On the Emergence of Spacetime and the Origin of
Gravity
Understanding E. Verlinde’s conjecture
Stefano Lorini
Amsterdam University College
Supervisor S. de Haro
May 15, 2012
CAPSTONE THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
BACHELOR OF SCIENCE
AMSTERDAM UNIVERSITY COLLEGE
Abstract
Despite the level of great precision at which we are able to describe gravitational attraction, thanks
first to Newton and later to Einstein, we still lack a complete theory able of telling us where this force
actually comes from. An original view on this matter has been introduced in 2010 by Verlinde, who tried
to explain gravity as an emerging phenomenon from an entropic force. He makes us of the parallelism
between a polymer in a heat reservoir pulled out of its equilibrium position and slowly allowed to return
to its natural state, and a holographic screen with a test particle at distance ∆x from it. The change in
available information due to the changing position of the particle with respect to the holographic screen
leads to a change in the entropy which causes an entropic force shown to be identifiable with gravity.
The content of this paper is based on understanding this conjecture and criticizing it where possible.To
do so, some physics tools used by Verlinde such as black-hole thermodynamics, the holographic principle
and statistical physics have been given sufficient attention in order to be understood, as they represent
core concepts of his work.
1
Contents
1 Introduction
3
2 Entropy and Entropic force
4
3 Towards E. Verlinde’s conjecture
10
3.1 Counting degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Black hole thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Holographic bound: 1 bit per Planck Area (Ads/CFT) . . . . . . . . . . . . . . . . 27
4 Heuristic and non-relativistic argument
30
4.1 Newtons law of gravitation and 2nd law of Newton . . . . . . . . . . . . . . . . . . 30
4.2 First generalization: The Newton potential . . . . . . . . . . . . . . . . . . . . . . 34
4.3 General matter distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Discussion
39
5.1 Problems with E. Verlinde proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2
1
Introduction
Isaac Newton was the first physicist capable of formulating a proper description of how an apple
falls from a tree, or how the Moon orbits around the Earth due to the interaction of their masses.
This description is known as Newtons theory of gravity and it describes how two objects are
attracted to each other. Nevertheless, this theory is incomplete. Newton himself was not entirely
satisfied with his own work because the inverse-square law that he formulated lacks a justification
of where the force comes from; he could not understand the true origin of gravity, in particular, he
could not determine the existence of some particles, such as gravitons, which would be responsible
for this interaction. Approximately two hundred years after Newtons great discovery, Albert
Einstein understood that the description of gravity needed to be reformulated and extended. In
his greatest work known as General Relativity, he proposed a new way of describing how two
objects interact with each other. One of the major changes that Einstein introduced is the nature
of spacetime. He believed that space and time should be unified and seen as one entity, and
that the presence of objects having mass would curve the spacetime. Taking this as his essential
starting point, he could write equations which describe how planets orbit around each other not
following straight lines, as believed by Newton, but following the curvature of spacetime due to
their masses. More specifically, in Newtonian spacetime, two events at two different positions
in space can occur at the same time; on the other hand, the revolution introduced by Einstein
imposes a limit on the speed at which a particle can travel, namely, slower or equal to the speed
of light. This rule has profound implications among which the fact that there is no absolute
meaning of two events occurring at the same time. By taking a point in space, an event, we can
construct a light cone which describes the allowed trajectory that light can travel. Unfortunately,
despite the great contribution of Einstein to understand gravity, his starting point is still the
assumption that gravity is just there as a fundamental force. In physics, there are four forces
which are considered to be fundamental: the strong nuclear force, the weak nuclear force, the
electromagnetic force and the gravitational force. For all these forces, some particles that cause
the force itself have been detected, except for gravity. This essential point well introduces the
purpose of this paper. We are still in need of a complete theory that explains the true origin
of gravity so that we can finally treat it as a fundamental force with certainty or perhaps, as
Verlinde believes, find out that gravity is an emergent force, and particularly, a manifestation of
entropy. Where does gravity actually come from? Is it really a fundamental force? These are
some of the questions we will try to answer in our research. In fact, solving this issue could have
3
very precious implications in order to obtain a complete theory of quantum gravity. Towards
this direction, an important piece of information comes from the fact that an accurate study of
black-hole thermodynamics suggest that there indeed might be a connection between gravity and
thermodynamical principles. If Verlinde’s ideas will turn out to be correct, the implications of
this work would have deep consequences on our understanding of the origin of gravity.
In our research, Erik Verlinde’s paper On the Origin of Gravity and the Laws of Newton[11] is
used as primary source and central paper of this research. In fact, our aim is to understand the
conjecture elaborated by the Dutch physicist as far as possible, with the tools of an ambitious
and advanced undergraduate student. Since I am fully aware of the great difficulties encountered
in the understanding process due to the highly technical material and the lack of knowledge on
highly technical concepts, the paper of the Verlinde has also been taken as a pretext and as a
starting point to dig into more advanced topics than those typically taught at an undergraduate
level.
2
Entropy and Entropic force
Entropy is one of the key ingredients in thermodynamics and it is at the core of Verlinde’s
conjecture; it is an unavoidable concept when dealing with the holographic principle of which we
will make an extensive use for the emergence of space and the origin of gravity as an entropic
force.
For a variety of systems, particles tend to rearrange themselves in such a way to reach the
biggest value of multiplicity (or very near). In order to do so, the system must be large, typically
in the order of 1020 oscillators, and must be in equilibrium. This statement is simply a general
expression of the second law of thermodynamics. Since the number of ways in which oscillators
can arrange themselves is most of the time a very large number, it is useful to take the logarithm
of the multiplicity. The obtained quantity is the entropy [6]:
S = κB logΩ
(2.1)
The more particle and energy a system have, the greater the multiplicity will be and therefore,
the entropy. Entropy can be seen as measure of chaos, disorder of a system. It is also possible to
increase entropy by breaking molecules, or by increase the space in which the molecule can move.
Knowing the concept of entropy, it is now straightforward to think of an entropic force as a force
4
that originates in a system with many degrees of freedom (DOF) due to the change (increase) ∆S
in entropy. We would like to give an example of an entropic force to contextualize this situation.
One of the most used examples for describing an entropic force is a polymer molecule immersed
in a heat bath. A heat bath is a system with substantially an infinite amount of energy; in
fact, when it is put in thermal contact with another object, its temperature remains substantially
the same. The polymer we want to take into consideration is composed by single monomers
all of the same length and the point of attachment with each other makes them free to move
in any direction. Under these conditions, once the polymer is immersed in the heat reservoir,
it will assume a random configuration with a higher value of entropy in comparison to other
possible configurations. When an external force is applied on the polymer and takes it out if
its equilibrium configuration (keeping one end fixed), the statistical tendency that the polymer
return to a configuration with a higher entropy will result in macroscopic force which points in
the opposite direction of the initially applied force (see figure 1). The direction of the (restoring)
entropic force is justified by the fact that there are more states at a lower entropy when the
polymer is stretched rather than when it is in a short configuration.
Figure 1: A polymer in a heat reservoir is pulled out of its equilibrium position by an external
force F.
The entropy for this system equal
S(E, x) = κB logΩ(E, x)
5
(2.2)
since we know that
Ω ∝ V Vp
(2.3)
where V is the volume of ordinary space (position space) and Vp is the volume of momentum
space. It is possible to determine the entropic force by analyzing the micro-canonical ensemble
which is given by Ω(E + F x, x), where the introduced external force depends on the length x of
the polymer. Using the first law of thermodynamics:
dU = T dS − P dV
(2.4)
T dS = P dV
(2.5)
we can write the force as
when the internal energy of the system does not change during our observations. Since our case
study is for the moment a linear displacement, the above identity can be rewritten as
T ds = F dx
(2.6)
where dV ⇒ dx, and from which we obtain an expression of the force as:
F =T
dS
dx
(2.7)
At this point, we would like to show that this force obeys Hooke’s law. In order to do so, we
are in need of an appropriate expression for the entropy and its derivative with respect to x, the
relative length of the polymer. The entropy of the polymer is given in eq.2.2 and according to
the Einstein model of a solid, the multiplicity Ω can be written as [6]:


N↑ !
 N 
Ω(N↑ ) = 
=
N↑ !N↓!
N↑
(2.8)
where the two directions of the arrow are meant to identify the direction (left or right) of each
link of the linear chain of the polymer, and N represents the total number of single links. We can
rewrite the expression above as
Ω(N↑ ) =
N↑ !
