#  Are Entropy Bounds Epistemic?

Emily Adlam The Rotman Institute of Philosophy, 1151 Richmond Street, London
N6A5B7<EMAIL_ADDRESS>
###### Abstract

Entropy bounds have played an important role in the development of holography
as an approach to quantum gravity, so in this article we seek to gain a better
understanding of the covariant entropy bound. We observe that there is a
possible way of thinking about the covariant entropy bound which would suggest
that it encodes an epistemic limitation rather than an objective count of the
true number of degrees of freedom on a light-sheet; thus we distinguish
between ontological and epistemic interpretations of the covariant bound. We
consider the consequences that these interpretations might have for physics
and we discuss what each approach has to say about gravitational phenomena.
Our aim is not to advocate for either the ontological or epistemic approach in
particular, but rather to articulate both possibilities clearly and explore
some arguments for and against them.

##  1 Introduction

Discussions about entropy bounds and holography often focus on the strong
entropy bound, which conjectures that the entropy associated with some
spacelike volume is upper bounded by a quantity proportional to the surface
area of the region. For example, holography is often described in terms of a
map from space to the spatial boundaries of the world[1]. But there are known
counterexamples to the strong entropy bound, including some which are believed
to occur in our actual world; to fix these problems we must turn to the
covariant entropy bound (also known as the Bousso bound), which tells us that
the entropy associated with a null surface known as a ‘light-sheet’ is upper
bounded by a quantity proportional to the surface area of the spacelike region
bounding the light-sheet. In Bousso’s words, the covariant bound ‘ _evades
these counterexamples because of the special properties of light-sheets. Thus,
the notion of light-sheets (rather than spatial volumes, or light-cones
lacking a nonexpansion condition), appears to be crucial._ ’

So it seems that if we are to understand the significance of entropy bounds
and holography we should be focusing on the light-sheet construction used in
the covariant bound. In this article we set out to understand the meaning of
this construction. We observe that there is a possible interpretation of the
covariant entropy bound which would suggest that it encodes an epistemic
limitation rather than an objective count of the true number of degrees of
freedom on a light-sheet; thus we distinguish between ontological and
epistemic interpretations of the covariant bound. We discuss how these
interpretations relate to Jacobson’s derivation of the Einstein equations, and
we consider the evidence for each interpretation. Our aim here is not to
advocate for either the ontological or epistemic approach in particular, but
rather to articulate both possibilities clearly and explore some arguments for
and against them.

Note that this paper will largely be focused on entropy bounds in a
semiclassical context - in a companion paper we will address the significance
of the entropy bounds in the context of quantum mechanics and quantum gravity.
Thus in this paper we will largely deal with entropy as a classical notion,
deferring discussion of entanglement entropy until the companion paper.

##  2 Background: Entropy Bounds

Entropy bounds have their origins in an argument due to Bekenstein[2] and
Hawking[3] suggesting that if the second law of thermodynamics is to be obeyed
in the vicinity of black holes, we must generalize it to include an entropy
associated with black holes equal to $\frac{A}{4G\hbar}$, where $A$ is the
surface area of the black hole; this value is known as the Bekenstein-Hawking
entropy. Subsequently Bekenstein argued that if the generalized second law is
to remain true when we lower systems into a black hole, it must be the case
that the maximum entropy contained in a spherical system of radius $R$ and
total gravitating energy $E$ is $S\leq\frac{2\pi ER}{\hbar}$[4]; this is known
as the Bekenstein bound. Furthermore, for a weakly gravitating system $E$ will
always be significantly less than $\frac{2\pi R}{4G}$, so for weakly
gravitating systems this bound reduces to the form $S\leq\frac{A}{4\hbar G}$,
where $A$ is the area of a spherical surface fully containing the system.

Although this result was derived originally for the weak gravity regime, it
inspired a more general conjecture known as the ‘strong entropy bound’ which
says that the maximum entropy which can be contained in any region of space is
upper bounded by $\frac{A}{4G\hbar}$, where $A$ is the region of any surface
bounding the region. For after all, as originally pointed out by t’Hooft[5],if
we could find a region with entropy greater than $\frac{A}{4G\hbar}$, we could
add energy to it adiabatically without changing its entropy, thus transforming
it into a black hole of area $A$ \- but then this black hole would have
entropy greater than the Bekenstein-Hawking entropy. So if the Bekenstein-
Hawking entropy formula is correct, the entropy of a black hole of surface
area $A$ must be the maximum possible entropy for any region of space with
surface area $A$. The strong bound appears to be correct in many physically
relevant situations[4], and it is part of the inspiration for a flourishing
research field aiming to understand quantum gravity in terms of
‘holography’[6, 7, 8].

The strong entropy bound initially seems quite surprising. First of all, we
may be surprised that there is any finite bound on the entropy inside a region
of spacetime, since quantum field theory tells us that the number of degrees
of freedom in any given region of space is infinite. There are ways we might
make sense of this - for example, in quantum gravity, states above a certain
energy can be expected to collapse to a black hole, and this mechanism might
be expected to lead to a finite space of physically possible states[9].
Alternatively, several approaches to quantum gravity postulate that spacetime
is discrete[10, 11, 12, 13], which would also explain why there should be a
finite bound on the entropy of a spacetime region. However, the natural way of
discretizing spacetime would seem to suggest that the degrees of freedom
inside a region should scale with its volume, not the surface area of its
surface, so why does the entropy content of spacetime seem to be so much lower
than we would naturally expect?

In fact, the strong bound is not universally true: the heuristic arguments
used to motivate it only work for sufficiently static scenarios, so it can be
violated in situations like cosmic inflation and gravitational collapse. These
scenarios are rare, but nonetheless they are important if one hopes to use the
bound to learn something profound about fundamental physics, as many
physicists apparently do - for a bound that is not universally valid cannot
translate straightforwardly to a universal constraint on fundamental physics.

To solve these problems Bousso proposed a ‘covariant entropy bound’ [14]:
instead of bounding the information inside a volume enclosed by the relevant
surface, it bounds the information on a ‘light-sheet’ associated with the
relevant surface. A light-sheet is a set of null geodesics leaving the surface
orthogonally such that the expansion of the set in the direction going away
from the surface is zero or negative, i.e. the geodesics are remaining
parallel or coming closer together as they get further from the surface. The
light-sheet continues up until the geodesics intersect at a ‘caustic’ (i.e.
crossing-point) or encounter a singularity of spacetime. Bousso’s bound then
simply says that the entropy on the light-sheet associated with a surface of
area $A$ is upper bounded by $\frac{A}{4G\hbar}$; and it currently seems
likely that this bound is satisfied everywhere in our actual universe, at
least at the semiclassical level[15].

In the case of a closed sphere we can think of the null geodesics leaving the
surface as the possible paths for rays of light leaving this surface; they
must be emitted at the time $t$ and must be orthogonal to the surface of the
sphere. In this case there are four possible directions in which the rays
could be emitted - inside the sphere into the past, outside of the sphere into
the past, inside the sphere into the future, or outside of the sphere into the
future. Clearly the rays emitted outside of the sphere are not converging, so
these sets of rays have positive expansion and thus do not define light-
sheets; but the rays emitted into the interior of the sphere are converging,
so the sphere will have one interior light-sheet oriented into the past and
another interior light-sheet oriented into the future. In the co-moving
reference frame (assuming locally flat spacetime), the light emitted inwards
will propagate towards the centre of the sphere and all of the rays of light
will meet at the centre of the sphere at the same time. This meeting point is
a ‘caustic’ and thus the light-sheet terminates at this point.

This example makes it easy to see why the covariant entropy bound implies the
strong entropy bound whenever the state inside the relevant surface is static.
For in the co-moving reference frame, every spatial point inside the sphere is
traversed exactly once by a ray belonging to the light-sheet, though different
points will in general be traversed at different times. Thus if the state
inside the sphere is static, the entropy traversed by these light rays is
simply the same as the entropy of the state inside the sphere on any spacelike
hypersurface, and thus the covariant bound implies the strong entropy bound in
this case.

##  3 Does the Covariant Bound Count Degrees of Freedom?

Entropy bounds have often been used to motivate claims about the number of
degrees of freedom in some region of space. But care must be taken with these
claims, because entropy does not straightforwardly correspond to an ‘amount of
information’ or ‘number of degrees of freedom.’ For a start, these terms are
quite vague - as Timpson stresses in ref [16], there are a number of different
things that the term ‘information content’ could correspond to, and even more
so in the context of quantum mechanics. In addition, the validity of these
claims seems to depend crucially on what kind of entropy one has in mind. The
notion that maximum entropy is a measure of a number of degrees of freedom
seems to draw on an information-theoretic notion of entropy, i.e. the Shannon
entropy, which is given by $\sum_{x}p(x)\log(p(x))$ where the sum is taken
over all possible values of a variable $x$ and $p(x)$ gives the probability
for some particular value. This is because if we maximize the Shannon entropy
for all possible probability distributions over the possible states of a given
physical system, the result can indeed be interpreted as counting the capacity
for information storage in that system - for example, given a system
consisting of $n$ bits (i.e. $n$ independent systems with exactly two states
each), the maximum Shannon entropy in units of bits is $n$. So talk of
‘degrees of freedom’ in this context may be inspired by the idea of literally
counting bits in simple systems, and indeed Bekinstein did suggest thinking
about the black hole entropy in terms of Shannon entropy in his original paper
on the subject[2]

However, there are reasons to think that the Bekinstein-Hawking entropy should
be understood primarily as a thermodynamic entropy, defined in terms of
macroscopic thermodynamical quantities[17, 18] \- in particular, one of
Bekinstein’s initial arguments for the black hole entropy formula was that
black holes must have this entropy in order to protect the second law of
thermodynamics, and one of the main pieces of (theoretical) evidence for the
correctness of Bekinstein-Hawking formula is the fact that black holes can be
shown to emit Hawking radiation at exactly the temperature predicted by the
first law of thermodynamics for a system whose entropy is the Bekinstein-
Hawking entropy[3]. And since the entropy bounds were originally a kind of
generalization of the Bekinstein-Hawking entropy, one might naturally think
that they too should be understood first and foremost as thermodynamic
entropies. Yet thermodynamical definitions of entropy in terms of macroscopic
thermodynamical properties like mass, volume, pressure and temperature do not
necessarily seem to have anything to do with degrees of freedom. Of course,
the thermodynamic entropy is usually assumed to be reducible to the Gibbs
and/or Boltzmann statistical-mechanical entropies (although even for
paradigmatic thermodynamical systems the details of the reduction remain
somewhat controversial[19, 20, 21]) and indeed there have been many proposals
for statistical-mechanical accounts of the Bekinstein-Hawking entropy within
various possible approaches to quantum gravity[22, 23, 18]. But since we don’t
yet have an empirically-verified theory of quantum gravity, there is still
some doubt about the correct state space for a black hole, and thus it is
still not certain that the black hole entropy can be thought of in a
statistical-mechanical way. And likewise for the covariant bound: we don’t yet
know the complete state space for arbitrary regions of spacetime, since this
state space may have to include quantum-gravitational degrees of freedom, so
perhaps we should not too quickly assume that the entropies featuring in
entropy bounds can always be explicated as statistical-mechanical
entropies111Of course, the entropy bounds might also be understood as relating
to quantum-mechanical von Neumann entropies; we will discuss this in Part II
of this paper..

