Goodmorning everyone. I would like to share with you a question that has been gripping me for some time, but which I have never been able to give a convincing answer. When representing the ergosphere or the external event horizon of a black hole, it is often not taken into account that the coordinates used (if space-time is Kerr, the most usual are those of Boyer-Lindquist) have no physical meaning, in the sense that they do not allow us to "see" what the real form of such spatial hypersurfaces would be if they could be "spied" from the earth.
Now, I tried to formulate the embedding, so that the line element of the metric was the Euclidean one IE $$ds ^ 2 = dx ^ 2 + dy ^ 2 + dz ^ 2;$$ the problem (which I found also in the literature) is that this process is not always possible (for example if the spin of the black hole exceeds a certain critical value).
My question is: imagining a rotating black hole with very high angular velocity (t.c. angular momentum at = 0.99 in natural units), what should I see? And how do I understand analytically what geometric shape would have external horizon and ergosphere (spied from the earth) if I cannot embed them in a 3D space?