Non singular black hole solutions

[George Jones]

The WP page geodesic entry after explaining the different null, timelike and spacelike types says: "Note that a geodesic cannot be spacelike at one point and timelike at another". Is there a difference in this respect between vectors and geodesics?

Yes, even though vectors and curves are related, there is, in principle, a difference in this respect. In particular, the integral curves associated with the Killing vector field \partial_t (curves of contant r, theta,a and phi) are not geodesics, so your result could possible fail for them. It, however, does not fail, i.e., an individual integral curve for \partial_t does not change its causal character, but different integral curves for \partial_t can different characters. Outside the event horizon (where spactime is static), the integral curves of \partial_t are all timelike; inside the event horizon (where spacetime is not even stationary), the integral curves of \partial_t are all spacelike.

The integral curves for \partial_t are the curved dashed lines on

http://www.google.com/imgres?imgurl...MkRTtjWJfTLsQKpl6i-Cg&ved=0CDcQ9QEwBQ&dur=235

These means that the vectors that make up the vector field \partial_t are tangent vectors for these curves. Note the different causal character outside and inside the event horizon.

 

[TrickyDicky]

Thanks.