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The static Killing vectors in Kruskal coordinates

  1. Apr 14, 2010 #1
    Hello

    In the Kruskal extension of the Schwarzschild metric, where the metric is now transformed into the new coordinates [tex](T,X,\theta,\phi)[/tex], the line element is

    [tex]ds^2=-\frac{32M^3e^{-r/2m} }{r}(-dT^2+dX^2)+r^2(d\theta^2+\sin^2(\theta)d\phi^2),[/tex]

    with

    [tex]\left (r/2m-1 \right ) e^{r/2m}=X^2-T^2 [/tex] (1)
    and [tex]t/2m=2\tanh^{-1}(T/X).[/tex] (2)

    From the equation (1) we see that [tex]\nabla_{\alpha}r=0[/tex] at [tex]X=T=0[/tex] and this is obvious. Okay, but we know that the static Killing field [tex]\xi^{\alpha}[/tex] becomes collinear with [tex]\nabla_{\alpha}r=0[/tex] thus requiring [tex]\xi^{\alpha}[/tex] to also vanish there. My question is that how is this possible? Looking at the static Killing vectors of the Schwarzschild metric in a general Cartesian-like coordinate system [tex]x^{\mu}[/tex],
    [tex]\xi^0=0[/tex] and [tex]{\xi}^{i}={\epsilon}^{{ik}}{x}^{k}[/tex]
    where
    [tex]\epsilon^{ik}=-\epsilon^{ki} =\left[ \begin {array}{ccc} 0&a&-b\\ \noalign{\medskip}-a&0&c
    \\ \noalign{\medskip}b&-c&0\end {array} \right], [/tex]

    with [tex]a,b,c[/tex] being all arbitrary constants, how come the above requirement [tex](X=T=0)[/tex] gets all Killing vectors to vanish? My own view on the problem is that we must find the explicit expressions for [tex]r[/tex] and [tex]t[/tex] from (1) and (2), respectively, and then calculate the Killing vectors and put [tex]X=T=0.[/tex] Am I on the right track or what?

    Thanks in advance
     
  2. jcsd
  3. Apr 15, 2010 #2

    George Jones

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    Sorry, I haven't tried to read your post in detail.

    I would use (1) and (2), and the chain rule for ordinary partial derivatives to calculate the static Killing vector field in Kruskal coordinates.
     
  4. Apr 15, 2010 #3
    Would you mind going a little bit deep into details of the calculation!?

    Thanks
    AB
     
  5. Apr 15, 2010 #4

    George Jones

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    In Schwarzschild coordinates, the static Killing vector is [itex]\partial/\partial t[/itex], and

    [tex]
    \frac{\partial}{\partial t} = \frac{\partial T}{\partial t} \frac{\partial}{\partial T} + \frac{\partial X}{\partial t} \frac{\partial}{\partial X},
    [/tex]

    so the components of Killing vector [itex]\partial/\partial t[/itex] with respect to the [itex]\left\{ T, X, \theta, \phi \right\}[/itex] coordinates are

    [tex]
    \left\{ \frac{\partial T}{\partial t}, \frac{\partial X}{\partial t}, 0, 0 \right\}.
    [/tex]

    Differentiating (1) and (2) with respect to [itex]t[/itex] gives two linear equations in the two non-zero components.
     
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