N↑ !(N − N↑ )!
6
(2.9)
Therefore, the entropy becomes
S
= κln
N↑ !
(N − N↑ )!N↑ !
= κ [lnN↑ ! − ln(N − N↑ )!N↑ !]
= κ [N lnN − N − (N − N↑ )ln(N − N↑ ) + (N − N↑ ) − N↑ lnN↑ + N↑ ]
(2.10)
Since lnN↑ ! ' N lnN − N and finally we can write
S = κ [N lnN − (N − N↑ )ln(N − N↑ ) − N↑ ln(N↑ )]
(2.11)
Before taking the derivative of the entropy, for reasons that will be clear in a moment, it is useful
to write an expression for the relative length of the polymer L in terms of its links l and their
directions:
L = l − N↓ − N↑ = l(N↑ − N↓ ) = l(N↑ − (N − N↑ )) = l(2N↑ − N )
(2.12)
Now we are ready to compute dS/dx by rewriting the force (eq.2.13) as
We know that
dL
dN↑
= 2l, then
dS
dN↑
F =T
dS
dS dN↑
dS dN↑
=T
=T
dx
N↑ dx
N↑ dL
dN↑
dL
1
2l
=
(2.13)
and
−1
1
= κ ln(N − N↑ ) − N↑
− ln(N↑ ) − N↑
N↑
N↑
= κ [ln(N − N↑ ) + 1 − ln(N↑ ) − 1]
= κ [ln(N − N↑ ) − ln(N↑ )]
(2.14)
At this point, the entropic force can be written as
7
F
κ
2l
κ
= T
2l
κ
= T
2l
= T
[ln(N − N↑ ) − ln(N↑ )]
(N − N↑ )
ln
N↑
N
ln(
)−1
N↑
(2.15)
Manipulating eq.2.12 we can obtain the following relation:
N
=
N↑
L
lN
2
+1
(2.16)
Hence, in the limit of L << N l the entropic force writes
κ
F = T ln
2l
2
L
lN + 1
!
(2.17)
Since,
1
'1−x
1+x
(2.18)
we can rewrite the force as
F
'
'
κ
L
ln 2(1 −
)
2l
lN
L
κ
ln(2) + ln 1 −
T
2l
Nl
T
(2.19)
(2.20)
For values of x << 1 ⇒ ln(1 + x) ' x(sinceL << N l). Thus,
κ
L
F 'T
ln(2) −
2l
lN
(2.21)
and finally,
F ' −κ
T
L+C
2l2 N
(2.22)
The above expression is exactly what we wanted to prove; the force does obey the Hooke’s law:
8
F = −kx
(2.23)
where in our case,
k = κ(T ) =
κT
2l2 N
(2.24)
We indeed showed that for the polymer, the force obeys Hooke’s law and therefore, it is
proportional to the stretched length. The entropic force is also possible to understand from the
first law of thermodynamics. In fact, this principle states that total change in energy of the system
is equal to the change in thermal energy, which is heated up by adding heat, minus the amount of
heat lost due to the work performed by the system on its surrounding. From this description, the
first law of thermodynamics appears to be a natural description of energy conservation. Now, as
we have previously described, our objects of interest are the heat bath, the polymer and of course,
the external system that surround our system; when the polymer is situated in its equilibrium
position, namely the one with the highest entropy, there is no entropic force appearing in the
system. In fact, the entropic force arises only when the polymer is ”artificially” extracted by
its natural configuration, assuming that the total internal energy of the system stays constant
during the change in entropy. When we apply an external force on the polymer to pull it out of
its equilibrium position, by the first law of thermodynamics, we provide energy to the heat bath
which must be equal to the work performed by the force (see eq.2.13), where F is the external force,
dx is the displacement of the polymer, and on the right hand side, T is the polymer’s temperature
and dS is the decrease in entropy of the polymer. In this way, the polymer loses heat by T dS
which is transfered to the heat bath, and this thermal loss is equal to the increase in energy gained
by the reservoir. During this process, where the external force acts on the polymer, the entropy
of the polymer decreases in favor of an entropic increase of the heat bath. Now, when we let the
polymer to be at a fixed displacement dx, the entropic force will be equal to the external force,
though their direction will be opposite to each other. Analyzing the opposite motion, if we let
the polymer to gradually go back to its equilibrium position, the entropic force will perform work
on the external system of an amount equal to the extracted energy of the heat bath; during this
process, the entropy of the heat reservoir will decrease and by energy conservation, the entropy of
the polymer will increase [4]. From the considerations we have carried out until now, we consider
the entropic force to emerge purely from the statistical tendency of increasing entropy and not
from any energy effects; this is an extremely important considerations, since it tells us something
9
more about the nature of the entropic force. In fact, it seems indeed strange that a force arises
without any real energetic effect, since the total energy of the system is conserved when the
entropic force is changing. The information that we can extract in light of this observation is that
the entropic force is not autonomous, but it strictly bases its existence on the presence of the heat
reservoir as a domain (if the polymer would be in a vacuum, there would be no entropic force);
although the entropic force is independent of the dynamics at a microscopic level, its existence
does depend on those microscopic interactions. As we will see in section 4, the discussion we
elaborated in this section has been used by Verlinde as analogy to argue that in the same way,
gravity is also a force emerging as an entropic one.
3
3.1
Towards E. Verlinde’s conjecture
Counting degrees of freedom
One of the most contemporary issues in theoretical physics finds its origins in the mid 70’s, when
Hawking introduced the so called black-hole information paradox [9]. The essence of this paradox
lays in Hawking’s belief that an object thrown into a black hole cannot be retrieved by any
means. In fact, the Hawking radiation tells us that a black hole could actually evaporate since
it emits radiation. Nevertheless, this radiation is a pure thermal emission and therefore, it does
not contain any information of the black hole itself and of whatever might be inside. Now, if
we consider the spontaneous creation of pair particles near the black-hole horizon which are in
a pure state, the one with negative energy might very likely fall into the black hole, while the
particle with a positive energy will fly away further from the black hole; in this situation, we
would be left only with one particle which is entangled with nothing, and therefore, the only
way we could describe it, would be using a density matrix, but that necessarily implies that
the ”left over particle” will no longer be in a pure state, but in a mix one. Unfortunately, this
violates the funding quantum mechanical principle of being a unitary theory. In contrast to this
view, such as the one proposed by Verlinde [5], takes into account some corrections to Hawking’s
calculation, attributing increasing importance to factors that have been neglected by Hawking
while computing the black-hole radiation; these corrections to Hawking’s radiation would solve
the black-hole information paradox, since not only thermal emission would be present, but actual
information could be extracted. It is clears that black holes represent a major challenge and one
the main difficulties encountered is to formulate with accuracy what are the degrees of freedom.
10
According to t’ Hooft [10][7], the father of the holographic principle, there is no need of a three
dimensional theory to fully describe nature, but only a two dimensional lattice at the spacial
boundary of the world. It is intriguing to think that according to the holographic principle, the
world we live in is, in some sense, two dimensional since all the information that it contains can be
stored on its boundary. The holographic principle finds its basis in Bekenstein Hawking formula
S=
area
area
log 2 =
lp2
4G
(3.1)
which poses a limit on the entropy of a black hole and therefore, to any region of volume V. Further,
Bekenstein’s thought experiment together with eq.3.1 led him to conclude that the maximum
entropy is proportional to the area and not the volume, as one might expect (see section 3.2).
