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About geodesically incomplete metrics

  1. Apr 9, 2010 #1

    Well I came up with this crucial point in the study of the Eddington-Finkelstein form of the Schwartzschild metric. Since the Eddington-Finkelstein splits the SM into two separate metrics of their own geodesical characteristics, I'd like to take the plus sign of the mixed term [tex](g_{01}dtdr)[/tex] of the metric so that [tex](c=1)[/tex]

    [tex]ds^2=(1-2m/r)dt^2-(1+2m/r)dr^2-r^2(d\theta^2+\sin^2(\theta)d\phi^2)+(4m/r)dtdr.[/tex] ($)

    As is apparant, this metric is geodesically incomplete in the directions defined by [tex]r=2m[/tex] and [tex]t\rightarrow -\infty[/tex] and this can be explained as follows. Take a radial null geodesic of this spacetime defined by

    [tex]\frac{dr}{dt}=-\frac{r-2m}{r+2m}.[/tex] (*)

    Now take [tex]\lambda[/tex] to be the parameter of this curve and assume that the affine parameter along this curve is denoted by [tex]s.[/tex] From the fact that for an affine parameter we must have it hold in the equation

    [tex]\frac{d^2s}{d\lambda^2}=f(\lambda)\frac{ds}{d\lambda},[/tex] (&)

    where we have used the notations above, it can be concluded that the replacements [tex]s=t[/tex] and [tex]\lambda=r[/tex] lead to the following value for [tex]f[/tex] in the above equation:

    [tex]-4\,{\frac { \left( {\frac {d}{dt}}r \left( t \right) \right) m}{\left( r \left( t \right) +2\,m \right) ^{2}}},[/tex]

    and thus confirming that the coordinate [tex]r[/tex] is the affine parameter on the null geodesic of the metric given and that the proper time along the geodesic is [tex]r[/tex]. This reasoning stands for a deficieny in the metric ($) in the sense that since between a point [tex]P[/tex] from a null geodesic with equation (*) in the region [tex]r>2m[/tex] and the hypersurface [tex]r=2m[/tex] the proper time [tex]r[/tex] along the geodesic is the finite value [tex]2m-r_P[/tex], therefore this contradicts the fact that the geodesic reaches asymptotically [tex]r=2m[/tex] for [tex]t\rightarrow -\infty[/tex].

    Now my question is that assuming the equation (&) as a yardstick to say whether [tex]r[/tex] is an affine parameter along any other curve or not, will the above result be correct for, say, a non-radial time-like geodesic, too? I.e. can we also take [tex]r[/tex] to be the affine parameter of a time-like geodesic, too? The motivation for asking this is that the book I'm reading says this is correct, but I cannot understand how this is possible while there is a term [tex]-(v^2_{\theta}+v^2_{\phi})[/tex] with [tex]v_{\theta}=(rd\theta/dt)[/tex] and [tex]v_{\phi}=\sin\theta (rd\phi/dt),[/tex] if was differentiated wrt [tex]t[/tex] in (&), would not give a term like [tex](...)(dr/dt).[/tex] Is this true?

    Last edited: Apr 9, 2010
  2. jcsd
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