LIGO · Listening to Spacetime

Mathematical model

Symbols

$L$

Reference separation with no wave

$h_{+}(t)$

Dimensionless plus-polarized strain

$L_x,\,L_y$

Proper separations along two transverse axes

Model boundaryThis is the standard long-wavelength, TT-gauge teaching picture. The animation exaggerates strain by roughly 10²⁰. Coordinate positions may remain fixed while proper distances change.

Mathematical model

Symbols

$L$

Reference arm length; 4 km at LIGO

$\Delta L_{\mathrm{diff}}$

Differential change between the arms

$h_{+}$

Plus strain aligned with the arms

Model boundaryThe equation shows the clearest case: normal incidence with plus polarization aligned to the arms. A general sky direction and polarization introduce detector responses F₊ and F×.

Mathematical model

Symbols

$\lambda$

Laser wavelength in vacuum

$\phi_{\mathrm{rt}}$

Optical phase accumulated in one round trip

$\Delta\phi$

Phase difference between the returning beams

Model boundaryThis scene begins with a simple Michelson interferometer. Real LIGO arm cavities make the light sample an arm repeatedly; scene five adds that gain.

Mathematical model

Symbols

$E_{\mathrm{out}}$

Optical field amplitude at the output port

$I_{\mathrm{out}}$

Light intensity measured by the photodetector

$\Delta\phi$

Phase difference between the returning beams

Model boundaryThe picture first shows an ideal dark fringe. Advanced LIGO introduces a tiny DARM offset so a small carrier field acts as a local oscillator and provides a linear DC readout.

Mathematical model

Symbols

$N$

Effective traversal count in this teaching approximation, about 300

$L_{\mathrm{arm}}$

Physical arm-cavity length, 4 km

$L_{\mathrm{effective}}$

Intuitive effective optical path

Model boundaryThe 1200 km figure is path-length intuition. Real cavity gain also depends on finesse, storage time, and gravitational-wave frequency; it is not a fixed N at every frequency.

Mathematical model

Symbols

$f_{\mathrm{GW}}$

Dominant quadrupole gravitational-wave frequency

$f_{\mathrm{orb}}$

Binary orbital frequency

$\Delta t$

Arrival-time difference between the sites

Model boundaryThe displayed chirp is a teaching waveform, not raw detector data. Real analysis calibrates strain, models noise, and performs statistical inference with full waveform models and the detector network.