🌌 Gravitational Lensing — Einstein Ring

Massive objects curve spacetime, bending light paths around them. When a background source, massive lens, and observer are perfectly aligned, light travels around all sides of the lens forming a complete Einstein ring. For off-axis sources, multiple images and arcs appear. Drag the source or lens to explore lensing geometry. The background field is computed pixel-by-pixel using the point-mass lens equation: β = θ − θE² / θ, where θE is the Einstein radius.

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Drag on the canvas to move the black hole • touch-drag on mobile

Parameters

Display

Einstein / photon ring emphasis

Stats

Mass1.0 M
Shadow radius ≈—
Lens positioncenter
Light bending:
deflection ∝ Rs/b

Schwarzschild radius:
Rs = 2GM/c²

Photon sphere:
r = 1.5 Rs

From Prediction to Discovery

Einstein predicted light deflection by the Sun in 1915; the 1919 Eddington expedition confirmed it during a solar eclipse. The first gravitationally lensed quasar (double quasar Q0957+561) was discovered in 1979. Today gravitational lensing is one of the most powerful tools in cosmology — mapping the distribution of dark matter in galaxy clusters (the "Bullet Cluster"), measuring the Hubble constant independently, and finding exoplanets via microlensing. In the SIS model (singular isothermal sphere) the lens equation becomes β = θ − θE·sgn(θ), producing two images of equal brightness flanking the lens.

About Gravitational Lensing

Gravitational lensing is the bending of light by massive objects, a direct consequence of general relativity. Einstein's field equations predict that mass curves spacetime, and light follows geodesics (shortest paths) in this curved spacetime—appearing to travel in a straight line locally but following a curved path from a distant observer's perspective. The bending angle for a light ray passing a mass M at closest approach distance b is α = 4GM/(c²b), exactly twice the Newtonian prediction, a discrepancy confirmed by Eddington's 1919 solar eclipse observations that provided the first empirical test of general relativity.

Three regimes of gravitational lensing are distinguished by the source-lens-observer geometry. Strong lensing occurs when source, lens, and observer are nearly perfectly aligned, producing Einstein rings, arcs, or multiple images of a single background galaxy. Weak lensing causes subtle distortions of background galaxy shapes, measurable only statistically over thousands of galaxies, and is used to map dark matter distributions. Microlensing occurs when a compact object (star, planet, black hole) passes in front of a background star, causing a temporary brightness amplification without producing resolved images—used to detect exoplanets and dark compact objects.

This simulator places a massive lens between source and observer and computes the deflected light ray paths using the gravitational lens equation, rendering the distorted image of a background source. You can adjust lens mass, source position, and impact parameter to observe the transition from weak distortion to strong arcing to Einstein ring formation, and explore how lensing maps to the lens mass distribution—demonstrating the techniques cosmologists use to weigh galaxy clusters and detect dark matter.

Frequently Asked Questions

What is an Einstein ring and when does it form?

An Einstein ring is a complete circular image of a background source formed when the source, gravitational lens, and observer are in perfect alignment. Light from the source passes around all sides of the lens equally, arriving at the observer from all directions around the lens—forming a ring of angular radius θ_E = √(4GM·D_LS/(c²·D_L·D_S)), the Einstein radius. Perfect alignment is rare; more commonly, near-alignment produces partial arcs. Einstein rings and arcs have been observed in hundreds of galaxy-galaxy and cluster-galaxy lens systems by the Hubble Space Telescope.

How is gravitational lensing used to weigh galaxy clusters?

Massive galaxy clusters act as gravitational lenses on background galaxies. The Einstein radius and degree of arc distortion are related to the projected mass distribution of the cluster by the lens equation. Strong lensing constrains the mass within the Einstein radius precisely. Weak lensing maps the mass distribution out to large radii through statistical alignment of background galaxy shapes. Combining strong and weak lensing data produces detailed mass maps of clusters, revealing that total cluster mass (including dark matter) typically exceeds the visible stellar mass by 50–100×.

