A Scientist Takes On Gravity
Elwood H. Smith
By DENNIS OVERBYE
Published: July 12, 2010
Kirsten Luce for The New York Times
ZERO GRAVITY Dr. Erik Verlinde says, “For me gravity doesn’t exist.” In a recent paper he expounded on his theory.
Zero Gravity Corp, via Associated Press
AFLOAT The astrophysicist Stephen Hawking goes weightless in a special jet.
Readers’ Comments
Readers shared their thoughts on this article.
But what if it’s all an illusion, a sort of cosmic frill, or a side effect of something else going on at deeper levels of reality?
So says Erik Verlinde, 48, a respected string theorist and professor of physics at the University of Amsterdam, whose contention that gravity is indeed an illusion has caused a continuing ruckus among physicists, or at least among those who profess to understand it. Reversing the logic of 300 years of science, he argued in a recent paper, titled “On the Origin of Gravity and the Laws of Newton,” that gravity is a consequence of the venerable laws of thermodynamics, which describe the behavior of heat and gases.
“For me gravity doesn’t exist,” said Dr. Verlinde, who was recently in the United States to explain himself. Not that he can’t fall down, but Dr. Verlinde is among a number of physicists who say that science has been looking at gravity the wrong way and that there is something more basic, from which gravity “emerges,” the way stock markets emerge from the collective behavior of individual investors or that elasticity emerges from the mechanics of atoms.
Looking at gravity from this angle, they say, could shed light on some of the vexing cosmic issues of the day, like the dark energy, a kind of anti-gravity that seems to be speeding up the expansion of the universe, or the dark matter that is supposedly needed to hold galaxies together.
Dr. Verlinde’s argument turns on something you could call the “bad hair day” theory of gravity.
It goes something like this: your hair frizzles in the heat and humidity, because there are more ways for your hair to be curled than to be straight, and nature likes options. So it takes a force to pull hair straight and eliminate nature’s options. Forget curved space or the spooky attraction at a distance described by Isaac Newton’s equations well enough to let us navigate the rings of Saturn, the force we call gravity is simply a byproduct of nature’s propensity to maximize disorder.
Some of the best physicists in the world say they don’t understand Dr. Verlinde’s paper, and many are outright skeptical. But some of those very same physicists say he has provided a fresh perspective on some of the deepest questions in science, namely why space, time and gravity exist at all — even if he has not yet answered them.
“Some people have said it can’t be right, others that it’s right and we already knew it — that it’s right and profound, right and trivial,” Andrew Strominger, a string theorist at Harvard said.
“What you have to say,” he went on, “is that it has inspired a lot of interesting discussions. It’s just a very interesting collection of ideas that touch on things we most profoundly do not understand about our universe. That’s why I liked it.”
Dr. Verlinde is not an obvious candidate to go off the deep end. He and his brother Herman, a Princeton professor, are celebrated twins known more for their mastery of the mathematics of hard-core string theory than for philosophic flights.
Born in Woudenberg, in the Netherlands, in 1962, the brothers got early inspiration from a pair of 1970s television shows about particle physics and black holes. “I was completely captured,” Dr. Verlinde recalled. He and his brother obtained Ph.D’s from the University of Utrecht together in 1988 and then went to Princeton, Erik to the Institute for Advanced Study and Herman to the university. After bouncing back and forth across the ocean, they got tenure at Princeton. And, they married and divorced sisters. Erik left Princeton for Amsterdam to be near his children.
He made his first big splash as a graduate student when he invented Verlinde Algebra and the Verlinde formula, which are important in string theory, the so-called theory of everything, which posits that the world is made of tiny wriggling strings.
You might wonder why a string theorist is interested in Newton’s equations. After all Newton was overturned a century ago by Einstein, who explained gravity as warps in the geometry of space-time, and who some theorists think could be overturned in turn by string theorists.
Over the last 30 years gravity has been “undressed,” in Dr. Verlinde’s words, as a fundamental force.
This disrobing began in the 1970s with the discovery by Jacob Bekenstein of the Hebrew University of Jerusalem and Stephen Hawking of Cambridge University, among others, of a mysterious connection between black holes and thermodynamics, culminating in Dr. Hawking’s discovery in 1974 that when quantum effects are taken into account black holes would glow and eventually explode.
In a provocative calculation in 1995, Ted Jacobson, a theorist from the University of Maryland, showed that given a few of these holographic ideas, Einstein’s equations of general relativity are just a another way of stating the laws of thermodynamics.
Those exploding black holes (at least in theory — none has ever been observed) lit up a new strangeness of nature. Black holes, in effect, are holograms — like the 3-D images you see on bank cards. All the information about what has been lost inside them is encoded on their surfaces. Physicists have been wondering ever since how this “holographic principle” — that we are all maybe just shadows on a distant wall — applies to the universe and where it came from.
In one striking example of a holographic universe, Juan Maldacena of the Institute for Advanced Study constructed a mathematical model of a “soup can” universe, where what happened inside the can, including gravity, is encoded in the label on the outside of the can, where there was no gravity, as well as one less spatial dimension. If dimensions don’t matter and gravity doesn’t matter, how real can they be?
Lee Smolin, a quantum gravity theorist at the Perimeter Institute for Theoretical Physics, called Dr. Jacobson’s paper “one of the most important papers of the last 20 years.”
But it received little attention at first, said Thanu Padmanabhan of the Inter-University Center for Astronomy and Astrophysics in Pune, India, who has taken up the subject of “emergent gravity” in several papers over the last few years. Dr. Padmanabhan said that the connection to thermodynamics went deeper that just Einstein’s equations to other theories of gravity. “Gravity,” he said recently in a talk at the Perimeter Institute, “is the thermodynamic limit of the statistical mechanics of “atoms of space-time.”
Dr. Verlinde said he had read Dr. Jacobson’s paper many times over the years but that nobody seemed to have gotten the message. People were still talking about gravity as a fundamental force. “Clearly we have to take these analogies seriously, but somehow no one does,” he complained.
His paper, posted to the physics archive in January, resembles Dr. Jacobson’s in many ways, but Dr. Verlinde bristles when people say he has added nothing new to Dr. Jacobson’s analysis. What is new, he said, is the idea that differences in entropy can be the driving mechanism behind gravity, that gravity is, as he puts it an “entropic force.”
That inspiration came to him courtesy of a thief.
As he was about to go home from a vacation in the south of France last summer, a thief broke into his room and stole his laptop, his keys, his passport, everything. “I had to stay a week longer,” he said, “I got this idea.”
Up the beach, his brother got a series of e-mail messages first saying that he had to stay longer, then that he had a new idea and finally, on the third day, that he knew how to derive Newton’s laws from first principles, at which point Herman recalled thinking, “What’s going on here? What has he been drinking?”
When they talked the next day it all made more sense, at least to Herman. “It’s interesting,” Herman said, “how having to change plans can lead to different thoughts.”
Think of the universe as a box of scrabble letters. There is only one way to have the letters arranged to spell out the Gettysburg Address, but an astronomical number of ways to have them spell nonsense. Shake the box and it will tend toward nonsense, disorder will increase and information will be lost as the letters shuffle toward their most probable configurations. Could this be gravity?
As a metaphor for how this would work, Dr. Verlinde used the example of a polymer — a strand of DNA, say, a noodle or a hair — curling up.
“It took me two months to understand polymers,” he said.
The resulting paper, as Dr. Verlinde himself admits, is a little vague.
“This is not the basis of a theory,” Dr. Verlinde explained. “I don’t pretend this to be a theory. People should read the words I am saying opposed to the details of equations.”
Dr. Padmanabhan said that he could see little difference between Dr. Verlinde’s and Dr. Jacobson’s papers and that the new element of an entropic force lacked mathematical rigor. “I doubt whether these ideas will stand the test of time,” he wrote in an e-mail message from India. Dr. Jacobson said he couldn’t make sense of it.
John Schwarz of the California Institute of Technology, one of the fathers of string theory, said the paper was “very provocative.” Dr. Smolin called it, “very interesting and also very incomplete.”
At a workshop in Texas in the spring, Raphael Bousso of the University of California, Berkeley, was asked to lead a discussion on the paper.
“The end result was that everyone else didn’t understand it either, including people who initially thought that did make some sense to them,” he said in an e-mail message.
“In any case, Erik’s paper has drawn attention to what is genuinely a deep and important question, and that’s a good thing,” Dr. Bousso went on, “I just don’t think we know any better how this actually works after Erik’s paper. There are a lot of follow-up papers, but unlike Erik, they don’t even understand the problem.”
The Verlinde brothers are now trying to recast these ideas in more technical terms of string theory, and Erik has been on the road a bit, traveling in May to the Perimeter Institute and Stony Brook University on Long Island, stumping for the end of gravity. Michael Douglas, a professor at Stony Brook, described Dr. Verlinde’s work as “a set of ideas that resonates with the community, adding, “everyone is waiting to see if this can be made more precise.”
Until then the jury of Dr. Verlinde’s peers will still be out.
Over lunch in New York, Dr. Verlinde ruminated over his experiences of the last six months. He said he had simply surrendered to his intuition. “When this idea came to me, I was really excited and euphoric even,” Dr. Verlinde said. “It’s not often you get a chance to say something new about Newton’s laws. I don’t see immediately that I am wrong. That’s enough to go ahead.”
He said friends had encouraged him to stick his neck out and that he had no regrets. “If I am proven wrong, something has been learned anyway. Ignoring it would have been the worst thing.”
The next day Dr. Verlinde gave a more technical talk to a bunch of physicists in the city. He recalled that someone had told him the other day that the unfolding story of gravity was like the emperor’s new clothes.
