Ergosphere in different coordinates

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.

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.