Gravitational Wave Chirp: The Quadrupole Formula, LIGO & Inspiral Waveforms

Spacetime stretches and squeezes

General relativity describes gravity not as a force but as spacetime curvature. When massive objects accelerate asymmetrically, the changing curvature propagates outward at the speed of light as gravitational waves. Unlike electromagnetic waves, they stretch space along one axis while compressing it along the perpendicular axis — the "plus" and "cross" polarisations. The amplitude, called strain h = ΔL/L, is minuscule: LIGO's first detection measured h ~ 10⁻²¹, meaning a 4 km arm changed length by about 10⁻¹⁸ m, roughly one-thousandth the diameter of a proton.

The quadrupole formula and the runaway inspiral

Einstein's quadrupole formula gives the power a binary system radiates as gravitational waves in terms of its total mass, reduced mass μ and orbital separation a:

P = −32/5 · G⁴/c⁵ · (m₁m₂)²(m₁+m₂) / a⁵
Chirp mass: M_c = (m₁m₂)^(3/5) / (m₁+m₂)^(1/5)   → GW150914: M_c ≈ 28.3 M☉

Because P scales as a⁻⁵, as the binary spirals inward it radiates more power, which shrinks the orbit further and increases the radiation still more — a runaway inspiral. This is exactly why binary neutron stars merge in finite time, a process that inspired the 1993 Nobel Prize for the Hulse-Taylor pulsar. The chirp mass is the single quantity most directly extracted from the waveform, since it controls how fast the frequency rises.

Inspiral, merger, ringdown — and how LIGO listens

A binary black hole signal unfolds in three phases: a slow inspiral whose frequency sweeps upward as f_GW = 2f_orbital, a brief merger lasting around 10 milliseconds where the strain peaks, and a ringdown where the newly merged black hole settles through damped quasi-normal modes, revealing its final mass and spin. LIGO detects this using Michelson interferometry: a laser is split into two perpendicular 4 km beams, bounced roughly 280 times between 40 kg fused-silica mirrors suspended on quadruple pendulums to isolate seismic noise, then recombined so a passing wave shifts the interference pattern. Two detectors 3,000 km apart (plus Virgo in Italy and KAGRA in Japan) enable coincident detection and sky localisation, and template-matched filtering cross-correlates the data against roughly 10⁵ precomputed waveform shapes.

Key detections and multi-messenger astronomy

GW150914 (2015) merged two black holes of 36 and 29 solar masses into a 62-solar-mass remnant about 430 megaparsecs away. GW170817 (2017) was historic for a different reason: the first binary neutron star merger detected in both gravitational waves and light. A gamma-ray burst arrived 1.7 seconds later, confirming that short gamma-ray bursts come from neutron star mergers, and weeks of optical follow-up revealed a kilonova — the radioactive afterglow of roughly 10 Earth-masses of freshly forged gold, platinum and other heavy elements. As of the third LIGO/Virgo/KAGRA observing run, over 90 compact-binary merger candidates have been catalogued, including GW190521's intermediate-mass 142-solar-mass remnant, a size ordinary stellar evolution alone cannot easily produce.

Frequently asked questions

Try it live

Everything above runs in your browser — open Gravitational Wave Chirp, set the binary masses and distance, and watch the h(t) strain waveform sweep and chirp through inspiral, merger and ringdown, alongside a spectrogram with LIGO noise overlay. Nothing is installed, nothing is uploaded.