In the preceding chapter the Kerr metric was introduced and its domain extended to Kerr-star spacetime K*, which includes the axis of rotation and the horizons (when the latter exist), but not the ring singularity, where curvature becomes infinite.
Is K* a maximally extended analytic spacetime? Yes, in the fast case, a2>m, but only then. One point that might arouse suspicion--other than experience with Schwarzschild spacetime--is the asymmetry in the construction of K*, where ingoing principal null geodesics were preferred over outgoing. In this chapter we begin by repeating the construction, but with the outgoers preferred. The resulting star-Kerr spacetime *K is built of the same Boyer-Lindquist blocks as K*, but they are not assembled in the same way. Thus the two spacetimes *K and K* can themselves be glued together, along a same-numbered block, to produce a spacetime that is an extension of both.
Continuing this assembly process--with the principal null geodesics as warp and woof--leads quickly to a maximal extension in the extreme case a2=m2.The slow case, as usual, is more ornate, and in its maximal extension, the horizons self-intersect.
With the maximal Kerr black Kerr isometries are constructed, analogously to the spacetimes
themselves, by assembling isometries of the Boyer-Lindquist blocks, and
we find the isometry groups for the three rotation types: fast, extreme,
slow.
The topology of the maximal Kerr black holes--particularly in the
slow case--is not simple. However, by retaining the ring singularity in
each block III of M, we get a smooth manifold M~ that is a 2-sphere
bundle over the (north or south) axis A of M--indeed, a product
manifold A ×S2. This makes it possible to find the
topology and in fact the homotopy type of M itself.
In relativity, chronology [causality] refers to the relationship
between events that can be joined by a timelike [nonspacelike] curve.
Chapter 2 has already shown that in Boyer-Lindquist block III chronology
is violated by the existence of closed timelike curves. In
Section 10 we consider global chronology of Kerr black holes, where
as usual the slow case is the most interesting.
The three standard maximal Kerr spacetimes are by no means the
only ones. In particular, there are "small" variants whose geometric
conciseness is paid for by radical violations of chronology.