# Gravitation as a Statistical Theory on the Light Cone

José M. Isidro  Instituto Universitario de Matemática Pura y Aplicada
Universidad Politécnica de Valencia
Valencia 46022
Spain<EMAIL_ADDRESS>,  Claudio F. Paganini  Fakultät für Mathematik
Universität Regensburg
D-93040 Regensburg
Germany<EMAIL_ADDRESS>and  Alessandro Pesci  INFN
Sezione di Bologna
Via Irnerio 46
40126 Bologna
ITALY<EMAIL_ADDRESS>
###### Abstract.

In this paper, we will explore Padmanabhan’s mesoscopic, statistical approach
to gravity [62] with a twist. The general picture of his approach is that
spacetime is made of large numbers of localized quantum degrees of freedom.
Padmanabhan assumed that the degrees of freedom of a given quantum state of
geometry contribute, after averaging over fluctuations, a vector degree of
freedom for spacetime at a point. For null vectors, this can be regarded as
corresponding to one single vector, i.e. a pure state, for the statistical
ensemble on the light cone at every point. In the present paper, we consider
instead the case where the states of the gravitational degrees of freedom are
spread out and overlap, with only probabilistic information on which of them
determines the actual spacetime at a point. In the continuum limit, this
corresponds to a mixed state for the statistical ensemble on the light cone at
every point.
This change in assumptions leads to some interesting observations. When we
define a statistical ensemble on the light cone, its variance “knows” about
the interior of the light cone. As an intriguing consequence, we find that the
cosmological constant can be related to the variance over the light cone.
With a mixed state, we can no longer derive the gravitational field equations
from an entropy functional. Here, instead, we show that a naive implementation
of the measure of a mixed state on the light cone in the variation principle
leads to modified measure theories (MMT) as the grand canonical ensemble and
allows one to reframe unimodular gravity as the canonical ensemble of a
statistical theory on the light cone.

###### Contents

  1. 1 Introduction
    1. 1.1 Organization
  2. 2 Mathematical Background
  3. 3 Padmanabhan’s Thermodynamic Considerations
  4. 4 A Probability Measure on the Light Cone
  5. 5 Some Observations
  6. 6 Modified Measure Theories from a Density of States
    1. 6.1 Unimodular Gravity from a Density of States.
  7. 7 Summary and Outlook
    1. 7.1 Summary
    2. 7.2 Outlook
  8. A Positivity Lemma
  9. B Average Vector Field for Specific Probability Distributions

##  1\. Introduction

The present paper is a result of a series of papers comparing the structures
and ideas of different approaches to fundamental physics [17, 19, 18] as well
as the upcoming articles [20, 21]. The goal of this project is to motivate the
community to establish an extensive collection of such articles as a sort of
”Rosetta stone” for approaches to fundamental physics. The hope is that such a
set of dictionaries of ideas helps the exchange across approaches and thereby
catalyzes progress in the foundations of physics. The present article
demonstrates the potential of this project, set in the context of
thermodynamic approaches to gravity, it takes inspiration from Causal Fermion
Systems (CFS) [1, 16] and links directly to Modified Measure Theories (MMT)
[28, 26, 27] and unimodular gravity (UG) [35, 9, 52, 81, 51, 42].

The original idea of spacetime thermodynamics came about in the context of
black holes. Bekenstein [6] realized that the black hole parameters satisfy
the same relations as conjugate variables in thermodynamics and boldly
suggested assigning an entropy to black holes proportional to the area of
their event horizon. This idea was given further credibility by Hawking’s
discovery [30, 31] that black holes indeed have a temperature, thus fixing the
proportionality constant to be $\frac{k_{B}}{4l_{p}^{2}}$. In his seminal
paper in 1995 Jacobson [38] then showed that in fact the equations of General
Relativity can be understood as thermodynamic balance equations across any
null surface through any point in spacetime. To obtain the full Einstein
equations in the bulk, one then needs to assume that the stress energy tensor
of the matter fields is divergence-free. This fixes the dynamics in the bulk
up to a constant of integration which can be identified with the cosmological
constant.

Ever since, there has been great interest in this approach to the puzzle of
(quantum) gravity. For recent work, see, for example, [5, 4, 31, 25, 38, 87,
13, 40, 8, 71, 83, 2, 3, 41, 78]. For our results here, the paper [2] is of
particular interest, as they argue that rather than giving rise to the
Einstein-Hilbert action, thermodynamic arguments actually favor UG which
features equivalent dynamics. Our main focus, however, will be on the
contributions by Padmanabhan [57, 59, 60, 68, 62] and his ideas of a
mesoscopic Boltzmann approach to gravity. Hence in his reasoning, he assumes
that there do exist internal degrees of freedom underlying every point of
spacetime; however, he remains agnostic to the concrete nature of these
microscopic degrees of freedom. The goal of his approach was to obtain a
variational principle for gravity such that the equations of motion are
invariant under $\mathcal{L}_{M}\rightarrow\mathcal{L}_{M}+C$ where
$\mathcal{L}_{M}$ is the matter Lagrangian and $C$ is a constant. To that end
he studied the small sphere limit $x\rightarrow y$ of Synge’s world function
modified by adding a minimal length $\sigma_{L}(x,y)=\sigma(x,y)+L^{2}$
111Synge’s world function $\sigma(x,y)$ is given by one half times the
geodesic interval between $x$ and $y$ squared. to derive a ”gravitational
density of states”.

His original derivation was performed in the Euclidean setting and carried
over to the Lorentzian setting via analytic continuation. One of the authors
of this paper later showed [72, 73, 74, 11] that the same result can be
obtained working directly in the Lorentzian setting. In the course of his
investigation, Padmanabhan frequently identified the light cone as the
configuration space for the internal degrees of freedom222The derivation for
the density of states works equally well for time-like or space-like vectors,
however the particular emphasis on null-vectors is due to their connection to
horizons and thereby the original thermodynamic considerations.. This is the
starting point for our present work. In his later work [62] he remarked that
one should keep in mind that the null vectors that enter the variation
principle for the entropy should be thought of as representing underlying
gravitational quantum degrees of freedom which are localized in a small volume
around a point $x$ with a sharply peaked distribution in momenta. He demands
that when averaging over this state in momentum space, the result should be a
null vector again. This implies that in the continuum limit where we consider
a state on the light cone at a point $x$ this corresponds to a pure state,
i.e. a Dirac delta in configuration space.

In the present paper, we take inspiration from Causal Fermion Systems. In the
continuum limit of the Minkowski vacuum, the underlying Hilbert space is given
by the Dirac sea [16]. It is therefore natural to associate these states with
the gravitational sector of the theory. Hegerfeldt’s theorem [34, 10, 24]
tells us that vectors in this Hilbert space cannot be localized in any
spatially compact region. In fact, if we consider the plane wave solutions,
eigenstates of the momentum operator, the Dirac sea is built from maximally
delocalized quantum states. As a result, we should consider that the volume at
a point $x$ is not sourced by a single (pure) gravitational state, but by a
collection of states; and, assuming to have only probabilistic information on
them, by a mixed of state. In the continuum limit, this corresponds to a
general measure on configuration space, i.e., on the light cone.

Padmanabhan [62] argued that for the (pure) states he considered, the variance
is small and hence can be neglected in a first-order approximation. This can
not hold true if we consider mixed states. Working with mixed states leads to
a number of interesting observations. In particular, we find that taking the
average over a probability field naturally gives rise to a time-like vector
field that is similar in nature to the regularizing vector field that plays a
key role in [22]. We also find that the variance can be linked to the
cosmological constant if we consider the validity of the Einstein field
equations as an observational fact.

Furthermore, it turns out that working with mixed states allows us to link
several lines of research in the context of modified/thermodynamic gravity. In
this paper, we explore the link with MMT and UG which is a subset of MMT, as
explained in [19, Section 2.2]. MMT was developed with the same goal in mind
that was driving Padmanabhan’s research: To find a variational principle for
which the equations of motion for the matter sector and the gravitational
sector are both invariant, to the addition of constants in the Lagrangian.
Instead of focusing on the light cone, MMT replace the metric measure
$\sqrt{-g}d^{4}x$ with a measure that is independent of the metric. As a
result, additive constants to the Lagrangian drop out of the equations of
motion as desired333We will touch briefly on the conjecture that these
approaches are indeed two sides of the same coin if one considers the
conformal factor of the metric as an independent degree of freedom. However, a
rigorous discussion of this idea is beyond the scope of this paper. .