Furthermore, even if we decide the entropy appearing in the covariant bound
_can_ be explicated as a statistical-mechanical entropy, there are still
difficulties with identifying maximum statistical-mechanical entropies with
degrees of freedom. For example, the Gibbs entropy is defined as
$-k_{B}\int_{x}p(x)\log(p(x))$ where $k_{B}$ is Boltzmann’s constant, and
$p(x)$ is a probability distribution over microstates $\\{x\\}$, which may be
understood as representing an observer’s knowledge about the true microstate
$x$, or some kind of coarse-graining, or relative frequencies in an ensemble.
This has a similar form to the Shannon formula, and indeed the maximum
theoretically possible Gibbs entropy for a given system is equal to the total
number of possible microstates for that system, so there is a sense in which
the maximum Gibbs entropy may be regarded as counting degrees of freedom - but
note that this maximum can be reached only with a probability distribution
assigning equal probability to all possible microstates, which may or may not
be a physically realistic probability distribution, depending on what we take
the distribution to encode. Similarly, the Boltzmann entropy is given by
$k_{B}\ln(W)$ where $W$ is the total number of microstates compatible with the
current macrostate of a system, so it is equal to the Gibbs entropy in the
case where the probability distribution $p$ assigns equal probability to all
the microstates compatible with the current macrostate (as for a system in
thermal equilibrium using the canonical ensemble[19]), and is also equal to
the Shannon entropy for a variable with $W$ equally probable values. Thus if
there exists a macrostate compatible with _all_ possible microstates, its
entropy will be the maximum possible Boltzmann entropy and that entropy will
indeed be something like a count of the total number of degrees of freedom -
but note that depending on how we identify macrostates, it may or may not be
physically realistic to suppose that there is a macrostate which is compatible
with all or nearly all possible microstates. So the maximum Gibbs and
Boltzmann entropies are not guaranteed to yield the actual total number of
microstates or ‘degrees of freedom’ - this depends on the way in which we
choose the probability distributions or macrostates that we are maximizing
over.

A further difficulty is presented by the fact that thermodynamic entropy is
arguably not an observer-independent quantity: there is a long tradition,
beginning with Maxwell[24], of understanding thermodynamics as a science which
involves designating a certain set of manipulable variables and then studying
the responses of physical systems to manipulations of those variables. For
example, Myrvold[25] argues for an an observer-dependent account of
thermodynamics, arguing that entropy ‘ _must be relativized to the means we
have available for gathering information about and manipulating physical
systems,_ ’[26]. The same can also be said of the statistical-mechanical
definitions of entropy, since both the probability distribution $p(x)$ for the
Gibbs entropy, and the choice of macrostates for the Boltzmann entropy, can
also potentially be observer-dependent. And if we look more closely at the
arguments for the strong entropy bound, it is striking that these arguments do
indeed proceed by considering the processes that can be performed by agents
who have access to only a certain set of ‘manipulable’ variables
(specifically, variables which can be accessed entirely outside of the black
hole or relevant spatial region) - for example, Bekenstein’s original argument
for the Bekenstein-Hawking entropy involved an assumption that entropy is
‘lost’ once it enters the black hole and becomes inaccessible to external
observers, and the argument for the strong entropy bound invoked a process in
which an external observer lowers a system into a black hole[4].

However, anticipating this criticism, ref [15] makes the following argument: ‘
_there is an apparent tension between the fact that these statements are
supposed to have the status of fundamental laws and the fact that entropy is a
quantity whose definition is coarse-graining dependent. However, this tension
is resolved by noting that the number of degrees of freedom should be an upper
bound for the entropy S, irrespective of choice of coarse-graining._ ’ The
idea here is that even though thermodynamical and statistical-mechanical
entropies may be observer-relative, nonetheless if we discover a universal
bound on one of these entropies which applies to all possible observers, we
have good reason to infer that this bound reflects the true number of degrees
of freedom of the system, since no observer can assign to a system an entropy
greater than its total number of degrees of freedom. Thus we will for now
entertain the possibility that the covariant bound can indeed be regarded as
providing a count of the total number of degrees of freedom in some region of
spacetime. We will therefore be agnostic about exactly which of the four types
of entropy discussed above is the one relevant to the covariant bound, since
it seems reasonable to expect the total number of degrees of freedom of a
given system to be an upper bound for all four types of entropy, so at least
some potential explanations of the bound would seem to apply regardless of
which type of entropy one has in mind.

###  3.1 Which region?

Even if we accept that the covariant bound is counting degrees of freedom in
some region of space, the question remains - _which_ region? Bousso
himself[27] infers from the covariant entropy bound that the total number of
degrees of freedom _on a light-sheet_ is proportional to the area of the
surface with which the light-sheet is associated; we will refer to this as the
light-sheet postulate. On the other hand, others[28, 29, 30, 22] advocate a
weaker principle which says only that the total number of degrees of freedom
_associated with a surface_ is proportional to the area of that surface; we
will refer to this as the surface postulate. For example, Smolin gives a
quantum formulation of the surface postulate that he calls the ‘weak entropy
bound’ which stipulates that for a system $\Sigma$ defined by the
identification of a fixed boundary $\delta\Sigma=S$, and a Hilbert space
$H_{S}$ defined as the the smallest faithful representation of the algebra of
observables measurable on the boundary only, then if the area $A(S)$ of the
boundary is fixed, we must have $\log(dim(H_{S}))\leq\frac{A(S)}{4\hbar
G}$[28].

Smolin’s argument for the surface postulate[28] is based on the fact that, as
we have already noted, the entropies featuring in the derivation of the
Bekenstein-Hawking entropy are thermodynamic ones; but rather than concluding
on this basis that the covariant bound does not count degrees of freedom at
all, he argues that since these thermodynamic arguments pertain specifically
to exchanges of matter and radiation between the black hole and external
observers, the resulting bound must pertain to the degrees of freedom on the
surface rather than the interior. Similarly, Sorkin notes that ‘ _the internal
dynamics of a black hole ought to be irrelevant to its exhibited entropy
because - almost by definition - the exterior is an autonomous system for
whose behavior one should be able to account without ever referring to
internal black hole degrees of freedom._ ’ On this basis Smolin[28], and also
Jacobson[29], Rovelli[30] and Sorkin[22] all contend that the Bekenstein-
Hawking entropy describes only the amount of information an observer external
to the black hole can gain about its interior from measurements made outside
the horizon, not the actual amount of information inside the black hole.
Smolin then concludes[28] that since the arguments for the strong entropy
bound are derived from the Bekenstein-Hawking entropy bound, it follows that
this bound also applies only to observables measurable on a boundary, not
observables measurable in the interior. And since the covariant bound is
intended as a generalization of the strong entropy bound, a similar argument
would suggest that it too applies only to observables measurable on a
boundary.

But if the surface postulate is in fact the only inference about degrees of
freedom justified by the covariant entropy bound, this would be a little
disappointing, for the surface postulate doesn’t seem to have much to do with
‘holography’ in the usual sense. After all, if for any reason we decide that
surfaces in spacetime should be associated with a finite number of degrees of
freedom, it would seem very natural to expect that this finite number will be
proportional to the area of the surface, so the surface postulate doesn’t seem
to give us much new information, other than possibly providing additional
support for the idea that spacetime regions generally contain a finite number
of degrees of freedom. Indeed, Smolin sees it as evidence for the hypothesis
that spacetime is discretized: ‘ _It is then difficult to escape the
conclusion that the holographic principle, in its weak form, is telling us
that nature is fundamentally discrete. The finiteness of the information
available per unit area of a surface is to be taken simply as an indication
that fundamentally, geometry must turn out to reduce to counting._ ’[28]222We
think Smolin’s conclusion seems a little too strong here - a discrete
spacetime could certainly explain why the number of degrees of freedom on a
surface is finite, but there are other possible explanations, such as the fact
that states of too high energy will collapse to form a black hole, so it’s
unclear that we can immediately infer the existence of spacetime discreteness
from finiteness. So if entropy bounds only tell us about the degrees of
freedom on a surface, they seem much less novel - as we have noted, the
entropy bounds attracted so much interest precisely because they seemed to be
pointing to something _more_ than mere finiteness or discreteness.

However, there are reasons to think that the surface postulate does not
exhaust the content of the entropy bounds. For the surface postulate does not
acknowledge the difference between the original strong entropy bound and the
covariant entropy bound, and yet we seem to have reasonably good evidence that
the covariant bound holds everywhere in our actual universe, at least in the
semiclassical regime, while there are counterexamples indicating that the
strong bound does not hold everywhere in our actual universe, even in the
semiclassical regime. So perhaps _both_ the surface postulate and the light-
sheet postulate are correct? Yet if they are both correct, we have some
explaining to do, because it surely cannot be a coincidence that the total
number of degrees of freedom on a light-sheet are always related in this
specific way to the total number of degrees of freedom on its bounding surface
- it seems natural to think that these bounds must be connected in some way.
We will now examine one interesting way in which they could be connected.

###  3.2 Thought Experiment: Learning About Physics Inside a Region

In this section we will be concerned with issues relating to information and
information transfer. So it should be emphasized that throughout this paper we
will be focused on information as a technical, physical notion - we will
concern ourselves with the ways in which physical systems exchange information
and not, for example, the extent to which information may be connected to
language or meaning. In addition, we will follow Timpson[16] in maintaining
that information ‘ _does not refer to a substance or to an entity’_ \- rather
we understand information as an abstract noun which can be used to
characterize various features of physical reality, including the ways in which
systems become correlated with each other, the inferences that can be made
about one system from knowledge of the state of some other system, the
efficiency with which certain kinds of communication systems convey messages,
and so on. So in this paper when we talk about information transfer or flux,
we are not describing how some _thing_ travels: rather we are characterising
physical processes in which physical systems move around and interact and
thereby the information stored in them becomes accessible at different
locations in spacetime. In particular, in this article we are largely
concerned with what Timpson refers to as _accessible information_ \- i.e.
information that can be accessed by a physically realistic observer performing
physically possible measurements - and thus when we say that information has
been transferred to some location, this entails that the information can be
accessed by observers at that location. On this understanding of ‘information
transfer,’ our best current physics makes it clear that information can be
transferred between spacetime points only by local processes: although we
observe non-local effects in quantum physics, those non-local effects cannot
be used to transfer accessible information (e.g. to send superluminal
signals), and in fact all processes which do transfer information in this
sense are mediated by physical signals travelling along continuous paths in
spacetime. For example, in order to use quantum teleportation to transfer
information via quantum entanglement to another observer, we must send a
classical message to the other observer, and the amount of information that
the other observer can ultimately obtain is no greater than the information
content of the classical message[31].

So imagine that we have some information which is initially located on the
‘inside’ of some boundary such as a sphere, in a locally flat spacetime: now
for any choice of reference frame we can identify a temporally extended 3D
region corresponding to the location of the boundary (assuming it to be
stationary in the chosen reference frame) and thus we can also identify a
temporally extended 4D region corresponding to the interior of the boundary.
And then, due to the locality of information transfer, the information
originally on the inside can only get out of the temporally extended interior
region (in that reference frame) by passing through the temporally extended
boundary (in that reference frame). In addition, the locality of information
transfer seems to suggest that this information transfer must involve a
process in which the information in question can be regarded as occupying
degrees of freedom located on or near a spacelike slice through the temporally
extended bounding surface, and therefore the number of degrees of freedom
associated with each spacelike slice through the surface will constrain the
rate at which information can move from the ‘interior’ to the ‘exterior.’ This
will be true in any reference frame, though of course different reference
frames will identify the interior and exterior regions differently, and then
we can perhaps imagine taking limits as the temporal extension goes to zero,
in order to arrive at an instantaneous rate of information transfer which does
not depend on a choice of reference frame. With this understanding of
information transfer in mind, the surface postulate has a natural corollary:
if the number of degrees of freedom associated with a given surface has an
upper bound proportional to the area of the surface, then presumably the rate
at which information can pass through a surface also has an upper bound
proportional to the area of the surface.

So now let us consider a thought experiment. Suppose you are presented with a
large opaque sphere covered with $n$ LEDs. You are told that the interior of
this sphere is not governed by the usual laws of physics - some completely new
physics is going on inside it. The LEDs are linked to detectors on the
interior, and they are reset at one second intervals, lighting up if their
detector has received a ‘signal’ from the inside during the most recent
interval. You have been assured that this is the only way in which information
can emerge from the sphere. Your task is to use the information displayed by
the LEDs to learn something about the physics on the inside.

Presumably, you will do something like the following. First, you will observe
the sphere for a time, creating a record of the state of the LEDs every
second. Then you will look at the series of states you have written down,
attempting to discern some pattern. Hopefully you will eventually be able to
write down some kind of ‘time evolution’ law which relates past states to
future states. Having done this, what more can you say about the physics
inside the sphere?

