The Four-Kilometer Yardstick

Rainer Weiss, Kip Thorne, Barry Barish, and the LIGO Gravitational Wave Detector

A gravitational wave arriving from a cataclysmic merger of black holes over a billion light-years away stretches and squeezes space itself along perpendicular axes by a dimensionless strain $h$ of roughly one part in a sextillion—a fractional length change of $10^{-21}$. Across the entire diameter of the Earth, a strain of $10^{-21}$ shifts the planet's crust by less than a hundredth of a nanometer, about a tenth the width of a single hydrogen atom. Across a four-kilometer laboratory baseline on the surface of the planet, it shifts the separation between two suspended mirrors by $4 \times 10^{-18}\text{ meters}$—four attometers, or one-thousandth the charge diameter of a single proton. When Albert Einstein first derived the quadrupole gravitational wave formula from general relativity in 1916 and 1918, he concluded that such infinitesimal ripples in the metric of spacetime could never play a measurable role in experimental physics.

In the spring of 1972, asked to teach an undergraduate general relativity seminar at the Massachusetts Institute of Technology at a time when Joseph Weber was claiming controversial gravitational-wave detections using solid aluminum cylinder bars, the experimental physicist Rainer Weiss decided to work out from first principles how a laser-driven Michelson interferometer with free-falling mirrors might detect a passing spacetime wave. Because a gravitational wave is quadrupolar—simultaneously stretching space along an $x$-arm of length $L$ by $+\frac{1}{2} h L$ while compressing the perpendicular $y$-arm by $-\frac{1}{2} h L$ before reversing on the next half-cycle—the differential path change between the two perpendicular interferometer arms is related directly to the gravitational strain $h$:

Staying up late in his MIT office in April 1972, Weiss wrote an unpublished sixty-three-page quarterly progress report for the Research Laboratory of Electronics (RLE Technical Report), systematically calculating every single source of environmental and quantum noise that stood between a terrestrial interferometer and $10^{-21}$ strain sensitivity. Around the peak gravitational-wave frequency band of $100\text{ Hz}$, the ambient ground beneath a quiet rural field shakes continuously from ocean microseisms, wind in trees, and distant highway traffic with an amplitude of $10^{-9}$ to $10^{-7}\text{ meters}$—ten billion times larger than the target gravitational-wave displacement. At the same time, room-temperature thermal energy ($k_B T$) makes the atoms on the mirror surface jiggle at their acoustic resonance frequencies, residual gas molecules crossing the laser beam alter the refractive index, and the discrete arrival of individual laser photons creates quantum shot noise at the photodetector.

Joining forces with Caltech gravitational theorist Kip Thorne in the late 1970s and later reorganized in 1994 under the management of high-energy particle physicist Barry Barish, the Laser Interferometer Gravitational-Wave Observatory (LIGO) constructed two identical L-shaped observatories with four-kilometer concrete-sheltered vacuum arms separated by 3,002 kilometers—one in the pine flats of Livingston Parish, Louisiana, and the other on the arid scrubland of Hanford, Washington—so that any local earthquake or truck vibration would be vetoed unless the exact same waveform struck both observatories within the ten-millisecond light-travel time between them.

To bridge ten orders of magnitude in noise for the Advanced LIGO upgrade completed in 2015, engineers layered four interlocking isolation and amplification systems inside each 1.2-meter-diameter stainless-steel vacuum tube, pumped down to one-trillionth of an atmosphere ($10^{-9}\text{ torr}$):

1. 1. Active Seismic Hydraulic Platforms and Quadruple Fused-Silica Pendulums: Each forty-kilogram mirror (test mass) hung at the bottom of a four-stage cascade of pendulums suspended by $0.4\text{ millimeter}$ fused-silica glass fibers welded directly to the mirror ears; above its $1\text{ Hz}$ resonant frequency, each pendulum stage filtered out ground vibration by $1/f^2$, so four stages in series attenuated $100\text{ Hz}$ seismic noise by more than $10^{-10}$, letting the mirrors hang free in the horizontal plane.
2. 2. Fabry-Pérot Arm Cavities (280 Bounces): Rather than bouncing the infrared $1,064\text{ nanometer}$ Nd:YAG laser beam once down each four-kilometer arm, partially transmissive input mirrors trapped the light inside resonant Fabry-Pérot cavities for roughly 280 round trips, extending the effective optical arm length from 4 kilometers to over 1,100 kilometers.
3. 3. Power Recycling to Defeat Photon Shot Noise: Because photon counting follows Poisson statistics ($\Delta N_{\gamma} \propto \sqrt{N_{\gamma}}$), resolving a $10^{-11}\text{ radian}$ phase shift required an enormous flux of photons; LIGO operated the beam splitter at destructive interference ("the dark port") so that nearly all returning laser light bounced back toward the laser source, where a power-recycling mirror reflected it back into the arms until circulating optical power reached 100 to 200 kilowatts from a 20-watt input laser.

At 09:50:45 UTC on September 14, 2015, just days before Advanced LIGO's first official observing run was scheduled to begin, a rising frequency chirp (designated GW150914) swept from $35\text{ Hz}$ to $250\text{ Hz}$ through the Livingston detector and arrived at Hanford $6.9\text{ milliseconds}$ later, matching general-relativistic numerical simulations of two black holes of thirty-six and twenty-nine solar masses spiraling together 1.3 billion light-years away. During the final two-tenths of a second before the two black holes merged on September 14, 2015, their collision radiated three solar masses of pure gravitational wave energy—briefly outshining all the stars in the observable universe combined and nudging LIGO's mirrors by four attometers.