Inspired by Bekenstein and Hawking discovery, t’ Hooft proposed a radical interpretation of these
findings suggesting that it must be possible to describe all the phenomenon taking place in a
region of volume V by using a set of degrees of freedom which are stored on the boundary of V. In
some sense, this means that all the phenomena taking place on a three dimensional space, might
be simply projected onto a two dimensional ”screen” without a single bit of information loss, with
a discrete lattice described by ”pixels” or ”bits”. At this point, it is in our interest to determine
how many degrees of freedom we can have. There are multiple ways to accomplish this goal and
we will do so by using thermodynamics. From eq. 3.1 we can write the entropy of a black hole as
S = 4πM 2 + C
(3.2)
where C is an unknown constant. If we consider a discrete quantum theory, the third law of
thermodynamics will hold and therefore, we can always relate the entropy of our system to the
total number of degrees of freedom. In particular, this relation looks like
eS = N = 2n
(3.3)
This represents the minimum amount of DOF, with n number of spins that can assume only two
values, as Boolean variables. We can solve for the number of Boolean DOF, n, in a region close
by a black hole to find
log eS = log 2n = n log 2
and thus
11
(3.4)
n=
S
4πM 2
A
=
=
log 2
log 2
4 log 2
(3.5)
with A being the horizon area. Now, we consider any closed surface with area A and we ask
how many possible field and metric configurations we can have inside this surface. If we want
our surface to be visible to an outer observer, then its total energy cannot overcome Bekenstein’s
limit, otherwise it would lie within the Schwarzschild radius. Given these limits on the volume
and the energy, we can assume that the most possible state (which will be soon justified) would
be (for an ordinary quantum field theory) a gas at a temperature T =
E = U = C1 ZV T 4
1
β
and the energy [10]:
(3.6)
where Z is the number of different fundamental particle types with m < T and C1 a constant of
order 1. The above expression for the energy has been obtained from statistical thermodynamics
and in particular, using the Planck distribution, for which the total energy density is given by
8π 5 (κT )4
U
=
V
15(hc)3
(3.7)
Before proceeding further, I would like to discuss a bit more in depth the origin of our energy
expression which naturally follows from eq 3.7. Since we considered a closed system and we are
interested in the probability of finding it in a particular state, by analogy, we can now consider the
(typical) situation in which out system is in thermal equilibrium with a reservoir at a particular
temperature. The microstates of our system correspond to the energy levels of the atom and for
each energy level, it is important to remember that there could be more states (degeneracy). If
our system would be isolated, then all its microstates corresponding to the energy at which the
atom is isolated would be equally possible. On the other hand, a more interesting case for our
purposes is to imagine our system interacting with many other atoms in the reservoir (at a fixed
temperature). Now, in principle, our system could now be in any of its microstates, depending on
their energies. We can now simplify the problem a little bit and we consider only the probability
of two particular microstates, s1 and s2 . Now, our system is not an isolated system, but the
atom together with the reservoir form an isolated system and therefore, the probability to find
our (combined) system in any of its microstates is equally distributed. In light of this, we can
express the ratio of the probability of the microstates s1 and s2 , namely P (s1 ) and P (s2 ), as
proportional to the number of microstates in the reservoir.
12
P (s2 )
ΩR (s2 )
=
P (s1 )
ΩR (s1 )
(3.8)
where ΩR (s2 ) and ΩR (s1 ) represent the mulitplicity of the reservoir when the atom is in that
particular microstate. Since
S = κlnΩ
(3.9)
P (s2 )
eSR (s2 )/κ
= S (s )/κ = eSR (s2 )−SR (s1 )/κ
P (s1 )
e R 2
(3.10)
we can rewrite the ratio as
By using the following thermodynamical identity,
dSR = 1/T (dUR + P dVR − µdNR )
(3.11)
we can express the very right hand side of eq.3.10 as [6]:
SR (s2 ) − SR (s1 ) = 1/T [UR (s2 ) − UR (s1 )] = −1/T [E(s2 ) − E(s1 )]
(3.12)
were E(s2 ) and E(s1 ) are the energy of the atom for a particular microstate. We did not consider
P dv since it is much smaller than the change in the energy of the reservoir, and µdN is really
zero for our system composed solely by one atom. By filling-in this expression into eq.3.10, we
get
P (s2 )
e−E(s2 )/κT
= e(SR (s2 )−SR (s1 ))/κ = −E(s )/κT
1
P (s1 )
e
(3.13)
We would like to rewrite this expression to have everything dependent on (s2 ) and (s1 ) on the
same side of the equation (N.B = fixed volume),
P (s2 )
P (s1 )
= −E(s )/κT
1
e−E(s2 )/κT
e
(3.14)
where the denominator of both fractions, e−Es is the so called Boltzmann factor. Since the left side
of eq.3.14 is independent of s1 and the right side of s2 , it means that they are both independent
of both s1 and s2 . This means that both the probability are equal to the same constant, namely
1/Z which transform the Boltzmann factor into a probability:
13
Ps = 1/Ze−Es /κT
(3.15)
known also as canonical or Boltzmann distribution, it represents the probability of a system to
be in a particular microstate. Since the total probability of finding our atom in one state is one,
then we can calculate Z as follows:
1=
X
P (s) =
X
e−E(s1 )/κT = 1/Z
X
e−E(s1 )/κT
(3.16)
and therefore,
Z=
X
e−E(s1 )/κT
(3.17)
where Z is called the partition function and it expresses the sum of all the Boltzmann factors.
Taking one step further, we can imagine our atom as a particle in a box, behaving like a harmonic
oscillator. Because of quantum mechanical principle, the harmonic oscillator cannot assume any
energy, but it has discrete packets of energy levels, such as En = 0, hf, 2hf, ... We can now express
the partition function for our oscillator in terms of the allowed energy levels:
Z = 1 + e−βhf + e−2βhf + .. =
1
1 − e−βhf
(3.18)
By definition, the average energy of a system is given by
Es e−βEs
Z
(3.19)
hf
1 dZ
= hf /κT
Z dβ
e
−1
(3.20)
P
Ē =
and it can be rewritten as
Ē = −
Since the energy levels are discrete, the average number of ”packets” of energy (occupation number) is given by
n̄ =
1
ehf /κT − 1
(3.21)
also known as Planck distribution, which tells us how many units of energy, photons, are in one
specific state. Finally, to compute the total energy of all the photons inside our surface, let
14
us imagine that in one dimension, the box has length L. Therefore, the photon wavelength and
momentum is given as
λ=
2L
n
(3.22)
p=
hn
2L
(3.23)
and since p = h/λ, in our case we obtain
where n indicates the mode we are considering. Since photons are relativistic, their energy is
= pc =
hcn
2L
(3.24)
We can now consider the fact that we are in 3 dimensions, and therefore the momentum becomes
a vector and the energy can be rewritten as
q
q
hc
hcn
λ = c (px )2 + (py )2 + (pz )2 =
(nx )2 + (ny )2 + (nz )2 =
2L
2L
(3.25)
To get the total average energy for all modes, we multiply the energy λ with the occupancy of
that mode, which is given by Planck distribution:
hcn
1
L ehcn/2LκT − 1
X
U=
nx ,ny ,nz
(3.26)
where the factor of 2 in the denominator of λ disappeared as it cancels out with an extra factor
of 2 that we added to account for the spin up and spin down of the photon. We can convert the
sums of the expression above into three integrals in spherical coordinates as follows:
Z
U=
∞
Z
π/2
Z
dn
0
π/2
dθ
0
dφn2 sin θ
0
hcn
1
hcn
L e 2LκT
−1
(3.27)
Since the angular integral gives an overall factor of π/2, we can reduce the triple integral to
Z
U=
∞
dn
0
hcn3
π
hcn
2L e 2LκT − 1
At this point, we would like to change variable to the photon energy d, where =
therefore, dn = 2L/hcd. Then, our energy expression can be written as
15
(3.28)
hcn
2L
and
∞
n3 π
e κT − 1
0
3 3
Z ∞
π 2L
hc
=
d κT − 1
e
0
Z ∞
8π3 /(hc)3 3
d
=
L
e κT − 1
0
Z
U=
d
(3.29)
where L3 = V and therefore
U
=
V
Z
∞
d
0
8π3 /(hc)3
e κT − 1
(3.30)
There is one last change of variables we would like to do which will allow us to express the energy
density in a more compact way. Let us consider x = /κT , where d = κT dx and thus
8π 3
U
=
(κT )4
V
hc
Z
∞
dx
0
x3
ex − 1
(3.31)
Now, using the following identity:
Z
∞
dx
0
xs−1
= Γ(s)ζ(s)
ex − 1
(3.32)
where Γ(s)ζ(s) are respectively the gamma function and the Riemann Zeta function, we obtain
U
8π 3
=
(κT )4 Γ(4)ζ(4)
V
hc
π4
8π 3
(κT )4 6
=
hc
90
(3.33)
and finally, we can write the energy density (eq.3.7)
=
8π 5 (κT )4
15(hc)3
(3.34)
Now that we have justified eq.3.7, we can use our energy expression (eq. 3.6) to write the entropy
as
16
S = C2 ZV T 3
(3.35)
[E] ∝ kT
(3.36)
since we know that
and that
[S] ∝
1
T
(3.37)
where C2 is another numerical constant.
