The radial equation for a null geodesic in Schwarzschild spacetime

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · Schwarzschild spacetime and the tests of general relativity

The Geometry of Gravity, Ch Orbits and light rays in Schwarzschild spacetime

Problem

The radial equation for a null geodesic in Schwarzschild spacetime, written with u=1/ru=1/r, is u′′+u=3GMc2u2,u''+u=\frac{3GM}{c^2}u^2, primes denoting d/dϕd/d\phi. ((a)) Solve this equation to first order in GM/bc2GM/bc^2 about the straight-line zeroth-order solution u0=sin⁡ϕ/bu_0=\sin\phi/b (bb the impact parameter), and show that the total deflection of the ray from a straight line is Δϕ=4GM/(bc2)\Delta\phi=4GM/(bc^2). ((b)) Sirius B, the white dwarf companion of Sirius A, has mass M=1.02 M⊙M=1.02\,M_\odot and radius R⋆=5850R_\star=5850 km. Using GM⊙/c2=1.48GM_\odot/c^2=1.48 km, find the deflection, in arcseconds, of a light ray that grazes its surface (b=R⋆b=R_\star), and compare it with the 1.751.75 arcsecond deflection at the solar limb.

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