“We’ve known for a long time gravity doesn’t exist,” Dr. Verlinde said, “It’s time to yell it.”
29
So says Erik Verlinde, 48, a respected string theorist and professor of physics at the University of Amsterdam, whose contention that gravity is indeed an illusion has caused a continuing ruckus among physicists, or at least among those who profess to understand it. Reversing the logic of 300 years of science, he argued in a recent paper, titled “On the Origin of Gravity and the Laws of Newton,” that gravity is a consequence of the venerable laws of thermodynamics, which describe the behavior of heat and gases.
“For me gravity doesn’t exist,” said Dr. Verlinde, who was recently in the United States to explain himself. Not that he can’t fall down, but Dr. Verlinde is among a number of physicists who say that science has been looking at gravity the wrong way and that there is something more basic, from which gravity “emerges,” the way stock markets emerge from the collective behavior of individual investors or that elasticity emerges from the mechanics of atoms.
Looking at gravity from this angle, they say, could shed light on some of the vexing cosmic issues of the day, like the dark energy, a kind of anti-gravity that seems to be speeding up the expansion of the universe, or the dark matter that is supposedly needed to hold galaxies together.
Dr. Verlinde’s argument turns on something you could call the “bad hair day” theory of gravity.
It goes something like this: your hair frizzles in the heat and humidity, because there are more ways for your hair to be curled than to be straight, and nature likes options. So it takes a force to pull hair straight and eliminate nature’s options. Forget curved space or the spooky attraction at a distance described by Isaac Newton’s equations well enough to let us navigate the rings of Saturn, the force we call gravity is simply a byproduct of nature’s propensity to maximize disorder.
Some of the best physicists in the world say they don’t understand Dr. Verlinde’s paper, and many are outright skeptical. But some of those very same physicists say he has provided a fresh perspective on some of the deepest questions in science, namely why space, time and gravity exist at all — even if he has not yet answered them.
“Some people have said it can’t be right, others that it’s right and we already knew it — that it’s right and profound, right and trivial,” Andrew Strominger, a string theorist at Harvard said.
“What you have to say,” he went on, “is that it has inspired a lot of interesting discussions. It’s just a very interesting collection of ideas that touch on things we most profoundly do not understand about our universe. That’s why I liked it.”
Dr. Verlinde is not an obvious candidate to go off the deep end. He and his brother Herman, a Princeton professor, are celebrated twins known more for their mastery of the mathematics of hard-core string theory than for philosophic flights.
Born in Woudenberg, in the Netherlands, in 1962, the brothers got early inspiration from a pair of 1970s television shows about particle physics and black holes. “I was completely captured,” Dr. Verlinde recalled. He and his brother obtained Ph.D’s from the University of Utrecht together in 1988 and then went to Princeton, Erik to the Institute for Advanced Study and Herman to the university. After bouncing back and forth across the ocean, they got tenure at Princeton. And, they married and divorced sisters. Erik left Princeton for Amsterdam to be near his children.
He made his first big splash as a graduate student when he invented Verlinde Algebra and the Verlinde formula, which are important in string theory, the so-called theory of everything, which posits that the world is made of tiny wriggling strings.
You might wonder why a string theorist is interested in Newton’s equations. After all Newton was overturned a century ago by Einstein, who explained gravity as warps in the geometry of space-time, and who some theorists think could be overturned in turn by string theorists.
Over the last 30 years gravity has been “undressed,” in Dr. Verlinde’s words, as a fundamental force.
This disrobing began in the 1970s with the discovery by Jacob Bekenstein of the Hebrew University of Jerusalem and Stephen Hawking of Cambridge University, among others, of a mysterious connection between black holes and thermodynamics, culminating in Dr. Hawking’s discovery in 1974 that when quantum effects are taken into account black holes would glow and eventually explode.
In a provocative calculation in 1995, Ted Jacobson, a theorist from the University of Maryland, showed that given a few of these holographic ideas, Einstein’s equations of general relativity are just a another way of stating the laws of thermodynamics.
Those exploding black holes (at least in theory — none has ever been observed) lit up a new strangeness of nature. Black holes, in effect, are holograms — like the 3-D images you see on bank cards. All the information about what has been lost inside them is encoded on their surfaces. Physicists have been wondering ever since how this “holographic principle” — that we are all maybe just shadows on a distant wall — applies to the universe and where it came from.
In one striking example of a holographic universe, Juan Maldacena of the Institute for Advanced Study constructed a mathematical model of a “soup can” universe, where what happened inside the can, including gravity, is encoded in the label on the outside of the can, where there was no gravity, as well as one less spatial dimension. If dimensions don’t matter and gravity doesn’t matter, how real can they be?
Lee Smolin, a quantum gravity theorist at the Perimeter Institute for Theoretical Physics, called Dr. Jacobson’s paper “one of the most important papers of the last 20 years.”
But it received little attention at first, said Thanu Padmanabhan of the Inter-University Center for Astronomy and Astrophysics in Pune, India, who has taken up the subject of “emergent gravity” in several papers over the last few years. Dr. Padmanabhan said that the connection to thermodynamics went deeper that just Einstein’s equations to other theories of gravity. “Gravity,” he said recently in a talk at the Perimeter Institute, “is the thermodynamic limit of the statistical mechanics of “atoms of space-time.”
Dr. Verlinde said he had read Dr. Jacobson’s paper many times over the years but that nobody seemed to have gotten the message. People were still talking about gravity as a fundamental force. “Clearly we have to take these analogies seriously, but somehow no one does,” he complained.
His paper, posted to the physics archive in January, resembles Dr. Jacobson’s in many ways, but Dr. Verlinde bristles when people say he has added nothing new to Dr. Jacobson’s analysis. What is new, he said, is the idea that differences in entropy can be the driving mechanism behind gravity, that gravity is, as he puts it an “entropic force.”
That inspiration came to him courtesy of a thief.
As he was about to go home from a vacation in the south of France last summer, a thief broke into his room and stole his laptop, his keys, his passport, everything. “I had to stay a week longer,” he said, “I got this idea.”
Up the beach, his brother got a series of e-mail messages first saying that he had to stay longer, then that he had a new idea and finally, on the third day, that he knew how to derive Newton’s laws from first principles, at which point Herman recalled thinking, “What’s going on here? What has he been drinking?”
When they talked the next day it all made more sense, at least to Herman. “It’s interesting,” Herman said, “how having to change plans can lead to different thoughts.”
Think of the universe as a box of scrabble letters. There is only one way to have the letters arranged to spell out the Gettysburg Address, but an astronomical number of ways to have them spell nonsense. Shake the box and it will tend toward nonsense, disorder will increase and information will be lost as the letters shuffle toward their most probable configurations. Could this be gravity?
As a metaphor for how this would work, Dr. Verlinde used the example of a polymer — a strand of DNA, say, a noodle or a hair — curling up.
“It took me two months to understand polymers,” he said.
The resulting paper, as Dr. Verlinde himself admits, is a little vague.
“This is not the basis of a theory,” Dr. Verlinde explained. “I don’t pretend this to be a theory. People should read the words I am saying opposed to the details of equations.”
Dr. Padmanabhan said that he could see little difference between Dr. Verlinde’s and Dr. Jacobson’s papers and that the new element of an entropic force lacked mathematical rigor. “I doubt whether these ideas will stand the test of time,” he wrote in an e-mail message from India. Dr. Jacobson said he couldn’t make sense of it.
John Schwarz of the California Institute of Technology, one of the fathers of string theory, said the paper was “very provocative.” Dr. Smolin called it, “very interesting and also very incomplete.”
At a workshop in Texas in the spring, Raphael Bousso of the University of California, Berkeley, was asked to lead a discussion on the paper.
“The end result was that everyone else didn’t understand it either, including people who initially thought that did make some sense to them,” he said in an e-mail message.
“In any case, Erik’s paper has drawn attention to what is genuinely a deep and important question, and that’s a good thing,” Dr. Bousso went on, “I just don’t think we know any better how this actually works after Erik’s paper. There are a lot of follow-up papers, but unlike Erik, they don’t even understand the problem.”
The Verlinde brothers are now trying to recast these ideas in more technical terms of string theory, and Erik has been on the road a bit, traveling in May to the Perimeter Institute and Stony Brook University on Long Island, stumping for the end of gravity. Michael Douglas, a professor at Stony Brook, described Dr. Verlinde’s work as “a set of ideas that resonates with the community, adding, “everyone is waiting to see if this can be made more precise.”
Until then the jury of Dr. Verlinde’s peers will still be out.
Over lunch in New York, Dr. Verlinde ruminated over his experiences of the last six months. He said he had simply surrendered to his intuition. “When this idea came to me, I was really excited and euphoric even,” Dr. Verlinde said. “It’s not often you get a chance to say something new about Newton’s laws. I don’t see immediately that I am wrong. That’s enough to go ahead.”
He said friends had encouraged him to stick his neck out and that he had no regrets. “If I am proven wrong, something has been learned anyway. Ignoring it would have been the worst thing.”
The next day Dr. Verlinde gave a more technical talk to a bunch of physicists in the city. He recalled that someone had told him the other day that the unfolding story of gravity was like the emperor’s new clothes.
“We’ve known for a long time gravity doesn’t exist,” Dr. Verlinde said, “It’s time to yell it.”
