Past exam of the relativity course of the University of Cambridge / 2022 / Section A

Use angular size under a collinear relativistic boost. For a small distant source, the two angular diameters have ratio $e^{|\eta |}$, where $\eta$ is the relative rapidity. Thus $e^{|\eta |}=2$, and the relative velocity satisfiesThe speed is therefore $0.6c$. The observers are momentarily coincident, so this is relativistic aberration, rather than a difference in their distance from the planet. The observer moving faster towards the planet measures the smaller angular diameter under the source-direction aberration convention.

The metric tensor and its inverse are diagonal: $g_{xx}=1$, $g_{yy}=f^2$, $g^{xx}=1$, $g^{yy}=f^{-2}$ on a patch where $f\ne 0$. Only ${\partial }_x{g}_{yy}=2f{f}^{\prime }$ is nonzero. The Christoffel symbol formula therefore giveswith all other components zero. Using the stated Riemann curvature tensor convention,so $\boxed{R^x{}_{yxy}=-f{f}^{\prime \prime }}$. This is the curvature of a two-dimensional warped metric; its Gaussian curvature is $-f^{\prime \prime }/f$.

For a grazing null geodesic, take the impact parameter as $b\simeq R=6.0\times 10^6\mathrm{m}$. The leading Schwarzschild light deflection isWith the gravitational constant $G\simeq 6.6743\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and speed of light $c\simeq 2.998\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$,Here $GM/(R{c}^2)\simeq 2.5\times 10^{-4}\ll 1$, so the weak-field approximation is appropriate. A ray passing at a larger impact parameter has a correspondingly smaller deflection.