The simplest option is to postulate a one-to-one correspondence between states
of the LEDs and states on the interior. Then the evolution law you arrived at
for the LED states can be regarded as the evolution law for the states on the
interior too; and clearly the states that you assign to the interior will be
describable in exactly $n$ bits, which is to say they will have a Shannon
entropy of no greater than $\log(n)$. You could perhaps do something more
complex - for example, you could come up with a hypothesis about the nature of
the signals which cause the LEDs to switch on or off, and thereby work
backwards from the surface states to interior states. In this case there may
no longer be a straightforward one-to-one mapping from the surface states to
the interior states; however, it is still quite likely that the states you
arrive at will be describable in $n$ bits or less, since you can get no more
than $n$ bits of information out of the sphere at any given time to determine
those states.

We might make the thought experiment more complex by allowing you to
‘intervene,’ as in a normal laboratory situation. This would involve a version
of the scenario where the LEDs on the outside can both receive signals and
also emit signals into the inside of the sphere, so you can prepare a ‘state’
of the interior by setting the LEDs to some desired configuration, then let
the system evolve and observe its response. But of course you will only be
able to prepare interior states up to a resolution of $n$ bits, since you only
have $n$ degrees of freedom on which you can intervene, and then just as
before you will only be able to learn $n$ bits about the succession of
interior states that follows, so even with the ability to intervene it would
not be surprising if you were to arrive at a description which assigns the
interior only $n$ degrees of freedom.

To be clear, we don’t intend to suggest that you could _never_ be justified in
assigning to the system a set of interior states which require more than $n$
bits to describe. After all, you are not just looking at a single observation
but rather at a series of observations, and you might be able to find
interesting correlations in the time-series data which you decide to explain
by postulating additional degrees of freedom inside the region333Thanks to an
anonymous reviewer for pressing this point.: for example, if you see non-
Markovian features in the data, you might be inclined to think that there must
be extra degrees of freedom inside the system storing additional information
about the past. However, if you really have no idea at all about what kind of
physics is going on inside the sphere, you will not know what kind of
correlations to look for in the temporal data, so it could be quite hard to
find them unless they take a particularly obvious form. In addition, even if
you do see non-Markovian behaviour, it will always be open to you to suppose
that the system just has non-Markovian evolution laws rather than additional
degrees of freedom, and arguably the former option is more ontologically
economical. So although it is certainly possible for you to end up with a
state space which requires more than $n$ bits to describe, nonetheless it
would not be in any way _surprising_ if you were to arrive at a description
which assigns the interior only $n$ degrees of freedom, regardless of how many
degrees of freedom there really are inside. Thus the point of this experiment
is to show that a bound of this kind would be a natural outcome of the
epistemic limitations you are subject to in this situation, even though it
might be possible to do better than the bound with sufficient ingenuity.

We also note that if one is concerned by the possibility of gaining extra
information from the temporal domain, it is possible to consider a
generalization of this scenario using a different kind of surface, e.g. a
temporally extended surface surrounding the entire sequence of observations
that you extract from the LED sphere. For example, consider Oeckl’s general
boundary formalism, which is motivated by considerations similar to those we
have discussed here: as Oeckl writes, for any arbitrary region in spacetime, ‘
_by locality, the interior of a region can interact with the rest of the
universe only through the region’s boundary_ ’[32]. Oeckl therefore argues
that experiments can always be thought as being enclosed by a boundary such
that the agent performing the experiment is outside of the boundary and can
only manipulate and observe variables on the boundary; thus he constructs a
reformulation of quantum mechanics designed to predict ‘transition amplitudes’
from one part of the boundary to another. Oeckl seems to suggest that one
might expect something like an entropy bound to hold here: ‘ _connected
boundaries of compact regions are the main focus of attention … in this sense
the formulation is holographic i.e. the information about the interior of a
region is encoded through the states on the boundary_’[33]. So it is possible
that the covariant entropy bound may ultimately be regarded as a special case
of a temporally extended boundary formalism like Oeckl’s, which would provide
a more realistic model of our circumstances as temporally extended observers.
However, in this article we will continue to focus on the covariant bound, as
we think the instantaneous case is sufficient to explore a number of
interesting ideas about the role of surfaces and the locality of information
transfer.

###  3.3 Light-sheets and accessible information

The upshot of all this is that your epistemic situation with regards to the
sphere leads to a natural ‘entropy bound’ - it is natural to expect that the
entropy you assign the interior may be upper bounded by the number of LEDs on
the surface. Moreover, it would seem that the sphere example is a fairly close
analogy for the situation described by the covariant entropy bound.

In light of this, consider again the specific role played by the light-sheet
construction in the covariant entropy bound. First note that a ‘null surface’
is really the appropriate relativistically covariant way of describing the
region to which we assign a state based on information received at a single
time. For example, suppose I observe a football field. Although I may have the
impression that I am observing the whole region simultaneously, that is not
really true; each ‘snapshot’ that I see of the field corresponds not to a
spacelike hyperplane but rather to something like a null surface, assuming
that all the information reaches me at the speed of light. Now, of course I do
not directly ‘see’ the whole null surface, since it’s possible for light to be
blocked by opaque objects: what I see is something like an infinitesimal two-
sphere encoding information from various different parts of the null surface.
So the null surface can be thought of as containing all the information that
would in principle be available to me in the best possible case, i.e. it is
the locus of all the information I could possibly obtain from my observation
if there were nothing blocking the light from reaching me.

What about light-sheets? Well, they are simply a special kind of null surface.
Specifically, a past-directed light-sheet is the locus of the information
available to me if instead of collecting information at a single point, I
collect it from a bounding surface. For example, if the surface is a sphere
and I collect all the information that arrives at the surface at a single
instant of time in the co-moving reference frame, then assuming that the
information all arrived at the surface at the speed of light and nothing
blocks it, I get a ‘snapshot’ of the interior of the sphere which corresponds
to the associated light-sheet. So in a sense, the role of the light-sheet
construction is simply to identify the appropriate null surface to which we
should assign a state based on information collected at a spacelike surface;
and as we have just seen in our thought-experiment, it is natural under these
circumstances to assign states from a state space which has no more than $n$
degrees of freedom, where $n$ is the number of bits which can pass through the
surface in one instant of time. And of course if the surface postulate is
correct, it follows that $n$ will be proportional to the area of the surface.

One might object that this argument only applies to a closed surface, whereas
the covariant entropy bound applies also to open surfaces. But we can
formulate a similar argument for an open surface. Suppose I am standing on one
side of a screen of area $A$ and trying to gain information about a system on
the other side of the screen: if the surface postulate is correct, the rate at
which information about the system can pass through the screen to me has an
upper bound proportional to the area of the surface. Of course, in this case
it’s possible for information to go around the screen in order to reach me,
but if I am to learn anything from that information I’d have to know about the
state of the region above around the screen so I can account for any further
interactions the signals might undergo on the way. Thus we need to consider
the process by which I learn the state of the surrounding regions as well,
which requires us to extend the screen to cover information coming from those
regions too, and so on, and therefore it’s hard to see how I could do any
better than learning a number of bits proportional to the area of the surface,
unless I’m allowed to assume that I already know the state of some region
around the screen. So the epistemic interpretation of the bound seems to work
for open surfaces too: the light-sheet of such a surface still represents the
locus of all the information which could pass through the screen in a given
instant, which can be expected to yield an upper bound proportional to the
area of the screen.

Our thought experiment is designed to highlight the the special role of
surfaces as interfaces through which observers access systems. This has been
pointed out by various authors. For example, Oeckl[33] has emphasized this
feature of experimental design: ‘ _Consider for example a scattering
experiment in high energy physics. A typical detector has roughly the form of
a sphere with the scattering happening inside (e.g. a collision of incoming
beams). The entries for particles and the individual detection devices are
arranged on the surface_.’ Similarly Crane[34] observed that ‘ _if we divide
the universe into “system” and “observer” the observer no longer measures the
state of the system, but only that part of it which impinges instantaneously
on the observer,_ ’ and thus proposed a system of observable algebras and
Hilbert spaces, one associated with every possible splitting of the universe
into system and observer. And May[35] has argued that holography can be
expressed as the requirement that the asymptotic tasks which are possible
using the bulk dynamics should coincide with tasks that are possible using the
boundary; here ‘asymptotic tasks’ are those which can be stated in terms of
inputs and outputs located on the spacetime boundary. Put this way, it does
not seem surprising that our physics should have this feature: for after all
we never really access a bulk directly, we only ever manipulate systems
through a boundary, i.e. the spatial boundary separating the the system from
us, so it is to be expected that the physical theories we come up with
describing possible tasks should be limited to the tasks which are possible
using the boundary.

##  4 Epistemic or Ontological

The thought experiment of section 3.2 leaves us with a dilemma. For it
suggests that, regardless of the number of degrees of freedom there really are
on light-sheets, it is natural for observers to end up describing the physics
on a light-sheet with a number of bits no greater than the amount of
information which can pass through the associated surface - which according to
the surface postulate will be proportional to the area of that surface. So how
can we ever know whether or not the covariant bound describes the true number
of degrees of freedom on a light-sheet? It seems that anyone who accepts the
surface postulate has good reason to think of the covariant bound not as
describing the true numbers of degrees of freedom on a light-sheet, but simply
as an epistemic restriction on the number of degrees of freedom accessible to
an external observer. We note that the possibility of the bound being
epistemic in this sense is implicit in our discussion of the Gibbs and
Boltzmann entropies in section 3 \- the probability distributions or
macrostates that we are maximizing over can be understood as being in some
sense ‘epistemic’ and we noted already that some ways of choosing the set of
probability distributions or macrostates that we maximize over will lead to
‘maximum entropies’ which are smaller than the total number of possible
microstates or degrees of freedom.

So we now set out several different attitudes that one might take to the
covariant bound, assuming that one accepts the surface postulate:

  1. 1.

Ontological: views which hold that the covariant entropy bound is a constraint
on the number of real number of ontological degrees of freedom on a light-
sheet:

    1. (a)

Pragmatic-ontological: views which invoke some form of pragmatism or
empiricism to say that since we can never have any information about degrees
of freedom beyond the covariant entropy bound, we should assume there aren’t
any further degrees of freedom, but the bound doesn’t have any deep meaning
beyond that.

    2. (b)

Positive-ontological: views which postulate that the covariant entropy bound
is a fundamental constraint or follows from aspects of the fundamental
ontology, so it tells us something profound about the nature of our reality.

  2. 2.

Epistemic: views which hold that the covariant entropy bound is merely
epistemic, and the real number of degrees of freedom on a light-sheet may be
greater than the amount specified by the bound:

    1. (a)

Pragmatic-epistemic: views which postulate that there may exist degrees of
freedom beyond those allowed by the covariant entropy bound, but we will
probably never know anything about them and they will have no impact on
anything we can observe, so we can get on with physics as if they don’t exist
at all.

    2. (b)

Positive-epistemic: views which postulate that there may exist degrees of
freedom beyond those allowed by the covariant entropy bound, and it may be
important to take them into consideration, even though we can’t directly
observe them.

Now, there may appear to be a natural connection between interpretations of
the covariant bound and the interpretation of the entropy appearing in the
covariant bound. In particular, if this entropy is understood to be a
thermodynamic or statistical-mechanical entropy which is regarded as being
observer-relative in the sense described by Myrvold, then it seems natural to
think of the covariant bound as a claim about the information which can be
obtained by an observer with access to certain manipulable variables, so it is
epistemic. Moreover, this identification may appear to settle the question in
favour of epistemic views, since as we have noted, the thermodynamic arguments
for the Bekenstein bound seem to pertain to observer-relative thermodynamic
entropies, and the covariant bound is a generalization of the Bekenstein
bound.

