My dissertation, finished at long last (illustration credit 6.1)
At first I couldn’t understand what the point was. Penrose had shown that once a dying star contracted to a certain radius, there would inevitably be a singularity, a point where space and time came to an end. Surely, I thought, we already knew that nothing could prevent a massive cold star from collapsing under its own gravity until it reached a singularity of infinite density. But in fact the equations had been solved only for the collapse of a perfectly spherical star, and of course a real star won’t be exactly spherical. If Lifshitz and Khalatnikov were right, the departures from spherical symmetry would grow as the star collapsed, and would cause different parts of the star to miss each other, thus avoiding a singularity of infinite density. But Penrose showed they were wrong: small departures from spherical symmetry will not prevent a singularity.
I realized that similar arguments could be applied to the expansion of the universe. In this case, I could prove there were singularities where space-time had a beginning. So again Lifshitz and Khalatnikov were wrong. General relativity predicted that the universe should have a beginning, a result that did not pass unnoticed by the Church.
The original singularity theorems of both Penrose and myself required the assumption that the universe had a Cauchy surface, that is, a surface that intersects every particle path once and only once. It was therefore possible that our first singularity theorems simply proved that the universe didn’t have a Cauchy surface. While interesting, this didn’t compare in importance with time having a beginning or end. I therefore set about proving singularity theorems that didn’t require the assumption of a Cauchy surface.
In the next five years, Roger Penrose, Bob Geroch, and I developed the theory of causal structure in general relativity. It was a wonderful feeling, having a whole field virtually to ourselves. How unlike particle physics, where people were falling over themselves to latch on to the latest idea. They still are.
I wrote up some of this in an essay that won an Adams Prize at Cambridge in 1966. This was the basis for the book The Large Scale Structure of Space-Time, which I wrote with George Ellis and which was published by Cambridge University Press in 1973. The book is still in print because it is virtually the last word on the causal structure of space-time: that is, which pole of space-time can affect events at other points. I would caution the general reader against attempting to consult it. It is highly technical and was written at a time when I was trying to be as rigorous as a pure mathematician. Nowadays I’m concerned to be right rather than righteous. Anyway, it is almost impossible to be rigorous in quantum physics, because the whole field is on very shaky mathematical ground.
7
BLACK HOLES
THE IDEA BEHIND BLACK HOLES GOES BACK MORE than two hundred years. In 1783 a Cambridge don, John Michell, published a paper in Philosophical Transactions of the Royal Society of London about what he called “dark stars.” He pointed out that a star that was sufficiently massive and compact would have such a strong gravitational field that light could not escape. Any light emitted from the surface of the star would be dragged back by the star’s gravitational attraction before it could get very far.
Michell suggested that there might be a large number of stars like this. Although we would not be able to see them, because the light from them would not reach us, we would still feel their gravitational attraction. Such objects are what we now call black holes, because that is what they are: black voids in space. A similar suggestion was made a few years later by a French scientist, the Marquis de Laplace, apparently independently of Michell. Interestingly enough, Laplace included it in only the first and second editions of his book The System of the World and left it out of later editions. Perhaps he decided that it was a crazy idea.
Both Michell and Laplace thought of light as consisting of particles, rather like cannonballs, that could be slowed down by gravity and made to fall back on the star. This was not consistent with the Michelson-Morley experiment, carried out in 1887, which showed that light always travels at the same speed. A consistent theory of how gravity affects light did not come until 1915, when Einstein formulated general relativity. Using general relativity, Robert Oppenheimer and his students George Volkoff and Hartland Snyder showed in 1939 that a star that had exhausted its nuclear fuel could not support itself against gravity if its mass was greater than a certain limit, about the order of the mass of the Sun. Burnt-out stars above this mass would collapse in on themselves and form black holes containing singularities of infinite density. Although they were a prediction of his theory, Einstein never accepted black holes or that matter could be compressed to infinite density.
Then the war intervened and diverted Oppenheimer to work on the atomic bomb. After the war, people were more interested in atomic and nuclear physics and neglected gravitational collapse and black holes for more than twenty years.
INTEREST IN gravitational collapse was reawakened in the early 1960s with the discovery of quasars, very distant objects that are very compact and powerful optical and radio sources. Matter falling into a black hole was the only plausible mechanism that could explain the production of so much energy in so small a region of space. Oppenheimer’s work was rediscovered and people began to work on the theory of black holes.
In 1967 Werner Israel produced an important result. He showed that unless the remnant from a non-rotating collapsing star was exactly spherical, the singularity it contained would be naked—that is, it would be visible to outside observers. This would have meant the breakdown of general relativity at the singularity of a collapsing star, destroying our ability to predict the future of the rest of the universe.
