Throughout differential geometry, geodesics are a geometric invariant second in importance only to curvature. For a spacetime, geodesics provide the description of light (future-pointing null geodesics) and freely falling material particles (future-pointing timelike geodesics). To understand the Kerr black holes we need to understand, in general terms at least, how geodesics behave with respect to their distinctive features--Boyer-Lindquist blocks and horizons, axis, ring singularity, infinite limits of r, and so on. There must certainly be bound orbits analogous to Keplerian ellipses in the Kerr exterior, Boyer-Lindquist block I, but are there any on "white hole" side r<0? or between the horizons in block II? Will an astronaut falling through the horizon r=a into a slow Kerr black hole inevitably be destroyed? or is it possible to travel from exterior through the core of the black hole to r<0?
It is rarely possible to find explicit solutions to the differential equations for geodesics. The best information is provided by first- integrals: expressions that involve only first derivatives of coordinates and are constant on each geodesic c. There is always one: q=<c',c'>, and every Killing vector field gives another, so for Kerr spacetime we get two more. The surprising fact is that there is a fourth, called the Carter constant Q. Four independent first-integrals on a (four-dimensional) spacetime--complete integrability--provide rich information about geodesics that make a thorough analysis possible.
A crucial preliminary task is to show that the maximal Kerr spacetimes are geodesically complete mod S, that is, every geodesic c can be extended (with affine parametrization) over the entire real line--unless c hits the ring singularity.
Next comes a classification of geodesics into various orbit types (e.g., bound, flyby). To relate orbit type and first integrals the fundamental tool is an r-L plot, which displays the relation between angular momentum L and the range of the radial coordinate r(s). (Fortunately, the behavior of the colatitude coordinate turns out to be rather simple for all Kerr geodesics.) To avoid a proliferation of special cases we often concentrate on theimportant one: timelike geodesics in the slow Kerr black hole. The qualitative character of the r-L relation (Sec.8) is then determined by an explicit formula relating energy E and Carter constant Q (Sec.7). Thereafter, knowledge of r-coordinates is used to find the global trajectories of geodesics through the pattern of Boyer-Linquist blocks that constitute the black hole.
Finally we consider some special classes of geodesics--including those in horizons, axis, or equatorial plane; those with Carter constant Q<0; those that approach the center r=0 of the Kerr black hole; and, in particular, those that approach the ring singularity.