 Ask
anybody what the physical world is made of, and you are likely
to be told "matter and energy."
Yet if we have learned anything from engineering, biology and
physics, information is just as crucial an ingredient. The robot
at the automobile factory is supplied with metal and plastic but
can make nothing useful without copious instructions telling it
which part to weld to what and so on. A ribosome in a cell in
your body is supplied with amino acid building blocks and is
powered by energy released by the conversion of ATP to ADP, but
it can synthesize no proteins without the information brought to
it from the DNA in the cell's nucleus. Likewise, a century of
developments in physics has taught us that information is a
crucial player in physical systems and processes. Indeed, a
current trend, initiated by John A. Wheeler of Princeton
University, is to regard the physical world as made of
information, with energy and matter as incidentals.
This viewpoint invites a new look at venerable questions. The
information storage capacity of devices such as hard disk drives
has been increasing by leaps and bounds. When will such progress
halt? What is the ultimate information capacity of a device that
weighs, say, less than a gram and can fit inside a cubic
centimeter (roughly the size of a computer chip)? How much
information does it take to describe a whole universe? Could
that description fit in a computer's memory? Could we, as
William Blake memorably penned, "see the world in a grain of
sand," or is that idea no more than poetic license?
Remarkably, recent developments in theoretical physics answer
some of these questions, and the answers might be important
clues to the ultimate theory of reality. By studying the
mysterious properties of black holes, physicists have deduced
absolute limits on how much information a region of space or a
quantity of matter and energy can hold. Related results suggest
that our universe, which we perceive to have three spatial
dimensions, might instead be "written" on a two-dimensional
surface, like a hologram. Our everyday perceptions of the world
as three-dimensional would then be either a profound illusion or
merely one of two alternative ways of viewing reality. A grain
of sand may not encompass our world, but a flat screen might.
A Tale of Two Entropies
Formal information theory originated in seminal 1948 papers by
American applied mathematician Claude E. Shannon, who introduced
today's most widely used measure of information content: entropy.
Entropy had long been a central concept of thermodynamics, the
branch of physics dealing with heat. Thermodynamic entropy is
popularly described as the disorder in a physical system. In
1877 Austrian physicist Ludwig Boltzmann characterized it more
precisely in terms of the number of distinct microscopic states
that the particles composing a chunk of matter could be in while
still looking like the same macroscopic chunk of matter. For
example, for the air in the room around you, one would count all
the ways that the individual gas molecules could be distributed
in the room and all the ways they could be moving.
When Shannon cast about for a way to quantify the information
contained in, say, a message, he was led by logic to a formula
with the same form as Boltzmann's. The Shannon entropy of a
message is the number of binary digits, or bits, needed to
encode it. Shannon's entropy does not enlighten us about the
value of information, which is highly dependent on context. Yet
as an objective measure of quantity of information, it has been
enormously useful in science and technology. For instance, the
design of every modern communications device--from cellular
phones to modems to compact-disc players--relies on Shannon
entropy.
Thermodynamic entropy and Shannon entropy are conceptually
equivalent: the number of arrangements that are counted by
Boltzmann entropy reflects the amount of Shannon information one
would need to implement any particular arrangement. The two
entropies have two salient differences, though. First, the
thermodynamic entropy used by a chemist or a refrigeration
engineer is expressed in units of energy divided by temperature,
whereas the Shannon entropy used by a communications engineer is
in bits, essentially dimensionless. That difference is merely a
matter of convention.
 Even when reduced to common units, however, typical values of
the two entropies differ vastly in magnitude. A silicon
microchip carrying a gigabyte of data, for instance, has a
Shannon entropy of about 1010 bits (one byte is eight bits),
tremendously smaller than the chip's thermodynamic entropy,
which is about 1023 bits at room temperature. This discrepancy
occurs because the entropies are computed for different degrees
of freedom. A degree of freedom is any quantity that can vary,
such as a coordinate specifying a particle's location or one
component of its velocity. The Shannon entropy of the chip cares
only about the overall state of each tiny transistor etched in
the silicon crystal--the transistor is on or off; it is a 0 or a
1--a single binary degree of freedom. Thermodynamic entropy, in
contrast, depends on the states of all the billions of atoms (and
their roaming electrons) that make up each transistor. As
miniaturization brings closer the day when each atom will store
one bit of information for us, the useful Shannon entropy of the
state-of-the-art microchip will edge closer in magnitude to its
material's thermodynamic entropy. When the two entropies are
calculated for the same degrees of freedom, they are equal.