However, we would caution against making this identification in an overly
simple way. For proponents of an ontological view of the bound need not deny
that the entropies employed in arguments for the Bekenstein-Hawking entropy
and the strong entropy bound are observer-relative entropies - it may be that
_in any particular case_ , the entropy that we assign to a region of spacetime
is a measure of the information available to some specific (possibly
hypothetical) observer. But the Bekenstein-Hawking entropy and the strong and
covariant bounds do not describe the entropy assigned in any particular case,
rather they represent a universal upper bound on the entropies assigned by all
possible observer in all possible cases. And the universality of this bound
means that it cannot be understood as a measure of the information available
to any one specific observer, so we still have the possibility of interpreting
it ontologically - for example, we have already mentioned the argument of
Flanagan et al [15] that the universality of the bound indicates that it is a
count of the true number of degrees of freedom. On the other hand, we have
noted that a universal entropy bound could also be explained by the fact that
all realistic agents are all subject to some kind of epistemic restriction
which prevents them from gaining more than a certain number of bits of
information about it. So really, nothing can be inferred one way or another
from the observer-relative nature of the entropies used in the derivation of
the bounds, because the nature of those entropies does not tell us the origin
on the _universal bound_ on those entropies. Thus we would urge that, contrary
to the arguments made by Smolin and others[28], the thermodynamical nature of
Bekenstein’s reasoning does not settle the question of whether the covariant
bound (or the strong entropy bound, or even the Bekenstein-Hawking entropy) is
epistemic or ontological.

We can classify some of the views we have previously examined within this
schema. Bousso clearly takes a positive-ontological view, writing of the
covariant entropy bound: ‘ _The bound’s simplicity, in addition to its
generality, makes the case for its fundamental significance compelling_ ’[27].
Other proponents of holography in the conventional sense, such as Susskind[1]
and t’Hooft[5], also seem to take this kind of view. On the other hand
Smolin[28], Jacobson[29] and Sorkin[22] all appear to take an epistemic view.
For example, these kinds of epistemic considerations are implicit in Smolin’s
criticisms of the original Bekenstein bound[28] \- he emphasizes that the
thermodynamic bounds on the entropy inside a black hole pertain specifically
to the _accessible_ entropy, and we shouldn’t assume that the information
accessible from outside a surface constitutes a bound on the information
inside that surface, so it isn’t clear that the arguments for Bekenstein’s
bound necessarily constrain the ‘real’ degrees of freedom.

Meanwhile, it seems likely that the idea that entropy is ‘lost’ when
information falls into a black hole is partly grounded on a pragmatic view,
since this would justify the practice of simply ignoring the possibility of
degrees of freedom beyond the accessible ones. For example, Dougherty and
Callender offer the following account of the reasons why physicists believe
that entropy is lost when it crosses a black hole horizon: ‘ _it is held that
the entropy vanishes when it passes behind the event horizon because we can’t
gain access to it. The system itself doesn’t vanish; indeed, it had better not
because its mass is needed to drive area increase. But for the ordinary
entropy, when it crosses the event horizon it’s “out of sight, out of world,”
or at least, out of physics._ ’[36] So possibly some physicists are really
agnostic about whether there is additional information in the black hole, but
are willing to disregard such information for practical purposes, and
presumably these physicists would also be inclined to take a pragmatic-
epistemic or pragmatic-ontological view to the covariant entropy bound.

However, there is potential for the pragmatic approach to lead to confusion:
if we simply ignore the existence of some degrees of freedom, we may get
strange effects where information appears to mysteriously vanish. Indeed, it
has been argued that this is exactly what is going on in the case of the black
hole information paradox. As noted by Rovelli[30], the Page argument for the
information loss paradox ‘ _is based on the fact that if the number of black
hole states is determined by the area, then there are no more available states
to be entangled with the Hawking radiation when the black hole shrinks_ ’ and
thus ‘ _if there are more states available in a black hole than $e^{A/4}$,
then the Page argument for the information loss paradox fails._’ Also, as
noted in section 3.2, it is possible that even though we cannot directly
access these extra degrees of freedom, we might be able to detect signatures
of their existence by careful analysis of time-series data, for example via
the detection of non-Markovian behaviour, so there may be important physical
insights to be gained by taking into account the possibility of additional
states. Thus although the pragmatic views have merits in certain contexts, we
will henceforth focus on the positive views.

###  4.1 Positive-Epistemic Views

It is common to understand the term ‘epistemic’ as connoting ‘not objective.’
For example, sometimes a distinction is made between ‘objective’ and
‘epistemic’ conceptions of entropy, as in ref[36]. But we emphasize that
entropy bounds could be ‘epistemic’ while still being objective: for example,
an entropy bound could arise from an objective description of the way in which
certain kinds of physically embodied observers are limited in their ability to
access certain kinds of information. That is to say, these bounds are
‘epistemic’ not because they are merely features of the knowledge of one
particular agent, but because they are objective descriptions of what it is
possible for a certain kind of physically embodied agent to know, as a
consequence of the nature of their physical embodiment and the means of
enquiry available to them. Moreover, we are using the term ‘epistemic’ in
quite a general way here: the putative epistemic interpretation of the
covariant bound involves the suggestion that the bound pertains to the number
of degrees of freedom which can be _accessed_ by an external observer, which
is to say, it describes what the observer can know about the system, but it
also describes the means available to the observer to act on, manipulate and
extract work from the system. As argued by Robertson and Prunkl[37], resource-
relativity of this kind is not ‘subjective,’ even if it is to some degree
anthropocentric. So really ‘epistemic’ here should be understood as meaning
something like ‘relational’ - if the covariant bound is epistemic, that simply
means it doesn’t describe the degrees of freedom of an individual physical
system, but rather says something about a relation between physical systems
and observers, which themselves are a kind of physical system. These sorts of
objective, relational epistemic facts are familiar from thermodynamics - for
example, Myrvold emphasizes that although thermodynamic concepts like entropy
are observer-relative, at the same time ‘ _it would be misleading to call them
subjective, as we are considering limitations on the physical means that are
at the agents’ disposal._ ’[26]

To understand what follows from an positive-epistemic view, let us disregard
established physics and ask what effects we might expect to observe if we did
in fact live in a world where every region of space contains some degrees of
freedom which are inaccessible to any external observers. For a start,
presumably we would observe phenomena which would look indeterministic, even
if the underlying processes are really deterministic: the inaccessible degrees
of freedom would essentially act as ‘hidden variables,’ since they would
determine the outcomes of various interactions but we would never be able to
observe them directly. Thus it may be tempting to elide the hidden degrees of
freedom with the ‘hidden variables’ invoked in some interpretations of quantum
mechanics, thus potentially providing a novel explanation for quantum
indeterminism. In addition, we might expect to see some apparently non-
Markovian behaviour as the inaccessible degrees of freedom could store extra
information about the past, and since it has been argued that quantum
mechanics itself has non-Markovian features[38], additional degrees of freedom
could again play an explanatory role.

Second, perhaps we might get behaviour superficially similar to the
uncertainty relations in quantum mechanics[39]. For suppose a particle is
localised with position uncertainty $\delta x$ in each direction. Thus the
particle is definitely to be found within a region of surface area $4\pi\delta
x^{2}$, and the surface postulate implies that the total amount of information
we can get out of this region per unit time is proportional to $4\pi\delta
x^{2}$. In particular, in order to find out the momentum of the particle we
must get the information about its momentum at a given time out of the region,
and thus the surface postulate implies that the smaller the region, the less
information we can obtain about its momentum, or equivalently the greater the
momentum uncertainty. So, roughly speaking, we’d expect that as the particle
becomes better localised in spacetime our uncertainty about its momentum would
increase, and vice versa.

These examples show that an epistemic interpretation of the entropy bound may
offer interesting new possibilities for understanding quantum mechanics.
Indeed, the bound on the amount of information which we can get out of a
spacetime region looks a lot like the kind of ‘epistemic limitation’
postulated by Spekkens, and as he shows in ref [40], the existence of
epistemic limitations can explain many of the puzzling features of quantum
mechanics. So there may indeed be value in thinking explicitly about the
possibility of additional degrees of freedom beyond the covariant bound,
rather than merely disregarding them on pragmatic grounds - even if we can’t
learn anything very concrete about these degrees of freedom, simply
acknowledging their existence can provide novel explanations of various
features of physics. Let us now consider whether the epistemic and ontological
views of the covariant bound can shed any light on the matter.

###  4.2 Positive-Ontological Views

The very possibility of an epistemic interpretation of the bound poses quite a
serious problem for attempts to understand it in an ontological way, because
proponents of such views will be vulnerable to the criticism that they are
simply mistaking an epistemic constraint for an ontological one. For example,
the fact that four different methods of calculating the black hole entropy all
give the Bekenstein-Hawking entropy formula[41] might initially be taken as
evidence that the formula reflects the true number of degrees of freedom in a
black hole, but once we start thinking about possible epistemic restrictions,
we might worry that the result is the same in all four cases not because it
reflects the true, ontological number of degrees of freedom of the black hole
but merely because all four approaches are based on physics developed by
agents subject to a certain kind of universal epistemic restriction.

How can proponents of an ontological view resist this? Well, they could call
into question some of the ontological and metaphysical presuppositions which
enter implicitly into the epistemic view. In particular, the argument that
there could be more degrees of freedom inside a given region than observers
are able to access is founded on the assumption that we inhabit a three-
dimensional space (defined by a foliation of a four-dimensional spacetime)
with independent autonomous degrees of freedom at each point of space. But the
thought experiment of section 3.2 makes it clear that this assumption far
outstrips our evidence - we don’t actually receive data in three-dimensional
chunks but rather on two-dimensional surfaces, so there is a significant
mismatch between our representation of the data in 3D space and the amount of
information we actually obtain about the contents of space. Thus rather than
concluding that there exists a whole realm of invisible, inaccessible physics,
perhaps we should adopt a different ontology which is a better match to the
data.

For example, one might imagine an ontology where information at distinct
spacetime points is not independent, so the total number of effective degrees
of freedom scales instead with the surface area. As t’Hooft puts it: ‘ _This
suggests that physical degrees of freedom in three-space are not independent
but, if considered at Planckian scale, they must be infinitely correlated_
’[5]. Alternatively, one could imagine an ontology composed entirely of two-
dimensional surfaces, or something isomorphic to them. After all, since we
always interact with bounding surfaces rather than the light-sheets associated
with those surfaces, it seems that only the bounding surfaces are strictly
necessary to account for our experiences, so we could potentially take it that
a bounding surface and its associated light-sheet are really just equivalent
representations of the same element of ontology. Indeed, there are some
indications from within General Relativity itself that this may be the right
way to think about its ontology - for diffeomorphism invariance together with
the equivalence principle prevent us from defining quantities like energy,
momentum and angular momentum at individual spacetime points, and thus these
quantities must instead be defined in terms of integrals over two dimensional
surfaces[42]; so it seems quite natural to associate the theory not with an
ontology of locally defined quantities at individual spacetime points, but
rather with an ontology in which all meaningful quantities are associated with
surfaces.

However, if we choose an ontology specifically to match the nature of the data
available to observers like us, we run the risk of arriving at on ontology
which works for the kinds of systems we observe but which depends on a split
between systems and observers and hence does not allow us to accommodate
observers or to explain the relation between systems and observers in physical
terms. For example, it is easy enough to say that whenever an observer
observes a region through a bounding surface, only the surface is really
ontological and the region inside is just a representation of that surface
created by the observer. But what if there is another observer inside the
region, observing a smaller region through another bounding surface? Should we
then say that the inner observer is not real? What about the case where two
observers observe one another through different bounding surfaces - whose
surface trumps whose? Moreover we can’t simply put all possible bounding
surfaces into the ontology, because then the ontology would contain the whole
of spacetime and we would be back to the 3D picture. So evidently if we don’t
wish to resort to operationalism we will need some principled approach which
allows us to identify an ontology of surfaces without appealing to external
observers, in order that we can integrate observed systems and observers into
a unified physical description.

