# About geodesically incomplete metrics

1. Apr 9, 2010

### Altabeh

Hello

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 $$(g_{01}dtdr)$$ of the metric so that $$(c=1)$$

$$ds^2=(1-2m/r)dt^2-(1+2m/r)dr^2-r^2(d\theta^2+\sin^2(\theta)d\phi^2)+(4m/r)dtdr.$$ ($) As is apparant, this metric is geodesically incomplete in the directions defined by $$r=2m$$ and $$t\rightarrow -\infty$$ and this can be explained as follows. Take a radial null geodesic of this spacetime defined by $$\frac{dr}{dt}=-\frac{r-2m}{r+2m}.$$ (*) Now take $$\lambda$$ to be the parameter of this curve and assume that the affine parameter along this curve is denoted by $$s.$$ From the fact that for an affine parameter we must have it hold in the equation $$\frac{d^2s}{d\lambda^2}=f(\lambda)\frac{ds}{d\lambda},$$ (&) where we have used the notations above, it can be concluded that the replacements $$s=t$$ and $$\lambda=r$$ lead to the following value for $$f$$ in the above equation: $$-4\,{\frac { \left( {\frac {d}{dt}}r \left( t \right) \right) m}{\left( r \left( t \right) +2\,m \right) ^{2}}},$$ and thus confirming that the coordinate $$r$$ is the affine parameter on the null geodesic of the metric given and that the proper time along the geodesic is $$r$$. This reasoning stands for a deficieny in the metric ($) in the sense that since between a point $$P$$ from a null geodesic with equation (*) in the region $$r>2m$$ and the hypersurface $$r=2m$$ the proper time $$r$$ along the geodesic is the finite value $$2m-r_P$$, therefore this contradicts the fact that the geodesic reaches asymptotically $$r=2m$$ for $$t\rightarrow -\infty$$.

Now my question is that assuming the equation (&) as a yardstick to say whether $$r$$ 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 $$r$$ 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 $$-(v^2_{\theta}+v^2_{\phi})$$ with $$v_{\theta}=(rd\theta/dt)$$ and $$v_{\phi}=\sin\theta (rd\phi/dt),$$ if was differentiated wrt $$t$$ in (&), would not give a term like $$(...)(dr/dt).$$ Is this true?

AB

Last edited: Apr 9, 2010
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