In our present paper, we show that even a naive implementation of a mixed
state on the light cone in the variation principle has interesting
consequences. It naturally leads to MMT as the grand canonical ensemble and
allows us to reframe UG as the canonical ensemble of a statistical theory on
the light cone. Despite the ad hoc nature of these action principles, on the
conceptual level they fit nicely with the considerations regarding
baryogenesis [22] in the context of CFS [16, 1]. In particular, they give a
complementary point of view on MMT. This perspective also fits with the
arguments in [2] as the derivation of UG requires the conventionally defined
matter stress-energy tensor to be divergence-free, an assumption that is
dropped in MMT. However, given the ad-hoc nature of the variation principles
brought forward in this paper, they likely will not be the final answer, but
at least they seem to be a good starting point for further investigations.

Finally, we mention that this mathematical setup has the potential to link to
Verlinde’s heuristic derivation [85, 86, 37] of modified Newtonian dynamics
from entropic considerations.

###  1.1. Organization

In Section 2 we introduce basic mathematical notations and useful facts to
keep in mind for the later constructions. In Section 3 we summarize the key
arguments in Padmanabhan’s papers. The mathematical core of the paper is in
Section 4 where we introduce the probability density on the light cone. Here
we recall the fact that averaging over the light cone always results in a
time-like vector except for pure states. Finally, we prove that the variance
is a positive definite tensor. The core physical results are presented in the
following two sections. In Section 5 we observe that the variance can be
connected to the cosmological constant. In Section 6 we show how theories with
modified measures can be obtained if we consider a density of states instead
of a probability measure on the light cone. Finally, in Section 7, we give an
overview of possible applications and generalizations of our mathematical
framework.

##  2\. Mathematical Background

In the present work we will always consider $(M,g)$ to be a spacetime, i.e. a
four-dimensional orientable Lorentzian manifold. The study of null-geodesics,
i.e. geodesics $\gamma$ who’s tangent vector $\dot{\gamma}^{\mu}$ satisfies

$g_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}=0$ (2.1)

at every point has been of considerable interest as they play a crucial role
in characterizing the causal structure of spacetime, see e.g., [49, 50].

From a physical perspective, null geodesics describe the path of light rays
and are thus, in principle, amendable to experiment/observation. For that
reason, they play a key role in the axiomatic reconstruction of spacetime from
physical structures by Ehlers, Schild and Piran [46, 15]. In the mathematical
literature, there has been recent progress in the study of the geometry of the
space of null-geodesics [47, 33, 12]. In the present work, we are interested
in a geometric structure closely associated with the space of null-geodesics,
namely the bundle of past light cones.

######  Definition 2.1 (Past Light Cone Bundle (PLCB)).

Let $(M,g)$ be a smooth spacetime, that is, a smooth time orientable
Lorentzian manifold. Then the PLCB is the subset of the tangent bundle given
by

$\mathcal{L}:=\\{(x,v)\in TM|g_{x}(v,v)=0,v\neq 0\ \text{ and }v\text{ past
directed}\\}.$ (2.2)

The fiber $\mathcal{L}_{x}$ of $\mathcal{L}$ at a point $x$ is just the past
light cone at this point. In [7] it was shown that the PCLB $\mathcal{L}$ is,
in fact, a submanifold of the tangent bundle. This is a first sanity check
that it makes sense to consider $\mathcal{L}$ as a configuration space for a
physical theory.

To make calculations more tractable, we will use a coordinate system based on
an orthonormal tetrad $e_{0}^{\mu},e_{1}^{\mu},e_{2}^{\mu},e_{3}^{\mu}$ where
$e_{0}^{\mu}$ is time-like and the other tetrad vectors are space-like. Using
the notation
$\bar{k}\cdot\bar{e}^{\mu}=k_{1}e_{1}^{\mu}+k_{2}e_{2}^{\mu}+k_{3}e_{3}^{\mu}$
we notice that any null vector can then be written as

$n^{\mu}=\lambda(e_{0}^{\mu}+\bar{k}\cdot\bar{e}^{\mu})$ (2.3)

where $\bar{k}=(k_{1},k_{2},k_{3})$ is a three vector with unit Euclidean norm
$|k|=1$. This makes the ${\mathord{\mathbb{R}}}\times S^{2}$ topology of the
light cone explicit. We make use of this split in the definition of the past
celestial sphere bundle.

######  Definition 2.2 (Past Celestial Sphere Bundle (PCSB)).

Let $(M,g)$ be a smooth spacetime and $X$ be a smooth, non-vanishing timelike
vector field. Then the PCSB is the subset of the tangent bundle given by

$CSM:=\\{v\in T_{x}M|g(v,v)=0,g(v,X)=1\\}.$ (2.4)

From the results in [7], we get the following corollary.

######  Corollary 2.3.

There is a global factorization of the PLCB by the PCSB.

$\mathcal{L}={\mathord{\mathbb{R}}}_{+}\times SM.$ (2.5)

This can be seen immediately from the definition of the PCSB and a global
rescaling of the vector field $X$ by a scalar $\lambda$. This fact will become
especially important in our follow-up paper [69], where we will remove most of
the structures that we assumed for simplicity in the present paper. This
concludes our collection of relevant notation and existing results.

##  3\. Padmanabhan’s Thermodynamic Considerations

In [38] Jacobson showed that the equations of motion of General Relativity on
the light cone

$(R_{\mu\nu}-\kappa T_{\mu\nu})n^{\mu}n^{\nu}=0,\qquad\forall n^{\mu}\text{
with }g(n,n)=0$ (3.1)

can be recovered as a thermodynamic balance equation. To obtain the full
Einstein equations from the equations on the light cone, one has to impose the
fact that the stress-energy tensor is divergence free
$\nabla^{\mu}T_{\mu\nu}=0$. This leads to

$R_{\mu\nu}-\frac{1}{2}R+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}$ (3.2)

with $\Lambda$ as a free constant of integration. Padmanabhan’s idea was to
extend on this argument and derive (3.1) from mesoscopic considerations. The
philosophy of this approach is to assume that there exist microscopic degrees
of freedom underlying spacetime and GR, which are macroscopic phenomena.
However, no assumptions are made about the detailed properties of these
microscopic degrees of freedom other than their existence. To this end
Padmanabhan postulated, e.g. in [61], a density of microscopic degrees of
freedom $\rho(x,\phi_{A})$, where $\phi_{A}$ is a yet to be specified variable
labeling the internal microscopic degrees of freedom. He arrived at this
conclusion by investigating the minimum-length metric (also quantum metric, or
q-metric for short), i.e., a matric description of spacetime with a built-in
minimal length [43, 45, 82, 67, 44]. The q-metric postulates that the squared
integral between two points $\sigma^{2}(x,y)$ (i.e., two times Synge’s world
function [84]) is modified on short scales444This allows for a covariant
implementation of a minimal length scale in the universe. See [36] for a
review of minimal length scenarios in quantum gravity and [80] for a
connection to holography.. The simplest implementation of this idea is the
addition of a constant to its square (the results derived from the q-metric do
not depend on the details of the modification)

$\sigma(x,y)^{2}\longrightarrow\sigma_{L}(x,y)^{2}=\sigma(x,y)^{2}+L^{2}.$
(3.3)

The q-metric corresponding to this modified world function is a bitensor, i.e.
it is defined in terms of two points $x$ and $y$, which is singular everywhere
in the coincidence limit (where $x\rightarrow y$). This singular behaviour is
a direct consequence of the minimal length in the corresponding distance
function.

