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Timelike geodesic

  1. Jan 7, 2017 #1
    1. The problem statement, all variables and given/known data

    The question is to find ##A## and ##B## such that the specified curve (we are given a certain parameterisation , see below) is a timelike geodesic , where we have ##|s| < 1 ##

    I am just stuck on the last bit really.

    So since the geodesic is affinely paramterised ##dL/ds=0## and so I can set ##L=constant##, ##L ## the Lagrangian of a freely-falling particle.

    Let ## L ## be this constant.

    And with the specified metric and parameterised curve, which are all given to us, this gives:

    ##B^2(\frac{A^2-s^2}{1-s^2}) = L ##

    This is all fine.

    MY QUESTION IS...



    2. Relevant equations


    see above

    3. The attempt at a solution

    MY QUESTION IS...

    From this I conclude that (since a null curve is given by ##L=0##, a space-like by ## L < 0 ## and a time-like by ##L>0##, since the metric signature in the question is ( +, - ) ) that we require ##|A|<1## since we have ##|s| < 1 ## , and ##B\neq 0 ##, however the solution gives:

    we need ##A=\pm 1 ## and ##B\neq 0 ##.
    I don't understand where ##A=\pm 1 ## comes from, I thought we just need it such that ##L > 0## and ##A=\pm 1 ## does this

    Many thanks in advance
     
  2. jcsd
  3. Jan 7, 2017 #2

    Orodruin

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    You have not given us the full problem. You have forgotten to specify the given curve and metric.
     
  4. Jan 9, 2017 #3
    I know, Im pretty sure theyre not needed, it is just the final conclusion described above that I am stuck on. but I will post them now.

    curve ## t= A tanh^{-1} s ## , ##x=B(1-s^{2})^{1/2}##
    metric : ##ds^2=x^2 dt^2 - dx^2 ##
     
    Last edited: Jan 9, 2017
  5. Jan 9, 2017 #4

    Orodruin

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    Did you try inserting the curve into the geodesic equations for the given metric?
     
  6. Jan 11, 2017 #5
    the question was completed using the euler-lagrange equations. One replaced with setting ##L## to a constant as above, the other the E-L equation for ##t## which gave no new constraints on ##A## and ##B## .

    It is just the conclusion as I say in OP that I am on stuck on.
     
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