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HOME > Classical Novels > A Brief History of Time > CHAPTER 7 BLACK HOLES AIN’T SO BLACK
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Before 1970, my research on general relativity hadconcentrated mainly on the question of whether or not therehad been a big bang singularity. However, one evening inNovember that year, shortly after the birth of my daughter,Lucy, I started to think about black holes as I was getting intobed. My disability makes this rather a slow process, so I hadplenty of time. At that date there was no precise definition ofwhich points in space-time lay inside a black hole and whichlay outside. I had already discussed with Roger Penrose theidea of defining a black hole as the set of events from which itwas not possible to escape to a large distance, which is nowthe generally accepted definition. It means that the boundary ofthe black hole, the event horizon, is formed by the light raysthat just fail to escape from the black hole, hovering foreverjust on the edge (Fig. 7.1). It is a bit like running away fromthe police and just managing to keep one step ahead but notbeing able to get clear away!
Suddenly I realized that the paths of these light rays couldnever approach one another. If they did they must eventuallyrun into one another. It would be like meeting someone elserunning away from the police in the opposite direction - youwould both be caught! (Or, in this case, fall into a black hole.)But if these light rays were swallowed up by the black hole,then they could not have been on the boundary of the blackhole. So the paths of light rays in the event horizon hadalways to be moving parallel to, or away from, each other.
Another way of seeing this is that the event horizon, theboundary of the black hole, is like the edge of a shadow - theshadow of impending doom. If you look at the shadow cast bya source at a great distance, such as the sun, you will see thatthe rays of light in the edge are not approaching each other.
If the rays of light that form the event horizon, theboundary of the black hole, can never approach each other,the area of the event horizon might stay the same or increasewith time, but it could never decrease because that would meanthat at least some of the rays of light in the boundary wouldhave to be approaching each other. In fact, the area wouldincrease whenever matter or radiation fell into the black hole(Fig. 7.2). Or if two black holes collided and merged togetherto form a single black hole, the area of the event horizon ofthe final black hole would be greater than or equal to the sumof the areas of the event horizons of the original black holes(Fig. 7.3). This nondecreasing property of the event horizon’sarea placed an important restriction on the possible behavior ofblack holes. I was so excited with my discovery that I did notget much sleep that night. The next day I rang up RogerPenrose. He agreed with me. I think, in fact, that he had beenaware of this property of the area. However, he had beenusing a slightly different definition of a black hole. He had notrealized that the boundaries of the black hole according to thetwo definitions would be the same, and hence so would theirareas, provided the black hole had settled down to a state inwhich it was not changing with time.
The nondecreasing behavior of a black hole’s area was veryreminiscent of the behavior of a physical quantity called entropy,which measures the degree of disorder of a system. It is amatter of common experience that disorder will tend to increaseif things are left to themselves. (One has only to stop makingrepairs around the house to see that!) One can create orderout of disorder (for example, one can paint the house), butthat requires expenditure of effort or energy and so decreasesthe amount of ordered energy available.
A precise statement of this idea is known as the second lawof thermodynamics. It states that the entropy of an isolatedsystem always increases, and that when two systems are joinedtogether, the entropy of the combined system is greater thanthe sum of the entropies of the individual systems. Forexample, consider a system of gas molecules in a box. Themolecules can be thought of as little billiard balls continuallycolliding with each other and bouncing off the walls of the box.
The higher the temperature of the gas, the faster the moleculesmove, and so the more frequently and harder they collide withthe walls of the box and the greater the outward pressure theyexert on the walls. Suppose that initially the molecules are allconfined to the left-hand side of the box by a partition. If thepartition is then removed, the molecules will tend to spread outand occupy both halves of the box. At some later time theycould, by chance, all be in the right half or back in the lefthalf, but it is overwhelmingly more probable that there will beroughly equal numbers in the two halves. Such a state is lessordered, or more disordered, than the original state in which allthe molecules were in one half. One therefore says that theentropy of the gas has gone up. Similarly, suppose one startswith two boxes, one containing oxygen molecules and the othercontaining nitrogen molecules. If one joins the boxes togetherand removes the intervening wall, the oxygen and the nitrogenmolecules will start to mix. At a later time the most probablestate would be a fairly uniform mixture of oxygen and nitrogenmolecules throughout the two boxes. This state would be lessordered, and hence have more entropy, than the initial state oftwo separate boxes.
The second law of thermodynamics has a rather differentstatus than that of other laws of science, such as Newton’s lawof gravity, for example, because it does not hold always, just inthe vast majority of cases. The probability of all the gasmolecules in our first boxfound in one half of the box at a later time is many millionsof millions to one, but it can happen. However, if one has ablack hole around there seems to be a rather easier way ofviolating the second law: just throw some matter with a lot ofentropy such as a box of gas, down the black hole. The totalentropy of matter outside the black hole would go down. Onecould, of course, still say that the total entropy, including theentropy inside the black hole, has not gone down - but sincethere is no way to look inside the black hole, we cannot seehow much entropy the matter inside it has. It would be nice,then, if there was some feature of the black hole by whichobservers outside the blackhole could tell its entropy, and which would increasewhenever matter carrying entropy fell into the black hole.
