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.