From the Schwarzschild limit obtainable from the Schwarzschild metric, we can write
4 1
2E < (V / π) 3
3
(3.38)
Now, combining eq.3.6 with the Schwarzschild limit, we obtain
1
2C1 ZV T 4 < (3V /4π) 3
1
T 4 < (3V /4π) 3 1/2GZV
1
T 4 < (3V /4π8C13 Z 3 V 3 ) 3
1
T 4 < (3/32C13 Z 3 V 2 ) 3
1
T < (3/32C13 Z 3 V 2 ) 12
(3.39)
and finally,
1
1
T < Z− 4 V − 6
(3.40)
with
1
C3 = (3/32C13 ) 12
and therefore,
17
(3.41)
1
S < C4 Z 4 V
1
2
1
3
= C5 Z 4 A 4
(3.42)
At this point, directly justified by Bekenstein [2], if we consider a set of N black holes with
masses Mi we know that their degraded energy is smaller than the sum of the degraded energies
of the black holes if considered separately. Therefore, we can write
X
Mi < C6 A1/2
(3.43)
i
while the total entropy of the black holes is given by
S = C7
X
Mi2
(3.44)
i
with a maximum entropy value given by Bekenstein-Hawking entropy (see next section 3.2):
1
Smax = C8 A
4
(3.45)
which tells us that the black hole is the only limit to the total number of possible states, which
is given by eq. 3.3 and eq. 3.5 in case we consider only boolean variables.
In the following section, we will describe and derive in an accurate way the black-hole entropy
formulated by Bekenstein. In fact, his results together with the ones obtained by Hawking are of
great impact in the modern physics and for our purposes, they tell us that the number of possible
states grows with the area and not with the volume as we might have expected; therefore, given
any close surface we can describe all the information stored inside it looking at the degrees of
freedom on the surface of the object itself.
3.2
Black hole thermodynamics
In Verlinde’s conjecture, the emergence of spacetime is substantially, and implicitly, described
using a black hole and the entropy that it carries. The purpose of this section is to resemble the
line of argument that led Bekenstein to formulate a rather precise definition of blackhole entropy
in a thermodynamical language.
The fact that a black hole area might increase is an effect that had already been noticed by
Flyod and Penrose by considering a Kerr black hole [3], which has the characteristic of having
no charge and to rotate around a central axis. The tentative of explaining this effect gave birth
18
to the so called Penrose effect, a process for which energy can be extracted from a rotating
black hole. The extraction of some energy from the Kerr black hole is made possible by the fact
that its rotational energy is not situated inside the event horizon, but in an outer region called
ergosphere, a region in which it is possible to enter and/or leave again, but it is not possible to
stay still. Now, if some matter enters this region, it splits into two in a manner for which the
momentum of the material leads one piece to fly off to infinity, and the other one to fall into the
hole, since one piece carries a positive mass-energy and the other negative. Further, it is possible
Figure 2: A particle that falls into the ergosphere of a black hole, splits into two. The one with
higher energy escapes, while the other one falls into the hole (Penrose process).
that the escaping piece has a greater mass-energy than the other one, and therefore, this process
causes a decrease in the angular momentum of the black hole, which is converted in extracted
energy (see figure 2). The predictions hypothesized by Penrose and Floyd were sustained by an
independent work conducted by Christodoulou, but the revolutionary approach on this matter
comes from Hawking, who proved that the area of a black hole cannot decrease by means of
any process. More specifically, Hawking’s theorem suggests that, if we consider a system of a
black hole, the area of each individual black holes cannot be bigger than the area of the resulting
black hole. From these results obtained by Hawking, it is possible to hazard treating black hole
physics from a thermodynamics approach; in fact, since changes in a black hole seem to take
place in the direction of increasing area, this process sounds like an alternative version of the
19
second law of thermodynamics, for which changes of a closed (thermodynamical) system occur
in the direction of increasing entropy. This apparent relationship, which is the principal starting
point of Bekenstein argument, nicely highlights the possible correspondence that here we would
like to show, between an increasing entropy and an increasing area. According to Bekenstein,
this parallelism between black-hole area and entropy is even deeper; in fact, the blach-hole area is
shown to be related to the degradation energy in the same way the entropy is. The degradation
energy is the energy that due to an increase in entropy cannot be converted into work anymore,
and also, it corresponds to the energy that cannot be extracted by a classical process, such as
Penrose process; this energy is also called irreducible mass [2] of a black hole and it is given by
Mir = (A/16π)1/2
(3.46)
Abh = 4πR2 = 16πM 2
(3.47)
recalling the fact that
Now, as we can notice from eq.3.46, if the black-hole area increases, so it does the amount of
degraded energy of a black hole. Since the irreducible mass of a Schwarzschild black hole is 100%
of its total mass, it means that no energy can be extracted. Nevertheless, it is possible to bypass
this issue by considering, for example, a system of two black holes. In fact, in this way, if we let
the two objects to merge, we can still retrieve energy in the form of gravitational waves; in the
merging process, as we have seen before, the only thing that matters is that the total black-hole
mass does not decrease, as it should be according to Hawking. On the other hand, if we write
down the degraded energy for our system of black holes, we obtain
Ed =
X
A/16π
1/2
=
X
2
Mir
1/2
(3.48)
and it appears evident that the degraded energy for our system of black hole turns out to be
smaller than the sum of the degraded energies if we consider them separately, since we would
obtain cross-terms. The great advantage of such a process it that, by combining Schwarzschild
black holes, which are substantially dead since all their mass is irreducible, we can still obtain
energy. In the same way, we imagine the same process from a thermodynamical point of view;
if we consider two closed systems in equilibrium and we let them interact, we can obtain work,
while originally, the two systems alone cannot do any work. In light of this fact, the parallelism
20
we are trying to formalize between the black-hole area and entropy comes out even stronger, and
then, the next step should to find the correspondent expression for the black hole to the first law
of thermodynamics
dE = T dS − P dV
(3.49)
In order to do so, Bekenstein starts considering a Kerr black hole, so that we consider only the
~ and the charge Q. Now, the so called ”rationalized”
black-hole mass M, the angular momentum L
area of the black hole is given by
2
α = r+
+ a2 = 2M r+ − Q2
(3.50)
~
a = L/M
(3.51)
r+ = M ± (M 2 − Q2 − a2 )1/2
(3.52)
with
and
Now, we are indeed ready to find the correspondence of the first law of thermodynamics for a
black hole. In fact, deriving eq. 3.50 with respect to dM and then solving for it, we obtain
~ L
~ + ΦdQ
dM = Θdα + Ωd
(3.53)
with the three parameters
1
(r+ − r− ) /α
4
Θ
=
~
Ω
= ~a/α
(3.55)
Φ
= Qr+ /α
(3.56)
(3.54)
(3.57)
~ + ΦdQ corresponds
Now, eq.3.53 already is the expression we were looking for since the term ΩdL
~ and charge Q, done by any external entity;
to an increase in respectively, angular momentum L
21
to do so, the external object needs to perform work, and therefore, the above term precisely
corresponds to the thermodynamical term −P dV . Further, from our previous observations on
the relationship between the area and entropy, we can conclude that dα corresponds to dS. Finally,
the correspondence of the temperature T in eq.3.49 is given by our expression for Ω, which again,
is non negative.
