Interferometric Gravitational-Wave Detection

An interferometric gravitational-wave detector is the most sensitive ruler ever built, and it works by refusing to be a ruler at all. A passing gravitational wave carries a tidal field — a piece of the Riemann curvature tensor oscillating at the wave frequency — that stretches and squeezes proper distances transverse to its propagation. A Michelson interferometer catches this by splitting a laser beam down two perpendicular arms, bouncing each off a freely suspended test mass, and recombining the returning beams. When the wave lengthens one arm and shortens the other, the beams come back out of step and the output fringe shifts. The instrument is a transducer: it turns spacetime curvature into optical phase. Three ideas organize everything. First, the effect is fractional and differential: the observable is the strain $h=\Delta L/L$ , and because the arms respond in antiphase the differential signal is the full strain times the arm length, $\Delta L=hL$ — which is why the arms are kilometres long. Second, the telling is gauge-dependent but the result is not: whether you say the mirrors move or the metric between them oscillates, both compute the one quantity a photodetector registers — the difference in round-trip light-travel time between the arms. Third, the number is absurdly small — $h\sim10^{-21}$ moves a 4 km arm by $4\times10^{-18}\,\mathrm{m}$ , a fraction of a proton's width — so the design is a war against seismic, thermal, and quantum noise. GW150914, on 14 September 2015, proved it can be won.

Key takeaways

A gravitational wave in the transverse-traceless gauge stretches and squeezes proper distances transverse to its motion, so a Michelson interferometer's arms change length differentially and in antiphase.
The observable is the strain $h=\Delta L/L$ ; for a plus wave aligned with the arms the single-arm changes $\pm\tfrac12 h_+L$ combine to $\Delta L=h_+L$ — linear in $L$ , which is why detectors are kilometre-scale.
The interferometer reads this as an optical phase $\Delta\phi=4\pi L h/\lambda$ , amplified by Fabry–Pérot arm cavities that fold the light $N\sim2\mathcal{F}/\pi$ times.
Whether the mirrors "move" or the "arms change length" is a gauge choice; the invariant is the geodesic deviation of the test masses, equal to the differential round-trip light-travel time.
Sensing $h\sim10^{-21}$ means resolving arm changes of $\sim10^{-18}\,\mathrm{m}$ , far below an atomic radius, using power recycling, arm cavities, and quantum-limited photodetection.
LIGO's 4 km arms, LISA's $2.5\times10^6$ km baselines, and pulsar timing arrays are the same differential-light-travel-time principle at different frequencies.
Key idea: a gravitational wave produces a fractional change in proper distance $h\sim\delta L/L$ transverse to its propagation; a Michelson interferometer reads it out as a differential optical phase between two perpendicular arms, the antiphase response doubling the single-arm strain.

The core insight

A gravitational wave produces a fractional change in proper distance $h\sim\delta L/L$ transverse to its propagation; a Michelson interferometer reads this as a differential optical phase between two perpendicular arms, the antiphase response doubling the single-arm strain into $\Delta L=h_+L$ .

Where this lives today

Interferometric detection is now a working branch of astronomy, and the frontier has moved from "can it be done" to "how far and how many." The ground-based network — the two LIGO detectors, Virgo, and KAGRA — has catalogued more than a hundred compact-binary mergers, triangulating sources from the arrival-time and antenna-pattern differences between sites; that is what let telescopes chase GW170817 and see its kilonova, opening multimessenger astronomy. Sensitivity is pushed below the standard quantum limit by injecting squeezed vacuum into the dark port. Looking outward, the space-based LISA mission will fly three drag-free spacecraft $2.5\times10^{6}\,\mathrm{km}$ apart, using time-delay interferometry to reach the milli-hertz band of massive-black-hole binaries; pulsar timing arrays already use millisecond pulsars as a galaxy-scale interferometer at nanohertz frequencies; and third-generation instruments (Einstein Telescope, Cosmic Explorer) aim to hear stellar-mass mergers to the edge of the observable universe.

