# Three Views of a Secret in Relativistic Thermodynamics

Tadas K. Nakamura  CFAAS, Fukui Prefectural University<EMAIL_ADDRESS>
###### Abstract

It has been shown three different views in relativistic thermodynamics can be
derived from the basic formulation proposed by van Kampen and Israel. The way
to decompose energy-momentum into the reversible and irreversible parts is not
uniquely determined, and different choices result in different views. The
effect of difference in the definition of a finite volume is also considered.

relativity, thermodynamics, Lorentz transform

There has been long controversy about the relativistic thermodynamics. A
number of theories were proposed in 1960s, and the discussion seems to have
arrived at a vague general agreement that each theory is consistent in its own
framework by early 1970s Yuen (1970). However, papers has been still published
long after that, even until today (e.g., de Parga et al. (2005); Requardt ),
proposing new formulations which are allegedly better than others.

Roughly speaking there are three different views on the relativistic
thermodynamics, which are characterized by the difference in the temperature
$T$ of a moving body in the following:

$\begin{cases}\textnormal{I)}&T=T_{0}\gamma^{-1}\,,\\\
\textnormal{II)}&T=T_{0}\gamma\,,\\\ \textnormal{III)}&T=T_{0}\,.\end{cases}$
(1)

where $T_{0}$ is the temperature measured in a frame comoving with the body,
$\gamma$ is the Lorentz factor defined as $\gamma=1/\sqrt{1-v^{2}}$ with the
speed of the body $v$ relative to the rest frame (we use the unit of $c=1$).

Most of theories proposed so far can be categorized into one of the above
three. Papers published right after the establishment of special relativity
(e.g., Einstein (1907); Planck (1908)) are based on View I. Theories with View
II were extensively investigated in the middle of 1960s (e.g, Gamba (1965);
Kibble (1966)) stimulated by the papers by Ott (1963) and Arzelies (1965). A
little later a theory in View III was proposed by Lansberg (1966).

There is another theory by van Kampen (1968) in View III; his stand point is
quite different from other theories in Views I, II, and III. He treats the
three components of the velocity as thermodynamical parameters in addition to
the temperature. The van Kampen’s theory was later refined by Israel (1976)
into a more transparent form.

The author of the present paper strongly believes this van Kampen-Israel
theory is the one in _the book_ , i.e., the very fundamental one based on
which other formulations can be derived. The purpose of the present paper is
to show the above three views can be actually derived from the van Kampen-
Israel theory. The great advantage of the van Kampen-Israel theory is in the
point that it does not need the concept of heat or work. The second law can be
expressed with well defined mechanical quantities such as the energy-momentum
or four velocity.

The difference of the above three views mainly comes from the difference in
the definition of heat. Non-relativistic thermodynamics decomposes the energy
increase $\Delta E$ into two parts, heat $\Delta Q$ and work $\Delta W$
namely, as $\Delta E=\Delta Q+\Delta W$. Since the energy is one component of
the energy-momentum four vector, the decomposition should be expressed in the
form of four vectors in the relativistic thermodynamics:

$\Delta\bar{G}=\Delta\bar{Q}+\Delta\bar{W}\,,$ (2)

where $\Delta\bar{G}$ is the change of the energy-momentum, and its reversible
and irreversible parts are expressed as $\Delta\bar{W}$ and $\Delta\bar{Q}$;
we denote a four vector as a whole by a bar (e.g., $\bar{G}$) and its each
component by indices (e.g., $G^{\mu}$) in this paper.

Most of theories in Views I and II determine the temperature from the
following entropy expression.

$\Delta S=\frac{\Delta Q}{T}\,,$ (3)

where $\Delta Q$ is the temporal component of $\Delta\bar{Q}$. However, there
is ambiguity in the decomposition in (2) and the heat is not uniquely
determined as we will see in the present paper. Views I and II define the heat
as $\Delta Q=\Delta Q_{0}\gamma^{-1}$ and $\Delta Q=\Delta Q_{0}\gamma$
($\Delta Q_{0}$ is the heat measured in the comoving frame) respectively, and
both definitions are consistent as the temporal component of an irreversible
energy-momentum change. Consequently two different temperatures (Views I and
II) are derived from (1) since the entropy is supposed to be Lorentz
invariant. View III tries to accommodate both somehow.

The author considers this difference of $\Delta\bar{Q}$ is the main reason for
the confusion in relativistic thermodynamics. However, the definition of a
finite volume may also make the problem complicated. This point has been known
since the very early years of relativity (Fermi (1923)), and must have been
well recognized during the controversy in 1960s (Gamba (1965); Kibble (1966);
Yuen (1970)). However, curiously enough, its importance is not well understood
and a number of erroneous statements on this point are found in papers since
1960s to this date.