Following the discovery, described above, that the area of theevent horizon increased whenever matter fell into a black hole,a research student at Princeton named Jacob Bekensteinsuggested that the area of the event horizon was a measure ofthe entropy of the black hole. As matter carrying entropy fellinto a black hole, the area of its event horizon would go up,so that the sum of the entropy of matter outside black holesand the area of the horizons would never go down.
This suggestion seemed to prevent the second law ofthermodynamics from being violated in most situations.
However, there was one fatal flaw. If a black hole has entropy,then it ought to also have a temperature. But a body with aparticular temperature must emit radiation at a certain rate. Itis a matter of common experience that if one heats up apoker in a fire it glows red hot and emits radiation, but bodiesat lower temperatures emit radiation too; one just does notnormally notice it because the amount is fairly small. Thisradiation is required in order to prevent violation of the secondlaw. So black holes ought to emit radiation. But by their verydefinition, black holes are objects that are not supposed to emitanything. It therefore seemed that the area of the eventhorizon of a black hole could not be regarded as its entropy.
In 1972 I wrote a paper with Brandon Carter and anAmerican colleague, Jim Bardeen, in which we pointed out thatalthough there were many similarities between entropy and thearea of the event horizon, there was this apparently fataldifficulty. I must admit that in writing this paper I wasmotivated partly by irritation with Bekenstein, who, I felt, hadmisused my discovery of the increase of the area of the eventhorizon. However, it turned out in the end that he wasbasically correct, though in a manner he had certainly notexpected.
In September 1973, while I was visiting Moscow, I discussedblack holes with two leading Soviet experts, Yakov Zeldovichand Alexander Starobinsky. They convinced me that, accordingto the quantum mechanical uncertainty principle, rotating blackholes should create and emit particles. I believed theirarguments on physical grounds, but I did not like themathematical way in which they calculated the emission. Itherefore set about devising a better mathematical treatment,which I described at an informal seminar in Oxford at the endof November 1973. At that time I had not done the calculationsto find out how much would actually be emitted. I wasexpecting to discover just the radiation that Zeldovich andStarobinsky had predicted from rotating black holes. However,when I did the calculation, I found, to my surprise andannoyance, that even non-rotating black holes should apparentlycreate and emit particles at a steady rate. At first I thoughtthat this emission indicated that one of the approximations Ihad used was not valid. I was afraid that if Bekenstein foundout about it, he would use it as a further argument to supporthis ideas about the entropy of black holes, which I still did notlike. However, the more I thought about it, the more it seemedthat the approximations really ought to hold. But what finallyconvinced me that the emission was real was that the spectrumof the emitted particles was exactly that which would be emittedby a hot body, and that the black hole was emitting particlesat exactly the correct rate to prevent violations of the secondlaw. Since then the calculations have been repeated in anumber of different forms by other people. They all confirmthat a black hole ought to emit particles and radiation as if itwere a hot body with a temperature that depends only on theblack hole’s mass: the higher the mass, the lower thetemperature.
How is it possible that a black hole appears to emit particleswhen we know that nothing can escape from within its eventhorizon? The answer, quantum theory tells us, is that theparticles do not come from within the black hole, but from the“empty” space just outside the black hole’s event horizon! Wecan understand this in the following way: what we think of as“empty” space cannot be completely empty because that wouldmean that all the fields, such as the gravitational andelectromagnetic fields, would have to be exactly zero. However,the value of a field and its rate of change with time are likethe position and velocity of a particle: the uncertainty principleimplies that the more accurately one knows one of thesequantities, the less accurately one can know the other. So inempty space the field cannot be fixed at exactly zero, becausethen it would have both a precise value (zero) and a preciserate of change (also zero). There must be a certain minimumamount of uncertainty, or quantum fluctuations, in the value ofthe field. One can think of these fluctuations as pairs ofparticles of light or gravity that appear together at some time,move apart, and then come together again and annihilate eachother. These particles are virtual particles like the particles thatcarry the gravitational force of the sun: unlike real particles,they cannot be observed directly with a particle detector.
However, their indirect effects, such as small changes in theenergy of electron orbits in atoms, can be measured and agreewith the theoretical predictions to a remarkable degree ofaccuracy. The uncertainty principle also predicts that there willbe similar virtual pairs of matter particles, such as electrons orquarks. In this case, however, one member of the pair will bea particle and the other an antiparticle (the antiparticles of lightand gravity are the same as the particles).
Because energy cannot be created out of nothing, one of thepartners in a particle/antiparticle pair will have positive energy,and the other partner negative energy. The one with negativeenergy is condemned to be a short-lived virtual particle becausereal particles always have positive energy in normal situations. Itmust therefore seek out its partner and annihilate with it.
However, a real particle close to a massive body has lessenergy than if it were far away, because it would take energyto lift it far away against the gravitational attraction of the body.
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