Hello(adsbygoogle = window.adsbygoogle || []).push({});

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?

AB

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# About geodesically incomplete metrics

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**