AMBRIDGE, England — In the fall of 1973 Dr. Stephen W. Hawking, who has spent his entire professional career at the University of Cambridge, found himself ensnared in a horrendous and embarrassing calculation. Attempting to investigate the microscopic properties of black holes, the gravitational traps from which not even light can escape, Dr. Hawking discovered to his disbelief that they could leak energy and particles into space, and even explode in a fountain of high-energy sparks.

Dr. Hawking first held off publishing his results, fearing he was mistaken. When he reported them the next year in the journal Nature, he titled his paper simply "Black Hole Explosions?" His colleagues were dazzled and mystified.

Nearly 30 years later, they are still mystified. When they gathered in Cambridge this month to mark Dr. Hawking's 60th birthday with a weeklong workshop titled "The Future of Theoretical Physics and Cosmology," the ideas spawned by his calculation and its aftermath often took center stage.

They are ideas that touch on just about every bone-jarring abstruse concept in modern physics.

"Black holes are still fundamentally enigmatic objects," said Dr. Andrew Strominger, a Harvard physicist, who attended. "In fundamental physics, gravity and quantum mechanics are the big things we don't understand. Hawking's discovery of black hole radiation was of fundamental importance to that connection."

Jonathan Player for The New York Times Dr. Stephen W. Hawking, with his wife, Elaine Mason, at a symposium at Cambridge held to honor his 60th birthday and to consider the consequences of his research. Essay: Cracking the Cosmic Code With a Little Help From Dr. Hawking (December 11, 2001) It's All in the Mathematics When Stephen Hawking startled cosmologists by asserting that energy and matter could leak out of black holes, his calculations did not say how particles escaped from the black hole, only that they could. The only truth is in the mathematics, he says. According to Heisenberg's uncertainty principle, a pillar of quantum theory, the so-called vacuum of space is not empty but rather foaming with virtual particles that flash into existence in particle-antiparticle pairs on borrowed energy and then meet and annihilate each other in a flash of energy that repays the debt of their existence. If only one member of a pair fell into a black hole, though, its mate would be free to wander away. To a distant observer it would appear to be coming out of the black hole, and, since the energy for its creation had been borrowed from the black hole's gravitational field and then not been paid back, the black hole would accordingly appear to shrink. As the black hole shrank it would get hotter and radiate faster, according to Dr. Hawking's calculations, until it finally exploded. The mortality of a black hole was of little practical concern. A typical black hole would last 1064 years, trillions of times the age of the universe. Jonathan Player for New York Times; Reuters (bottom) Dr. Leonard Susskind of Stanford, top, has sparred with Dr. Stephen Hawking on the properties of black holes. The Hawking workshop drew leading theoretical physicists, including, from second photo, Dr. Andrew Strominger of Harvard, Dr. Raffael Bousso of California at Santa Barbara and Dr. Gerard 't Hooft of Utrecht.

Black holes are the prima donnas of Einstein's general theory of relativity, which explains the force known as gravity as a warp in space-time caused by matter and energy. But even Einstein could not accept the idea that the warping could get so extreme, say in the case of a collapsing star, that space could wrap itself completely around some object like a magician's cloak, causing it to disappear as a black hole.

Dr. Hawking's celebrated breakthrough resulted partly from a fight. He was hoping to disprove the contention of Jacob Bekenstein, then a graduate student at Princeton and now a professor at the Hebrew University in Jerusalem, that the area of a black hole's boundary, the point of no return in space, was a measure of the entropy of a black hole. In thermodynamics, the study of heat and gases, entropy is a measure of wasted energy or disorder, which might seem like a funny concept to crop up in black holes. But in physics and computer science, entropy is also a measure of the information capacity of a system — the number of bits that it would take to describe its internal state. In effect, a black hole or any other system was like a box of Scrabble letters — the more letters in the box the more words you could make, and the more chances of gibberish.

According to the second law of thermodynamics, the entropy of a closed system always stays the same or increases, and Dr. Hawking's own work had shown that the hole's surface area always increased, a process that seemed to ape that law.