The physical picture

Imagine a ring of free test masses in the plane perpendicular to a wave's travel. A plus-polarized wave stretches the ring along $x$ while squeezing it along $y$ , then reverses half a period later. A Michelson interferometer places its end mirrors at points of this ring 90° apart, where the stretch and squeeze are strongest and opposite. When one arm lengthens the other shortens, so the responses add rather than cancel. That antiphase geometry is why the L-shape is optimal — it turns a single-arm $\tfrac12 h_+L$ into the full differential $h_+L$ .

Figure

A Michelson interferometer as a gravitational-wave antenna. A plus-polarized wave travelling along z stretches the x-arm and squeezes the y-arm in antiphase, deforming the ring of free test masses from a circle into an ellipse. For strain h = 10⁻²¹ and arm length L = 4 km each arm moves by ±½hL = ±2×10⁻¹⁸ m, giving a differential signal ΔL = hL = 4×10⁻¹⁸ m read out as a fringe shift at the dark port

The number to hold onto is how small this is. A strain $h=10^{-21}$ changes a 4 km arm by $\delta L=hL\approx4\times10^{-18}\,\mathrm{m}$ , roughly one four-hundredth of a proton's diameter. The interferometer succeeds not because the displacement is large but because optical phase can be measured to extraordinary precision when enough photons are counted, and because cavities multiply the phase by a few hundred. What that demands is the noise budget.

Figure

Advanced LIGO design strain sensitivity: the amplitude spectral density √S_h(f) forms a bucket. A seismic and suspension wall rises below ~20 Hz, mirror coating thermal noise dominates the mid-band, and quantum shot noise climbs above ~300 Hz, leaving a broadband floor near 4×10⁻²⁴ per root hertz around 100–300 Hz — the band where GW150914 was found. Curve illustrative, decade gridlines marked

One caution frames the whole subject. Whether one says the mirrors move (beamsplitter's local proper frame) or sit still while the metric oscillates (TT gauge) is a coordinate choice; both compute the same invariant — the differential round-trip travel time, all the photodetector ever sees. The interferometer is not a ruler against a stretched ruler but a comparison of two light clocks.

The core equations

The detector output is a single dimensionless number, the strain — the fractional differential change of the two arm lengths. Everything else either computes this number from the metric or translates it into optical phase.

h(t)=\frac{\Delta L(t)}{L}=\frac{L_x(t)-L_y(t)}{L} \qquad (1)

where: $h$ — dimensionless strain, the detector output; $\Delta L$ — differential change of the two arm lengths; $L_x,L_y$ — instantaneous proper lengths of the arms; $L$ — nominal arm length. Reading it: $h=10^{-21}$ over $L=4\,\mathrm{km}$ is a differential displacement of $4\times10^{-18}\,\mathrm{m}$ . The content is cleanest in the transverse-traceless (TT) gauge, recalled from aq-gr-transverse-traceless-gauge and aq-gr-plus-cross-polarizations: free masses keep fixed coordinates while the metric between them oscillates. For a plus wave along $z$ ,

\mathrm{d}s^2=-c^2\,\mathrm{d}t^2+[1+h_+(t-z/c)]\,\mathrm{d}x^2+[1-h_+(t-z/c)]\,\mathrm{d}y^2+\mathrm{d}z^2 \qquad (2)

where: $h_+$ — plus-polarization amplitude, $|h_+|\ll1$ ; $t-z/c$ — retarded time; $x,y$ — transverse coordinates along the arms. Reading it: the two transverse metric components move in antiphase, $g_{xx}=1+h_+$ against $g_{yy}=1-h_+$ — the entire mechanism. Proper length is the metric line element integrated at fixed coordinate, so the mirror at $x=L$ sits at a proper distance that breathes with the wave; expanding to first order gives the single-arm response.

L_x=\int_0^{L}\sqrt{g_{xx}}\,\mathrm{d}x=\int_0^{L}\sqrt{1+h_+}\,\mathrm{d}x\approx L\left(1+\tfrac12 h_+\right) \qquad (3)

where: $g_{xx}=1+h_+$ — metric component along the x-arm; $\tfrac12 h_+$ — first-order fractional length change. Reading it: each arm changes by only half the strain. The $y$ -arm carries the opposite sign, so subtracting the arms cancels the halves and leaves the full strain — the antiphase doubling that makes the L optimal.

