by Jacob D. Bekenstein
Scientific American August 2003
from
Essentia Website
An
astonishing theory called the holographic principle
holds that the universe is like a hologram: just as a
trick of light allows a fully threedimensional image to
be recorded on a flat piece of film, our seemingly
threedimensional universe could be completely
equivalent to alternative quantum fields and physical
laws "painted" on a distant, vast surface.
The physics of black holesimmensely dense
concentrations of massprovides a hint that the
principle might be true. Studies of black holes show
that, although it defies common sense, the maximum
entropy or information content of any region of space is
defined not by its volume but by its surface area.
Physicists hope that this surprising finding is a clue
to the ultimate theory of reality. 
An astonishing theory called the
holographic principle holds that the universe is like a hologram:
just as a trick of light allows a fully threedimensional image to
be recorded on a flat piece of film, our seemingly threedimensional
universe could be completely equivalent to alternative quantum
fields and physical laws "painted" on a distant, vast surface.
The physics of black holes  immensely dense concentrations of
mass  provides a hint that the principle might be true. Studies of
black holes show that, although it defies common sense, the maximum
entropy or information content of any region of space is defined not
by its volume but by its surface area.
Physicists hope that this surprising finding is a clue to the
ultimate theory of reality.
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
OUR INNATE
PERCEPTION that the world is threedimensional could be an
extraordinary illusion. 
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 twodimensional surface, like a
hologram.
Our everyday perceptions of the world as threedimensional 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.
The
Entropy of a Black Hole
THE ENTROPY
OF A BLACK HOLE is proportional to the area of its event
horizon, the surface within which even light cannot
escape the gravity of the hole. Specifically, a hole
with a horizon spanning A Planck areas has A/4 units of
entropy. (The Planck area, approximately 1066 square
centimeter, is the fundamental quantum unit of area
determined by the strength of gravity, the speed of
light and the size of quanta.) Considered as
information, it is as if the entropy were written on the
event horizon, with each bit (each digital 1 or 0)
corresponding to four Planck areas. 
The Entropy of a Black Hole is
proportional to the area of its event horizon, the surface within
which even light cannot escape
the gravity of the hole. Specifically, a
hole with a horizon spanning A Planck areas has A/4 units of
entropy. (The Planck area, approximately 1066 square centimeter, is
the fundamental quantum unit of area determined by the strength of
gravity, the speed of light and the size of quanta.)
Considered as
information, it is as if the entropy were written on the event
horizon, with each bit (each digital 1 or 0) corresponding to four
Planck areas.
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 devicefrom cellular phones to modems to
compactdisc playersrelies 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.
Limits
of Functional Density
The thermodynamics of black holes allows one to deduce limits on the
density of entropy or information in various circumstances. The
holographic bound defines how much information can be contained in a
specified region of space. It can be derived by considering a
roughly spherical distribution of matter that is contained within a
surface of area A. The matter is induced to collapse to form a black
hole (a). The black hole’s area must be smaller than A, so its
entropy must be less than A/4 [see illustration]. Because entropy
cannot decrease, one infers that the original distribution of matter
also must carry less than A/4 units of entropy or information. This
resultthat the maximum information content of a region of space is
fixed by its areadefies the commonsense expectation that the
capacity of a region should depend on its volume.
The universal entropy bound defines how much information can be
carried by a mass m of diameter d. It is derived by imagining that a
capsule of matter is engulfed by a black hole not much wider than it
(b). The increase in the black hole’s size places a limit on how
much entropy the capsule could have contained. This limit is tighter
than the holographic bound, except when the capsule is almost as
dense as a black hole (in which case the two bounds are equivalent).
The holographic and universal information bounds are far beyond the
data storage capacities of any current technology, and they greatly
exceed the density of information on chromosomes and the
thermodynamic entropy of water (c).
THE
THERMODYNAMICS OF BLACK HOLES allows one to deduce
limits on the density of entropy or information in
various circumstances.
The holographic bound defines how much information can
be contained in a specified region of space. It can be
derived by considering a roughly spherical distribution
of matter that is contained within a surface of area A.
The matter is induced to collapse to form a black hole
(a). The black hole’s area must be smaller than A, so
its entropy must be less than A/4 [see illustration].
Because entropy cannot decrease, one infers that the
original distribution of matter also must carry less
than A/4 units of entropy or information. This
resultthat the maximum information content of a region
of space is fixed by its areadefies the commonsense
expectation that the capacity of a region should depend
on its volume.