What are the ultimate degrees of freedom? Atoms, after all, are
made of electrons and nuclei, nuclei are agglomerations of
protons and neutrons, and those in turn are composed of quarks.
Many physicists today consider electrons and quarks to be
excitations of superstrings, which they hypothesize to be the
most fundamental entities. But the vicissitudes of a century of
revelations in physics warn us not to be dogmatic. There could
be more levels of structure in our universe than are dreamt of
in today's physics.
One cannot calculate the ultimate information capacity of a
chunk of matter or, equivalently, its true thermodynamic entropy,
without knowing the nature of the ultimate constituents of
matter or of the deepest level of structure, which I shall refer
to as level X. (This ambiguity causes no problems in analyzing
practical thermodynamics, such as that of car engines, for
example, because the quarks within the atoms can be ignored--they
do not change their states under the relatively benign
conditions in the engine.) Given the dizzying progress in
miniaturization, one can playfully contemplate a day when quarks
will serve to store information, one bit apiece perhaps. How
much information would then fit into our one-centimeter cube?
And how much if we harness superstrings or even deeper, yet
undreamt of levels? Surprisingly, developments in gravitation
physics in the past three decades have supplied some clear
answers to what seem to be elusive questions.
THE INFORMATION CONTENT of a pile of computer chips
increases in proportion with the number of chips or,
equivalently, the volume they occupy. That simple rule must
break down for a large enough pile of chips because eventually
the information would exceed the holographic bound, which
depends on the surface area, not the volume. The "breakdown"
occurs when the immense pile of chips collapses to form a black
hole.
Black Hole Thermodynamics
A central player in these developments is the black hole. Black
holes are a consequence of general relativity, Albert Einstein's
1915 geometric theory of gravitation. In this theory,
gravitation arises from the curvature of spacetime, which makes
objects move as if they were pulled by a force. Conversely, the
curvature is caused by the presence of matter and energy.
According to Einstein's equations, a sufficiently dense
concentration of matter or energy will curve spacetime so
extremely that it rends, forming a black hole. The laws of
relativity forbid anything that went into a black hole from
coming out again, at least within the classical (nonquantum)
description of the physics. The point of no return, called the
event horizon of the black hole, is of crucial importance. In
the simplest case, the horizon is a sphere, whose surface area
is larger for more massive black holes.
It is impossible to determine what is inside a black hole. No
detailed information can emerge across the horizon and escape
into the outside world. In disappearing forever into a black
hole, however, a piece of matter does leave some traces. Its
energy (we count any mass as energy in accordance with
Einstein's E = mc2) is permanently reflected in an increment in
the black hole's mass. If the matter is captured while circling
the hole, its associated angular momentum is added to the black
hole's angular momentum. Both the mass and angular momentum of a
black hole are measurable from their effects on spacetime around
the hole. In this way, the laws of conservation of energy and
angular momentum are upheld by black holes. Another fundamental
law, the second law of thermodynamics, appears to be violated.
The second law of thermodynamics summarizes the familiar
observation that most processes in nature are irreversible: a
teacup falls from the table and shatters, but no one has ever
seen shards jump up of their own accord and assemble into a
teacup. The second law of thermodynamics forbids such inverse
processes. It states that the entropy of an isolated physical
system can never decrease; at best, entropy remains constant,
and usually it increases. This law is central to physical
chemistry and engineering; it is arguably the physical law with
the greatest impact outside physics.
As first emphasized by Wheeler, when matter disappears into a
black hole, its entropy is gone for good, and the second law
seems to be transcended, made irrelevant. A clue to resolving
this puzzle came in 1970, when Demetrious Christodoulou, then a
graduate student of Wheeler's at Princeton, and Stephen W.
