* As early as 1916, in his paper, ‘Contributions to Einstein’s Theory of Gravitation’, Ludwig Flamm had anticipated much of the Einstein–Rosen solution, but he does not appear to have identified the ‘bridge’.
† Kruskal told John Wheeler about his coordinates that covered the entirety of the maximally extended Schwarzschild spacetime smoothly but didn’t bother to publish the idea. Wheeler wrote up a short paper on the matter and sent it for publication with Kruskal as sole author, originally without Kruskal’s knowledge. George Szekeres discovered the same coordinate system, also in 1960.
‡ We might arrange them in a rectangular grid.
§ These slices do not correspond to constant time slices using an array of identical clocks all at rest with respect to each other. It is a feature of flat spacetime that such a network of clocks can be conceived of, but the warping of spacetime makes it impossible to arrange in general. Rather they are slices of constant ‘Kruskal’ time. Nevertheless, the slices are spacelike in the sense that they have the property that no object can travel along any of the five curves (they are everywhere tilted at less than 45 degrees to the horizontal). These slices through spacetime are the best we can do to define the notion of a snapshot of space at some moment in time.
¶ We should really say ‘Kruskal time’ here, as described in Box 6.1.
7
The Kerr Wonderland
In 1963, New Zealand mathematician Roy Kerr succeeded in finding the unique solution to Einstein’s equations for a spinning black hole. Perhaps you might have expected that adding spin to Schwarzschild’s 1916 solution should not be particularly taxing, but the fact that it took almost half a century to be achieved is testament to the complexity Kerr discovered. Kerr’s solution, like Schwarzschild’s, corresponds to an eternal black hole: an immortal twisting in empty space. But unlike Schwarzschild’s, it is no longer spherically symmetric. Like most spinning objects, including the Sun and the Earth, a Kerr black hole bulges at the equator and is symmetric only about its axis of spin. This lack of symmetry has dramatic consequences.
There are two main types of Kerr black hole, which differ according to how fast they spin. We’ll consider slowly spinning black holes first and get to the faster ones later. A slowly spinning Kerr black hole is illustrated in Figure 7.1. Compared to the Schwarzschild black hole, it has three new features. Firstly, the singularity is a ring.* The plane of the ring is aligned at a right-angle to the spin axis, which means that only trajectories in the equatorial plane will encounter it. All other trajectories will miss it. An astronaut could therefore fall into a Kerr black hole and dodge the end of time. Secondly, the hole has two event horizons, which we’ve labelled the inner and outer horizons. Thirdly, there is a region outside of the outermost horizon in which space is being dragged around so violently that it is impossible for anything to stand still.† This region is known as the ergosphere.
To appreciate the wonders of a spinning black hole, let’s once again follow the adventures of an immortal astronaut. On descending towards the black hole, the first new feature our astronaut encounters is the ergosphere.‡ The outer surface of the ergosphere is the place where a light ray travelling radially outwards will freeze. In the Schwarzschild case, this is also the event horizon of the black hole: the place where the river of space is falling inwards at the speed of light, trapping the outward-swimming photon ‘fish’ forever. For the Kerr geometry, this place does not correspond to the event horizon – the place of no return. Our astronaut could pass into the ergosphere and then decide to turn around and escape back into the Universe. How can it be that a light ray heading radially outwards cannot escape, but an astronaut can? When the astronaut enters the ergosphere, it is impossible for her to avoid being swept around in the direction of rotation of the black hole. Space is being dragged around with the spin and no amount of rocketry can prevent the astronaut, or anything else, from being dragged around with it. This drag is the reason why an astronaut can beat a radially outgoing light ray and escape. We investigate this in more detail in Box 7.1.
Figure 7.1. Schematic representation of a slowly spinning black hole.
BOX 7.1. A place where it is impossible to stand still
Figure 7.2 illustrates a rotating black hole, viewed along the axis of spin. Look at the little circles with nearby dots. The dots correspond to places where a flash of light is emitted, and the circles show the outgoing light front a few moments later. Far away from the hole, the dot lies pretty much at the centre of the circle, but as we move closer to the hole the dots become increasingly displaced. The circles are shifted inwards and also in the direction of the spin. Inside the ergosphere, the dots lie outside of the circles, and that is the important feature. For a Schwarzschild black hole, a similar thing happens inside the horizon: the dots lie outside of the circles, but in this case the circles are only pulled inwards. For a Kerr black hole, they are also ‘dragged around’ in the direction of the spin.
A way to picture this is to appreciate that, for a non-rotating black hole, a dot (flash) on the horizon produces an expanding spherical shell of light that must fall inwards because none of the light can ever travel beyond the horizon. In the language of the river model, the light is being swept inwards by the flow of space. For a rotating black hole, the same is true, but there is also a swirling effect which drags the circles around. It is possible for a dot to lie outside of a circle, as shown in the diagram, which means that it is not possible for someone who emitted the flash to stand still at the position of the dot. If they did so, they would have outrun the light they emitted. They therefore are forced to swirl around with the black hole. This is the same idea we discussed when considering observers falling into a non-rotating black hole. In that case, observers cannot stand still inside the horizon. In the rotating case, the same role reversal is true in the ergosphere, which is a region from which it is possible to escape.
Figure 7.2. The ergosphere of a rotating black hole.
We can see how it is possible to escape from the ergosphere by focusing on the circle that straddles the ergosphere (the third circle from the left). A little portion of the circle lies outside the ergosphere. If we think of the circle as the future light cone of the person who emitted the flash, then we see that it is possible to draw a worldline for that person that crosses the boundary of the ergosphere and heads back outwards into the Universe beyond.
On crossing the ergosphere, our astronaut decides to continue inwards and cross the outer event horizon. As for the Schwarzschild black hole, this is a featureless place that marks the point of no return. The astronaut must now head inwards, and must cross the second, inner, event horizon. But there is a fascinating difference between the Schwarzschild and Kerr cases. For the spinning black hole, the astronaut regains her freedom to navigate once she has crossed the inner horizon. The singularity does not lie inexorably in her future, so time does not have to end. We can appreciate all of this by considering the spacetime diagram in Figure 7.3. Close to the outer event horizon, in region I, the geometry is similar to the Schwarzschild case. On crossing the outer horizon into region II, the light cones tip inwards, which means that the astronaut inexorably travels towards the inner horizon. However, on crossing the inner horizon into region III, the light cones tip back again and our astronaut can navigate around without encountering the singularity. Forever is still available. What, then, becomes of an astronaut who chooses to dodge the end of time?