††institutetext: Dept of Physics, College of Exact Sciences, Hamma Lakhdar
University, El Oued, Algeria.††institutetext: Lab of Linear Operators and
Partial Diff Equations, Theory and Application, Hamma Lakhdar Univeristy.

#  On Horizon Molecules and Entropy in Causal Sets

D. Dou<EMAIL_ADDRESS>
###### Abstract

We review the different proposals and attempts to identify the “horizon
molecules” that would give a kinematical estimation for the black hole entropy
in causal set theory. The proposals are presented according to their
chronological appearance in scientific literature. The review is neither very
technical nor merely descriptive; it is aimed to provide the reader with a
lucid introduction to the necessary concepts and mathematical background, and
give him or her a broad view on the subject, by focusing on the main technical
and conceptual issues that summarize the progress made in the last two
decades.

###### Keywords:

Causal Sets, Quantum Gravity, Black Holes, Entropy, Horizon Molecules,
Statistical Geometry.

##  1 Introduction

Although the energy scale at which quantum effects on spacetime are expected
to show up is well beyond the range of any foreseeable laboratory-based
experiments, the theoretical consequences of quantum mechanics and general
relativity have been major reasons for studying quantum gravity and searching
for a more fundamental structure of spacetime. Most importantly, the discovery
of the close relationship between certain laws of black hole physics and the
ordinary laws of thermodynamics, on one hand, and the discovery of the quantum
induced radiation by black hole (BH), on the other hand, appear to be two
major pieces of a puzzle that fit together so perfectly that there can be
little doubt that this “fit” is of deep significance Wald:1999vt ;
Sorkin:1997ja ; Carlip:2014pma ; Sorkin:2005qx ; Bekenstein:1994bc .

Today, well into its fifth decade of the development, this merger remains
intellectually stimulating and puzzling at once.

One of the most puzzling aspects is the fact black hole possesses an entropy
equal to one quarter of its horizon area expressed in units of Planck area.
And in spite of five decades of intensive research, debates and genuine
advances in different directions, especially within the context of string
theory and 2+1 gravity Strominger:1996sh ; Horowitz:1996qd ; Maldacena:1997re
; Carlip:2014pma ; Carlip:1994gy , see also Ashtekar:1997yu  for loop quantum
gravity results, it is fair to say that the physical origin of this entropy
and all questions accompanying the thermodynamic of BH are still lacking
satisfactory answers, and the debate is far from being settled . In
particular, it remains uncertain what “degrees of freedom” or microstates the
entropy refers to, or what unavailable information it quantifies. Moreover, it
can be said that a well accepted criterion to select one approach out of the
different approaches to quantum gravity or a fundamental theory of nature is
its success in solving black hole thermodynamics puzzles in a satisfactory and
general manner, in particular revealing the statistical mechanics behind BH
entropy .

It also is generally believed that all the puzzles of the BH are not
independent and will be solved once we really solve one of them. For this and
other reasons, providing a controllable calculation of BH entropy has been a
prime target of all theories and proposals to quantum gravity.

Indeed, in the current climate the role being played by BH thermodynamics in
this connection looks more and more analogous to the role played historically
by the thermodynamics of a box of gas and black body radiation in revealing
the underlying atomicity and quantum nature of everyday matter and radiation.
This analogy can be brought out more clearly by recalling some facts about
thermodynamics in the presence of event horizons.

A well accepted definition of entropy is as a measure of missing or
“unavailable” information about a physical system, and from this point of
view, one would have to expect some amount of entropy to accompany an event
horizon, since it is by definition an information hider par excellence, and
therefore the BH entropy could be understood as a response of having an event
horizon which hides information about a region of space time, and here the
notion of entanglement entropy comes into play. This originates from the well
known observation that an observer outside the horizon has no access to the
degrees of freedom behind the horizon. For this reason the outside observer
would describe the world with a reduced density matrix obtained by tracing out
the inaccessible degrees of freedom behind the horizon. If the exterior modes
and the external modes are correlated “entangled” the resulting density
operator is thermal even if the global state of the system is pure
PhysRevLett.56.1885 ; Dowker:1994fi .

Now, what modes or missing information the BH entropy refers to generally
remains a mystery. Nevertheless, in the presence of a horizon, in principle
one should associate to each quantum field an “entanglement entropy” that
necessarily results from tracing out the interior modes of the field, given
that these modes are necessarily correlated with the exterior ones. In the
continuum, this entanglement entropy turns out to be infinite, at least when
calculated for a free field on a fixed background spacetime. However, if one
imposes a short distance cutoff on the field degrees of freedom, one obtains
instead a finite entropy; and if the cutoff is chosen around the Planck length
then this entropy has the same order of magnitude as that of the horizon
Bombelli:1986rw ; Srednicki:1993im . Based on this appealing result, there
have been many speculations attributing the black hole entropy to the sum of
all the entanglement entropies of the fields in nature Bekenstein:1994bc .
Whether or not the entanglement of quantum fields furnishes all of the entropy
or part of it, contributions of this type must be present, and any consistent
theory must provide for them in its thermodynamic accounting.

It is not, of course, the aim of this introduction to give an account of the
developments in different directions that have surrounded the entanglement
entropy in connection with black holes, and reader is referred for instance to
Nishioka:2018khk  and references therein. However; there is a growing
consensus that entanglement entropy, and in general quantum entanglement and
holography, will play a central role in revealing a finer structure of
spacetime and possibly leading to a radical revision of our perception of the
universe.

At present, and without having at hand a viable and more fundamental theory of
spacetime, it is hard to expect a resolution of the problem of the divergence
of entanglement entropy, which is very likely deeply linked to other issues of
BH thermodynamics. Nevertheless, the finiteness of the BH entropy on one hand,
the behavior of the entanglement entropy in the continuum picture, on the
other hand, seem to point directly towards an underlying discrete structure of
spacetime. The situation actually appears to be similar to that of an ordinary
box of gas, where we know that, fundamentally, the finiteness of the entropy
rests on the finiteness of the number of molecules, and to lesser extent on
the discreteness of their quantum states. Indeed, at temperatures high enough
to avoid quantum degeneracy, the entropy is, up to a logarithmic factor,
merely the number of molecules composing the gas. The similarity with the BH
becomes evident when we remember that the picture of the horizon as composed
of discrete constituents gives a good account of the entropy if we suppose
that each such constituent occupies roughly one unit of Planck area and
carries roughly one bit of entropy Sorkin:1997ja .

A proper statistical derivation along these lines would require a knowledge of
the dynamics of these constituents, of course. However, in analogy with the
gas, one may still anticipate that the horizon entropy can be estimated by
counting suitable discrete structures, analogs of the gas molecules, without
referring directly to their dynamics. Clearly, this type of estimation can
succeed only if well defined discrete entities can be identified which are
available to be counted. Within a continuum theory, it is hard to think of
such entities. However, in causal set theory PhysRevLett.59.521 , the elements
of the causal set serve as “spacetime atoms”, and one can ask whether these
elements, or some related structures, are suited to play the role of “horizon
molecules”.

The idea of considering a certain causal set structure as a potential
candidate for the horizon molecules was first taken up in $1999$ using causal
links. This proposal was partially successful and gave promising results in 2
-dimensions. It was subsequently followed by other proposals to refine it or
look for more suitable definitions for the horizon molecules that would work
in higher spacetime dimensions .

In this review, we go through the different horizon molecules proposals that
emerged in the last two decades or so within the causal set approach to
quantum gravity. The different proposals will be presented according to their
chronological appearance in literature. We therefore shall first focus on the
causal links proposal that appeared in Dou:1999fw ; Dou:2003af , which
historically was the first proposal and so far seems to be the simplest one,
and in spite of the fact that it has turned out to be unsuccessful beyond
2-dimensions, this proposal remains pedagogically useful and conceptually
stimulating . As a consequence of the failure of the links proposal in higher
dimensions, other horizon molecules proposals were put forward in subsequent
and recent years aiming to succeed where the first proposal failed Sarah ;
Barton:2019okw ; Machet:2020uml . These subsequent and recent proposals will
then be reviewed, their main results will be reported and discussed.

This review is not intended to be a full comprehensive survey on this subject,
however, we hope that the material presented herein will offer the beginner
researcher in the subject, or the interested theoretical physicist in general,
an accessible introduction to the subject, enough background, tools and
concepts that enable him or her to understand the above-mentioned efforts and
developments to identify the horizon molecules in causal set theory, and
direct the reader to the still open issues.

##  2 Background and Terminology

In this section we give the essential mathematical definitions and terminology
related to the causal set picture of spacetime. We shall limit ourselves to
the necessary background relevant to this review. For more comprehensive and
extensive introduction to causal set hypothesis we refer the reader to Meyer ;
Luca , for a recent and broad review with a fuller set of references see
Surya:2019ndm .

Definition 1 A causal set (or a causet for short) $\mathcal{C}$ is a set
endowed with an order relation $\prec$ satisfying the following axioms:

  1. 1.

Acyclic (antisymmetric): $\forall p,q\in\mathcal{C},p\prec
q\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ q\prec p\Rightarrow p=q$ ,

  2. 2.

Transitive:$\forall p,q,r\in\mathcal{C},p\prec q\prec r\Rightarrow p\prec r$ ,

  3. 3.

Reflexive : $\forall p\in\mathcal{C},p\prec p$ ,

  4. 4.

Locally finite: $\forall p,q\in\mathcal{C},|I[p,q]|<\infty$, where
$I[p,q]=\text{Fut}(p)\cap\text{Past}(q)$, $|.|$ stands for the cardinality of
the set, Fut and Past denote the future and the past of a given point,

$\text{Fut}(p)=\\{q\in\mathcal{C}|p\prec q,q\neq p\\}$

$\text{Past}(p)=\\{q\in\mathcal{C}|q\prec p,q\neq p\\}\ .$

Notice here that the reflexivity axiom is a matter of convention and we could
instead have used the irreflexive convention.

Fut$(p)$ and Past$(p)$ are to be compared with the notion of chronological
future and past, $I^{+}(p)$ and $I^{-}(p)$, in continuum Lorentzian geometry.
$I[p,q]$ is referred to as the causal or order interval, the analogue of
Alexandrov interval in the continuum.

The discreteness of the causal set is encoded in the local finiteness axiom.

The acyclicity axiom ensures that causets do not have closed causal loops. An
important concept for the description of causets and that we shall frequently
need is the _Link_.

Definition 2: Let $p$ and $q$ $\in\mathcal{C}$, $p\prec q$, $q\neq p$ . If
$|I[p,q]|=0$, we say there is link between $p$ and $q$ and write
$p\prec\\!\\!\cdot\,q$.

The knowledge of all links is equivalent to knowledge of all relations among
elements: $p\prec q$ iff there are elements $q_{1},q_{2},,......q_{n}$ such
that $p\prec\\!\\!\cdot\,q_{1}\prec\\!\\!\cdot\,q_{2}\prec\\!\\!\cdot,.....$
$\prec\\!\\!\cdot\,q_{n}\prec\\!\\!\cdot\,q$. Therefore links are irreducible
relations and in some sense are the building blocks of the causet.

Definition 3: Let $\mathcal{C}^{\prime}\subset\mathcal{C}$,
$p\in\mathcal{C}^{\prime}$ is said to be _maximal_ (resp._minimal_) in
$\mathcal{C}^{\prime}$ iff it is in the past (resp. future) of no other
element in $\mathcal{C}^{\prime}$.

An extended notion of maximality and minimality condition that will later be
needed is the notion of maximal and minimal-but-$n$.

Definition 4: Let $\mathcal{C}^{\prime}\subset\mathcal{C}$,
$p\in\mathcal{C}^{\prime}$ is said to be _maximal-but- $n$_ (resp._minimal-
but- $n$_) in $\mathcal{C}^{\prime}$ iff it is in the past (resp. future) of
exactly $n$ elements in $\mathcal{C}^{\prime}$.

The basic hypothesis of the causal set approach to quantum gravity is that
_“spacetime, ultimately, is discrete and its underlying structure is that of a
locally finite, partial ordered set which continues to make sense even when
the standard geometrical picture ceases to do so”_. The macroscopic spacetime
continuum we experience must be recovered as an approximation to the causet.
The causal set proposal can roughly be summarized in the following two points

  1. 1.

Quantum Gravity is a quantum theory of causal sets.

  2. 2.

A continuum spacetime $(\mathcal{M},g)$ is an approximation of an underlying
causal set $C\sim(\mathcal{M},g)$, where

(a) Order $\sim$ Causal Order

(b) Number $\sim$ Spacetime Volume

Point or step (2) is not to be viewed as independent of step (1). Actually the
quantum theory of causal set should dictate how the continuum picture emerge
as an approximation, and this could ultimately involve a more sophisticated
notion of approximation. For instance, in view of the fact that not all
causets admit a realization as spacetimes with a given dimension while
respecting conditions (2a) and (2b), the process by which the continuum 4-d
spacetime picture, or that of higher dimensional spacetimes with compactified
extra-dimensions, is reached may involve some sort of coarse-graining in which
the manifold picture would be a scale dependent approximation of the causal
set. However, in the absence of a quantum dynamics of causet, a systematic way
of defining a coarse-graining that would fit automatically our expectations is
yet to be discovered. Nevertheless, we may use point (2) as a stepping stone
(given) to investigate possible kinematical consequences of the causet
approach. In short and without expanding too much around this point, the
intuitive idea at work here is that of a _faithful embedding_ which we define
below.

Definition 5 If $(\mathcal{M},g)$ is a $d$-dimensional Lorentzian manifold and
$\mathcal{C}$ a causet, then a faithful embedding of $\mathcal{C}$ into
$\mathcal{M}$ is an injection map $f:\mathcal{C}\hookrightarrow\mathcal{M}$ of
the causet into the manifold that satisfies the following requirements:

  1. 1.