One can calculate the volume and surface of an equi-geodesic ball with
$\sigma_{L}(x,y)^{2}=C$. Taking the small-sphere limit, i.e. $x\rightarrow y$,
one can compare the result from the modified distance with that from the
original metric. In the ordinary spacetime metric volumes and areas vanish, of
course, in the coincidence limit. For the q-metric, however, only the volume
vanishes in the coincidence limit, while the area does not. This paves the way
to introduce the density of gravitational degrees of freedom. Padmanabhan then
defines the density of microscopic degrees of freedom by the limit

$\rho(x,\phi_{A})=\lim_{x\rightarrow
y}\frac{\sqrt{h_{\sigma_{L}}}}{\sqrt{h_{\sigma_{L,\text{flat}}}}}=1-\frac{1}{6}L^{2}R_{\mu\nu}n^{\mu}n^{\nu}.$
(3.4)

where $\sqrt{h_{\sigma_{L}}}$ and and $\sqrt{h_{\sigma_{L,\text{flat}}}}$ are
the area elements associated with the modified distance function in curved
space and in flat space respectively and $n^{\mu}$ is the surface normal
vector. Working in a Euclidean setting, Padmanabhan interpreted the zero point
as the light cone after Wick rotation and $n^{\mu}$ in the limit accordingly
as null vectors. This calculation led Padmanabhan to identify the internal
degrees of freedom $\phi_{A}$ in his density of microscopic degrees of freedom
by the set of null vectors $n^{\mu}$ at a point.

In his later papers [62, 63, 66], Padmanabhan then conceived a given quantum
state of spacetime as the product of a collection of elementary quantum
degrees of freedom over elemental volumes throughout all of spacetime. To go
from the quantum state of spacetime to the classical description of spacetime
he defined in [62] the average over quantum fluctuations at every point $x$
representing an elemental volume through the functional integral

$\langle
n_{\mu}\rangle=\int\mathcal{D}n\;n(x)_{\mu}P(n(x)^{\mu},x)=l^{\mu}(x)$ (3.5)

where $P(n(x)^{\mu},x)$ is parameterized by some null vector field
$l^{\mu}(x)$, and $P$ is the probability that the actual quantum degree of
freedom is given by $n(x)^{\mu}$ at $x$. Here, in principle, one should think
of $P$ as a sharply peaked Gaussian in the variable $[n(x)^{\mu}-l^{\mu}(x)]$
supported in a small volume around the point $x$. This suggests, that $\langle
n_{\mu}\rangle$ in (3.5) should be regarded as the expectation value of the
momentum over a single quantum degree of freedom treated essentially as a
pure. In the continuum limit, which we consider subsequently, such a state
corresponds to a Dirac measure on the light cone.

Following Padmanabhan, we will treat the light cone itself as the
configuration space of the internal variables $n^{\mu}$. However, inspired by
CFS we now consider a situation where the states of the underlying
gravitational quantum degrees of freedom overlap in spacetime, with only
probabilistic information of which state actually determines spacetime at $x$.
In the present set-up this motivates the consideration of mixed states
described by general invariant probability measures on the light cone
$dP(n^{\mu})$. Given (2.3), if the measure is absolutely continuous with
respect to the Lebesgue measure of the tetrad coordinates, we can write this
probability measure as $dP(n^{\mu})=P(\lambda,\bar{k})d\lambda d\Omega$, where
$d\Omega$ is the canonical measure on $S^{2}$ and $d\lambda$ the canonical
measure on ${\mathord{\mathbb{R}}}_{+}$. We define the average over the light
cone at a point $x$ by

$\langle n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}=\int_{R_{+}\times
S^{2}}\lambda(e_{0}^{\mu}+\bar{k}\cdot\bar{e}^{\mu})P(\lambda,\bar{k})d\lambda
d\Omega.$ (3.6)

Following Padmanabhan we define the variance analogously.

######  Definition 3.1 (Gravitational Dissipation).

We define the gravitational dissipation as the variance over the light cone

$\sigma(P)^{\mu\nu}=\langle
n^{\mu}n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}-\langle
n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}\langle
n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}.$ (3.7)

Here $\langle\cdot\rangle_{\mathcal{L}_{x}}$ again refers to the average over
the configuration space of the internal variable $n^{\mu}$ (i.e. the light
cone). In analogy to fluid mechanics, where

$\Sigma^{ab}=\langle p^{a}p^{b}\rangle_{p}-\langle p^{a}\rangle_{p}\langle
p^{b}\rangle_{p}$ (3.8)

is the dissipation tensor and $\langle\cdot\rangle_{p}$ denotes the average
over momentum space with $P^{a}=\langle p^{a}\rangle_{p}$ being the flow
velocity of the macroscopic fluid. For the collection of (pure) states
considered by Padmanabhan [62] $\sigma^{\mu\nu}$ is a small correction
originating from quantum gravity (as diffusion is a small effect compared to
the overall flow of a fluid) and $l^{\mu}=\langle n^{\mu}\rangle$ is a null
vector depending on the single quantum degree of freedom at $x$ through the
specific state $P(n(x)^{\mu},x)$. Therefore a variation of the physical system
with respect to $P(n(x)^{\mu},x)$ is equivalent to a variation of the null
vector $l^{\mu}$. In Padmanabhan’s approach, requiring extremization of total
entropy (of spacetime and matter degree of freedom) with respect to
$P(n(x)^{\mu},x)$ hence leads to

$\delta[(R_{\mu\nu}-\kappa T_{\mu\nu})\langle
n^{\mu}n^{\nu}\rangle]\approx\delta[(R_{\mu\nu}-\kappa
T_{\mu\nu})l^{\mu}l^{\nu}]=0,$ (3.9)

subject to the constraint $l^{\mu}l_{\mu}=0$. Where we used that $\delta
P\sim\delta l^{\mu}$. To leading order one thus gets that

$(R_{\mu\nu}-\kappa T_{\mu\nu})\langle
n^{\mu}n^{\nu}\rangle\approx(R_{\mu\nu}-\kappa T_{\mu\nu})\langle
n^{\mu}\rangle\langle n^{\nu}\rangle=(R_{\mu\nu}-\kappa
T_{\mu\nu})l^{\mu}l^{\nu}=0.$ (3.10)

has to hold for all states $P$ and hence for all null vectors $l^{\mu}$.
Demanding again, that the stress-energy tensor be divergence free
$\nabla^{\mu}T_{\mu\nu}=0$, this leads to the Einstein field equations with a
cosmological constant as a constant of integration, along the lines of the
original argument by Jacobson.

As we shall see in the following, extracting expectation values of mixed
states strongly impacts the argument just mentioned. Indeed, it is a well-
known fact in Lorentzian geometry that the average of any two future (past)
directed light-like vectors can only be null if the vectors are linearly
dependent. This is the starting point for our results below.

##  4\. A Probability Measure on the Light Cone

In this section we will assume two things:

  1. (1)

The configuration space for the internal degrees of freedom for gravitation is
the PLCB.

  2. (2)

In analogy to Boltzmann’s phase-space density $f(x,p^{\mu})$ for fluids, we
will assume a field of invariant Borel measures $dP(x,n^{\mu})$ on the PLCB.
If, in a fiber ${\mathcal{L}_{x}}$ at a point $x$, the measure is absolutely
continuous with respect to the Lebesgue measure of the tetrad coordinates,
then it can be written in terms of the coordinates introduced above as
$P(\lambda,\bar{k})d\lambda d\Omega$, where $P(\lambda,\bar{k})\geq 0$ is
nonnegative.

######  Remark 4.1.

In an abuse of notation, we will also write Dirac measures in terms of
$P(\lambda,\bar{k})$ with Dirac distributions.

The goal of this section is to establish the properties of the statistical
quantities required for Padmanabhan’s formalism, in the context of mixed
states. For now, we ignore the spatial dependence and consider $dP(n^{\mu})$
to be a probability measure on the past light cone at a point. This means that
we assume

$\int_{R_{+}\times S^{2}}P(\lambda,\bar{k})d\lambda d\Omega=1,$ (4.1)

and for the variance to be finite

$\int_{R_{+}\times S^{2}}\lambda^{2}P(\lambda,\bar{k})d\lambda
d\Omega<\infty.$ (4.2)

As a warmup, we show the following lemma which is a well-known fact in
Lorentzian geometry.

