetry that changes in time. Two black holes whipping around each other are about as good a time-varying quadrupole as the universe builds.

$$\frac{dE}{dt} = \frac{G}{5c^5}\sum_{ij}\left(\frac{d^3 I_{ij}}{dt^3}\right)^2$$

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Look at the prefactor: $G/c^5$. That $c^5$ in the denominator is astronomical, about $2.7 \times 10^{42}$ in SI units, and dividing by it is what makes gravitational radiation so feeble. It's why you can swing your arms all day and not lose a measurable joule to gravitational waves. To radiate anything detectable you need masses comparable to the Sun, moving at a sizeable fraction of $c$, with a quadrupole that changes shape on a timescale of milliseconds. Only the most violent objects in the cosmos qualify.

How two test masses feel the wave

So the wave arrives. What does it actually do to matter? Nothing, in one sense: it doesn't push individual particles. What it changes is the distance between particles that are otherwise free. The right tool is the geodesic deviation equation, which tracks the separation vector $\xi$ between two nearby free-falling test masses as spacetime curvature acts on them.

$$\frac{d^2\xi_j}{dt^2} = -c^2\, R_{0j0k}\,\xi^k$$

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The left side is the relative acceleration of the two masses; the right side is curvature, the Riemann tensor, doing the pushing. A passing gravitational wave makes $R_{0j0k}$ oscillate, so $\xi$ oscillates: the masses drift apart, then together, then apart, in step with the wave. Here's the geometry that makes the whole detection scheme work. The wave has two polarisations, and the dominant one stretches space along one axis while squeezing it along the perpendicular axis, then reverses. A ring of free test masses with a wave passing through it pulses into an ellipse, taller then wider then taller, like a balloon someone is gently rolling between two hands.

That is exactly why LIGO has two arms at right angles. When one arm gets longer, the other gets shorter, and a laser bouncing down both can compare their lengths by watching its own light go in and out of step. The differential design is not an engineering nicety; it's matched to the literal shape of the distortion equation (3) describes.

The chirp: an orbit dying in your ear

The shape in the lab has a name, and the name is the whole insight. As the binary loses orbital energy to gravitational waves, equation (2) feeds back on itself: radiating shrinks the orbit, a smaller orbit radiates more, so the frequency and amplitude both run away upward in the final moments. That rising sweep, low to high in a fraction of a second, is acoustically a chirp, the sound a bird makes. Shift GW150914 into audio and it's a swift "whoop" rising from a low note to a high one and stopping. The signal swept from 35 Hz to 250 Hz in about 0.2 seconds.

The strain you measure during that inspiral has a beautifully compact form. It depends on distance, on frequency, and on a single combination of the two masses called the chirp mass.

$$h(t) = \frac{4G}{c^4}\,\frac{m_c^{5/3}}{r}\left(\frac{\pi f}{c}\right)^{2/3}$$

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The chirp mass is $m_c = (m_1 m_2)^{3/5}/(m_1+m_2)^{1/5}$, a particular weighted blend of the two masses. The reason it matters so much is that the rate at which the chirp sweeps upward in frequency depends on $m_c$ alone. Measure how fast the note rises and you have weighed the binary, without ever seeing it, from a wiggle in a detector. Once you know the masses, the amplitude in equation (4) tells you the distance, because a heavier-sounding chirp that arrives faint must be far away. From 0.2 seconds of signal LIGO read off two masses, a distance, and a final mass. That is what the chirp is: an orbit converting itself into a signature you can decode.

Strain at peak

~10⁻²¹

fractional length change ΔL/L the wave imposed on the arms

Arm displacement

4 × 10⁻¹⁸ m

about 1/1000 of a proton's diameter, on 4 km arms

Energy radiated

3.0 M☉c²

three solar masses turned to gravitational waves in 0.2 s

Pause on that last number, because it's the most violent statistic in modern physics. Three solar masses of energy, $3 \times M_\odot c^2$, were radiated as gravitational waves in roughly a fifth of a second. For that brief instant the merger outshone the combined light of every star in the observable universe, by a factor of fifty, in a channel that emits no light at all. None of it reached us as heat or radiation. It reached us as geometry, a passing flex in the shape of space, and even at that absurd luminosity it moved a mirror by less than a proton's width because it had spread across 1.3 billion light-years of sky.