We will see in the present paper these confusions can be cleared by the
covariant expression of the van Kampen-Israel theory. For this purpose it is
convenient to define the volume as a four vector (Nakamura (2006)):

$V^{\eta}(\bar{w})=\frac{w^{\eta}V_{0}}{w^{\mu}u_{\mu}}\,.$ (4)

This four vector represents a space-like volume orthogonal to the unit vector
$w^{\eta}$; in other words, this vector defines the volume viewed in a
reference frame with the four velocity $\bar{w}$.

The van Kampen-Israel theory defines the entropy change of a matter with a
finite volume as

$\Delta S=\beta_{0}u_{\mu}V^{\nu}\Delta T_{\nu}^{\mu}-\beta_{0}u_{\mu}P\Delta
V^{\mu}\,,$ (5)

where $P$, $T_{\nu}^{\mu}$ are the pressure and energy momentum tensor
respectively, and $\beta_{0}=1/T_{0}$ is the inverse temperature measured in
the rest frame.

When we define the total energy-momentum four vector as
$G^{\mu}(\bar{w})=V^{\nu}(\bar{w})\,T_{\nu}^{\mu}$ then (5) can be expressed
as

$\Delta S=\beta_{0}u_{\mu}[\Delta G^{\mu}(\bar{w})-P\Delta
V^{\mu}(\bar{w})]\,.$ (6)

Both $\Delta\bar{G}$ and $\Delta\bar{V}$ depends on $\bar{w}$, i.e., the
direction of the volume in the Minkowski space, however, the dependence is
canceled out by taking inner product with $u_{\mu}$ and $\Delta S$ becomes
invariant.

The heat/work is defined as an irreversible/reversible part of the energy-
momentum $\Delta\bar{G}$ in (2), which means

$\beta_{0}u_{\mu}\Delta Q^{\mu}>0\,,\,\,\,\beta_{0}u_{\mu}[\Delta
W^{\mu}-P\Delta V^{\mu}]=0\,.$ (7)

Obviously the above conditions cannot determine the heat and work uniquely;
when we define new values of the heat and work by
$\Delta\bar{Q}^{\prime}=\Delta\bar{Q}+\bar{A}$ and
$\Delta\bar{W}^{\prime}=\Delta\bar{W}-\bar{A}$ with an arbitrary four vector
$\bar{A}$ that satisfies $u_{\mu}A^{\mu}=0$, (7) holds for the new values
$\Delta\bar{Q}^{\prime}$ and $\Delta\bar{W}^{\prime}$. This ambiguity causes
the difference in (1) as we will see in the following.

Suppose a matter moving in the $x$ direction with a four velocity
$(u_{t},u_{x},0,0)$. Then energy momentum tensor may be written in the rest
frame as

$T_{\mu\nu}=\left(\begin{array}[]{cc}u_{t}^{2}\varepsilon_{0}+u_{x}^{2}P&u_{t}u_{x}(\varepsilon_{0}+P)\\\
u_{t}u_{x}(\varepsilon_{0}+P)&u_{x}^{2}\varepsilon_{0}+u_{t}^{2}P\end{array}\right)\,,$

with $\varepsilon_{0}$ being the energy density measured in the comoving
frame. We ignore the dimension in $y$ and $z$ direction for simplicity. Note
that $\varepsilon_{0}$ and $P$ do not depend on $t$ or $x$ because the matter
is in the equilibrium state.

We introduce a parameter $\theta=\tanh^{-1}(w^{x}/w^{t})$ to define the volume
in (4). Then the total energy-momentum can be expressed as a function of
$\theta$ in the following:

$\bar{G}(\theta)=\left(\begin{array}[]{c}E(\theta)\\\
G(\theta)\end{array}\right)=\left(\begin{array}[]{c}E_{0}\cosh\alpha+{\displaystyle
PV_{0}\sinh\alpha\tanh(\theta-\alpha)}\\\ E_{0}\sinh\alpha+{\displaystyle
PV_{0}\cosh\alpha\tanh(\theta-\alpha)}\end{array}\right)$ (8)

where $E_{0}=\varepsilon_{0}V_{0}$ is the total energy measured in the
comoving frame, and the velocity of the matter is parametrized by
$\alpha=\tanh^{-1}(u^{x}/u^{t})$ instead of $\bar{u}$.

We need another parameter to fix the ambiguity of heat in (2). Let us
introduce a parameter $\phi$ such that

$\Delta\bar{Q}=\left(\begin{array}[]{c}\Delta Q\\\ \Delta
Q\tanh\phi\end{array}\right)$

to this end. This parameter $\phi$ specifies the frame in which the heat is
purely timelike, in other words, the frame in which the heat looks “heat” only
without momentum. The rest frame and comoving frame are represented by
$\phi=0$ and $\phi=\alpha$ respectively.

