1.1 Bound and unbound trajectories

The specific energy of a moving particle remains constant; in figure 1.1 this motion is shown by a horizontal line. The intersection of the horizontal line

Figure 1.1. An example of the effective black hole potential for ˜L > √3.

with the effective potential determines the turning points. The horizontal a with Ẽ₁ < 1 corresponds to the motion in a bound region in space between r₁ and r₂; this is an analogue of elliptic motion in Newtonian theory. The corresponding trajectory is not a conic section, and, in general, is not closed. If the orbit as a whole lies far from the black hole, it is an ellipse which slowly rotates in the plane of motion.

The segment b with Ẽ₂ > 1 corresponds to a particle coming from infinity and then moving back to infinity (an analogue of hyperbolic motion). Finally, the segment c with Ẽ₃ does not intersect the potential curve but passes above its maximum Ẽ_max. It corresponds to a particle falling into the black hole (gravitational capture). This type of motion is impossible in Newtonian theory and is typical for the black hole. Gravitational capture becomes possible because the effective potential has a maximum. No such maximum appears in the effective potential of Newtonian theory.

In addition, another type of motion is possible in the neighborhood of a black hole. This line d (with energy Ẽ₄) may lie below or above unity, stretching from r_S to the intersection with the curve V (r ). This segment represents the motion of a particle which, for example, first recedes from the black hole and reaches rmax (at the point of intersection of Ẽ₄ and V (r )), and then again falls toward the black hole and is absorbed by it. Examples of different types of trajectory are shown in figure 1.2.

Figure 1.2. Different types of particle trajectory.

A body can escape to infinity if Ẽ ≥ 1. From equation

 (1.1)

Figure 1.3. Effective black hole potential for different values of ˜ L : (a) ˜L = 0, (b)  ˜L = √3, (c) ˜L = 2, and (d) ˜L = √6.

we find the escape velocity as

 (1.2)

which coincides with the Newtonian expression. Note that in Newtonian theory in the gravitational field of a pointlike mass, the escape velocity guarantees the escape to infinity regardless of the direction of motion. The case of the black hole is different. Even if a particle has the escape velocity, it can be trapped by the black hole, the latter occurring if the particle moves towards the black hole. We have already mentioned this effect, calling it gravitational capture.

1.2 Circular motion

For circular motion around a black hole dr/dτ ≡ 0. This motion is represented in figure 1.1 by a point at the extremum of the effective potential curve. A point at the minimum corresponds to a stable motion, and a point at the maximum to an unstable motion. The latter motion has no analogue in Newtonian theory and is specific to black holes. If the motion of a particle is represented by a horizontal line Ẽ= constant very close to Ẽ_max, then the particle makes many turns around the black hole at a radius close to r corresponding to Ẽ_max before the orbit moves far away from this value of r . The shape and the position of the potential V (r ) are different for different ˜L : the corresponding curves for some values of ˜L are shown in figure 1.3.

The maximum and minimum appear on V(r ) curves when ˜L > √3. If ˜L < √3 the V(r ) curve is monotone. Hence, the motion on circular orbits is possible only if ˜L > √3. The minima of the curves then lie at r > 3r_S. Stable circular orbits thus exist only for r > 3r_S. At smaller distances, there are only unstable circular orbits corresponding to the maximum of Ẽ_max curves. If ˜L →∞, the position of the maximum of the Ẽ_max curve decreases to r = 1.5rS. Even unstable inertial circular motion becomes impossible at r less than 1.5r_S.

The critical circular orbit that separates stable motions from unstable ones corresponds to r = 3r_S. Particles move along it at a velocity v = c/2, the energy of a particle being %E = √8/9 ≈ 0.943. This is the motion with the maximum possible binding energy E ≈ 0.057 mc².

Let us emphasize the importance of this result for black hole astrophysics. Suppose a non-rotating black hole is surrounded by a thin accretion disk. Let us follow the time evolution of a matter element of the disc. It is moving along a practically circular orbit slowly losing its energy and angular momentum until it reaches the position of the last stable circular orbit. After this, it falls almost freely into the black hole. This means that the maximum efficiency of the energy release by matter falling into a non-rotating black hole is 5.7%.

The velocity on (unstable) orbits, with r < 3r_S, increases as r decreases from c/2 to c on the last circular orbit with r = 1.5r_S. When r = 2r_S, the particle's energy is Ẽ= 1, that is, the circular velocity is equal to the escape velocity. If r is still smaller, the escape velocity is smaller than the circular velocity. There is no paradox in it, since the circular motion here is unstable and even the tiniest perturbation (supplying momentum away from the black hole) transfers the particle to an orbit moving it to infinity, that is, an orbit corresponding to hyperbolic motion.