On the basis of the observations we have given above, Bekenstein proposed that indeed, the
black-hole entropy is proportional to some constant with its horizon area
Sbh = cAhor
(3.58)
where c is a constant. Bekenstein’a belief of such a deep relationship between a geometrical
quantity and a thermodynamical one led him to take even a step further and trying to analyze
this situation from the point of view of information theory. In this field, the correspondence
between entropy and information is well described and understood. For instance, let us consider
an isothermal compression of an ideal gas; because of the compression, we are able of better
determining the position of the molecules with more accuracy, since they are more localized, but
contrarily, the compression also reduces the thermal entropy of the gas, which now is not in its
most favorable state. According to Shannon [2], this dynamics can be well expressed as
∆I = −∆S
(3.59)
From this approach, it follows that an increase of information available for a system, automatically
leads to a decrease in its entropy, but nevertheless, acquiring information is never a free process,
since overall, the total entropy of the universe would increase by a factor which is accordance with
the second law, must be greater than ∆I. In this way, it is possible to describe the entropy as the
measure of a lack of information of a given system; the entropy of a thermodynamical system not in
equilibrium increases since in time, the available information concerning the internal configuration
of the system is ”wiped away”. A formalization of Shannon’s view has been given by Boltzmann
as it follows:
S=−
X
pn ln(pn )
(3.60)
n
where Boltzmann constant κb has been set to 1 and pn expresses the probability of occurrence
of a particular internal configuration, labeled by n. Conventionally, the unit of information is
22
given by the ”bit” which is defined as the information associated to a situation for which the
answer of a yes-or-no question is precisely known (zero entropy). Now, if we want to relate this
with Boltzmann’s equation (3.60), we see that the entropy function reaches is maximum when
pn → pyes = pno = 21 ; this, corresponds to a maximum entropy of ln2 which can be considered
as the unitary value of entropy. We can justify this maximum value by computing explicitly the
entropy in eq.3.60:
S = −p1 ln(p1 ) − p2 p2 = −pln(p) − (1 − p)ln(1 − p)
(3.61)
where p1 and p2 represents the two probability to a yes-or-no question; also, since we are considering pn → pyes = pno = 21 , then we can rewrite p1 and p2 as we have done in the above expression.
Now, if we take the derivative of the above entropy expression and set it to zero we obtain:
1−p
δS
= −ln(p) − 1 + ln(1 − p) +
= ln
δp
1−p
1−p
p
=0
(3.62)
and therefore,
p=
1
2
(3.63)
Filling this back into eq.3.61, we obtain
1
1
1
1
1
S = − ln( ) − ln( ) = −ln( ) = ln2
2
2
2
2
2
(3.64)
In this context, we can conclude that the black-hole entropy Sbh arises as the consequence of
the lack of information about its gravitational collapse, the inward fall of an object due to its
own gravitational force. As it has already been anticipated before, the corresponding second law
of thermodynamics for the black hole is characterized by three parameters (see eq.3.53) M, Q
~ these parameters are all we need to know about the black hole and they described the
and L;
post-collapse configuration of it, while on the other hand, they encode the information concerning
the different sets of events that occurred during the gravitational collapse; in the same way
that a thermodynamical system is characterized by some quantities such as volume, pressure
and temperature which encode information about the microstates of the system. At this point,
Bekenstein makes an essential remarks in understanding what goes on here, in fact ”black holes in
equilibrium having the same set of three parameters may still have different internal configurations,
[..] in fact, our black-hole entropy refers to the equivalence class of all black holes which have the
23
same mass, charge and angular momentum” [2]. What Bekenstein really means to say is that we
should not consider the black-hole entropy as the thermal entropy inside the black hole horizon,
but a class of black holes which share certain properties.
To compute a step further in order to obtain a precise expression for the black-hole entropy,
we can refer to eq.3.58 which is first generalized by Bekenstein as
Sbh = f (α)
(3.65)
where we recall that α = Ahor /4π. This expression of the entropy assumes that the functionf is
a monotonically increasing function. In support of this choice, we need to remember Hawking’s
theorem, for which the black-hole entropy cannot decrease by means of any process. In fact,
eq.3.65 is in total agreement with this, since a possible loss of information about the initial
conditions of the black hole naturally reflects an increase of its entropy. Now, in order to find
out which value we should assign to f (α) several options have been taken into consideration and
successively thrown away since they were falling into logical contradictions on the nature of the
black hole itself and its properties. The choice adopted by Bekenstein is
f (α) = γα
(3.66)
where γ is a constant. Further, Bekenstein noted that the correct units that γ had to have are
the ones of (length)−2 since the entropy is a dimensionless quantity (if we do not consider the
boltzmann constant), and since α represents an area. Now, there is no such a quantity in classical
general relativity that would give that order of units, and therefore, to obviate to this issue,
quantum physics has been taken into play. In fact, the only way in which Sbh and Ahor can be
proportional is the case in which
γ = µ/~
(3.67)
where µ is a dimensionless constant. After having identify the best candidate for the value of
f (α), the last step to take consists in trying to obtain a precise value for µ/~ and to do so, we
are in need of the following argument.
Let us consider a Kerr black hole and a particle which is approaching it. At some point, the
particle will fall into the black hole and some information will be lost. According to Boltzmann’s
equation (3.60) the loss of information will cause an increase in the entropy by an appropriate
24
order of magnitude. Now, the amount of information loss depends on the internal state of the
particle, which we know nothing about. On the other hand, what we CAN know is the minimum
amount of information lost which is an answer to the question of whether the particle exists or
not. Following Christodolou’s technique, Bekenstein argues that the minimum increase in the
area of a black hole due to a particle that falls in is given by
(∆α)min = 2µbmin
(3.68)
where µ is the mass of the infalling particle and b its proper radius. Obviously, b cannot be
smaller than the Compton wavelength ~/µ or than its gravitational radius 2µ; since the Compton wavelength and the gravitational radius reach respectively their maximum size for values of
(∆α)min equal to
µ ≤ (~/2)1/2
(3.69)
µ ≥ (~/2)1/2
(3.70)
(3.71)
According to these limits, the minimum increase in the area of a black hole 2µbmin can be as
small as 2~ when we consider the Compton wavelength, while concerning the gravitational radius,
2~ cannot be smaller than 4µ2 > 2~. Therefore, since we are interested in the minimum value of
(∆α)min , this is indeed the case when it is equal to 2~. Finally, we need to recall the fact that
the minimum loss of information is a binary bit that corresponds to an increase in entropy of ln2.
Thus, equating the minimum increase in black-hole entropy with ln2 we obtain
df
(∆α)min = ln2
dα
(3.72)
and integrating the above expression gives
f (α) =
1α
ln2
2~
(3.73)
Retrieving all the constants, such as G, c and κb which were originally set to 1, and inserting
back the value of the rationalized area α = A/4π we obtain
25
Sbh =
1
ln2κb c3 Ahor
8πG~
(3.74)
This is the expression we were looking for which has been proposed by Bekenstein, and it is equal
to the one proposed by Hawking, apart for the factor of ln2.
An alternative, though very similar way to retrieve the black-hole entropy goes as follows. Once
again, we consider the first law of thermodynamics
dE = Tbh dS
(3.75)
1
dS
8πM GκB
=
,
=
dE
Tbh
~c3
(3.76)
Hence,
with Th =
~c3
8πM GκB .
Integrating the above expression and since for a black hole dS = c2 dM ,
dS =
8πM GκB
dM
~c
dS =
8πGκB
~c
Z
(3.77)
Z
M dM
(3.78)
And then,
Sbh
=
=
8πGκB 1 2
M
~c 2
4πM 2 GκB
~c
(3.79)
(3.80)
(3.81)
Now, remembering the expression for the Scwarzschild radius,
Rs =
2GM
c2
⇒ M = Rs
2
c
2G
(3.82)
4πGκB R2 c4
4G2 ~c
(3.83)
Hence,
S=
26
Which taking A = 4πRs2 become
S=
Ac3 κB
AκB
=
4G~
4lp2
(3.84)
which is the classical expression of the black-hole entropy.
