LIGO and the Global Detector Network

A gravitational wave carries no ruler with it. It does not push test masses through a fixed space; it modulates the proper distance between masses in free fall. In the transverse-traceless gauge a passing wave stretches one transverse direction while squeezing the perpendicular one, and the fractional change of separation between two freely-falling masses along $\hat n$ is $\delta L/L = \tfrac12 h_{ij}^{TT}\hat n^i\hat n^j$ . Lay two 4-km arms at right angles, and the difference of these strains is exactly the plus polarization $h_+$ — an L-shaped Michelson interferometer is therefore a differential geodesic-deviation meter, reading the wave as a shift of interference fringes at its dark port. The engineering is a study in the almost-impossible. A strain $h\sim10^{-21}$ moves a 4-km arm by $\Delta L = hL \sim 4\times10^{-18}\,\mathrm{m}$ — a thousandth of a proton's diameter. LIGO reaches it with Fabry–Perot arm cavities that fold a stabilized $1064\,\mathrm{nm}$ laser hundreds of times, $40\,\mathrm{kg}$ fused-silica test masses hung from multi-stage pendula so they float freely above a few hertz, ultra-high vacuum, and squeezed light. But one L cannot say where a source sits: its antenna pattern is a smooth quadrupole over the sky. The cure is a planet-spanning network — the two LIGO sites, Virgo near Pisa, KAGRA in the Kamioka mine, GEO600 — whose arrival-time differences triangulate the source, whose relative amplitudes fix its polarization, and whose coincidence requirement vetoes glitches. GW150914 turned this apparatus into the first direct detection of gravitational waves and a laboratory for strong-field general relativity; GW170817 added a neutron-star merger seen across the electromagnetic spectrum, opening multi-messenger astronomy.

Key takeaways

An L-shaped Michelson interferometer measures the difference of the tidal strains along its two perpendicular arms; for plus polarization this is $\Delta L = L_x - L_y = hL$ , read out as an optical phase at the dark port.
A strain $h\sim10^{-21}$ moves LIGO's 4-km arms by only $\Delta L\sim4\times10^{-18}\,\mathrm{m}$ , about $10^{-3}$ of a proton diameter — measurable because Fabry–Perot cavities fold the light $\sim300$ times and $\sim10^{20}$ photons beat down the phase noise.
The differential readout doubles the signal and cancels common-mode laser and length noise; the price is a broad quadrupolar antenna pattern.
No single detector can localize a source; a network triangulates the sky position from arrival-time differences ( $\Delta t = \hat N\cdot(\vec r_a-\vec r_b)/c$ , up to $\sim10\,\mathrm{ms}$ for Hanford–Livingston).
On 14 September 2015 both LIGO detectors recorded GW150914, a binary black hole of $\sim36$ and $\sim29\,M_\odot$ at $\sim410\,\mathrm{Mpc}$ — the first direct detection.
GW170817 (17 August 2017), a binary-neutron-star merger with a coincident short gamma-ray burst, launched multi-messenger astronomy and bounded the wave speed to $|v_{gw}/c - 1|\lesssim10^{-15}$ .
Key idea: a kilometre-scale Michelson interferometer is a differential geodesic-deviation meter — a wave of strain $h$ changes the difference of the two arm lengths by $\Delta(L_x-L_y)=hL$ , read out as an optical phase, and a network of such antennas localizes sources by triangulating arrival times.

The core insight

A kilometre-scale Michelson interferometer is a differential geodesic-deviation meter: a gravitational wave of strain $h$ changes the difference of the two arm lengths by $\Delta(L_x - L_y) = h L$ , read out as an optical phase, and a network of such antennas localizes sources by triangulating arrival times.

