Ergoregions and Energy Extraction

[adsquestion]

Yes--or at least one that covers enough of the spacetime to include the horizon and the region inside it.
That's what is not possible in Kerr spacetime (whereas its analogue is in the black string spacetime).

Right and we can show it's possible in the black string case since $\partial_T$ is hypersurface orthogonal for all r.

Still though, for Kerr, suppose I take the metric in Boyer-Lindquist coordinates and show that $\partial_t$ doesn't satisfy the Frobenius equation - what does that tell me? Surely it just says that $\partial_t$ isn't hypersurface orthogonal in B-L coordinates? Wouldn't I need to check this for the infinite number of possible coordinate systems in order to establish no such hypersurface orthogonal timelike KVF exists?

 

[PeterDonis]

Wouldn't I need to check this for the infinite number of possible coordinate systems in order to establish no such hypersurface orthogonal timelike KVF exists?

No. $\partial_t$ in Boyer-Lindquist coordinates is the only timelike KVF in Kerr spacetime*. Proving that it is not hypersurface orthogonal can be done in any chart, and since hypersurface orthogonality is a coordinate-free geometric property, if it holds in one chart, it holds in any chart.

* - strictly speaking, this isn't true; any linear combination of $\partial_t$ and $\partial_\phi$ with constant coefficients that is timelike is also a timelike KVF. But proving that $\partial_t$ is not hypersurface orthogonal is sufficient to prove that none of those other timelike KVFs are hypersurface orthogonal either.