Hawking of the University of Cambridge independently proved that
in various processes, such as black hole mergers, the total area
of the event horizons never decreases. The analogy with the
tendency of entropy to increase led me to propose in 1972 that a
black hole has entropy proportional to the area of its horizon [see
illustration on preceding page]. I conjectured that when matter
falls into a black hole, the increase in black hole entropy
always compensates or overcompensates for the "lost" entropy of
the matter. More generally, the sum of black hole entropies and
the ordinary entropy outside the black holes cannot decrease.
This is the generalized second law--GSL for short.
The GSL has passed a large number of stringent, if purely
theoretical, tests. When a star collapses to form a black hole,
the black hole entropy greatly exceeds the star's entropy. In
1974 Hawking demonstrated that a black hole spontaneously emits
thermal radiation, now known as Hawking radiation, by a quantum
process [see "The Quantum Mechanics of Black Holes," by Stephen
W. Hawking; Scientific American, January 1977]. The
Christodoulou-Hawking theorem fails in the face of this
phenomenon (the mass of the black hole, and therefore its
horizon area, decreases), but the GSL copes with it: the entropy
of the emergent radiation more than compensates for the
decrement in black hole entropy, so the GSL is preserved. In
1986 Rafael D. Sorkin of Syracuse University exploited the
horizon's role in barring information inside the black hole from
influencing affairs outside to show that the GSL (or something
very similar to it) must be valid for any conceivable process
that black holes undergo. His deep argument makes it clear that
the entropy entering the GSL is that calculated down to level X,
whatever that level may be.
OUR INNATE PERCEPTION that the world is three-dimensional
could be an extraordinary illusion.
Hawking's radiation process allowed him to determine the
proportionality constant between black hole entropy and horizon
area: black hole entropy is precisely one quarter of the event
horizon's area measured in Planck areas. (The Planck length,
about 10-33 centimeter, is the fundamental length scale related
to gravity and quantum mechanics. The Planck area is its square.)
Even in thermodynamic terms, this is a vast quantity of entropy.
The entropy of a black hole one centimeter in diameter would be
about 1066 bits, roughly equal to the thermodynamic entropy of a
cube of water 10 billion kilometers on a side.
The World as a Hologram
The GSL allows us to set bounds on the information capacity of
any isolated physical system, limits that refer to the
information at all levels of structure down to level X. In 1980
I began studying the first such bound, called the universal
entropy bound, which limits how much entropy can be carried by a
specified mass of a specified size [see box on opposite page]. A
related idea, the holographic bound, was devised in 1995 by
Leonard Susskind of Stanford University. It limits how much
entropy can be contained in matter and energy occupying a
specified volume of space.
In his work on the holographic bound, Susskind considered any
approximately spherical isolated mass that is not itself a black
hole and that fits inside a closed surface of area A. If the
mass can collapse to a black hole, that hole will end up with a
horizon area smaller than A. The black hole entropy is therefore
smaller than A/4. According to the GSL, the entropy of the
system cannot decrease, so the mass's original entropy cannot
have been bigger than A/4. It follows that the entropy of an
isolated physical system with boundary area A is necessarily
less than A/4. What if the mass does not spontaneously collapse?
In 2000 I showed that a tiny black hole can be used to convert
the system to a black hole not much different from the one in
Susskind's argument. The bound is therefore independent of the
constitution of the system or of the nature of level X. It just
depends on the GSL.
We can now answer some of those elusive questions about the
ultimate limits of information storage. A device measuring a
centimeter across could in principle hold up to 1066 bits--a
mind-boggling amount. The visible universe contains at least
10100 bits of entropy, which could in principle be packed inside
a sphere a tenth of a light-year across. Estimating the entropy
of the universe is a difficult problem, however, and much larger
numbers, requiring a sphere almost as big as the universe itself,
are entirely plausible.
But it is another aspect of the holographic bound that is truly
astonishing. Namely, that the maximum possible entropy depends
on the boundary area instead of the volume. Imagine that we are
piling up computer memory chips in a big heap. The number of
transistors--the total data storage capacity--increases with the
volume of the heap. So, too, does the total thermodynamic
entropy of all the chips. Remarkably, though, the theoretical
ultimate information capacity of the space occupied by the heap
increases only with the surface area. Because volume increases
more rapidly than surface area, at some point the entropy of all
the chips would exceed the holographic bound. It would seem that
either the GSL or our commonsense ideas of entropy and
information capacity must fail. In fact, what fails is the pile
itself: it would collapse under its own gravity and form a black
hole before that impasse was reached. Thereafter each additional
memory chip would increase the mass and surface area of the
black hole in a way that would continue to preserve the GSL.