Where this lives today

The network is now an observatory. Advanced LIGO, Advanced Virgo, and KAGRA run coordinated campaigns and issue public alerts within seconds, so telescopes can chase counterparts while the sky patch is fresh — the workflow that caught GW170817's kilonova in NGC 4993. Catalogues now list roughly a hundred compact-binary mergers, mapping the mass distribution of stellar black holes (including surprising $\sim80\,M_\odot$ objects) and giving standard-siren measurements of the Hubble constant independent of the distance ladder. Sensitivity keeps climbing as squeezed-vacuum injection beats the standard quantum limit across the band. Ahead lie the space-based LISA mission in the millihertz band, the nanohertz stochastic background reported by pulsar-timing arrays, and third-generation instruments — the Einstein Telescope and the 40-km Cosmic Explorer. Open problems abound: tests of the no-hair theorem from ringdown spectra, the neutron-star equation of state from tidal deformability, and whether the polarization content ever departs from the two tensor modes general relativity allows.

The physical picture

Set a ring of freely-falling test masses in the plane transverse to a plus-polarized wave. As the wave passes, the ring breathes between a horizontal and a vertical ellipse: one diameter lengthens while the perpendicular one shortens, in antiphase. Put a Michelson interferometer's two arms along those diameters and it measures exactly the difference between the long and short axes. Because the arms move oppositely, the differential signal is twice what either arm sees alone — and anything that lengthens both arms together (laser drift, common seismic motion) cancels. The first figure is that machine: laser, beamsplitter, two orthogonal arms with suspended mirrors, and the dark port.

Figure

LIGO as a differential strain meter: a plus-polarized wave stretches the x-arm by +½hL and squeezes the y-arm by −½hL, so the measured differential is ΔL = L_x − L_y = hL. For h = 10⁻²¹ and L = 4 km this is only ΔL = 4×10⁻¹⁸ m, about 10⁻³ of a proton diameter, read as an optical phase at the dark port

Numbers make the difficulty vivid and the network necessary. For $h\sim10^{-21}$ and $L=4\,\mathrm{km}$ the arm motion is $\Delta L\sim4\times10^{-18}\,\mathrm{m}$ ; the cavities bounce the light $\sim300$ times, and the readout beam carries enough photons that its phase noise falls below this signal. Yet a single L is nearly direction-blind: its response is a broad quadrupole with two nodes, so one strain record is consistent with a whole band of sky positions. The escape is geometry. Detectors thousands of kilometres apart see the same wavefront at different instants, and those arrival-time differences — up to $\sim10\,\mathrm{ms}$ for Hanford–Livingston, $\sim42\,\mathrm{ms}$ across the full Earth — pin the source to rings whose intersection is a localized patch. The second figure is that act: two sites, one wavefront, and the timing ring it carves.

Figure

Network triangulation: a wavefront from direction N̂ reaches detectors a and b at times differing by Δt = N̂·(r_a − r_b)/c. The Hanford–Livingston baseline d ≈ 3000 km caps this at Δt ≤ d/c ≈ 10 ms; a measured Δt fixes the angle to the baseline through cos α = c Δt / d, placing the source on a ring, and a third detector cuts the ring to a patch

The core equations

The starting point is kinematic and gauge-invariant: how much does one arm's proper length change when the wave passes? With the mirrors at fixed TT coordinates the metric along the arm is $g_{ij}=\delta_{ij}+h_{ij}^{TT}$ , and integrating $\sqrt{g_{ij}}$ gives the fractional stretch below.

\frac{\delta L}{L} = \frac{1}{2}\, h_{ij}^{TT}\, \hat n^i \hat n^j \qquad (1)

where: $L$ — proper length of the arm; $\delta L$ — its change induced by the wave; $h_{ij}^{TT}$ — transverse-traceless perturbation at the detector; $\hat n^i$ — unit vector along the arm. Reading: one arm tracks the tidal strain projected onto it, at half the raw $h$ . Two perpendicular arms sample the strain with opposite sign for the plus polarization, since $h_{xx}^{TT}=-h_{yy}^{TT}=h_+$ . Subtracting them doubles the single-arm effect and yields the measured quantity, the differential arm length.

