Is the Schwarzschild spacetime with negative mass geodesically complete?

[LAHLH]

Hi,

If you look at the Schwarzschild geodesics for negative mass, I believe that radial null rays can hit the naked singularity in finite value of affine parameter? but if L <>0 then the null rays get repelled away from r=0 no matter what their energy?

does this mean the spacetime is null geodesically incomplete because radial null rays bump into the singularity at finite affine parameter?

But if you look at the geodesic equations for massive particles, the potential (with M<0) is such that they are always repelled from r=0, no matter if L is zero or not, and not even radial timelike geodesics can reach r=0 in finite proper time. Hence the spacetime timelike geodesically complete?

Thanks

 

[PeterDonis]

I believe that radial null rays can hit the naked singularity in finite value of affine parameter?

I don't think this is correct. The effective potential for null geodesics is always positive, and increases without bound as $r \rightarrow 0$, which would indicate that it is impossible for any null geodesic, regardless of its energy, to reach $r = 0$ at all.

Hence the spacetime timelike geodesically complete?

The effective potential for timelike geodesics has the same properties as above, so I believe this is correct, yes.