On the Origin of Gravity
and the Laws of Newton
Erik Verlinde1
Institute for Theoretical Physics
University of Amsterdam
Valckenierstraat 65
1018 XE, Amsterdam
The Netherlands
Abstract
Starting from rst principles and general assumptions Newton's law of grav-
itation is shown to arise naturally and unavoidably in a theory in which space
is emergent through a holographic scenario. Gravity is explained as an entropic
force caused by changes in the information associated with the positions of ma-
terial bodies. A relativistic generalization of the presented arguments directly
leads to the Einstein equations. When space is emergent even Newton's law of
inertia needs to be explained. The equivalence principle leads us to conclude
that it is actually this law of inertia whose origin is entropic.
1 e.p.verlinde@uva.nl
arXiv:1001.0785v1 [hep-th] 6 Jan 2010
1 Introduction
Of all forces of Nature gravity is clearly the most universal. Gravity inuences and
is inuenced by everything that carries an energy, and is intimately connected with
the structure of space-time. The universal nature of gravity is also demonstrated by
the fact that its basic equations closely resemble the laws of thermodynamics and
hydrodynamics2. So far, there has not been a clear explanation for this resemblance.
Gravity dominates at large distances, but is very weak at small scales. In fact, its
basic laws have only been tested up to distances of the order of a millimeter. Gravity is
also considerably harder to combine with quantum mechanics than all the other forces.
The quest for uni cation of gravity with these other forces of Nature, at a microscopic
level, may therefore not be the right approach. It is known to lead to many problems,
paradoxes and puzzles. String theory has to a certain extent solved some of these, but
not all. And we still have to gure out what the string theoretic solution teaches us.
Many physicists believe that gravity, and space-time geometry are emergent. Also
string theory and its related developments have given several indications in this direction.
Particularly important clues come from the AdS/CFT, or more generally, the
open/closed string correspondence. This correspondence leads to a duality between
theories that contain gravity and those that don't. It therfore provides evidence for
the fact that gravity can emerge from a microscopic description that doesn't know
about its existence.
The universality of gravity suggests that its emergence should be understood from
general principles that are independent of the speci c details of the underlying microscopic
theory. In this paper we will argue that the central notion needed to derive
gravity is information. More precisely, it is the amount of information associated with
matter and its location, in whatever form the microscopic theory likes to have it, measured
in terms of entropy. Changes in this entropy when matter is displaced leads to
an entropic force, which as we will show takes the form of gravity. Its origin therefore
lies in the tendency of the microscopic theory to maximize its entropy.
The most important assumption will be that the information associated with a
part of space obeys the holographic principle [8, 9]. The strongest supporting evidence
for the holographic principle comes from black hole physics [1, 3] and the AdS/CFT
correspondence [10]. These theoretical developments indicate that at least part of
the microscopic degrees of freedom can be represented holographically either on the
boundary of space-time or on horizons.
The concept of holography appears to be much more general, however. For instance,
in the AdS/CFT correspondence one can move the boundary inwards by exploiting a
holographic version of the renormalization group. Similarly, in black hole physics there
exist ideas that the information can be stored on stretched horizons. Furthermore, by
thinking about accelerated observers, one can in principle locate holographic screens
2An incomplete list of references includes [1, 2, 3, 4, 5, 6, 7]
2
anywhere in space. In all these cases the emergence of the holographic direction is
accompanied by redshifts, and related to some coarse graining procedure. If all these
combined ideas are correct there should exist a general framework that describes how
space emerges together with gravity.
Usually holography is studied in relativistic contexts. However, the gravitational
force is also present in our daily non-relativistic world. The origin of gravity, whatever
it is, should therefore also naturally explain why this force appears the way it does,
and obeys Newton law of gravitation. In fact, when space is emergent, also the other
laws of Newton have to be re-derived, because standard concepts like position, velocity,
acceleration, mass and force are far from obvious. Hence, in such a setting the laws of
mechanics have to appear alongside with space itself. Even a basic concept like inertia
is not given, and needs to be explained again.
In this paper we present a holographic scenario for the emergence of space and
address the origins of gravity and inertia, which are connected by the equivalence
principle. Starting from rst principles, using only space independent concepts like
energy, entropy and temperature, it is shown that Newton's laws appear naturally and
practically unavoidably. Gravity is explained as an entropic force caused by a change
in the amount of information associated with the positions of bodies of matter.
A crucial ingredient is that only a nite number of degrees of freedom are associated
with a given spatial volume, as dictated by the holographic principle. The energy, that
is equivalent to the matter, is distributed evenly over the degrees of freedom, and thus
leads to a temperature. The product of the temperature and the change in entropy due
to the displacement of matter is shown to be equal to the work done by the gravitational
force. In this way Newton's law of gravity emerges in a surprisingly simple fashion.
The holographic principle has not been easy to extract from the laws of Newton
and Einstein, and is deeply hidden within them. Conversely, starting from holography,
we nd that these well known laws come out directly and unavoidably. By reversing
the logic that lead people from the laws of gravity to holography, we will obtain a
much sharper and even simpler picture of what gravity is. For instance, it clari es why
gravity allows an action at a distance even when there in no mediating force eld.
The presented ideas are consistent with our knowledge of string theory, but if correct
they should have important implications for this theory as well. In particular, the
description of gravity as being due to the exchange of closed strings can no longer be
valid. In fact, it appears that strings have to be emergent too.
We start in section 2 with an exposition of the concept of entropic force. Section
3 illustrates the main heuristic argument in a simple non relativistic setting. Its generalization
to arbitrary matter distributions is explained in section 4. In section 5 we
extend these results to the relativistic case, and derive the Einstein equations. The
conclusions are presented in section 7.
3
2 Entropic Force.
An entropic force is an e
ective macroscopic force that originates in a system with
ective macroscopic force that originates in a system with
many degrees of freedom by the statistical tendency to increase its entropy. The force
equation is expressed in terms of entropy di
erences, and is independent of the details
erences, and is independent of the details
of the microscopic dynamics. In particular, there is no fundamental eld associated
with an entropic force. Entropic forces occur typically in macroscopic systems such as
in colloid or bio-physics. Big colloid molecules suspended in an thermal environment
of smaller particles, for instance, experience entropic forces due to excluded volume
e
ects. Osmosis is another phenomenon driven by an entropic force.
ects. Osmosis is another phenomenon driven by an entropic force.
Perhaps the best known example is the elasticity of a polymer molecule. A single
polymer molecule can be modeled by joining together many monomers of xed length,
where each monomer can freely rotate around the points of attachment and direct itself
in any spatial direction. Each of these con gurations has the same energy. When the
polymer molecule is immersed into a heat bath, it likes to put itself into a randomly
coiled con guration since these are entropically favored. There are many more such
con gurations when the molecule is short compared to when it is stretched into an
extended con guration. The statistical tendency to return to a maximal entropy state
translates into a macroscopic force, in this case the elastic force.
By using tweezers one can pull the endpoints of the polymer apart, and bring it
out of its equilibrium con guration by an external force F, as shown in gure 1. For
de niteness, we keep one end xed, say at the origin, and move the other endpoint
along the x-axis. The entropy equals
S(E; x) = kB log (E; x) (2.1)
where kB is Boltzman's constant and (E; x) denotes the volume of the con guration
space for the entire system as a function of the total energy E of the heat bath and
the position x of the second endpoint. The x dependence is entirely a con gurational
e
ect: there is no microscopic contribution to the energy E that depends on x.
ect: there is no microscopic contribution to the energy E that depends on x.
€
F
€
x
€
T
Figure 1: A free jointed polymer is immersed in a heat bath with temperature T and pulled
out of its equilibrium state by an external force F. The entropic force points the other way.
4
In the canonical ensemble the force F is introduced in the partition function3
Z(T; F) =
Z
dEdx (E; x) e(E+Fx)=kBT : (2.2)
as an external variable dual to the length x of the polymer. The force F required to
keep the polymer at a xed length x for a given temperature E can be deduced from
the saddle point equations
1
T
=
@S
@E
;
F
T
=
@S
@x
: (2.3)
By the balance of forces, the external force F should be equal to the entropic force, that
tries to restore the polymer to its equilibrium position. An entropic force is recognized
by the facts that it points in the direction of increasing entropy, and, secondly, that it
is proportional to the temperature. For the polymer the force can be shown to obey
Hooke's law
Fpolymer const kBT hxi:
This example makes clear that at a macroscopic level an entropic force can be conservative,
at least when the temperature is kept constant. The corresponding potential
has no microscopic meaning, however, and is emergent.
It is interesting to study the energy and entropy balance when one gradually lets
the polymer return to its equilibrium position, while allowing the force to perform work
on an external system. By energy conservation, this work must be equal to the energy
that has been extracted from the heat bath. The entropy of the heat bath will hence be
reduced. For an in nite heat bath this would be by the same amount as the increase in
entropy of the polymer. Hence, in this situation the total entropy will remain constant.
This can be studied in more detail in the micro-canonical ensemble, because it
takes the total energy into account including that of the heat bath. To determine the
entropic force, one again introduces an external force F and examines the balance of
forces. Speci cally, one considers the micro canonical ensemble given by (E +Fx; x),
and imposes that the entropy is extremal. This gives
d
dx
S(E+Fx; x) = 0 (2.4)
One easily veri es that this leads to the same equations (2.3). However, it illustrates
that microcanonically the temperature is in general position dependent, and the force
also energy dependent. The term Fx can be viewed as the energy that was put in to
the system by pulling the polymer out of its equilibrium position. This equation tells
us therefore that the total energy gets reduced when the polymer is slowly allowed to
return to its equilibrium position, but that the entropy in rst approximation stays
the same. Our aim in the following sections is to argue that gravity is also an entropic
force, and that the same kind of reasonings apply to it with only slight modi cations.
