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

On Thu, 13 Feb 2025 1:06:12 +0000, rhertz wrote:

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

I add one more challenge with your beloved Schwarzschild metric.
In its full expression, it's known as being
ds² = -(1 - 2GM/c² r) c² dt² + 1/(1 - 2GM/c² r) dr² + r² (dɸ² + sin² ɸ
dϴ²)
Can you derive, for GPS satellites, the alleged shift in frequency:
Δf/f ≈ GM/c²  [1/r_E − 1/(r_E + h)]− GM/[2 c² (r_E + h)]
which contains gravitational and relativistic time dilation terms,
assuming that orbits are circular?
Can you use it to binary stars as well? They have distances r1 and r2
between them, with masses M1 and M2. What are the limitations applying
Schwarzchild here, like gravitational waves?
I'm proposing these problems to you, so you can show off the shit that
you accumulated in more than 30 years defending relativity. Let us know
if the time you wasted paid off, or if you can't work beyond 1905
Lorentz shit and rockets going forward and back at near speed c.