Beyond the singularity: negative space
The ring singularity acts a little like the magical door of fantasy worlds: by going through it we end up in a different place than if we go around it. If we go around it, nothing remarkable happens. But if we go through it (I mean through the disk bounded by the ring, of course: it is best to avoid the ring itself, where curvature is infinite, if we do not wish to be crushed to something unnatural), the r coordinate becomes negative, and starts measuring the opposite of the distance to the black hole (which, from that side, no longer looks like a black hole): this region of space-time is known as negative space.
This negative space region is infinite, and, if we go far away in it in any direction (for very negative values of r, that is, away from the black hole), then space becomes flat again: so there is, in effect, an infinite world tucked beyond the black hole's ring singularity. This can be thought of as the black hole's flip side, or evil twin brother; but it should be emphasized that there is nothing strange about negative space in itself (deep negative space is just a flat region of space, and, of course, from the point of view of negative space, it is positive space which is lies beyond the black hole's ring singularity). It is the negative side of the black hole (or negative black hole—
not to be confused with the
white hole which will be described later), and the region near it, which is strange, not the deep negative space.
One question I am not addressing is whether we reach the
same negative space by crossing the ring singularity from north side or from the south side (or, in a somewhat similar line of thought, if we enter negative space from the north, go around the ring in negative space, then cross it again from the north, do we re-enter the same positive space as we left). Mathematically, the most natural answer to this question is yes, because for the Kerr manifold to be an algebraic variety in a certain sense demands it; but general relativity, and the Kerr solution, is agnostic about this, becaues it only makes prescriptions about the local geometric properties of space-time, not about its global topology. (In a certain sense, the question is meaningless, because the Kerr metric is a mathematical abstraction and is
empty, and it is meaningless to ask whether two empty and identical regions of space-time are actually
the same or not. Real life black holes might not have a negative space anyway.)
A first surprise is that, in the negative side, the black hole is repulsive: there are no orbits around it with negative r, and it takes a considerable amount of energy to enter deep negative space (essentially, the particle's rest mass energy divided by the black hole's fraction of maximality, so the particle must be relativistic). Another surprise is that that there are no horizons on the negative side: seen from the deep negative space, the black hole is a
naked singularity. This is one reason why it is believed that there are no negative black holes in our universe (i.e., black holes for which
we would be in the deep negative sapce). A third surprising characteristic of the negative side of the black hole is that it contains the Carter time machine: there is a region, rougly in the shape of a torus having the ring singularity as its inner equator, in which a material particle can travel at
infinite speed along the ring, thus returning to initial position in space
and time after a finite amount of proper time, or even go back in time. It takes a tremendous amount of energy and calibration to do this (I know it because I've tried to produce a geodesic that does so, and failed), but in principle it is possible. And, of course, this is one of the points where it should be emphasized that the interior region of the Kerr metric is a mathematical abstraction which probably does not describe real life black holes accurately (
more about this later).