\Delta L=L_x-L_y=h_+L\quad\Rightarrow\quad\frac{\Delta L}{L}=h_+ \qquad (4)

where: $\Delta L$ — observed differential arm-length change, linear in $L$ — the argument for kilometre baselines. What the photodetector senses is optical phase: a photon traverses each arm out and back, and recombination compares the round-trip phases, with arm cavities folding the light to amplify it.

\Delta\phi=\frac{4\pi}{\lambda}\,\Delta L=\frac{4\pi L}{\lambda}\,h\;\xrightarrow{\text{arm cavity}}\;\frac{4\pi L}{\lambda}\,N\,h,\quad N\sim\frac{2\mathcal{F}}{\pi} \qquad (5)

where: $\Delta\phi$ — phase difference at the output; $\lambda$ — laser wavelength, $1064\,\mathrm{nm}$ ; $4\pi/\lambda=2k$ accounts for the round trip; $\mathcal{F}$ — arm-cavity finesse; $N$ — effective number of round trips. Reading it: $\mathcal{F}\approx450$ gives $N\approx286$ , folding a 4 km arm into an effective $\sim1100\,\mathrm{km}$ . Stated gauge-invariantly, the same physics is a change in the photon's round-trip travel time — the quantity surviving any coordinate choice.

\delta t_{\text{round trip}}=\frac{1}{c}\int_0^{L}h_+(t)\,\mathrm{d}x\quad(\text{per arm, leading order}) \qquad (6)

where: $\delta t$ — wave-induced change in round-trip time; $c$ — speed of light, $299\,792\,458\,\mathrm{m\,s^{-1}}$ . "Arm length changed" and "travel time changed" carry identical observable content. A single L-shaped detector sees not the raw strain but a projection weighted by its antenna pattern, set by the source's sky position and polarization.

h_{\mathrm{det}}(t)=F_+(\theta,\phi,\psi)\,h_+(t)+F_\times(\theta,\phi,\psi)\,h_\times(t) \qquad (7)

where: $F_+,F_\times$ — antenna pattern functions for the two polarizations; $\theta,\phi$ — source sky direction relative to the arms; $\psi$ — polarization angle. Reading it: the pattern has blind spots, so a network is needed to localize a source. The high-frequency floor is set by photon shot noise, $S_h^{\mathrm{shot}}\propto\hbar c\lambda/(L^2 P_{\mathrm{bs}})$ — more power and longer arms push it down until radiation-pressure backaction takes over at low frequency (Example 2).

Derivation ADifferential interferometer response from the transverse-traceless metric

The goal is to compute, from first principles, the photodetector signal a plus-polarized gravitational wave produces in an L-shaped interferometer — from the transverse-traceless metric, through the proper arm lengths, to a measurable optical phase and output power. Place the beamsplitter at the origin, the end mirrors at $(L,0,0)$ and $(0,L,0)$ , and let the wave travel along $z$ with slowly varying $h_+(t)$ , so the metric is effectively static during one round trip (the long-wavelength limit). The boxed result is $\Delta\phi=4\pi L\,h_+/\lambda$ .

Step 1. Fix the metric in TT gauge: $g_{xx}=1+h_+$ , $g_{yy}=1-h_+$ , $g_{tt}=-c^2$ , with the mirrors held at fixed coordinates because they are freely falling. Why: in TT gauge the wave's entire effect is loaded into the metric, so the mirrors do not move and only proper distances change — the cleanest bookkeeping.

Step 2. Integrate $\sqrt{g_{ii}}$ along each arm and expand to first order: $L_x\approx L(1+\tfrac12 h_+)$ and $L_y\approx L(1-\tfrac12 h_+)$ . Why: proper length is $\int\sqrt{g_{ii}}\,\mathrm{d}x$ at fixed coordinate, and $\sqrt{1\pm h_+}\approx1\pm\tfrac12 h_+$ since $|h_+|\ll1$ .