The causal relations induced by the embedding agree with those of
$\mathcal{C}$ itself,

i.e. $x\prec y\Leftrightarrow f(x)\in J^{-}(f(y))$

where $J^{-}(p)$ stands for the causal past of $p$ in $\mathcal{M}$;

  2. 2.

The embedded points are distributed uniformly at density
$\varrho_{c}=l_{c}^{-d}$ with respect to the spacetime volume measure of
$(\mathcal{M},g)$.

  3. 3.

The characteristic length over which the geometry varies appreciably is
everywhere much greater than the mean spacing between the embedded points.

$l_{c}$ is referred to as the discreteness scale.

When these conditions are satisfied, the spacetime $(\mathcal{M},g)$ is said
to be a continuum approximation to $\mathcal{C}$ and we write
$\mathcal{C}\sim(\mathcal{M},g).$

To ensure covariance the above embedding is realized by randomly sprinkling in
points until the required density is reached. Therefore from the point of view
of $\mathcal{M}$ the causet resembles a “random lattice”, e.g “a regular”
lattice cannot do the job since it is not uniform in all frames or coordinate
systems.

A natural choice for obtaining or creating a faithfully embedded causet is via
a Poisson point process; under which the probability to find $n$ elements in a
spacetime region of volume $V$ is given by

$(\varrho_{c}V)^{n}\frac{e^{-\varrho_{c}V}}{n!}\ .$ (1)

This makes $f(\mathcal{C})$ a random causet and thereby any function
$F:\mathcal{C}\rightarrow\mathbb{R}$ is a random variable.

For more detailed discussion of the issue of faithful embedding and the
probabilistic nature of the process we refer the reader to Surya:2019ndm  and
references therein.

##  3 Horizons molecules as causal links

As discussed in the introduction, the expectation is that the BH entropy can
be understood as entanglement in a sufficiently generalized sense, and we may
hope to estimate its leading behavior by counting suitable discrete structures
that measure the potential entanglement in some way between in-outside
discrete structures. Moreover, and owing to the fact that the entropy
essentially measures the horizon area in Planck units, the problem is reduced
to coming up with this measure in the causal set picture.

It is worthy of note here that it seems far from obvious that such structures
must exist. If they do, then they provide a relatively simple order theoretic
measure of the area of a cross section of a null surface, and, unlike what
one’s Euclidean intuition might suggest, it is known that such measures are
not easy to come by. For example, no one knows such a measure of spacelike
distance between two sprinkled points that works in general, though some
progress has been made in such Minkowski spacetime Rideout_2009 .

It follows from the above discussion that a natural and the simplest candidate
for the structure we seek is a _link_ crossing the horizon. Indeed, we may
think heuristically of “information flowing along links” and producing
entanglement when it flows across the horizon during the course of the
causet’s growth (or “time development”). Since links are irreducible causal
relations (in some sense the building blocks of the causet), it seems natural
that by counting links between elements that lie outside the horizon and
elements that lie inside, one would measure the degree of entanglement between
the two regions. Equally, it seems natural that the number of such causal
links if supplemented with extra conditions might turn out to be proportional
to the horizon area and play the role of the Horizon molecules.

In what follows we discuss with some detail the links proposal for horizon
molecules and its applications in different 1+1 geometrical setups.

###  3.1 The general Setup

Let us consider a causet $\mathcal{C}$ obtained via Poisson random sprinkling
in a black hole background $\mathcal{M}$ with density $\varrho_{c}$, so this
causal set is faithfully embeddable in this geometrical background by
definition. Let $\mathcal{H}$ be a BH horizon and let $\Sigma$ be an achronal
hypersurface intersecting the horizon, Figure 1.

Figure 1: A typical geometrical setting showing a typical causal link crossing
the horizon.

The goal is to come up with a measure of the area of the resulting cross
section between $\mathcal{H}$ and $\Sigma$, which in turn would measure the
horizon entropy and define _Horizon molecules_.

A natural and intuitive candidate for such molecules is to take them made of
pairs of points $(p,q)$, with $p$ lying outside the black hole and to the past
of $\Sigma$, while $q$ is inside the black hole and to the future of $\Sigma$,
and $p\prec\\!\\!\cdot\ q$, i.e. $p\prec q$ is a link .

If no further conditions are imposed on $p$ and $q$, the expected number of
such links can easily be shown to diverge.

To see what conditions must be imposed on the pairs $(p,q)$, let us remember
that, intuitively, what we are trying to estimate is not the total sum of all
“lost information” but only that corresponding “to a given time”, meaning in
the vicinity of the given hypersurface $\Sigma$. Hence, to associate the same
causal link with more than one hypersurface would be to “ overcount” it in
forming our estimate, and it is this overcounting that seems to be the source
of the above mentioned divergence. Therefore further conditions are needed to
be imposed to give a definition of the horizon molecules which is truly proper
to $\Sigma$ rather to some earlier or later hypersurface. Several
possibilities suggest themselves for this purpose, but none seems to be
clearly best, as the end result (the leading order) will be shown to be
insensitive to which choice one makes. Below we pick up a specific choice or
definition of horizon molecules, which will be referred to as the “ _causal
links proposal_ ”, and the general issue will be discussed further in
subsection 3.5 . Actually working out explicitly with this particular choice,
seeing its success in 1+1 and failure in higher dimensions, due to IR
divergence, will be instructive for the reader to conceive the motivations
behind the re-definitions of the horizon molecules that subsequently departed
from the original links proposal.

The causal links proposal (Dou-Sorkin 1999): A horizon molecule with respect
to a given hypersurface $\Sigma$ is a pair $(p,q)$ satisfying the following
conditions

  1. 1.

$p\in I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$

  2. 2.

$q\in I^{+}(\Sigma)\cap I^{+}(\mathcal{H})$

  3. 3.

$|I[p,q]|=0$, i.e $p\prec q$ is a link.

  4. 4.

$p$ is maximal in $I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$ and $q$ is minimal in
$I^{+}(\mathcal{H})$ .

The $4^{th}$ condition may seem asymmetric, as one would have expected
symmetric Max and Min conditions between $p$ and $q$ to be more natural,
however, the reason that we do not impose a similar condition on $q$ is
because this would give zero for a null hypersurface case, but the result
should agree for null or spacelike if both intersect the horizon in the same
time, moreover for stationary black the results should agree in all cases.

Before we move on, we draw the reader’s attention that throughout this section
and the next one $p$ will stand for points in $I^{-}(\Sigma)\cap
I^{-}(\mathcal{H})$ and $q$ for the ones in $I^{+}(\Sigma)\cap
I^{+}(\mathcal{H})$.

Let us now see how to count the expected number of these horizon molecules by
reducing it to the calculation of an integral over the manifold $\mathcal{M}$.

Remember that the probability of finding or sprinkling $n$ points in some
region of spacetime, $\mathcal{R}$, is given by the Poisson distribution

$P(n,\mathcal{R})=\frac{(\varrho_{c}\text{vol}(\mathcal{R}))^{n}}{n!}e^{-\varrho_{c}\text{vol}(\mathcal{R})}\
,$

where $\text{vol}(\mathcal{R})$ is the spacetime volume of $\mathcal{R}$.

Consider first an infinitesimal region $\Delta\mathcal{R}$, the probability of
sprinkling a single point in it is follows from

$P(1,\Delta\mathcal{R})\approx\varrho_{c}\text{vol}\Delta\mathcal{R}\equiv\varrho_{c}\Delta
V\ .$ (2)

Consider now two infinitesimal regions $\Delta\mathcal{R}_{p}\in
I^{+}(\Sigma)\cap I^{+}(\mathcal{H})$ and $\Delta\mathcal{R}_{q}\in
I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$. The probablity of having a pair of
points $(p,q)$ with $p\in I^{+}(\Sigma)\cap I^{+}(\mathcal{H})$ and $q\in
I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$ sprinkled in $\Delta\mathcal{R}_{p}$ and
$\Delta\mathcal{R}_{q}$ resp. is given by

$P(p,q|\Delta\mathcal{R}_{p},\Delta\mathcal{R}_{q})=\varrho_{c}\Delta
V_{p}\varrho_{c}\Delta V_{q}\ .$ (3)

If we further require the relation between $p$ and $q$ to be a link then the
Alexandrov interval $A(p,q)$ between $p$ and $q$ must contain no point and
therefore the probability becomes

$P(p\prec\\!\\!\cdot\,q|\Delta\mathcal{R}_{p},\Delta\mathcal{R}_{q})=P(0,\text{vol}(A(p,q))\varrho_{c}^{2}\Delta
V_{p}\Delta V_{q}=\varrho_{c}^{2}e^{-\varrho_{c}\text{vol}(A(p,q))}\Delta
V_{p}\Delta V_{q}\ .$ (4)

In addition to the link condition Max and Min conditions must be imposed on
$p$ and $q$. The Max and Min conditions are just statements about an extra
region in $\mathcal{M}$ being empty, with no sprinkled points. If we denote by
$\mathcal{R}(p,q)$ the region resulting from the union of $A(p,q)$,
$I^{+}(p)\cap I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$ and $I^{-}(q)\cap
I^{-}(\mathcal{H})$ the probability for the above link to become a horizon
molecule reduces to

$P(\mathbf{H}(p,q);\Delta\mathcal{R}_{p},\Delta\mathcal{R}_{q})=\varrho_{c}^{2}e^{-\varrho_{c}V(p,q)}\Delta
V_{p}\Delta V_{q}\ ,$ (5)

where $V(p,q)=\text{vol}(\mathcal{R}(p,q))$.

To count the expected number of horizon molecules we remember that the
existence of horizon molecule is a random variable generated by a function
whose value is $1$ if the horizon molecule conditions are fulfilled and $0$
otherwise. With this in mind, it follows that expected number of horizon
molecules is obtained by summing in (5) over all $p\in I^{-}(\Sigma)\cap
I^{-}(\mathcal{H})$ and $q\in I^{+}(\Sigma)\cap I^{+}(\mathcal{H})$ in the
limit $\Delta V_{p}$ and $\Delta V_{q}$ go to zero. In this limit the sums are
replaced by integrals over the domain of $p$ and $q$ to obtain the following
final expression for the expected number of horizon molecules

$<\mathbf{H}_{link}>=\varrho_{c}^{2}\int_{I^{-}(\Sigma)\cap
I^{-}(\mathcal{H})}dV_{p}\int_{I^{+}(\Sigma)\cap
I^{+}(\mathcal{H})}dV_{q}\leavevmode\nobreak\ e^{-\varrho_{c}V(p,q)}\ .$ (6)

For a more systematic derivation of the above integral formula see Sarah  .

For horizon molecules as such to be successful, one has to show that in the
limit of large density, or $l_{c}$ is much smaller than the geometrical length
scales of the setting, $<\mathbf{H}_{link}>$ has the asymptotic form

$\varrho_{c}^{\frac{d-2}{d}}<\mathbf{H}_{link}>=a^{(d)}\int_{\mathcal{J}}dV_{\mathcal{J}}+\cdots\
,$ (7)

where the dots refer to terms vanishing in the continuum limit.
$\mathcal{J}:=\Sigma\cap\mathcal{H}$ and $dV_{\mathcal{J}}$ is the surface
measure on $\mathcal{J}$. $a^{(d)}$ is constant that depends on the dimension
of the spacetime but, in principle, not on the nature of $\Sigma$, null or
spacelike. In two dimensions the leading term in $<\mathbf{H}_{link}>$ should
be just a constant.

###  3.2 Horizon molecules and the area law in 2-dimensions

Ideally one would have used (6) to the evaluate the expected number of horizon
molecules, $<\mathbf{H}_{links}>$, in a full four dimensional BH background,
e.g Schwarschild BH, however, historically and for technical reasons
(difficulties) a simplified version was first worked out. This consisted in
considering a “ dimensionally reduced” two dimensional metric instead of the
true four dimensional one. The hope was twofold; it would first be a warm up
exercise for a more realistic four dimensional BH; second the establishment of
the area law in 2-d models would give strong evidence for the validity of this
proposal in the full four dimensional case. Stated differently, the four-
dimensional answer would differ from the two-dimensional one only by a fixed
proportionality coefficient of order one, together with a factor of the
horizon area.

Now, although the above defined horizon molecules proposal did not work beyond
$2$-d, in contrast to what had first been hoped, due to IR divergences, the
establishment of the area law in 2-d using the above defined horizon molecules
makes the calculation worth discussing . Beside this obvious reason, it will
be seen that in $1+1$ the resulting expected number of links seems to exhibit
some interesting features: a sort of universality, giving exactly the same
answer for two different geometrical backgrounds, in equilibrium and far from
equilibrium, and remaining finite in the strict continuum limit,
$\varrho_{c}\rightarrow\infty$.

In the sequel two cases will explicitly be worked out, a 2-d reduced
Schwarschild geometry and collapsing null shell. We shall set $\varrho_{c}=1$
in all 2-d models discussed in this section; because the leading term is a
dimensionless constant and the subleading ones are easy to express and control
in these units.

###  3.3 An equilibrium black hole: 2-d reduced model

Consider a dimensionally reduced Schwarzschild spacetime obtained from the
realistic $4$-dimensional BH spacetime, outside a collapsing spherically
symmetric star, by identifying each $2$-sphere $S^{2}$ to a point. The
resulting two dimensional spacetime has exactly the same causal structure as
the S-sector of the 4-dimensional one. The Penrose diagram for this spacetime
is depicted in Figure 2 . For simplicity the presence of the collapse has been
ignored; this of course will not change the argument, since the detail of the
collapse should be irrelevant, or one can choose the hypersurface to intersect
the horizon far from the collapse and the result will not be affected by the
presence of collapse.

