In Schwarzschild spacetime of mass

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

In Schwarzschild spacetime of mass MM, with Schwarzschild radius rs=2GM/c2r_s = 2GM/c^2, a freely falling test particle of nonzero rest mass moves on a geodesic that, by spherical symmetry, can be taken to lie in the equatorial plane θ=π/2\theta = \pi/2, where ds2=−(1−rs/r)c2dt2+dr2/(1−rs/r)+r2dϕ2ds^2 = -(1-r_s/r)c^2dt^2 + dr^2/(1-r_s/r) + r^2d\phi^2. ((a)) Using the Killing vectors ∂t\partial_t and ∂ϕ\partial_\phi of this metric, derive the conserved quantities E=(1−rs/r) dt/dτE = (1-r_s/r)\,dt/d\tau (energy per unit rest energy) and L=r2 dϕ/dτL = r^2\,d\phi/d\tau (angular momentum per unit rest mass). ((b)) Using the four-velocity normalisation gμνuμuν=−c2g_{\mu\nu}u^\mu u^\nu = -c^2, eliminate dt/dτdt/d\tau and dϕ/dτd\phi/d\tau to obtain the radial equation (dr/dτ)2=c2E2−(1−rs/r)(c2+L2/r2)(dr/d\tau)^2 = c^2E^2 - (1-r_s/r)(c^2 + L^2/r^2). ((c)) A GPS satellite orbits the Earth at areal radius r=2.656×107r = 2.656\times10^7 m. Using GM⊕/c2=4.44GM_\oplus/c^2 = 4.44 mm, evaluate the ratio rs/rr_s/r that measures the relative size of the term the Schwarzschild radial equation adds beyond the Newtonian one, and say whether this correction to the shape of the orbit is significant at the metre level.

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