\Delta L = L_x - L_y = L\,h_+ \qquad (\text{arms along } \hat x,\hat y) \qquad (2)

where: $L_x, L_y$ — proper lengths of the two arms; $h_+$ — plus-polarization amplitude; $\Delta L$ — differential arm length. Reading: the L-shape converts the spin-2 tidal field into a length difference, common-mode noise subtracted. That length difference is sensed as an optical phase. Each arm is a Fabry–Perot cavity storing the light for $\sim\mathcal F$ round trips, multiplying the phase a given $\Delta L$ imprints, and the arms interfere at the dark port so that $\Delta L$ shifts the fringe.

\Delta\phi = \frac{4\pi \mathcal{F}}{\lambda_L}\,\Delta L \qquad (3)

where: $\Delta\phi$ — differential phase at the dark port; $\lambda_L$ — laser wavelength, $1064\,\mathrm{nm}$ ; $\mathcal{F}$ — effective round trips, $\sim300$ for LIGO. Reading: folding the light amplifies the phase response to a given displacement. A real wave carries both polarizations, and the detector projects them onto its own geometry as a weighted sum, the weights set by where the source sits and how its polarization is oriented.

h(t) = F_+(\theta,\varphi,\psi)\,h_+(t) + F_\times(\theta,\varphi,\psi)\,h_\times(t) \qquad (4)

where: $h(t)$ — strain time series recorded by the detector; $F_+, F_\times$ — antenna-pattern functions; $\theta,\varphi$ — source sky position; $\psi$ — polarization angle. Reading: the pattern functions encode directional sensitivity — why one L cannot localize. Projecting the detector tensor $D^{ij}=\tfrac12(\hat x^i\hat x^j-\hat y^i\hat y^j)$ onto the polarization tensors and rotating into the wave frame gives the explicit angular response — a spin-2 function of position with genuine nodes.

F_+ = \tfrac12(1+\cos^2\theta)\cos2\varphi\cos2\psi - \cos\theta\sin2\varphi\sin2\psi \qquad (5)

where: $\theta$ — polar angle from the detector zenith (arm-plane normal); $\varphi$ — azimuth from one arm; $\psi$ — polarization angle. Reading: smooth and broad — good for all-sky monitoring, useless for pinpointing. Localization comes from timing across the network: a plane wavefront from direction $\hat N$ sweeps the Earth, reaching two detectors at times whose difference is fixed by the projected baseline between them.

\Delta t_{ab} = \frac{\hat N \cdot (\vec r_a - \vec r_b)}{c} \qquad (6)

where: $\Delta t_{ab}$ — arrival-time difference between detectors $a$ and $b$ ; $\hat N$ — unit vector toward the source; $\vec r_a,\vec r_b$ — detector positions; $c$ — speed of light (= the wave speed in GR). Reading: each pair constrains the source to a ring; three or more cut the rings to a patch. Finally, the deepest limit is quantum: counting photons at the dark port carries an irreducible Poisson (shot) noise, which converted to strain scales inversely with arm length and as the inverse square root of circulating power.

S_{shot}^{1/2}(f) \propto \frac{1}{L}\sqrt{\frac{\hbar c\,\lambda_L}{P}} \qquad (7)

where: $S_{shot}^{1/2}$ — amplitude spectral density of strain noise from photon counting; $P$ — circulating laser power; $\hbar$ — reduced Planck constant; $L$ — arm length. Reading: longer arms, higher power, and squeezed light lower the floor — until pushing $P$ up trades shot noise for radiation-pressure noise at the SQL.

Derivation AThe interferometer as a differential geodesic-deviation meter

The goal is to show that the measured quantity $\Delta L = hL$ follows directly from the transverse-traceless metric, without any force acting on the mirrors, and to make explicit why the two-arm difference is both twice a single arm and immune to common-mode noise. We compute the proper length of one arm, evaluate it for each polarization, and subtract — reducing the whole detection principle to one gauge-invariant length measurement.