There are two obvious ways to achieve this. The first is to accept that the
choice of surface is always relativized to an observer, and adopt a relational
approach similar to relational quantum mechanics (RQM)[43, 44] in which we say
that _all_ valid physical descriptions must be relative to an observer. RQM
avoids collapsing to operationalism because it tells us that every physical
system counts as an observer and thus each physical system defines its own
observer-system split. So we could potentially add to RQM the specification
that for every physical system, the description of the world relative to that
system at a given time is defined on a two-dimensional surface, and that
surface contains everything that belongs to the ontology defined relative to
that observer444It is not entirely clear how one arrives at a specification of
which observers exist in the first place within this kind of approach, given
that everything is supposed to be relativized to an observer so there can’t be
observer-independent facts about the set of observers. But this is a generic
problem for all relational and perspectival approaches to quantum mechanics -
adding the specification that data is defined on surfaces doesn’t make the
problem worse, although it doesn’t offer a solution either.. Moreover,
provided that we include a postulate ensuring the existence of links between
the perspectives of different observers, as described in ref [45], RQM still
gives rise to a shared macroscopic reality common to large classes of
observers, and thus we can potentially arrive at something close to an
integrated description of ‘the universe as a whole,’ even though this approach
does not allow such a thing as a completely observer-independent gods-eye view
of the universe.

Alternatively, we could define a unique surface, or small set of surfaces, to
which all of spacetime can be reduced - that surface thus acting as something
like a ‘hologram’ (in Part II we will take a more detailed look at
holography). For example, a picture like this arises within Kent’s solution to
the Lorentzian quantum reality problem[46, 47] which is intended as a solution
to the measurement problem: in this model the wavefunction undergoes its usual
unitary evolution until the end of time, and then we imagine something like a
measurement being performed on the final state, with the actual course of
history being determined by the result of the measurement (the ‘measurement’
is to be understood as a mathematical device for extracting probabilities,
rather than a literal physical operation)555Kent also makes allowance for the
possibility that there is no end of time - in this case we simply take a limit
as $t\rightarrow\infty$, making some assumptions which ensure that the limit
is well-defined.. This approach looks like it could be a good fit with the
covariant bound because it has the consequence that the information in a
region (e.g. on a light-sheet) depends on the state of its bounding surfaces
in its lightlike or timelike future: a beable can exist at some spacetime
point only if there is a record of its existence somewhere in the final state,
and thus the total amount of information in a given region cannot be greater
than the amount of information which can escape that region to arrive at
timelike infinity, so something like the covariant entropy bound will
naturally arise666The version of this approach presented by Kent in ref [48]
is slightly different because it has the consequence that the information at a
point is determined only by parts of the final state which are _outside_ the
future light-cone of that point, so this formulation of Kent’s view does not
postulate the same tight connection between the information in a region and
its bounding surfaces in the lightlike or timelike future. In future work we
hope to consider in more detail the relationship between entropy bounds and
these two formulations of Kent’s approach.. And in this picture we have no
need to appeal to external observers to decide which surfaces should feature
in the ontology, because in a sense the ontology includes only the surface on
which the result of the final measurement is defined; the contents of the rest
of spacetime, including observers themselves, are merely a projection from
that surface.

##  5 Gravity

Clearly if we had arrived at the covariant entropy bound purely by counting
the number of degrees of freedom in the observations that we make about
various physical systems, the thought-experiment in section 3.2 would be a
strong argument against taking the bound too seriously - after all, if you
know in advance that regardless of how many degrees of freedom there are in a
system you will never be able to extract more than $n$ bits of information
about it, then when you subsequently observe that you extract $n$ bits of
information, you have grounds to conclude only that the system has at least
$n$ degrees of freedom, not that it has exactly $n$ degrees of freedom.
However, we did not actually arrive at the covariant entropy bound by directly
counting degrees of freedom, but rather by a variety of theoretical arguments.
This seems quite puzzling - our purely theoretical arguments are not subject
to epistemic restrictions, so why should they have led us to the exact same
bound that our epistemic restrictions would have imposed if we had literally
counted degrees of freedom?

In the case of the original thermodynamic arguments for the bound this puzzle
is quite easily resolved: we have already noted that thermodynamic entropy is
relativized to a choice of manipulable variables, so it would not be
surprising if the thermodynamic arguments for the Bekinstein-Hawking entropy
and strong entropy bound were to lead us to a general bound on _accessible_
degrees of freedom in regions of spacetime. However, we now have evidence for
the bound based not on thermodynamics but on gravitational physics - for in
many cases it is phenomena like gravitational collapse and gravitational
focussing which protect the covariant bound. In Bousso’s own words: ‘
_Gravitational backreaction plays a crucial role in preventing violations of
this bound. In realistic systems, an increase in entropy is accompanied by an
increase in energy. Energy focusses light rays by an amount proportional to G.
Thus it hastens the termination of a light-sheet at caustic points, preventing
it from “seeing” too much entropy._ ’[49] This leaves us with something of a
dilemma: as we saw in section 3.2 there is a sense in which the covariant
bound looks like it could correspond to an epistemic restriction, and yet some
of the evidence for it comes from gravitational effects, which are surely not
epistemic. Does this disprove the epistemic view, or can we argue that some
epistemic considerations are somehow being smuggled into the gravitational
evidence?

###  5.1 Focussing

To answer this question, let us take a closer look a the focussing theorem.
The theorem arises from Raychauduri’s equation[50], which describes the
evolution of a ‘null congruence’ (i.e. a family of geodesics) with affine
parameter $\lambda$ in a D-dimensional manifold:

$\frac{d\theta}{d\lambda}=-\frac{1}{D-2}\theta^{2}-\sigma_{ab}\sigma^{ab}-R_{ab}k^{a}k^{b}$

Here, $\theta$ is the expansion of the bundle, representing the change in the
area spanned by infinitesimally neighbouring geodesics:
$\theta=\lim_{A\rightarrow 0}\frac{1}{A}\frac{dA}{d\lambda}$. $\sigma_{ab}$ is
the shear, representing the change in the shape of the area spanned by
infinitesimally neighbouring geodesics. Using Raychauduri’s equation, we can
arrive at the focussing theorem[51]: in a spacetime satisfying the null
curvature condition ($R^{ab}k^{a}k^{b}\geq 0$ for all null vectors $k^{a}$)
the expansion is non-increasing at all points where $\theta$ is finite (i.e.
points which are not singularities or caustics) of a surface-orthogonal null
congruence. That is to say, if the null curvature condition is satisfied
geodesics can focus, i.e. come together to a point, but they can never anti-
focus, i.e. diverge away from one another.

The focussing theorem is a theorem of differential geometry, so it does not
depend on the Einstein equations or on any features of general relativity.
However, Einstein’s equations imply that the null curvature condition is
equivalent to the null energy condition, i.e. $T_{ab}k^{a}k^{b}\geq 0$. So in
a general relativistic spacetime, we can conclude that if the null _energy_
condition is satisfied then geodesics can focus but not anti-focus. And since
the null energy condition is true at least on average for all realistic
classical matter, we can conclude that in classical general relativity
geodesics will always focus.

It is evident that the covariant bound would not work without gravitational
focussing. In Bousso’s own words, ‘ _Entropy costs energy, energy focusses
light, focussing leads to the formation of caustics, and caustics prevent
light-sheets from going on forever._ ’[27] This leads to a natural speculation
- could we turn this reasoning around and explain gravitational phenomena as a
consequence of an entropy bound? For if we are prepared to take for granted
that the information on a light-sheet (or at least, the _accessible_
information on a light-sheet) is upper bounded by the information that can
pass through its bounding surface, then Bousso’s reasoning can be employed to
argue that geodesics must be focussed by the presence of mass-energy, so the
existence of some gravity-like behaviour is an inevitable consequence of the
covariant entropy bound.

One may wonder exactly how far we can get with this approach: does the
covariant bound single out the Einstein equations specifically, or does it
merely lead us to a class of theories all exhibiting some sort of focussing of
geodesics? In fact there have been suggestions that entropy-area relations can
be used to derive the Einstein equations - most famously Jacobson’s 1995
‘thermodynamical’ derivation of the Einstein equations[52]. Jacobson’s
argument was published four years before Bousso published his covariant bound,
so his derivation associates an entropy with a surface rather than a light-
sheet; however, in appendix A we suggest a way of rewriting Jacobson’s
derivation using a light-sheet formulation where we suppose that the covariant
bound is the consequence of a restriction on the number of degrees of freedom
on a light-sheet, The derivation proceeds by assuming that the resulting
inequality relating the curvature of spacetime and area is saturated, i.e.
spacetime is always curved just enough to ensure that the entropy bound is
obeyed, and no more. The derivation is not fully general - in appendix A.1 we
discuss some of its limitations - but it provides at least some support for
the idea that Einstein’s equations could be regarded as a consequence of the
covariant entropy bound, interpreted either ontologically or epistemically.

In what follows we will not assume that the exact form of the Einstein
equations can be derived from the covariant entropy bound, but we will take it
that, as indicated by Bousso’s comments above, the covariant bound does at
least entail the existence of some kinds of gravitational phenomena. So we
have a two-way relation: the covariant bound implies the existence of certain
gravitational phenomena, and meanwhile certain gravitational phenomena imply
the covariant bound, or at least play a major role in protecting the bound in
many cases. It seems natural, therefore, to suppose that one of these things
can be regarded as explaining the other - but how do we decide in which
direction the explanation should go?

Ref [53] addresses this question, criticizing Jacobson’s proof on the grounds
that we must invoke the Einstein equations to arrive at the universal relation
between entropy and area on which the proof is based, so the derivation is
circular. The authors assume that the entropy is an entanglement entropy and
then note that the entanglement is usually thought to depend on the types of
fields present, via the relation $S=\frac{nA}{L^{2}}$, where $n$ is the number
of types of fields and $L$ is the cutoff length at which we reach the Planck
regime. But matter fields interact gravitationally and thus if there are more
fields present the energy density will be higher, so we will reach the Planck
scale at a higher cutoff length $L$; it can be shown that the two effects
exactly cancel out, so we end up with a universal relation between area and
entropy that does not depend on the number of fields. Evidently if this is the
right way to think about the relation between entropy and area, then this
relation is a _consequence_ of the Einstein equations, and thus it would
indeed be circular to use it to explain gravitational effects.

In part II of this paper we will examine the relationship between entropy
bounds and entanglement entropy in greater detail, but for now it is enough to
note that this criticism is begging the question: someone who believes that
the covariant bound should be understood as explaining the existence of
gravitational phenomena would presumably argue that the entropy bounds should
_not_ be understood as a consequence of the Einstein equations, but should be
accepted as true for some other reason - for example, because they are
regarded as a fundamental fact about ontology. From this point of view the
reasoning of ref [53] is the wrong way round: it is precisely _because_ there
is a universal relation between entropy and area that the gravitational
interaction between matter fields must act in such a way as to cancel the
dependence on the number of fields, and indeed this can be regarded as simply
an alternative version of the argument that a universal entropy-area relation
entails the existence of gravitational phenomena.

In fact, explaining gravitational phenomena by appeal to an entropy-area
relation rather than vice versa arguably leads to a better explanation. For if
we start from the Einstein equations and give an explanation similar to the
one in ref [53], we have to conclude that it just happens to turn out that
gravitational effects manage to cancel the dependence on the number of
species, in a way that one might regard as suspiciously coincidental. The
coincidence disappears if we run the explanation in the other direction, with
gravity required to work in such a way as to respect the universality of the
entropy bound. Moreover, if we start from the Einstein equations and give an
explanation similar to the one in ref [53], we have to accept that a large
range of physical mechanisms are responsible for protecting the entropy bound
in different circumstances: ‘ _they differ according to the physical situation
studied, and they can involve combinations of different effects more
reminiscent of a conspiracy than of an elegant mechanism._ ’[14] This
piecewise explanation seems quite unsatisfactory, whereas we can give a more
unifying explanation if we see the entropy bound as a fundamental feature of
the ontology and derive all of these effects from it. Therefore we do not
think it can be taken for granted that the correct direction of explanation is
from Einstein’s equations to the covariant bounds rather than vice versa. Let
us now consider what the epistemic and ontological views of the covariant
bound have to say about the matter.

