For particles moving in the equatorial plane of a rotating black hole, the expressions for dr/dτ and dφ/dτ can be written in the form

 (1.1)

 (1.2)

They are analogous to the corresponding equations for a Schwarzschild black hole. An analysis of the peculiarities of motion is performed in the same way as before by using the effective potential.

1.1 Circular orbits

The most important class of orbits is circular orbits. For given Ẽ and l_z , the radius r₀ of a circular orbit can be found by solving simultaneously the equations

 (1.3)

Figure 1.1. Trajectories of particles in the equatorial plane. In each case two trajectories are shown. Both trajectories have the same initial conditions. The particle is moving in the Kerr metric with a = |a| and a = −|a|, respectively.

One can also use these equations to obtain the expressions for the specific energy Ẽ_circ and specific angular momentum l_circ as functions of the radius r of the circular motion [15],

 (1.4)

 (1.5)

The upper signs in these and the subsequent formulas correspond to direct orbits (i.e. co-rotating with l_z > 0), and the lower signs correspond to retrograde orbits (counter-rotating with l_z < 0). We always assume that a ≥ 0.

The coordinate angular velocity of a particle on the circular orbit is

 (1.6)

Figure 1.2. r_photon, r_bind, and r_bound as functions of the rotation parameter a/M. The quantities corresponding to the direct and retrograde motions are shown by dashed and dotted lines, respectively.

1.2 Last stable circular orbits

Circular orbits can exist only for those values of r for which the denominator in the expressions for Ẽ_circ and lcirc is real, i.e.

 (1.7)

The radius of the circular orbit closest to the black hole (the motion along it occurs at the speed of light) is

 (1.8)

This orbit is unstable. For a = 0, we have r_photon = 3M, while for a = M, we

find r_photon = M (direct motion) or r_photon = 4M (retrograde motion).

The circular orbits with r > r_photon and Ẽ ≥ 1 are unstable. A small perturbation directed outwards forces the particle to leave its orbit and escape to infinity on an asymptotically hyperbolic trajectory.

The radius of the unstable circular orbit, on which Ẽ_circ = 1, is given by

 (1.9)

These values of the radius are the minima of periastra of all parabolic orbits. A particle in the equatorial plane, coming from infinity where its velocity is v_∞ << c, is captured if it passes the black hole closer than rbind.

Finally the radius of the boundary circle separating stable circular orbits from unstable ones is given by the expression

 (1.10)

Table 1.1. The radii rphoton, rbind, and rbound (in units of r_S = 2M) for a non-rotating (a = 0) and an extremely rotating (a = M) black hole.

Table 1.2. Specific energy Ẽ, specific binding energy 1 − Ẽ, and specific angular momentum |l_z |/M of a test particle at the last stable circular orbit.

where

 (1.11)

 (1.12)

The quantities rphoton, rbind, and rbound as the functions of the rotation parameter a/M are shown in figure 1.2.

Table 1.1 lists rphoton, rbind, and rbound for the black hole rotating at the limiting angular velocity, a = M, and gives a comparison with the case of a = 0 (in units of r_S = 2M). As a → M, the invariant distance from a point r to the horizon r₊,

 (1.13)

diverges. This does not mean that all orbits coincide in this limit and lie at the horizon, although at L > 0 the radii r of all three orbits tend to the same limit r₊ [15]. Finally, we will give the values of specific energy Ẽ, specific binding energy 1 − Ẽ, and specific angular momentum |l_z |/M of a test particle at the last stable circular orbit, rbound (see table 1.2).

The binding energy has a maximum for an extremely rotating black hole with a = M. It is equal to

 (1.14)

Thus, the maximum efficiency of the energy release by matter falling into a rotating black hole is 42%. This is much higher than in a non-rotating case.

1.3 Motion with negative Ẽ

It is easy to show that orbits with negative Ẽ are possible within the ergosphere for any θ ≠ 0, π. This follows from the fact that the Killing vector ξ_(t) is spacelike inside the ergosphere. The specific energy Ẽ is defined as Ẽ = −u^μξ^μ _(t). Local analysis shows that for a fixed spacelike vector ξ_(t) it is always possible to find a timelike or null vector u^μ representing the four-velocity of a particle or a photon so that Ẽ is negative. Orbits with Ẽ < 0 make it possible to devise processes that extract the ‘rotational energy’ of the black hole. Such processes were discovered  by Penrose [16].