3.3
Holographic bound: 1 bit per Planck Area (Ads/CFT)
In this section, I will follow Susskind and Witten [8] line of argument to prove the holographic
bound of 1 bit of information per Planck area, which is one of the most important assumptions in
Verlinde’s conjecture. Below, I carry out a computation which has been left implicit by Susskind
and Witten in their proof. As we have seen before, Bekenstein’s bound on the maximum value of
the black-hole entropy varies with respect to the area and not to the volume. If we consider any
standard quantum field theory, this bound is not valid and this is precisely where the holographic
principle comes into play. The holographic principle tells us that in a quantum gravity theory,
the physics contained in a given volume can be represented using some theory on the surface
boundary of the object itself, as long as it has less than one degree of freedom per Planck area
[1]. In Ads/CFT correspondence, physics is described in the bulk of Ads, using a field theory
with one less dimension which lives on the boundary. This type of theory is a conformal one and
therefore, there are an infinite number of degrees of freedom and the boundary of the Ads also
have an infinite area. Thus, we need to introduce a cutoff, so that we can limit the number of
degrees of freedom available in the field theory, and see what the correspondence would be in the
gravity theory. Now, in order prove the holographic bound, we need to calculate the area A of a
given surface, namely a sphere, starting from an AdS space, so that in this way we will be able of
checking what are the degrees of freedom there and prove the holographic bound. To do so, we
employ a four dimensional metric used by Susskind & Witten [8] in their derivation [13][1]:
"
2
dS = R
2
2
2
4dxi dxi
21 + r
−
dt
(1 − r2 )
1 − r2
#
(3.85)
with i = 1, ..., 4 and r2 = xi xi where the AdS space is the ball of radius r < 1 and then the
boundary conformal theory lives on the sphere r = 1. Since we want to obtain the area of a three
dimensional object, we use the boundary of our AdS sphere, at r = 1, to reduce the dimensions of
the metric. If we fill in for r = 1, we see that the metric diverges. In order to avoid this problem,
we can regulate the area of the boundary by replacing it with a sphere infinitesimally smaller
27
than r = 1, namely at r = 1 − δ, where δ << 1. Now, we can write the volume element of the
metric as [3]
p
|g|dn x =
p
|g|d0 x ∧ · · · ∧ dxn−1
(3.86)
where ∧ is the wedge product and g = |gµν | is the metric tensor which determinant does not
vanish. Then, we can calculate the volume taking the integral of the above expression:
V =
Z p
|g|dn x
(3.87)
and since we are interested in the area A, we can modify the above expression taking one less
dimension and evaluating the integral with respect to our boundaries to obtain
∞
Z
dn−1 x
A=
p
Z
∞
|gn−1 | =
δ
d3 x
p
|g3 |
(3.88)
δ
since in our case n = 4. In our calculation of the area, we can forget about the time dependence
in eq. 3.85 which is now irrelevant, and the metric reduces to
dS 2 = R2
4dxi dxi
(1 − r2 )
(3.89)
Finally, we need to calculate the determinant of the metric which can be written as
dS 2 = R2
4r2 dΩ2(3)
(3.90)
(1 − r2 )
since
dxi dxi = r2 dΩ2(3)
(3.91)
Hence,

gµν


=



4
(1−r 2 )2
0
0
4
(1−r 2 )2
0
0
0
4
(1−r 2 )2
0

2
 2
 R = 4R
dΩ2 matrixg̃

2
2
(1 − r )

(3.92)
Therefore,
p
|g3 | =
4R2
(1 − r2 )2
28
23
p
g̃3
(3.93)
p
|g3 | =
8R3
[1 − (1 − δ)2 ]
3
=
8R3
R3 p
8R3
=
=
g̃3
(1 − 1 + 2δ + ...)3
(2δ)3
δ3
(3.94)
Finally, we obtain the expression for the area as
Z
A=
∞
d3 x
p
|g3 | =
δ
R3
δ3
Z
∞
d3 x
p
|g̃3 |
(3.95)
δ
At this point we are almost done; what we need to do is to get an expression for the degrees
of freedom of our cutoff boundary theory. In order to do so, we need to make some assumptions
concerning the ”storage capacity of a cutoff field theory” [8]. When we introduce a cutoff in
the filed theory, it is like if we would replace the space in which the theory lives in by discrete
little cells with a dimension corresponding to the cutoff size. In our case, the regulated sphere is
replaced by cells of the size δx (which is dimensionless). Susskind & Witten have also assumed
that each quantum field is replaced by one single degree of freedom for each cell and that each
degree of freedom can store only one bit of information. Now, thanks to eq. 3.95, we can assert
that the total number of cells that form the regulating sphere is in the order of δ −3 and number
of field degrees of freedom in a U(N)1 theory is in the order of N 2 . Therefore, the number of
DOF can be written as
Ndof =
N2
δ3
Now, since we calculated before that the area is proportional to A '
(3.96)
R3
δ3 ,
then we can rewrite 3.96
as
Ndof =
N 2A
R3
(3.97)
and inserting the expression for the radius R in terms of the string coupling gs , and the string
lenght ls [1] gives us
Ndof =
AR5
ls8 gs2
(3.98)
where ls8 gs2 R−5 is the 5 dimensional gravitational constant. Therefore, our ultimate expression
1 In our situation, the field theory used is a gauge field theory, a type of theory in which the Langrangian L, a function that
summarize the behavior of a system, is invariant under a continuous group of local transformations. Possible transformations
between gauges, the so called gauge transformations, form the symmetry group (or gauge group) at which is associated the
group generators. As Susskind & Witten refer to [8] ”[..] we saw that regulating the boundary area is equivalent to regulating the
super Yang-Mills theory.” In a Yang-Mills theory, the degrees of freedom are in a non-abelian gauge field, and the gauge field
transforms according to a gauge group U(N), of which generators of the group are N XN matrices with order of N 2 components.
29
for the number of DOF is
A
G5
(3.99)
c3 lp2
,
~
(3.100)
Ndof =
Since
G=
then eq.3.99 is indeed the expected result which proves the holographic bound of 1 bit per Planck
area.
4
4.1
Heuristic and non-relativistic argument
Newtons law of gravitation and 2nd law of Newton
At this point, we are ready to work out Verlinde’s argument in order to retrieve the laws of
Newton in analogy with the example of the polymer that we have treated in section 2. The
starting point of the argument and of the analogy with the polymer refers directly to Bekenstein’s
thought experiment (see section 3.2); in the attempt to find a precise value to the constant of
proportionality between the black-hole entropy and the area, he hypothesized a situation in which
a particle approaches a black hole until when it disappear with it and the information that it carries
is lost in the black hole. As we discussed previously, this loss of information (eq. 3.59) has to
reflect a proportional increase in black-hole entropy which already takes place when the particle
is found only one Compton length away from the black-hole horizon. Verlinde’s argument makes
use of this reasoning, substituting the black hole by employing the usage of a piece of holographic
screen and still a particle of mass m that approaches it (from the part of space that has already
emerged). The switch between using a black hole and just a piece of holographic screen appears a
bit fictitious since by the holographic principle, a black hole is indeed itself a holographic sphere,
where all the information is stored on its boundary. What we would like to show here, is how the
force that causes the particle to merge with the holographic screen arises.
According to the Dutch physicist, the particle of mass m will behave in the same way that the
polymer does when it is out pulled from its equilibrium position, in fact ”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 effective force [..] ” [11]. Indeed, from previous observations, we can understand
30
that the particle has an entropic reason to merge with the holographic screen since when ∆x → 0,
then ∆S → max. According to Bekenstein’s solution and the classical expression of the entropy
(eq.2.1), we can then express the change in entropy and its corresponding distance from the screen
as
∆S
=
2πκB
(4.1)
∆x
=
~
mc
(4.2)
(4.3)
where the expression for ∆x arises from the Compton limit, since λc =
~
mc
and the factor of 2π in
the change in entropy has been introduced by E. Verlinde in order to properly retrieve Newton’s
law. Now, to work out the units properly, we would like to rewrite ∆S in a slightly different form,
since we assume that the change in entropy is linearly dependent on the distance ∆x from the
screen and inserting eq.4.1
∆S = 2πκb
mc
∆x
~
(4.4)
Now, being faithful to Verlinde’s assumption, since the particle does have an entropic reason to
go towards the screen, it will experience an entropic force exactly as the polymer does:
F ∆x = T ∆S
(4.5)
The last piece of information we need to perfectly reconstruct Newton’s law comes from the fact
that, if we want to have a non-zero force, we need a non-zero temperature. Since according to
Newton a non-zero force means a non-zero acceleration, we can find a bridging equation by using
Unruh’s temperature
T =
1 ~a
2π c
(4.6)
This temperature arises from Unruh’s prediction that an accelerating particle detector in empty
space will eventually encounter a distribution of particle with a thermal spectrum (a heat bath);
this thermal effect is expressed in the Unruh’s temperature. Now, rewriting eq. 4.5 as
31
F =T
∆S
,
∆x
(4.7)
we can fill in for the expression of ∆S and T given in eq. 4.4 and eq. 4.6 obtaining
F =
~a
mc
1
2πκb
∆x
2πκb c
~
∆x
(4.8)
which is indeed equal to
F = ma
(4.9)
Figure 3: A test particle of mass m interacts with the Holographic sphere at temperature T. F
is the entropic force which drags the particle towards the screen which with it will merge.
After having precisely recovered Newton’s second law, we would like to continue following
Verlinde’s proposal and retrieve also Newton’s law of gravitation. To do so, we need a few more
pieces of information to have a complete and coherent set of equations which allow us to do so.
