There have only been a few occasions where we have discussed the properties of a rotating black hole. Back in module five for example, we discovered that the rotation of a black hole changes the location of the innermost stable circular orbit. For a non-rotating black hole, the ISCO was three times farther from the center of the black hole than its Schwarzschild event horizon. But for a rotating black hole, the ISCO can shrink until it exactly matches up with the black holes event horizon. It should be noted that as a black hole spins faster and faster, it also pulls the event horizon inwards. Both the ISCO and the event horizon can be as small as half of a Schwarzschild radius for a maximally rotating black hole. In fact, now would be a good time for us to fess up about a little white lie we've been telling you. Any realistic black hole will have angular momentum and will therefore be spinning. We know that black holes must be spinning because in-falling particles carry angular momentum, and we have a pesky law of conservation of angular momentum, which tells us that particles will contribute angular momentum to the black hole. While it's convenient for scientists to learn about the properties of black holes from non-rotating solutions, the reality is that the perfect Schwarzschild black hole is unlikely to exist. We should also note something very strange about black hole rotation. There is a limit to how quickly they can spin. We'll talk a little bit more about the mathematics behind this concept shortly, but the basic idea is that the rotation of a black hole drags spacetime along with it. Similar to the way that water spirals down a drain, spacetime rotates around a rotating black hole. This is a process called frame-dragging. The rotation of a black hole depends on its original spin and the cumulative effects brought about by all of the material that has fallen into it. A problem arises when we begin to talk about angular momentum. Previously, we had taken an object's mass and multiplied it by the distance from the point of rotation to calculate its moment of inertia. Do you see the problem here? If all the mass of a black hole is trapped within a zero volume singularity at its center, how is it possible for a black hole to have a moment of inertia? Well, in 1963, a scientist named Roy Kerr developed a solution to Einstein's field equations which precisely described the properties of a rotating black hole. In the original research, Kerr described how he characterized the angular momentum of a black hole, but mentions how the moment of inertia cannot be characterized except to say that they are very small. The rotation of black holes is usually characterized by a number between zero and one called a, which is calculated by the equation, a equals J times c divided by M squared G. J is the angular momentum of the black hole, M is its mass, and G is Newton's gravitational constant. If you haven't figured it out already, lowercase c is always used to describe the speed of light. If a black hole is not rotating at all, unlikely I know, a takes on the value of zero. If a black hole is spinning maximally, meaning it has reached the upper limit we mentioned before, a takes on a value of one. As the black hole spins faster, the event horizon and the ISCO are pulled inwards. The rotating black holes event horizon shrinks to about half the size of a non-rotating Schwarzschild black hole with the same mass, and the ISCO decreases from three times the Schwarzschild radius to coincide with the event horizon at half the Schwarzschild radius from maximum rotation. With that said, I now feel comfortable showing you how the radius of the ISCO changes from three times the Schwarzschild radius down to one-half as rotation of the black hole changes from a equals zero to a equals one. In case you're wondering, the maximum allowed spin frequency at the event horizon of a solar mass black hole is 16,000 Hertz. The event horizon of a black hole with the same mass as the sun can spin as fast as 16,000 times per second. For other masses, we can simply calculate the maximum spin rate as 16,000 Hertz times the mass of the sun divided by the mass of a black hole. Therefore, higher mass black holes must spin at a slower rate. For a black hole about 36 times the mass of the sun, you would find the maximum rotation corresponds with about 440 Hertz or for you music aficionados, the same as a concert 'a' note. The underlying mathematics of Kerr solution would take an entire course to discuss, but its impact within the scientific community is characterized best by Nobel Prize winning physicist, Chandrasekhar who said, "In my entire scientific life extending over 45 years, the most shattering experience has been the realization that an exact solution of Einstein's equations of general relativity discovered by New Zealand mathematician, Roy Kerr provide an absolute exact representation of the untold number of massive black holes that populate the universe". As a rotating black hole twist the spacetime around it like the ripples in a whirlpool, the twisting and warping of spacetime itself begins influencing the particles and objects within it. Far from the black hole, these forces gently swirl objects around the black hole. The closer you approach the event horizon of a rotating black hole, the more extreme this interaction becomes eventually pulling anything falling past the event horizon into complete lockstep with the black hole. Let's have a top-down look at a non-rotating Schwarzschild black hole. In this diagram, there is a central non-rotating black hole and each point here represents some source of light. When a light source is far from the black hole, the light propagates outwards in all directions. As the sources approach the event horizon, the light sphere begins to distort towards the black hole center. When one of these light sources crosses the event horizon, the light becomes trapped inside. Now, let's impart some spin on this black hole changing it from a Schwarzschild black hole into a rotating Kerr black hole. Since the Kerr black hole just drag spacetime, light sources far from the