Precession of relativistic orbit in pure inverse-square force

[RylonMcknz]

PROBLEM:

Show that Special Relativity predicts a precession of π(GMm/cl)² radians per orbit for any elliptic orbit under a pure inverse-square force.

where G is gravitational constant, M is mass of larger body, m is mass of smaller orbiting body, c is speed of light and l is angular momentum.​

HINTS:

Differential equation for Newtonian inverse-square force orbit: d²u/dθ² + u = mk/l² (Eq.1), where u = 1/r, k = GMm.

To make this relativistic: mk→ϒmk, where ϒ = 1/√(1 - (v²/c²)).

Also E = ϒmc² + U.

Then right side of (Eq. 1) becomes ⇒ϒmk/l² = (E - U)k/c²l² = (E + ku)k/c²l² (Eq. 2).

Precession causes the following change of phase angle (in radians/orbit):

Δθ - Δθ' = 2π/ω - 2π = 2π(1/ω - 1) (Eq. 3).​

ATTEMPT:

(Eq. 1) & (Eq. 2) ⇒ d²u/dθ² + u = (E + ku)k/c²l²

⇒ d²u/dθ² + u - k²u/c²l² = Ek/c²l²

⇒ d²u/dθ² + (1 - k²/c²l²)u = Ek/c²l²,

where ω² = (1 - k²/c²l²) (Eq. 4),
and let A = Ek/c²l².

Then substitute y = u - A/ω².

Then we have d²y/dθ² + ω²[y + A/ω²] - A = 0

⇒ d²y/dθ² + ω²y = 0

Solution is then y = y₀cos(ωθ - θ₀).

I then try substituting (Eq. 4) into (Eq. 3) to find the resulting precession in radians per orbit, but this doesn't not give the desired result. This is where I'm stuck, not sure what I'm doing wrong. Thank you for your considerations.​

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?