Step 1. In the TT gauge the mirrors stay at fixed coordinates $x=0$ and $x=L$ ; only the metric changes. The proper length of the $\hat x$ -arm is $L_{\rm prop}=\int_0^L\sqrt{1+h_{xx}^{TT}}\,dx$ . Why: coordinates are just labels — the physical, measurable arm length is the proper length, and it is this that the light samples.

Step 2. Linearize for $|h|\ll1$ : $\sqrt{1+h_{xx}^{TT}}\approx1+\tfrac12 h_{xx}^{TT}$ , so $\delta L_x/L=\tfrac12 h_{xx}^{TT}$ ; likewise $\delta L_y/L=\tfrac12 h_{yy}^{TT}$ . Why: the strain is tiny, so first order is exact to one part in $10^{21}$ , and each arm reads the strain projected onto it.

Step 3. For a plus-polarized wave along $\hat z$ the transverse metric is traceless with $h_{xx}^{TT}=-h_{yy}^{TT}=h_+$ , so $\delta L_x=+\tfrac12 h_+ L$ and $\delta L_y=-\tfrac12 h_+ L$ : one arm lengthens while the other shortens equally. Why: the tracelessness of a spin-2 field makes the two transverse directions move in antiphase — the geometric origin of the L-shape.

Step 4. Subtract the arms: $\Delta L=L_x-L_y=h_+L$ . The differential is twice the single-arm change, and any perturbation that stretches both arms equally cancels identically. For a wave off the principal axes, replace $h_+$ by $h_{ij}^{TT}D^{ij}$ with $D^{ij}=\tfrac12(\hat x^i\hat x^j-\hat y^i\hat y^j)$ , recovering the antenna pattern of equation (5). Why: a Michelson topology, not a single cavity, is the right instrument — the physics you want adds while the noise you fear subtracts.

Go deeper · Derivation B · 1 minPhoton round-trip time in a passing wave

Derivation BPhoton round-trip time in a passing wave

The proper-length picture invites a natural objection: if the wave stretches space, does it not also stretch the light's wavelength by the same factor, so arm and ruler grow together and nothing is measured? The goal here is to answer that by computing the gauge-invariant observable — the round-trip travel time of a photon in the time-dependent metric — and showing it is modulated by $h$ , giving an effective $\tfrac12 hL$ per arm that matches the proper-length result. We integrate the null condition along the arm and take the long-wavelength limit.

Step 1. A photon moves on a null path. For a wave with $h_{xx}^{TT}=h(t)$ along the $\hat x$ -arm, $0=-c^2dt^2+(1+h(t))dx^2$ , so $c\,dt\approx(1+\tfrac12 h)\,dx$ . Why: the observable is not "how long is the ruler" but "how long does light take" — a coordinate-independent clock reading, which is what the interferometer compares between arms.

Step 2. Integrate outbound from beamsplitter to mirror: $c(t_1-t_0)=L+\tfrac12\int_0^L h(t(x))\,dx$ , the return leg adding an equal contribution. Why: the perturbation accumulates along the whole arm, so it is the path-integrated strain, not a local value, that shifts the photon's arrival.

Step 3. Take the long-wavelength limit. LIGO's round-trip time $2L/c\approx27\,\mu\mathrm{s}$ is far shorter than a gravitational period ( $\gtrsim1\,\mathrm{ms}$ ), so $h$ is essentially frozen during a round trip. The round-trip time becomes $\tau_{\rm round}=\tfrac{2L}{c}(1+\tfrac12 h)$ , i.e. an effective length change $\delta L_{\rm eff}=\tfrac12 hL$ . Why: because $\lambda_{gw}\gg L$ the wave is quasi-static across the arm and the response amplitude is frequency-independent — why LIGO's band is flat in strain over much of its range.

