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."
Player for The New York TimesDr. 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
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
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
Player for New York Times; Reuters
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
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
"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
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 mιlange 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
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
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
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
Dr. Susskind added: "We don't read the hologram. We are the
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 "naοve 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
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
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