The line element of the resulting spacetime is obtained by omitting the
angular coordinates from the four dimensional line element, namely

$d^{2}s=-\frac{4a^{3}}{r}e^{-r/a}dudv\ ,$ (8)

where $a=2M$ is the radius of the BH and $u$ and $v$ are the usual Kruskal-
Szekeres coordinates, with $r$ defined implicitly by the equation

$uv=(1-\frac{r}{a})e^{r/a}\ .$ (9)

The associated volume element is

$dV=\sqrt{-g}dudv=\frac{2a^{3}}{r}e^{-r/a}dudv\ .$ (10)

Our signs convention is such that $u\sim t-r$, $v\sim t+r$, and the horizon
$\mathcal{H}$ coincides with $u=0$.

Figure 2: An equilibrium BH obtained from real the 4-d Shwarszchild BH by
dimensional reduction, keeping only the radial section. The shaded region is
required to be free from any sprinkled points and with volume $V(p,q)$.

Let now $\Sigma$ be an ingoing null hypersurface defined by the equation
$v=v_{0}$ . The shaded region depicted in Figure 2 is the region
$\mathcal{R}(p,q)$ with no sprinkled point, its $V(p,q)$ volume can readily be
evaluated using (9)

$V=a^{2}+r_{pq}^{2}-r_{pp}^{2}-r_{qq}^{2}\ ,$ (11)

where we have introduced the following notation

$u_{i}v_{j}=\left(1-\frac{r_{ij}}{a}\right)e^{r_{ij}/a}\ .$ (12)

Let us note that in two dimension and for a null $\Sigma$ the maximality
condition on $p$ is actually redundant and insured by the link condition, but
it would be needed with spaclike $\Sigma$.

Using (6) and (11), the expected number of horzion molecules is given by

$<\mathbf{H}_{link}>=(2a^{3})^{2}\int_{0}^{v_{0}}dv_{p}\int_{-\infty}^{0}du_{p}\int_{v_{0}}^{\infty}dv_{q}\int_{0}^{1/v_{q}}du_{q}\frac{e^{-r_{pp}/a-r_{qq}/a}}{r_{pp}r_{qq}}\;e^{-V}\
.$ (13)

%ِِA change of integration variables

A change of integration variables from $(u_{p},v_{p},u_{q},v_{q})$ to
$(r_{pp},r(u_{p},v_{0})\equiv r_{p0},r_{pq},r_{qq})$, followed by the
notational substitutions $x=r_{pq}$, $y=r_{p0}$, $z=r_{pp}$, reduces
$<\mathbf{H}_{link}>$ to the form,

$<\mathbf{H}_{link}>=4\,I(a)\,J(a)\ ,$

where

$I(a)=\int_{a}^{\infty}dx\frac{x}{x-a}e^{-x^{2}}\int_{a}^{x}dy\frac{y}{y-a}\int_{a}^{y}e^{z^{2}}dz\
,$ (14)

and

$J(a)=e^{-a^{2}}\int_{0}^{a}e^{r_{qq}^{2}}dr_{qq}\ .$ (15)

It is worth noting here that the initial explicit dependence of
$<\mathbf{H}_{link}>$ on $v_{0}$ has disappeared, reflecting the stationarity
of the black hole.

Now, inasmuch as comparison with the Bekenstein-Hawking entropy is meaningful
only for macroscopic black holes, it is natural to assume that $a\gg 1$, and
under this condition, $I(a)$ can be shown to have the following asymptotic
behavior Dou:1999fw :

$I(a)={\pi^{2}\over 12}\;a+\mathcal{O}\left({1\over a}\right)\ .$

On the other hand it is not difficult to see that

$J(a)=\frac{1}{2a}+\mathcal{O}\left({1\over a^{3}}\right)\ .$

Putting everything together, we end up with

$<\mathbf{H}_{link}>=\frac{\pi^{2}}{6}+\mathcal{O}\left(\frac{1}{a^{2}}\right)\
.$ (16)

As the intersection of $\Sigma$ and $\mathcal{H}$ in two dimension is just a
point, the area law, if finite, should naturally turn out to be a pure number,
therefore (16), or the expected number of horizon molecules, is proportional
to the area of the horizon in $1+1$.

Some remarks about the above derivation of the area law in 2-d using this
horizon molecules proposal are in order.

The first remark concerns the locations of the pairs forming the molecules
that give the dominant contribution to $<\mathbf{H}_{link}>$. It is easy to
see that the dominant contribution to the integral $J(a)$ plainly comes from
$r_{qq}\approx{a}$, but since $r_{qq}$ is the radial coordinate $r$ of
sprinkled point $q$, and since $r=a$ is the horizon, this implies that $q$
resides near the horizon. Similarly, an inspection of the integral $I(a)$
shows that the dominant contribution to the integral $I(a)$ comes as well from
$z\approx y\approx{a}$, which, since $z=r_{pp}$ and $y=r_{p0}$, implies in
turn that sprinkled point $p$ resides near the horizon as well Dou:1999fw .
Consequently this counting can be said to be controlled by the near horizon
geometry.

It should be noted too that from the unboundedness of the region
$I^{+}(\Sigma)\cap I^{+}(\mathcal{H})$ and the finitness of
$<\mathbf{H}_{link}>$, we can infer that points $q$ sitting arbitrarily close
to the horizon but far from the $\Sigma\cap\mathcal{H}$ cannot continue to
contribute indefinitely to $<\mathbf{H}_{link}>$. Moreover, the fact that
$<\mathbf{H}_{link}>$ turns out to be just a pure number strongly suggests
that the pairs which give the dominant contribution are not only residing near
the horizon but are hovering near $\Sigma\cap\mathcal{H}$ too .

It is interesting to look at this result and its features from another point
of view. If we inspect the integral $I(a)$ we note that what makes the near
horizon molecules special is the vanishing of the denominators in $I(a)$ when
the dummy integration variables $x$ and $y$ tend to $a$. To the extent that it
is this divergence which makes the horizon such a strong source for the links,
and here we may be reminded of the analogous fact that the strong redshift in
the vicinity of the horizon allows modes of arbitrarily high (local) frequency
to contribute to the entanglement entropy without influencing the energy as
seen from infinity. Notice also that the clustering of $p$ and $q$ near the
horizon is not simply a consequence of the maximality and minimality
conditions we imposed on them. For instance, pairs $(p,q)$ sitting arbitrarily
close to the hypersurface $\Sigma$, with $q$ arbitrarily close to the horizon,
still do not contribute to the leading term in $I(a)$ if $q$ is far from the
horizon, namely with coordinate $\left|u_{p}\right|\gg 1$.

###  3.4 A black hole far from equilibrium: 2-reduced collapsing null matter

We now turn to another case which, though still spherically symmetric, is very
far from equilibrium, namely that of a spherically collapsing null shell of
matter with stress energy tensor given by

$T_{vv}=\frac{M\delta(b-v)}{4\pi r^{2}}\ ,$

and the other components are identically zero.

The collapsing shell forms a Schwarzschild BH. The Penrose diagram for the
resulting spacetime (after dimensional reduction $S^{2}\rightarrow$ point) is
shown in Figure 3. Let the shell sweep out the world sheet $v=b$ and let us
choose for our hypersurface $\Sigma$ a second ingoing null surface defined by
$v=a$, with $a<b$ so that $\Sigma$ lies wholly in the flat region. Here $a$ is
of course generally different from $a$ defined in the Schwarzschild case, $u$
and $v$ are null coordinates, chosen so that the horizon first forms at
$u=v=0$ and normalized for convenience such that the line element in the flat
region is given by

$ds^{2}=-2dudv+r^{2}d\Omega^{2}\ .$

Since our interest is again in macroscopic black holes, we will assume as
before that the horizon radius at $\Sigma\cap\mathcal{H}$ is large in units
such that $\varrho_{c}=1$, which amounts to $a{\gg}1$; and to simplify matters
further, we will also restrict ourselves to a time well before the infalling
matter arrives (as judged in the center of mass frame). One thus has the
double inequality, $b{\gg}a{\gg}1$. Once again, the calculation will be
performed for the two dimensional radial section rather than the full four
dimensional spacetime.

Figure 3: A non-stationary BH. The region to the past of the world line of the
infalling matter is flat space with an expanding event horizon, whereas the
one to its future is Shwarzschild region. We have depicted an extra null
hypersurface $\Sigma_{0}$ for later reference.

Since we are assuming that the infalling matter is far to the future of the
hypersurface $\Sigma$, points $q$ sprinkled into that region should not
contribute significantly when our minimality and link conditions are taken
into account. For this reason, we shall, for convenience, restrict the
counting to pairs $(p,q)$ with $v_{q}<b$.

Using the definition of the horizon molecules we introduced above, one obtains
for the expected number of horizon molecules

$<\mathbf{H}_{links}>=\int_{a}^{b}dv_{q}\int_{0}^{v_{q}}du_{q}\int_{-\infty}^{0}du_{p}\int_{0}^{a}dv_{p}e^{-V}\
,$ (17)

where $V=u_{q}v_{q}-u_{p}(v_{q}-v_{p})-u_{q}^{2}/2$, the volume of shaded
region in Figure 3.

Note here that the contribution of the points $p$ with $v_{p}<0$, i.e. to the
past of $\Sigma_{0}$, has been ignored; we will return to its justification
below.

The integration over $v_{p}$ and $u_{p}$ is easy to perform, followed by
change of variables, $x=v_{q},y=v_{q}-u_{q}$, we end up with

$<\mathbf{H}_{link}>=\int_{a}^{b}\ln(\frac{x}{x-a})e^{-x^{2}/2}dx\int_{0}^{x}e^{y^{2}/2}dy\
.$ (18)

At this stage it is not difficult to show that the leading behavior of this
integral for large $a$ is given by

$<\mathbf{H}_{link}>={\pi^{2}\over 6}-l\left({a\over
b}\right)+\mathcal{O}(1/a^{2})\ ,$ (19)

where $l(x)\equiv\sum_{k=1}^{\infty}{x}^{k}/k^{2}$, a convergent series that
vanishes in the limit $x\to{}0$.

Originally the correction to the leading term in (19) were set to be of the
order of $1/a$ in Dou:1999fw  and Dou:2003af , but a careful repetition of the
calculation due to Marr showed that the correction is of the order $1/a^{2}$
Sarah .

Since we have assumed that $a{\ll}b$, we can write this more simply as

$<\mathbf{H}_{link}>={\pi^{2}\over 6}+\mathcal{O}(a/b)+\mathcal{O}(1/a^{2})\
.$ (20)

Notice that the presence of a negative contribution like $-l(a/b)$ was to be
expected, since we have omitted to count molecules that extend past the shell
into the Schwarzschild region. For $\Sigma$ near to the shell, one obviously
should not neglect such links, and this counting is incomplete. However, if
the collapse is pushed far away from $\Sigma$, in particular to future
infinity, we can safely restrict the counting to the flat region without
worrying about the presence of the Schawrzschild region and therefore reducing
the problem (even in higher dimension) to a counting in flat background
geometry.

Now, what is striking about the above result is is the occurrence of the same
numerical coefficient ${\pi^{2}}/{6}$ in both (20) and (16). This agreement
seems at first sight to furnish a nontrivial consistency check of the
suggestion that one can attribute the horizon entropy to the horizon molecules
made of “causal links” crossing it.

As mentioned above, in writing (19) we implicitly ignored the contribution of
pairs $(p,q)$ with negative $v_{p}$. No justification for this was given in
Dou:1999fw  nor in Dou:2003af . However, this point was raised and briefly
discussed by Marr in Sarah .

It is easy to write an integral formula for this type of contribution, and
maybe compute it, however, it is not difficult to argue that it should not be
considered as part of the horizon molecules associated with
$\Sigma\cap\mathcal{H}$. This kind of contribution counts the expected number
of horizon molecules associated with a hypersurface $\Sigma_{0}$, Figure 3,
which is not intersecting the horizon, or they occur before the horizon
formation. Therefore they must be taken as sort of random statistical
fluctuations extraneous and not genuine horizon molecules associated to
$\Sigma$. Actually, if we remember that the geometrical setting we are using
is $2$-d reduced of a $4$-dimensional one, this extra contribution would turn
out to be just of order one in genuine $4$-dimensional counting, thus a
negligible fluctuation around the mean value.

###  3.5 On the Min/Max conditions

As we briefly discussed before picking up the particular choice for
the“Max/Min” conditions we adopted in the definition the causal links
proposal, this choice did not seem unique or particularly sacred and other
variants were possible. Of course, one must be careful not to use something
like “$q$ minimal in $I^{+}(\Sigma)$”, which would drive $<\mathbf{H}_{link}>$
to zero in the limit of null $\Sigma$, but this does not rule out, for
example, a condition like “$p$ maximal in $I^{-}(\Sigma)$”.

Let us note that it turns out that there are at least two variants of the
“Max/Min” condition that seem to be equivalent, as long as the leading term is
concerned. These variants are

Variant 1: $p$ is Max in $I^{-}(\Sigma)$ and $q$ is Min in
$I^{+}(\mathcal{H})\cap I^{+}(\Sigma)$ .

Variant 2: $p$ is Max in $I^{-}(\Sigma)\cap I^{-}(\Sigma)$ and $q$ is Min in
$I^{+}(\mathcal{H})\cap I^{+}(\Sigma)$ .

If we consider for instance the first variant, it is easy to show that the
resulting expected number of horizon molecules has the same asymptotic
behavior as the one resulting from the original causal links
proposalDou:1999fw , namely

$<\mathbf{H}_{link}>={\pi^{2}\over 6}+O(1/a^{2})\ .$

Thus, for this variant at least, one obtains exactly the same numerical answer
as the original proposal we started with. As for the second variant, although
it has not been worked out explicitly, we do not expect the slight change in
the volume $V(p,q)$ should alter the leading term.

Figure 4: Two continuations of a hypersurface to the interior region.