Step 4. Resolve the objection. The naive "the wavelength stretches too" argument is gauge-dependent bookkeeping; the travel-time calculation is gauge-invariant and gives a real, non-zero phase modulation. Differencing the two arms doubles it to $hL$ for plus polarization, matching equation (2). Why: physical predictions must be gauge-invariant, and the round-trip time is exactly such an observable — it settles the paradox without a preferred ruler.

Go deeper · Derivation C · 1 minReading the chirp mass out of GW150914

Derivation CReading the chirp mass out of GW150914

The goal is to show how the single most important source parameter — the chirp mass that sets the inspiral rate — is extracted from nothing but the observed frequency and its rate of change, reproducing the published value for GW150914 to leading post-Newtonian order. The quadrupole formula shrinks the orbit, sweeping the wave frequency upward; inverting the sweep law at any instant fixes the chirp mass, and the rise from tens to hundreds of hertz is the audible "chirp" the detectors recorded. We state the inspiral law, invert it, and insert the numbers.

Step 1. Energy balance between orbital binding energy and quadrupole luminosity gives the leading sweep $\dot f=\frac{96}{5}\pi^{8/3}\left(\frac{G\mathcal M}{c^3}\right)^{5/3}f^{11/3}$ , controlled entirely by the chirp mass $\mathcal M=(m_1 m_2)^{3/5}/(m_1+m_2)^{1/5}$ , with $f$ the wave frequency (twice the orbital frequency). Why: to leading order every other parameter drops out of the sweep — the chirp mass is the one combination the inspiral measures cleanly, and the best-determined quantity in the data.

Step 2. Invert for the mass at a single instant: $\mathcal M=\frac{c^3}{G}\left[\frac{5}{96}\pi^{-8/3}f^{-11/3}\dot f\right]^{3/5}$ . One reading of $f$ and $\dot f$ suffices. Why: the law is monotone in $\dot f/f^{11/3}$ , so the measurement is a direct read-out, not a fit requiring the whole waveform.

Step 3. Insert GW150914. Near $f\approx150\,\mathrm{Hz}$ the signal swept at $\dot f\approx1.6\times10^{4}\,\mathrm{Hz/s}$ (climbing from $\sim35$ to $\sim250\,\mathrm{Hz}$ in about $0.2\,\mathrm{s}$ ). Plugging in, with $GM_\odot/c^3=4.93\times10^{-6}\,\mathrm{s}$ , gives $\mathcal M\approx30\,M_\odot$ (source-frame $\approx28.6\,M_\odot$ ). Why: a back-of-envelope inversion reproduces the published chirp mass, confirming a pair of tens-of-solar-mass compact objects.

Step 4. Break the degeneracy. The chirp mass alone gives $\mathcal M=30\,M_\odot$ but not $m_1,m_2$ separately; higher-order terms and the merger frequency supply the mass ratio, yielding $m_1\approx36\,M_\odot$ , $m_2\approx29\,M_\odot$ , a final black hole of $\sim62\,M_\odot$ , and $\sim3\,M_\odot c^2$ radiated. Why: leading order fixes the loudest combination; sub-leading structure and ringdown pin the individual masses — how LIGO reported $36+29\to62$ .

Limiting cases

The response is governed by the ratio of the gravitational wavelength $\lambda_{gw}$ to the arm length $L$ , and by the source direction relative to the arms:

Case	Result	Meaning
Long-wavelength regime	$\lambda_{gw}\gg L\Rightarrow\Delta L\approx hL$	For $L=4\,\mathrm{km}$ and $f\lesssim$ a few kHz, $\lambda_{gw}\sim c/f\gtrsim100\,\mathrm{km}\gg L$ , so the quasi-static tidal picture holds and the strain response is flat in amplitude.
Optimal orientation	$F_+=1,\ h(t)=\Delta L/L$	A wave along the arm-plane normal with polarization aligned to the arms gives maximal response; the detector reads the strain directly.
Blind direction	$F_+=F_\times=0$	For certain positions and polarizations the antenna pattern vanishes; one detector is deaf to such sources, forcing a diverse network.
High frequency	$f\sim c/(2L)\Rightarrow$ response rolls off	When the gravitational period nears the light round-trip time (tens of kHz for $L=4\,\mathrm{km}$ ), the strain averages over the arm and the response falls — well above the astrophysical band.