3We like to thank B. Nienhuis and M. Shigemori for enlightning discussions on the following part.
5
3 Emergence of the Laws of Newton
Space is in the rst place a device introduced to describe the positions and movements
of particles. Space is therefore literally just a storage space for information. This
information is naturally associated with matter. Given that the maximal allowed information
is nite for each part of space, it is impossible to localize a particle with
in nite precision at a point of a continuum space. In fact, points and coordinates
arise as derived concepts. One could assume that information is stored in points of a
discretized space (like in a lattice model). But if all the associated information would
be without duplication, one would not obtain a holographic description. In fact, one
would not recover gravity.
Thus we are going to assume that information is stored on surfaces, or screens.
Screens separate points, and in this way are the natural place to store information
about particles that move from one side to the other. Thus we imagine that this
information about the location particles is stored in discrete bits on the screens. The
dynamics on each screen is given by some unknown rules, which can be thought of
as a way of processing the information that is stored on it. Hence, it does not have
to be given by a local eld theory, or anything familiar. The microscopic details are
irrelevant for us.
Let us also assume that like in AdS/CFT, there is one special direction corresponding
to scale or a coarse graining variable of the microscopic theory. This is the direction
in which space is emergent. So the screens that store the information are like stretched
horizons. On one side there is space, on the other side nothing yet. We will assume
that the microscopic theory has a well de ned notion of time, and its dynamics is time
translation invariant. This allows one to de ne energy, and by employing techniques
of statistical physics, temperature. These will be the basic ingredients together with
the entropy associated with the amount of information.
3.1 Force and inertia.
Our starting assumption is directly motivated by Bekenstein's original thought experiment
from which he obtained is famous entropy formula. He considered a particle with
mass m attached to a ctitious "string" that is lowered towards a black hole. Just
before the horizon the particle is dropped in. Due to the in nite redshift the mass
increase of the black hole can be made arbitrarily small, classically. If one would take a
thermal gas of particles, this fact would lead to problems with the second law of thermodynamics.
Bekenstein solved this by arguing that when a particle is one Compton
wavelength from the horizon, it is considered to be part of the black hole. Therefore,
it increases the mass and horizon area by a small amount, which he identi ed with one
bit of information. This lead him to his area law for the black hole entropy.
We want to mimic this reasoning not near a black hole horizon, but in at non-
6
€
m
€
Δx
€
€ ΔS
T
Figure 2: A particle with mass approaches a part of the holographic screen. The screen
bounds the emerged part of space, which contains the particle, and stores data that describe
the part of space that has not yet emerged, as well as some part of the emerged space.
relativistic space. So we consider a small piece of an holographic screen, and a particle
of mass m that approaches it from the side at which space time has already emerged.
Eventually the particle merge with the microscopic degrees of freedom on the screen,
but before it does so, it already inuences the amount of information that is stored on
the screen. The situation is depicted in gure 2.
Motivated by Bekenstein's argument, let us postulate that the change of entropy
associated with the information on the boundary equals
S = 2 kB when x =
~
mc
: (3.5)
The reason for putting in the factor of 2 , will become apparent soon. Let us rewrite
this formula in the slightly more general form by assuming that the change in entropy
near the screen is linear in the displacement x.
S = 2 kB
mc
~
x: (3.6)
To understand why it is also proportional to the mass m, let us imagine splitting the
particle into two or more lighter sub-particles. Each sub-particle then carries its own
associated change in entropy after a shift x. Since entropy and mass are both additive,
it is therefore natural that the entropy change is proportional to the mass. How does
force arise? The basic idea is to use the analogy with osmosis across a semi-permeable
membrane. When a particle has an entropic reason to be on one side of the membrane
and the membrane carries a temperature, it will experience an e
ective force equal to
ective force equal to
F x = T S: (3.7)
This is the entropic force. Thus, in order to have a non zero force, we need to have
a non vanishing temperature. From Newton's law we know that a force leads to a
7
non zero acceleration. Of course, it is well known that acceleration and temperature
are closely related. Namely, as Unruh showed, an observer in an accelerated frame
experiences a temperature
kBT =
1
2
~a
c
; (3.8)
where a denotes the acceleration. Let us take this as the temperature associated with
the bits on the screen. Now it is clear why the equation (3.6) for S was chosen to be
of the given form, including the factor of 2 . It is picked precisely in such a way that
one recovers the second law of Newton
F = ma: (3.9)
as is easily veri ed by combining (3.8) together with (3.6) and (3.7).
Equation (3.8) should be read as a formula for the temperature T that is required
to cause an acceleration equal to a. And not as usual, as the temperature caused by
an acceleration.
3.2 Newton's law of gravity.
Now suppose our boundary is not in nitely extended, but forms a closed surface. More
speci cally, let us assume it is a sphere. For the following it is best to forget about
the Unruh law (3.8), since we don't need it. It only served as a further motivation for
(3.6). The key statement is simply that we need to have a temperature in order to
have a force. Since we want to understand the origin of the force, we need to know
where the temperature comes from.
One can think about the boundary as a storage device for information. Assuming
that the holographic principle holds, the maximal storage space, or total number of
bits, is proportional to the area A. In fact, in an theory of emergent space this how
area may be de ned: each fundamental bit occupies by de nition one unit cell.
Let us denote the number of used bits by N. It is natural to assume that this
number will be proportional to the area. So we write
N =
Ac3
G~
(3.10)
where we introduced a new constant G. Eventually this constant is going to be identi
ed with Newton's constant, of course. But since we have not assumed anything
yet about the existence a gravitational force, one can simply regard this equation as
the de nition of G. So, the only assumption made here is that the number of bits is
proportional to the area. Nothing more.
Suppose there is a total energy E present in the system. Let us now just make the
simple assumption that the energy is divided evenly over the bits N. The temperature
8
€
m
€
F
€
M
€
R
€
T
Figure 3: A particle with mass m near a spherical holographic screen. The energy is evenly
distributed over the occupied bits, and is equivalent to the mass M that would emerge in the
part of space surrounded by the screen.
is then determined by the equipartition rule
E =
1
2
NkBT (3.11)
as the average energy per bit. After this we need only one more equation:
E = Mc2: (3.12)
Here M represents the mass that would emerge in the part of space enclosed by the
screen, see gure 3. Even though the mass is not directly visible in the emerged space,
its presence is noticed though its energy.
The rest is straightforward: one eliminates E and inserts the expression for the
number of bits to determine T. Next one uses the postulate (3.6) for the change of
entropy to determine the force. Finally one inserts
A = 4 R2:
and one obtains the familiar law:
F = G
Mm
R2 : (3.13)
We have recovered Newton's law of gravitation, practically from rst principles!
These equations do not just come out by accident. It had to work, partly for
dimensional reasons, and also because the laws of Newton have been ingredients in the
steps that lead to black hole thermodynamics and the holographic principle. In a sense
we have reversed these arguments. But the logic is clearly di
erent, and sheds new
erent, and sheds new
light on the origin of gravity: it is an entropic force! That is the main statement, which
is new and has not been made before. If true, this should have profound consequences.
9
3.3 Naturalness and robustness of the derivation.
Our starting point was that space has one emergent holographic direction. The additional
ingredients were that (i) there is a change of entropy in the emergent direction
(ii) the number of degrees of freedom are proportional to the area of the screen, and
(iii) the energy is evenly distributed over these degrees of freedom. After that it is
unavoidable that the resulting force takes the form of Newton's law. In fact, this reasoning
can be generalized to arbitrary dimensions4 with the same conclusion. But how
robust and natural are these heuristic arguments?
Perhaps the least obvious assumption is equipartition, which in general holds only
for free systems. But how essential is it? Energy usually spreads over the microscopic
degrees of freedom according to some non trivial distribution function. When the lost
bits are randomly chosen among all bits, one expects the energy change associated with
S still to be proportional to the energy per unit area E=A. This fact could therefore
be true even when equipartition is not strictly obeyed.
Why do we need the speed of light c in this non relativistic context? It was necessary
to translate the mass M in to an energy, which provides the heat bath required for
the entropic force. In the non-relativistic setting this heat bath is in nite, but in
principle one has take into account that the heat bath loses or gains energy when the
particle changes its location under the inuence of an external force. This will lead to
relativistic redshifts, as we will see.
Since the postulate (3.5) is the basic assumption from which everything else follows,
let us discuss its meaning in more detail. Why does the entropy precisely change like
this when one shifts by one Compton wave length? In fact, one may wonder why
we needed to introduce Planck's constant in the rst place, since the only aim was
to derive the classical laws of Newton. Indeed, ~ eventually drops out of the most
important formulas. So, in principle one could multiply it with any constant and still
obtain the same result. Hence, ~ just serves as an auxiliary variable that is needed
for dimensional reasons. It can therefore be chosen at will, and de ned so that (3.5)
is exactly valid. The main content of this equation is therefore simply that there is
an entropy change perpendicular to the screen proportional to the mass m and the
displacement x. That is all there is to it.
If we would move further away from the screen, the change in entropy will in general
no longer be given by the same rule. Suppose the particle stays at radius R while the
screen is moved to R0 < R. The number of bits on the screen is multiplied by a factor
(R0=R)2, while the temperature is divided by the same factor. E
ectively, this means
ectively, this means
that only ~ is multiplied by that factor, and since it drops out, the resulting force will
stay the same. In this situation the information associated with the particle is no longer
concentrated in a small area, but spreads over the screen. The next section contains a
proposal for precisely how it is distributed, even for general matter con gurations.