Reading the merger and the ringdown

GW150914 was two black holes of 36 and 29 solar masses, give or take a few each, that merged into a single black hole of 62 solar masses. Add the parents: 36 plus 29 is 65. The remnant is 62. The missing 3 solar masses are exactly the energy radiated, by $E = mc^2$, the most expensive accounting in the sky. The signal records all three acts. The long inspiral chirp, building over the visible part of the wave. The merger, where the two horizons touch and the equations stop being solvable by hand and need supercomputers. Then the third act, the part that makes the whole event sound like an instrument.

There's a subtlety in the duration worth getting right. The signal lasted about 0.2 seconds, but the merger was not a 0.2-second event. The black holes had been spiralling in for billions of years. What lasted 0.2 seconds was the slice of the inspiral that happened to sweep through LIGO's sensitive band, from 35 Hz up to 250 Hz. Below 35 Hz the wave was real but too low and slow for the detector to pick out of the seismic noise. The "fifth of a second" is the window in which the dying orbit was loud enough and fast enough to register, not the lifetime of the system. The black holes had been falling toward this for longer than the Earth has existed.

We did not photograph the collision. We felt its shockwave in the geometry of our own laboratory, 1.3 billion years after the event, as a chirp lasting a fifth of a second.

We had a witness before LIGO

None of this should suggest the 2015 detection came out of nowhere. There was indirect proof on the books for forty years. In 1974, Russell Hulse and Joseph Taylor found a pulsar, a spinning neutron star sweeping a radio beam past Earth like a lighthouse, locked in orbit with another neutron star. A pulsar is a clock of extraordinary precision, so they could track the orbit with exquisite accuracy. And the orbit was shrinking.

If general relativity is right, that binary must be radiating gravitational waves, bleeding orbital energy by exactly the amount equation (2) prescribes, which means the two stars should spiral together and the orbital period should shorten at a calculable rate. Hulse and Taylor measured the orbit decaying by about 76.5 microseconds per year. The quadrupole formula predicted that decay to better than 0.2 percent, with no free parameters. The stars were falling together precisely as fast as the energy carried off by gravitational waves required. It won the 1993 Nobel Prize, and it told everyone the waves were real two decades before anyone caught one directly. LIGO confirmed what the Hulse-Taylor binary had already all but proven: spacetime radiates, and you can hear an orbit die.

A new sense

For all of recorded history, astronomy meant catching light: visible, radio, X-ray, infrared, every window onto the electromagnetic spectrum. GW150914 opened a channel that has nothing to do with light. Gravitational waves pass through dust, through stars, through the opaque early universe that no telescope can see past, because they couple to mass-energy itself and almost nothing absorbs them. The thing that makes them maddening to detect, that they barely interact, is the thing that makes them priceless: they arrive unscattered, carrying the unfiltered motion of the densest objects in existence.

What we caught was a collision of two black holes, an event that emits no light at all, invisible to every telescope ever built, and we caught it cleanly. Since then the detectors have logged neutron stars merging (with a gamma-ray flash and a kilonova that forged gold and platinum), dozens more black hole pairs, and the slow groan of supermassive binaries showing up in the timing of pulsars across the galaxy. We spent a century deaf to it. Now there's a second way to listen to the universe, and the first thing it told us was the precise weight of two black holes that died before the Earth was born. Some food for thought, the next time you read that a number can't matter much because it's small.

Reading further

• Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger (Phys. Rev. Lett. 116, 061102, 2016). The discovery paper for GW150914; the masses, the strain, the chirp and ringdown all come straight from here. It led to the 2017 Nobel Prize.
• Abbott et al., Properties of the binary black hole merger GW150914 (Astrophys. J. Lett. 818, L22, 2016). The companion paper that nails down the precise masses (36 and 29 M☉), the distance, and the ringdown analysis.
• Hulse & Taylor, Discovery of a pulsar in a binary system (Astrophys. J. Lett. 195, L51, 1975). The binary pulsar whose decaying orbit confirmed the quadrupole formula to 0.2 percent, forty years before a direct detection.
• Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy (W. W. Norton, 1994). The Nobel laureate's narrative of the physics and the people behind LIGO; the best long-form account of how the detectors came to be.