Worked examples

1 · The smallest measurable length (fully worked)

A wave of strain $h=1\times10^{-21}$ passes optimally through LIGO ( $L=4\,\mathrm{km}$ , $\lambda_L=1064\,\mathrm{nm}$ , $\mathcal F\approx300$ ). Find the differential arm motion, compare it to a proton, and estimate the optical phase.

Step 1. Differential arm length from equation (2): $\Delta L=hL=(1\times10^{-21})(4\times10^{3}\,\mathrm{m})=\mathbf{4\times10^{-18}\ m}$ . Each arm moves half this, $\pm2\times10^{-18}\,\mathrm{m}$ , in antiphase.

Step 2. Scale it. A proton is about $1.7\times10^{-15}\,\mathrm{m}$ across, so $\Delta L/d_{\rm proton}\approx(4\times10^{-18})/(1.7\times10^{-15})\approx\mathbf{2\times10^{-3}}$ — the arms move by roughly a thousandth of a proton diameter.

Step 3. Optical phase from equation (3): $\Delta\phi=\dfrac{4\pi\mathcal F}{\lambda_L}\,\Delta L=\dfrac{4\pi(300)}{1.064\times10^{-6}}(4\times10^{-18})\approx\mathbf{1.4\times10^{-8}\ rad}$ . This sounds hopeless, but shot noise falls as $1/\sqrt{N}$ , and $N\sim10^{20}$ photons give a phase resolution $\sim10^{-10}\,\mathrm{rad}$ . ΔL = 4×10⁻¹⁸ m, Δφ ≈ 1.4×10⁻⁸ rad, measurable because the photon flux is enormous.

2 · Chirp mass of GW150914 (fully worked)

Near $f=150\,\mathrm{Hz}$ the GW150914 signal swept upward at $\dot f\approx1.6\times10^{4}\,\mathrm{Hz/s}$ . Recover the chirp mass. Use $GM_\odot/c^3=4.93\times10^{-6}\,\mathrm{s}$ .

Step 1. Invert the leading inspiral law (Derivation 3): $\mathcal M=\dfrac{c^3}{G}\left[\dfrac{5}{96}\pi^{-8/3}f^{-11/3}\dot f\right]^{3/5}$ . With $f=150$ and $\dot f=1.6\times10^{4}$ the bracket is $4.13\times10^{-7}\,\mathrm{s}$ (this is $G\mathcal M/c^3$ ).

Step 2. Raise to the $3/5$ power: $(4.13\times10^{-7})^{3/5}=1.48\times10^{-4}\,\mathrm{s}$ ; divide by $GM_\odot/c^3=4.93\times10^{-6}\,\mathrm{s}$ to get $\mathcal M\approx\mathbf{30\ M_\odot}$ .

Step 3. Interpret. A chirp mass of $\sim30\,M_\odot$ (source-frame $\sim28.6\,M_\odot$ ) is far above any neutron star, so the source is a binary of massive compact objects; the mass ratio and merger frequency then give $m_1\approx36\,M_\odot$ and $m_2\approx29\,M_\odot$ . 𝓜 ≈ 30 M_⊙ straight from f and ḟ — the first weighing of a stellar-mass binary black hole.

3 · Where on the sky (triangulation ring)

The Hanford–Livingston baseline is $d\approx3000\,\mathrm{km}$ ; a signal arrives $\Delta t=7\,\mathrm{ms}$ earlier at one site. On what ring does the source lie? The maximum delay is $d/c\approx\mathbf{10\ ms}$ (equation 6), so $7\,\mathrm{ms}$ is within range. The angle to the baseline satisfies $\cos\alpha=c\,\Delta t/d=(3.00\times10^{8})(7\times10^{-3})/(3.00\times10^{6})=0.70$ , so $\alpha\approx\mathbf{46^\circ}$ : the source lies on a ring $46^\circ$ from the baseline axis. Two detectors give a ring; a third intersects a second ring and collapses the region to tens of deg².