###  5.2 Epistemic View

If we take an epistemic view of the covariant bound, we know that
gravitational phenomena do not explain the covariant bound, because the bound
is already fully explained by the facts about how much information observers
can obtain out of various regions of spacetime. So, if we accept that there is
an explanatory relation between the covariant bound and gravitational
phenomena, we can straightforwardly conclude that gravitational phenomena must
be explained by the bound. And furthermore, since it is plausible that
scientific explanations are usually transitive (as argued by Lange in ref
[54]), it would seem to follow that gravitational phenomena must ultimately be
explained by the facts about how much information observers can obtain out of
various regions of spacetime. This provides an answer to the question of how
we could possibly have obtained a bound which is really just an epistemic
restriction on the amount of information we can obtain out of various regions
of spacetime, even though we arrived at this bound by consideration of
gravitational effects rather than explicitly counting the number of states in
various regions. For if we understand gravitational phenomena as a consequence
of the same epistemic restrictions which give rise to the covariant bound
itself, then reasoning based on gravitational phenomena can be expected to
lead to the same results as we would have obtained had we explicitly counted
states.

Now, it is likely that explaining gravitational phenomena as a consequence of
an epistemic restriction in this way would invite criticisms similar to
Albert’s emphatic reaction to the idea that the thermodynamic entropy could be
epistemic: ‘ _Can anybody seriously think that it is somehow necessary . . .
that the particles that make up the material world must arrange themselves in
accord with what we know, with what we happen to have looked into? Can anybody
seriously think that our merely being ignorant of the exact microconditions of
thermodynamic systems plays some part in bringing it about, in making it the
case, that (say) milk dissolves in coffee? How could that be?_’[55]

However, in response to this kind of criticism it is important to recall that,
as noted in section 4.1, ‘epistemic’ is not the same as ‘subjective.’ If
indeed it is the case that only a certain amount of information about a system
can emerge through its bounding surface, that is an objective fact about
reality: so if gravitational phenomena are to be understood as a consequence
of limitations on the amount of information that can pass through a surface,
then they are indeed an objective fact about a certain regime of reality. From
this point of view gravitational focusing is in a sense an illusion, but the
fact that all realistic observers are subject to that illusion is very much
objective.

One might also criticize this approach on the grounds that the argument we
have given leans too heavily on some questionable assumptions about
explanation: in addition to assuming the transitivity of scientific
explanation, we have assumed that once we have a complete explanation of some
physical phenomenon, we shouldn’t posit another one. But many philosophers
believe that explanation must be relativized to an audience or context of
explanation[56, 57], so really this assumption should say that we shouldn’t
posit another explanation _relative to the same audience or context of
explanation_ , and thus one might hope to avoid the conclusion by arguing that
in one context of explanation the covariant bound is explained by epistemic
limitations, and in another context of explanation the covariant bound is
explained by gravitational phenomena, so there is no single context of
explanation within which we can apply the transitivity of explanation to infer
that gravitational phenomena are explained by epistemic limitations.

However, this response may be blocked by bypassing the transitivity argument
and showing directly that epistemic limitations on the information obtainable
out of various regions of spacetime can give rise to ‘illusory’ gravitational
effects. To do so, we will make an argument inspired by Susskind’s ideas in
ref [1]. Take a screen with area $A$, and an object $O$ behind the screen with
surface area $A^{\prime}$; assume the surface postulate is correct, so the
information on the surface of $O$ has an upper bound proportional to
$A^{\prime}$, and the information passing through the screen has an upper
bound proportional to $A$. Following Susskind, we will also assume the
constant of proportionality is the same for both the surface and screen and
that information on the surface of $O$ saturates the bound. Now let light rays
carry the information about the state of the surface of the object towards the
screen. Suppose all the information in some area element $dA^{\prime}$ is
carried by a bundle of light rays $B$, which intersect the screen orthogonally
and which have cross-sectional area $dA$ by the time they reach the screen.
Suppose that gravitational focusing does not occur for some particular area
element $dA^{\prime}_{x}$ on the object, i.e. $\frac{d\theta}{d\lambda}$ is
positive for the corresponding bundle $B_{x}$. Since $\theta$ is zero at the
screen, this implies that $\theta$ is negative for $B_{x}$ between the object
and the screen, and hence the corresponding area element $dA_{x}$ on the
screen is smaller than $dA_{x}^{\prime}$, so some of the information on the
area element $dA^{\prime}_{x}$ will not be able to pass through the screen.
Thus we may perhaps make the case that the _perceived_ area of
$dA_{x}^{\prime}$ will be less than its real area, since the external observer
can only judge the area of $dA_{x}^{\prime}$ on the basis of the information
that emerges about $dA_{x}^{\prime}$ through the screen; so an external
observer will always perceive the area of any area element $dA^{\prime}$ on
the surface of the object as being less than or equal to the corresponding
area $dA$ on the screen, and thus it will always look as though gravitational
focusing has occurred. This argument suggests a way in which gravitational
focusing could indeed be somewhat illusory: it might be understood as a
function of the fact that we must always perceive objects as having less
surface area than the screens through which we observe them.

###  5.3 Ontological View

The question of whether gravitational phenomena are explained by the covariant
bound or vice versa in an ontological picture will ultimately hinge on the
nature of the ontological picture one adopts. But as a concrete example,
suppose we adopt an ontological view where the ontology is composed entirely
of bounding surfaces. In that case, the covariant bound is explained by the
fact that light-sheets are nothing more than an alternative encoding of the
information on bounding surfaces, and then gravitational phenomena are
explained by the ontology of light-sheets: geodesics must be focussed by
matter because otherwise the light-sheets would contain more information than
their bounding surfaces, which is impossible if the light-sheets are just
alternative representations of bounding surfaces.

Smolin proposes something of this kind when he describes the significance of
his own formulation of the entropy bound: ‘ _Its role is to constrain the
quantum causal structure of a quantum spacetime in a way that connects the
geometry of the surfaces on which measurements may be made with a measure of
the information that those measurements may produce … the notion of area is
reduced fundamentally to a measure of the flow of quantum information_ ’[28].
It seems that Smolin is imagining something like the following: spacetime
should be understood as a construction - presumably on the part of human
observers, or subconscious information-processing going on in their brains -
whose geometry and dimensionality is determined by the nature of the
information that the observers receive from the outside world, which need not
itself be spatiotemporal in the ordinary sense. So in this picture, it is
indeed the case that light-sheets are just representations of the information
on surfaces. Markopoulou and Smolin[7] have constructed a model of this kind,
based on information flowing in a network of ‘holographic screens’ which take
the form of a causal set, with screens composed by a quadruple of events.

It should be noted that the derivation of the Einstein equations from the
covariant bound relies on various approximations about entropy (particularly
that it can be regarded as a fluid) and it is likely that these approximations
do not hold at the most fundamental level. Moreover, the classical focussing
theorem does not carry over precisely to quantum mechanics, because it relies
on the null energy condition, which can be violated by physically reasonable
states in a quantum field theory. Thus if gravitational phenomena are really
to be explained as a consequence of the covariant bound, understood in an
ontological sense, it is likely that gravity and spacetime structure should be
thought of as not fundamental but emergent - the fundamental description of
reality pertains to the underlying degrees of freedom, and the curvature of
spacetime emerges in the limit as the entropy-fluid approximation and the
classical focussing theorem become valid, as a kind of higher-level
description of the way in which these degrees of freedom depend on each other.
The ontological picture thus falls in line with a long tradition of proposals
about ‘emergent gravity’ and ‘emergent spacetime’ within both physics[58, 59,
52] and philosophy[60].

##  6 Philosophical issues

Having examined the empirical evidence for the different possible views of the
covariant entropy bound, we will now consider if there are philosophical
arguments which might support one view or another. Philosophical arguments are
relevant here because the epistemic and ontological views are naturally
associated with certain metaphysical pictures. In particular, an epistemic
view of the bound follows naturally if we are committed to a view that might
be described as ‘orthodox reductionism,’ (i.e. the idea that spacetime is
completely filled with autonomous degrees of freedom from which macroscopic
phenomena emerge in some appropriate limit) because the assumption of autonomy
at each spacetime point leads naturally to the idea that there may be
information inside any given spacetime region which is unable to get out of
the region. Conversely an ontological view of the bound requires us to accept
that the content of a spacetime region is somehow dependent on its bounding
surfaces in the past and future, which is hard to reconcile with the orthodox
reductionist picture - it seems to entail that the degrees of freedom at
different spacetime points are not generically autonomous, since they depend
in some sense on higher-level phenomena pertaining to bounding surfaces. So it
seems that an orthodox reductionist would have little option but to take an
epistemic view.

Now, one might feel that there is something worrying about this dialectic. A
metaphysical conviction (i.e. orthodox reductionism) combined with certain
empirical results (i.e. the evidence that spacetime is discretized and/or that
information flux through surfaces is bounded) apparently compels us to
postulate a realm of unobservable physics which we can never observe or learn
anything definite about. It may seem worrying that we end up committed to a
view so ontologically excessive - from a naturalistic point of view, one might
argue that rather than simply accepting the existence of invisible,
undetectable physics, we should consider abandoning the metaphysical
convictions that led to this conclusion. A historical comparison may help make
the point: in the case of the ether hypothesis, it was the metaphysical
conviction that light must be propagated in some substance which forced
scientists of the day to postulate an invisible omnipresent substance which
was undetectable in all experiments, and in hindsight it is easy to see that
the right thing to do was to abandon the metaphysical conviction.

That said, some care must be taken here, for the reason to get rid of the
ether was not simply the fact that it was not directly detectable - we can
have good theoretical reasons to believe in things that are not directly
detectable. Rather, the ether was ultimately discarded because it served no
theoretical function: there was no reason to believe in it other than the
metaphysical conviction that light must be propagated in a substance, so this
metaphysical conviction was forcing scientists of the day to make their
theories more complex for no increase in explanatory or predictive power. This
suggests that in order to decide whether or not we should accept the
unobservable degrees of freedom postulated by an epistemic account of the
bound, it is important to assess what theoretical function they might serve.
Evidently they do not _currently_ serve any theoretical function, since our
best current theories obey the covariant bound and thus they don’t include any
unobservable degrees of freedom, but that leaves open the question of whether
we could increase the explanatory or predictive power of our theories by
taking these degrees of freedom into account.

On the one hand, we saw in section 4.1 that the existence of these degrees of
freedom could potentially explain some features of our current physics. On the
other hand, no _specific_ details about those degrees of freedom can possibly
be explanatory, since ex hypothesi the world outside the relevant surfaces
cannot depend on those details in any way. And one might question whether an
entity really has theoretical power if only its existence and no specific
features of its state or configuration are relevant to the predictions of the
theory. So arguably the _literal_ understanding of these degrees of freedom
required by orthodox reductionism is indeed not serving any theoretical
function - we would perhaps be better off thinking of these degrees of freedom
as something like gauge, in that their existence is a necessary feature of our
representation of the theory but their specific configuration is unimportant.
That said, ultimately an answer to this question must await a better
understanding of what a theory which explicitly includes unobservable degrees
of freedom might look like - it’s possible that the existence of specific
states and configurations of these degrees of freedom might play some crucial
theoretical function that is not immediately evident in advance of writing
down the theory, and if that were the case then we would clearly be justified
in being committed to their existence despite the fact that they are not
directly detectable.

On the other hand, abandoning reductionism comes with its own conceptual
problems. In particular, when $A$ can be derived from $B$ but also $B$ can be
derived from $A$, how do we know the correct direction of explanation?
Orthodox reductionism provides a simple answer - the smaller always explains
the larger and not vice versa - so if we are moving away from reductionism we
will need alternative ways to settle these questions. We saw something of this
dialectic in the discussion of Jacobson’s derivation in section 5.3: the
argument of ref [53] implicitly invokes the reductionist intuition that
higher-level effects like the universality of the entropy-area relation must
be given constructive explanations based on smaller-scale phenomena, such as
the gravitational behaviour of fundamental fields, and yet a supporter of the
view that entropy bounds explain gravitational phenomena would presumably want
to insist that the gravitational behaviour of the fundamental fields is a
consequence of the universal entropy bound, which would mean that a higher-
level feature of reality actively constrains the behaviour of the smaller-
scale entities. If this kind of reasoning is to become accepted in science, we
will need better criteria for judging the fitness of non-reductionist
explanations and for deciding the correct direction of explanation in disputed
cases.