The universal entropy bound defines how much information
can be carried by a mass m of diameter d. It is derived
by imagining that a capsule of matter is engulfed by a
black hole not much wider than it (b). The increase in
the black hole’s size places a limit on how much entropy
the capsule could have contained. This limit is tighter
than the holographic bound, except when the capsule is
almost as dense as a black hole (in which case the two
bounds are equivalent).
The holographic and universal information bounds are far
beyond the data storage capacities of any current
technology, and they greatly exceed the density of
information on chromosomes and the thermodynamic entropy
of water (c). 
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 crystalthe
transistor is on or off; it is a 0 or a 1a 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 stateoftheart 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 ignoredthey 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 onecentimeter 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 = mc^{2})
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.
Holographic SpaceTime
Two universes of different dimension and obeying disparate physical
laws are rendered completely equivalent by the holographic
principle. Theorists have demonstrated this principle mathematically
for a specific type of fivedimensional spacetime ("antide Sitter")
and its fourdimensional boundary. In effect, the 5D universe is
recorded like a hologram on the 4D surface at its periphery.
Superstring theory rules in the 5D spacetime, but a socalled
conformal field theory of point particles operates on the 4D
hologram. A black hole in the 5D spacetime is equivalent to hot
radiation on the hologramfor example, the hole and the radiation
have the same entropy even though the physical origin of the entropy
is completely different for each case. Although these two
descriptions of the universe seem utterly unalike, no experiment
could distinguish between them, even in principle.
TWO
UNIVERSES of different dimension and obeying disparate
physical laws are rendered completely equivalent by the
holographic principle. Theorists have demonstrated this
principle mathematically for a specific type of
fivedimensional spacetime ("anti–de Sitter") and its
fourdimensional boundary. In effect, the 5D universe
is recorded like a hologram on the 4D surface at its
periphery.
Superstring
theory rules in the 5D spacetime, but a socalled
conformal field theory of point particles operates on
the 4D hologram. A black hole in the 5D spacetime is
equivalent to hot radiation on the hologramfor
example, the hole and the radiation have the same
entropy even though the physical origin of the entropy
is completely different for each case. Although these
two descriptions of the universe seem utterly unalike,
no experiment could distinguish between them, even in
principle. 
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. 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 lawGSL for short.
Our innate perception that the world is threedimensional 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
1033 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 bitsa mindboggling
amount. The visible universe contains at least 10100 bits of
entropy, which could in principle be packed inside a sphere a tenth
of a lightyear 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
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. 
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 transistorsthe total data storage
capacityincreases 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 resultthat information capacity depends on surface
areahas a natural explanation if the holographic principle
(proposed in 1993 by Novelist 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 threedimensional image when it is illuminated in
the right manner. All the information describing the 3D scene is
encoded into the pattern of light and dark areas on the
twodimensional 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 3D region: it proposes that
another physical theory defined only on the 2D boundary of the
region completely describes the 3D physics. If a 3D system can be
fully described by a physical theory operating solely on its 2D
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 4D 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 3D boundary of
spacetime somewhere, that would be equivalent to our known 4D
physics. We do not yet know of any such 3D 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 socalled antide 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 antide 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 antide Sitter spacetime, theorists have devised a concrete
example of the holographic principle at work: a universe described
by superstring theory functioning in an antide 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 antide 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 5D
antide 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 theoriesnot
even acting in spaces of the same dimensionare equivalent.
Creatures living in one of these universes would be incapable of
determining if they inhabited a 5D universe described by string
theory or a 4D 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
4D boundary spacetime, such as the behavior of quarks and gluons,
to be traded for another, easier calculation in the highly
symmetric, 5D antide Sitter spacetime. The correspondence works
the other way, too. Witten has shown that a black hole in antide
Sitter spacetime corresponds to hot radiation in the alternative
physics operating on the bounding spacetime. The entropy of the
holea deeply mysterious conceptequals the radiation’s entropy,
which is quite mundane.
The
Expanding Universe
Highly symmetric and empty, the 5D antide Sitter universe is
hardly like our universe existing in 4D, 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 antide Sitter universe but
rather a "FriedmannRobertsonWalker" 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 2D 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
surfaceone 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
HooftSusskind 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
oversteppedas for a collapsing sphere of matter already inside a
black holethese bounds eventually fail, whereas Bousso’s bound
continues to hold. Bousso has also shown that his strategy can be
used to locate the 2D surfaces on which holograms of the world can
be set up.
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.