Physics of the portrait

Interferometric gravitational-wave detection has many parents, but the one who turned it into a buildable machine — with a hard-nosed accounting of every noise source that would bury the signal — was Rainer Weiss.

Origins · 1 minRainer Weiss and how this was discovered

Rainer Weiss

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

Born in Berlin and raised in New York after his family fled Nazi Germany, Rainer Weiss came to gravitational waves through precision measurement and a knack for finding the noise that ruins an experiment. Joseph Weber's resonant-bar detectors of the 1960s had made the first serious attempts, and Gertsenshtein and Pustovoit had floated interferometry as early as 1962, but it was Weiss who, in a 1972 MIT quarterly report, laid out the kilometre-scale laser interferometer in full and — crucially — estimated every fundamental noise term: seismic, thermal, shot, radiation-pressure, and residual-gas. That noise budget made the field credible, because it showed the signal was not hopelessly buried but sat just below a floor better engineering could lower. Weiss prototyped at MIT while Ronald Drever and Kip Thorne pushed parallel efforts at Caltech; Barry Barish then reorganized the project into the collaboration that could build and run two 4-km detectors. The gamble paid off on 14 September 2015, when GW150914 appeared in both instruments within their mutual light-travel time; Weiss, Barish, and Thorne shared the 2017 Nobel Prize. His enduring lesson is methodological: to measure the almost-immeasurable, first write down honestly everything that will stop you.

History

1916

— Einstein predicts gravitational waves from the linearized field equations

1972

— Weiss details the kilometre-scale laser interferometer and its full noise budget at MIT

1974

— Hulse and Taylor find the binary pulsar, indirect proof that orbits lose energy to waves

2015

— the two LIGO detectors record GW150914, the first direct detection

2017

— GW170817 opens multi-messenger astronomy; the Nobel Prize honours Weiss, Barish, and Thorne

Going further

Texts. Misner, Thorne and Wheeler, Gravitation, ch. 37 · Schutz, A First Course in General Relativity, ch. 9 · Carroll, Spacetime and Geometry, ch. 7 · Hartle, Gravity, ch. 16 · Maggiore, Gravitational Waves, Vol. 1, ch. 9 · Saulson, Fundamentals of Interferometric Gravitational Wave Detectors.

Key papers. Weiss, MIT RLE Quarterly Progress Report No. 105 (1972) — the interferometric detector and its noise budget · Abbott et al. (LIGO/Virgo), Phys. Rev. Lett. 116, 061102 (2016) — GW150914 · Abbott et al. (LIGO/Virgo), Phys. Rev. Lett. 119, 161101 (2017) — GW170817. Key numerical benchmarks: the proper-length result $\delta L/L=\tfrac12 h_{ij}^{TT}\hat n^i\hat n^j$ and differential $\Delta L=h_+L$ ; the photon round-trip modulation and its long-wavelength limit; the GW150914 chirp-mass inversion giving $\mathcal M\approx30\,M_\odot$ , $m_1\approx36\,M_\odot$ , $m_2\approx29\,M_\odot$ , $d_L\approx410\,\mathrm{Mpc}$ , $E_{\rm rad}\approx3\,M_\odot c^2$ ; and the GW170817 values $\Delta t_{\gamma}\approx1.7\,\mathrm{s}$ , sky area $\sim28\,\mathrm{deg}^2$ , $|v_{gw}-c|/c\lesssim10^{-15}$ . Detector parameters used throughout: 4-km arms, $1064\,\mathrm{nm}$ laser, $40\,\mathrm{kg}$ mirrors, $\sim10\,\mathrm{ms}$ Hanford–Livingston light-travel delay; antenna-pattern functions $F_+,F_\times$ as given in equation (5).