At first, most people, including Israel himself, thought this implied that because real stars aren’t exactly spherical, their collapse would give rise to naked singularities and a breakdown of predictability. However, a different interpretation was put forward by Roger Penrose and John Wheeler: that the remnant from the gravitational collapse of a non-rotating star would rapidly settle down to a spherical state. They suggested that there is cosmic censorship: nature is a prude and hides singularities in black holes, where they can’t be seen.
I used to have a bumper sticker that read BLACK HOLES ARE OUT OF SIGHT on the door of my office in DAMTP. This so irritated the head of the department that he engineered my election to the Lucasian Professorship, moved me to a better office on the strength of it, and personally tore the offending notice off the door of the old office.
MY WORK on black holes began with a eureka moment in 1970, a few days after the birth of my daughter, Lucy. While getting into bed, I realized that I could apply to black holes the causal structure theory I had developed for singularity theorems. In particular, the area of the horizon, the boundary of the black hole, would always increase. When two black holes collide and merge, the area of the final black hole is greater than the sum of the areas of the original holes. This, and other properties that Jim Bardeen, Brandon Carter, and I discovered, suggested that the area was like the entropy of a black hole. This would be a measure of how many states a black hole could have on the inside for the same appearance on the outside. But the area couldn’t actually be the entropy, because if black holes had entropy, they would also have a temperature and would glow like a hot body. As everyone thought, black holes were completely black and didn’t emit light or anything else.
There was an exciting period culminating in the Les Houches summer school in 1972 in which we solved most of the major problems in black hole theory. In particular, David Robinson and I proved the no-hair theorem, which said that a black hole would settle down to a state characterized by only two numbers, the mass and the rotation. This again suggested that black holes had entropy, because many different stars could collapse to produce a black hole of the same mass and rotation.
All this theory was developed before there was any observational evidence for black holes, which shows that Feynman was wrong when he said an active re
search field has to be experimentally driven. The one problem that was never solved was to prove the cosmic censorship hypothesis, though a number of attempts to disprove it failed. It is fundamental to all work on black holes, so I have a strong vested interest in its being true. I therefore have a bet with Kip Thorne and John Preskill on the outcome of this problem. It is difficult for me to win this bet, but quite possible for me to lose if anyone finds a counterexample with a naked singularity. In fact, I lost an earlier version of the bet, by not being careful enough about the wording. Thorne and Preskill were not amused by the T-shirt I offered in settlement.
Cosmology humor, part one:
I had this printed on a T-shirt to settle a bet
WE WERE so successful with the classical general theory of relativity that I was at a bit of a loose end in 1973, after the publication of The Large Scale Structure of Space-Time. My work with Penrose had shown that general relativity would break down at singularities. So the obvious next step would be to combine general relativity, the theory of the very large, with quantum theory, the theory of the very small. I had no background in quantum theory, and the singularity problem seemed too difficult for a frontal assault at that time. So as a warm-up exercise, I considered how particles and fields governed by quantum theory would behave near a black hole. In particular, I wondered, can one have atoms in which the nucleus is a tiny primordial black hole, formed in the early universe?
To answer this, I studied how quantum fields would scatter off a black hole. I was expecting that part of an incident wave would be absorbed and the remainder scattered. But to my great surprise, I found that there seemed to be emission from the black hole. At first I thought this must be a mistake in my calculation. What finally persuaded me that it was real was that the emission was exactly what was required to identify the area of the horizon with the entropy of a black hole. It is summed up in this simple formula: where S is the entropy and A is the area of horizon. This expression contains the three fundamental constants of nature: c, the speed of light; G, Newton’s constant of gravitation; and ħ bar, Planck’s constant. It reveals that there is a deep and previously unsuspected relationship between gravity and thermodynamics, the science of heat.
The radiation from a black hole will carry away energy, so the black hole will lose mass and shrink. Eventually, it seems, the black hole will evaporate completely and disappear. This raised a problem that struck at the heart of physics. My calculation suggested that the radiation was exactly thermal and random, as it has to be if the area of the horizon is to be the entropy of the black hole. So how could the radiation left over carry all the information about what made the black hole? Yet if information is lost, this is incompatible with quantum mechanics.
Cosmology humor, part two: a bet with John Preskill (illustration credit 7.1)
This paradox had been argued for thirty years, without much progress, until I found what I think is its resolution. Information is not lost, but it is not returned in a useful way. It is like burning an encyclopedia: the information contained in the encyclopedia is not technically lost if one keeps all the smoke and ashes, but it is very hard to read. In fact, Kip Thorne and I had a bet with John Preskill on the information paradox. When John won the bet, I gave him a baseball encyclopedia, but maybe I should have just given him the ashes.
8
CALTECH
IN 1974 I WAS ELECTED A FELLOW OF THE ROYAL Society. The election came as a surprise to members of my department because I was young and only a lowly research assistant. But within three years I had been promoted to professor.