To begin with, we would like to extend the piece of holographic screen that we just used to a
complete holographic sphere, despite the fact that by doing so, it seems that we are indeed fully
evoking a black hole and therefore the type of physics that it carries (see section 5). Now, since
our sphere is a holographic one (see figure 3), all its information and therefore energy, is stored
on its boundary; we can assume that total energy of the system-sphere E is evenly distributed
over the bits N on its surface area A. Therefore, by the equipartition principle, the energy of the
sphere reads
32
Ekin =
1
mv 2
2
=
3
1
kT = N κB T
2
2
(4.10)
where T is the temperature of the surface of the holographic sphere and N is the number of
bits. It is precisely of the latter that we need to find an appropriate expression. Since we are
basing all the argument on the holographic principle, we can assume that the total number of
bits is proportional to the area A, as we have seen discussing Bekenstein’s results. In the previous
subsection (3.3) we gave a justification for the holographic bound of 1 bit per Planck area (eq.3.99)
Thus, in light of this, we can express the total number of bits as
N=
where
c3
G~
=
1
2.
lp
Ac3
G~
(4.11)
We are almost ready, we need only one more equation that Verlinde introduces
without much explanation. This equation is
E = M c2
(4.12)
In fact, it seems confusing introducing a purely relativistic expression in an argument that so far
has been substantially classical. On the other hand, since the Newtonian limit is for c → ∞,
this means that the rest energy is much bigger than any other energy terms and therefore, it can
be neglected. Now we have all we need to retrieve Newton’s law of gravitation starting from a
holographic scenario and following a reversed logic that brought to ”discover” holography. The
two energy expression, eq. 4.10 and eq. 4.12 can be equated,
1
N κb T = M c2
2
(4.13)
2
N κb T
(4.14)
and solving for T,
T = M c2
If we now insert the expression for the total number of bits (eq. 4.11), we obtain
T =
2M c2
2M
=
Ac
Ac3
κb G~
κb G~
(4.15)
Remembering the expression we postulated for the change in entropy which is linearly dependent
on the displacement (eq.4.4) and inserting it in eq.4.7, together with the above one and trivially
33
A = 4πR2 , we get
2M
F =
2c
κb 4πR
G~
2πκb
mc
4M m
= R2
~
G
(4.16)
and therefore, the expression we were looking for:
F =G
Mm
R2
(4.17)
According to the father of this conjecture, the fact that we have been able of reconstructing
Newton’s law mainly from first principles, using holography and black-hole thermodynamics as
starting points, should reveal a new face on the origin of gravity as an entropic force. In fact,
he sustains that we have not only reverse these arguments, but we have more in depth changed
the logic that brought to the construction of Newton’s law. But on the other hand, what else
could have we expected to obtain? In some sense, it seems natural to obtain the above equation
according to the steps we have taken and therefore, it is indeed mainly a matter of the logic and
the approach to the matter.
4.2
First generalization: The Newton potential
We would like to continue the above argument and try to generalize it as far as we can. Previously,
we were looking at the situation in which a particle with mass m approaches and merges with
holographic sphere. By doing so, the information corresponding to the particle will be stored and
represented by the bits on the screen. Now, in line with our purpose, we can find an expression
for the entropy change in relation with the acceleration by first combining eq. 4.4 with eq. 4.6
and then inserting eq. 4.13 as follows
∆S = 2πκb
mc
1 ~a
∆x ← T =
~
2π c
(4.18)
Thus,
∆S = 2π
1 ~a mc
∆x
2π cT ~
And inserting eq. 4.13,
34
(4.19)
∆S
1 ~a 1
1
∆x nκb T
2π cT ~
2c
an∆x
= κb
2c2
=
2π
(4.20)
(4.21)
Which gives us:
∆s
a∆x
= κb 2
n
2c
(4.22)
With the above expression, we manage to find a relation between the acceleration a and the
entropy gradient. Therefore, it seems appropriate to write the acceleration in its gradient form,
introducing the Newton potential Φ:
a = −∇Φ
(4.23)
Now the gradient of the Newton potential is given by
∇Φ =
δΦ~ δΦ~ δΦ ~
i+
j+
k
δx
δy
δz
(4.24)
but since we are interested in one specific direction (and the entropy change is perpendicular to
the screen proportional to the displacement ∆x), we can simplify the above expression as
∇Φ =
∆Φ
∆x
(4.25)
and therefore, eq.4.22 can be written as
∆S
∆Φ
= −κB 2
n
2c
(4.26)
As Verlinde highlights himself, this is indeed an important equation, since we can understand
the role that the Newton potential plays here; it is proportional to the change in entropy of the
system and keeps track of it. In some sense, it seems to me that the essence of this equations
resemble a more specific situation of the one expressed by Shannon’s formula (eq. 3.59) that we
used in section 3.2. In fact, the physical meaning of Shannon’s formula tells us that when the
amount of available information decreases, on the other hand the entropy increases. Now2 , in
2I
would like to thank S. de Haro for the insightful discussion on this matter.
35
our situation we are considering what happens to the entropy in relationship with the distance
between the test particle and the screen; if we want to increase ∆x, we need to perform work
in order to do so, and therefore, the entropy will be reduced. In contrary, if we put the particle
closer to the screen, the entropy will increase since we go to a more likely thermodynamical state.
In light of this, it seems logical to introduce a minus sign in eq.4.4, and if we would carry it all
along in this section (4.2), eq.4.26 would become:
∆S
∆Φ
= κB 2
n
2c
(4.27)
which testimony the fact that the entropy changes anytime that the Newton’s potential changes,
in accordance to what have been said previously.
4.3
General matter distributions
In this section, we would like to take a step further in the generalization of section 4.1 considering
a situation in which we have a random matter distribution on the holographic screen, where
spacetime is not yet emerged. In order to do so, E. Verlinde introduces the concept of coarse
graining, a process of rescaling of a phenomenon into fewer and larger units of size close to the
uncertainty of the measuring device (see figure 4). Now, due to coarse graining, we reduce the
amount of information which is visible at a macroscopic level. In fact, the entropy measures
precisely this amount of lost microscopic information, which would look random at a macroscopic
level, and that has been ”neglected” by coarse graining; if we want to apply this technique on our
holographic sphere, we are obliged to reduce its radius so that we reduce the amount of bits, and
by doing so, we trade some loss of information on the boundary in exchange of some extra part
of emerged spacetime. From a thermodynamical point of view, a coarse graining process seems
a natural choice, since the decrease of available information on the screen leads to an increase
in entropy. Therefore, Verlinde came to the conclusion that, in light of eq. (3.16 Verlinde) the
Newton potential is the responsible element that keeps track of the change in entropy: since
the amount of information ”invisible” at macroscopic level due to coarse graining is dependent
on the entropy change and on a general matter distribution on the screen, then the Newton
potential, which we showed to be related to the change in entropy, well described the degree
at which the screen present coarse graining. Since our aim is to obtain a generalization of the
dynamics expressed in section 4.1, let us go back to observe the change in entropy; in fact, if the
latter indeed changes in relation to the location of the matter distribution and the coarse graining
36
Figure 4: Representation of coarse-grained approximation of polymer segments from the atomic
representation to the ideal chain representation and bead-spring model.
process, then this entropic change would again generate an entropic force. As we have done
before, we are interested in calculating the gravitational force due to the displacement ∂x and the
corresponding entropic change, but this time, we would like to us a collection of test particle of
mass mi . Since we have a general matter distribution, the force we want to get will depend on it,
since the change in entropy might not be homogeneous. Specifically, Verlinde describes a virtual
displacement ∂xi related to the test particles (m = mi ) and therefore, forgetting for the moment
of time effects, we can consider the matter density as a static scalar value. Further, he considers
again an holographic screen S where the matter distribution is contained inside the volume of the
screen, while all the test particles are in the outside region where spacetime is already emerged.