######  Lemma 4.2.

The vector $\langle n^{\mu}\rangle_{\mathcal{L}_{x}}$555It is tempting to
associate $-t^{\mu}=-\langle n^{\mu}\rangle_{\mathcal{L}_{x}}$ with the flow
of time, given it also shows up in the “classical” thermodynamic arguments
[79]. Therefore, it would be tempting to identify $-t^{\mu}$ as the arrow of
time.  is time-like except in the case where
$P(\lambda,\bar{k})=P(\lambda)\delta(\bar{k}-\bar{k}_{0})$.

###### Proof.

For $P(\lambda,\bar{k})=P(\lambda)\delta(\bar{k}-\bar{k}_{0})$ it is clear
that

$\langle
n^{\mu}\rangle_{\mathcal{L}_{x}}=\int_{R_{+}}\lambda(e_{0}^{\mu}+\bar{k}_{0}\cdot\bar{e}^{\mu})P(\lambda)d\lambda=(e_{0}^{\mu}+\bar{k}_{0}\cdot\bar{e}^{\mu})\int_{R_{+}}\lambda
P(\lambda)d\lambda.$ (4.3)

and therefore $\langle n^{\mu}\rangle_{\mathcal{L}_{x}}$ is still a null
vector. Now we treat the case of a general $P(\lambda,\bar{k})$. For that, it
is convenient to introduce the marginal probability density

$P(\lambda)=\int_{S^{2}}P(\lambda,\bar{k})d\Omega.$ (4.4)

$P(\bar{k})$ is then defined analogously. In addition, it is convenient to
define

$\lambda_{\text{avg}}=\int_{R_{+}}\lambda P(\lambda)d\lambda.$ (4.5)

This allows us to write the expectation value over the light cone as

$\langle
n^{\mu}\rangle_{\mathcal{L}_{x}}=\lambda_{\text{avg}}\left(e_{0}^{\mu}+\int_{R_{+}\times
S^{2}}\frac{\lambda P(\lambda,\bar{k})\bar{k}}{\lambda_{\text{avg}}}d\Omega
d\lambda\cdot\bar{e}^{\mu}\right)$ (4.6)

Now it is clear that $\int_{S^{2}}P(\lambda,\bar{k})\bar{k}d\Omega$ is a
convex combination over $S^{2}$ for more than one $\lambda$ and therefore that
the inequality

$\displaystyle\left|\int_{{\mathord{\mathbb{R}}}_{+}\times S^{2}}\frac{\lambda
P(\lambda,\bar{k})\bar{k}}{\lambda_{\text{avg}}}d\Omega d\lambda\right|$
$\displaystyle\leq\int_{{\mathord{\mathbb{R}}}_{+}}\frac{\lambda}{\lambda_{\text{avg}}}\left|\int_{S^{2}}P(\lambda,\bar{k})\bar{k}d\Omega\right|d\lambda$
(4.7)
$\displaystyle\leq\int_{{\mathord{\mathbb{R}}}_{+}}\frac{\lambda}{\lambda_{\text{avg}}}\left(\int_{S^{2}}P(\lambda,\bar{k})|\bar{k}|d\Omega\right)d\lambda$
(4.8)
$\displaystyle=\int_{{\mathord{\mathbb{R}}}_{+}}\frac{\lambda}{\lambda_{\text{avg}}}P(\lambda)d\lambda=1,$
(4.9)

where $|\cdot|$ is the Euclidean scalar product, in three dimensions is strict
unless $P(\lambda,\bar{k})$ is of the form
$P(\lambda,\bar{k})=P(\lambda)\delta(\bar{k}-\bar{k}_{0})$. This finishes the
argument. ∎

Note, that the exception of course includes the pure states considered by
Padmanabhan with
$P(\lambda,\bar{k})=\delta(\lambda-\lambda_{0})\delta(\bar{k}-\bar{k}_{0})$.
The following corollary will be useful for further calculations.

######  Corollary 4.3.

For a probability measure over the light cone $dP(n^{\mu})$ that is not of the
form $P(\lambda,\bar{k})=P(\lambda)\delta(\bar{k}-\bar{k}_{0})$, we can choose
an orthonormal basis $e_{0}^{\mu},e_{1}^{\mu},e_{2}^{\mu},e_{3}^{\mu}$, that
is, a coordinate system such that

$\int_{{\mathord{\mathbb{R}}}_{+}\times S^{2}}\frac{\lambda
P(\lambda,\bar{k})\bar{k}}{\lambda_{avg}}d\lambda d\Omega=0$ (4.10)

and thus

$\langle
n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}=\lambda_{\text{avg}}e_{0}^{\mu}.$ (4.11)

In the next step, we will show that every state that is not of the form
$P(\lambda,\bar{k})=P(\lambda)\delta(\bar{k}-\bar{k}_{0})$ can be normalized.

######  Lemma 4.4.

Let $dP(n^{\mu})$ be a probability measure that is not of the form
$P(\lambda,\bar{k})=P(\lambda)\delta(\bar{k}-\bar{k}_{0})$, and
$e_{0}^{\mu},e_{1}^{\mu},e_{2}^{\mu},e_{3}^{\mu}$ a tetrad choosen according
to Corollary 4.3, then we can always find a related probability measure
$d\tilde{P}(n^{\mu}):=dP(\alpha n^{\mu})$ such that $\langle
n^{\mu}\rangle_{\mathcal{L}_{x}}$ is a unit timelike vector.

Let $\tilde{P}(\lambda,\bar{k})=P(\alpha\lambda,\bar{k})$ with $\alpha>0$ then
we have

$\int_{R_{+}\times
S^{2}}\lambda(e_{0}^{\mu}+\bar{k}\cdot\bar{e}^{\mu})\tilde{P}(\lambda,\bar{k})d\lambda
d\Omega=\int_{R_{+}\times
S^{2}}\lambda(e_{0}^{\mu}+\bar{k}\cdot\bar{e}^{\mu})P(\alpha\lambda,\bar{k})d\lambda
d\Omega\phantom{\frac{1}{\alpha^{2}}}$ (4.12)

Replacing $\frac{\tilde{\lambda}}{\alpha}=\lambda$ and
$\frac{d\tilde{\lambda}}{\alpha}=d\lambda$ we get

$\displaystyle=\frac{1}{\alpha^{2}}\int_{R_{+}\times
S^{2}}\tilde{\lambda}(e_{0}^{\mu}+\bar{k}\cdot\bar{e}^{\mu})P(\tilde{\lambda},\bar{k})d\tilde{\lambda}d\Omega$
(4.13) $\displaystyle=\frac{\lambda_{\text{avg}}}{\alpha^{2}}e_{0}^{\mu}$
(4.14)

Setting $\alpha^{2}=\lambda_{\text{avg}}$ gives us

$\langle n^{\mu}\rangle_{{\mathcal{L}_{x}}(\tilde{P})}=e_{0}^{\mu}.$ (4.15)

This finished the argument.

######  Definition 4.5.

We will call a probability measure $d\tilde{P}$ with $\langle
n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}=e_{0}^{\mu}$ a normalized state.

The above lemma gives us a relation between the normalization of the resulting
timelike vector field and the change in the underlying probability measure.

We are now ready to study the variance of a probability measure on the light
cone. For that, we need the following definition.

######  Definition 4.6.

We call $P(\lambda,\bar{k})$ degenerate of order $n$ if there exist $n$
linearly independent vectors $k_{i}\in S^{2}$ such that

$supp(P(\lambda,\bar{k}))\subset span\\{k_{i}|i\leq n\in\mathbb{N}\\}^{\perp}$
(4.16)

or if there are $n-1$ such vectors and

$P(\lambda)=\delta(\lambda-\lambda_{0}).$ (4.17)

######  Theorem 4.7.

For any non-degenerate $P(\lambda,\bar{k})$

$\sigma(P)^{\mu\nu}=\langle
n^{\mu}n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}-\langle
n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}\langle
n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}$ (4.18)

is strictly positive.