Jane became depressed after my election, feeling I had achieved my goals and that it was going to be downhill after that. Her depression was lifted somewhat when my friend Kip Thorne invited us and a number of others working in general relativity to the California Institute of Technology (Caltech).
Our house in Pasadena (illustration credit 8.1)
For the past four years, I had been using a manual wheelchair as well as a blue electric three-wheeled car, which went at a slow cycling speed, and in which I sometimes illegally carried passengers. When we went to California, we stayed in a Caltech-owned colonial-style house near the campus, and there I used an electric wheelchair for the first time. It gave me a considerable degree of independence, especially as in the United States buildings and sidewalks are much more accessible for the disabled than they are in Britain. I also had one of my research students live with us. He helped me with getting up and going to bed and some meals, in return for accommodation and a lot of my academic attention.
Jane, Lucy, Robert, and me at home in Pasadena (above and below) (illustration credit 8.2)
Our two children at that time, Robert and Lucy, loved California. The school they attended there was afraid its students would be kidnapped, so one couldn’t just collect one’s child from the school gate in the normal way. Instead one had to drive around the block and come to the gate one by one. The child in question would then be summoned over a bullhorn. I’d never encountered anything like this before.
The house was equipped with a color television set. In England, we’d had only a black-and-white set that hardly worked. So we watched a lot of television, particularly British series such as Upstairs, Downstairs and The Ascent of Man. We had just watched the episode of The Ascent of Man in which Galileo is tried by the Vatican and condemned to house arrest for the rest of his life when I heard that I had been awarded the Pius XI Medal by the Pontifical Academy of Sciences. At first I felt like indignantly refusing it, but then I had to admit that the Vatican had ultimately changed its mind about Galileo. So I flew to England to meet up with my parents, who then accompanied me to Rome. While visiting the Vatican, I made a point of demanding to be shown the account of the trial of Galileo in the Vatican library.
(illustration credit 8.3)
At the award ceremony, Pope Paul VI got down from his throne and knelt by my side. After the ceremony I met Paul Dirac, one of the founders of quantum theory, to whom I had not talked while he was a professor at Cambridge because I had not at that time been interested in matters quantum. He told me he had originally proposed another candidate for the medal but in the end had decided I was better and had told the academy to award it to me.
THE TWO principal stars of the Caltech physics department at that time were the Nobel Prize winners Richard Feynman and Murray Gell-Mann, and there was great rivalry between them. At the first of Gell-Mann’s weekly seminars, he said, “I’m just going to repeat some talks I gave last year,” whereupon Feynman got up and walked out. Gell-Mann then said, “Now that he’s gone, I can tell you what I really wanted to talk about.”
This was an exciting time in particle physics. New “charmed” particles had just been discovered at Stanford, and the discovery helped confirm Gell-Mann’s theory that protons and neutrons were made of three more fundamental particles called quarks.
While at Caltech, I bet Kip Thorne that the binary star system Cygnus X-1 did not contain a black hole. Cygnus X-1 is an X-ray source in which a normal star is losing its outer envelope to an unseen compact companion. As matter falls toward the companion, it develops a spiral motion and gets very hot, emitting X-rays. I was hoping to lose this bet, as I obviously had made a big intellectual investment in black holes. But if they were shown not to exist, at least I would have had the consolation of winning a four-year subscription to Private Eye magazine. On the other hand, if Kip won, he would receive one year of Penthouse magazine. In the years following the bet, the evidence for black holes became so strong that I conceded and gave Kip a subscription to Penthouse, much to the displeasure of his wife.
WHILE IN California, I worked with a research student at Caltech, Don Page. Don had been born and brought up in a village in Alaska where his parents were schoolteachers and the three of them were the only non-Inuits. He was an evangelical Christian, and he did his best to convert me when he later came to live with us in Cambridge. He used to read me Bible stories at breakfast, b
ut I told him I knew the Bible well from my time in Majorca, and because my father used to read the Bible to me. (My father was not a believer but thought the King James Bible was culturally important.)
Don and I worked on whether it might be possible to observe the emission from black holes that I had predicted. The temperature of the radiation from a black hole of the mass of the Sun would be only about a millionth of a kelvin, barely above absolute zero, so it would be swamped by the cosmic background of microwaves, which has a temperature of 2.7 kelvin. However, there might be much smaller black holes left over from the Big Bang. A primordial black hole with the mass of a mountain would be emitting gamma rays and would now be ending its lifetime, having radiated away most of its original mass. We looked for evidence of such emissions in the background of gamma rays but found no sign. We were able to place an upper limit on the number density of black holes of this mass, which indicates that we are not likely to be close enough to one to detect it.
My Brief History Page 4