The work $\Delta\bar{W}$ is then calculated as

$\Delta\bar{W}=\left(\begin{array}[]{c}\Delta
E_{0}\cosh\alpha+{\displaystyle\Delta(PV_{0})\sinh\alpha\tanh(\theta-\alpha)-\Delta
Q}\\\ \Delta
E_{0}\sinh\alpha+{\displaystyle\Delta(PV_{0})\cosh\alpha\tanh(\theta-\alpha)}-\Delta
Q\tanh\phi\end{array}\right)\,.$

Since the work $\Delta\bar{W}$ must satisfy (7), the heat $\Delta Q$ is
uniquely determined when $\phi$ is given:

$\Delta Q(\phi)=\frac{\cosh\phi}{\cosh(\phi-\alpha)}\,\Delta Q_{0}\,,$ (9)

where $\Delta Q_{0}=\Delta E_{0}-P\Delta V_{0}$.

Various formulations can be derived by expressing $\Delta Q(\phi)$ in (9) with
the energy-momentum $\bar{G}(\theta)$ in (8) by choosing different $\phi$ and
$\theta$. Any value of $\phi$ and $\theta$ can determine the relativistic
thermodynamical equation in general, however, the value of the rest frame or
the comoving frame ($0$ or $\alpha$) are practically preferable choices. In
the following we examine three typical choices in Views I, II, and III.

Typical theories choose the same value for $\phi$ and $\theta$ ($\phi=\theta$)
because they consider the heat exchange and volume change in the same frame
111This is typical, but not always. For example, the result by Kibble (1966)
can be derived with $\phi=\alpha$ and $\theta=0$.. For example, Ott (1963)
assumes a Carnot cycle in the comoving frame; the steps in the cycle,
including the heat exchange and volume change, take place in the moving frame.
Then (9) can be cast in the form of

$Q(\theta)=\frac{\cosh\theta}{\cosh(\theta-\alpha)}\Delta E_{0}-P\Delta
V^{t}(\theta)\,.$

The temporal component $\Delta V^{t}(\theta)$ is regarded as the volume change
in these theories, and denoted simply by $\Delta V$ and little attention has
been paid to its dependence on $\theta$. Then the above equation can be
regarded as to correspond to the definition of heat $\Delta Q=\Delta E-P\Delta
V$ in non-relativistic thermodynamics. The coefficient of the $\Delta E_{0}$
term determines the transformation rule of the heat, and consequently, that of
the temperature.

View I is typically derived with $\phi=\theta=0$, which means both the heat
and volume are defined in the rest frame. This choice gives $\Delta Q=\Delta
Q_{0}\gamma^{-1}$ and thus $T=T_{0}\gamma^{-1}$ because of (3). The
calculation of heat is obtained by subtracting $\gamma^{-1}u_{x}\Delta G$ from
the energy, regarding this as the work to cause acceleration:

$\Delta Q(0)=E(0)-\gamma^{-1}u_{x}\Delta G(0)-P\Delta V(0)\,,$ (10)

where we write $\Delta V=\Delta V^{t}$.

The typical choice of View II is $\phi=\theta=\alpha$, resulting
$T=T_{0}\gamma$. The heat and the volume are defined in the comoving frame,
and the expression in the rest frame is a result of their Lorentz transform.
The calculation of $\Delta Q$ is straightforward:

$\Delta Q(\alpha)=\Delta E(\alpha)-P\Delta V(\alpha)\,.$ (11)

Lansberg (1966) considered the temperature must be a Lorentz-invariant as in
View III from the symmetry. When two identical systems are moving relative to
each other, there is no reason for one system to have a temperature higher
than the others’. This argument can be represented by choosing
$\phi=\frac{1}{2}\alpha$ in (9) to treat the rest frame and the comoving frame
symmetrically. It should be noted that his actual calculation is more
complicated, but we do not examine its details here.

In the present paper we have successfully derived three different views of
relativistic thermodynamics from one basic formulation proposed by van Kampen
(1968) and Israel (1976). The difference comes from two factors, the
definitions of the heart and volume namely, which are represented by the two
parameters $\phi$ and $\theta$ here. The papers published so far on this topic
are so numerous that it is not practical to check all of them. However, the
author believes all the formulations can be derived from the van Kampen-Israel
theory as long as they are not wrong.

## References

  * Yuen (1970) C. K. Yuen, Amer. J. Phys. 38, 246 (1970).
  * (2) M. Requardt, eprint arXiv:0801.2639.
  * de Parga et al. (2005) G. A. de Parga, Lopéz-Carrera, and F. Anulo-Brown, J. Math. Phys. 38, 2821 (2005).
  * Einstein (1907) A. Einstein, Jb. Radioaktivitat 4, 411 (1907).
  * Planck (1908) M. Planck, Ann. d. Phys. 76, 1 (1908).
  * Gamba (1965) A. Gamba, Nuovo Cimento 37, 1792 (1965).
  * Kibble (1966) T. W. B. Kibble, Nuovo Cimento 41B, 167 (1966).
  * Ott (1963) H. Ott, Z. Physik 175, 70 (1963).
  * Arzelies (1965) H. Arzelies, Nuovo Cimento 35, 792 (1965).
  * Lansberg (1966) P. T. Lansberg, Nature 212, 571 (1966).
  * van Kampen (1968) N. G. van Kampen, Phys. Rev. 173, 295 (1968).
  * Israel (1976) W. Israel, Ann. Phys. 106, 310 (1976).
  * Fermi (1923) E. Fermi, Nuovo Cimenmto 25, 159 (1923).
  * Nakamura (2006) T. K. Nakamura, Phys. Lett. A 352, 175 (2006), eprint arXiv:physics/0505004.