4In d dimensions (3.10) includes a factor 1
2
d2
d3 to get the right identi cation with Newton's constant.
10
3.4 Inertia and the Newton potential.
To complete the derivation of the laws of Newton we have to understand why the symbol
a, that was basically introduced by hand in (3.8), is equal to the physical acceleration
x. In fact, so far our discussion was quasi static, so we have not determined yet how
to connect space at di
erent times. In fact, it may appear somewhat counter-intuitive
erent times. In fact, it may appear somewhat counter-intuitive
that the temperature T is related to the vector quantity a, while in our identi cations
the entropy gradient S= x is related to the scalar quantity m. In a certain way it
seems more natural to have it the other way around.
So let reconsider what happens to the particle with mass m when it approaches the
screen. Here it should merge with the microscopic degrees of freedom on the screen,
and hence it will be made up out of the same bits as those that live on the screen.
Since each bit carries an energy 1
2kBT, the number of bits n follows from
mc2 =
1
2
n kBT: (3.14)
When we insert this in to equation (3.6), and use (3.8), we can express the entropy
change in terms of the acceleration as
S
n
= kB
a x
2c2 (3.15)
By combining the above equations one of course again recovers F = ma as the entropic
force. But, by introducing the number of bits n associated with the particle, we
succeeded in making the identi cations more natural in terms of their scalar versus
vector character. In fact, we have eliminated ~ from the equations, which in view of
our earlier comment is a good thing.
Thus we conclude that acceleration is related to an entropy gradient. This will be
one of our main principles: inertia is a consequence of the fact that a particle in rest
will stay in rest because there are no entropy gradients. Given this fact it is natural to
introduce the Newton potential and write the acceleration as a gradient
a = r :
This allows us to express the change in entropy in the concise way
S
n
= kB
2c2 (3.16)
We thus reach the important conclusion that the Newton potential keeps track of
the depletion of the entropy per bit. It is therefore natural to identify it with a coarse
graining variable, like the (renormalization group) scale in AdS/CFT. Indeed, in the
next section we propose a holographic scenario for the emergence of space in which the
Newton potential precisely plays that role. This allows us to generalize our discussion
to other mass distributions and arbitrary positions in a natural and rather beautiful
way, and give additional support for the presented arguments.
11
4 Emergent Gravity for General Matter Distributions.
Space emerges at a macroscopic level only after coarse graining. Hence, there will be
a nite entropy associated with each matter con guration. This entropy measures the
amount of microscopic information that is invisible to the macroscopic observer. In
general, this amount will depend on the distribution of the matter. The information is
being processed by the microscopic dynamics, which looks random from a macroscopic
point of view. But to determine the force we don't need the details of the information,
nor the exact dynamics, only the amount of information given by the entropy, and the
energy that is associated with it. If the entropy changes as a function of the location
of the matter distribution, it will lead to an entropic force.
Therefore, space can not just emerge by itself. It has to be endowed by a book
keeping device that keeps track of the amount of information for a given energy distribution.
It turns out, that in a non relativistic situation this device is provided by
Newton's potential . And the resulting entropic force is called gravity.
We start from microscopic information. It is assumed to be stored on holographic
screens. Note that information has a natural inclusion property: by forgetting certain
bits, by coarse graining, one reduces the amount of information. This coarse graining
can be achieved through averaging, a block spin transformation, integrating out, or
some other renormalization group procedure. At each step one obtains a further coarse
grained version of the original microscopic data. The gravitational or closed string side
of these dualities is by many still believed to be independently de ned. But in our
view these are macroscopic theories, which by chance we already knew about before we
understood they were the dual of a microscopic theory without gravity. We can't resist
making the analogy with a situation in which we would have developed a theory for
elasticity using stress tensors in a continuous medium half a century before knowing
about atoms. We probably would have been equally resistant in accepting the obvious.
Gravity and closed strings are not much di
erent, but we just have not yet got used
erent, but we just have not yet got used
to the idea.
The coarse grained data live on smaller screens obtained by moving the rst screen
further into the interior of the space. The information that is removed by coarse
graining is replaced by the emerged part of space between the two screens. In this way
one gets a nested or foliated description of space by having surfaces contained within
surfaces. In other words, just like in AdS/CFT, there is one emerging direction in
space that corresponds to a "coarse graining" variable, something like the cut-o
scale
scale
of the system on the screens.
A priori there is no preferred holographic direction in at space. However, this is
where we use our observation about the Newton potential. It is the natural variable
that measures the amount of coarse graining on the screens. Therefore, the holographic
direction is given by the gradient r of the Newton potential. In other words, the
holographic screens correspond to equipotential surfaces. This leads to a well de ned
12
Figure 4: The holographic screens are located at equipotential surfaces. The information on
the screens is coarse grained in the direction of decreasing values of the Newton potential .
The maximum coarse graining happens at black hole horizons, when =2c2 = 1.
foliation of space, except that screens may break up in to disconnected parts that each
enclose di
erent regions of space. This is depicted in gure 4.
erent regions of space. This is depicted in gure 4.
The amount of coarse graining is measured by the ratio =2c2, as can be seen
from (3.16). Note that this is a dimensionless number that is always between zero and
one. It is only equal to one on the horizon of a black hole. We interpret this as the
point where all bits have been maximally coarse grained. Thus the foliation naturally
stops at black hole horizons.
4.1 The Poisson equation for general matter distributions.
Consider a microscopic state, which after coarse graining corresponds to a given mass
distribution in space. All microscopic states that lead to the same mass distribution
belong to the same macroscopic state. The entropy for each of these state is de ned
as the number of microscopic states that ow to the same macroscopic state.
We want to determine the gravitational force by using virtual displacements, and
calculating the associated change in energy. So, let us freeze time and keep all the
matter at xed locations. Hence, it is described by a static matter density (~r). Our
aim is to obtain the force that the matter distribution exerts on a collection of test
particles with masses mi and positions ~ri.
We choose a holographic screen S corresponding to an equipotential surface with
xed Newton potential 0. We assume that the entire mass distribution given by (x) is
contained inside the volume enclosed by the screen, and all test particles are outside this
volume. To explain the force on the particles, we again need to determine the work that
is performed by the force and show that it is naturally written as the change in entropy
multiplied by the temperature. The di
erence with the spherically symmetric case is
erence with the spherically symmetric case is
that the temperature on the screen is not necessarily constant. Indeed, the situation
13
is in general not in equilibrium. Nevertheless, one can locally de ne temperature and
entropy per unit area.
First let us identify the temperature. We do this by taking a test particle and
moving it close to the screen, and measuring the local acceleration. Thus, motivated
by our earlier discussion we de ne temperature analogous to (3.8), namely by
kBT =
1
2
~r
kc
: (4.17)
Here the derivative is taken in the direction of the outward pointing normal to the
screen. Note at this point is just introduced as a device to describe the local acceleration,
but we don't know yet whether it satis es an equation that relates it to the
mass distribution.
The next ingredient is the density of bits on the screen. We again assume that
these bits are uniformly distributed, and so (3.10) is generalized to
dN =
c3
G~
dA: (4.18)
Now let us impose the analogue of the equipartition relation (3.11). It is takes the
form of an integral expression for the energy
E =
1
2
kB
Z
S
TdN: (4.19)
It is an amusing exercise to work out the consequence of this relation. Of course, the
energy E is again expressed in terms of the total enclosed mass M. After inserting our
identi cations for the left hand side one obtains a familiar relation: Gauss's law!
M =
1
4 G
Z
S
r dA: (4.20)
This should hold for arbitrary screens given by equipotential surfaces. When a bit of
mass is added to the region enclosed by the screen S, for example, by rst putting it
close to the screen and then pushing it across, the mass M should change accordingly.
This condition can only hold in general if the potential satis es the Poisson equation
r2 (~r) = 4 G (~r): (4.21)
We conclude that by making natural identi cations for the temperature and the information
density on the holographic screens, that the laws of gravity come out in a
straightforward fashion.
14
€
m1
€
F 1
€
δ
r 1
€
m2
€
F 2
€
m3
€
F 3
€
Φ = Φ0
Figure 5: A general mass distribution inside the not yet emerged part of space enclosed by
the screen. A collection of test particles with masses mi are located at arbitrary points ~ri in
the already emerged space outside the screen. The forces ~Fi due to gravity are determined
by the virtual work done after in nitesimal displacement ~ri of the particles.
4.2 The gravitational force for arbitrary particle locations.
The next issue is to obtain the force acting on matter particles that are located at
arbitrary points outside the screen. For this we need a generalization of the rst postulate
(3.6) to this situation. What is the entropy change due to arbitrary in nitesimal
displacements ~ri of the particles? There is only one natural choice here. We want to
nd the change s in the entropy density locally on the screen S. We noted in (3.16)
that the Newton potential keeps track of the changes of information per unit bit.
Hence, the right identi cation for the change of entropy density is
s = kB
2c2 dN (4.22)
where is the response of the Newton potential due to the shifts ~ri of the positions
of the particles. To be speci c, is determined by solving the variation of the Poisson
equation
r2 (~r) = 4 G
X
i
mi ~ri ri (~r ~ri) (4.23)
One can verify that with this identi cation one indeed reproduces the entropy shift
(3.6) when one of the particles approaches the screen.
Let us now determine the entropic forces on the particles. The combined work done
by all of the forces on the test particles is determined by the rst law of thermodynamics.
However, we need to express in terms of the local temperature and entropy
15
variation. Hence, X
i
~Fi ~ri =
Z
S
T s (4.24)
To see that this indeed gives the gravitational force in the most general case, one simply
has to use the electrostatic analogy. Namely, one can redistribute the entire mass M as
a mass surface density over the screen S without changing the forces on the particles.
