Rotating Black Holes

The Schwarzschild solution corresponds to the simplest black hole, which can be characterized by a single parameter, the mass. It is spherically symmetric, and carries neither charge nor angular momentum. However, other solutions to the field equations of general relativity permit black holes with more complexity.

The Kerr Solution
Of particular interest are those solutions where we permit the black hole to be characterized by angular momentum in addition to mass (we continue to assume that charged black holes are unlikely to be of interest in astronomy). Rotating black holes are described by the Kerr solution of the Einstein field equations, named for New Zealand mathematician Roy Kerr who discovered this solution in 1963. As shown in the above figure, rotating black holes can be viewed as having two event horizons. The inner horizon is the boundary inside of which nothing can escape. The outer horizon marks the region inside of which an observer cannot avoid being dragged by rotating spacetime (see the discussion of frame dragging below). It is possible to escape from the region between the inner and outer horizon, which is called the ergosphere. Furthermore, while a particle sucked inside a spherical black hole cannot avoid the singularity at its center, a particle could enter a rotating black hole and avoid the singularity (which forms a ring instead of a single point for the rotating black hole).

Importance of Rotating Black Holes

Rotating black holes are of great interest in astrophysics. First, most stars possess angular momentum. Therefore, if they collapse to form black holes the black hole will possess angular momentum too. Second, supermassive rotating black holes are thought to power quasars and other active galaxies. We shall look at this in more detail in Chapter 25 when we consider quasars and active galaxies.

A Source of Energy and Angular Momentum
It is possible to extract energy and angular momentum from a rotating black hole by allowing particles to follow particular trajectories near them (see the right panel). Theory indicates that in principle all the angular momentum of a rotating black hole could be extracted in this way. This would convert the rotating, deformed black hole (Kerr solution) into a simple non-rotating, spherical black hole (Schwarzschild solution). Once the hole has become spherical, no more energy or angular momentum can be extracted from it by this method. It has been suggested that if we ever attained the technology to manipulate black holes, a rotating one would be the ultimate recycler. We could get rid of our garbage by throwing it into the ergosphere of a Kerr black hole in just the right way, and get energy out in exchange!

Frame Dragging
One of the strangest predictions of the general theory of relativity concerning black holes is called frame dragging. For a rotating black hole, the theory indicates that spacetime can be dragged and wound up by the rotating black hole. The adjacent figure shows an artist's conception of this idea.

It is important to understand the meaning of this figure. We do not mean that matter is simply swirling around the black hole. That would not be such a startling possiblity. What we mean is that space and time itself is in a certain sense swirling around the rotating black hole. A more detailed description of frame dragging and motion in the ergosphere is given in the box below.

Pendulums and Frame Dragging

The idea of frame dragging may be illustrated by considering the motion of pendulums. Imagine a pendulum placed at the North Pole of the Earth. If we are careful to isolate the pendulum from the Earth (for example, by hanging it from a wire supported by the edge of a knife) the plane of the swinging pendulum appears to rotate once in 24 hours. This precession of the pendulum is in fact due to the rotation of the Earth under it in the opposite direction. The pendulum keeps the same direction of oscillation with respect to the distant stars as the Earth rotates. Many museums of natural history have such demonstrations, which are commonly called Foucault pendulums. They were the first way that the rotation of the Earth could be demonstrated conclusively (however, if the pendulum is not at a Pole, the apparent motion is more complex than that described above and the precession period is longer than 24 hours).

General relativity suggests a deviation from this picture. Because spacetime itself is dragged by the rotating mass of the Earth, the plane of the pendulum should precess relative to the distant stars because of the rotation of the spacetime in which it is embedded. This is a tiny effect for the Earth (theory predicts a precession of only 0.04 arc seconds per year for a pendulum on Earth), but is much greater for a rotating black hole with its enormous gravitational field. Inside the ergosphere of a rotating black hole, the precession of spacetime is so large that it is impossible to maintain the same position in space because it is impossible to avoid being dragged by the rotating spacetime. The basic reason is that the frame dragging is so large that just to maintain its current position a particle would have to be accelerated to speeds exceeding that of light, which is forbidden by relativity.