How LIGO Detected Gravitational Waves

Equivalent forms

chirp mass

\mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1+m_2)^{1/5}}

frequency sweep

\dot{f} = \frac{96}{5}\pi^{8/3}\!\left(\frac{G\mathcal{M}}{c^3}\right)^{5/3} f^{11/3}

Measuring a length change a thousand times smaller than a proton turned spacetime itself into something we can hear ring.

Symbol	Name	SI unit	Dimension	Range
$h$	strain (fractional length change)output	dimensionless	1	—
$\Delta L$	arm length change	m	L	—
$L$	interferometer arm length	4000 m	L	—
$\mathcal{M}$	chirp mass	kg	M	—
$f_{GW}$	gravitational-wave frequency	Hz	T^{-1}	—

Discovery

LIGO Scientific Collaboration (Weiss, Barish, Thorne) · 2015

Predicted by Einstein in 1916, gravitational waves resisted direct detection for a century. On 14 September 2015 the twin LIGO interferometers recorded GW150914 — the merger of two black holes (36 and 29 solar masses) radiating ~3 solar masses of energy as gravitational waves. The 2017 Nobel Prize honored Weiss, Barish, and Thorne.

Research status: active

Real-world applications

Opened gravitational-wave astronomy: dozens of black-hole and neutron-star mergers now catalogued.
GW170817 (neutron-star merger) seen with light too — launching multi-messenger astronomy and measuring H0.
Tests GR in the strong-field, dynamical regime (ringdown 'no-hair' tests).

Common misconceptions

'Gravitational waves move matter through space.' — They stretch space itself; everything embedded co-moves.
'Bigger detectors just need longer arms.' — Sensitivity also demands seismic isolation, vacuum, and quantum-squeezed light.
'A single detector localizes the source.' — Sky position needs timing across multiple detectors (LIGO-Virgo-KAGRA).

Experimental verification

GW150914 detected at >5 sigma by both LIGO sites; >90 events through O3; waveforms match numerical-relativity templates; Hulse-Taylor binary pulsar orbital decay (1974) gave the earlier indirect proof.

Derivation

A passing gravitational wave imposes a quadrupole strain h: one arm lengthens while the perpendicular arm shortens.
Michelson interferometry converts $\Delta L = h L$ into a measurable shift in the laser interference fringe.
For $h \sim 1e-21$ and $L = 4\,\mathrm{km}$ , $\Delta L \sim 4e-18\,\mathrm{m}$ — a thousandth of a proton diameter, reached via Fabry-Perot cavities and 100 kW circulating power.
Inspiral dynamics fix the waveform: frequency and amplitude sweep upward (the chirp) governed by the chirp mass M_chirp.
Matched filtering against template banks extracts the signal from noise and reads off the masses and distance.

Limiting cases

Early inspiral⟶ slow, low-frequency waveWide orbit radiates weakly; frequency creeps up.

Merger / ringdown⟶ peak strain, frequency cutoffFinal plunge produces the loudest, highest-frequency chirp then damps.

What if…

What if the black holes were lighter?

Lower chirp mass means the merger frequency rises into LIGO's band later; very light binaries chirp faster and weaker.

What if there were only one detector?

You confirm a wave but cannot triangulate the sky position; multiple detectors give timing-based localization.

1

Arm length change for GW150914

Given · Strain

h \sim 1e-21

, arm length

L = 4000\,\mathrm{m}

.

Find · Physical length change Delta L

Steps

$\Delta L = h L = 1e-21 * 4000$ .
$= 4e-18\,\mathrm{m}$ .
Compare a proton diameter $\sim 1.7e-15\,\mathrm{m}$ : $\Delta L is \sim 1/400$ of that.

Result ·

\sim 4e-18\,\mathrm{m}

— about a thousandth of a proton's width, which LIGO measures via laser interferometry.

Gravitational-Wave Strain→Schwarzschild Radius→Frame Dragging (Lense–Thirring Effect)→Kerr Metric for Rotating Black Holes→