But Dr. Hawking, citing classical physics, argued that an object with entropy had to have a temperature, and anything with a temperature — from a fevered brow to a star — must radiate heat and light with a characteristic spectrum. If a black hole could not radiate, it could have no temperature and thus no entropy. But that was before gravity, which shapes the cosmos, met quantum theory, the paradoxical rules that describe the behavior of matter and forces within it. When Dr. Hawking added a touch of quantum uncertainty to the standard Einsteinian black hole model, particles started emerging. At first he was annoyed, but when he realized this "Hawking radiation" would have the thermal spectrum predicted by thermodynamic theory, he concluded his calculation was right.

But there was a problem. The radiation was random, Dr. Hawking's theory said. As a result, all the details about whatever had fallen into the black hole could be completely erased — a violation of a hallowed tenet of quantum theory, which holds that it should always be possible to run the film backwards and find out the details of how something started — whether an elephant or a Volkswagen had been tossed into the black hole, for example. If he was right, Dr. Hawking suggested, quantum theory might have to be modified. Black holes, he said in his papers and talks in the late 1970's, were ravagers of information, spewing indeterminacy and undermining law and order in the universe.

"God not only plays dice with the universe," Dr. Hawking said, inverting the phrase by which Einstein had famously rejected quantum uncertainty, "but sometimes throws them where we can't see them." Such statements aroused the attention of particle physicists. Weird as it may be, quantum theory is nonetheless the foundation on which much of the modern world is built, everything from transistors to CD's, and it is the language in which all of the fundamental laws of physics, save gravity, are expressed. "This cannot be," Dr. Leonard Susskind, a theorist at Stanford, recalled saying to himself.

It was the beginning of what Dr. Susskind calls an adversarial relationship. "Stephen Hawking is one of the most obstinate people in the world; no, he is the most infuriating person in the universe," Dr. Susskind told the birthday workshop, as Dr. Hawking grinned in the back row.

In the ensuing 20 years, opinions have split mostly along party lines. Particle physicists like Dr. Susskind and Dr. Gerard 't Hooft, a physicist at the University of Utrecht and the 1999 Nobel Prize winner, defend quantum theory and say that the information must get out somehow, perhaps subtly encoded in the radiation. Another possibility — that the information was left behind in some new kind of elementary particle when the black hole evaporated — seems to have fallen from favor.

Relativity experts like Dr. Hawking and his friend the Caltech physicist Dr. Kip Thorne were more likely to believe in the power of black holes to keep secrets. In 1997, Dr. Hawking and Dr. Thorne put their money where the black hole mouth was, betting Dr. John Preskill, a Caltech particle physicist, a set of encyclopedias that information was destroyed in a black hole.

To date neither side has felt obliged to pay up.

Writing on the Wall

Dr. Susskind and others have argued that nothing ever makes it into the black hole to begin with because, in accord with Einstein, everything at the boundary, where time slows, would appear to an outside observer to "freeze" and then fade, spreading out on the surface where it could produce subtle distortions in the Hawking radiation.

In principle, then, information about what had fallen onto the black hole could be read in the radiation and reconstructed; it would not have disappeared.

The confusion had arisen, Dr. Susskind explained, because physicists had been trying to imagine the situation from the viewpoint of God rather than that of a particular observer who had to be either in the black hole or outside, but not both places at once. When the accounting is done properly, he said, "No observer sees a violation of the laws of physics."

The information paradox made it important for theorists to try to go beyond thermodynamic analogies and actually calculate how black holes store information or entropy. But there was a catch. According to a well-known formula developed by the Austrian physicist Ludwig Boltzmann (and engraved on his tombstone), the entropy of a system could be determined by counting the number of ways its contents could be arranged.

In order to enumerate the possible ways of arranging the contents of a black hole, physicists needed a theory of what was inside. By the mid- 1990's they had one: string theory, which portrays the forces and particles of nature, including those responsible for gravity, as tiny vibrating strings.

In this theory, a black hole is a tangled mélange of strings and multidimensional membranes known as "D-branes." In a virtuoso calculation in 1995, Dr. Strominger and Dr. Cumrun Vafa, also of Harvard, untangled the innards of an "extremal" black hole, in which electrical charge just balanced gravity.

Such a hole would stop evaporating and would thus appear static, allowing the researchers to count its quantum states. They calculated that the entropy of a black hole was its area divided by four — just as Dr. Hawking and Dr. Bekenstein said it would be.