Another related feature the links counting must have if it is to yield the
horizon area is that, within reason, the expected number of horizon molecules
should depend only on the intersection $\mathcal{H}\cap\Sigma$, and not on how
the surface $\Sigma$ is prolonged outside or (especially) inside the horizon
$\mathcal{H}$. For example one should get the same answer for both of the
continuations shown in Figure 4. The case where the difference is confined to
the interior black hole region is of particular significance for the
entanglement interpretation of horizon entropy, since such a difference
cannot, by definition, influence the effective density operator for the
external portion of $\Sigma$ (at least to the extent that unitary quantum
field theory is a good guide). For instance, we note that the volume $V(p,q)$
needed to insure $p$ maximal in $I^{-}(\mathcal{H})\cap I^{-}(\Sigma)$ and $q$
be minimal in $I^{+}(\mathcal{H})$ is the same for both
$\Sigma_{ext}\cup\sigma_{1}$ and $\Sigma_{ext}\cup\sigma_{2}$, therefore from
this perspective the definition we have so far adopted seems to have advantage
over the other two variants, at least in the case of null $\Sigma$. However,
in view of the fact that the leading order is controlled by contributions
coming from links residing near the horizon, we expect the different variants
to have the same leading behaviors no matter how $\Sigma$ is prolonged inside
or outside the horizon. Indeed, the issue of which Max/Min condition is
favored cannot be settled nor properly discussed unless we settle the central
issue of how to define the horizon molecules in a way that works in higher
dimensions and for both types of hypersurfaces, null and spacelike.

###  3.6 The spacelike hypersurface in 2-d reduced

So far the counting of the horizon molecules has been restricted to null
hypersurfaces in 2-d reduced black hole geometries. However, no proposal can
be considered fully successful, even in two dimensions, unless it correctly
reproduces the same result for both null and spacelike hypersurfaces.

In this section, we look at this issue by discussing the previous links
counting when $\Sigma$ is a spacelike hypersurface crossing the horizon under
the same Max/Min conditions.

It is first intriguing to discuss one of the heuristic arguments that is
sometimes invoked in this context to conclude that the null and spacelike
counting should be expected to yield the same result.

This argument generally goes as follows, Dou:2003af ; Machet:2020uml .

Consider a one-parameter family of spacelike hypersurfaces $\Sigma_{t}$ which
continuously deform to a null hypersurface,
$\Sigma=\lim_{t\rightarrow\infty}\Sigma_{t}$. On other hand, the region one
sprinkles into and so the probability measure is also continuous with respect
to the deformations. Now, because all spacelike hypersurfaces give the same
result then the null hypersurface $\Sigma$ which can be casted as a limit of a
sequence of spacelike hypersurface should also give the same result. In the
flat case one can equally use Lorentz invariance of the counting and the fact
any spacelike hypersurface must give the same result, as any other related to
it by a boost, and in the limit of tilting, a spacelike line becomes null.
Note that a similar argument can also be made in the Schwarzschild case, using
time-translation Killing vector instead of the boost killing vector.

Stated mathematically, one would expect the following limit to hold

$\lim_{t\rightarrow\infty}<\mathbf{H}(\Sigma_{t})>=<\mathbf{H}(\Sigma)>\ .$
(21)

In the above equation we do not of course require strict equality, but it
would be enough to hold in the limit $l_{c}\rightarrow 0$ modulo some
statistical deviations from the leading mean value. In Machet:2020uml , it was
for instance argued that it is the non-commutativity of the two limits,
$t\rightarrow\infty$ and $l_{c}\rightarrow 0$, which causes the above identity
to fail, namely

$\lim_{l_{c}\rightarrow
0}\lim_{t\rightarrow\infty}l_{c}^{d-2}<\mathbf{H}(\Sigma_{t})>\neq\lim_{t\rightarrow\infty}\lim_{l_{c}\rightarrow
0}l_{c}^{d-2}<\mathbf{H}(\Sigma_{t})>\ .$ (22)

However, we shall see in the fourth section that in some horizon molecules
counting the limit $l_{c}\rightarrow 0$ is not at all required for the
derivation of the area law, nonetheless the null and spacelike hypersurfaces
give two different results. Therefore, we conclude that the above heuristic
argument invoking the non-ommutativity of the two limits is at best not
generally sustainable, as some counting could be inherently discontinuous and
depend on the nature of the hypersurface crossing the horizon.

Figure 5: Different arrangement contributing in the spacelike case, accorrding
to different volume epxressions. For the arrangement type-A1 the point $q$
does not necessary lie to the future of $\Sigma^{\prime}$, it could be to the
past of it as well .

Let us now consider a spacelike $\Sigma$ given by $t=\frac{a}{2}$. To
facilitate the discussion it is convenient to introduce a null hypersurface
$\Sigma^{\prime}$ defined by $v=a$. Again, we shall push the collapse to
future infinity and restrict the counting to the flat region, Figure 5.

It is not difficult to see that one has to distinguish five cases, each having
a different expressions for the volume $V(p,q)$. These cases are depicted in
Figure 5.

The contributions $A1,A2,A3$ can be easily seen to be qualitatively of the
same order of magnitude as the null contribution we already evaluated, thus
they will just give constants of order one, but surely each is less then
$\frac{\pi^{2}}{6}$.

Contributions of type $(B)$ and $(C)$ are different and one can not directly
conclude that are finite or of the order one. For instance contributions from
pairs $(p,q)$ with $u_{q}\rightarrow 0$ and $t_{p}\rightarrow a/2$ could lead
to IR divergences. However, it was explicitly shown in Dou:1999fw  that both
contributions are finite and of order one.

Now, although it was possible to show that the expected number of such horizon
molecules give a constant of order one, the question whether the spacelike and
null cases yield the same result has so far remained open due to the
analytical intractability of the integrals involved . But what should be noted
in this context is that the difficulty to settle this issue in two dimensions
may not solely be of technical nature, due to the intractability of the
integrals, but there could be an other issue of conceptual nature at work
here. For example in the non-equilibrium case and for a null hypersurface
counting, we argued that contributions coming from the pairs $(p,q)$ with
$v_{p}<0$ are to be viewed as random fluctuations around the mean value not
genuinely associated with $\Sigma\cap\mathcal{H}$ not to be included in the
counting, despite the fact that they are not zero, but in higher dimension
they are easily seen to be negligible for a macroscopic horizon. For the same
reasons we expect similar fluctuations to be present in the above different
contributions; in particular we do not expect arrangement of type-$(B)$ and
$(C)$ to be fully associated with $\Sigma\cap\mathcal{H}$. However, as it will
be shown in the next section, the present links counting fails to work beyond
two dimensions due to IR divergences, and therefore pursuing this issue would
be no more than mathematical curiosity without physical guidance nor
relevance.

###  3.7 The failure of the causal links proposal in higher dimensions

In view of the success and promising results of the causal links counting in
two dimensional models, the natural step would of course be to try to apply it
on a more realistic four dimensional black hole background. Any direct attempt
to do this counting in Schwarzschild geometry will inevitably be encountered
by mathematical complications that are almost impossible to surpass. However,
the results obtained previously in two dimensions enable us to transform the
whole problem into calculation in 4- dimensional (or $d$-dimensional ) flat
spacetimes using the collapsing null-shell model by pushing the collapse
world-line to future infinity. Therefore, one is in principle entitled to
consider a flat spacetime with the future light-cone of the origin being the
horizon, take a hypersuface $\Sigma$ (spacelike or null) intersecting the
horizon and compute the expected number of horizon molecules, Figure 6 .

Although working in flat spacetimes drastically simplify the problem, in $d>2$
the calculation of $<\mathbf{H}_{link}>$ is still complicated enough and a
much more elaborated technique is needed to do the counting explicitly, for
both the null and spacelike hypersurfaces. The calculation of the volumes
needed to insure the link and Max/Min conditions is lengthy; and it turns out
that one has to distinguish many cases depending on the relative positions of
the points $p$ and $q$, each case making its own contribution to
$<\mathbf{H}_{link}>$ Dou:1999fw . For instance, for spacelike $\Sigma$ and to
the exception of one contribution, coming from an arrangement similar to type
A.1 in two dimensions, Figure 5, which could be evaluated and was reported in
Dou:1999fw ; Dou:2003af ; the remaining arrangements turned out to be either
very complicated or intractable. Nonetheless, at least for one non-trivial
arrangement the volume was computed exactly in Dou:1999fw . The arrangement in
question is of the type-B , depicted in Figure 5 (its four dimensional
analogue) . For this particular arrangement it was later realized by the
author that its corresponding contribution to $<\mathbf{H}_{link}>$ diverges .
It is unnecessary to give the detail of this calculation, but it is not
difficult to qualitatively understand the source of this IR divergence .

Figure 6: The red curve shows the intersection of $\dot{I}^{+}(p)$ with the
horizon, this curve extends to future infinity and by considering sprinkled
points $q$ asymptotically approaching this curve, an arbitrarily number of
$p\prec\\!\\!\cdot\,q$ links can be found.

Let us first take a step back and consider the arrangement Type-B depicted in
Figure 5, in its 1+1 version. The only potential source of IR divergences
comes from the contributions of points $q$ arbitrarily far away from the
intersection point $\Sigma\cap\mathcal{H}$; however the $e^{-V}$ term
appearing in the integrand exponentially suppresses all contributions except
those when $p$ is arbitrarily close to the intersection point and $q$ bound to
the horizon, which is not enough to produce any IR divergences.

The situation in higher dimensions is quite different and this can be grasped
by considering the 2+1 case depicted in Figure 5, and using the qualitative
argument given in Sarah .

In Figure 6 the horizon light cone $\mathcal{H}$ is intersected by a spacelike
hypersurface $\Sigma$, $t=a$ say. Consider a point $p$ and let
$\dot{I}^{+}(p)$ be the boundary of its future light-cone . Unlike 1+1, in 2+1
dimensions (or higher) the intersection of $\mathcal{H}$ is no longer a point,
but rather a curve ( $d-2$ dimensional surface in general). This adds a new
degree of freedom and allows the existence of new links formed with points
$q$, asymptotically close to $\dot{I}^{+}(p)\cap\mathcal{H}$ and arbitrarily
far form $\Sigma\cap\mathcal{H}$. In addition, and unlike the 1+1 case, the
point $p$ is not at all required to be arbitrarily close to the
$\Sigma\cap\mathcal{H}$ for the volume $V$ to vanish; it is enough to be
arbitrarily close to $\Sigma$ and anywhere far from the intersection of
$\mathcal{H}$ and $\Sigma$. For these distant pairs $(p,q)$, the interval
between $p$ and $q$ remains small and highly likely to be free from additional
sprinkled points. For these reasons the $e^{-V}$ is not enough to suppress the
contributions of such pairs, as there is an infinite number of potential pairs
$(p,q)$ with vanishing volume in the analytic limit. As a consequence the
expected number of causal links, no matter which Max/Min conditions are
imposed, will diverge like $\Lambda_{r}^{d-2}$, in $d$-dimensions, with
$\Lambda_{r}$ an appropriate IR cutoff. This IR divergence is incurable within
the causal links proposal and a real departure from the links definition is
therefore inevitable .

##  4 The triplet proposal

The failure of the causal links proposal beyond 1+1 subsequently led soon to a
different and modified causet structures as a new candidates for the horizon
molecules.

We first note that it is almost obvious that the day can not be saved by
simple modifications of the links proposal, for instance by modifying the
Max/Min conditions, as we have exhausted all acceptable variants. The first
attempt to depart from the links structure was taken by Marr Sarah . This
attempt was mainly based on ”triad” structure or ”Triplet”. Although there are
some hints that this modified horizon molecular structure may fail to cure all
the IR divergences we observed in the causal links proposal in higher
dimensions, we find that the triplet proposal worthy of brief discussion. We
shall omit all technical details, as it is similar in spirit to the link-
counting, and only focus on the main results and their discussion.

Let us start by noting that there actually some suggestion that a certain type
of triplets are naturally related to the kind of correlation responsible for
entanglement entropy in a quantum field theory framework Sorkin:1995nj .

$\Lambda$-Triplet (Marr 2007):

A horizon molecule with respect to a given hypersurface $\Sigma$ is a triplet
$(p,q,r)$ satisfying the following conditions

  1. 1.

$p\in I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$ ,

  2. 2.

$q\in I^{+}(\Sigma)\cap I^{+}(\mathcal{H})$,

  3. 3.

$r\in I^{-}(\Sigma)\cap I^{+}(\mathcal{H})$,

  4. 4.

$|I[p,q]|=0$, $|I[r,q]|=0$ .

  5. 5.

$p$ is maximal in $I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$ , $q$ is minimal-but-
one in $I^{+}(\mathcal{H})$ and $r$ is minimal in $I^{-}(\Sigma)\cap
I^{+}(\mathcal{H})$ .

Notice that the $4^{th}$ condition automatically requires $p$ and $r$ to be
causally unrelated or spacelike related.

The condition $q$ minimal-but-one in $I^{+}(\mathcal{H})$ is meant to ensure
that the only point in $I^{-}(q)$ is $r$, Figure 7.

Figure 7: The regions shaded grey are required to be free from any sprinkled
points, whilst the green ones have only one point .

Marr first applied the $\Lambda$-triplet proposal for the collapsing shell
model in 1+1 - pushing the collapsing shell to future infinity- to obtain an
expected number of horizon molecules of order one, more precisely she obtained

$<\mathbf{H}_{\Lambda-triplet}>={\pi^{2}\over 6}-1+\mathcal{O}((a/b)^{2})\ .$
(23)

Using a triplet instead of a simple link (doublet) has therefore reduced the
expected number of horizon molecules by one.

In contrast to the collapsing shell model in $1+1$, the case of $2$-d reduced
Schwarzschild BH turned out to be analytically intractable for the
$\Lambda$-triplet.