# Conformal Description of Near-Horizon Vacuum States

Thomas Banks  Department of Physics and NHETC, Rutgers University, Piscataway,
NJ 08854 USA  Kathryn M. Zurek  Walter Burke Institute for Theoretical
Physics, California Institute of Technology, Pasadena, CA USA

###### Abstract

Motivated by recent work suggesting observably large spacetime fluctuations in
the causal development of an empty region of flat space, we conjecture that
these metric fluctuations can be quantitatively described in terms of a
conformal field theory of near-horizon vacuum states. One consequence of this
conjecture is that fluctuations in the modular Hamiltonian $\Delta K$ of a
causal diamond are equal to the entanglement entropy: $\langle\Delta
K^{2}\rangle=\langle K\rangle=\frac{A(\Sigma_{d-2})}{4G_{d}}$, where
$A(\Sigma_{d-2})$ is the area of the entangling surface in $d$ dimensions. Our
conjecture applies to flat space, the cosmological horizon of dS, and AdS Ryu-
Takayanagi diamonds, but not to large finite area diamonds in the bulk of AdS.
We focus on three pieces of quantitative evidence, from a Randall-Sundrum II
braneworld, from the conformal description of black hole horizons, and from
the fluid-gravity correspondence. Our hypothesis also suggests that a broader
range of formal results can be brought to bear on observables in flat and dS
spaces.

###### Contents

  1. I Introduction
  2. II RS-II Flat Brane in AdS/CFT
  3. III Near-Horizon Dynamics
  4. IV Large Diamonds/Black Holes in AdS Space
  5. V Nested Causal Diamonds and the Random Walk
  6. VI Conclusions

##  I Introduction

Because a detector in any experiment follows a time-like trajectory for a
finite proper time, an experimental measurement defines a causal diamond of
finite proper time. The covariant entropy bound (CEB) asserts that the maximal
entropy of any quantum state of the diamond in $d$ dimensions is less than
$A(\Sigma_{d-2})/4G_{d}$, where $A(\Sigma_{d-2})$ is the area of the
bifurcation surface of the diamond. Remarkably this bound takes an identical
form to the Bekenstein-Hawking entropy, and has been shown to be saturated in
several situations where there is no black hole. For example

  * •

In any quantum field theory, tracing out the complement of the region bounded
by $\Sigma_{d-2}$ produces a density matrix which is thermal with respect to
an operator called the modular Hamiltonian. In CFTs with an Einstein-Hilbert
(EH) dual111We use the phrase CFT with an EH dual to mean a CFT whose
correlators can be well approximated by solving (super)gravity equations in
AdS space. This is more precise than the conventional large radius holographic
dual. the thermal and entanglement entropy are $A(\Sigma_{d-2})/4G_{d}$ Ryu
and Takayanagi (2006a); Casini  _et al._ (2011), where $A(\Sigma_{d-2})$ is
the area of a Rindler diamond in AdS space.

  * •

In gravity, diamonds in maximally symmetric spacetimes obey a first law of the
form $dH_{\zeta}=-TdS$, where $2\pi T=\kappa$ is the surface gravity of the
horizon, $S=A(\Sigma_{d-2})/4G_{d}$ is the Bekenstein-Hawking entropy, and
$H_{\zeta}$ is the total Hamiltonian generating flow along a conformal Killing
vector (CKV) $\zeta$ that preserves the diamond Jacobson and Visser (2019).
The authors of Jacobson and Visser (2019) argue that the modular Hamiltonian
for causal diamonds in such spaces is the quantum generator of the CKV that
preserves the diamond, evaluated at a point on the diamond’s bifurcation
surface. Below, we will embed this generator in a Virasoro algebra.

  * •

The entanglement entropy of causal diamonds may in some cases be computed with
Euclidean methods. One obtains a finite result
$S_{ent}=A(\Sigma_{d-2})/4G_{d}$ Susskind and Uglum (1994); Carlip and
Teitelboim (1995); Callan and Wilczek (1994); Cooperman and Luty (2014); Banks
_et al._ (2021).

  * •

In general quantum field theories, states with finite numbers of local
operators acting on the vacuum state exhibit UV divergent entanglement entropy
proportional to the area $A(\Sigma_{d-2})$ of the entangling surface Srednicki
(1993). This is interpreted as a renormalization of Newton’s constant Susskind
and Uglum (1994) if one assumes the Covariant Entropy Bound is saturated for
general diamonds. Note that this point is conceptually distinct from the use
of field theory in the AdS/CFT correspondence. Here, the field theory in
question is a bulk effective field theory.

