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In the thin-lens approximation for gravitational lensing, consider a ray crossing the lens plane at transverse distance $b$ above the aligned source-observer axis. With the observer at axial position $0$, the black hole at $L$ and the source at $D$, the segment joining the lens plane to the source has slope $-b/(D-L)$, while the segment joining the observer to the lens plane has slope $b/L$. The observer sees the ray at apparent angle $\theta \simeq b/L$ to the axis. When following the photon toward the observer, the two small angles to the axis have magnitudes $b/(D-L)$ and $b/L$. The physical bend is their sum:Substitute the Schwarzschild light deflection, $\delta \simeq 4GM/(b{c}^2)$, and use $b\simeq L\theta$:Every azimuth around the axis is equivalent by rotational symmetry. Isotropic emission therefore supplies rays at all these azimuths, and the ideal image is an Einstein ring centered on the black hole, with Einstein angular radius $\theta$ and angular diameter $2\theta$. A finite source size or imperfect alignment broadens or breaks the ideal ring.

The stated expression uses small angles, distant endpoints and weak-field approximation $b\gg 2GM/c^2$, with ordinary Euclidean distances. For cosmological gravitational lensing, use angular diameter distances $D_l,D_s,D_{ls}$ in ${\theta }^2=4GM{D}_{ls}/(c^2{D}_l{D}_s)$; generally $D_{ls}\ne D_s-D_l$.

A galaxy cluster deflects background light through its projected gravitational field. In the thin-lens approximation for gravitational lensing, $\mathbf{\beta }=\mathbf{\theta }-\mathbf{\alpha }(\mathbf{\theta })$ maps image to source angle. Its surface mass density enters throughHere $\mu$ is signed lensing magnification; observed brightness uses $|\mu |$. The $D$'s are angular diameter distances to lens, source and between them; ${\gamma }_s$ denotes lensing shear, not a Lorentz factor. strong gravitational lensing creates arcs and multiple images near critical curves. For an axisymmetric lens, a tangential Einstein ring satisfies mean enclosed convergence one, givingThis is projected enclosed mass, not a general three-dimensional mass profile inferred from a single arc.

weak gravitational lensing uses coherent distortions of many background galaxies. Their intrinsic orientations average approximately to zero, allowing an estimate of reduced shear $g={\gamma }_s/(1-\kappa )$ and a projected mass map. The method traces all gravitating material without assuming hydrostatic equilibrium, which is valuable in merging clusters. Accurate work requires correction of the point spread function, calibrated shape measurement, source redshifts, foreground-member rejection and selection effects. Intrinsic shape noise, unrelated line-of-sight structures and lens-model degeneracies remain. The mass-sheet degeneracy leaves reduced shear unchanged under $\kappa \mapsto \lambda \kappa +(1-\lambda )$ and ${\gamma }_s\mapsto \lambda {\gamma }_s$, so absolute mass normalization needs additional information.

In the Bullet Cluster, weak gravitational lensing places the main projected mass peaks near the collisionless galaxies, displaced from the X-ray-emitting gas that contains most observed baryon mass. Comparing the lensing, optical and X-ray maps supports a dominant weakly collisional dark component under the usual gravity model; lensing alone does not identify its particle species. Cluster lensing measures projected gravitational mass independently of gas temperature or dynamical equilibrium.^[ref]