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Parameterize a geodesic using one of the coordinates

  1. Oct 13, 2013 #1
    I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the one I found, and shall present bellow, has actually a simplification that my problem doesn't, so please ignore that simplification (I shall indicate it when it comes up).

    Given a Riemannian manifold with metric
    $$\tag{1} ds^{2}=\frac{dr^2}{a+r^2}+r^2\left(d \theta^2+\sin^2(\theta) \, d\phi^{2}\right),$$
    consider a curve [itex]c(\lambda)=(r(\lambda),\phi(\lambda),\pi/2)[/itex] whose tangent vector is [itex]c'(\lambda)=(r'(\lambda),\phi'(\lambda),0)[/itex]. The geodesic equations (if I didn't mess up) are given by:
    \begin{cases}
    \tag{2} r''-\frac{r'^{2}}{2(a+r)}-r(a+r)\phi'^{2}=0,\\
    \phi''+2\frac{r'}{r}\phi'=0.
    \end{cases}

    The second equation of Eq.[itex](2)[/itex] can be integrated, such that:
    $$\tag{3} \phi'(\lambda)=\frac{C}{r^{2}(\lambda)},$$
    where [itex]C[/itex] is a constant of integration.

    Now I introduce a new equation:
    $$\tag{4}r'^{2}\left(\frac{1}{a+r^{2}}\right)+\phi'^{2}r^{2}=1,$$
    which is basically impose unit speed. Substituting Eq.[itex](3)[/itex] in Eq.[itex](4)[/itex] we have that:
    $$\tag{5} r'=\sqrt{(a+r^{2})\left(1-\frac{C^{2}}{r^{2}}\right)}.$$
    (Here is the difference from my case since Eq.[itex](5)[/itex] can be integrated analytically and in my case, the congener equation can't).

    Dividing Eq.[itex](3)[/itex] by Eq.[itex](5)[/itex] we have that
    $$\tag{6} \frac{d\phi}{dr}=\frac{C}{r^{2}(\lambda)}\frac{1}{\sqrt{(a+r^{2})\left(1-\frac{C^{2}}{r^{2}}\right)}}.$$

    So my question is:Does Eq.[itex](6)[/itex] allow me to write the curve [itex]c[/itex] as [itex]c(r)=(r,\phi(r))[/itex], where [itex]\phi(r)[/itex] is given by [itex](6)[/itex] and [itex]c[/itex] is a geodesic?

    Note: This treatment is based on the Clairaut parametrization but I'm not sure if I can do it in this kind of problem where the [itex]g_{rr}[/itex] component of the metric varies...
     
    Last edited: Oct 13, 2013
  2. jcsd
  3. Oct 17, 2013 #2
    I'm not sure if you can simply introduce a new equation when its not given. Have you tried to find an expression which yields the r equation after substitution? Such as an integrating factor so you could put it into sturm-liouville form? I don't have a pencil and paper right now but it just looking at it, seems that there should be some expression like maybe [(a+r)^2*r']' yields the rest of the equation when dividing by some other expression of a+r it seems like it would resemble bessel's equation... If you don't think so, then you are probably right. I'm just diving into Tensors and Diff Geometry. But, that is what I see when looking at that equation because it seems to have some kind of symmetry.

    However, I don't know the exact context of the problem, but from what is given to you, I would think that the method of characteristics would be the best approach. I found a really good demonstration of the method on google last year when I needed to learn it for a mathematical modeling class, since they didn't teach it in PDE's at my school.
     
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