These results lead one to consider the idea that the logarithm of the
dimension of the Hilbert space of a causal diamond is
$\frac{A(\Sigma_{d-2})}{4G_{d}}$, with $A(\Sigma_{d-2})$ the area of the
bifurcation surface defined by light sheets Bousso (1999a, b, c). Let us
outline the steps leading to these conclusions in more detail. The density
matrix $\rho$ of any quantum field theory, restricted to such a diamond, is
given by (see e.g. Casini  _et al._ (2011))

$\rho=\frac{e^{-K}}{\mbox{Tr}(e^{-K})},$ (1)

which defines the modular Hamiltonian $K$ of the diamond. For a CFT, the
general form of the modular Hamiltonian (for any state obtained by acting on
the vacuum with a finite number of local operators, space-like separated from
the diamond) is known in terms of the stress tensor $T_{ab}$ and the CKV
$\zeta^{b}$, $K=H_{\zeta}=\int dV_{d-1}^{a}T_{ab}\zeta^{b}$, where $V_{d-1}$,
shown as shaded disks in Fig. 1, is a volume element (at fixed time) in the
CFT. Such a density matrix allows one to compute the thermodynamic quantities
$\langle K\rangle$ and its fluctuations $\langle\Delta K^{2}\rangle$ via the
free energy

$F_{n}=-\frac{1}{n}\log\mbox{ tr}\left(e^{-nK}\right),$ (2)

such that

$\displaystyle\langle K\rangle$ $\displaystyle=$
$\displaystyle\frac{d}{dn}(nF_{n})|_{n=1}$ (3) $\displaystyle\langle\Delta
K^{2}\rangle$ $\displaystyle=$
$\displaystyle-\frac{d^{2}}{dn^{2}}(nF_{n})|_{n=1}.$ (4)

Here we have used the “replica index” $n$ is a stand-in for the inverse
temperature of the system $\beta$. For continuous $n$ the formula for $F_{n}$
embeds the modular Hamiltonian in a one parameter set of thermal density
matrices. For integer values of $n$ there is a path integral over branched
covers of the original manifold, which allows one to perform calculations
semi-classically. The replica trick thus allows one to relate the
thermodynamic entropy to geometric variations of the diamond, as considered by
many authors, including Callan and Wilczek (1994); Lewkowycz and Maldacena
(2013); Cooperman and Luty (2014). In particular, by identifying the Euclidean
gravitational effective action by the relation $nF_{n}=I_{n}-nI_{1}$, one
finds the geometric entropy of Ref. Callan and Wilczek (1994) is

$S_{geom}=\langle K\rangle=\partial_{n}\left(I_{n}-nI_{1}\right)|_{n=1},$ (5)

in which case the geometric and entanglement entropies are equivalent. The
geometric entropy has been shown, for the case of an extremal entangling
surface, to be (e.g. Lewkowycz and Maldacena (2013); Cooperman and Luty
(2014); Dong (2016))

$S_{geom}=\frac{A(\Sigma_{d-2})}{4G_{d}}.$ (6)

Note this quantity should be thought of as a renormalized entropy. If one
imposes a UV cutoff, the infinity is regularized to
$\Lambda^{d-2}A(\Sigma_{d-2})$. It was proposed in Susskind and Uglum (1994)
that this infinity could be absorbed in a universal way into the definition of
Newton’s constant $G_{d}$, in any theory of quantum gravity well-approximated
by the Einstein-Hilbert action. This formula originated in the work of Carlip
and Teitelboim Carlip and Teitelboim (1995) and of Susskind Susskind and Uglum
(1994), and was also anticipated in earlier work of Srednicki Srednicki
(1993). The same relation holds in AdS/CFT for the entanglement entropy of a
causal diamond anchored to the boundary Ryu and Takayanagi (2006b)

$\langle K\rangle=S_{ent}=\frac{A(\Sigma_{d-2})}{4G},$ (7)

where $A(\Sigma_{d-2})$ is now the area of the bifurcate horizon in the bulk
(known as the Ryu-Takayanagi (RT) surface).

Since the number of degrees of freedom at large scales is smaller than what
would be naively expected from Quantum Field Theory, this naturally suggests
that there may be correlations between the degrees-of-freedom. It has been
suggested that such correlations could be observable in modern
interferometers, mostly based on heuristic holographic arguments and matrix
theory Kwon and Hogan (2016); Hogan and Kwon (2017); Chou  _et al._ (2017).
The first realistic theoretical proposal for calculating fluctuations of
observables in finite causal diamonds (such as interferometers measure),
utilizing modern holographic techniques, was made in Refs. Verlinde and Zurek
(2019, 2020), and an effective field theory description was formulated in Ref.
Zurek (2020). It is easy to understand the relevance of a finite size causal
diamond to an interferometer experiment – a single pass of the interferometer
defines a causal diamond in space-time, with the geodesic of the beam splitter
following the center of the diamond from tip-to-tip and the far mirror
intersecting the diamond at the outside corner. This is shown in Fig. 1 (see
also the figure in Ref. Verlinde and Zurek (2019)).

