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Christoffel symbols and Geodesic equations.

  1. Apr 25, 2009 #1

    trv

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    1. The problem statement, all variables and given/known data

    (a) Consider a 2-dimensional manifold M with the following line element

    ds2=dy2+(1/z2)dz2

    For which values of z is this line element well defined.

    (b) Find the non-vanishing Christoffel symbols

    (c) Obtain the geodesic equations parameterised by l.

    (d) Solve the geodesic equations and suggest an improved coordinate system. What is the metric in the new coordinates? What lines describ the geodesic geometrically?

    (e) What can you say about the Riemann curvature tensor, the Ricci tensor and the Ricci scalar of this manifold.


    2. Relevant equations



    3. The attempt at a solution

    (a) The line element is well defined for all values of y and z other then z=0

    (b) gzz,z= -2/z3

    The only non vanishing christoffel symbol is,

    Czzz= -1/z

    (c) The geodesic equations are given by,

    d2z/(dl2)-(1/z)(dz/dl2)=0

    (d)
    d2z/dl2=(1/z)(dz/dl2)

    Stuck here.
     
    Last edited: Apr 25, 2009
  2. jcsd
  3. Apr 25, 2009 #2

    Dick

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    To solve the differential equation in (d), try the change of variables z=e^v. Think about applying this change of variables to the original coordinate system.
     
  4. Apr 25, 2009 #3

    trv

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    Hey Dick thanks for the hint. Haven't managed to get anywhere with it yet, but will give it a go again tomorrow.

    If I may ask you another question however,....is there a general method for solving the differential equations/geodesic equations? If so it would be really useful if you guide me to an online resource for the same.
     
  5. Apr 26, 2009 #4

    Dick

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    For some forms of differential equations there are methods specific to that form. But there is no one method. There's tons of stuff online. Just google 'solving differential equations' and pick your favorite.
     
  6. May 4, 2009 #5

    trv

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    Hi dick, could you possibly have meant e^l rather than e^v?
     
  7. May 4, 2009 #6

    Dick

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    No, I meant substitute z(l)=e^(v(l)). What does the differential equation for v(l) look like?
     
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