Earth's orbit has mean orbital speed m s at mean distance m from the Sun ( m s );…

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

Earth's orbit has mean orbital speed

v = 2.978\times10^4

m s

^{-1}

at mean distance

r = 1.496\times10^{11}

m from the Sun (

GM_\odot = 1.327\times10^{20}

^3

^{-2}

); treat the orbit as circular for this problem. ((a)) Starting from the timelike radial equation

c^2E^2 = (1-r_s/r)(c^2+L^2/r^2)

(with

\dot r=0

for a circular orbit), and using the Newtonian circular-orbit relation

L^2/r^2 = v^2 = GM/r

, expand

E

to first order in

GM/rc^2

and

v^2/c^2

and show it reduces to

E \approx 1 - GM/2rc^2

. ((b)) Evaluate

E-1

numerically for Earth's orbit. ((c)) The weak-field metric has

g_{00} = -(1+2\Phi/c^2)

with

\Phi=-GM/r

, the same

g_{00}

used here to leading order. Using the equivalence-principle-test logic that any metric theory reproducing the measured redshift shares this

g_{00}

, and Birkhoff's theorem that the Schwarzschild metric is the unique vacuum solution of the full field equations, explain why matching

E

to the Newtonian energy in this way tests only the weak-field limit shared by many metric theories, and not the field equations themselves, in contrast with the perihelion advance.

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