廣義相對論: 水星近日點徑動

by allenlu2007

Reference:  Schutz 

Sweetser: “24 Steps to the Precession of The Perihelion Of Mercury”

 

前文提到廣義相對論在弱重力和 slow speed (v << 1).

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先說結論

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∆Φ = advance / revolution = 42.9” per century.

 

24 Steps to get the result

1. Start with a spherically symmetric, non-rotating, uncharged mass. 

The spherical Schwarzchild solution (no approximation, compared with 8.50)

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dx^2 + dy^2 + dz^2 = (dR^2 + R^2 dθ^2 + R^2 sinθ^2 dΦ^2)/c^2 

dt 就是 proper time.  dτ = ds   

𝝫 = – GM/R is the static potential function.

 

2-4. 假設 θ=90 度。就是大家都在黃道面上。 dϴ=0

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再來就是用 Killing vectors (Gauge transformation?) to simplify the above equation.

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要把 dt 和 dΦ 消除!!!

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Ignore the rest steps:

 

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