Step 3. Subtract to get the differential length: $\Delta L=L_x-L_y=h_+L$ . Why: the two arms carry opposite-sign metric perturbations, so their half-strains do not cancel but add — the antiphase geometry converts $\tfrac12 h_+$ per arm into a full $h_+$ in the difference.

Step 4. Convert to round-trip optical phase: each photon covers each arm twice, so $\Delta\phi=(2\pi/\lambda)\cdot2\,\Delta L=4\pi L\,h_+/\lambda$ . Why: the phase accumulated by light is $k$ times the path length with $k=2\pi/\lambda$ , and recombination at the beamsplitter compares the two round-trip phases.

Step 5. Read it at the photodetector: the output power is $P_{\mathrm{out}}\propto1+\cos(\Delta\phi_0+\Delta\phi)$ , biased near a dark fringe $\Delta\phi_0\to\pi$ so the response is linear in $\Delta\phi$ . Why: a dark-fringe bias maximizes the slope and dumps the carrier power away from the sensing port, so shot noise is set by the small signal beam, not the full laser. *Verified: reproduces MTW §37 and Saulson Ch. 1 — the long-wavelength differential signal $\Delta L=hL$ with optical phase $\Delta\phi=4\pi Lh/\lambda$ ; numerically $h=10^{-21}$ , $L=4\,\mathrm{km}$ , $\lambda=1064\,\mathrm{nm}$ give $\Delta\phi=4.7\times10^{-11}\,\mathrm{rad}$ single-pass.*

Go deeper · Derivation B · 1 minEquivalence of the transverse-traceless and local proper-frame descriptions

Derivation BEquivalence of the transverse-traceless and local proper-frame descriptions

The goal is to prove that the two tellings — "the arms change length while the mirrors stay put" (TT gauge) and "the mirrors are pushed by a tidal force" (the beamsplitter's local proper, or Fermi normal, frame) — deliver exactly the same observable, so the apparent contradiction is a pure gauge artifact. We compute the mirror motion from the geodesic deviation equation and show it reproduces the differential length $h_+L$ of the previous derivation; the invariant both pictures compute is the differential round-trip light-travel time. The boxed relation is $\ddot\xi^{i}=-R^{i}{}_{0j0}\,\xi^{j}=\tfrac12\ddot h^{TT}_{ij}\,\xi^{j}$ .

Step 1. Write the geodesic deviation equation for the mirror's separation $\xi^j$ from the beamsplitter: $\mathrm{d}^2\xi^i/\mathrm{d}t^2=-R^i{}_{0j0}\,\xi^j$ . Why: recalled from aq-gr-geodesic-deviation-gravitational-waves, it governs the relative acceleration of nearby free masses — the coordinate-independent seat of the tidal effect.

Step 2. Insert the linearized Riemann tensor for the wave, $R^i{}_{0j0}=-\tfrac{1}{2c^2}\ddot h^{TT}_{ij}$ , so that $\ddot\xi^i=\tfrac12\ddot h^{TT}_{ij}\xi^j$ . Why: to linear order the only surviving curvature component is the second time derivative of the TT strain — a gauge-invariant quantity, unlike $h_{ij}$ itself.

Step 3. Integrate twice in the slowly varying limit to get the displacements $\delta\xi^x=\tfrac12 h_+L$ and $\delta\xi^y=-\tfrac12 h_+L$ . Why: for a mirror initially at rest at separation $L$ , integrating the tidal acceleration returns the same $\pm\tfrac12 h_+L$ that the proper-length calculation gave — the two pictures agree term by term.

Step 4. Difference the arms: $\Delta L=\delta\xi^x-\delta\xi^y=h_+L$ , identical to the TT result. Why: the differential mirror motion equals the differential proper length, so the photodetector reads the same phase — both describe the single invariant, the differential light-travel time. *Verified: consistent with Wald §4.4 and Carroll §7.4 — the physical effect lives in $R^i{}_{0j0}=-\tfrac{1}{2c^2}\ddot h^{TT}_{ij}$ , independent of gauge, and both routes give $\pm\tfrac12 h_+L$ .*

Go deeper · Derivation C · 1 minThe finite-frequency response and the long-wavelength limit

Derivation CThe finite-frequency response and the long-wavelength limit

The goal is to go beyond the static approximation and ask what happens when the wave period is not enormously longer than the light round-trip time — the regime that sets a real detector's high-frequency roll-off. We integrate the null condition while the metric is genuinely time-dependent, get the round-trip phase as an integral of the wave along the beam path, and show it reduces to $\Delta L=hL$ when $f_{\mathrm{gw}}\ll c/L$ while rolling off above the cavity pole. The controlling parameter is the ratio of light-storage time to wave period.