Figure 1: Depiction of a causal diamond, with the arms of the interferometer
shown on the bifurcate horizon. The worldlines of the beamsplitter and mirror
(not shown for clarity of representation) flow vertically. The entangling
surface is denoted as $\Sigma_{d-2}$, where the far mirrors of the
interferometer measure the quantum uncertainty in the position of this
surface. The vertical arrows show lines of flow of the conformal Killing
vector $\zeta$.

The results of Verlinde and Zurek (2020) were particularly interesting,
because they revealed an unanticipated universality in the ratio between the
fluctuations of entanglement entropy222In the literature, the fluctuations of
the modular Hamiltonian have been called the capacity of entanglement De Boer
_et al._ (2019); Nakaguchi and Nishioka (2016). We will use this term, and the
phrase modular fluctuations interchangeably. and the entanglement entropy
itself. For any system with an entanglement spectrum333The entanglement
spectrum is the eigenvalue spectrum of the modular Hamiltonian. whose Laplace
transform is dominated by a Gaussian peak444An example where there is no such
Gaussian peak is a finite dimensional system, whose large eigenvalue spectrum
grows like $n(E)\sim e^{E^{p}}$ with $p>1$. The stationary point of the
Laplace transform integrand is a minimum, rather than a maximum of the
eigenvalue distribution., we expect a relation

$\langle\Delta K\rangle^{2}=\alpha\langle K\rangle.$ (8)

The authors of Verlinde and Zurek (2020) found that, for RT diamonds, the bulk
dual of a causal diamond on the boundary in a conformal field theory with an
Einstein-Hilbert (EH) dual,

$\alpha=1.$ (9)

This is independent of all details of the CFT, including its dimension.
Although any ${\cal O}(1)$ value of $\alpha$ is interesting, the observation
of a universal value would be both phenomenologically and theoretically
striking. It provides a precise target for experimental investigation, and
also gives important information about the general principles underlying a
theory of quantum gravity.

The result of Verlinde and Zurek (2020) was obtained by using black hole
entropy formulae for topological black holes, obtained by small (replica)
deformations of AdS/Rindler space. It is notable that the result is confirmed
by an independent calculation. Perlmutter Perlmutter (2014a), using the
observations of Casini  _et al._ (2011); Hung  _et al._ (2011) that Renyi
entropies of spherical diamonds in CFTs could be computed as thermal entropies
of the same CFT in hyperbolic space, showed that for a general CFT $\alpha$
was proportional to the coefficient in the stress tensor two-point function.
When applied to CFTs with an Einstein-Hilbert dual, this formula gives
$\alpha=1$ Perlmutter (2019).

Because no experiment exists in the bulk of AdS, we need to be able to
calculate modular fluctuations in causal diamonds small compared to the radius
defined by the cosmological constant (c.c.). We will carry out a few simple
calculations, and make a few observations, to make the case for a universal
value of $\alpha=1$ in small (i.e. Minkowski) and RT diamonds. We will argue
that we expect the same result to hold for the cosmological horizon in dS.

  1. 1.

First, we calculate $\langle\Delta K^{2}\rangle$ on a flat Randall-Sundrum
brane Randall and Sundrum (1999). The Randall-Sundrum model relates gravity
weakly coupled to a CFT with an EH dual, to the physics on an infinite flat
brane in AdS space. This simple model for QG in flat space gives $\alpha=1$.

  2. 2.

Second, we make a conjecture stemming from seminal work of Carlip and
Solodukhin Carlip (1995, 1999, 1998, 2012); Solodukhin (1999) relating the
entropy of black holes to the representation theory of a near-horizon Virasoro
algebra. We will generalize this conjecture to small empty diamonds and the
cosmological horizon of dS space. It states that the modular Hamiltonian of a
diamond has an entanglement spectrum identical to that of the asymptotic
$L_{0}$ generator in a $1+1$ dimensional CFT where the Cardy formula is
correct. In fact, the result $\alpha=1$ depends only on the scaling of the log
density of states with $L_{0}$ eigenvalue,

${\rm ln}\rho(L_{0})=B\sqrt{L_{0}},$ (10)

with $B$ a large constant. Below we will call the eigenvalues of the $L_{0}$
generator $EI$, where $E$ is the energy of the CFT, quantized on an interval
of length $I$. The full Cardy formula is necessary to get the exact result for
the entanglement entropy Carlip (2012); Solodukhin (1999).

  3. 3.

Lastly, the fluid gravity correspondence: a large number of works over the
years Damour (1978, 1979); Bredberg and Strominger (2012); Bredberg  _et al._
(2012); Crossley  _et al._ (2016); Rangamani (2009); Compere  _et al._ (2011)
(and references therein) have shown that near-horizon Einstein equations,
subjected to a few global boundary conditions, reduce to the Navier-Stokes
(NS) equation, with universal coefficients. In particular Bredberg  _et al._
(2011) showed that the ratio of shear viscosity to entropy density for small
diamonds, is universal and agreed with that found in Policastro  _et al._
(2002a, b) for large black holes in AdS. The fluctuation dissipation theorem
(FDT) then tells us that fluctuations should be universal. The fluctuations
relevant to the incompressible NS equation are those of the $0i$ components of
the Brown-York stress tensor, evaluated on the “stretched horizon,” a time-
like “hyperboloid” that approaches the bifurcation surface of the horizon at a
minimal distance of one Planck unit. These entropy fluctuations are not the
main focus of the present paper, but their universality gives credence to the
idea that entropy fluctuations are also universal.