###### Proof.

We choose an orthonormal basis according to Corollary 4.3. Then we have the
corresponding co-tangent basis
$e^{0}_{\mu},e^{1}_{\mu},e^{2}_{\mu},e^{3}_{\mu}$. Due to orthogonality for
$i=\\{1,2,3\\}$ we have

$\displaystyle\left(\langle
n^{\mu}n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}\right.$
$\displaystyle-\left.\langle n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}\langle
n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}\right)e^{i}_{\mu}e^{i}_{\nu}=\langle
n^{\mu}n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}e^{i}_{\mu}e^{i}_{\nu}$ (4.19)
$\displaystyle=\int_{R_{+}\times
S^{2}}\lambda^{2}(e_{0}^{\mu}+\bar{k}\cdot\bar{e}^{\mu})(e_{0}^{\nu}+\bar{k}\cdot\bar{e}^{\nu})P(\lambda,\bar{k})d\lambda
d\Omega e^{i}_{\mu}e^{i}_{\nu}$ (4.20) $\displaystyle=\int_{R_{+}\times
S^{2}}\lambda^{2}(k_{i})^{2}P(\lambda,\bar{k})d\lambda d\Omega$ (4.21)

This is positive by our assumption, that $P(\lambda,\bar{k})$ be non-
degenerate. For $e_{0}$ we need a different argument. We calculate

$\displaystyle\left(\langle
n^{\mu}n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}\right.$
$\displaystyle-\left.\langle n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}\langle
n^{\nu}\rangle_{{\mathcal{L}_{x}}(P)}\right)e^{0}_{\mu}e^{0}_{\nu}=\int_{R_{+}}\lambda^{2}P(\lambda)d\lambda-\left(\int_{R_{+}}\lambda
P(\lambda)d\lambda\right)^{2}$
$\displaystyle=\int_{R_{+}}P(\kappa)d\kappa\int_{R_{+}}\lambda^{2}P(\lambda)d\lambda-\left(\int_{R_{+}}\lambda
P(\lambda)d\lambda\right)\left(\int_{R_{+}}\kappa P(\kappa)d\kappa\right)$
$\displaystyle=\int_{R_{+}\times
R_{+}}\lambda(\lambda-\kappa)P(\lambda)P(\kappa)d\lambda d\kappa\geq 0$ (4.22)

The positivity follows from Lemma A.1 in Appendix A due to the fact that
$P(\lambda)P(\kappa)$ is positive and symmetric. This concludes the proof. ∎

##  5\. Some Observations

In the following we apply the results of the previous section to the variation
of entropy. Clearly, in general we can not apply straightforwardly
Padmanabhan’s extremization of entropy setting the variation of the state $P$
equal with the variation of the null vector $l^{a}$. Instead we always have to
carry the variation with respect to the state $P$ along which leads to

$\displaystyle 0=\delta[(R_{\mu\nu}-\kappa T_{\mu\nu})\langle
n^{\mu}n^{\nu}\rangle]$ $\displaystyle=\delta[(R_{\mu\nu}-\kappa
T_{\mu\nu})(\langle n^{\mu}\rangle\langle n^{\nu}\rangle+\sigma(P)^{\mu\nu}]$
(5.1) $\displaystyle\Updownarrow$ $\displaystyle\delta[(R_{\mu\nu}-\kappa
T_{\mu\nu})\sigma(P)^{\mu\nu}]$ $\displaystyle=-\delta[(R_{\mu\nu}-\kappa
T_{\mu\nu})\langle n^{\mu}\rangle\langle n^{\nu}\rangle]$ (5.2)

at equilibrium. We now restrict to variations such that

$\delta[(R_{\mu\nu}-\kappa T_{\mu\nu})\langle n^{\mu}\rangle\langle
n^{\nu}\rangle]=(R_{\mu\nu}-\kappa T_{\mu\nu})\delta[\langle
n^{\mu}\rangle\langle n^{\nu}\rangle]$ (5.3)

under the constraint $\langle n_{\nu}\rangle\langle n^{\nu}\rangle=-C$.
Following [73] this implies that the Einstein equation hold

$R_{\mu\nu}-\frac{1}{2}R+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}$ (5.4)

with the cosmological constant originating from the Lagrange multiplier. If we
then formally integrate equation (5.2) we get

$(R_{\mu\nu}-\kappa
T_{\mu\nu})\sigma(P)^{\mu\nu}=-\lambda_{\text{avg}}^{2}(R_{\mu\nu}-\kappa
T_{\mu\nu})e_{0}^{\mu}e_{0}^{\nu}+G$ (5.5)

with $G=(R_{\mu\nu}-\kappa T_{\mu\nu})\langle n^{\mu}n^{\nu}\rangle$ following
from (5.1), a constant independent of $P$. We now make use of the fact that
the Einstein field equations hold for this system (with the trace term either
on the geometric side or on the matter side)

$\displaystyle R_{\mu\nu}-\kappa T_{\mu\nu}$
$\displaystyle=\left(\Lambda-\frac{\kappa}{2}T\right)g_{\mu\nu}$ (5.6)
$\displaystyle R_{\mu\nu}-\kappa T_{\mu\nu}$
$\displaystyle=\left(\frac{1}{2}R-\Lambda\right)g_{\mu\nu}$ (5.7)

to replace the relevant term on the right-hand side of (5.2) we get the
following intriguing result

$(R_{\mu\nu}-\kappa
T_{\mu\nu})\frac{\sigma(P)^{\mu\nu}}{\lambda_{\text{avg}}^{2}}=\Lambda-\frac{\kappa}{2}T+\left(\Lambda-\frac{\kappa}{2}T\right)g_{\mu\nu}\langle
n^{\mu}n^{\nu}\rangle=\Lambda-\frac{\kappa}{2}T=\frac{1}{2}R-\Lambda$ (5.8)

where we used $g_{\mu\nu}\langle n^{\mu}n^{\nu}\rangle=\langle
g_{\mu\nu}n^{\mu}n^{\nu}\rangle=0$ in the second step. Similarly we get

$(R_{\mu\nu}-\kappa
T_{\mu\nu})\sigma(\tilde{P})^{\mu\nu}=\Lambda-\frac{\kappa}{2}T=\frac{1}{2}R-\Lambda$
(5.9)

for normalized states. At the present state of our understanding we do not
want to interpret too much into this relation but report it as something
reassuring and promising. Nevertheless, it is tempting that the cosmological
constant shows up either way, especially in light of the fact that so much of
Padmanabhan’s work was geared towards explaining the cosmological constant.

As a final remark, it is interesting to observe that a mixed state gives rise
to a vector field $t^{\mu}=\langle n^{\mu}\rangle_{\mathcal{L}}$ relevant for
the computation similar to the regularizing vector fields that appear in
Causal Fermion Systems in the context of baryogenesis [22]. In light of the
results in [39], it is interesting to note that this vector field seems to
have no dynamical relevance, but is more of a sort of a bookkeeping device. In
accordance with classical thermodynamic treatments, there is a plethora of
states $P(\lambda,\bar{k})$ that lead to the same right-hand side in (5.9).
However, as a bookkeeping device, it might be relevant in the context of
Padmanabhan’s earlier considerations. In [70, 54] he derived GR in terms of
thermodynamically conjugate variables, and in [58, 59, 56, 55, 53] he derived
what he calls the “holographic equipartition” for stationary spacetimes. All
of these considerations feature the choice of a vector field. In Appendix B we
calculated this vector field for several forms of $P(\lambda,\bar{k})$ that
are relevant in Causal Fermion Systems, and it turns out that
$\lambda_{\text{avg}}$ is proportional to the inverse of the regularization
length $\varepsilon$, which can be thought to be of the order of the Planck
length.