The first modification we introduce in comparison with the previous case where we considered an
holographic sphere (now an arbitrary holographic screen) is the change in the density of bits. If
we assume again that the bits are homogeneously distributed, and since we are considering an
infinitesimal displacement, the relationship between the number of bits and the area reads:
dN =
c3
dA
G~
(4.28)
The second modification we introduce regards the expression for the temperature T that we previously identified with the Unruh’s temperature. In this case, not having a spherically symmetric
holographic screen, the temperature T might not be in equilibrium, but nobody stops us to identify it locally per unit area. By doing so, and keeping in mind the role we attributed to the
Newton potential, we can write Unruh’s temperature as
37
κB T =
1 ~∇Φ
2π c
(4.29)
while in Verlinde’s paper, this formula appears with an extra factor κ in the denominator,
which is probably a typo. Finally, the last piece which is object to modifications regards the
equipartition of the energy expression (eq. 4.10) which assumes the form of:
E=
1
κB
2
Z
dN T
(4.30)
S
Now, working out these last three rearranged equations together with the relativistic energy we
obtain the following:
2
Mc
=
=
Z
1
κB dN T
2
S
Z
1
c3
κB dAT
2
g~
S
(4.31)
(4.32)
(4.33)
M
=
=
Z
c3 κB
dAT
2c2 G~ S
Z
c
dAκB T
2G~ S
(4.34)
(4.35)
(4.36)
and inserting the modified Unruh’s temperature,
M
=
=
Z
c
1 ~∇Φ
dA
2G~ S 2π c
Z
1
dA∇Φ
4πG S
(4.37)
(4.38)
(4.39)
we obtain precisely a very well known equation: Gauss’s law for gravity.
38
5
5.1
Discussion
Problems with E. Verlinde proposal
In this paper we have extensively discussed Verlinde’s argument that sustains the fact that gravity
is an entropic force. At the basis of his argument, he makes use of a parallelism to explain that
the gravitational interaction between the two systems that he considers, namely an holographic
screen and a test particle at a distance ∆x, is an entropic phenomenon. To do so, Verlinde
started considering a polymer, specifically its end point, as resembling the test particle that
slowly is allowed to return to its equilibrium position in the heat bath, which corresponds to the
holographic screen. One of the first problem with this view, which has been raised by Verlinde
himself, lays in the fact that it is not totally correct to consider the screen as a heat bath; a heat
reservoir is a system with such a large heat capacity that even when it is placed in thermal contact
with another system, its temperature remains substantially constant. On the other hand, this is
not entirely correct for the holographic screen [4]; if we assume that the screen at an equipotential
surface is in equilibrium, then the Unruh temperature could lead to a too large entropy which
would break the Bekenstein bound of:
S≤
2πκB RE
~c
(5.1)
which means that a system contained in a region with radius R and total energy E, cannot have
an entropy larger than the above equation. Of course, we could choose a holographic screen in
such a way that the acceleration in the Unruh’s temperature, and therefore the potential, would
not break the Bekenstein’s bound, but this is against Verlinde’s idea that it should be possible to
take any holographic surface. Verlinde himself tries to solve this problem arguing that it might be
possible to rescale the value of ~; this process would not alter the entropic nature of the force, but
it would affect the value of the temperature and the entropy in opposite ways, since the entropy
S=
AκB c3
4G~
(5.2)
would be divided by the rescaling factor of ~, while the temperature
κB T =
1 ~c
2π a
(5.3)
would be multiplied by the same factor. Despite this proposal Verlinde does not go beyond it,
39
claiming that ”there is something to be understood ”[11]. This inconsistency seems to suggest that
there might be a problem with the identification of the temperature in terms of the Unruh effect;
in fact, as Visser [12] points out, let us start from the definition of the entropic force:
F =T
∆S
= T ∇S
∆x
(5.4)
What we want to do is to identify the above equation with the conservative Newtonian gravitational law:
F = −∇Φ
(5.5)
Now, if we are considering a single body which is interacting with a specific potential , then eq.5.4
and eq.5.5 imply that [12]:
∇Φ = −T ∇S
(5.6)
This is an important expression that we obtain; in fact, it tells us that in the case of a single bodyexternal potential interaction, the temperature will certainly be some function of the potential Φ.
On the other hand, this is not precisely want Verlinde wants, since he demands the temperature
to be related to the Unruh effect, and consequently to be some function of |a|, which must also
be some function of the potential ∆Φ. Thus, the problem is that these requirements are mutually
inconsistent, since in general, the level sets of the potential Φ are not the same of the |∇Φ|, where
the level sets are surfaces at a fixed potential. More specifically, Visser [12] argues that, given the
Unruh’s temperature,
T =
~|a|
2πκB c
(5.7)
then, since F = ma, in light of eq. 5.4, we can write
∇S =
2πκB mc
~a
~
(5.8)
Now, according to what we just said, the expression for the temperature and entropy can be
rewritten as
T =
~|∇Φ|
2πκb mc
40
(5.9)
and
∇S =
F
2πκB mc ∇Φ
=−
T
~
|∇Φ|
(5.10)
where
F = −∇Φ
(5.11)
Equation 5.10 has solution only for specific values of the potential Φ, while for general values this
is not true [12]. This argument is in accordance to what we previously said about the choice that
Verlinde makes to consider the temperature in relation to the Unruh effect.
Furthermore, it seems to me that some inconsistencies also exist in the parallelism between the
particle and the polymer: the polymer has a well-defined temperature and entropy, while the
same does not hold for a test particle, does it? If this is true, than the thermodynamical behavior
of the polymer when pulled out equilibrium would become meaningless in relationship with the
test particle. Furthermore, I would like to reproduce a short argument against the view of gravity
as an entropic force proposed by Gao [4]. His starting point concerns the energy increase of the
screen; in fact, when the particle interacts with the screen, the energy of the screen increases
along with the increase of its entropy and more specifically, the amount of increase of entropy and
energy would be the same. According to Gao, this is already an indication that gravity cannot
be an entropic effect and he extends the argument considering the fact that an increase in energy
can indeed cause an increase in entropy, but on the contrary, a spontaneous increase in entropy
does not lead to an increase in energy. In fact, if the ”entropy-increasing system” does work, then
the energy would decrease. This view, is indeed in complete antithesis with the one proposed
by Verlinde; the Dutch physicists sustains that the increase in entropy of the screen is due to its
statistical tendency of increasing which is in the nature of entropy and that is required in order
to obtain an entropic force. Gao’s view just tells us that the entropy increases merely because
more energy is put into the system. We still need to answer the question of where the energy
increase comes from; when we look at the system holographic screen-test particle, we assumed
that a gravitational field exists between them, and this potential is precisely the energy source of
the screen; the gravitational field does work through the force to provide energy to the screen. It
appears that a number of problems have been identified with Verlinde’s proposal, but nevertheless,
his intuition might have opened a new route that looks at fundamental problems from a different
prospective, where different views have often been the leading drivers of scientific progress.
41
References
[1] Ofer Aharony, Steven S. Gubser, Juan Martin Maldacena, Hirosi Ooguri, and Yaron Oz.
Large N field theories, string theory and gravity. Phys.Rept., 323:183–386, 2000.
[2] Jacob D. Bekenstein. Black holes and entropy. Phys. Rev. D, 7:2333–2346, Apr 1973.
[3] Sean Caroll. Spacetime and Geometry. An introduction to general relativity. Addison Wesley,
2004.
[4] Shan Gao. Is gravity an entropic force? Phys. Rev. D, 1:936–948, Jul 2011.
[5] Youngjai Kiem, Herman L. Verlinde, and Erik P. Verlinde. Black hole horizons and complementarity. Phys.Rev., D52:7053–7065, 1995.
[6] Daniel V. Schroeder. An Introduction to Thermal Physics. Addison Wesley Longman, 2000.
[7] Leonard Susskind. The World as a hologram. J.Math.Phys., 36:6377–6396, 1995.
[8] Leonard Susskind and Edward Witten. The Holographic bound in anti-de Sitter space. 1998.
[9] Lindesay J. Susskind L. Black holes, information, and the string theory revolution. Singapore:
World Scientific Publishing Co. Pte. Ltd., 2005.
[10] Gerard ’t Hooft. Dimensional reduction in quantum gravity. 1993. Essay dedicated to Abdus
Salam.
[11] Erik P. Verlinde. On the Origin of Gravity and the Laws of Newton. JHEP, 1104:029, 2011.
[12] Matt Visser. Conservative entropic forces. JHEP, 1110:140, 2011. V1: 21 pages/ no figures.
V2: now 24 pages. Two new sections (reduced mass formulation, decoherence). Many small
clarifying comments added throughout the text. Several references added. V3: Three more
references added. V4: now 25 pages. Some extra discussion on the relation between Verlinde’s
scenario and the Jacobson and Padmanabhan scenarios. This version accepted for publication
in JHEP.
[13] Edward Witten. Anti-de Sitter space and holography. Adv.Theor.Math.Phys., 2:253–291,
1998.
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