As mentioned earlier, the underlying motivation that led to the departure from
the link structure to the triplets was to kill off the contributions coming
from $p-q$ links generated by points $q$ arbitrarily close to
$\mathcal{H}\cap\dot{I}(p)$ but arbitrarily far from $\mathcal{H}\cap\Sigma$.
However, it has been argued in Sarah  that although the introduction of a
third element in the horizon molecule structure, i.e. $r$, seems to cure this
IR divergences, the $\Lambda$-triplet counting still suffers from another IR
divergences, of course already present in the links proposal. This IR
divergence arises as a result of having unsuppressed contributions coming now
from points $p$ asymptotically approaching $\Sigma\cap\dot{I}(q)$ as they move
further into the past, simultaneously keeping $p-q$ interval small, and as $r$
does not bound $p$ from $\Sigma$, they are spacelike related, thus avoiding
any exponential suppression .

The above qualitative argument seemingly rules out the $\Lambda$-triplet as
possible alternative candidate that would work in higher dimensions, and led
Marr to consider other possible arrangements for the triplet. The guide of
course was to cure the IR divergences which plagued the link-counting and
persisted in the $\Lambda$-triplet counting.

To that end two different arrangements were considered in Sarah , the $z$ and
$l$-triplet.

The $z$-triplet is obtained from the definition of the $\Lambda$-triplet by
keeping the first three conditions, moving $r$ to the future of $p$ to form a
link with it, keeping its link relation with $q$, so the three points form a
path or a maximal chain, i.e $p\prec\\!\\!\cdot\,r\prec\\!\\!\cdot\,q$, and
removing the minimality condition on $r$ from the $5^{th}$ condition, Figure
7.

As for the $l$-triplet, it is a re-arrangement of the $z$-triplet by moving
$r$ to region $\in I^{+}(\Sigma)\cap I^{-}(\mathcal{H})$, requiring $p$ to be
maximal in $I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$ and $q$ maximal in
$I^{+}(\mathcal{H})$, Figure 7.

Marr used both the $l$ and $z$-triplet to count the expected number of horizon
molecules for the collapsing null shell 1+1 reduced model and obtained the
following results

$<\mathbf{H}_{z-triplet}>=2-{\pi^{2}\over 6}+\mathcal{O}((a/b)^{2})\ ,$ (24)
$<\mathbf{H}_{l-triplet}>=1+\mathcal{O}(a/b)\ .$ (25)

Moreover, the $l$-triplet turned out to be manageable analytically for $1+1$
Schwarzschild static model and gave the same result (leading term) as the
collapsing null shell setting, which is a promising result.

Let us remember that both the $z$ and $l$-triplet were introduced with an eye
on their use as a candidate for horizon molecules in higher dimensions.
Although Marr did not report any analytical results concerning the triplet
structures in 1+2 or 1+3, she gave qualitative argument suggesting that the
$z$ and $l$-triplet are in principle free of the IR divergences we discussed
above. Her argument goes as follows: The presence of a third element $r$ in
the $z$-triplet bounds $p$ away from $\Sigma$ and $q$ from $\mathcal{H}$,
whereas in the $l$-triplet the role of $r$ is reversed, it bounds $p$ away
from $\mathcal{H}$ and $q$ from $\Sigma$. Hence for both triplets an
arbitrarily large number of $p-q$ links is unlikely to build up in higher
dimensions .

Let us note that Marr’s qualitative argument regarding the would-be role
played by $r$ in killing off the IR divergences in higher dimensions does not
guarantee the finiteness of $<\mathbf{H}>$ nor the emergence of the area law.
For there could exist other less obvious and more subtle sources of
divergences. Moreover, the finiteness of the result does not either guarantee
that the resulting $<\mathbf{H}>$ will scale like the area. Therefore the
matter can only be settled by explicit calculation and this brings in a
technical difficulties that one has to deal with when considering higher
dimensions, and in particular 3+1. These technical difficulties come from the
necessity to evaluate the volumes needed to insure links and max/min
conditions, which turned out to be complicated and in some cases intractable
even in the flat case and for the link structure Dou:1999fw , let alone the
triplet structure . However, there are indications that the case $2+1$ could
be easier to handle analytically. Thus it would be interesting to explicitly
test the triplets proposals, in particular the $z$-triplet in $2+1$ is worth
revisiting .

It is also worth mentioning that $z$ or $l$-triplets could work in $2+1$ but
fail beyond this dimension, and one may be led to consider “diamond” structure
containing both types of triplet simultaneously.

##  5 An extended notion of horizon molecules

The definition of the horizon molecules as simple causal links crossing the
horizon, supplemented with certain Max/Min conditions, worked nicely and gave
promising results in $2$-d reduced spacetimes, at least for null
hypersurfaces. However, this success did not carry over to higher dimensions
due to pathological IR divergences. The higher cardinalilty molecule
definitions, namely the triplets, proposed by Marr to cure these divergences
turned out to be mathematically cumbersome and challenging beyond two
dimensions, and so far no one has devised a technique that would allow
analytical investigation of the triplets proposal in higher dimensions. This
calculational impasse, the failure of the causal links proposal and the desire
to extend the concept of horizon molecule to all causal horizons including
black hole, acceleration, and cosmological horizons stimulated two recent
sequential and related works by Barton et al Barton:2019okw  and by Machet and
Wang Machet:2020uml . These two works will be the subject of the present
section.

Our discussion will not cover the technical details presented in
Barton:2019okw ; Machet:2020uml , but will be limited to introducing the key
technical ideas, the results obtained and their discussion.

###  5.1 The spacelike hypersurface case

The extended proposal put forward by Barton et al was devised to extend the
notion of horizon molecule to more general causal horizons, thus it does not
refer to any particular black hole geometry, and to produce the area law in
higher dimensions.

The definition goes as follows. Let $(\mathcal{M},g)$ be a globally hyperbolic
spacetime with a Cauchy surface $\Sigma$. Let $\mathcal{H}$ be a causal
horizon, defined as the boundary of the past of a future inextendible timelike
curve $\gamma$, i.e.$\mathcal{H}:=\dot{I}^{-}(\gamma)$, and consider a causet
$\mathcal{C}$ generated on $\mathcal{M}$ through a random sprinkling
sprinkling with a density $\varrho_{c}$.

Barton et al proposal (2019)  A horizon molecule with respect to space-like
hypersurface $\Sigma$ ( a Cauchy surface) is a pair of elements of
$\mathcal{C}$, $\\{p_{-},p_{+}\\}$, such that

  1. 1.

$p_{-}\prec p_{+}$ ,

  2. 2.

$p_{-}\in I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$ ,

  3. 3.

$p_{+}\in I^{-}(\Sigma)\cap I^{+}(\mathcal{H})$ ,

  4. 4.

$p_{+}$ is the only element in $I^{-}(\Sigma)\cap I^{+}(p_{-})$ .

Theses conditions imply that the horizon molecule is a link. An illustration
of this type of horizon molecules is depicted in Figure 8.

Figure 8: A typical geometrical setting showing a typical horizon molecule
$(p_{-},p_{+})$ ($n=1$).

The above definition is easily seen to be generalizable to $n$-molecule
$\\{p_{-},p_{1,+},p_{2,+},\cdot\cdot\cdot,p_{n,+}\\}$ by requiring $p_{-}\prec
p_{+,k}$ and $\\{p_{1,+},p_{2,+},\cdot\cdot\cdot,p_{n,+}\\}$ to be the only
elements in $I^{-}(\Sigma)\cap I^{+}(p_{-})$ . However, our discussion of this
proposal will be limited to the horizon molecule of minimal size $n=1$.
Actually the cardinality of the horizon molecules plays no essential technical
role in the derivation of the results obtained in Barton:2019okw .

Before we move to the discussion of the derivation of the results of Barton et
al, we find it instructive to compare the above definition with the original
definition of horizon molecules as causal links with certain Max/Min
conditions.

The extended horizon definition requires $p_{-}$ to be a maximal element in
$I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$, or maximal-but-one in $I^{-}(\Sigma)$,
and no similar maximality condition is imposed on $p_{+}$. The requirement
that $p_{-}$ be maximal-but-one (or but-$n$) would for instance derive the
expected number of horizon molecules directly to zero if the hypersufrace was
null (straight null plane) and the future of the horizon is unbounded, as it
is the case for Rindler space for example. For this and others reasons the
null case has motivated an independent work by Machet and Wang Machet:2020uml
, to which we will come lastly.

It was first shown in Barton:2019okw  that $p_{-}$ is in the chronological
past of $\mathcal{J}=\Sigma\cap\mathcal{H}$. Using similar steps we used to
arrive to (6), it is not difficult to obtain the following integral
representation for the expected number of such horizon molecules,

$<\mathbf{H}_{1}>=\varrho_{c}\int_{I^{-}(\mathcal{J})}\rho_{c}V_{+}(p)e^{-\varrho_{c}V(p)}dV_{p}\
,$ (26)

where

$V_{+}(p):=\text{vol}(I^{-}(\Sigma)\cap I^{+}(\mathcal{H})\cap
I^{+}(p)),\leavevmode\nobreak\ \leavevmode\nobreak\
V_{+}(p):=\text{vol}(I^{-}(\Sigma)\cap I^{+}(p))\ ,$

and $p$ is used to denote the point $p_{+}$. Figure 8 illustrates the volumes
involved in the counting.

The main result shown by Barton et al is that under certain conceivable
assumptions and in the continuum limit, the expected number of such defined
horizon molecules, suitably rescaled, is equal to the area of $\mathcal{J}$,
the intersection of $\Sigma$ and $\mathcal{H}$, up to a dimension dependent
constant of order one. Mathematically stated we have the following limit

$\lim_{\varrho_{c}\rightarrow\infty}\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}_{1}>=a^{(d)}\int_{\mathcal{J}}dV_{\mathcal{J}}\
,$ (27)

where $dV_{\mathcal{J}}$ is the area measure on $\mathcal{J}$, and $a^{(d)}$
is a constant that only depends on the dimension $d$. Moreover, the approach
to the limit involves finite $\varrho_{c}$ corrections forming a derivative
expansion of local geometric quantities on $\mathcal{J}$ and increasing powers
of $l_{c}$, the discreteness length. In what follows we shall therefore
explicitly keep track of $\varrho_{c}$ and $l_{c}$.

####  5.1.1 Rindler Horizon with a flat hypersurface

Before outlining the explicit calculation of Barton:2019okw , it would be
instructive to present their heuristic argument supporting the validity of
(27) in general. This heuristic argument actually summarizes the motivation
behind defining the horizon molecules as such.

Consider Figure 8, the fact that $p_{-}$ lies in $I^{-}(\mathcal{J})$ and is
required to be maximal in this region means that $p_{-}$ is close to
$\mathcal{H}$, and as $\varrho_{c}\rightarrow\infty$ it gets closer. The
requirement that $p_{-}$ is maximal-but-one in $I^{-}(\Sigma)\cap
I^{+}(\mathcal{H})$ pushes $p_{-}$ towards $\Sigma$ and prevents it from
moving to the past of $\mathcal{J}$.

This tendency can be seen by inspecting the integrand of (27) in which the
exponential will suppress any contribution from regions with
$\varrho_{c}V(p)\gg 1$. Contributions coming from points far from the horizon
are suppressed by the $e^{-\varrho_{c}V_{-}(p)}$, whereas those close to the
horizon but far from $\Sigma$ are suppressed by $e^{-\varrho_{c}V_{+}(p)}$.
Therefore the only region which gives a non-negligible contribution is a small
and decreasing subregion of $I^{-}(\mathcal{J})$, immediately to the past of
$\mathcal{J}$. This strongly suggests that in the limit, the integral will
only depend on geometric quantities intrinsic to $\mathcal{J}$. On dimensional
ground, the only geometric quantity that can appear on the RHS of (27) is the
area of $\mathcal{J}$ times a dimensionless constant, $a^{(d)}$, which is
independent of the geometry.

To prove (27) Barton et al first proceeded by probing their defintion on
Minkowski $d$-dimnsional spacetime, flat $\Sigma$ and Rindler horizon
$\mathcal{H}$, in short all-flat.

First an inertial coordinates system is set up,
$(x^{0},x^{1},y^{\alpha}),\alpha=2,3,\cdots d-1$ , the hypersurface $\Sigma$
is chosen at $x^{0}=0$. $\mathcal{H}$ is given by $x^{0}=-x^{1}$. For
technical convenience a past-futue sawpped set up was instead used, so the
domain of integration is $p\in I^{+}(\mathcal{J})$. The integrand is
independent of $y^{\alpha}$ and the scaled expected number of horizon
molecules takes the form

$\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}_{1}>=\int_{\mathcal{J}}d^{d-2}yI^{d,flat}(l_{c})\
,$ (28)

where $I^{d,flat}(l_{c})$ is a dimensionless function given by

$I^{(d,flat)}(l_{c})=l_{c}^{-(d+2)}\int_{0}^{\infty}dx^{0}\int_{-x^{0}}^{x^{0}}dx^{1}\tilde{V}_{+}(x)e^{-\varrho_{c}\tilde{V}(x)}\
,$ (29)

where $l_{c}=\varrho_{c}^{-1/d}$ is the discreteness length. In the limit
$l_{c}\rightarrow 0$, the function $I^{(d,flat)}(l_{c})$ determines the
constant $a^{(d)}$.