##  6\. Modified Measure Theories from a Density of States

The Hawking-King-McCarthy and Malament theorem [32, 48] states that under very
weak causality conditions the causal order $(M,\prec)$ determines the metric
$(M,g)$ up to a conformal factor. Formally, this implies that $(M,g)$ is
equivalent to $(M,\prec)$ plus a volume form $\Phi$, or colloquially:
spacetime is equal to causal order plus volume. The attentive reader might
notice that, by working on the light cone, we already fixed the causal
structure, so the only variable left is the conformal factor of the metric. If
we drop the requirement typically invoked in thermodynamic derivations of
gravity that the stress-energy tensor of matter needs to be divergence free
$\nabla^{\mu}T_{\mu\nu}=0$, then this allows for MMT. Instead of integrating
the Lagrangian against the measure $\sqrt{-g}d^{4}x$ MMT promote the measure
to be an independent quantity. The total action is then given by

$S=\int_{M}L\Phi(A)d^{4}x\qquad\text{ with }\qquad
L=\frac{-1}{\kappa}R(\Gamma,g)+L_{m},$ (6.1)

where $\kappa$ is the gravitational coupling constant, the scalar curvature is
given in terms of the connection and the metric, and the measure by

$\Phi(A)=\frac{1}{6}\varepsilon^{\alpha\beta\mu\nu}\partial_{\alpha}A_{\beta\mu\nu},$
(6.2)

where $A_{\beta\mu\nu}$ is the tensor gauge potential of a non-singular exact
$4$-form $\omega=dA$. Thus, the non-Riemannian volume element density
$\Phi(A)$ is given by the scalar density of the dual field-strength associated
with that potential. As a result, $\Phi(A)d^{4}x$ is invariant under general
coordinate transformations. This introduces an extra scalar degree of freedom
as can be easily seen by rewriting $\Phi(A)$ as $\chi(x)\sqrt{-g}$ where
$\chi(x)=\frac{\Phi(A)}{\sqrt{-g}}$.

In the following, we will show that when working with a measure on the light
cone, it is straightforward to implement it in such a way that it naturally
leads to MMT. For that we will relax assumption (4.1) and reinstate the
dependence of the state $dP(x,n^{\mu})$ on the spacetime point $x$. In the
following, we assume a global time-like vector field $t^{\mu}$ such that the
condition from Corollary 4.3 is satisfied at every point. We then make the
following definition.

######  Definition 6.1 (Density of gravitational states).

Given a measure $dP(x,n^{\mu})$ on the PLCB which can be represented in terms
of coordinates by $P(x,\lambda,\bar{k})d\lambda d\Omega\sqrt{|g|}d^{4}x$ the
density of gravitational states is given by

$\chi(x)=\int_{R_{+}\times S^{2}}P(x,\lambda,\bar{k})d\Omega d\lambda.$ (6.3)

While in the previous discussion we have fixed $\chi(x)=1$ we now allow it to
take any positive value. This definition is in perfect analogy with the number
density of particles in Boltzmann’s treatment of fluids 666 In this case we
have $n(x)=\int_{\Omega(p^{\mu})}f(x,p^{\mu})d\omega$, where $p^{\mu}$ are the
internal degrees of freedom of the microscopic degrees of freedom,
$\Omega(p^{\mu})$ is the domain of these internal degrees of freedom and
$d\omega$ is a suitable measure on $\Omega(p^{\mu})$.  and $\chi(x)$ counts
the number of gravitational states that contribute to the volume density at a
point $x$.

We are now ready to formulate an action on $\mathcal{L}$ as

$S[g_{\mu\nu},L_{m},P(x,\lambda,\bar{k})]=\int_{\mathcal{L}}\left(\frac{-1}{\kappa}R+L_{m}\right)dP(x,n^{\mu})=\int_{\mathcal{M}}\left\langle\frac{-1}{\kappa}R+L_{m}\right\rangle_{\mathcal{L}_{x}}\sqrt{-g}d^{4}x$
(6.4)

which is to be minimized. A short calculation

$\displaystyle S[g_{\mu\nu},L_{m},P(x,\lambda,\bar{k}]$
$\displaystyle=\int_{\mathcal{L}}\left(\frac{-1}{\kappa}R+L_{m}\right)P(x,\lambda,\bar{k})\sqrt{-g}d^{4}xd\lambda
d\Omega$ (6.5)
$\displaystyle=\int_{M}d^{4}x\sqrt{-g}\left[\left(\frac{-1}{\kappa}R+L_{m}\right)\int_{R_{+}\times
S^{2}}P(x,\lambda,\bar{k})d\Omega d\lambda\right]$ (6.6)
$\displaystyle=\int_{M}\left(\frac{-1}{\kappa}R+L_{m}\right)\chi(x)\sqrt{-g}d^{4}x=\int_{M}\left(\frac{-1}{\kappa}R+L_{m}\right)\Phi_{A}d^{4}x.$
(6.7)

shows that this gives rise to a spacetime action with a modified measure. Now
what happens if we minimize (6.7) with respect to $P(x,\lambda,\bar{k})$ (i.e.
if we vary $P(x,\lambda,\bar{k})$ itself)777Conceptually a variation of
$P(x,\lambda,\bar{k})$ represents a variation of the distribution of the
momenta of the underlying gravitational states.? The answer is straightforward

$\displaystyle\int_{M}$
$\displaystyle\left\langle\frac{-1}{\kappa}R+L_{m}\right\rangle_{{\mathcal{L}_{x}}(P+\delta
P)}\sqrt{-g}d^{4}x$ (6.8)
$\displaystyle=\int_{M}d^{4}x\sqrt{-g}\left[\left(\frac{-1}{\kappa}R+L_{m}\right)\int_{R_{+}\times
S^{2}}P(x,\lambda,\bar{k})+\delta P(x,\lambda,\bar{k})d\Omega d\lambda\right]$
(6.9)
$\displaystyle=\int_{M}\left(\frac{-1}{\kappa}R+L_{m}\right)[\chi(x)+\delta\chi(x)]\sqrt{-g}d^{4}x,$
(6.10)

where $\delta\chi(x)=\int_{R_{+}\times S^{2}}\delta
P(x,\lambda,\bar{k})d\Omega d\lambda$. Therefore, we see immediately that a
variation of the state $P(x,\lambda,\bar{k})$ induces a variation in the
modified measure, and hence, if we vary the state under the requirement that
the variation respect a volume constraint, as is usually required in MMT, the
dynamics obtained from this variation principle are identical to that of MMT
with a single modified measure. Following [28] we get the following system of
equations

$\displaystyle
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{\kappa}{2}\left(T_{\mu\nu}+Mg_{\mu\nu}\right)+\frac{1}{\chi}\left(\chi_{,\mu;\nu}-\right.$
$\displaystyle\left.g_{\mu\nu}\,\Box\,\chi\right),$ (6.11)
$\displaystyle\Box\,\chi-\frac{\kappa}{D-1}\left[\left(M+\frac{1}{2}T\right)+\frac{(D-2)}{2}L_{m}\right]$
$\displaystyle\chi=0.$ (6.12)

Where $T$ is the trace of the stress energy tensor and the integration
constant $M=-\frac{2\Lambda}{\kappa}$ takes the role of the cosmological
constant. We see that this system of equations can only give rise to solutions
compatible with the equations of Einstein’s General Relativity if
$\frac{1}{\chi}\left(\chi_{,\mu;\nu}-g_{\mu\nu}\Box\chi\right)=0$. This holds,
e.g., if $\chi=\text{const.}$ which requires
$\left(M+\frac{1}{2}T\right)+\frac{(D-2)}{2}L_{m}=0$. That is, for the right
hand side of (3.7) to vanish and for $L_{m}=0$, which can be satisfied if we
minimize the action with respect to the fields in $L_{m}$. An interesting
consequence of the above equations of motion is that we get a non-conservation
of the conventionally defined matter stress energy that does not include a
contribution from the field $\chi$,

$\nabla^{\mu}T_{\mu\nu}=-2\,\frac{\partial L_{m}}{\partial
g^{\mu\nu}}\,g^{\mu\alpha}\nabla_{\alpha}\ln{\chi}.$ (6.13)

This explains why the conventional derivation of the Einstein field equations
from thermodynamic considerations never considered MMT as a possible
alternative solution compatible with the usual arguments on the light cone.

Now what happens if instead of general variations of the state
$P(x,\lambda,\bar{k})$, we constrain the variation to a subclass of states
with $\chi(x)=\text{const.}$? This leads us directly to unimodular gravity.