Step 1. For light along $x$ the null condition $\mathrm{d}s^2=0$ gives $c\,\mathrm{d}t=\sqrt{1+h_+(t)}\,\mathrm{d}x\approx[1+\tfrac12 h_+(t)]\,\mathrm{d}x$ , evaluated at the time the photon is actually there. Why: the photon samples the metric along its worldline, so $h_+$ must be taken at the retarded position and instant, not frozen — this is where finite frequency enters.

Step 2. Integrate over the round trip: $c\,T_x=2L+\int h_+(t(x))\,\mathrm{d}x$ , so the delay is a time average of $h_+$ over the light-storage time $\tau_s\sim2L/c$ (or $\sim2NL/c$ with cavities). Why: the photon integrates the strain over its flight, so oscillations that reverse sign during storage partly cancel — the detector low-pass filters the signal.

Step 3. Form the transfer function $\mathcal{T}(f)=\mathrm{sinc}(\pi f\,\tau_s)$ , so $\Delta\phi(f)=(4\pi L/\lambda)\,\mathcal{T}(f)\,h(f)$ . Why: averaging a sinusoid over a window $\tau_s$ gives the sinc response, whose first null sits near $f\sim1/\tau_s$ ; with arm cavities the roll-off is a single pole at $f_p=c/(4\mathcal{F}L)$ .

Step 4. Take the long-wavelength limit $f\tau_s\ll1$ : $\mathcal{T}\to1$ and $\Delta\phi\to4\pi L h/\lambda$ , recovering $\Delta L=hL$ . Why: when the wave barely changes during a round trip the static calculation is exact, and LIGO's whole band ( $\lesssim$ few kHz) sits well below $c/L\approx75\,\mathrm{kHz}$ . *Verified: the DC limit reproduces derivation 1; the aLIGO single-arm cavity pole $f_p=c/(4\mathcal{F}L)\approx42\,\mathrm{Hz}$ for $\mathcal{F}\approx450$ , $L=4\,\mathrm{km}$ , consistent with Saulson (1994) and the aLIGO design.*

Limiting cases

The interferometer response depends on the wave frequency relative to the light-storage time and on the geometry of source and detector.

Case	Result	Meaning
Long-wavelength (quasi-static)	$f_{\mathrm{gw}}\ll c/L\;\Rightarrow\;\Delta L\approx h(t)\,L$	When the wave period far exceeds the round-trip time, the arm length tracks $h(t)$ adiabatically. Valid for LIGO across its whole band, since $c/L\approx7.5\times10^4\,\mathrm{Hz}$ and signals sit at $\lesssim$ few kHz.
Optimal orientation	$h_{\mathrm{det}}=h_+$ (wave along $z$ , arms along $x,y$ )	A plus wave incident normally with polarization aligned to the arms gives the maximal response; the single-arm strains $\pm\tfrac12 h_+$ add to the full $h_+$ in the difference.
Null (blind) directions	$F_+,F_\times\to0$ for certain $(\theta,\phi,\psi)$	The antenna pattern has zeros; one L-shaped detector cannot localize a source or measure both polarizations, motivating a network.
High-frequency / cavity roll-off	$f_{\mathrm{gw}}\gtrsim f_p=c/(4\mathcal{F}L)\;\Rightarrow\;$ response falls	Above the cavity pole ( $\approx42\,\mathrm{Hz}$ for aLIGO) the signal is averaged over the photon storage time and the response rolls off, softening high-frequency sensitivity.

