General Relativity: An Introduction for Physicists · Problem 9.20

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · apparent-size-massive-sphere

Problem

All massive objects look larger than they really are. Show that a light ray grazing the surface of a massive sphere of coordinate radius

r > 3GM/c^2

will arrive at infinity with impact parameter

b = r\left(\frac{r}{r - 2GM/c^2}\right)^{1/2}.

Hence show that the apparent diameter of the Sun (

M_\odot = 2\times 10^{30}\,\text{kg}

R_\odot = 7\times 10^8\,\text{m}

) exceeds the coordinate diameter by nearly 3 km.