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Difficult general relativity problem - Weyl's axisymmetric solution

  1. Apr 23, 2013 #1
    I'm currently doing a course on general relativity, and I'm struggling with the following exercise - I would greatly appreciate the help anyone might offer. Whilst not technically homework (not assessed or in a tutorial set) it is from a collection of difficult questions from a recommended resource. It is as follows:

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

    Weyl's solution to the field equations in a vacuum has that

    $$ ds^{2} = e^{2U}dt^{2} -e^{-2U}(r^{2}d\phi ^{2} + e^{2V}(dr^{2} +dz^2)) $$ $$ where \; \; U=U(r,z),\; V=V(r,z); $$

    and the field equations reduce to Laplace's equation in cylindrical coordinates, that being

    $$ \nabla^{2}U = U_{,rr} + r^{-1}U_{,r} + U_{,zz}=0, $$

    as given U we can find V by integration:

    $$ V_{,r}=r((U_{,r})^{2}-(U_{,z})^2)\;, \; \; \; V_{,z}=2rU_{,r}U_{,z} \;. $$

    This suggests a way of generating solutions to the Einstein Fields equations: Laplace's
    equation is linear, so just add arbitrary (axially symmetric) solutions of it to get U
    and then find V .


    (a) Why is the Weyl metric static?


    (b) Explain why adding a constant to either (or both) U and V is geometrically
    (and hence physically) irrelevant.


    (c) If $$U\equiv 0 $$ we clearly have flat space. Show that if $$U = V$$ we also have flat space. [Hint: perhaps we might change coordinates (something about hyperbolic trig functions and rapidity?)] This suggests generating GR solutions from Newtonian ones will not be as easy as it seems.


    (d)The simplest non-trivial solution to Laplace's equation is the (spherically symmetric) potential of a point particle: $$U=-\frac{m}{\sqrt{r^{2}+z^{2}}}$$. Show this is a solution to $$\nabla^{2}U=0$$, find the corresponding V and explain why the metric is asymptotically flat.


    (e) The solution is part (d) is known as the Curzon-Chazy solution. Prove it is not Schwarzschild and so not spherically symmetric. (Thus once again demonstrating that the Newtonian solution and the Einsteinian solutions are not simply related.)


    *Nota bene c=1 here.




    2. Relevant equations
    See question statement


    3. The attempt at a solution
    I've struggled a bit, there's some obvious substitution work in here verifying things are solutions but I'm not very confident with the rest of the material.
     
  2. jcsd
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