Sunday, October 12, 2014

homework and exercises - Textbook disagreement on geodesic deviation on a 2-sphere


Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant.



I'm trying to use the geodesic equation


D2ξμDλ2+Rμμβαγξαdxβdλdxγdλ=0

to find the geodesic deviation on the surface of a unit 2-sphere. My question is are the following calculations correct?


I start with the Riemann tensor components: Rθθϕθϕ=sin2θ, Rθθϕϕθ=sin2θ, Rϕθθθϕ=1, Rϕθθϕθ=1.


Let uσdxσdλ. Then expand out the Riemann components to get: (Rμμθθθuθuθ+Rμμθθϕuθuϕ+Rμμϕθθuϕuθ+Rμμϕθϕuϕuϕ)ξθ+(Rμμθϕθuθuθ+Rμμθϕϕuθuϕ+Rμμϕϕθuϕuθ+Rμμϕϕϕuϕuϕ)ξϕ.


Set μ=θ to give D2ξθDλ2+(Rθμθθθuθuθ+Rθμθθϕuθuϕ+Rθμϕθθuϕuθ+Rθμϕθϕuϕuϕ)ξθ+(Rθμθϕθuθuθ+Rθμθϕϕuθuϕ+Rθμϕϕθuϕuθ+Rθμϕϕϕuϕuϕ)ξϕ=0


D2ξθDλ2+(sin2θ)(uϕuϕ)ξθ(sin2θ)(uϕuθ)ξϕ=0

D2ξθDλ2=(sin2θ)(uϕuθ)ξϕ(sin2θ)(uϕuϕ)ξθ


Set μ=ϕ to give D2ξϕDλ2+(Rϕμθθθuθuθ+Rϕμθθϕuθuϕ+Rϕμϕθθuϕuθ+Rϕμϕθϕuϕuϕ)ξθ+(Rϕμθϕθuθuθ+Rϕμθϕϕuθuϕ+Rϕμϕϕθuϕuθ+Rϕμϕϕϕuϕuϕ)ξϕ=0


D2ξϕDλ2+(1)(uθuϕ)ξθ+(1)(uθuθ)ξϕ=0


D2ξϕDλ2ξθ(uθuϕ)+ξϕ(uθuθ)=0

D2ξϕDλ2=ξθ(uθuϕ)ξϕ(uθuθ).


My reason for asking is that Misner et al give the geodesic deviation equation (the one I use in my question) as D2ξμDλ2+Rμμβαγξαdxβdλdxγdλ=0,

whilst in another of my textbooks it's (spot the difference!) D2ξμDλ2+Rμμαβγξαdxβdλdxγdλ=0,
with α being the first lower index on the Riemann tensor and the upper index on the right-hand-side ξ. When I use the second equation in my 2-sphere calculations everything seems to come out to zero, which doesn't seem right. So I'm wondering if the second equation is wrong.





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