###  6.1. Unimodular Gravity from a Density of States.

Unimodular gravity was originally developed with a goal similar to MMT. If one
limits the class of admissible variations of the metric to those that leave
$\sqrt{-g}$ invariant, then adding a constant to the Lagrangian will not
change the gravitational equations of motion. UG has been shown to appear as a
special case of MMT with two measures when formulated in a generally covariant
form [35]

$\mathcal{S}=\int d^{4}x\,\sqrt{-g}\,(R+2\Lambda+\mathcal{L}_{m})-\int
d^{4}x\,\Phi(A)\,2\Lambda\>.$

Here, a priori, $\Lambda$ is a dynamical scalar field, and $\Phi(A)$ is as
above. However, variations with respect to $A$ imply that $\Lambda$ is a
constant, whereas variations with respect to $\Lambda$ yield
$\Phi(A)=\sqrt{-g}$.

This variation principle can be reformulated in terms of a measure on the PLCB
as a minimization of

$\mathcal{S}=\int
d^{4}x\sqrt{-g}\left\langle(R+2\Lambda+\mathcal{L}_{m})\right\rangle_{\mathcal{L}_{x}}$
(6.14)

under the constraint

$\int
d^{4}x\sqrt{-g}\left\langle\Lambda\right\rangle_{\mathcal{L}_{x}}=const.$
(6.15)

If we vary the constraint with respect to $P(x,\lambda,\bar{k})$ then we get
as above, that $\Lambda$ is a constant, while variations with respect to
$\Lambda$ yields that $\chi(x)$ must be constant. All other variations proceed
as usual. Hence, the constraint restricts us to the subclass of measures
$dP(x,n^{\mu})$ for which the density of states is constant in spacetime. Note
that from a thermodynamic perspective, the constraint (6.15) now has a clear
interpretation: it is nothing else than the average energy $\langle E\rangle$
per state. This is again remeniscent of Padmanabhan’s holographic
equipartition [55] and allows us to interpret the cosmological constant as the
average energy per gravitational degree of freedom and the constraint as
Boltzmann’s equipartition of energy per degree of freedom888Alternatively one
could try to interpret it as the average spacetime volume per gravitational
degree of freedom..

At this point it is worth noting again that none of the variation principles
in this section depend on the time-like vector field that we can associate
with $dP(x,n^{\mu})$ and accordingly this is truly just a bookkeeping device
and we need not worry about the phenomenological constraints discussed in
[39]. We would also remind the reader that the physically relevant quantity
derived from $dP(x,n^{\mu})$ is Lorentz invariant and that, in general, there
is a huge number of measures $dP(x,n^{\mu})$ with the same $\chi(x)$ and
$\langle n^{\mu}\rangle_{\mathcal{L}_{x}}$.

##  7\. Summary and Outlook

For better readability, we split this section into three parts. A short
summary, an outlook that ranges from concrete to speculative, and the
conclusion.

###  7.1. Summary

On the mathematical side, starting from an invariant probability measure on
the light cone, we presented in full detail the well-known fact from
Lorentzian geometry that an average over the light cone always leads to a
time-like vector field. Furthermore, we showed that the variance associated
with any nondegenerate measure is a positive two tensor.

On the physical side, we showed that the same setup can give rise to modified
measures in a straightforward manner. Furthermore, we showed that unimodular
gravity (UG), which is a subset of modified measure theories (MMT), can be
characterized in terms of thermodynamic concepts. While MMT corresponds to the
grand canonical ensemble where the number of gravitational degrees of freedom
is free to change, UG corresponds to the canonical ensemble where the density
of gravitational states is constant throughout spacetime. This aligns neatly
with the observation in [22] in the context of causal fermion systems (CFS)
that the matter/antimatter asymmetry originated in a shift of states from the
gravitational sector to the matter sector. The present work strengthens the
case for modified measures playing a key role in this mechanism. The fact that
the stress energy tensor is not conserved in MMT is, in this context, not a
bug but an essential feature.

###  7.2. Outlook

In the present work, we established the basic viability of a theory assuming a
field of probability measure or a density of states. We will now show that
this rather simple idea actually opens a window to a wide variety of
possibilities for future avenues of research, some more mathematical in
nature, some more physical.

First, as noted in the body of the text, working on the light cone a priori
assumes a lot of structure, as the light cone fixes the metric except for the
conformal factor. If we intend for the formalism to have any bearing on the
problem of quantum gravity, this is clearly not satisfactory. For this we
would want to be able to derive the metric structure of spacetime from the
fundamental ingredients alone. In an upcoming paper [69] some of the present
authors will show that we can indeed strip away most of the structure assumed
here and reconstruct it instead from more fundamental assumptions.

From the thermodynamic viewpoint, the present paper raises several interesting
questions. One is whether we can gain a deeper understanding of the
formulation of GR with respect to conjugate variables, cf. [70, 59], where a
vector field $\xi$ plays an important role. It is tempting to identify this
vector field as the one derived from the state on the light cone. This is
connected to another question we have left unanswered in the present work,
namely how to connect $dP(x,n^{\mu})$ to some notion of entropy of spacetime.

As a follow-up the same question can be asked in the context of Padmanabhan’s
holographic equipartition in static spacetimes [58, 59, 56, 55, 53]. Here, two
aspects seem particularly interesting. First, his argument only works for a
particular vector field, i.e., in the present language a particular class of
measures. Second, for static spacetimes we can make a global split into
positive and negative frequencies for the Dirac equation. For CFS this implies
that we can define the Dirac sea globally. If, as suggested by the derivation
of the Dirac equation in Minkowski space [16], we assume the gravitational
degrees of freedom to be in the Dirac sea, then in this setting we get a clean
split between the matter degrees of freedom and the gravitational ones. On a
more speculative note, this line of investigation could suggest that, in this
setting, holography only applies to static or stationary spacetimes.

Furthermore, it would be interesting to see whether the present formalism can
be connected to the line of reasoning developed in [75, 76, 77, 78].
Especially the ideas in the most recent paper in this series [78] might
connect to the results some of the present authors are preparing in the
aforementioned follow-up paper [69].

Finally, it seems possible to connect the formalism presented here with
Verlinde’s derivation of Modified Netwonian Dynamics from entropic arguments
[86, 85]. In particular, suppose that we consider the interacting region of de
Sitter space in static coordinates.

$ds^{2}=\left(1-\frac{1}{3}\Lambda
r^{2}\right)dt^{2}-\left(1-\frac{1}{3}\Lambda
r^{2}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$ (7.1)

If we start with a probability measure $P(x,n^{\mu})$ on the PLCB such that
$\partial_{t}=\xi=\langle n^{\mu}\rangle_{P}$ at all points $x\in\mathcal{M}$
with $r(x)\leq r_{\mathcal{H}_{C}}$ where $r_{\mathcal{H}_{C}}$ is the
coordinate radius of the cosmological horizon. That is, the time-like vector
field $\xi$ associated with the measure $dP(x,n^{\mu})$ is the hypersurface
normal with respect to a $t=const$ surface in that coordinate system. In this
setup Padmanabhan’s holographic equipartition of states [58, 59, 56, 55, 53]
applies, which fits with Verlinde’s assumption that the number of bulk degrees
of freedom is equal to the number of surface degrees of freedom calculated
from the cosmological horizon entropy.

Now if one wants to realize Verlinde’s perturbation of de Sitter space by
adding a small mass $M$ at the center of the interacting region, one has to
look for a suitable perturbation of the measure $dP(x,n^{\mu})$. From the
calculations in Section 4 a variation $\delta P$ corresponds to a variation of
the time-like vector field associated with it

$\langle n^{\mu}\rangle_{{\mathcal{L}_{x}}(P+\delta P)}=\xi+\delta u.$ (7.2)

Given that $\langle n^{\mu}\rangle_{{\mathcal{L}_{x}}(P)}$ is a hypersurface
normal vector field, its variation $\delta u$ will necessarily be tangential
to the hypersurface $t=const$ for any possible variation. Choosing $\delta P$
to be spherically symmetric around the center of the interacting region seems
to be a good starting point for an attempt to reproduce his results from our
formalism. One key step is to relate such a variation $\delta P$ to the
addition of a perturbative mass in the center of the interacting region used
by Verlinde in his argument. Another question is whether we can associate
$\delta P$ with a change in entropy in a suitable sense to match Verlinde’s
argument. It seems important to note here that Padmanabhan’s holographic
equipartition breaks down for the perturbed vector field.