This surprising result--that information capacity depends on
surface area--has a natural explanation if the holographic
principle (proposed in 1993 by Nobelist Gerard 't Hooft of the
University of Utrecht in the Netherlands and elaborated by
Susskind) is true. In the everyday world, a hologram is a
special kind of photograph that generates a full three-dimensional
image when it is illuminated in the right manner. All the
information describing the 3-D scene is encoded into the pattern
of light and dark areas on the two-dimensional piece of film,
ready to be regenerated. The holographic principle contends that
an analogue of this visual magic applies to the full physical
description of any system occupying a 3-D region: it proposes
that another physical theory defined only on the 2-D boundary of
the region completely describes the 3-D physics. If a 3-D system
can be fully described by a physical theory operating solely on
its 2-D boundary, one would expect the information content of
the system not to exceed that of the description on the boundary.
A Universe Painted on Its Boundary
Can we apply the holographic principle to the universe at large?
The real universe is a 4-D system: it has volume and extends in
time. If the physics of our universe is holographic, there would
be an alternative set of physical laws, operating on a 3-D
boundary of spacetime somewhere, that would be equivalent to our
known 4-D physics. We do not yet know of any such 3-D theory
that works in that way. Indeed, what surface should we use as
the boundary of the universe? One step toward realizing these
ideas is to study models that are simpler than our real universe.
A class of concrete examples of the holographic principle at
work involves so-called anti-de Sitter spacetimes. The original
de Sitter spacetime is a model universe first obtained by Dutch
astronomer Willem de Sitter in 1917 as a solution of Einstein's
equations, including the repulsive force known as the
cosmological constant. De Sitter's spacetime is empty, expands
at an accelerating rate and is very highly symmetrical. In 1997
astronomers studying distant supernova explosions concluded that
our universe now expands in an accelerated fashion and will
probably become increasingly like a de Sitter spacetime in the
future. Now, if the repulsion in Einstein's equations is changed
to attraction, de Sitter's solution turns into the anti-de
Sitter spacetime, which has equally as much symmetry. More
important for the holographic concept, it possesses a boundary,
which is located "at infinity" and is a lot like our everyday
spacetime.
Using anti-de Sitter spacetime, theorists have devised a
concrete example of the holographic principle at work: a
universe described by superstring theory functioning in an
anti-de Sitter spacetime is completely equivalent to a quantum
field theory operating on the boundary of that spacetime [see
box above]. Thus, the full majesty of superstring theory in an
anti-de Sitter universe is painted on the boundary of the
universe. Juan Maldacena, then at Harvard University, first
conjectured such a relation in 1997 for the 5-D anti-de Sitter
case, and it was later confirmed for many situations by Edward
Witten of the Institute for Advanced Study in Princeton, N.J.,
and Steven S. Gubser, Igor R. Klebanov and Alexander M. Polyakov
of Princeton University. Examples of this holographic
correspondence are now known for spacetimes with a variety of
dimensions.
This result means that two ostensibly very different theories--not
even acting in spaces of the same dimension--are equivalent.
Creatures living in one of these universes would be incapable of
determining if they inhabited a 5-D universe described by string
theory or a 4-D one described by a quantum field theory of point
particles. (Of course, the structures of their brains might give
them an overwhelming "commonsense" prejudice in favor of one
description or another, in just the way that our brains
construct an innate perception that our universe has three
spatial dimensions; see the illustration on the opposite page.)
The holographic equivalence can allow a difficult calculation in
the 4-D boundary spacetime, such as the behavior of quarks and
gluons, to be traded for another, easier calculation in the
highly symmetric, 5-D anti-de Sitter spacetime. The
correspondence works the other way, too. Witten has shown that a
black hole in anti-de Sitter spacetime corresponds to hot
radiation in the alternative physics operating on the bounding
spacetime. The entropy of the hole--a deeply mysterious concept--equals
the radiation's entropy, which is quite mundane.
