The Theory of Everything: The Origin and Fate of the Universe
Chapter 4 - FOURTH LECTURE - BLACK HOLES AIN’T SO BLACKBefore 1970, my research on general relativity had concentrated mainly onthe question of whether there had been a big bang singularity. However,one evening in November of that year, shortly after the birth of my daughter,Lucy, I started to think about black holes as I was getting into bed. My disabil-ity made this rather a slow process, so I had plenty of time. At that date therewas no precise definition of which points in space-time lay inside a black holeand which lay outside.I had already discussed with Roger Penrose the idea of defining a black hole asthe set of events from which it was not possible to escape to a large distance.This is now the generally accepted definition. It means that the boundary ofthe black hole, the event horizon, is formed by rays of light that just fail to getaway from the black hole. Instead, they stay forever, hovering on the edge ofthe black hole. It is like running away from the police and managing to keepone step ahead but not being able to get clear away.Suddenly I realized that the paths of these light rays could not be approachingone another, because if they were, they must eventually run into each other. Itwould be like someone else running away from the police in the opposite direc-tion. You would both be caught or, in this case, fall into a black hole. But ifthese light rays were swallowed up by the black hole, then they could not havebeen on the boundary of the black hole. So light rays in the event horizon hadto be moving parallel to, or away from, each other.Another way of seeing this is that the event horizon, the boundary of the blackhole, is like the edge of a shadow. It is the edge of the light of escape to a greatdistance, but, equally, it is the edge of the shadow of impending doom. And ifyou look at the shadow cast by a source at a great distance, such as the sun, youwill see that the rays of light on the edge are not approaching each other. Ifthe rays of light that form the event horizon, the boundary of the black hole,can never approach each other, the area of the event horizon could stay thesame or increase with time. It could never decrease, because that would meanthat at least some of the rays of light in the boundary would have to beapproaching each other. In fact, the area would increase whenever matter orradiation fell into the black hole.Also, suppose two black holes collided and merged together to form a singleblack hole. Then the area of the event horizon of the final black hole wouldbe greater than the sum of the areas of the event horizons of the original blackholes. This nondecreasing property of the event horizon’s area placed animportant restriction on the possible behavior of black holes. I was so excitedwith my discovery that I did not get much sleep that night.The next day I rang up Roger Penrose. He agreed with me. I think, in fact, thathe had been aware of this property of the area. However, he had been using aslightly different definition of a black hole. He had not realized that theboundaries of the black hole according to the two definitions would be thesame, provided the black hole had settled down to a stationary state.THE SECOND LAW OFTHERMODYNAMICSThe nondecreasing behavior of a black hole’s area was very reminiscent of thebehavior of a physical quantity called entropy, which measures the degree ofdisorder of a system. It is a matter of common experience that disorder willtend to increase if things are left to themselves; one has only to leave a housewithout repairs to see that. One can create order out of disorder-for example,one can paint the house. However, that requires expenditure of energy, and sodecreases the amount of ordered energy available.A precise statement of this idea is known as the second law of thermodynam-ics. It states that the entropy of an isolated system never decreases with time.Moreover, when two systems are joined together, the entropy of the combinedsystem is greater than the sum of the entropies of the individual systems. Forexample, consider a system of gas molecules in a box. The molecules can bethought of as little billiard balls continually colliding with each other andbouncing off the walls of the box. Suppose that initially the molecules are allconfined to the left-hand side of the box by a partition. If the partition is thenremoved, the molecules will tend to spread out and occupy both halves of thebox. At some later time they could, by chance, all be in the right half or all beback in the left half. However, it is overwhelmingly more probable that therewill be roughly equal numbers in the two halves. Such a state is less ordered,or more disordered, than the original state in which all the molecules were inone half. One therefore says that the entropy of the gas has gone up.Similarly, suppose one starts with two boxes, one containing oxygen moleculesand the other containing nitrogen molecules. If one joins the boxes togetherand removes the intervening wall, the oxygen and the nitrogen molecules willstart to mix. At a later time, the most probable state would be to have athoroughly uniform mixture of oxygen and nitrogen molecules throughout thetwo boxes. This state would be less ordered, and hence have more entropy,than the initial state of two separate boxes.The second law of thermodynamics has a rather different status than that ofother laws of science. Other laws, such as Newton’s law of gravity, forexample, are absolute law-that is, they always hold. On the other hand, thesecond law is a statistical law-that is, it does not hold always, just in the vastmajority of cases. The probability of all the gas molecules in our box beingfound in one half of the box at a later time is many millions of millions to one,but it could happen.However, if one has a black hole around, there seems to be a rather easier wayof violating the second law: Just throw some matter