The result was a huge triumph for string theory. "If string theory had been wrong, that would have been deadly," Dr. Strominger said.

The success of the Harvard calculation has encouraged some particle physicists to conclude that black holes can be analyzed with the tools of quantum mechanics, and thus that the information issue has been resolved. But others say this has yet to be accomplished — among them Dr. Strominger, who added, "It remains an unsettled issue."

Degrees of Freedom

Perhaps the most mysterious and far-reaching consequence of the exploding black hole is the idea that the universe can be compared to a hologram, in which information for a three-dimensional image can be stored on a flat surface, like an image on a bank card.

In the 1980's, extending his and Dr. Hawking's work, Dr. Bekenstein showed that the entropy and thus the information needed to describe any object were limited by its area. "Entropy is a measure of how much information you can pack into an object," he explained. "The limit on entropy is a limit on information."

This was a strange result. Normally you might think that there were as many choices — or degrees of freedom about the inner state of an object — as there were points inside that space. But according to the so- called Bekenstein bound, there were only as many choices as there were points on its outer surface.

The "points" in this case are regions with the dimensions of 10^-33 centimeters, the so-called Planck length that physicists believe are the "grains" of space. According to the theory, each of these can be assigned a value of zero or one — yes or no — like the bits in a computer.

"What happens when you squeeze too much information into an object is that you pack more and more energy in," Dr. Bousso said. But if it gets too heavy for its size, it becomes a black hole, and then "the game is over," as he put it. "Like a piano with lots of keys but you can't press more than five of them at once or the piano will collapse."

The holographic principle, first suggested by Dr. 't Hooft in 1993 and elaborated by Dr. Susskind a year later, says in effect that if you can't use the other piano keys, they aren't really there. "We had a completely wrong picture of the piano," explained Dr. Bousso. The normal theories that physics uses to describe events in space-time are redundant in some surprising and as yet mysterious way. "We clearly see the world the way we see a hologram," Dr. Bousso said. "We see three dimensions. When you look at one of those chips, it looks pretty real, but in our case the illusion is perfect."

Dr. Susskind added: "We don't read the hologram. We are the hologram."

The holographic principle, these physicists say, can be applied to any space-time, but they have no idea why it works.

"It really should be mysterious," Dr. Strominger said. "If it's really true, it's a deep and beautiful property of our universe — but not an obvious one."

The Frontiers of Beauty

That beauty, however, comes at a price, said Dr. 't Hooft, namely cause and effect. If the information about what we are doing resides on distant imaginary walls, "how does it appear to us sitting here that we are obeying the local laws of physics?" he asked the audience at the Hawking birthday workshop.

Quantum mechanics had been saved, he declared, but it still might need to be supplanted by laws that would preserve what physicists call "naïve locality."

Dr. 't Hooft acknowledged that there had been many futile attempts to eliminate quantum mechanics' seemingly nonsensical notions, like particles that can instantaneously react to one another across light- years of space. In each case, however, he said there were assumptions, or "fine print," that might not hold up in the end.

Recent observations have raised the stakes for ideas like holography and black hole information. The results suggest that the expansion of the universe is accelerating. If it goes on, astronomers say, distant galaxies will eventually be moving away so fast that we will not be able to see them anymore.

Living in such a universe is like being surrounded by a horizon, glowing just like a black hole horizon, over which information is forever disappearing. And since this horizon has a finite size, physicists say, there is a limit to the amount of complexity and information the universe can hold, ultimately dooming life.

Physicists admit that they do not know how to practice physics or string theory in such a space, called a de Sitter space after the Dutch astronomer Willem de Sitter, who first solved Einstein's equations to find such a space. "De Sitter space is a new frontier," said Dr. Strominger, who hopes that the techniques and attention that were devoted to black holes in the last decade will enable physicists to make headway in understanding a universe that may actually represent the human condition.

Dr. Bousso noted that it was only in the last few years, with the discovery of D-branes, that it had been possible to solve black holes. What other surprises await in string theory? "We have no idea how small or large a piece of the theory we haven't seen yet," he said.

In the meantime, perhaps in imitation of Boltzmann, Dr. Hawking declared at the end of the meeting that he wanted the formula for black hole entropy engraved on his own tombstone.