The variation of the Newton potential can be obtained from the Greens function for
the Laplacian. The rest of the proof is a straightforward application of electrostatics,
but then applied to gravity. The basic identity one needs to prove is
X
i
~Fi ~ri =
1
4 G
Z
S
r r
dA (4.25)
which holds for any location of the screen outside the mass distribution. This is easily
veri ed by using Stokes theorem and the Laplace equation. The second term vanishes
when the screen is chosen at a equipotential surface. To see this, simply replace by
0 and pull it out of the integral. Since is sourced by only the particles outside the
screen, the remaining integral just gives zero.
The forces we obtained are independent of the choice of the location of the screen.
We could have chosen any equipotential surface, and we would obtain the same values
for ~Fi, the ones described by the laws of Newton. That all of this works is not just
a matter of dimensional analysis. The invariance under the choice of equipotential
surface is very much consistent with idea that a particular location corresponds to an
arbitrary choice of the scale that controls the coarse graining of the microscopic data.
The macroscopic physics, in particular the forces, should be independent of that choice.
5 The Equivalence Principle and the Einstein equations.
Since we made use of the speed of light c in our arguments, it is a logical step to try
and generalize our discussion to a relativistic situation. So let us assume that the microscopic
theory knows about Lorentz symmetry, or even has the Poincar e group as a
global symmetry. This means we have to combine time and space in to one geometry.
A scenario with emergent space-time quite naturally leads to general coordinate invariance
and curved geometries, since a priori there are no preferred choices of coordinates,
nor a reason why curvatures would not be present. Speci cally, we would like to see
how Einstein's general relativity emerges from similar reasonings as in the previous
section. We will indeed show that this is possible. But rst we study the origin of
inertia and the equivalence principle.
16
5.1 The law of inertia and the equivalence principle.
Consider a static background with a global time like Killing vector a. To see the
emergence of inertia and the equivalence principle, one has to relate the choice of this
Killing vector eld with the temperature and the entropy gradients. In particular, we
like to see that the usual geodesic motion of particles can be understood as being the
result of an entropic force.
In general relativity5 the natural generalization of Newton's potential is [11],
- =
1
2
log( a a): (5.26)
Its exponent e- represents the redshift factor that relates the local time coordinate to
that at a reference point with - = 0, which we will take to be at in nity.
Just like in the non relativistic case, we like to use - to de ne a foliation of space,
and put our holographic screens at surfaces of constant redshift. This is a natural
choice, since in this case the entire screen uses the same time coordinate. So the
processing of the microscopic data on the screen can be done using signals that travel
without time delay.
We want to show that the redshift perpendicular to the screen can be understood
microscopically as originating from the entropy gradients6 . To make this explicit, let
us consider the force that acts on a particle of mass m. In a general relativistic setting
force is less clearly de ned, since it can be transformed away by a general coordinate
transformation. But by using the time-like Killing vector one can given an invariant
meaning to the concept of force [11].
The four velocity ua of the particle and its acceleration ab uaraub can be expressed
in terms of the Killing vector b as
ub = e- b; ab = e2- ara b:
We can further rewrite the last equation by making use of the Killing equation
ra b + rb a = 0
and the de nition of -. One nds that the acceleration can again be simply expressed
the gradient
ab = rb-: (5.27)
Note that just like in the non relativistic situation the acceleration is perpendicular
to screen S. So we can turn it in to a scalar quantity by contracting it with a unit
outward pointing vector Nb normal to the screen S and to b.
5In this subsection and the next we essentially follow Wald's book on general relativity (pg. 288-
290). We use a notation in which c and kB are put equal to one, but we will keep G and ~ explicit.
6In this entire section it will be very useful to keep the polymer example of section 2 in mind, since
that will make the logic of our reasoning very clear.
17
The local temperature T on the screen is now in analogy with the non relativistic
situation de ned by
T =
~
2
e-Nbrb-: (5.28)
Here we inserted a redshift factor e-, because the temperature T is measured with
respect to the reference point at in nity.
To nd the force on a particle that is located very close to the screen, we rst use
again the same postulate as in section two. Namely, we assume that the change of
entropy at the screen is 2 for a displacement by one Compton wavelength normal to
the screen. Hence,
raS = 2
m
~
Na; (5.29)
where the minus sign comes from the fact that the entropy increases when we cross
from the outside to the inside. The comments made on the validity of this postulate
in section 3.3 apply here as well. The entropic force now follows from (5.28)
Fa = TraS = me-ra- (5.30)
This is indeed the correct gravitational force that is required to keep a particle at xed
position near the screen, as measured from the reference point at in nity. It is the
relativistic analogue of Newton's law of inertia F = ma. The additional factor e- is
due to the redshift. Note that ~ has again dropped out.
It is instructive to rewrite the force equation (5.30) in a microcanonical form. Let
S(E; xa) be the total entropy associated with a system with total energy E that contains
a particle with mass m at position xa. Here E also includes the energy of the particle.
The entropy will in general also dependent on many other parameters, but we suppress
these in this discussion.
As we explained in the section 2, an entropic force can be determined microcanonically
by adding by hand an external force term, and impose that the entropy is
extremal. For this situation this condition looks like
d
dxa S
E+e-(x)m; xa
= 0: (5.31)
One easily veri es that this leads to the same equation (5.30). This xes the equilibrium
point where the external force, parametrized by -(x) and the entropic force statistically
balance each other. Again we stress the point that there is no microscopic force acting
here! The analogy with equation (2.4) for the polymer, discussed in section 2, should
be obvious now.
Equation (5.31) tells us that the entropy remains constant if we move the particle
and simultaneously reduce its energy by the redshift factor. This is true only when the
particle is very light, and does not disturb the other energy distributions. It simply
18
serves as a probe of the emergent geometry. This also means that redshift function
-(x) is entirely xed by the other matter in the system.
We have arrived at equation (5.31) by making use of the identi cations of the
temperature and entropy variations in space time. But actually we should have gone
the other way. We should have started from the microscopics and de ned the space
dependent concepts in terms of them. We chose not to follow such a presentation, since
it might have appeared somewhat contrived.
But it is important to realize that the redshift must be seen as a consequence
of the entropy gradient and not the other way around. The equivalence principle
tells us that redshifts can be interpreted in the emergent space time as either due
to a gravitational eld or due to the fact that one considers an accelerated frame.
Both views are equivalent in the relativistic setting, but neither view is microscopic.
Acceleration and gravity are both emergent phenomena.
5.2 Derivation of the Einstein equations.
We now like to extend our derivation of the laws of gravity to the relativistic case, and
obtain the Einstein equations. This can indeed be done in natural and very analogous
fashion. So we again consider a holographic screen on closed surface of constant redshift
-. We assume that it is enclosing a certain static mass con guration with total mass
M. The bit density on the screen is again given by
dN =
dA
G~
(5.32)
as in (4.18). Following the same logic as before, let us assume that the energy associated
with the mass M is distributed over all the bits. Again by equipartition each bit carries
a mass unit equal to 1
2T. Hence
M =
1
2
Z
S
TdN (5.33)
After inserting the identi cations for T and dN we obtain
M =
1
4 G
Z
S
e-r- dA (5.34)
Note that again ~ drops out, as expected. The equation (5.34) is indeed known to
be the natural generalization of Gauss's law to General Relativity. Namely, the right
hand side is precisely Komar's de nition of the mass contained inside an arbitrary
volume inside any static curved space time. It can be derived by assuming the Einstein
equations. In our reasoning, however, we are coming from the other side. We made
identi cations for the temperature and the number of bits on the screen. But we don't
19
know yet whether it satis es any eld equations. The key question at this point is
whether the equation (5.34) is su cient to derive the full Einstein equations.
An analogous question was addressed by Jacobson for the case of null screens. By
adapting his reasoning to this situation, and combining it with Wald's exposition of
the Komar mass, it is straightforward to construct an argument that naturally leads
to the Einstein equations. We will present a sketch of this.
First we make use of the fact that the Komar mass can be re-expressed in terms of
the Killing vector a as
M =
1
8 G
Z
S
dxa^dxb abcdrc d (5.35)
The left hand side can be expressed as an integral over the enclosed volume of certain
components of stress energy tensor Tab. For the right hand side one rst uses Stokes
theorem and subsequently the relation
rara b = Rb
a a (5.36)
which is implied by the Killing equation for a. This leads to the integral relation [11]
2
Z
Tab
1
2
Tgab
na bdV =
1
4 G
Z
Rabna bdV (5.37)
where is the three dimensional volume bounded by the holographic screen S and
na is its normal. The particular combination of the stress energy tensor on the left
hand side can presumably be xed by comparing properties on both sides, such as for
instance the conservation laws of the tensors that occur in the integrals.
This equation is derived in a general static background with a time like Killing
vector a. By requiring that it holds for arbitrary screens would imply that also the
integrands on both sides are equal. This gives us only a certain component of the
Einstein equation. In fact, we can choose the surface in many ways, as long as its
boundary is given by S. This means that we can vary the normal na. But that still
leaves a contraction with the Killing vector a.
To get to the full Einstein we now use a similar reasoning as Jacobson [7], except now
applied to time-like screens. Let us consider a very small region of space time and look
also at very short time scales. Since locally every geometry looks approximately like
Minkowski space, we can choose approximate time like Killing vectors. Now consider
a small local part of the screen, and impose that when matter crosses it, the value of
the Komar integral will jump precisely by the corresponding mass m. By following the
steps described above then leads to (5.37) for all these Killing vectors, and for arbitrary
screens. This is su cient to obtain the full Einstein equations.
