effect

gravitational lensing

Gravitational lensing is the bending of light rays by massive objects, distorting and magnifying images of distant sources.
27 related articles · 6 formulas · 6 who worked here

Description

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.

History

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.

How it works

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.

Fun fact

A gravitational lens can magnify light so much that we see the same distant galaxy in multiple places — like in a hall of mirrors!

Concept graph →

Formulas

$$\theta_E = \sqrt{\frac{4GM}{c^2} \frac{D_{\rm LS}}{D_{\rm L} D_{\rm S}}}}$$
Angular size of gravitational lensing depends on mass and distances.
What is in the formula
\theta_EvariableEinstein ring angular radius — angle radian
MvariableMass of the lensing object — mass kilogram
D_{\rm LS}variableAngular diameter distance from lens to source — distance metre
D_{\rm L}variableAngular diameter distance from observer to lens — distance metre
D_{\rm S}variableAngular diameter distance from observer to source — distance metre
GconstantUniversal constant of gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
cconstantSpeed of light in vacuum — speed of light = 299792458 metre per second
\sqrt{}operatorTakes the square root of the argument — square root
θ_e=\sqrt{\frac{4gm_l}{c^2}\frac{d_s-d_l}{d_sd_l}} · 2607.01853v1
θ_e=\sqrt{\frac{4gmd_{ls}}{c^2d_ld_s}} · 2607.02663
$$\tilde{h}_L(f) = \sum_j |\mu_j|^{1/2} e^{2\pi i f \Delta t_j - i\pi n_j} \tilde{h}_{\rm src}(f)$$
Lensed gravitational wave signal sum over images.
What is in the formula
h̃_LvariableLensed gravitational wave signal in frequency domain — gravitational wave strain
fvariableGravitational wave frequency — frequency hertz
μ_jvariableMagnification factor of image j — magnification
Δt_jvariableTime delay of image j relative to source — time delay second
n_jvariableMorse index of image j — morse index
h̃_srcvariableUnlensed source gravitational wave signal — gravitational wave strain
∑operatorSum over lensed images — summation
e^{...}operatorComplex exponential encoding phase shifts — exponential
\tilde{h}_l(f)=\sum_{j=1}^{n_{\rmimg}}|μ_j|^{1/2}\exp[2πifδt_j-iπn_j]\tilde{h}_{\rmsrc}(f;θ_s) · 2607.03885
$$h_L(f) = \mu(f) h_{unL}(f)$$
Lensed wave equals unlensed wave multiplied by magnification factor.
What is in the formula
h_LvariableLensed gravitational wave strain — gravitational wave strain
h_unLvariableUnlensed gravitational wave strain — gravitational wave strain
μvariableMagnification factor — magnification
h_l(f)=f(f)h_{unl}(f) · 2607.21834v1
$$\Delta t = \frac{D_{\Delta t}}{c} \Delta\phi$$
Time delay is proportional to the time delay distance and lensing potential difference.
What is in the formula
\Delta tvariabletime delay between images — time delay second
D_{\Delta t}variabletime delay distance — distance metre
\Delta\phivariabledifference in lensing potential — lensing potential difference
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
δt_{ij}=\frac{d_{δt}}{c}δφ_{ij} · 2606.03634v1
$$\Delta t = \frac{D_{\Delta t}}{c} \Delta\tau$$
Time delay from path difference in lensing.
What is in the formula
Δtvariabletime delay between two images or signals — time delay second
D_Δtvariabletime-delay distance, a combination of angular diameter distances in the lens system — time delay distance metre
Δτvariabledifference in the Fermat potential between two light paths — fermat potential difference
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
δt_{ij}=\frac{d_{δt}}{c}[τ(θ_i)-τ(θ_j)] · 2606.25205
$$\theta_E = \sqrt{\frac{4GM}{c^2} \frac{D_{LS}}{D_L D_S}}$$
Angular radius of gravitational lensing Einstein ring.
What is in the formula
\theta_EvariableAngular radius of the Einstein ring — einstein radius angle radian
GvariableGravitational constant — gravitational constant cubic metre per kilogram per second squared
MvariableMass of the lensing object — mass kilogram
cvariableSpeed of light in vacuum — speed of light metre per second
D_{LS}variableAngular diameter distance between lens and source — distance metre
D_LvariableAngular diameter distance from observer to lens — distance metre
D_SvariableAngular diameter distance from observer to source — distance metre
\sqrt{}operatorTakes the square root of the argument — square root
θ_{\rme}=\sqrt{\kappamπ_{\rmrel}},\quad\kappa=\frac{4g}{c^2{\rmau}}\approx8.144~{\rmmas}m_\odot^{-1} · 2607.04594v1
θ_e=\sqrt{\kappam_lπ_{{rel}}} · 2607.04955

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