Gravitational-Wave Strain
Also known as: Quadrupole strain · GW amplitude · Spacetime ripple
Accelerating masses ripple spacetime itself, and those ripples stretch and squeeze every length they pass through by a fractional amount h — the strain. The catch is the factor c⁴ in the denominator: it makes h staggeringly tiny. Two merging black holes a billion light-years away wobble LIGO's 4 km arms by less than a thousandth the width of a proton, which is exactly what it detected.
A ring of free test masses deformed by a passing + polarized gravitational wave — stretching along one axis while squeezing the other, oscillating in time.
Equivalent forms
Gravity has no dipole radiation — mass can't be pushed around like charge — so the leading term is the quadrupole, and that single fact sets the entire scale and waveform of gravitational-wave astronomy.
Einstein predicted gravitational waves in 1916 from the linearized field equations, then doubted they were real for decades. The quadrupole formula was put on firm footing by the 1970s, indirectly confirmed by the orbital decay of the Hulse–Taylor binary pulsar (Nobel 1993), and directly detected on 14 Sept 2015 when LIGO caught GW150914 — two ~30-solar-mass black holes merging 1.3 billion light-years away (Nobel 2017).
- LIGO/Virgo/KAGRA interferometers measure interfering laser light in km-scale arms.
- Binary-black-hole and binary-neutron-star mergers: the 'chirp' (rising frequency + amplitude) directly encodes the masses and distance.
- GW170817 + its gamma-ray and optical counterpart launched multi-messenger astronomy and measured the speed of gravity to equal .
- Pulsar timing arrays (NANOGrav) sense the nanohertz stochastic background from supermassive black-hole binaries across the galaxy.
- Hulse–Taylor binary pulsar's orbit decays at exactly the rate the quadrupole luminosity predicts — the first proof GWs carry energy.
- “Gravitational waves are like electromagnetic waves.” — They're tensor (spin-2) ripples of geometry itself, with two polarizations rotated by , not .
- “They move the detector's mirrors.” — They change the *proper distance* between freely-falling mirrors; it's spacetime stretching, not a force pushing the mass.
- “A spinning symmetric object radiates GWs.” — A perfectly axisymmetric spin has a constant quadrupole and radiates nothing; you need a *changing* quadrupole.
- Linearize the Einstein equations around flat space, with |h| ≪ 1.
- In the transverse-traceless gauge the wave equation becomes □ .
- Solve with a retarded potential; the leading multipole that radiates is the mass quadrupole Q_{ij} (mass conservation kills the monopole, momentum conservation kills the dipole).
- The far-field strain is \ddot{Q}_{ij}, evaluated at retarded time .
- The radiated power follows from the quadrupole luminosity ⟨\dddot{Q}_{ij}\dddot{Q}^{ij}⟩.
Limiting cases
What if…
There isn't — momentum conservation forbids it — so GWs are weaker (start at quadrupole order) and harder to detect than electromagnetic dipole radiation.
h scales as 1/d, so the strain grows the detectable volume grows — distance is the single biggest lever on detection rates.
Why LIGO needs sub-proton precision
- Compare to a proton radius : of a proton width.
Order-of-magnitude strain from a binary
- ;