black hole begin to see a shift in the direction their light spheres propagate. But there's an even more interesting change when a black hole is rotating. Not only does the event horizon shrink from the non-rotating Schwarzschild radius down to about half of its normal size for maximum rotation, but particles falling directly inward begin spiraling around the black hole even though there are no forces acting on them. There's a special distance from a rotating black hole that defines a region called the ergosphere. The outer boundary of the ergosphere is called the stationary limit and outside of the stationary limit, a spacecraft can park with respect to the black hole. But within the stationary limit, no spacecraft can ever appear at rest to a distant observer. Even spacecraft entering the ergosphere orbiting the opposite direction to the rotation of the black hole will eventually be pulled by the spiraling spacetime into a co-rotating trajectory. Although the word sphere is part of ergosphere, the ergosphere is not actually spherical but rather an ellipsoid. While the event horizon is still spherical, the ergosphere envelops the event horizon only touching at the spin axis of the event horizon. It's good to remind ourselves that the ergosphere and the event horizon are boundaries and not objects, so they don't interact with each other in the same way that particles interact with them. I'll emphasize now that a clever spaceship captain can still escape from the ergosphere, and can in fact steal rotational energy from the black hole. The word ergosphere comes from the Greek root ergon, which means work. The ergosphere is so-named because it's theoretically possible to extract the energy from the black hole's rotation with some clever tricks. For example, from within the ergosphere, you could throw a ship's garbage against the rotation of the black hole, accelerating the ship forward and in the spiraled spacetime, end up with more kinetic energy than you started out with. In a case like this, you're stealing energy from a black hole's rotation. Roger Penrose first described this process of stealing energy from a rotating black hole in 1971, which is why we call it the Penrose process. Without going into detail, within the ergosphere, it's possible for the energy of a particle to become negative, a consequence of the change in coordinate system at the stationary limit. Ultimately, what this means is a super-advanced civilization could survive around a rotating black hole, extracting a surplus of energy using the Penrose process until a black hole's rotational energy has been sapped. They could also do the reverse, storing energy as the angular momentum of a black hole and extracting it at a later time. It might seem far-fetched to you to be talking about spiralling spacetime and frame dragging, but it's possible to measure the gravitational effects of a rotating body without a black hole at all. In fact, a space probe aptly called Gravity Probe B was launched back in 2004 to investigate just how strong the frame dragging effects are here on earth. Gravity Probe B carried four incredibly precise gyroscopes in order to measure these effects. At the time of their construction, these gyroscopes were the most spherical objects ever made, differing from perfectly round by no more than 40 atoms on a sphere roughly the size of a ping pong ball. Since the effects are quite a bit weaker around a planet like Earth compared to black holes, it took four years of operation before NASA reported agreement with Einstein's theory of general relativity. We've been hiding a few details of the Kerr black hole behind the veil as it were. The event horizon of a Kerr black hole should really be called its outer horizon because the mathematics tell us that there must be another inner horizon hidden inside. The outer horizon is basically the same as the event horizon; it's the boundary from which nothing can escape. Even if you've fallen through the outer horizon, it's still possible to receive information from beyond the event horizon right up until you fall through the inner horizon often called the Cauchy horizon. The Cauchy horizon marks the boundary within a black hole where information from the entire history of the universe is compressed. An observer approaching the Cauchy horizon would see more and more of the history of the universe essentially being battered by the extreme energies that are compressed within that region. Crossing the Cauchy horizon would be perilous enough simply due to the incredible energy densities one would need to survive. But there's yet another mathematical danger lurking within the Cauchy horizon, the Kerr black hole's ring singularity. Unlike the point-like singularities we've been discussing for Schwarzschild black holes, the singularity of a rotating Kerr black hole is a ring instead of a point. The Cauchy horizon may be the universe's last stand at preventing observers from violating cosmic censorship and glimpsing the singularity. What happens if the black hole spin increases? The distance between the outer and inner horizons become smaller, and the two horizons will coincide if the black hole has maximal rotation. If the black hole spins faster than maximal rotation, the equations predict that the horizons will disappear and the singularity will become visible to the whole universe. Since there will be no event horizon, the resulting thing will not be a black hole, instead we call it a naked singularity. Cosmic censorship predicts that it is impossible to spin a black hole faster than the maximal amount. One final note about rotating black holes. If it were possible to pass through the Cauchy horizon and if it were possible to survive the cosmic censor, it might be possible that an extremely talented astronaut could pilot their ship past the ring singularity and emerge into another universe. What that universe might look like or what you would find there is still unknown. While it may be tempting to plunge into a Kerr black hole hoping to survive the journey to a new universe, there may be a less dangerous possibility which we'll talk about next: worm holes.