In this flat case $\tilde{V}(x)$ is just the $d$-dimensional volume of a solid
null cone of height $x^{0}$. For $\tilde{V}_{+}(x)$, it was not possible to
derive a formula for general dimension $d$, but it was possible to compute it
explicitly for the lower dimensions, $d=2,3$ and $4$, and they are given
respectively by

$\displaystyle d=2$ $\displaystyle:$
$\displaystyle\tilde{V}_{+}(x)=\frac{1}{4}(x^{0}-x^{1})^{2}\ ,$ $\displaystyle
d=3$ $\displaystyle:$
$\displaystyle\tilde{V}_{+}(x)=\frac{2}{3}(x^{0})^{3}\tan^{-1}\big{(}\sqrt{\frac{x^{0}-x^{1}}{x^{0}+x^{1}}}\big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{9}(2x^{0}-x^{1})(2x^{0}+x^{1})\sqrt{(x^{0}-x^{1})(x^{0}+x^{1})}\
,$ $\displaystyle d=4$ $\displaystyle:$
$\displaystyle\tilde{V}_{+}(x)=\frac{\pi}{48}(x^{0}-x^{1})^{3}(5x^{0}+3x^{1})\
.$

A direct, but not straightforward, calculation leads to the following limits

$\lim_{l_{c}\rightarrow 0}I^{(2,flat)}(l_{c})=a^{(2)}=\frac{1}{3}\ ,$

$\lim_{l_{c}\rightarrow
0}I^{(3,flat)}(l_{c})=a^{(3)}=\frac{1}{4}\big{(}\frac{3}{\pi}\big{)}^{2/3}\ ,$

$\lim_{l_{c}\rightarrow 0}I^{(4,flat)}(l_{c})=a^{(4)}=\frac{\sqrt{3}}{10}\ .$

Actually the constants $a^{(d)}$ were given in Barton:2019okw  for arbitrary
$n$.

It should be noted here that although Barton et al computed the constants
$a^{(d)}$ in the limit $l_{c}\rightarrow 0$ using Watson’s lemma, the function
$I^{(d,flat)}(l_{c})$ is independent of the discreteness length $l_{c}$ and
equals $a^{(d)}$ at any discreteness scale . In other words we have

$I^{(d,flat)}(l_{c}):=I^{(d,flat)}=a^{(d)}=l_{c}^{-(d+2)}\int_{0}^{\infty}dx^{0}\int_{-x^{0}}^{x^{0}}dx^{1}V_{+}(x)e^{-\varrho_{c}V(x)}\
,$ (30)

and the above particular numerical values for $a^{(d)}$ are the exact values
of the integrals $I^{(d,flat)}(l_{c})$ for different $d$ regardless of the
value of $l_{c}$.

There are indeed two ways to see why $I^{(d,flat)}(l_{c})$ must be independent
of density of the sprinkling $\varrho_{c}$ or $l_{c}$. The first is purely
technical and based on a simple dimensional analysis of the integral (29). As
$I^{(d,flat)}(l_{c})$ is dimensionless, and there is no length scale which can
pair with $l_{c}$ to form a dimensionless quantity, thus the result must be a
pure number.

Another heuristic, but more intuitive, argument to understand why the
derivation of the above area law should be independent of the density of the
sprinkling in the all-flat setting, is the following.

In an all-flat setup, the two regions $I^{-}(\Sigma)\cap I^{-}(\mathcal{H})$
and $I^{-}(\Sigma)\cap I^{+}(\mathcal{H})$ are flat and infinite (unbounded).
If one randomly sprinkles in points in both regions with a given density,
$\varrho_{1}$ say, and considers another sprinkling with density
$\varrho_{2}$, both sprinklings should give the same result. The situation is
just a matter of zoom -in and zoom -out, the trade-off here is simply that the
number of molecules one loses by decreasing the density of sprinkling, gains
by moving further to the past ( away from the intersection of the horizon and
$\Sigma$). The density only tells us how far into the past we should go for
the value of the constant $a^{(d)}$ to get effectively saturated, and in the
very large density limit the molecules contributing to $<\mathbf{H}_{1}>$ are
the ones located infinitesimally close to $\Sigma\cap\mathcal{H}$ . In other
words, if the integral over $x^{0}$ in (29) were cut off at some upper limit
$\tau\gg l_{c}$, the value of the $l_{c}\rightarrow 0$ limit would not be
affected, the deviation from the above limiting values tends to zero
exponentially fast. This locality property, the fast exponential vanishing of
the difference, will be crucial for the discussion of the general curvature
case to which we now turn.

####  5.1.2 The general curvature case

Our discussion of the general curvature case will more or less be sketchy,
escaping technical detail and just highlighting the crucial steps of the
calculation of Barton et al.

The key elements in proving the limit (26) in the general curvature setting
were first the construction of Florides-Synge Normal Corrdinates (FSNC’S)
based on the co-dimension $2$ spacelike submanifold $\mathcal{J}$ and second
the locality argument. Such coordinates system construction is always possible
in tubular neighborhood about a submanifold of any co-dimension in any
Riemannian or pseudo-Riemannian manifold doi:10.1098/rspa.1971.0085 .

For $d>2$, let $z^{a}=(x^{A},y^{\alpha})(A=0,1,\alpha=2,\cdots,d-1)$ denote
the FSNC’s corrdinates constructed within a small enough tubular neighborhood
$\mathcal{N}$ about $\mathcal{J}$. For $d=2$ FSBC’s are just the Riemann
Normal Corrdinates based on the intersection point $\mathcal{J}$.

The next step is to assume the existence of a length scale $\tau$ such that
$l_{c}\ll\tau\ll L_{G}$, where $L_{G}$ is the smallest geometric scale in the
setup. This assumption is reasonable, because the continuum approximation of
causal set is only valid when the curvature length scales involved in the
problem are much larger than the discreteness scale $l_{c}$.

Consider now the region $\mathcal{R}_{\tau}$ defined as

$\mathcal{R}_{\tau}:=\\{p\in
I^{-}(\mathcal{J})\cap\mathcal{N}\leavevmode\nobreak\ :-\tau<x^{0}(p)<0\\}\ ,$

where $\tau$ is assumed to be small enough that this region is inside the
tubular neighborhood $\mathcal{N}$.

Let $\bar{\mathcal{R}}_{\tau}:=I^{-}(\mathcal{N})\setminus\mathcal{R}_{\tau}$
denote the complement of $\mathcal{R}_{\tau}$, then the integral (27)
naturally splits into a part over $\mathcal{R}_{\tau}$, and another over
$\bar{\mathcal{R}}_{\tau}$.

Now, in Barton:2019okw  it was argued, using the locality argument, that the
integral over $\bar{\mathcal{R}}_{\tau}$ tends to zero faster than any power
of $l_{c}$, actually exponentially suppressed, and hence its contribution can
be ignored. Therefore the surviving part of the expected value can be written
as a local integral over $\mathcal{R}_{\tau}$

$\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}_{1}>=\varrho_{c}^{\frac{2-d}{d}+2}\int_{\mathcal{R}_{\tau}}V_{+}(p)e^{-\varrho_{c}V(p)}dV_{p}\
.$ (31)

In view of the fact that the region $\mathcal{R}_{\tau}$ lies by choice within
the tubular neighbourhood, and hence the constructed FSNC’s can be used to
express the expectation value explicitly as

$\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}_{1}>=\varrho_{c}^{\frac{2-d}{d}+2}\int_{\mathcal{J}}d^{d-2}y\int_{-\tau}^{0}dx^{0}\int_{x^{0}}^{-x^{0}}dx^{1}\sqrt{-g(x,y)}V_{+}(x,y)e^{-\varrho_{c}V(x,y)}\
,$ (32)

where $g(x,y)$ is the determinant of the metric.

Let $\sigma_{\alpha\beta}$ denote the induced metric on $\mathcal{J}$, then
(32) can be written as

$\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}_{1}>=\int_{\mathcal{J}}d^{d-2}y\sqrt{-\sigma(y)}I^{(d)}(y;l_{c},\tau)+\cdots\
,$ (33)

where $I^{(d)}(y;l_{c},\tau)$ is defined by

$I^{(d)}(y;l_{c},\tau):=l_{c}^{-(d+2)}\int_{-\tau}^{0}dx^{0}\int_{x^{0}}^{-x^{0}}dx^{1}\sqrt{\frac{-g(x,y)}{\sigma(y)}}V_{+}(x,y)e^{-\varrho_{c}V(x,y)}\
,$ (34)

where $\sigma(y)$ is the determinant of the induced metric.

The factor $\sqrt{\frac{-g(x,y)}{\sigma(y)}}$ makes $I^{(d)}(y;l_{c},\tau)$ a
scalar on $\mathcal{J}$ and can be rewitten in a free coordinate notation as
$I^{(d)}(q;l_{c},\tau)$, $q\in\mathcal{J}$.

The next crucial step in the calculation of Barton et al is to show that
$I^{(d)}(q;l_{c},\tau)$ admits the following local expansion

$I^{(d)}(q;l_{c},\tau)=a^{(d)}+l_{c}\sum_{i}b_{i}^{(d)}\mathcal{G}_{i}(q)+O(l_{c}^{2})\
,$ (35)

where $a^{(d)}$ and $b_{i}^{(d)}$ are constants that only depend upon the
dimension $d$. For instance $a^{(d)}$ is the same constant obtained in the
flat case. $\mathcal{G}_{i}(q)$ is the largest set of mutually independent
geometric scalars of length dimension $L^{-1}$, like the extrinsic curvature
$K$ or the null expansion $\theta$ evaluated at $q$.

Again, switching to an order-reversed setup $I^{(d)}(q;l_{c},\tau)$ is written
as

$I^{(d)}(y;l_{c},\tau)=l_{c}^{-(d+2)}\int_{0}^{\tau}dx^{0}\int_{-x^{0}}^{x^{0}}dx^{1}\sqrt{\frac{-g(x,y)}{\sigma(y)}}V_{+}(x,y)e^{-\varrho_{c}V(x,y)}\
.$ (36)

At this stage one is free to choose any coordinates on $\mathcal{J}$, and a
suitable choice is RNC’s $y^{\alpha}$ centered about $q\in\mathcal{J}$;
$y^{\alpha}(a)=0$. As all expressions appearing in (36) are evaluated at
$y^{\alpha}(a)=0$, the argument $y$ will be dropped entirely to write

$I^{(d)}(q;l_{c},\tau)=l_{c}^{-(d+2)}\int_{0}^{\tau}dx^{0}\int_{-x^{0}}^{x^{0}}dx^{1}\sqrt{-g(x)}V_{+}(x)e^{-\varrho_{c}V(x)}\
.$ (37)

Note that $\sigma(0)=1$ in these RNC’s on $\mathcal{J}$,
$\sigma_{\alpha\beta}(0)=\delta_{\alpha\beta}$.

Now spacetime RNC’s $Z^{a}=(X^{A},Y^{\alpha})$ can be introduced within a
neighbourhood $\mathcal{U}$ about $q$ , such that $X^{A}=x^{A}$, and such that
the coordinate vectors $\frac{\partial}{\partial
Y^{\alpha}}=\frac{\partial}{\partial y^{\alpha}}$ at $q$. With this choice the
determinant of the metric, evaluated at at $q$ keeps the same form in terms of
the coordinates $x^{A}$ and $X^{A}$; and we have

$I^{(d)}(q;l_{c},\tau)=l_{c}^{-(d+2)}\int_{0}^{\tau}dX^{0}\int_{-X^{0}}^{X^{0}}dX^{1}\sqrt{-g(X)}V_{+}(X)e^{-\varrho_{c}V(X)}\
.$ (38)

The determinant $g(X)$ can be expanded in small $X^{A}$ relative to the
curvature scales of spacetime at $q$:

$\sqrt{=g(X)}=1-\frac{1}{6}R_{AB}X^{A}X^{B}+O(Z^{3})\ ,$ (39)

where $R_{AB}$ is the Ricci tensor with indices restricted to $A,B=0,1$. To
bring out the role of the different length scales of the problem, the smallest
length scale $L_{G}$ is used to define a dimensionless tensor
$\hat{R}_{ab}:=L_{G}^{2}R_{ab}$, and $\tau$ is used to re-express the above
expansion in terms of dimensionless coordinates $\hat{Z}_{a}:=Z^{a}/\tau$

$\displaystyle\sqrt{-g(X)}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{6}(\frac{\tau}{L_{G}})^{2}\hat{R}_{AB}\hat{X}^{A}\hat{X}^{B}+O(Z^{3})$
(40) $\displaystyle=$ $\displaystyle
1-\frac{1}{6}\varepsilon^{2}\hat{R}_{AB}\hat{X}^{A}\hat{X}^{B}+O(\varepsilon^{3})\
.$

In view of the fact that $\varepsilon=\tau/L_{G}\ll 1$, and $L_{G}$ is the
smallest geometric scale, the correction $\frac{1}{6}R_{AB}X^{A}X^{B}$ is of
order $\varepsilon^{2}$.

The volumes $V(X)$ and $V_{+}(X)$ can similarly be expanded around the flat
ones in the neighborhood $\mathcal{U}$. Using different explicit geometric
setups; in particular different choices for the hypersurface $\Sigma$, Barton
et al suggested the following general expansion for the volumes

$\displaystyle V(X_{p})$ $\displaystyle=$
$\displaystyle\tilde{V}(X_{p})\big{[}1+\sum_{i}\mathcal{G}_{i}(q)f_{i}(X_{p})+O(\varepsilon^{2})\big{]}\
,$ $\displaystyle V(X_{p})_{+}$ $\displaystyle=$
$\displaystyle\tilde{V}_{+}(X_{p})\big{[}1+\sum_{i}\mathcal{G}_{i}(q)f_{+,i}(X_{p})+O(\epsilon^{2})\big{]}\
,$ (41)

where $\tilde{V}(X_{p})$ and $\tilde{V}_{+}(X_{p})$ are the volumes from the
all-flat case discussed previously. $f_{i}(X_{p})$ and $f_{+,i}(X_{p})$ are
functions of length dimension $L$.