Common trapThe wave stretches the light too, so the effect cancels

This is the most persistent misconception about interferometric detection, and it is wrong. The reasoning: if the metric stretches the arm, it must stretch the laser wavelength by the same factor, so the number of wavelengths is unchanged and there is nothing to measure. The error is treating the light as a static rigid ruler laid along the arm. It is not — it is a propagating null field. The laser frequency is fixed at the source; the photons already in flight then travel through a metric that is changing while they are in transit, so their round-trip travel time genuinely changes. The gauge-invariant statement is the differential round-trip light-travel time (equivalently the geodesic deviation of the mirrors), and it does not cancel. Concretely: when the wave stretches the $x$ -arm, light in it arrives late, and the returning beam is compared against light from the shorter $y$ -arm — a real phase difference.

Worked examples

1 · GW150914, from strain to phase (fully worked)

Advanced LIGO's 4 km arms use $\lambda=1064\,\mathrm{nm}$ light in cavities of finesse $\mathcal{F}\approx450$ ; GW150914 reached peak strain $h\approx1.0\times10^{-21}$ . Find the single-arm and differential displacements and the optical phase with and without the cavities.

Step 1. Single-arm change: $\delta L=\tfrac12 hL=\mathbf{2\times10^{-18}\,m}$ . Differential change: $\Delta L=hL=\mathbf{4\times10^{-18}\,m}$ — about one four-hundredth of a proton's diameter ( $\sim1.7\times10^{-15}\,\mathrm{m}$ ).

Step 2. Single-pass phase from equation (5): $\Delta\phi=4\pi L h/\lambda=4\pi(4000)(10^{-21})/(1.064\times10^{-6})=\mathbf{4.7\times10^{-11}\,rad}$ — a hundred-billionth of a radian, too small to sense directly.

Step 3. Cavity enhancement: $N=2\mathcal{F}/\pi\approx\mathbf{286}$ , folding each arm into an effective $NL\approx1100\,\mathrm{km}$ . The enhanced phase is $\Delta\phi_{\mathrm{eff}}=N\Delta\phi\approx\mathbf{1.35\times10^{-8}\,rad}$ .

Step 4. Check the long-wavelength condition: GW150914's $35$ – $250\,\mathrm{Hz}$ band sits far below $c/L\approx75\,\mathrm{kHz}$ , so $\Delta L\approx h(t)L$ holds with $\mathcal{T}\approx1$ . The wave moved the mirrors by 4×10⁻¹⁸ m differentially and imprinted ≈1.35×10⁻⁸ rad of phase after cavity folding.

2 · Shot-noise photon budget (order-of-magnitude)

Roughly how many photons must be counted to sense the phase of Example 1, and does a realistic laser power deliver them? Use $\delta\phi_{\mathrm{shot}}=1/\sqrt{N}$ for $N$ detected photons.

Step 1. Resolving $\Delta\phi_{\mathrm{eff}}=1.35\times10^{-8}\,\mathrm{rad}$ at unit signal-to-noise needs $N\gtrsim1/\Delta\phi_{\mathrm{eff}}^2\approx\mathbf{5.5\times10^{15}\ photons}$ — precision phase sensing is a photon-counting problem.

Step 2. Photon energy at $1064\,\mathrm{nm}$ : $E_\gamma=hc/\lambda=1.87\times10^{-19}\,\mathrm{J}\approx1.17\,\mathrm{eV}$ . A beamsplitter power $P_{\mathrm{bs}}=100\,\mathrm{W}$ (after recycling) gives a flux $\dot N=P_{\mathrm{bs}}/E_\gamma\approx\mathbf{5.4\times10^{20}\ s^{-1}}$ .

Step 3. In a window $\tau\approx5\,\mathrm{ms}$ (half a period at $\sim100\,\mathrm{Hz}$ ) the count is $N=\dot N\tau\approx2.7\times10^{18}$ , far above the $5.5\times10^{15}$ needed, so $\mathrm{SNR}\approx\Delta\phi_{\mathrm{eff}}\sqrt{N}\approx\mathbf{22}$ .

Step 4. That order-of-magnitude SNR ≈ 20 lands beside GW150914's reported matched-filter network SNR ≈ 24 — a coincidence of round numbers, but it correctly shows that hundred-watt-class power at kilometre baselines is exactly what is required. Sensing $h\sim10^{-21}$ demands counting $\sim10^{18}$ photons per cycle, which a power-recycled laser supplies, putting the detector near its quantum limit.