20
5.3 The force on a collection of particles at arbitrary locations.
We close this section with explaining the way the entropic force acts on a collection
of particles at arbitrary locations xi away from the screen and the mass distribution.
The Komar de nition of the mass will again be useful for this purpose. The de nition
of the Komar mass depends on the choice of Killing vector a. In particular also in its
norm, the redshift factor e-. If one moves the particles by a virtual displacement xi,
this will e
ect the de nition of the Komar mass of the matter con guration inside the
ect the de nition of the Komar mass of the matter con guration inside the
screen. In fact, the temperature on the screen will be directly a
ected by a change in
ected by a change in
the redshift factor.
The virtual displacements can be performed quasi statically, which means that
the Killing vector itself is still present. Its norm, or the redshift factor, may change,
however. In fact, also the spatial metric may be a
ected by this displacement. We are
ected by this displacement. We are
not going to try to solve these dependences, since that would be impossible due to the
non linearity of the Einstein equations. But we can simply use the fact that the Komar
mass is going to be a function of the positions xi of the particles.
Next let us assume that in addition to this xi dependence of the Komar mass that
the entropy of the entire system has also explicit xi dependences, simply due to changes
in the amount of information. These are de di
erence that will lead to the entropic
erence that will lead to the entropic
force that we wish to determine. We will now give a natural prescription for the entropy
dependence that is based on a maximal entropy principle and indeed gives the right
forces. Namely, assume that the entropy may be written as a function of the Komar
mass M and in addition on the xi. But since the Komar mass should be regarded
as a function M(xi) of the positions xi, there will be an explicit and an implicit xi
dependence in the entropy. The maximal entropy principle implies that these two
dependences should cancel. So we impose
S
M(xi+ xi); xi+ xi
= S
M(xi); xi
(5.38)
By working out this condition and singling out the equations implied by each variation
xi one nds
riM + TriS = 0 (5.39)
The point is that the rst term represents the force that acts on the i-th particle due
to the mass distribution inside the screen. This force is indeed equal to minus the
derivative of the Komar mass, simply because of energy conservation. But again, this
is not the microscopic reason for the force. In analogy with the polymer, the Komar
mass represents the energy of the heat bath. Its dependence on the position of the
other particles is caused by redshifts whose microscopic origin lies in the depletion of
the energy in the heat bath due to entropic e
ects.
ects.
Since the Komar integral is de ned on the holographic screen, it is clear that like
in the non relativistic case the force is expressed as an integral over this screen as well.
We have not tried to make this representation more concrete. Finally, we note that
21
this argument was very general, and did not really use the precise form of the Komar
integral, or the Einstein equations. So it should be straightforward to generalize this
reasoning to higher derivative gravity theories by making use of Wald's Noether charge
formalism [12], which is perfectly suitable for this purpose.
6 Conclusion and Discussion
The ideas and results presented in this paper lead to many questions. In this section
we discuss and attempt to answer some of these. First we present our main conclusion.
6.1 The end of gravity as a fundamental force.
Gravity has given many hints of being an emergent phenomenon, yet up to this day
it is still seen as a fundamental force. The similarities with other known emergent
phenomena, such as thermodynamics and hydrodynamics, have been mostly regarded
as just suggestive analogies. It is time we not only notice the analogy, and talk about
the similarity, but nally do away with gravity as a fundamental force.
Of course, Einstein's geometric description of gravity is beautiful, and in a certain
way compelling. Geometry appeals to the visual part of our minds, and is amazingly
powerful in summarizing many aspects of a physical problem. Presumably this explains
why we, as a community, have been so reluctant to give up the geometric formulation
of gravity as being fundamental. But it is inevitable we do so. If gravity is emergent,
so is space time geometry. Einstein tied these two concepts together, and both have to
be given up if we want to understand one or the other at a more fundamental level.
The results of this paper suggest gravity arises as an entropic force, once space and
time themselves have emerged. If the gravity and space time can indeed be explained
as emergent phenomena, this should have important implications for many areas in
which gravity plays a central role. It would be especially interesting to investigate
the consequences for cosmology. For instance, the way redshifts arise from entropy
gradients could lead to many new insights.
The derivation of the Einstein equations presented in this paper is analogous to
previous works, in particular [7]. Also other authors have proposed that gravity has
an entropic or thermodynamic origin, see for instance [14]. But we have added an
important element that is new. Instead of only focussing on the equations that govern
the gravitational eld, we uncovered what is the origin of force and inertia in a context
in which space is emerging. We identi ed a cause, a mechanism, for gravity. It is driven
by di
erences in entropy, in whatever way de ned, and a consequence of the statistical
erences in entropy, in whatever way de ned, and a consequence of the statistical
averaged random dynamics at the microscopic level. The reason why gravity has to
keep track of energies as well as entropy di
erences is now clear. It has to, because
erences is now clear. It has to, because
this is what causes motion!
The presented arguments have admittedly been rather heuristic. One can not expect
22
otherwise, given the fact that we are entering an unknown territory in which space does
not exist to begin with. The profound nature of these questions in our view justi es
the heuristic level of reasoning. The assumptions we made have been natural: they t
with existing ideas and are supported by several pieces of evidence. In the following
we gather more supporting evidence from string theory, the AdS/CFT correspondence,
and black hole physics.
6.2 Implications for string theory and relation with AdS/CFT.
If gravity is just an entropic force, then what does this say about string theory? Gravity
is seen as an integral part of string theory, that can not be taken out just like that.
But we do know about dualities between closed string theories that contain gravity
and decoupled open string theories that don't. A particularly important example is
the AdS/CFT correspondence.
The open/closed string and AdS/CFT correspondences are manifestations of the
UV/IR connection that is deeply engrained within string theory. This connection
implies that short and long distance physics can not be seen as totally decoupled.
Gravity is a long distance phenomenon that clearly knows about short distance physics,
since it is evident that Newton's constant is a measure for the number of microscopic
degrees of freedom. String theory invalidates the "general wisdom" underlying the
Wilsonian e
ective eld theory, namely that integrating out short distance degrees
ective eld theory, namely that integrating out short distance degrees
of freedom only generate local terms in the e
ective action, most of which become
ective action, most of which become
irrelevant at low energies. If that were completely true, the macroscopic physics would
be insensitive to the short distance physics.
The reason why the Wilsonian argument fails, is that it makes a too conservative
assumption about the asymptotic growth of the number of states at high energies. In
string theory the number of high energy open string states is such that integrating them
out indeed leads to long range e
ects. Their one loop amplitudes are equivalent to the
ects. Their one loop amplitudes are equivalent to the
tree level contributions due to the exchange of close string states, which among other
are responsible for gravity. This interaction is, however, equivalently represented by
the sum over all quantum contributions of the open string. In this sense the emergent
nature of gravity is also supported by string theory.
The AdS/CFT correspondence has an increasing number of applications to areas of
physics in which gravity is not present at a fundamental level. Gravitational techniques
are used as tools to calculate physical quantities in a regime where the microscopic
description fails. The latest of these developments is the application to condensed
matter theory. No one doubts that in these situations gravity emerges only as an
e
ective description. It arises not in the same space as the microscopic theory, but in
ective description. It arises not in the same space as the microscopic theory, but in
a holographic scenario with one extra dimension. No clear explanation exists of where
this gravitational force comes from. The entropic mechanism described in this paper
should be applicable to these physical systems, and explain the emergence of gravity.
23
Open
IR
Closed
IR
a) b)
Figure 6: The microscopic theory in a) is e
ectively described by a string theory consisting
ectively described by a string theory consisting
of open and closed strings as shown in b). Both types of strings are cut o
in the UV.
in the UV.
The holographic scenario discussed in this paper has certainly be inspired by the
way holography works in AdS/CFT and open closed string correspondences. In string
language, the holographic screens can be identi ed with D-branes, and the microscopic
degrees of freedom on these screens represented as open strings de ned with a cut o
in the UV. The emerged part of space is occupied by closed strings, which are also
de ned with a UV cut o
, as shown in gure 6. The open and closed string cut o
s
, as shown in gure 6. The open and closed string cut o
s
are related by the UV/IR correspondence: pushing the open string cut o
to the UV
to the UV
forces the closed string cut o
towards the IR, and vice versa. The value of the cut o
s
towards the IR, and vice versa. The value of the cut o
s
is determined by the location of the screen. Integrating out the open strings produces
the closed strings, and leads to the emergence of space and gravity. Note, however,
that from our point of view the existence of gravity or closed strings is not assumed
microscopically: they are emergent as an e
ective description.
ective description.
In this way, the open/closed string correspondence supports the interpretation of
gravity as an entropic force. Yet, many still see the closed string side of these dualities
is a well de ned fundamental theory. But in our view gravity and closed strings are
emergent and only present as macroscopic concept. It just happened that we already
knew about gravity before we understood it could be obtained from a microscopic
theory without it. We can't resist making the analogy with a situation in which we
would have developed a theory for elasticity using stress tensors in a continuous medium
half a century before knowing about atoms. We probably would have been equally
resistant in accepting the obvious. Gravity and closed strings are not much di
erent,
erent,
but we just have to get used to the idea.
24
6.3 Black hole horizons revisited
We saved perhaps the clearest argument for the fact that gravity is an entropic force
to the very last. The rst cracks in the fundamental nature of gravity appeared when
Bekenstein, Hawking and others discovered the laws of black hole thermodynamics. In
fact, the thought experiment mentioned in section 3 that led Bekenstein to his entropy
law is surprisingly similar to the polymer problem. The black hole serves as the heat
bath, while the particle can be thought of as the end point of the polymer that is
gradually allowed to go back to its equilibrium situation.
