Past exam of the general relativity course of the University of Oxford / 2022 / 3 / d

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 $D_{ls}\ne D_s-D_l$.