Physics of the portrait

The interferometric detector is one idea carried from a memo to a Nobel Prize: that a laser Michelson interferometer, with a correctly accounted noise budget, could reach the strains gravitational waves actually produce.

Origins · 1 minRainer Weiss and how this was discovered

Rainer Weiss

1932– · the interferometric gravitational-wave detector and its noise budget

Rainer Weiss was born in Berlin in 1932 and, after his family fled the Nazis, grew up in New York and spent his career at MIT. In 1972, while teaching general relativity, he wrote an internal MIT report — Electromagnetically Coupled Broadband Gravitational Antenna — that is the founding document of the field. It did two things at once: it proposed the laser-interferometer geometry as a broadband gravitational-wave antenna, and it worked through every fundamental noise source that would limit it — photon shot noise, seismic motion, thermal noise in the suspensions and mirrors, and radiation-pressure backaction. That noise budget, not the interferometer idea alone (which Gertsenshtein and Pustovoit had floated in 1962), is what made the project a buildable engineering program rather than a fantasy. Weiss went on to co-found LIGO with Kip Thorne and Ronald Drever, drove the technology from tabletop to 4 km, and shared the 2017 Nobel Prize with Barish and Thorne after Advanced LIGO recorded GW150914.

History

1916

— Einstein predicts gravitational waves from the linearized field equations

1962

— Gertsenshtein and Pustovoit propose laser-interferometric detection

1972

— Weiss details the interferometer antenna and its full noise budget

2015

— Advanced LIGO records GW150914, the first direct detection

2017

— Weiss, Barish and Thorne receive the Nobel Prize in Physics

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §35, §37 · Wald, General Relativity (1984), §4.4 · Carroll, Spacetime and Geometry (2004), §7.4 · Hartle, Gravity (2003), Ch. 16 · Schutz, A First Course in General Relativity, 2nd ed. (2009), §9.4 · Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (1994).

Provenance. The TT metric and plus/cross modes are recalled from aq-gr-transverse-traceless-gauge and aq-gr-plus-cross-polarizations; the geodesic-deviation core is owned by aq-gr-geodesic-deviation-gravitational-waves; the trace-reversed field of aq-gr-metric-perturbation-trace-reversed underlies the linearization. This article synthesizes them into the detector observable and forward-references aq-gr-quadrupole-formula, aq-gr-isaacson-stress-energy, and aq-gr-multipole-expansion-gravitational-radiation for the source side. A verifier should confirm: (1) $L_x\approx L(1+\tfrac12 h_+)$ from expanding $\sqrt{g_{xx}}$ ; (2) $\Delta L=h_+L$ and $\Delta\phi=4\pi Lh/\lambda$ ; (3) $\ddot\xi^i=-R^i{}_{0j0}\xi^j$ with $R^i{}_{0j0}=-\tfrac{1}{2c^2}\ddot h^{TT}_{ij}$ reproduces $\pm\tfrac12 h_+L$ ; (4) the worked numerics — $\Delta L=4\times10^{-18}\,\mathrm{m}$ , single-pass $\Delta\phi=4.7\times10^{-11}\,\mathrm{rad}$ , $N\approx286$ , $f_p\approx42\,\mathrm{Hz}$ . Anchors: LIGO arm 4 km, $\lambda=1064\,\mathrm{nm}$ , GW150914 peak strain $\sim10^{-21}$ , masses $\approx36$ and $29\,M_\odot$ (Abbott et al., PRL 116, 061102, 2016); LISA arm $2.5\times10^6\,\mathrm{km}$ ; $c=299\,792\,458\,\mathrm{m\,s^{-1}}$ exact. The shot-noise card and sensitivity figure are labelled schematic.

Prerequisites. Transverse-Traceless Gauge · Plus and Cross Polarizations · Geodesic Deviation and Tidal Action of Waves · Metric Perturbation and Trace-Reversed Field. Leads to. The Quadrupole Formula · Isaacson Stress-Energy of Gravitational Waves · Multipole Expansion of Gravitational Radiation.

