Re: Challenge for Paul; Probe that with Mercury ds^2>0 and the solution is spacelike

Some help here:
Starting with the line element ds in the Schwarzschild metric
(describing spacetime around a massive object like the Sun):
ds² = -(1 - 2GM/c² r) c² dt² + 1/(1 - 2GM/c² r) dr² + r² dɸ²
You have to prove that ds² > 0  and that
ds² ≈  1/(1 - 2GM/c² r) dr² + r² dɸ²
or, using the equation of the ellipse
r(ɸ) = a (1 - e²)/(1 + e cos ɸ)
that
ds² ≈ 1/(1-2GM/c²(1+e cos ɸ)/[a(1-e²)]) [a(1-e²) sin ɸ]²/(1+e cos ɸ)² +
[a(1-e²)/(1+e cos ɸ)]² dɸ
and that
S = ∫ds      [between 0 and 2π]
S ≈ 2πa + ΔS  , being ΔS ≈ k Δθ
Δθ = 6πGM/[c²a(1−e²)] ≈ 0.10367 arcseconds/orbit (43"/century)
This is a novel approach to the problem, using your beloved
Schwarzschild metric.
Enjoy, Paul, and show us what you are made with.