Of course, there is no polymer in the gravity system, and there appears to be no
direct contact between the particle and the black hole. But here we are ignoring the
fact that one of the dimensions is emergent. In the holographic description of this same
process, the particle can be thought of as being immersed in the heat bath representing
the black hole. This fact is particularly obvious in the context of AdS/CFT, in which a
black hole is dual to a thermal state on the boundary, while the particle is represented
as a delocalized operator that is gradually being thermalized. By the time that the
particle reaches the horizon it has become part of the thermal state, just like the
polymer. This phenomenon is clearly entropic in nature, and is the consequence of a
statistical process that drives the system to its state of maximal entropy.
Upon closer inspection Bekenstein's reasoning can be used to show that gravity
becomes an entropic force near the horizon, and that the equations presented in section
3 are exactly valid. He argued that one has to choose a location slightly away from the
black hole horizon at a distance of about the order of the Compton wave length, where
we declare that the particle and the black hole have become one system. Let us say
this location corresponds to choosing a holographic screen. The precise location of this
screen can not be important, however, since there is not a natural preferred distance
that one can choose. The equations should therefore not depend on small variations of
this distance.
By pulling out the particle a bit further, one changes its energy by a small amount
equal to the work done by the gravitational force. If one then drops the particle in
to the black hole, the mass M increases by this same additional amount. Consistency
of the laws of black hole thermodynamics implies that the additional change in the
Bekenstein Hawking entropy, when multiplied with the Hawking temperature TH, must
be precisely equal to the work done by gravity. Hence,
Fgravity = TH
@SBH
@x
: (6.40)
The derivative of the entropy is de ned as the response of SBH due to a change in the
distance x of the particle to the horizon. This fact is surely known, and probably just
regarded as a consistency check. But let us take it one or two steps further.
Suppose we take the screen even further away from the horizon. The same argument
applies, but we can also choose to ignore the fact that the screen has moved, and lower
25
the particle to the location of the previous screen, the one closer to the horizon. This
process would happen in the part of space behind the new screen location, and hence
it should have a holographic representation on the screen. In this system there is
force in the perpendicular direction has no microscopic meaning, nor acceleration. The
coordinate x perpendicular to the screen is just some scale variable associated with
the holographic image of the particle. Its interpretation as an coordinate is a derived
concept: this is what it means have an emergent space.
The mass is de ned in terms the energy associated with the particle's holographic
image, which presumably is a near thermal state. It is not exactly thermal, however,
because it is still slightly away from the black hole horizon. We have pulled it out of
equilibrium, just like the polymer. One may then ask: what is the cause of the change
in energy that is holographically dual to the work done when in the emergent space we
gradually lower the particle towards the location of the old screen behind the new one
. Of course, this can be nothing else then an entropic e
ect, the force is simply due to
ect, the force is simply due to
the thermalization process. We must conclude that the only microscopic explanation
is that there is an emergent entropic force acting. In fact, the correspondence rules
between the scale variable and energy on the one side, and the emergent coordinate x
the mass m on the other, must be such that F = TrS translates in to the gravitational
force. It is straightforward to see that this indeed works and that the equations for the
temperature and the entropy change are exactly as given in section 3.
The horizon is only a special location for observers that stay outside the black
hole. The black hole can be arbitrarily large and the gravitational force at its horizon
arbitrarily weak. Therefore, this thought experiment is not just teaching us about
black holes. It teaches us about the nature of space time, and the origin of gravity.
Or more precisely, it tells us about the cause of inertia. We can do the same thought
experiment for a Rindler horizon, and reach exactly the same conclusion. In this case
the correspondence rules must be such that F = TrS translates in to the inertial force
F = ma. Again the formulas work out as in section 3.
6.4 Final comments
This brings us to a somewhat subtle and not yet fully understood aspect. Namely, the
role of ~. The previous arguments make clear that near the horizon the equations are
valid with ~ identi ed with the actual Planck constant. However, we have no direct
con rmation or proof that the same is true when we go away from the horizon, or
eespecially when there is no horizon present at all. In fact, there are reasons to believe
that the equations work slightly di
erent there. The rst is that one is not exactly
erent there. The rst is that one is not exactly
at thermal equilibrium. Horizons have well de ned temperatures, and clearly are in
thermal equilibrium. If one assumes that the screen at an equipotential surface with
= 0 is in equilibrium, the entropy needed to get the Unruh temperature(3.8) is given
26
by the Bekenstein-Hawking formula, including the factor 1=4,
S =
c3
4G~
Z
S
dA: (6.41)
This value for this entropy appears to be very high, and violates the Bekenstein bound
[15] that states that a system contained in region with radius R and total energy E
can not have an entropy larger than ER. The reason for this discrepancy may be that
Bekenstein's argument does not hold for the holographic degrees of freedom on the
screen, or because of the fact that we are far from equilibrium.
But there may also be other ways to reconcile these statements, for example by
making use of the freedom to rescale the value of ~. This would not e
ect the nal
ect the nal
outcome for the force, nor the fact that it is entropic. In fact, one can even multiply ~
by a function f( 0) of the Newton potential on the screen. This rescaling would a
ect
ect
the values of the entropy and the temperature in opposite directions: T gets multiplied
by a factor, while S will be divided by the same factor. Since a priori we can not
exclude this possibility, there is something to be understood. In fact, there are even
other modi cations possible, like a description that uses weighted average over many
screens with di
erent temperatures. Even then the essence of our conclusions would
erent temperatures. Even then the essence of our conclusions would
not change, which is the fact that gravity and inertia are entropic forces.
Does this view of gravity lead to predictions? The statistical average should give
the usual laws, hence one has to study the uctuations in the gravitational force. Their
size depends on the e
ective temperature, which may not be universal and depends on
ective temperature, which may not be universal and depends on
the e
ective value of ~. An interesting thought is that uctuations may turn out to
ective value of ~. An interesting thought is that uctuations may turn out to
be more pronounced for weak gravitational elds between small bodies of matter. But
clearly, we need a better understanding of the theory to turn this in to a prediction.
It is well known that Newton was criticized by his contemporaries, especially by
Hooke, that his law of gravity acts at a distance and has no direct mechanical cause
like the elastic force. Ironically, this is precisely the reason why Hooke's elastic force is
nowadays not seen as fundamental, while Newton's gravitational force has maintained
that status for more than three centuries. What Newton did not know, and certainly
Hooke didn't, is that the universe is holographic. Holography is also an hypothesis, of
course, and may appear just as absurd as an action at a distance.
One of the main points of this paper is that the holographic hypothesis provides
a natural mechanism for gravity to emerge. It allows direct "contact" interactions
between degrees of freedom associated with one material body and another, since all
bodies inside a volume can be mapped on the same holographic screen. Once this
is done, the mechanisms for Newton's gravity and Hooke's elasticity are surprisingly
similar. We suspect that neither of these rivals would have been happy with this
conclusion.
27
Acknowledgements
This work is partly supported by Stichting FOM. I like to thank J. de Boer, B.
Chowdhuri, R. Dijkgraaf, P. McFadden, G. 't Hooft, B. Nienhuis, J.-P. van der Schaar,
and especially M. Shigemori, K. Papadodimas, and H. Verlinde for discussions and
comments.
References
[1] J. D. Bekenstein, \Black holes and entropy," Phys. Rev. D 7, 2333 (1973).
[2] J. M. Bardeen, B. Carter and S. W. Hawking, \The Four laws of black hole
mechanics," Commun. Math. Phys. 31, 161 (1973).
[3] S . W. Hawking , \Particle Creation By Black Holes," Commun Math. Phys. 43,
199-220, (1975)
[4] P. C. W. Davies, "Scalar particle production in Schwarzschild and Rindler metrics,"
J. Phys. A 8, 609 (1975)
[5] W. G. Unruh, \Notes on black hole evaporation," Phys. Rev. D 14, 870 (1976).
[6] T. Damour, "Surface e
ects in black hole physics, in Proceedings of the Second
ects in black hole physics, in Proceedings of the Second
Marcel Grossmann Meeting on General Relativity", Ed. R. Ru ni, North
Holland, p. 587, 1982
[7] T. Jacobson, \Thermodynamics of space-time: The Einstein equation of state,"
Phys. Rev. Lett. 75, 1260 (1995)
[8] G. 't Hooft, \Dimensional reduction in quantum gravity," arXiv:gr-qc/9310026.
[9] L. Susskind, \The World As A Hologram," J. Math. Phys. 36, 6377 (1995)
[arXiv:hep-th/9409089].
[10] J. M. Maldacena, \The large N limit of superconformal eld theories and supergravity,"
Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113
(1999)]
[11] R. M. Wald, "General Relativity," The University of Chicago Press, 1984
[12] R. M. Wald, \Black hole entropy is the Noether charge," Phys. Rev. D 48, 3427
(1993) [arXiv:gr-qc/9307038].
[13] L. Susskind, \The anthropic landscape of string theory," arXiv:hep-th/0302219.
28
[14] T. Padmanabhan, \Thermodynamical Aspects of Gravity: New insights,"
arXiv:0911.5004 [gr-qc], and references therein.
[15] J. D. Bekenstein, \A Universal Upper Bound On The Entropy To Energy Ratio
For Bounded Systems," Phys. Rev. D 23 (1981) 287.
29
No comments:
Post a Comment