If a piece of the in-falling object breaks away inside the ergosphere and carries negative energy into the black hole, the energy of the black hole will decrease. Energy must be conserved overall, however, and this means the outgoing part must carry away more energy than was thrown in.
Figure 7.8. Black hole mining. From Misner, Thorne and Wheeler’s 1973 book, Gravitation. (Figure 33.2 from Gravitation by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, page 908. Published by Princeton University Press in 2017. Reproduced here by permission of the publisher.)
In Figure 7.8 we reproduce a figure from Misner, Thorne and Wheeler showing how an advanced civilisation living around a Kerr black hole could exploit the Penrose process to dispose of their garbage and generate electrical power for their civilisation. The ultimate green energy scheme.
We’ve spent quite some time exploring the ergosphere because there’s an important postscript. It transpires that the area of the black hole’s event horizon always increases after a Penrose process. At first sight this is surprising because a rotating black hole loses mass through the Penrose process. We might therefore expect the horizon to shrink. We are considering rotating black holes, however, and the area of the outer event horizon can increase even if the mass of the black hole decreases, provided that the spin of the black hole also decreases. Using the equations of general relativity, it can be shown that the spin of a black hole always decreases in a Penrose process, and by enough to guarantee that the area of the outer horizon always increases. This ‘area always increases’ rule doesn’t just apply to the Penrose process. In 1971, Stephen Hawking proved that, according to general relativity, the area of the horizon of a black hole must always increase, no matter what.†† This is a result of great importance. It is our first encounter with the laws of black hole thermodynamics.
Before we delve into this important subject, we will take a step back from the purely theoretical to explore the formation of the real black holes we observe to be dotted throughout the Universe.
* The radius of the ring is J/c where J is the angular momentum of the hole divided by its mass and c is the speed of light. For the Earth J/c is roughly 10 metres and for the Sun it is roughly 1 kilometre.
† With respect to the distant stars, say.
‡ ‘Encounter’ is a bit misleading because the freely falling astronaut won’t notice anything as she crosses into the ergosphere – as ever, spacetime for her will be locally flat.
§ What our astronaut experiences in region III is not something we can appreciate from the Penrose diagram, but this is what the equations tell us.
¶ A horizontal, spacelike singularity like the one in a Schwarzschild black hole.
** Using notions of energy and time defined by an observer far from the hole.
†† It will be allowed to decrease when we come to consider quantum physics.
8
Real Black Holes from Collapsing Stars
‘In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity, discovered by the New Zealand mathematician, Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe. This shuddering before the beautiful, this incredible fact that a discovery motivated by a search after the beautiful in mathematics should find its exact replica in Nature, persuades me to say that beauty is that to which the human mind responds at its deepest and most profound.’
Subrahmanyan Chandrasekhar24
The black holes we’ve explored so far have inhabited the mathematical landscape of general relativity. These remarkable universes were known to and broadly speaking dismissed by physicists for a large part of the twentieth century, Einstein included, on the very reasonable grounds that we shouldn’t conclude that something exists just because a physical theory allows it. If black holes are to exist in the real sky rather than the mathematical one, Nature must construct them. Real black holes, formed from stellar collapse, are the focus of this chapter. We will learn that the solutions to Einstein’s equations of general relativity discovered by Schwarzschild and Kerr are of extraordinary significance in the real Universe because they are the only possible solutions for the spacetime in the region outside of every black hole. Nowhere else in physics is something apparently so complicated as a collapsing star reduced to something so simple and with such precision. The Schwarzschild solution depends on just one number (the mass) and the Kerr solution adds a second number (the spin). Knowing these two numbers alone, we can compute the gravitational landscape in the region outside of real black holes, exactly. That is an astonishing claim – it does not matter what collapsed to form the black hole, nor does it depend on how it fell in. All that remains outside the horizon is a spacetime perfect in its simplicity. This is what moved Chandrasekhar to write the powerful prose quoted above. Quoting Chandrasekhar again: ‘The black holes of nature are the most perfect macroscopic objects there are in the universe … and since the general theory of relativity provides only a single unique family of solutions for their descriptions, they are the simplest objects as well.’
John Wheeler, as ever, found a more pithy phrasing: ‘Black holes have no hair.’ In his memoir Geons, Black Holes and Quantum Foam, Wheeler recounts an exchange with Richard Feynman in which the often irreverent Feynman accused him of using language ‘unfit for polite company’. ‘I tried to summarize the remarkable simplicity of a black hole by saying a black hole has no hair. I guess Dick Feynman and I had different images in mind. I was thinking of a room full of bald-pated people who were hard to identify individually because they showed no differences in hair length, style, or colour. The black hole, as it turned out, shows only three characteristics to the outside world: its mass, its electric charge (if any) and its spin (if any). It lacks the “hair” that more conventional objects possess that give them their individuality … No hair stylist can arrange for a black hole to have a certain colour or shape. It is bald.’
A series of papers throughout the late 1960s and early 1970s established the magnificent simplicity of black holes, as viewed from the outside. Once formed, according to general relativity, the horizon shields us from the complexities within. Even if something as large as a planet or star falls across the horizon, the American physicist Richard H. Price proved, in 1972, that the black hole quickly settles down again into oblivious perfection. For a Schwarzschild black hole, the horizon will reassume the shape of a perfect sphere, and any disturbances caused by the in-falling body will be smoothed out by the emission of gravitational waves. The conclusion is that the spacetime outside of all black holes in the Universe is either Schwarzschild or Kerr.*