##  7 Conclusion

So, assuming that the covariant bound is in some sense correct at least at the
semiclassical level, is it epistemic or ontological? For those who are happy
to take a purely operational view where we aim only to describe the degrees of
freedom which are accessible to observers, this question may not be very
important, but for those who hope to do cosmology, describe the universe as a
whole or simply understand what is really going on, the issue must be
resolved.

We have seen that on the one hand, the thermodynamic arguments for the bound
can easily be understood in an epistemic way, but on the other hand the
arguments based on gravitational focussing don’t seem particularly epistemic,
and thus to maintain the epistemic interpretation it would seem that we would
have to adopt quite a radically revisionary view which sees gravity itself as
being in some sense epistemic. This provides motivation for an ontological
view, but on the other hand we have argued that an ontological view will
likely also be quite revisionary in its account of the nature of spacetime -
in order to avoid being undermined by the possibility of an epistemic view,
ontological approaches must show what is wrong with the assumptions
underpinning the epistemic account, which will most likely involve proposing
an alternative ontology which is a better fit to the nature of the data
available to us, such as an ontology based on surfaces rather than autonomous
degrees of freedom in a three-dimensional space.

The issues discussed in this article also have interesting consequences for
philosophy of science more generally. For we have emphasized the way in which
the physics we arrive at is limited by features of our epistemic situation,
and the ‘epistemic’ interpretation of the covariant bound presents us with a
vision of a world in which these limits are actually very severe, so
significant portions of reality may be forever beyond our grasp. This poses
some new challenges for scientific realism. First, how do we distinguish
between parts of our theories which reflect features of reality and parts of
our theories which are just consequences of our epistemic situation? And
having done so, what kind of attitude should we take to parts of physics that
we suspect are in fact consequences of our epistemic situation? For example,
if we conclude that the entropy bound is just epistemic, it is certainly a
part of reality and in that sense is compatible with scientific realism, and
yet it would seem that we should not adopt a naively realist attitude towards
its proposed count of degrees of freedom, since that count is known to be
objectively wrong. We hope to explore some of these issues in future work.

##  8 Acknowledgements

Thanks to Jacob Barandes, Niels Linneman, Ted Jacobson, and the UWO philosophy
of physics reading group for very helpful comments on this article. This
publication was made possible through the support of the ID #62312 grant from
the John Templeton Foundation, as part of the project ‘The Quantum Information
Structure of Spacetime’ (QISS). The opinions expressed in this
project/publication are those of the author(s) and do not necessarily reflect
the views of the John Templeton Foundation.

##  Appendix A Deriving Einstein’s Equations

Here we make a small modification to Jacobson’s 1995 derivation of the
Einstein equations from a horizon entropy bound, in order to argue that the
covariant entropy bound can be used to derive Einstein’s equations. Note that
this derivation ultimately requires us to assume that the inequality relating
the curvature of spacetime and the area is saturated, i.e. spacetime is always
curved just enough to ensure that the entropy bound is satisfied, and no more.
This allows us to turn the covariant bound into an equality, so we can use
basically the same mathematics as Jacobson’s original proof, which assumed an
entropy-area relation of the form $S=\frac{A}{4\hbar G}$. Of course the
derivation would not work without some such assumption, since we cannot derive
an equality from an inequality unless we assume the inequality is saturated.

###### Proof.

At a given point in spacetime, consider a 2-surface $\Omega$ of area $A$ whose
past directed null normal congruence $\\{k^{a}\\}$ to one side has vanishing
expansion and shear at the surface (as noted by Jacobson, it is always
possible to choose such a surface provided it is sufficiently small). The
past-directed null normal congruence has negative or zero expansion, and thus
is a light-sheet $L$; we parametrize the rays $\\{k^{a}\\}$ on $L$ with an
affine parameter $\lambda$ which vanishes at $A$. Because $\Omega$ is small,
we will take it that the Ricci tensor $R_{ab}$ and the stress-energy tensor
$T_{ab}$ are approximately constant in the neighbourhood of $\Omega$. We will
assume that $D$, the number of degrees of freedom on $L$, is less than or
equal to $\frac{A}{4\hbar G}$, thus explaining the fact that the entropy on
$L$ is upper-bounded by $\frac{A}{4\hbar G}$.

Now let $\Omega^{\prime}$ be the infinitesimally smaller surface obtained from
$\Omega$ by moving an affine parameter distance $d\lambda$ along each light-
ray. $\Omega^{\prime}$ has area $A^{\prime}$ and bounds the light-sheet
$L^{\prime}$ which is infinitesimally smaller than $L$. We will similarly
assume that $D^{\prime}$, the number of degrees of freedom on $L^{\prime}$, is
less than or equal to $\frac{A^{\prime}}{4\pi G}$. Then we will make a
locality assumption to the effect that the curvature of spacetime in the
region sandwiched by $\Omega^{\prime}$ and $\Omega$ depends only on the
contents of this region, and in particular, it is independent of the number of
degrees of freedom on $L^{\prime}$. Thus the curvature of spacetime between
$\Omega^{\prime}$ and $\Omega$ must be large enough that the entropy bound is
satisfied even if the number of degrees of freedom on $L^{\prime}$ takes its
maximal value, $\frac{A^{\prime}}{4\pi G}$. Therefore the number of degrees of
freedom between $\Omega^{\prime}$ and $\Omega$ must obey the bound
$D-D^{\prime}\leq\frac{A-A^{\prime}}{4G\hbar}$.

Note that $D-D^{\prime}$ acts as an upper bound for the entropy that could be
assigned by any external observer to this region, so this expression can be
recognised as a form of the generalized version of Bousso’s bound specified in
ref [15]: $S"\leq\frac{A-A^{\prime}}{4G\hbar}$, where we cut off the light-
sheet associated with a surface of area $A$ before it terminates, at a point
where it has area $A^{\prime}$, and $S"$ is the entropy on the cut-off light-
sheet. This bound is also known to be true in many circumstances, although not
universally - its validity depends on the assumption that entropy can be
regarded as something like a locally-defined fluid, so it will not necessarily
hold in scenarios where the nonlocal nature of the entropy becomes important -
and of course, in such scenarios our locality assumption about the number of
degrees of freedom would not be expected to hold, so this is exactly the
domain in which the derivation we suggest here would no longer be reliable.

Let $\theta(x)$ denote the expansion of the light-rays in $L$ at value $x$ of
the affine parameter $\lambda$; thus
$A-A^{\prime}=-\int_{\Omega^{\prime}}\theta(d\lambda)dAd\lambda$.

Now we may employ the Raychauduri equation. Having chosen $\Omega$ such that
the shear on it is zero, we have that
$\frac{d\theta}{d\lambda}=-\frac{1}{2}\theta^{2}-R_{ab}k^{a}k^{b}$, where
$R_{ab}$ is the Ricci tensor and $k$ is a unit vector field which at every
point in $L$ is tangent to the light-ray in $L$ passing through that point,
oriented from $\Omega$ towards $\Omega^{\prime}$. Since $\theta$ is zero on
the surface $\Omega$, we can work in the limit of small $\theta$, and then
since we have assumed that the Ricci tensor is approximately constant over the
region, we infer that everywhere on $\Omega^{\prime}$ we have
$\theta=-R_{ab}k^{a}k^{b}d\lambda$.

Hence $A-A^{\prime}=-\int_{\Omega^{\prime}}-R_{ab}k^{a}k^{b}d^{2}\lambda
dA=A^{\prime}R_{ab}k^{a}k^{b}d^{2}\lambda$.

Now we must derive a lower bound on $D-D^{\prime}$. We will do this by
assuming that the number of degrees of freedom in this region at least as
large as any entropy that could be assigned to the region by an external
observer. It is evident that the entropy assigned by an external observer
should be connected to the stress-energy tensor in some way; roughly speaking,
higher energy systems have more possible states and hence higher entropy. But
writing down an explicit expression will require some approximations. Here we
will follow Flanagan, Marolf and Wald[15] in assuming that the relevant
entropy can be expressed as an entropy flux $s^{a}(x)$, so that the entropy
between $\Omega^{\prime}$ and $\Omega$ is given by
$S=\int_{x\in\Omega^{\prime}}s^{a}(x)k^{a}dxd\lambda$. As argued by ref [15],
this is a good approximation in many regimes, though not all. Bousso writes: ‘
_Generally speaking, the notion of an entropy flux assumes that entropy can be
treated as a kind of local fluid. This is often a good approximation, but it
ignores the non-local character of entropy and does not hold at a fundamental
level_ ’[27].

Moreover, from thermodynamics we have that $dS=\frac{dQ}{T}$, so we will take
it that $s^{a}(x)$ can be written in the form of a heat flux divided by a
temperature. Since $D-D^{\prime}$ is at least as large as the entropy assigned
by any observer, let us consider some particular observer of acceleration
$\kappa$ whose Rindler horizon coincides approximately with $L$ in this
region. This observer sees a temperature given by the Unruh temperature
$\frac{\hbar\kappa}{2\pi}$, as well as an energy flux given by the boost-
energy current of matter, $T_{ab}\chi^{b}$, where $\chi^{b}$ is the
(approximate) Killing vector field generating Lorentz boosts in the Rindler
frame, which will coincide with a suitably defined null vector on the light-
sheet. As noted by refs [61, 52], in any spacetime, around any event, there
exists a class of (possible) local Rindler observers who will perceive this
boost-energy current and Unruh temperature, so we can be sure that such a
possible observer will indeed be well-defined in the relevant region.

As explained in refs [62, 52], $\chi^{b}$ and $k^{b}$ can be related as
follows. Since we are performing an infinitesimal calculation to first order
only, we can work on a stationary background. Then we have $\lambda
k^{a}=\lambda(\frac{dv}{d\lambda})\chi^{a}$, where $v$ is the Killing
parameter. In general the relation between affine and Killing parameters on a
Killing horizon is $\lambda=ae^{\kappa v}$, where $a$ and $b$ are arbitrary
constants, and $b$ can be chosen to be zero since we are free to shift the
affine parameter. So we find that
$\lambda\frac{dv}{d\lambda}=\frac{1}{\kappa}$ and hence we conclude that
$\chi^{a}=\lambda\kappa k^{a}$; thus at $\Omega^{\prime}$ we have that
$\chi^{a}=d\lambda\kappa k^{a}$.

Thus we infer that the entropy assigned by this specific observer is
$S=2\pi\int_{\Omega^{\prime}}\frac{T_{ab}k^{a}k^{b}}{\hbar}dAd^{2}\lambda$.
Since we have assumed that the stress-energy tensor is approximately constant
over the region, we conclude that $S=2\pi
A^{\prime}\frac{T_{ab}k^{a}k^{b}}{\hbar}d^{2}\lambda$, and hence
$D-D^{\prime}\geq 2\pi A^{\prime}\frac{T_{ab}k^{a}k^{b}}{\hbar}d^{2}\lambda$.

Thus in order to have $D-D^{\prime}\leq\frac{A-A^{\prime}}{4G\hbar}$, we
require:

$2\pi A^{\prime}\frac{T_{ab}k^{a}k^{b}}{\hbar}d^{2}\lambda\leq
A^{\prime}\frac{R_{ab}k^{a}k^{b}}{4G\hbar}d^{2}\lambda$

and thus $8\pi GT_{ab}k^{a}k^{b}\leq R_{ab}k^{a}k^{b}$.

Let us now assume that the the bound is in fact saturated - i.e. space is
always focussed exactly enough to ensure that the number of degrees of freedom
obeys the covariant entropy bound, and no more. Thus $8\pi
GT_{ab}k^{a}k^{b}=R_{ab}k^{a}k^{b}$. As noted by Jacobson, this implies that
$8\pi GT_{ab}=R_{ab}+fg_{ab}$ for some $f$. Local conservation of energy and
momentum implies that $T_{ab}$ is divergence free and therefore, using the
contracted Bianchi identity, that $f=-\frac{R}{2}+\Lambda$ for some constant
$\Lambda$. So we get $8\pi GT_{ab}=R_{ab}-\frac{Rg_{ab}}{2}+\Lambda g_{ab}$
which is the Einstein equation.