The Expanding Universe
Highly symmetric and empty, the 5-D anti-de Sitter universe is
hardly like our universe existing in 4-D, filled with matter and
radiation, and riddled with violent events. Even if we
approximate our real universe with one that has matter and
radiation spread uniformly throughout, we get not an anti-de
Sitter universe but rather a "Friedmann-Robertson-Walker"
universe. Most cosmologists today concur that our universe
resembles an FRW universe, one that is infinite, has no boundary
and will go on expanding ad infinitum.
Does such a universe conform to the holographic principle or the
holographic bound? Susskind's argument based on collapse to a
black hole is of no help here. Indeed, the holographic bound
deduced from black holes must break down in a uniform expanding
universe. The entropy of a region uniformly filled with matter
and radiation is truly proportional to its volume. A
sufficiently large region will therefore violate the holographic
bound.
In 1999 Raphael Bousso, then at Stanford, proposed a modified
holographic bound, which has since been found to work even in
situations where the bounds we discussed earlier cannot be
applied. Bousso's formulation starts with any suitable 2-D
surface; it may be closed like a sphere or open like a sheet of
paper. One then imagines a brief burst of light issuing
simultaneously and perpendicularly from all over one side of the
surface. The only demand is that the imaginary light rays are
converging to start with. Light emitted from the inner surface
of a spherical shell, for instance, satisfies that requirement.
One then considers the entropy of the matter and radiation that
these imaginary rays traverse, up to the points where they start
crossing. Bousso conjectured that this entropy cannot exceed the
entropy represented by the initial surface--one quarter of its
area, measured in Planck areas. This is a different way of
tallying up the entropy than that used in the original
holographic bound. Bousso's bound refers not to the entropy of a
region at one time but rather to the sum of entropies of locales
at a variety of times: those that are "illuminated" by the light
burst from the surface.
Bousso's bound subsumes other entropy bounds while avoiding
their limitations. Both the universal entropy bound and the 't
Hooft-Susskind form of the holographic bound can be deduced from
Bousso's for any isolated system that is not evolving rapidly
and whose gravitational field is not strong. When these
conditions are overstepped--as for a collapsing sphere of matter
already inside a black hole--these bounds eventually fail,
whereas Bousso's bound continues to hold. Bousso has also shown
that his strategy can be used to locate the 2-D surfaces on
which holograms of the world can be set up.
Augurs of a Revolution
Researchers have proposed many other entropy bounds. The
proliferation of variations on the holographic motif makes it
clear that the subject has not yet reached the status of
physical law. But although the holographic way of thinking is
not yet fully understood, it seems to be here to stay. And with
it comes a realization that the fundamental belief, prevalent
for 50 years, that field theory is the ultimate language of
physics must give way. Fields, such as the electromagnetic field,
vary continuously from point to point, and they thereby describe
an infinity of degrees of freedom. Superstring theory also
embraces an infinite number of degrees of freedom. Holography
restricts the number of degrees of freedom that can be present
inside a bounding surface to a finite number; field theory with
its infinity cannot be the final story. Furthermore, even if the
infinity is tamed, the mysterious dependence of information on
surface area must be somehow accommodated.
Holography may be a guide to a better theory. What is the
fundamental theory like? The chain of reasoning involving
holography suggests to some, notably Lee Smolin of the Perimeter
Institute for Theoretical Physics in Waterloo, that such a final
theory must be concerned not with fields, not even with
spacetime, but rather with information exchange among physical
processes. If so, the vision of information as the stuff the
world is made of will have found a worthy embodiment.
JACOB D. BEKENSTEIN has contributed to the foundation of
black hole thermodynamics and to other aspects of the
connections between information and gravitation. He is Polak
Professor of Theoretical Physics at the Hebrew University of
Jerusalem, a member of the Israel Academy of Sciences and
Humanities, and a recipient of the Rothschild Prize. Bekenstein
dedicates this article to John Archibald Wheeler (his Ph.D.
supervisor 30 years ago). Wheeler belongs to the third
generation of Ludwig Boltzmann's students: Wheeler's Ph.D.
adviser, Karl Herzfeld, was a student of Boltzmann's student
Friedrich Hasenöhrl.
Istanbul-
August
10th
2003
http://gulizk.com
Scientific American
August 2003
|