Using equations (40) and (5.1.2) the following expansion for
$I^{(d)}(q;l_{c},\tau)$, it is easy to obtain

$\displaystyle I^{(d)}(q;l_{c},\tau)$ $\displaystyle=$ $\displaystyle
l^{-(d+2)}\int_{0}^{\tau}dX^{0}e^{-\rho_{c}\tilde{V}(X^{0})}\bigg{\\{}\int_{-X^{0}}^{X^{0}}dX^{1}\tilde{V}_{+}(X)$
(42) $\displaystyle+$
$\displaystyle\sum_{i}\mathcal{G}_{i}(q)\bigg{[}\int_{-X^{0}}^{X^{0}}dX^{1}\tilde{V}_{+}(X)f_{+,i}(X)$
$\displaystyle-$
$\displaystyle\rho_{c}\tilde{V}(X^{0})\int_{-X^{0}}^{X^{0}}dX^{1}\tilde{V}_{+}(X)f_{i}(X)\bigg{]}+O(\epsilon^{2})\bigg{\\}}\
,$

where the fact that the flat cone volume $\tilde{V}$ only depends on $X^{0}$
was used, and the subscript $p$ from the coordinates $X^{A}$ have been
removed.

The integral in the first line is just the flat contribution $I^{(d,flat)(l)}$
given by (36) up to a difference which vanishes exponentially fast in the
limit $l_{c}\rightarrow 0$.

By dimensional argument and using Watson’s lemma again, the expression in
square bracket of (42) can be shown to evaluate to a term of the form
$Cl_{c}$, for some constant $C$, as $l_{c}\rightarrow 0$. Similarly, the
$O(\varepsilon^{2})$ corrections tend to a function of order $O(l^{2})$.
Therefore the expansion of (35) follows.

The constants $a^{(d)}$ are given by their flat values,
$I^{(d,flat)}=a^{(d)}$. The explicit form of the constants $b_{i}^{(d)}$ can
determined once a geometric setup is chosen. For instance Barton et al have
explicitly evaluated these constants for two different geometric setups
Barton:2019okw .

###  5.2 The null hypersurface case

The horizon molecules proposal of Barton et al was specially devised to work
when hypersurfaces of spacelike nature are considered. However, and as we
mentioned earlier, there are good reasons for requiring any horizon molecule
definition to be also valid in the case of null hypersurfaces. This issue was
not raised nor discussed in Barton:2019okw , but a subsequent recent work by
Machet and Wang addressed this question and investigated in detail the
extension of this definition to encompass null hypersurfaces intersecting the
horizon. The goal of the following subsection is to give a concise report of
the main results and conclusions of Machet and Wang.

Let us first give a general look at the problem to see how the success of
Barton et al proposal is tied to the spacelike nature of the hypersurface.

To that end consider the all-flat case of figure 9, a Rindler horizon in
Minlowski space, with $\Sigma$ being a straight null plane. As can easily be
seen the region $I^{-}(\Sigma)\cap I^{+}(\mathcal{H})\cap I^{+}(p_{-})$ is
unbounded with infinite volume for any randomly selected point $p_{-}$, and
the expected number of such horizon molecules is thus directly derived to
zero. Therefore, before any sensible calculation of the expected number is
started, one has to first bound this domain, at least for the all-flat case.
This can be done by either considering a folded null plane instead of the
straight one, or by taking a null hypersurface with different shape like a
downward light-cone. These two configurations were probed in Machet:2020uml
to compute the expected number of horizon molecules, although the motivation
there for bounding the region $I^{-}(\Sigma)\cap I^{+}(\mathcal{H})\cap
I^{+}(p_{-})$ was to avoid any potential IR divergence .

Let us note that, as we discussed in subsection 3.6, one can not approach the
null case by invoking a continuity argument, like the one given by equation
(21), by continuously deforming the spacelike result to obtain that of a null
hypersurface. Therefore the issue can only be settled by explicit calculation.

####  5.2.1 Rindler horizon in Minkowski spacetime

Machet and Wang applied Barton et al definition first to a Rindler horizon in
Minckowski space. As we have already mentioned, in the all-flat case and due
to the unboundedness of the region $I^{-}(\Sigma)\cap I^{+}(\mathcal{H})\cap
I^{+}(p_{-})$ the expected number of Barton et al horizon molecules is
identically zero, therefore for any sensible calculation to get started with
such definition and geometric setup one has first to introduce some IR
regulator to bound the domain $I^{-}(\Sigma)\cap I^{+}(\mathcal{H})\cap
I^{+}(p_{-})$. This can be achieved for instance by taking $\Sigma$ to be a
folded null plane or a downward light-cone.

Figure 9: An all-flat setting: a folded null plane crossing in a Rindler
horizon .

The setup of a folded null plane is depicted in Figure 9. A global coordinates
$(v,u,y^{\alpha})$ is set up, the null hypersurface $\Sigma$ is the $u$-axis,
$v=0$ and the horizon $\mathcal{H}$ is the $v$-axis, $u=0$. Another null
hpersurface $\Sigma^{\prime}$ given by $u=\lambda$ , with $\lambda>0$. The
union $\Sigma^{\prime}\cup\Sigma$ is the folded null plane with respect to
which the expected number of horzion molecules is to be counted.

In this setup the volumes $V_{+}(p)$ and $V(p)$ can explicitly be computed for
arbitrary dimension, and are given by

$V_{f}(u,v,\lambda)=\alpha_{d}(2v(u+\lambda))^{d/2},$
$V_{+,f}(u,v,\lambda)=\alpha_{d}(2v)^{d/2}\big{(}(u+\lambda)^{d/2}-u^{d/2}\big{)}\
.$

where $\alpha_{d}$ is a constant dependent on the dimension.

It follows again that the expected number of horizon molecules can be written
as

$\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}_{1}>=\int_{\mathcal{J}}d^{d-2}yI^{(d,flat)}_{null}(l_{c},\lambda)\
,$ (43)

where $I^{(d,flat)}_{null}(l_{c},\lambda)$ is given by

$I^{(d,flat)}_{null}(l_{c},\lambda)=l_{c}^{-(d+2)}\int_{0}^{\infty}dv\int_{0}^{\infty}duV_{+,f}(u,v,\lambda)e^{-\rho_{c}V_{f}}\
.$ (44)

Using the explicit formulas for $V_{+,f}$ and $V_{f}$ one gets

$I^{(d,flat)}_{null}(l_{c},\lambda)=\frac{\alpha_{d}^{-2/d}}{d}\Gamma[2/d+1]\int_{0}^{\infty}du\big{(}(u+\lambda)^{d/2}-u^{d/2}\big{)}(u+\lambda)^{-1-d/2}\
.$ (45)

It is noticeable that any dependence on the discreteness scale has disappeared
and therefore, by dimensional analysis, $I^{(d,flat)}_{null}(l_{c},\lambda)$
should be independent of $\lambda$. The above integral can be evaluated and
one obtains

$I^{(d,flat)}_{null}(l_{c},\lambda):=I^{(d,flat)}_{null}=a^{(d,flat)}_{null}=\frac{\alpha_{d}^{-2/d}}{d}\Gamma[2/d+1]H_{d/2}\
,$ (46)

where $H_{k}$ is the $k^{th}$ harmonic number.

Actually, Machet and Wang also carried out the calculation for $n$-molecule
and obtained a formula which can be exactly evaluated for each $n$.

It follows then

$\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}_{1}>=a^{(d,flat)}_{null}\int_{\mathcal{J}_{t}}dV_{\mathcal{J}}\
.$ (47)

Some comments on this result are in order.

It is first interesting to note that the resulting constants
$a^{(d,flat)}_{null}$ are different from the constants $a^{(d,flat)}$ obtained
in the spacelike case, as can be checked by substituting for particular values
of $d$.

Moreover, the final result is independent of the position of the
$\Sigma^{\prime}$, the parameter $\lambda$ . Therefore one may take the IR
regulator to infinity without changing the result, hence going back to null
plane case, and this goes in contradiction with the fact that if one started
with a null plane the expected number of horizon molecules would be
identically zero. Again, we see that this horizon molecules counting is
sensitive to how some limits are taken.

The independence of $a^{(d,flat)}$ from $l_{c}$ is actually related to its
independence of $\lambda$ and flat setup used to do the counting. As it was
pointed out in Machet:2020uml , because of Lorentz invariance of the counting
and the fact that one can always boost the system in the $u$ direction to pull
the surfrace $\Sigma^{\prime}$ arbitrarily close to $\mathcal{H}$ the result
should not depend on $\lambda$. If $\mathcal{J}$ has no geometrical quantity
associated to it, e.g intrinsic curvature, then $l_{c}$ has no length scale to
couple with and therefore $a^{(d,flat)}$ can only be a pure number. Similarly
to the spacelike case, to establish the area law in the all-flat setup using
this counting the continuum limit plays no role, equation (47) is valid for
any finite $\varrho_{c}$.

Another setup probed in Machet:2020uml  was again a Rindler horizon in
Minkowski space but with a null hypersurface $\Sigma$ having a different
shape, namely a downward light-cone.

Figure 10: A downward light-cone intersecting a Rindler horizon.

The downward light-cone $\Sigma$ is defined to be the boundary of the causal
past of a point $q\in\mathcal{H}$ , i.e $\dot{I}^{-}(q)$, Figure 10.

The volumes $V_{+}$ and $V(p)$ are now given by

$V_{+,q}+(p):=\text{vol}(I(p,q)\cap I^{+}(\mathcal{H})),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
V_{q}(p):=\text{vol}(I(p,q))\ .$

Machet and Wang could compute these volumes explicitly in $d=4$, in terms of
the null coordinates of $p$ and the affine distance between the horizon and
the point $q$, and obtained a general formula for $I^{(4,flat)}_{cone}$ which
reduces for $n=1$ to

$I^{(4,flat)}_{cone}\frac{1}{12}\sqrt{\frac{3}{2}}\bigg{(}\frac{27}{4}-{}_{2}F_{1}(1,-1,-2;\frac{3}{2})\bigg{)}=a^{(4)}\
.$ (48)

We notice that $I^{(4,flat)}_{cone}$ turns out to be again independent of the
discreteness scale and of the other length scale provided by the affine
distance between the horizon and the point $q$. The explanation of this
independence is similar to the folded null plane.

It is noteworthy that here too the constant $a^{(4)}$ is different from its
counterpart in the spacelike hypersurface.

If one insists on the necessity that the null and spacelike hypersurface must
give the same proportionality constant to the horzion area, and take it as a
sanity condition of any horizon molecules proposal, then we see that the
horizon molecules definition introduced by Barton et al does not meet this
requirement.

A final remark about the flat counting with a null hypersurface is that it is
free from any IR divergences. This IR divergences could have arisen from
points $p_{-}$ arbitrarily close to the $\Sigma$ and to the far past of
$\mathcal{J}$, e.g with $v\sim 0$ , $u\rightarrow-\infty$ and volume $V_{f}$
close to zero in the folded plane case, hence not exponentially suppressed. It
is not actually difficult to see how this IR divergence is cured within this
horizon molecule counting. The requirement that $p_{-}$ is max-but-$n$ (at
least $n=1$) bounds it away from $\Sigma$, this is realized in the general
integral formula of $<\mathbf{H}_{1}>$ by multiplying the exponential term,
$e^{-\varrho_{c}V}$, by the extra volume term $V_{+,f}$, or $V_{+}^{n}$ for
$n$-molecule, which has first to vanish before $V_{f}$ approaches zero.
Therefore, the term $V_{+,f}$ accompanying the exponential kills off this IR.

####  5.2.2 Curved case

To investigate the curved case with a null hypersurface the author in
Machet:2020uml  took a path similar in spirit to that taken by Barton et al,
but now by setting up a local Guassian Null Corrdinates (GNC) adapted to the
study of the null hypersurface case.

For folded null planes, a local coordinates $(v,u,y^{\alpha})$ is constructed
in a tubular neighborhood $\mathcal{N}\supset\mathcal{J}$. A region
$\mathcal{R}_{\Lambda}$ analogue to $\mathcal{R}_{t}$ in spacelike
hypersurface case is defined as follows (in time-reversed corrdinates )

$\mathcal{R}_{\Lambda}:=\\{p\in
I^{+}(\mathcal{J})\cap\mathcal{N}|0<v(p)<\Lambda,0<u(p)<\Lambda\\}\ ,$ (49)

where $\Lambda$ is an intermediate scale between the discreteness and the
geometric length of the setting, i.e $l_{c}\ll\Lambda\ll L_{G}$. For the
argument used in the spacelike hypersurface case to carry over to the null
setting one has to show that the rescaled expected number of horizon molecules
can be reduced to a local integral on $\mathcal{R}_{\Lambda}$

$\varrho_{c}^{\frac{2-d}{d}}<\mathbf{H}>=\varrho_{c}^{\frac{2-d}{d}+2}\int_{\mathcal{R}_{\Lambda}}V_{+}(p,\lambda)e^{-\varrho_{c}V(p,\lambda)}dV_{p}+\cdots\
,$ (50)

where $``....."$ refer to terms decaying exponentially fast as we go to the
limit $l_{c}\rightarrow 0$, and $\lambda$ is the parameter of the folding
hypersurface $\Sigma^{\prime}$.

Based on the discussion of the flat setting, it is not difficult to argue that
the above local expansion can not in general be true. This can be seen by
first noticing that although the region ( in future-past swapped setting)
$v(p)>\Lambda$ poses no problem for all values of $u(p)$, as $\varrho_{c}V\gg
1$, so that all contributions from this region will be exponentially
suppressed in the continuum limit. However, when $u(p)>\Lambda$ this argument
fails, contributions coming from points $p$ close to $u$-axis, with $v(p)\sim
0$, are not exponentially suppressed, as the volume $V(p)$ is now close to
zero for values of $u$ arbitrarily far in the future (or the past in the
original setting ) of $\mathcal{J}$. Thus, contributions coming from far away
along the past light-cone of the intersection hypersurface can not be
neglected. It follows then that the above local integral can not in general
count for the dominant contributions to the expected number of horizon
molecules in the continuum limit. Therefore Machet and Wang concluded that
this failure is a first indication that the proposal of Barton et al to count
horizon molecules with a null hypersurface is a flawed way to define entropy
on causal set setting.