Many of our considerations in this paper were informed by the CFS theory. In
particular, the fact that we chose to define the measure $dP(x,n^{\mu})$ as a
mixed state on the past light cone bundle (PLCB) was motivated by the
prominent role that the Dirac sea plays in CFS. Given the emergent nature of
spacetime in CFS it is natural to think about a thermodynamic interpretation
for the theory. A first step in that direction will be taken in our
forthcoming comparison paper between the two approaches [20] where the focus
will be on the role of a minimal length scale [65]. Another approach is to try
and interpret the weight of the measure $P(x,\lambda,\bar{k})$ as a weight in
the symbol of the fermionic projector. Dropping the spacetime dependence, this
motivated the particular choices of $P(\lambda,\bar{k})$ in Appendix B where
$P_{1}$ corresponds to the so-called hard cut off regularization and $P_{2}$
corresponds to the $i\varepsilon$ regularization. One could then try to
interpret the fermionic projector as the Wigner-Weyl quantization cf.[14] of
the density of states and relate it to the ideas in [64] concerning spacetime
correlations.

To complete the Boltzmannian picture, the missing piece in the present work is
an evolution equation for the measure $P(x,\lambda,\bar{k})$. Taking again
inspiration from CFS where it was shown in [23] that the regularization
satisfies a transport equation along null geodesics, this would be the natural
first candidate. Similarly, one could postulate for $P(x,\lambda,\bar{k})$ to
satisfy the massless Vlasov equation.

Finally, there are many choices in this paper that one might make differently.
For example, given that our framework is defined on the PLCB, instead of the
action principles (6.7) and (6.14) one could try building an action involving
expressions of the form of the left-hand side of (3.1) or (5.8) depending
explicitly on the null vectors $n^{\mu}$. Given that such a variation
principle contains terms of the form $n^{\mu}n^{\nu}$, it could give rise to
equations of motion quadratic in the vector fields as they appear in [37] for
a covariant version of Verlinde’s emergent gravity.

##  Appendix A Positivity Lemma

First we will prove a the following Lemma

######  Lemma A.1.

Let $P(x,y)$ be positive and symmetric, then

$\int_{R_{+}\times R_{+}}x(x-y)P(x,y)dxdy\geq 0$ (A.1)

###### Proof.

We simply split the integral along the diagonal

$\displaystyle\int_{R_{+}\times R_{+}}x(x-y)P(x,y)dxdy$ (A.2)
$\displaystyle=\int_{0}^{\infty}\int_{y}^{\infty}x(x-y)P(x,y)dxdy+\int_{0}^{\infty}\int^{\infty}_{x}x(x-y)P(x,y)dydx$
(A.3)
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}(l+y)lP(l+y,y)dydl+\int_{0}^{\infty}\int_{0}^{\infty}x(-k)P(x,x+k)dxdk$
(A.4)
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\left[(l+y)lP(l+y,y)-lyP(l,l+y)\right]dydl$
(A.5) $\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}l^{2}P(l+y,y)dydl\geq
0$ (A.6)

Where in (A.4) we set $l=x-y$ in the first integral and $k=y-x$ in the second
integral. In (A.5) we simply relabel the variables in the second integral
$x=l$ and $k=y$ and sum the integrals.

∎

##  Appendix B Average Vector Field for Specific Probability Distributions

In the following, we will calculate $\lambda_{\text{avg}}$ for three
particularly relevant probability measures characterized by their weight
function with respect to a particular choice of tetrad
$P_{0}=\frac{1}{4\pi}\delta(\varepsilon^{-1}-\lambda)$,
$P_{1}=\frac{\varepsilon}{4\pi}\theta(\lambda)\theta(\varepsilon^{-1}-\lambda)$,
and $P_{2}=\frac{\varepsilon}{4\pi}e^{-\varepsilon\lambda}$. Here,
$\theta(\lambda)$ corresponds to the Heaviside function and $P_{1}$
accordingly to the cutoff regularization in CFS. $P_{2}$ on the other hand
corresponds to the $i\varepsilon$ regularization.

It is clear that for $i=\\{0,1,2\\}$

$\int_{R_{+}\times S^{2}}P_{i}(\lambda,\bar{k})d\Omega d\lambda=1$ (B.1)

holds. We get for $P_{0}$

$\displaystyle\lambda_{avg}$ $\displaystyle=\int_{R_{+}\times S^{2}}\lambda
P_{0}(\lambda,\bar{k})d\Omega d\lambda$ (B.2) $\displaystyle=\epsilon^{-1}$
(B.3)

Note that this measure is degenerate of order one, as the zero-zero component
of the variance vanishes. Such states are of interest in our follow-up paper
[69] as they can be related to Hartle-Hawking’s no-boundary proposal [29].

We get for $P_{1}$

$\displaystyle\lambda_{avg}$ $\displaystyle=\int_{R_{+}\times S^{2}}\lambda
P_{1}(\lambda,\bar{k})d\Omega d\lambda$ (B.4)
$\displaystyle=\int_{0}^{\varepsilon^{-1}}\varepsilon\lambda d\lambda$ (B.5)
$\displaystyle=\left[\frac{\varepsilon\lambda^{2}}{2}\right]_{0}^{\varepsilon^{-1}}$
(B.6) $\displaystyle=\frac{\varepsilon^{-1}}{2}.$ (B.7)

We get for $P_{2}$

$\displaystyle\lambda_{avg}$ $\displaystyle=\int_{R_{+}\times S^{2}}\lambda
P_{2}(\lambda,\bar{k})d\Omega d\lambda$ (B.8)
$\displaystyle=\int_{0}^{\infty}\varepsilon\lambda
e^{-\varepsilon\lambda}d\lambda$ (B.9) $\displaystyle=-\left[\lambda
e^{-\varepsilon\lambda}\right]_{0}^{\infty}+\int_{0}^{\infty}e^{-\varepsilon\lambda}d\lambda$
(B.10)
$\displaystyle=-\left[\varepsilon^{-1}e^{-\varepsilon\lambda}\right]_{0}^{\infty}$
(B.11) $\displaystyle=\varepsilon^{-1}$ (B.12)

Therefore all three states are naturally associated with a vector field
$\varepsilon^{-1}e_{0}^{\mu}$ which we can identify with the regularizing
vector field of the locally rigid regularization used for the baryogenesis
result. There is a certain constant rescaling of $\lambda_{avg}$ between
different regularizations that might matter in detailed phenomenological
calculations. Over all this gives a tempting connection to CFS.

Note that it is a curious observation that due to the $S^{2}$ factor in the
light cone we always need to normalize by a factor of $4\pi$, which
Padmanabhan needs to determine the current value of the cosmological constant.
This need not mean much, as it will obviously show up in any calculation that
involves $S^{2}$, however, due to the fact that we demonstrated in (5.8) and
(5.9) that there is a direct connection between $dP$ and the cosmological
constant, there might be something more to it.

Note that if we make any of these states dependent on $x$ by
$P_{0}=\frac{1}{4\pi}\delta(\varepsilon(x)^{-1}-\lambda)$,
$P_{1}=\frac{\varepsilon_{0}}{4\pi}\theta(\lambda)\theta(\varepsilon(x)^{-1}-\lambda)$
and $P_{2}=\frac{\varepsilon_{0}}{4\pi}e^{-\varepsilon(x)\lambda}$ then for
$P_{1}$ and $P_{2}$ we immediately get $\nabla_{\mu}\chi(x)\neq 0$ when we
allow the regularization length $\varepsilon(x)$ to vary, while for $P_{0}$ we
get $\chi(x)=\text{const}$.

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