What is microlensing and how is it used to detect exoplanets?

Gravitational microlensing occurs when a foreground star passes in front of a more distant background star. The lens star's gravity focuses background star light, causing a temporary symmetric brightness amplification. If the lens star hosts a planet, the planet's additional gravity creates a brief anomaly superimposed on the main lensing light curve. By fitting the anomaly's shape and duration with planet-star mass ratio and separation models, planet masses and orbital separations can be inferred. Microlensing is particularly sensitive to planets near the Einstein ring radius (a few AU), complementing transit and radial velocity methods.

How did gravitational lensing provide evidence for dark matter in the Bullet Cluster?

The Bullet Cluster (1E 0657-558) is two galaxy clusters that passed through each other ~100 million years ago. X-ray observations show the hot intracluster gas concentrated between the two clusters—gas was slowed by electromagnetic interactions during collision while the galaxies (collisionless stars) passed through. Gravitational lensing mass maps show the mass peaks coinciding with the galaxies, not the gas. Since the mass follows the collisionless component, it cannot be ordinary matter—it must be dark matter that, like the galaxies, passed through without interacting electromagnetically. The Bullet Cluster is considered one of the strongest direct evidence for particle dark matter.

How is gravitational lensing used in cosmology?

Gravitational lensing is a powerful cosmological tool because it responds to all matter—dark and baryonic—without assumptions about its physical state. Cosmic shear (weak lensing by the large-scale structure of the universe) measures the growth of structure and the geometry of spacetime, constraining dark energy density and equation of state. Lensing magnification boosts background galaxy brightness, enabling observations of galaxies at redshifts otherwise too faint to detect. Cross-correlating lensing maps with galaxy surveys and CMB measurements places independent constraints on the Hubble constant and neutrino mass.

What does the Black-hole mass slider actually change in the rendering?

The Black-hole mass slider (0.2 to 3 solar masses in the shader's units) directly scales the Schwarzschild-like radius uRs = 0.10 × mass, which in turn sets the size of the rendered event-horizon shadow, the photon-sphere ring, and the Einstein-ring radius (uRs × 1.9). Every one of these visual features grows or shrinks together as you drag the slider, since they all derive from the same mass-dependent scale.

How is the deflection angle actually computed in the shader?

For each rendered pixel the shader computes the impact parameter b (distance from the lens centre) and bends the sampled background direction toward the lens by deflect = uRs / b, a Schwarzschild-like 1/b falloff — the same functional form as the real deflection angle α = 4GM/(c²b). The result is clamped to a maximum of 2.5 to prevent a singular blow-up exactly at the lens centre, where the true formula would otherwise diverge.

What does the Star density slider control, and why does it matter for lensing?

Star density (10-100%) sets how many background stars are scattered across the field behind the lens. A higher density means more stars pass close enough to the lens's Einstein radius at any given moment, so you are statistically more likely to catch one being stretched into an arc or momentarily brightened — this mirrors how real microlensing and cluster-lensing surveys need large numbers of background sources to catch lensing events.

Why does dragging the black hole on the canvas change which stars get lensed?

Dragging repositions the lens centre used to compute the impact parameter b for every point in the field. Because the deflection strength uRs/b depends only on distance from that centre, moving the lens brings a completely different set of background stars within the critical Einstein-radius band, which is why the ring and arcs appear to "follow" the black hole as you drag it across the star field.

Why did Eddington's 1919 eclipse expedition matter so much for confirming this effect?

Einstein's general relativity predicted light bending by the Sun's mass at exactly twice the value predicted by Newtonian gravity alone. Stars are normally invisible near the Sun in daylight, so Arthur Eddington's team had to photograph a total solar eclipse in 1919 to measure the tiny apparent shift in star positions near the solar limb. The measured deflection matched Einstein's relativistic prediction rather than the Newtonian one, providing the first direct experimental confirmation of general relativity and making Einstein internationally famous almost overnight.