Deflection of light to second order: A tool for illustrating principles of general relativity

We calculate the deflection of light by a spherically symmetric body in general relativity, to second order in the quantity GM/dc2, where M is the mass of the body and d is a measure of the distance of closest approach of the ray. Using three different coordinate systems for the Schwarzschild metric we show that the answers for the deflection, while the same at order GM/dc2, differ at order (GM/dc2)2. We demonstrate that all three expressions are really the same by expressing them in terms of measurable, coordinate-independent quantities. These results provide concrete illustrations of the meaning of coordinates and coordinate invariance, which may be useful in teaching general relativity.