Gravitational Lensing & the Einstein Radius

Equivalent forms

deflection

\hat{\alpha} = \frac{4GM}{c^2 b}

ring condition

\beta = 0 \;\Rightarrow\; \text{full Einstein ring at } \theta_E

Gravity turns the whole universe into a telescope, weighing invisible matter by the shape of light it deflects.

Symbol	Name	SI unit	Dimension	Range
$\theta_E$	Einstein radius (angle)output	rad	1	—
$M$	lens mass	kg	M	—
$D_L$	observer-lens distance	m	L	—
$D_S$	observer-source distance	m	L	—
$D_{LS}$	lens-source distance	m	L	—

Discovery

Albert Einstein / Fritz Zwicky · 1936

Einstein computed the ring in 1936 but thought it unobservable. Zwicky argued in 1937 that galaxies (far more massive than stars) would lens detectably and could weigh dark matter. The first lensed quasar (twin QSO 0957+561) was found in 1979; today lensing maps dark matter and magnifies the earliest galaxies.

Research status: active

Real-world applications

Weighing galaxy clusters and mapping dark-matter distributions (e.g. the Bullet Cluster).
Microlensing detects exoplanets and dark compact objects by transient brightening.
Cluster lenses magnify the most distant galaxies — JWST's deepest fields exploit this 'cosmic telescope'.
Time delays between lensed quasar images give an independent Hubble-constant measurement.

Common misconceptions

'Lensing bends light like glass via refraction.' — It is spacetime curvature; all wavelengths bend equally (achromatic).
'You need a black hole to lens.' — Any mass lenses; galaxies and clusters give the strong rings.
'The Einstein radius is a physical size.' — It is an angle on the sky set by mass and the distance ratio.

Experimental verification

Eddington's 1919 stellar deflection; twin quasar (1979); thousands of arcs and rings from HST/JWST; cosmic-shear weak-lensing surveys (DES, KiDS, Euclid).

Derivation

GR doubles Newton's light deflection: a ray with impact parameter b bends by $\alpha = 4GM/(c^2\,\mathrm{b})$ .
Lens geometry relates source angle beta, image angle theta, and deflection: $\beta = \theta - (D_LS/D_S) \alpha (\theta )$ .
For a point mass $\alpha (\theta ) = 4GM/(c^2\,\mathrm{D}_L \theta )$ .
Perfect alignment $(\beta = 0)$ gives a ring at $\theta _E^2 = (4GM/c^2)(D_LS/(D_L D_S))$ .
Off-axis sources split into two (point lens) or multiple arcs (extended lens), magnified by the lensing Jacobian.

Limiting cases

\beta = 0

(aligned)⟶ full Einstein ringSymmetric source-lens-observer line.

beta >> theta_E⟶ weak deflection, single faint imageFar off-axis lensing is just slight distortion (weak lensing).

What if…

What if the source is perfectly aligned?

You see a complete Einstein ring of radius theta_E; any misalignment breaks it into arcs or discrete images.

What if the lens has more (dark) mass than its stars suggest?

The ring is larger than visible matter predicts — exactly how lensing reveals dark matter.

1

Einstein radius of a galaxy

Given · Lens

M = 1e11

solar masses (2e41 kg),

D_L = D_LS \sim D_S/2

with

D_S = 2\,\mathrm{Gpc} (6.2e25\,\mathrm{m})

.

Find · Approximate theta_E in arcseconds

Steps

$4GM/c^2 = 4*6.674e-11*2e41/(9e16) \sim 5.9e14\,\mathrm{m}$ .
$D_LS/(D_L D_S) = (3.1e25)/((3.1e25)(6.2e25)) = 1/(6.2e25) m^-1$ .
$\theta _E = \sqrt{5.9e14/6.2e25} = \sqrt{9.5e-12} \sim 3.1e-6\,\mathrm{rad} \sim 0.64$ arcsec.

Result · About 0.6 arcsecond — the typical scale of galaxy-galaxy strong lenses.

General Relativity: Light Bending & Curved Spacetime→Schwarzschild Radius→Physics of the 1919 Eclipse→Shapiro Time Delay (Fourth Test of GR)→