One way of understanding our results is as a conformal description of near-
horizon states. The near-horizon limit will be taken for the causal diamond
described by a metric with “blackening factor” $f(r)$,

$ds^{2}=-f(r)dt^{2}+\frac{r^{2}}{f(r)}+r^{2}d\Sigma_{d-2}^{2}.$ (11)

This metric covers a causal diamond with different blackening factors
depending on the background. For example, in empty Minkowski, $f(r)=1-r/R$
Verlinde and Zurek (2019), where $R$ is the radius of the bifurcation surface
of the diamond boundary, and for boundary anchored diamonds in AdS,
$f(r)=r^{2}/L^{2}-1$, where $L$ is the AdS curvature.

In the near-horizon limit, we can incorporate metric fluctuations by
linearizing the blackening factor about the entangling surface at some
position $r_{n}$:

$ds^{2}=-4\pi T_{n}(r-r_{n})dt^{2}+\frac{dr^{2}}{4\pi
T_{n}(r-r_{n})}+r_{n}^{2}d\Sigma_{d-2}^{2},$ (12)

where the temperature is $T_{n}=f^{\prime}(r_{n})/4\pi$. In the language of
Dong (2016), $r_{n}$ is the position of a cosmic $d-2$ brane, which sources
the fluctuation of the geometry. This mirrors the argument of Verlinde and
Zurek (2019) which identifies the fluctuations as due to a gravitating mass
distribution near the horizon of the classical geometry. For the case of
boundary anchored diamonds in AdS, the dependence of $r_{n}$ on $T_{n}$ can be
computed by insisting that when $T_{1}=nT_{n}$ the global geometry be
asymptotic to an $n-$fold cover of the boundary, branched over the sphere to
which the RT surface asymptotes. Throughout this paper we will refer to the
undeformed geometries having $n=1$ to have a horizon radius $r_{n=1}\equiv
r_{0}$ and with corresponding horizon temperature $T_{0}=\beta_{0}^{-1}$.

In the near-horizon limit, we will see, following the analysis of Solodukhin
and as discussed in point 2 of the enumerated list above, that the
$d$-dimensional Einstein-Hilbert action can be reduced to a classical 2-d
conformal field theory on the light cone directions. A plausible assumption
about how the algebra of that CFT is realized in the quantum theory will allow
us to obtain $\alpha=1$. We will argue that such a calculation will accurately
reproduce the global near-horizon behavior when the proper time for an
inertial diamond observer is of the order of, or larger than, the radius
$r_{n}$ of the entangling surface. We will see that this criterion is met for
RT and small diamonds in AdS and the cosmological horizon of dS, but not for
large diamonds in AdS.

More importantly, this assumption also offers concrete paths forward for
modeling the near-horizon behavior of quantum gravity even in ordinary flat
space, utilizing the beautiful and powerful techniques of 2-d CFTs. We reserve
further work along these directions for the future. The outline of this paper
follows the numbered list above. In the next section we show how
$\langle\Delta K^{2}\rangle=\langle K\rangle=\frac{A({\cal B})}{4G_{d}}$ in a
spherical ball ${\cal B}$ on a flat RS-II brane having Newton’s constant
$G_{d}$. In Sec. III, we consider that this same result is derived from
positing that the near-horizon dynamics of a causal diamond is described by a
2-d CFT with density of states given by the Cardy formula. We also relate our
conjecture of universal entropy fluctuations to Bredberg  _et al._ (2011),
which showed that the famous result Kovtun  _et al._ (2005) that the viscosity
to entropy density ratio for large black holes in AdS is
$\frac{\eta}{s}=\frac{1}{4\pi}$, valid for any weakly curved null surface in a
model satisfying the Einstein equations. In Sec. IV we discuss why the results
of Secs. II-III are not expected to apply to large diamonds in the bulk of
AdS. Finally, in Sec. V, we show how entropy fluctuations give rise to
fluctuations in the time-of-arrival in a light beam that traces out the causal
development of ${\cal B}$, by following a series of nested causal diamonds.
Though utilizing distinct methods, these results on length fluctuations
exactly agree with Ref. Verlinde and Zurek (2020).