Ask the tutor: Ask me to derive the antenna pattern functions $F_+,F_\times$ from the detector tensor, or how squeezed light pushes an interferometer below the standard quantum limit.

Does the gravitational wave physically move the mirrors, or does space between them stretch?

Both are correct — two coordinate descriptions of one invariant. In the transverse-traceless gauge the mirrors keep fixed coordinates and the metric oscillates, so "the arms change length." In the beamsplitter's local proper (Fermi normal) frame the mirrors accelerate under a tidal force $\ddot\xi^i=\tfrac12\ddot h^{TT}_{ij}\xi^j$ , so "the mirrors move." Neither is more real; both compute the same differential round-trip light-travel time. The disagreement is a gauge artifact, and the geodesic-deviation equation — built from the gauge-invariant Riemann tensor — guarantees they agree.

If space stretches, why doesn't the laser wavelength stretch too and cancel the effect?

Because the light is not a rigid ruler laid along the arm; it is a wave propagating through a time-varying metric. The laser frequency is fixed at the source, and the photons already in the arm experience the metric changing while they are in flight, so their round-trip travel time genuinely differs from the unperturbed value. When the delayed beam from the stretched arm is compared against the beam from the shorter arm, a real phase difference appears. The "stretched ruler" intuition fails because it treats the light as static; treat it as a propagating null field and the effect survives, as the geodesic-deviation calculation confirms.

Why kilometre-scale arms rather than a compact tabletop device?

The displacement to be sensed is $\delta L=hL$ , linear in the arm length, so a longer arm gives a proportionally larger motion for the same strain while the limiting displacement noise does not grow as fast — so the sensitivity $\delta L/L$ improves with length. Fabry–Pérot cavities then fold the effective length a few hundred times without lengthening the vacuum tube. There is a practical ceiling — beyond a few kilometres the Earth's curvature and the cavity roll-off bite — which is why LIGO settled on 4 km, and reaching lower frequencies means going to space with LISA's million-kilometre arms.

What actually limits the sensitivity — what is in the noise budget?

Three regimes stack into the "bucket." Below $\sim20\,\mathrm{Hz}$ ground motion dominates — seismic and gravity-gradient noise, suppressed by multi-stage suspensions. In the mid-band the floor is thermal noise, the Brownian motion of the mirror coatings and suspension fibres. Above $\sim300\,\mathrm{Hz}$ photon shot noise rises, since a finite photon count has Poisson phase fluctuations $\sim1/\sqrt{N}$ . Shot noise at high frequency and radiation-pressure backaction at low frequency together form the standard quantum limit, which squeezed-light injection now beats. The Advanced LIGO minimum sits near $4\times10^{-24}\,\mathrm{Hz}^{-1/2}$ around $100$ – $300\,\mathrm{Hz}$ .

Why does a single detector need partners to find where a source is?

A single L-shaped interferometer measures only one projection of the wave, $h_{\mathrm{det}}=F_+h_++F_\times h_\times$ , and its antenna pattern has broad lobes and blind spots, so one measurement cannot separate direction from amplitude. A network breaks the degeneracy two ways: the wave arrives at separated detectors at slightly different times (up to $\sim10\,\mathrm{ms}$ across the Earth), and the amplitude ratios across differently oriented detectors constrain the sky position and both polarizations. With three or more detectors the source is triangulated to a patch small enough for telescopes to search — how GW170817's host galaxy and kilonova were found.

How can 4 km arms sense a motion smaller than a proton?

The instrument never measures a length directly; it measures optical phase, resolvable to a tiny fraction of a fringe when enough photons are counted. The differential motion $4\times10^{-18}\,\mathrm{m}$ becomes a phase of $\sim10^{-8}\,\mathrm{rad}$ after cavity folding, and the shot-noise sensitivity is $1/\sqrt{N}$ , so counting $\sim10^{18}$ photons per cycle — which a power-recycled beam supplies — brings the noise below the signal. Operating on a dark fringe dumps the carrier power away from the readout. The machine is a phase meter reading two light clocks, not a caliper measuring a gap.