Machet and Wang further argued that a truncated (by hand) local integral in
the form (50), in which contributions from the far past of the inetsection are
excluded, yields a small $l_{c}$ expansion of the following form for
$I^{(d)}(y;\lambda_{c},\lambda,\Lambda)$

$I^{(d)}(y;l_{c},\lambda,\Lambda)=a^{(d)}+\sum_{i}c_{i}^{(d)}\mathcal{F}_{i}(y,\lambda,\Lambda)+l_{c}\sum_{i}b_{i}^{(d)}\mathcal{G}_{i}(y,\lambda,\Lambda)+O(l_{c}^{2})\
,$ (51)

where $a^{(d)}$, $b_{i}^{(d)}$ and $c_{i}^{(d)}$ are dimensionless constants
dependent on $d$. The set ${\mathcal{G}_{i}(y,\lambda,\Lambda)}$ is a set of
mutually independent geometrical scalars of length $L^{-1}$ evaluated along a
geodesic segment. The extra set ${\mathcal{F}_{i}(y,\lambda,\Lambda)}$ is set
of independent geometrical dimensionless scalar evaluated along a geodesic
segment.

The presence of extra terms ${\mathcal{F}_{i}(y,\lambda,\Lambda)}$ carrying
geometrical information evaluated along the geodesic segment which, in
contrast to the spacelike hypersurface case, survive in the limit
$l_{c}\rightarrow 0$ can be given the heuristic explanation based on the above
discussion and dimensional analysis.

To further support their claim Machet and Wang worked out an explicit
geometrical setting in which $\Sigma$ is a downward light-cone, the past-
pointing light-cone of a point $q$ which is of affine distance $\lambda$ away
from $\mathcal{J}$. Within the relevant region, in which the $V$ and $V_{+}$
tend to a skinny causal interval or diamond, a Null Fermi Normal coordinates
system $(b,u,y^{\alpha})$ is set up. Figure 11 gives a sketch of the
coordinates system.

Figure 11: A sketch of the coordinates system in the relevant skinny diamond
setup. $V_{+}$ is shaded.

Both $V$ and $V_{+}$ can be expanded around the flat ones and admit the
following expansions in $d=4$,

$V^{(4)}(u,v,y;\lambda)=V_{f}^{(4)}(u,v,y;\lambda)\bigg{(}1+F(\lambda,u)+\mathcal{O}((u+\lambda)^{3},v^{3})\bigg{)}\
,$ (52)

$V^{(4)}(u,v,y;\lambda)=V_{f}^{(4)}(u,v,y;\lambda)\bigg{(}1+\tilde{F}(\lambda,u)+\mathcal{O}((u+\lambda)^{3},v^{3})\bigg{)}\
,$ (53)

where $F(\lambda,u)$ and $\tilde{F}(\lambda,u))$ are two dimensionless
function involving the integration over Ricci tensor along the $u$ direction.
The assumption here is of course that $\lambda$ is small relative to the local
curvature scales, $u$ is to be cutoff at distance $\Lambda$ small compared to
the local curvature scales.

Under the above assumptions along with an extra assumption about the Ricci
tensor (an assumption generic enough to support the claim) it was possible to
show that $I^{(4)}(y;\lambda_{c},\lambda,\Lambda)$ admits the following
continuum limit

$\lim_{l_{c}\rightarrow
0}I^{(4)}(y;l_{c},\lambda,\Lambda)=a^{(4)}+\mathcal{R}(y)c^{(4)}(\lambda,\Lambda)+\cdots\
.$ (54)

One can see that the limit is not local to the intersection $\mathcal{J}$, and
the area law is distorted. Machet and Wang then concluded that the horizon
molecules proposal of Barton et al does not yield a well behaved area law for
when evaluated on a null hypersurface intersecting the horizon.

Now, whether the above argument is conclusive or just an artifact of the
limitation of the expansion adopted by Machet and Wang, which relies on an ad
hoc truncation by hand of the integral $I^{(4)}$ and certainly neglecting
relevant contributions, remains in our view an unsettled issue. One, for
instance, cannot exclude the possibility that in a realistic BH model the area
law might be restored. A hint for this possibility is offered by the links
counting in 2-dimensional reduced Schwarzschild BH discussed in section 3.2,
where the area law is established in the null surface case in a quite subtle
way, and at the end the dominant contribution turns out to plainly comes from
the near horizon links.

###  5.3 Discussion and outlook

In this survey, we have tried to take the reader through the different
attempts to identify the right horizon molecules that would give a good
kinematical account for BH entropy within the causal set approach . Despite of
the fact that there have only been few scattered efforts and practitioners who
have devoted their time to this issue, it is undeniable that some progress has
been made along different directions. The simplicity, the success in
2-dimensions and the failure in higher dimensions of the causal links proposal
stimulated further investigations and proposals based on triplets, and
recently has sparked interest in the subject.

The early proposal based on causal links gave some promising results in two
dimensions, some of which may seem surprising . Prominent among them is the
fact that one finds a universal answer which took the same value in two quite
different black hole backgrounds, that of equilibrium and non-equilibrium
cases, and that the bulk of the links always reside in the close proximity to
the horizon, meaning that the result is controlled by the near horizon
geometry . However, a seemingly surprising result is the that this value
remains finite even in the continuum limit where the fundamental length
$l_{c}$ is sent to zero. In this sense, the replacement of continuous
spacetime by a causal set may appear in two dimensions as more of a
regularization device then something fundamental. Whether this has any deeper
meaning, or whether it might be related to some of the other properties that
both quantum field theory and quantum gravity possess in two dimensions
Fiola:1994ir , that remains an open question. Of note is also the fact that
the causal set approach to quantum gravity has been unique in attempting to
account for the statistical mechanics of the non-equilibrium horizon.

It must be added that the above features of the causal links counting are
shared by the triplet proposals, i.e. universality of the result and the
finiteness in the continuum limit in two dimensions.

Two questions about the links and triplets proposals have so far remained
open. The first is to find a way to decide whether the spacelike hypersurface
gives the same result as the null one in two dimensions, regardless of the
validity of these proposals beyond two dimension. But as we early mentioned
this a mere mathematical curiosity with little, if any, physical relevance.
The second is of importance and concerns the triplet proposal. Although Marr
Sarah  argued that the $\Lambda$-triplet would suffer from IR divergences in
higher dimensions, the issue is unsettled for $l$ and $z$-triplets. Therefore,
it would be interesting exercise to investigate this triplets counting at
least in three dimensional flat setting. However, as we already mentioned,
going beyond two dimensions would make the calculation cumbersome, but if
enough time and effort is devoted to this problem, some approximation methods
could possibly be devised to extract the leading contributions or settle the
divergence issue. Of course, one could also use numerical methods to approach
this counting.

Unlike the links and the triplets attempts, the Barton et al proposal has
succeeded in giving an expected number of horizon molecules proportional to
the area of the horizon intersecting a spacelike hypersurface, for almost all
reasonable geometrical settings. However, this proposal has some drawbacks.
Firstly, it seems to be inherently discontinuous as one moves from the
spacelike hypersurfaces to null ones, giving two different values, i.e
different proportionality constants. Moreover, if one accepts the conclusion
of Machet and Wang Machet:2020uml , in a curved geometrical background and for
a null hypersurface, the continuum limit of the expected number of Barton et
al horizon molecules is not local to the intersection of the horizon and the
hypersurface, yielding to an ill-behaved area law. Nonetheless, if one sticks
with spacelike hypersurfaces, the Barton et al counting provides a good
measure for the area of the intersection of the horizon (a null surface) and
spacelike hypersurface, which is a promising aspect of such horizon molecules
proposal.

Another weakness of Barton el al definition, in our view, is that it lacks the
physical intuitive and heuristic picture shared by the links and the triplet
(and the diamond) proposals. The elements that underpinned the Barton et al
horizon molecules are points exterior to the black hole and to the past of the
hypersurface, with no reference to the future of the hypersurface, hence it
would be hard to view such molecules as heuristically producing entanglement
during the course of the causal set growth (or time development).

Finally, the common weakness of all proposals discussed in this review, of
course, is that they remain at purely kinematical level, and even if a fully
successful kinematical identification of the horizon molecules is achieved, no
successful proposal can be substantiated or refuted before we possess a fully
quantum dynamics of causets. And despite a very important step made by Rideout
and Sorkin in developing classical stochastic dynamics for causets, _classical
sequential growth_ models Rideout:1999ub , building a viable quantum
sequential growth dynamics has so remained challenging Sorkin:2011sp ;
Martin:2004xi ; Surya:2019ndm .

Acknowledgments

While writing this review I have benefited from several discussions with Fay
Dowker and Ludovico Machet, I am grateful to them for answering my questions
and discussing several issues regarding their works.

## References

  * (1) R. M. Wald, _The thermodynamics of black holes_ , _Living Rev. Rel._ 4 (2001) 6, [gr-qc/9912119].
  * (2) R. D. Sorkin, _The statistical mechanics of black hole thermodynamics_ , in _Symposium on Black Holes and Relativistic Stars (dedicated to memory of S. Chandrasekhar)_ , 5, 1997.  gr-qc/9705006.
  * (3) S. Carlip, _Black Hole Thermodynamics_ , _Int. J. Mod. Phys. D_ 23 (2014) 1430023, [1410.1486].
  * (4) R. D. Sorkin, _Ten theses on black hole entropy_ , _Stud. Hist. Phil. Sci. B_ 36 (2005) 291–301, [hep-th/0504037].
  * (5) J. D. Bekenstein, _Do we understand black hole entropy?_ , in _7th Marcel Grossmann Meeting on General Relativity (MG 7)_ , pp. 39–58, 7, 1994.  gr-qc/9409015.
  * (6) A. Strominger and C. Vafa, _Microscopic origin of the Bekenstein-Hawking entropy_ , _Phys. Lett. B_ 379 (1996) 99–104, [hep-th/9601029].
  * (7) G. T. Horowitz, _The Origin of black hole entropy in string theory_ , _Astrophys. Space Sci. Libr._ 211 (1997) 46, [gr-qc/9604051].
  * (8) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231–252, [hep-th/9711200].
  * (9) S. Carlip, _The Statistical mechanics of the (2+1)-dimensional black hole_ , _Phys. Rev. D_ 51 (1995) 632–637, [gr-qc/9409052].
  * (10) A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, _Quantum geometry and black hole entropy_ , _Phys. Rev. Lett._ 80 (1998) 904–907, [gr-qc/9710007].
  * (11) R. D. Sorkin, _Toward a proof of entropy increase in the presence of quantum black holes_ , _Phys. Rev. Lett._ 56 (May, 1986) 1885–1888.
  * (12) J. S. Dowker, _Remarks on geometric entropy_ , _Class. Quant. Grav._ 11 (1994) L55–L60, [hep-th/9401159].
  * (13) L. Bombelli, R. K. Koul, J. Lee and R. D. Sorkin, _A Quantum Source of Entropy for Black Holes_ , _Phys. Rev. D_ 34 (1986) 373–383.
  * (14) M. Srednicki, _Entropy and area_ , _Phys. Rev. Lett._ 71 (1993) 666–669, [hep-th/9303048].
  * (15) T. Nishioka, _Entanglement entropy: holography and renormalization group_ , _Rev. Mod. Phys._ 90 (2018) 035007, [1801.10352].
  * (16) L. Bombelli, J. Lee, D. Meyer and R. D. Sorkin, _Space-time as a causal set_ , _Phys. Rev. Lett._ 59 (Aug, 1987) 521–524.
  * (17) D. Dou, _Causal sets, a possible interpretation for the black hole entropy and related topics_.  PhD thesis, SISSA, Trieste, 1999.  gr-qc/0106024.
  * (18) D. Dou and R. D. Sorkin, _Black hole entropy as causal links_ , _Found. Phys._ 33 (2003) 279–296, [gr-qc/0302009].
  * (19) S. Marr, _Black hole entropy from causal sets._ , PhD thesis, Imperial College, 2007.
  * (20) C. Barton, A. Counsell, F. Dowker, D. S. W. Gould, I. Jubb and G. Taylor, _Horizon molecules in causal set theory_ , _Phys. Rev. D_ 100 (2019) 126008, [1909.08620].
  * (21) L. Machet and J. Wang, _On the horizon entropy of a causal set_ , _Class. Quant. Grav._ 38 (2021) 085004, [2012.06212].
  * (22) D.Meyer, _PhD dessertation, MIT, 1988_ , .
  * (23) Luca Bombelli, _PhD dessertation, Syracuse, 1987_ , .
  * (24) S. Surya, _The causal set approach to quantum gravity_ , _Living Rev. Rel._ 22 (2019) 5, [1903.11544].
  * (25) D. Rideout and P. Wallden, _Spacelike distance from discrete causal order_ , _Classical and Quantum Gravity_ 26 (jul, 2009) 155013.
  * (26) R. D. Sorkin, _Quantum measure theory and its interpretation_ , in _3rd Workshop on Physics and Experiments with e+ e- Linear Colliders (LCWS 95)_ , 7, 1995.  gr-qc/9507057.
  * (27) P. S. Florides and J. L. Synge, _Coordinate conditions in a riemannian space for coordinates based on a subspace_ , _Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences_ 323 (1971) 1–10, [https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1971.0085].
  * (28) T. M. Fiola, J. Preskill, A. Strominger and S. P. Trivedi, _Black hole thermodynamics and information loss in two-dimensions_ , _Phys. Rev. D_ 50 (1994) 3987–4014, [hep-th/9403137].
  * (29) D. P. Rideout and R. D. Sorkin, _A Classical sequential growth dynamics for causal sets_ , _Phys. Rev. D_ 61 (2000) 024002, [gr-qc/9904062].
  * (30) R. D. Sorkin, _Toward a ’fundamental theorem of quantal measure theory’_ , 1104.0997.
  * (31) X. Martin, D. O’Connor and R. D. Sorkin, _The Random walk in generalized quantum theory_ , _Phys. Rev. D_ 71 (2005) 024029, [gr-qc/0403085].