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GR:KvFs and geodesics

  • Thread starter binbagsss
  • Start date
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Problem Statement
see below and question attached , parts c to e
Relevant Equations
see below
I am a bit confused with parts c,d,e . In general does ' write down ' the geodesics mean you can leave it in a form ##d/ds( ..) =0## for example here I get 3 equations given by :## \dot x^c =\frac{ k_c z^2}{R^2}## (for## c =t,x,y ##and k is a constant) , and let me use the lagrangian to replace the equation and then I get :## L=K -\frac{\dot{z}^2 R^2}{z^2}## where L will be set## >0 <0 ## or ##=0## depending on whether s-l , t- l etc and K is a sum of the constants ##k_c ^2##

d) the above seems to leave me with a pretty trivial answer to this part which is partly why I'm wondering whether the above is correct or not , since ##L <0 ## and the second term is positive . ( I thought perhaps I may need to look at the ##z## geodesic equation more explicitly and maybe solve for ##z(s) ## or replace it with## \dot{z}## but my ## z ## equation looks non linear and second order ... is the idea to solve for ##z(s) ## here before making a deduction on the conditions for the conserved quantities ?


e) again this question makes me question my part c ) since , this again seems pretty trivial :

I'm wondering what the difference between ' write down explicitly ' and ' write down ' . I would interpret explicit as perhaps solving the Euler Lagrange equations and taking the second derivative wrt ##s## etc . But , using this method for all four equations , the value of ##L## is not specified - I.e which type of geodesic . Whereas if I use three Euler Lagrange equations and replace one of them with the lagrangian ( plugging in the other variables that have been solved for etc - Kvfs and constants ) I can specify the nature of the geodesic via the value of ##L## . But then in this case c) and e) are pretty much the same - the only different is specifying the value of L and computing the derivative wrt ##s## for the constant equations via kvfs I would have wrote down for part c) .


Many thanks , really appreciated.
 

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PAllen

Science Advisor
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I would interpret "write down the geodesic equation for ..." as referring to a differential equation simplified maximally using symmetries and conserved quantities. For the null geodesics, I would think (without doing the exercise) that they expect an explicit formula in the stated coordinates, e.g. that for the exterior Schwarzschild metric in Schwarzschild coordinates, there is a family of spacelike geodesics defined by constant t and angular coordinates, with varying r. I would be expecting that null geodesics simplify to the extent of writing actual equation, rather than diff.eq.
 

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