with a lot of entropy, suchas a box of gas, down the black hole. The total entropy of matter outside theblack hole would go down. One could, of course, still say that the total entropy,including the entropy inside the black hole, has not gone down. But sincethere is no way to look inside the black hole, we cannot see how much entropythe matter inside it has. It would be nice, therefore, if there was some featureof the black hole by which observers outside the black hole could tell itsentropy; this should increase whenever matter carrying entropy fell into theblack hole.Following my discovery that the area of the event horizon increased whenevermatter fell into a black hole, a research student at Princeton named JacobBekenstein suggested that the area of the event horizon was a measure of theentropy of the black hole. As matter carrying entropy fell into the black hole,the area of the event horizon would go up, so that the sum of the entropy ofmatter outside black holes and the area of the horizons would never go down.This suggestion seemed to prevent the second law of thermodynamics frombeing violated in most situations. However, there was one fatal flaw: If a blackhole has entropy, then it ought also to have a temperature. But a body with anonzero temperature must emit radiation at a certain rate. It is a matter ofcommon experience that if one heats up a poker in the fire, it glows red hotand emits radiation. However, bodies at lower temperatures emit radiation,too; one just does not normally notice it because the amount is fairly small.This radiation is required in order to prevent violations of the second law. Soblack holes ought to emit radiation, but by their very definition, black holesare objects that are not supposed to emit anything. It therefore seemed that thearea of the event horizon of a black hole could not be regarded as its entropy.In fact, in 1972 I wrote a paper on this subject with Brandon Carter and anAmerican colleague, Jim Bardeen. We pointed out that, although there weremany similarities between entropy and the area of the event horizon, there wasthis apparently fatal difficulty. I must admit that in writing this paper I wasmotivated partly by irritation with Bekenstein, because I felt he had misusedmy discovery of the increase of the area of the event horizon. However, itturned out in the end that he was basically correct, though in a manner he hadcertainly not expected.BLACK HOLE RADIATIONIn September 1973, while I was visiting Moscow, I discussed black holes withtwo leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. Theyconvinced me that, according to the quantum mechanical uncertainty princi-ple, rotating black holes should create and emit particles. I believed their argu-ments on physical grounds, but I did not like the mathematical way in whichthey calculated the emission. I therefore set about devising a better mathemat-ical treatment, which I described at an informal seminar in Oxford at the endof November 1973. At that time I had not done the calculations to find outhow much would actually be emitted. I was expecting to discover just the radi-ation that Zeldovich and Starobinsky had predicted from rotating black holes.However, when I did the calculation, I found, to my surprise and annoyance,that even nonrotating black holes should apparently create and emit particlesat a steady rate.At first I thought that this emission indicated that one of the approximationsI had used was not valid. I was afraid if Bekenstein found out about it, he woulduse it as a further argument to support his ideas about the entropy of blackholes, which I still did not like. However, the more I thought about it, themore it seemed that the approximations really ought to hold. But what finallyconvinced me that the emission was real was that the spectrum of the emittedparticles was exactly that which would be emitted by a hot body.The black hole was emitting particles at exactly the correct rate to preventviolations of the second law.Since then, the calculations have been repeated in a number of different formsby other people. They all confirm that a black hole ought to emit particles andradiation as if it were a hot body with a temperature that depends only on theblack hole’s mass: the higher the mass, the lower the temperature. One canunderstand this emission in the following way: What we think of as emptyspace cannot be completely empty because that would mean that all the fields,such as the gravitational field and the electromagnetic field, would have to beexactly zero. However, the value of a field and its rate of change with time arelike the position and velocity of a particle. The uncertainty principle impliesthat the more accurately one knows one of these quantities, the less accuratelyone can know the other.So in empty space the field cannot be fixed at exactly zero, because then itwould have both a precise value, zero, and a precise rate of change, also zero.Instead, there must be a certain minimum amount of uncertainty, or quantumfluctuations, in the value of a field. One can think of these fluctuations as pairsof particles of light or gravity that appear together at some time, move apart,and then come together again and annihilate each other. These particles arecalled virtual particles. Unlike real particles, they cannot be observed directlywith a particle detector. However, their indirect effects, such as small changesin the energy of electron orbits and atoms, can be measured and agree with thetheoretical predictions to a remarkable degree of accuracy.By conservation of energy, one of the partners in a virtual particle pair willhave positive energy and the other partner will have negative energy. The onewith negative energy is condemned to be a short-lived virtual particle. This isbecause real particles always have positive energy in normal situations. It musttherefore seek out its partner and annihilate it. However, the gravitationalfield inside a black hole is so