Ergosphere in different coordinates

Question

I am currently working with the concept of an ergosphere and I was wondering if it has any meaning to consider the ergosphere after changing coordinates. I mean if someone looks only on the sign of g_{tt} then, with a coordinate transformations of the form of t-> t + a /phi , there will be a different surface that will have null timelike killing vector. Am I mistaken? If there another coordinate invariant way to find the ergosphere? or one should only go to a specific set of coordinates in order to realize it?

My problem has to do with BTZ black hole where all the metric components depend on the radial coordinate.

Thanks in advance!

Answer 1

I think the choice of the coordinate set is based uniquely on simplicity: in fact in those coordinate set you easily understand what is the ergosphere. In fact the definition of the Ergosphere is coordinate invariant; Taken the timelike killing vector of the kerr metric $$ \xi = k + \Omega_H m $$ where $k$ is the timelike killing vector of usual non rotating solution, usually written in ad hoc coordinate set as $k= \partial / \partial t$, and where $m$ is the spatial killing vector which has closed orbit, and where $\Omega_H$ is the angular velocity of the horizon as see from the infinity, $$ \Omega_H = \frac{a}{r_\pm^2 +a^2}, \quad a=J/M $$ The ergoregion is the region pf the spacetime where, (using $\eta = (-1,+1,...,+1)$), $$ g_{\mu\nu} \xi^\mu \xi^\nu > 0, \quad g_{\mu\nu}k^\mu k^\nu <0 $$ The ergosphere is defined as the boundary of this region.

See e.g. Black Holes, Lectures by P. K- Townsend, DAMTP Cambridge Univ.

http://arxiv.org/abs/gr-qc/9707012

PS: Check that the definition is coordinate independent.