gravitational wave strain — from formula anatomyGravitational wave strain is a dimensionless measure of how much a passing gravitational wave stretches and squeezes spacetime. It is defined as the fractional change in distance between two free-falling test masses: if the original separation is L and the wave changes it by ΔL, the strain is h = ΔL / L. Because spacetime itself is being distorted, the strain is not a force pushing objects apart; rather, it is a geometric effect that alters the very metric of space.
The concept of gravitational wave strain was formalized in the early 20th century through Einstein's general relativity, with the first explicit derivation of gravitational waves by Einstein in 1916. The first direct detection of strain was made by the LIGO collaboration in 2015.
A gravitational wave is a ripple in the curvature of spacetime, produced by accelerating massive objects (like merging black holes or neutron stars). As the wave passes, it alternately stretches space along one axis while compressing it along a perpendicular axis. The strain quantifies the amplitude of this distortion. For a wave traveling along the z-axis, the two polarizations (plus and cross) cause the separation between test masses to oscillate with a characteristic pattern. The strain is typically extremely small — for events detected by LIGO, h is on the order of 10^-21, meaning a change of one part in a billion trillion.
Gravitational wave detectors like LIGO and Virgo measure strain directly using laser interferometry, where the tiny changes in arm lengths (on the order of 10^-18 meters) are detected as phase shifts in the laser light.
The strain from the first detected gravitational wave (GW150914) was smaller than the diameter of a proton, yet it carried information about the merger of two black holes 1.3 billion light-years away.
| h_{ij}^{\text{TT}} | variable | Transverse-traceless gravitational wave strain tensor — gravitational wave strain |
| D | variable | Distance from source to observer — distance metre |
| I_{ij} | variable | Mass quadrupole moment tensor — mass quadrupole moment kilogram metre squared |
| t | variable | Time — time second |
| G | constant | Constant of gravitation in Newton's law — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| c | constant | Universal speed limit and speed of light in vacuum — speed of light = 299792458 metre per second |
| d^2/dt^2 | operator | Second derivative with respect to time — second time derivative |
| h̃_L | variable | Lensed gravitational wave signal in frequency domain — gravitational wave strain |
| f | variable | Gravitational wave frequency — frequency hertz |
| μ_j | variable | Magnification factor of image j — magnification |
| Δt_j | variable | Time delay of image j relative to source — time delay second |
| n_j | variable | Morse index of image j — morse index |
| h̃_src | variable | Unlensed source gravitational wave signal — gravitational wave strain |
| ∑ | operator | Sum over lensed images — summation |
| e^{...} | operator | Complex exponential encoding phase shifts — exponential |
| h_L | variable | Lensed gravitational wave strain — gravitational wave strain |
| h_unL | variable | Unlensed gravitational wave strain — gravitational wave strain |
| μ | variable | Magnification factor — magnification |



