Gravitational Wave Polarization 2D — Ring of Test Masses

Polarization

Polarization angle ψ 0°

Strain amplitude h₀ (exaggerated) 0.16

Wave frequency 0.40 Hz

Live Readouts

h₊(t)

0.000

h×(t)

0.000

ΔL/L, x-arm

0.00%

ΔL/L, y-arm

0.00%

How it works

A weak gravitational wave travelling along z in the transverse-traceless (TT) gauge perturbs the flat metric by δg_ij = h_ij, leaving free-falling test masses' coordinate positions fixed but changing the proper distance between them. For a ring of masses at radius R₀ in the x–y plane, the physical displacement of the mass initially at angle θ is:

δx(θ,t) =  (h₊(t)/2) R₀cosθ + (h×(t)/2) R₀sinθ
δy(θ,t) = -(h₊(t)/2) R₀sinθ + (h×(t)/2) R₀cosθ

h₊(t) = h₀cos(ωt)·cos(2ψ)
h×(t) = h₀cos(ωt)·sin(2ψ)

This 2D view computes the same linearized geodesic-deviation formula directly on a flat canvas — every dot's on-screen position is (x₀+δx, y₀+δy) evaluated live from h₊(t) and h×(t), so a pure plus (+) wave (ψ = 0°) stretches the ring along x while squeezing it along y, then reverses, tracing the classic "plus" cross pattern; a pure cross (×) wave is the identical pattern rotated 45°. Any other polarization angle ψ is a coherent superposition of the two, because gravitational waves have only these two independent tensor polarizations.

Polarization angle ψ — rotates the deformation pattern continuously between pure + and pure ×, exactly as a real GW's polarization is set by the source geometry and detector orientation.
Strain amplitude h₀ — real astrophysical strains from LIGO-detected mergers are ~10⁻²¹; this display multiplies that by roughly 10²⁰ so the ring's deformation is visible to the eye.
ΔL/L readouts — the fractional length change along the x- and y-arms, the same quantity a laser interferometer like LIGO measures directly as differential arm-length change.

Test masses (ring) x / y arms