Prerequisites. Transverse-Traceless Gauge · Geodesic Deviation and Gravitational Waves · Plus and Cross Polarizations · Quadrupole Formula. Leads to. Isaacson Stress-Energy · Multipole Expansion of Gravitational Radiation.

Ask the tutor: Ask me how a strain of $10^{-21}$ becomes a measurable phase, or how three detectors turn arrival times into a patch of sky.

If a gravitational wave stretches space, why doesn't it stretch the laser light too and cancel the signal?

It does stretch the light while the light is in the arm — but that is not what the interferometer measures. The physical, gauge-invariant observable is the round-trip travel time of the photons through the time-dependent metric, with the mirrors in free fall. That travel time is modulated by about $hL/c$ and does not cancel against anything. The "ruler versus wavelength" argument mixes up two coordinate descriptions; once you compute an actual observable — the dark-port phase — a real, measurable shift remains.

How can anyone measure a motion of 10⁻¹⁸ metres?

By never relying on a single photon. The displacement is a thousandth of a proton's diameter, but the readout beam delivers roughly $10^{20}$ photons per second, and the phase uncertainty of a coherent beam falls as $1/\sqrt{N}$ . Fabry–Perot cavities fold the light $\sim300$ times, power recycling builds the circulating power to hundreds of kilowatts, and squeezed vacuum cuts the quantum phase noise below the shot-noise limit. The net strain sensitivity reaches near $10^{-23}/\sqrt{\mathrm{Hz}}$ — enough to lift the wiggle above the noise.

Why do we need a whole network instead of one very good detector?

Because a single L-shaped detector has essentially no sense of direction: its antenna pattern is a broad quadrupole with two blind spots, so one strain record fits a large swath of sky positions. A network fixes this three ways: arrival-time differences between separated sites triangulate the source onto rings whose intersection is a small patch; the relative amplitudes across differently-oriented detectors pin down polarization and distance; and demanding a coincident, consistent signal rejects the glitches that plague any one detector. Localization is what lets telescopes chase electromagnetic counterparts, as in GW170817.

Is the wave moving the mirrors, or stretching the space between them?

In the transverse-traceless gauge the cleanest description is the second: the mirrors sit at fixed coordinates and it is the proper distance between them that oscillates, because the metric is time-dependent — no force acts in this frame. You can equally choose a local-Lorentz "proper detector" frame in which the mirrors feel a tidal force $\ddot\xi^i=-R^i{}_{0j0}\xi^j$ and physically move; the two are related by a coordinate change and predict the same phase shift. The TT frame is simplest for computing, the proper-detector frame closest to Newtonian intuition, and neither is "more real".

What actually limits LIGO's sensitivity?

Different noises dominate different bands. Below $\sim10\,\mathrm{Hz}$ seismic and Newtonian (gravity-gradient) noise dominate, which is why the test masses hang from multi-stage isolation and third-generation detectors go underground. In the mid-band, thermal noise of the coatings and fused-silica suspensions sets the floor. At high frequency, quantum shot noise takes over as $1/\sqrt{P}$ ; push the power higher and radiation-pressure noise grows, so the two meet at the standard quantum limit, which squeezed light re-partitions to beat across the band.

How do you know a detection is a real astrophysical wave and not an instrumental glitch?

By coincidence and consistency. A genuine wave appears in two or more widely separated detectors within their mutual light-travel time, with waveforms matching the same template up to the antenna-pattern weighting — a glitch in one instrument cannot fake that in another. Matched filtering against banks of inspiral–merger–ringdown templates assigns each candidate a false-alarm rate estimated by time-sliding the data streams past each other; GW150914's was below one per $200{,}000$ years. Data-quality vetoes flag known disturbances, so surviving candidates are astrophysical with high confidence.