Gravitational lensing is the bending of light rays by massive objects, distorting and magnifying images of distant sources.Imagine the Universe as a giant aquarium, where massive objects bend space like a heavy ball on a rubber sheet. Light from distant stars, as it passes by, gets its path bent, and we see distorted, magnified images, as if through a warped lens.
The idea of light bending by a massive body had been discussed since the 18th century, but within Newtonian theory. In 1915 Einstein derived the correct deflection angle (twice the Newtonian value). In 1919 Eddington's expedition confirmed the deflection during a solar eclipse. In the 1930s Fritz Zwicky proposed using lensing by galaxy clusters as 'telescopes'. The first multiple system (double quasar) was found in 1979 by Dennis Walsh and colleagues. Today it is a key tool for cosmology, e.g., surveys by the Vera Rubin Observatory.
According to general relativity, Einstein's equations describe spacetime curvature. Light follows null geodesics, and its trajectory is described by an effective refractive index n = 1 - 2Φ/c², where Φ is the gravitational potential. The deflection angle for a point mass is α = 4GM/(c²ξ), where ξ is the impact parameter. When source, lens, and observer align, an Einstein ring forms with angular radius θ_E = sqrt((4GM)/(c²) * (D_ls)/(D_l D_s)), where D are distances. For distributed mass, lensing yields distortions from which a density map of matter, including dark matter, is constructed.
A gravitational lens can magnify light so much that we see the same distant galaxy in multiple places — like in a hall of mirrors!
| \theta_E | variable | Einstein ring angular radius — angle radian |
| M | variable | Mass of the lensing object — mass kilogram |
| D_{\rm LS} | variable | Angular diameter distance from lens to source — distance metre |
| D_{\rm L} | variable | Angular diameter distance from observer to lens — distance metre |
| D_{\rm S} | variable | Angular diameter distance from observer to source — distance metre |
| G | constant | Universal constant of gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| c | constant | Speed of light in vacuum — speed of light = 299792458 metre per second |
| \sqrt{} | operator | Takes the square root of the argument — square root |
| 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 |
| \Delta t | variable | time delay between images — time delay second |
| D_{\Delta t} | variable | time delay distance — distance metre |
| \Delta\phi | variable | difference in lensing potential — lensing potential difference |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| Δt | variable | time delay between two images or signals — time delay second |
| D_Δt | variable | time-delay distance, a combination of angular diameter distances in the lens system — time delay distance metre |
| Δτ | variable | difference in the Fermat potential between two light paths — fermat potential difference |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| \theta_E | variable | Angular radius of the Einstein ring — einstein radius angle radian |
| G | variable | Gravitational constant — gravitational constant cubic metre per kilogram per second squared |
| M | variable | Mass of the lensing object — mass kilogram |
| c | variable | Speed of light in vacuum — speed of light metre per second |
| D_{LS} | variable | Angular diameter distance between lens and source — distance metre |
| D_L | variable | Angular diameter distance from observer to lens — distance metre |
| D_S | variable | Angular diameter distance from observer to source — distance metre |
| \sqrt{} | operator | Takes the square root of the argument — square root |















