Gravitational Lensing

The core idea

In Newton's picture light travels in straight lines and gravity acts on masses, so it is not obvious that light should bend at all. Einstein's general relativity reframes the question entirely: mass and energy curve spacetime, and light always follows the straightest available path — a geodesic — through that curved geometry. Near a concentration of mass, "straight" is no longer Euclidean-straight, so a passing light ray is deflected. The mass acts as a gravitational lens, and the resulting spacetime curvature reshapes the image of anything behind it.

The lensing analogy is imperfect but useful. Unlike a glass lens, a gravitational lens has no single focal point: rays passing closer to the mass bend more, so the "optical power" increases toward the center. This is why lenses produce rings, arcs, and multiple images rather than a single clean focus.

The deflection angle

For a light ray passing a point mass M at impact parameter b (closest approach distance), in the weak-field limit the GR deflection angle is

α = 4GM / (c² b)

The factor of 4 is the signature of general relativity. A naive Newtonian calculation — treating light as a fast particle falling through the potential — gives exactly half this value, 2GM/(c²b). The extra factor of 2 comes from the curvature of space (not just time): both the time-time and space-space parts of the metric contribute equally. Measuring which prediction nature obeys was the decisive 1919 test.

For a light ray grazing the Sun's limb (M ≈ 2×10³⁰ kg, b = R_☉ ≈ 7×10⁸ m):

α = 4 × (6.67e-11) × (2e30) / ((3e8)² × 7e8)
  ≈ 8.5e-6 rad ≈ 1.75 arcseconds

Prediction	Deflection at Sun's limb
Newtonian (light as corpuscle)	0.87 arcsec
General relativity	1.75 arcsec
Eddington 1919 (Sobral & Príncipe)	≈1.6 arcsec (favored GR)
Modern radio (VLBI)	1.75 arcsec to ~0.01% — GR confirmed

The Einstein ring and lens equation

Consider a source, a lens of mass M, and an observer. Let D_L be the observer–lens distance, D_S the observer–source distance, and D_LS the lens–source distance (these are angular-diameter distances, not simple subtractions in an expanding universe). The geometry connects the true source angle β, the observed image angle θ, and the deflection through the lens equation:

β = θ − α(θ) · (D_LS / D_S)

When the source sits exactly behind the lens (β = 0), symmetry sends light around every side of the lens, and the image becomes a complete ring — the Einstein ring — with angular radius

θ_E = √( (4GM / c²) · (D_LS / (D_L D_S)) )

Real alignments are rarely perfect, so we more often see partial arcs or two/four discrete images (a quad). The Einstein radius sets the characteristic angular scale of all lensing by a given mass, and measuring θ_E directly weighs the lens — independent of how much light it emits.

System	Lens	Einstein radius θ_E
Stellar microlensing in our galaxy	1 M_☉ star	~1 milliarcsecond (unresolvable)
Galaxy-scale strong lens	~10¹¹–10¹² M_☉	~0.5–2 arcseconds
Galaxy cluster	~10¹⁴–10¹⁵ M_☉	~10–50 arcseconds

Magnification and multiple images

Lensing does not create or destroy photons — it conserves surface brightness — but it changes the apparent area a source subtends on the sky, and hence its total observed flux. The magnification is the ratio of image area to source area, given by the inverse Jacobian of the lens mapping:

μ = 1 / |det(∂β/∂θ)|

For a point-mass lens with a source at angular separation u = β/θ_E (in Einstein radii), the two images have total magnification

μ_total = (u² + 2) / (u √(u² + 4))

which diverges as u → 0 (perfect alignment). Where det(∂β/∂θ) = 0, magnification formally goes to infinity along curves called caustics (on the source plane) and critical curves (on the image plane). Sources crossing a caustic flare dramatically — the basis of the most spectacular microlensing events and of cluster "cosmic telescopes" that have magnified individual stars at redshift z > 1 by factors of thousands (e.g. the star Earendel).

Strong, weak, and microlensing

Regime	What you see	Typical lens / use
Strong lensing	Multiple images, arcs, full Einstein rings	Massive galaxies & clusters; mass measurement, H₀, cosmic telescopes
Weak lensing	~1% coherent shear of galaxy shapes — only statistical	Large-scale structure; dark matter maps; dark energy surveys (Euclid, LSST)
Microlensing	Time-varying brightening, no resolvable image	Single stars/planets; exoplanets, MACHOs, free-floating planets

The dividing line is set by how the source position compares to the Einstein radius. Inside θ_E you get strong, multiply-imaged lensing; far outside it you get only the faint statistical stretching of weak lensing; and when the images are too small to resolve (stellar masses at galactic distances) you are left with the photometric signature of microlensing.

Mapping dark matter

Because lensing responds to total mass — it only cares how strongly spacetime is curved, not whether the mass shines — it is the cleanest way to find matter we cannot see. Weak-lensing surveys measure the slight, coherent tangential alignment of millions of faint background galaxies and invert it into a projected mass map. The famous Bullet Cluster (1E 0657-56) clinched the case: two clusters collided, the hot X-ray gas (most of the ordinary matter) was slowed and left in the middle, but the lensing mass sailed ahead with the galaxies. The mass and the visible baryons were spatially offset, exactly as expected if most of the mass is collisionless dark matter.

Where lensing shows up

• Weighing galaxies and clusters. θ_E gives the enclosed mass directly, including dark matter, with no assumptions about light-to-mass ratio.
• Dark matter mapping. Weak-lensing shear reconstructs the cosmic web; the Bullet Cluster separates dark matter from gas.
• Cosmic telescopes. Clusters magnify background galaxies by 10–100×, letting JWST and Hubble study the earliest galaxies and even individual high-redshift stars.
• Exoplanets via microlensing. A planet around the lens star adds a brief spike to the light curve — sensitive to cold, distant, and free-floating planets other methods miss.
• Measuring H₀. In a multiply-imaged variable quasar, light paths have different lengths; the time delay between flickers (days to years) yields the Hubble constant (time-delay cosmography).
• Testing general relativity. The 1.75″ Solar deflection, now measured to ~0.01% by radio interferometry, remains a precision test of GR in the weak field.

Common misconceptions

• "Light has mass, so gravity pulls it." Photons are massless. They bend because they follow geodesics of curved spacetime — and the curvature of space is exactly what doubles the Newtonian estimate.
• "A gravitational lens has a focal point." It does not. Bending increases toward the center, so lenses smear images into arcs and rings rather than focusing them to a point.
• "Lensing makes things brighter by adding light." No — surface brightness is conserved. The apparent solid angle grows, so total flux increases, but no photons are created.
• "Einstein rings prove the source is round." The ring traces the lens geometry, not the source. Even a point source produces a ring when perfectly aligned behind a circular lens.
• "You need a black hole to lens light." Any mass lenses light — the Sun, ordinary galaxies, and clusters do it routinely. Black holes are just an extreme case.
• "Cosmological lensing is the same as cosmological redshift." Redshift stretches wavelengths via expansion; lensing bends and magnifies images via spacetime curvature. Both occur over cosmic distances but are distinct effects.