∎

###  A.1 Limitations of the derivation

The derivation given above has a number of limitations. In particular,
although it is clear that entropy must be related to the stress-energy tensor
in some way, there is no universally valid expression giving the relation
between them, so the ‘entropy flux’ expression used in this calculation will
not be valid in all situations - for example, it may fail to hold when the
non-local character of entropy is relevant, or when matter is not in
thermodynamic equilibrium. One might also question whether the Unruh
temperature used in the calculation of the entropy is the right temperature to
use here - what if the matter in the region between $S$ and $S^{\prime}$ is
just ordinary classical matter with a normal thermodynamical temperature not
associated with any acceleration? Perhaps the Unruh temperature should be
thought as encoding something to do with the maximum amount of entropy which
could possibly be present for a given value of $T_{uv}$ (since after all it
includes Planck’s constant) but the details are hazy. But nonetheless, it
seems reasonable to think that a relation of a similar form must hold in other
scenarios, since it is clear that entropy should be linked to the stress-
energy tensor, so it seems plausible to think the argument should generalize.
(Note that Jacobson did subsequently generalize his argument to the case of
non-equilibrium thermodynamics[63] so one might think that similar strategies
will work here).

Also, we assumed at the final stage that the bound $8\pi GT_{ab}k^{a}k^{b}\leq
R_{ab}k^{a}k^{b}$ is in fact saturated. This assumption is somewhat
questionable because we justified our entropy calculation on the basis that
the entropy observed by the Rindler observer must be a lower bound on the true
number of degrees of freedom in the region - we have not ruled out the
possibility that some other observer could see more entropy in the region, and
if that were the case the true number of degrees of freedom would have to be
larger. However, note that the specific choice of acceleration $\kappa$
cancels out during the entropy calculation, so the same calculation will work
for any accelerated observer in the same location; so if one assumes that non-
accelerated observers necessarily see less entropy than accelerated observers,
one may argue that the entropy observed by the Rindler observer also provides
a suitable _upper bound_ in this situation.

In addition, we have assumed that the entropy inside the region is given only
by the entropy of matter; but in principle there should also be entropy
associated with gravitational degrees of freedom. Flanagan et al make this
point in ref [15], but they then leave out gravitational contributions because
it’s unclear how to calculate them. However, one might worry that once
gravitational degrees of freedom are included, a derivation of this kind will
potentially be subject to some kind of infinite regression, and thus leaving
out gravitational degrees of freedom is a non-trivial approximation. In Part
II of this paper we will look in more detail at the question of whether or not
an entropy bound can still hold once gravitational degrees of freedom become
nontrivial.

## References

  * [1] Leonard Susskind.  The world as a hologram.  Journal of Mathematical Physics, 36(11):6377–6396, 1995.
  * [2] Jacob D. Bekenstein.  Black holes and entropy.  Phys. Rev. D, 7:2333–2346, Apr 1973.
  * [3] S. W. Hawking.  Particle creation by black holes.  Communications in Mathematical Physics, 43(3):199 – 220, 1975.
  * [4] Jacob D. Bekenstein.  Universal upper bound on the entropy-to-energy ratio for bounded systems.  Phys. Rev. D, 23:287–298, Jan 1981.
  * [5] G. ’t Hooft.  Dimensional reduction in quantum gravity, 1993.  https://arxiv.org/abs/gr-qc/9310026.
  * [6] Juan Maldacena.  International Journal of Theoretical Physics, 38(4):1113–1133, 1999\.
  * [7] Fotini Markopoulou and Lee Smolin.  Holography in a quantum spacetime, 1999.
  * [8] Sanved Kolekar and T. Padmanabhan.  Holography in action.  Physical Review D, 82(2), jul 2010.
  * [9] Ning Bao, Sean M. Carroll, and Ashmeet Singh.  The Hilbert space of quantum gravity is locally finite-dimensional.  International Journal of Modern Physics D, 26(12):1743013, oct 2017\.
  * [10] C. Rovelli and F. Vidotto.  Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2015.
  * [11] Carlo Rovelli.  Quantum Gravity.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004.
  * [12] Rodolfo Gambini and Jorge Pullin.  Discrete space-time, 2005.
  * [13] R. Sorkin.  Geometry from order: Causal sets.  Einstein Online, 2006.
  * [14] Raphael Bousso.  A covariant entropy conjecture.  Journal of High Energy Physics, 1999(07):004–004, jul 1999.
  * [15] journal = Physical Review D Eanna E. Flanagan and Donald Marolf and Robert M. Wald, title = Proof of classical versions of the Bousso entropy bound and of the generalized second law.  62(8), sep 2000.
  * [16] Christopher G. Timpson.  Quantum Information Theory and the Foundations of Quantum Mechanics.  Oxford University Press, 04 2013.
  * [17] Carina E. A. Prunkl and Christopher G. Timpson.  Black hole entropy is thermodynamic entropy, 2019.
  * [18] David Wallace.  The case for black hole thermodynamics, Part I: phenomenological thermodynamics, 2018.
  * [19] Craig Callender.  Reducing thermodynamics to statistical mechanics: The case of entropy.  The Journal of Philosophy, 96(7):348–373, 1999.
  * [20] Wayne C. Myrvold.  Probabilities in statistical mechanics, September 2014.  Forthcoming in Christopher Hitchcock and Alan Hájek, eds., The Oxford Handbook of Probability and Philosophy, Oxford University Press.
  * [21] David Wallace.  The Necessity of Gibbsian Statistical Mechanics.  In Statistical Mechanics and Scientific Explanation, chapter 15, pages 583–616.
  * [22] Rafael D. Sorkin.  The statistical mechanics of black hole thermodynamics.  1997\.  https://arxiv.org/abs/gr-qc/9705006.
  * [23] Andrew Strominger and Cumrun Vafa.  Microscopic origin of the Bekenstein-Hawking entropy.  Physics Letters B, 379(1-4):99–104, jun 1996.
  * [24] J.C. Maxwell, E. Garber, S.G. Brush, and C.W.F. Everitt.  Maxwell on Heat and Statistical Mechanics: On ‘Avoiding All Personal Enquiries’ of Molecules.  Lehigh University Press, 1995.
  * [25] Wayne C. Myrvold.  The science of ${\Theta\Delta}^{\text{cs}}$.  Foundations of Physics, 50(10):1219–1251, Aug 2020.
  * [26] Wayne C. Myrvold.  Statistical mechanics and thermodynamics: A maxwellian view, May 2011\.
  * [27] Raphael Bousso.  The holographic principle.  Reviews of Modern Physics, 74(3):825–874, aug 2002.
  * [28] Lee Smolin.  The strong and weak holographic principles.  Nuclear Physics, 601:209–247, 2001.
  * [29] Ted Jacobson.  On the nature of black hole entropy.  In Eighth Canadian conference on general relativity and relativistic astrophysics. ASCE, 1999.
  * [30] Carlo Rovelli.  Black holes have more states than those giving the bekenstein-hawking entropy: a simple argument, 2017.
  * [31] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.  Phys. Rev. Lett., 70:1895–1899, Mar 1993.
  * [32] Robert Oeckl.  A first-principles approach to physics based on locality and operationalism, 2014.  https://arxiv.org/abs/1412.7731.
  * [33] Robert Oeckl.  A “general boundary” formulation for quantum mechanics and quantum gravity.  Physics Letters B, 575(3):318–324, 2003.
  * [34] Louis Crane.  Clock and category: Is quantum gravity algebraic?  Journal of Mathematical Physics, 36(11):6180–6193, 1995.
  * [35] Alex May.  Quantum tasks in holography.  Journal of High Energy Physics, 2019(10):233, October 2019.
  * [36] John Dougherty and Craig Callender.  Black Hole Thermodynamics: More Than an Analogy?, October 2016.  http://philsci-archive.pitt.edu/13195/.
  * [37] Katie Robertson and Carina Prunkl.  Is thermodynamics subjective?  Philosophy of Science, page 1–16, 2023.
  * [38] A. Montina.  Exponential complexity and ontological theories of quantum mechanics.  Physical Review A, 77(2):022104, February 2008.
  * [39] W. Heisenberg.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.  Zeitschrift fur Physik, 43(3-4):172–198, March 1927.
  * [40] R. W. Spekkens.  Evidence for the epistemic view of quantum states: A toy theory.  Phys Rev A, 75(3):032110, March 2007.
  * [41] David Wallace.  The case for black hole thermodynamics, Part I: phenomenological thermodynamics, 2017.  https://arxiv.org/abs/1710.02724.
  * [42] Tullio Regge and Claudio Teitelboim.  Role of surface integrals in the Hamiltonian formulation of general relativity.  Annals of Physics, 88(1):286–318, 1974.
  * [43] Federico Laudisa and Carlo Rovelli.  Relational Quantum Mechanics.  In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2021 edition, 2021.
  * [44] C. Rovelli.  Relational quantum mechanics.  International Journal of Theoretical Physics, 35:1637–1678, August 1996.
  * [45] Emily Adlam and Carlo Rovelli.  https://arxiv.org/abs/2203.13342.
  * [46] A. Kent.  Solution to the Lorentzian quantum reality problem.  Phys Rev A, 90(1):012107, July 2014.
  * [47] Adrian Kent.  Lorentzian quantum reality: postulates and toy models.  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 373(2047):20140241, Aug 2015.
  * [48] Adrian Kent.  Quantum reality via late-time photodetection.  Physical Review A, 96(6), Dec 2017.
  * [49] Raphael Bousso.  Flat space physics from holography.  Journal of High Energy Physics, 2004(05):050–050, 2004.
  * [50] George F. R. Ellis.  On the Raychaudhuri equation.  Pramana, 69(1):15, July 2007.
  * [51] Raphael Bousso, Zachary Fisher, Stefan Leichenauer, and Aron C. Wall.  Quantum focusing conjecture.  Phys. Rev. D, 93:064044, Mar 2016.
  * [52] Ted Jacobson.  Thermodynamics of Spacetime: The Einstein Equation of State.  Physical Review Letters, 75(7):1260–1263, 1995.
  * [53] Goffredo Chirco, Hal M. Haggard, Aldo Riello, and Carlo Rovelli.  Spacetime thermodynamics without hidden degrees of freedom.  Physical Review D, 90(4), aug 2014.
  * [54] Marc Lange.  Transitivity, Self-Explanation, and the Explanatory Circularity Argument Against Humean Accounts of Natural Law.  Synthese, 195(3):1337–1353, 2018.
  * [55] David Z. Albert.  Time and Chance.  Harvard University Press, 2000.
  * [56] B.C. van Fraassen, Oxford University Press, and P.P.B.C. Van Fraassen.  The Scientific Image.  Clarendon Library of Logic and Philosophy. Clarendon Press, 1980.
  * [57] Henk W. de Regt.  A Contextual Theory of Scientific Understanding.  In Understanding Scientific Understanding. Oxford University Press, 11 2017.
  * [58] Erik Verlinde.  Emergent gravity and the dark universe.  SciPost Physics, 2(3), 2017.
  * [59] Erik Verlinde.  On the origin of gravity and the laws of newton.  Journal of High Energy Physics, 2011(4), apr 2011.
  * [60] Eleanor Knox.  Effective spacetime geometry.  Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 44(3):346–356, 2013.
  * [61] T. Padmanabhan.  Exploring the nature of gravity.  2016\.  https://arxiv.org/abs/1602.01474.
  * [62] Ted Jacobson and Renaud Parentani.  Horizon entropy.  Foundations of Physics, 33(2):323–348, 2003.
  * [63] Christopher Eling, Raf Guedens, and Ted Jacobson.  Nonequilibrium thermodynamics of spacetime.  Physical Review Letters, 96(12), mar 2006.