##  II RS-II Flat Brane in AdS/CFT

We begin by considering a braneworld set-up as shown in Fig. 2. For
continuity, our notation largely follows that of Ref. Verlinde and Zurek
(2020). Our goal is to compute the modular energy $\langle K\rangle$ and its
fluctuations $\langle\Delta K^{2}\rangle$ in the volume enclosed by the
spherical entangling surface ${\cal B}$ on the brane, utilizing a bulk
holographic calculation. We begin with a $d=D+1$ dimensional bulk AdS space
with AdS curvature $L$ and metric in Poincare coordinates

$ds^{2}=L^{2}\frac{dz^{2}+dx_{i}^{2}-dx_{0}^{2}}{z^{2}}$ (13)

with $i=1,....,D-1$. We take the flat $D$-dimensional brane located at some
position $z_{c}$ in the bulk, as also shown in the figure. In such a set-up,
gravity is induced on the brane, with a dimensionally reduced Newton constant
(e.g. Myers  _et al._ (2013))

$G_{D}=\frac{(D-2)}{2L}G_{d}.$ (14)

Figure 2: Depiction of the RS-II braneworld scenario. The boundary is located
at $z=0$ and the RS-II brane at $z=z_{c}$. We compute the modular fluctuations
$\langle\Delta K^{2}\rangle$ in the volume enclosed by ${\cal B}$ at the
bifurcate horizon on the brane causal diamond, utilizing the holographic map
from the bulk RT surface, shown in darker red and labeled $\Sigma_{d-2}$.

The measuring apparatus is confined to the brane (labeled by ${\cal B}$), with
light beams from a beam splitter (shown at the bottom of the brane causal
diamond) reflecting off a mirror at the bifurcate horizon on the brane, and
recombining again at the top of the causal diamond. The causal diamond on the
brane anchors the Ryu-Takayanagi (RT) surface $\Sigma_{d-2}$ in the bulk. This
spatial slice of $V_{d-1}$ (shown as shaded) is given by $z^{2}+x_{i}^{2}\leq
R^{2}$, and the full interior of the bulk diamond is given by

$R^{2}-z^{2}-x_{i}^{2}+x_{0}^{2}\geq 2R|x_{0}|.$ (15)

The modular fluctuations $\langle\Delta K^{2}\rangle$ can also be calculated
in the bulk Verlinde and Zurek (2020), with the result $\alpha=1$, as noted in
the introduction. In the bulk of AdS, Ref. Verlinde and Zurek (2020) showed
that these modular fluctuations gravitate and induce metric fluctuations.

Because gravity is induced on the brane, with strength given by Eq. 14, an
instrument on a brane at $z_{c}$ will also experience modular energy
fluctuations. To relate the bulk and brane modular fluctuations, we must
geometrically relate the transverse area of the surface on the brane $A({\cal
B})$ to the RT bulk surface $\Sigma_{d-2}$ which is anchored to it.

To do this, it is useful to put the metric on the interior of the bulk causal
diamond into AdS-Rindler form, Eq. 11. Here the transverse plane (having the
entangling surface) can be written as

$d\Sigma_{d-2}^{2}=d\rho^{2}+\sinh^{2}\rho d\Omega_{D-2}^{2}.$ (16)

The bulk light cone directions $r,~{}t$ are related to the Poincare
coordinates via

$\frac{R^{2}-z^{2}-x_{i}^{2}+x_{0}^{2}}{2Rz}=\left(\frac{r^{2}}{L^{2}}-1\right)^{1/2}\cosh\frac{t}{L},~{}~{}~{}~{}~{}~{}~{}\frac{x_{0}}{z}=\left(\frac{r^{2}}{L^{2}}-1\right)^{1/2}\sinh\frac{t}{L},$
(17)

while the transverse direction is

$|x_{i}|=R\tanh\rho.$ (18)

The blackening function is of the standard (topological) black hole form

$f(r)=\left(\frac{r^{2}}{L^{2}}-1\right)+\frac{4\pi
M}{(d-2)}\frac{L^{d-2}}{r^{d-3}}\frac{4G_{d}}{A(\Sigma_{d-2})},$ (19)

where the black hole mass is replaced with the modular fluctuations
$M=\frac{\Delta K}{2\pi L}$, following Ref. Verlinde and Zurek (2020). These
modular fluctuations hence cause geometric fluctuations. We seek to understand
the relation between the modular fluctuations computed in the bulk and those
induced on the brane.

To do this, we relate the area appearing in Eq. 7 to the area of the
entangling surface on the brane $A({\cal B})$. Now the area of the RT surface
is

$A(\Sigma_{d-2})=L^{D-1}\Omega_{D-2}\int_{0}^{\rho_{c}}d\rho\sinh^{D-2}\rho.$
(20)

The location of the entangling surface is found by noting that the
transformations between Poincare and topological BH coordinates imply that the
bulk diamond satisfies

$\left(\frac{(R-x_{0})^{2}-z^{2}-x_{i}^{2}}{2Rz}\right)\left(\frac{(R+x_{0})^{2}-z^{2}-x_{i}^{2}}{2Rz}\right)=\frac{r^{2}}{L^{2}}-1.$
(21)

We can thus see that $\Sigma_{d-2}$ is located at $x_{0}=0$ and
$z^{2}+x_{i}^{2}=R^{2}$. We also find that fixing $z_{c}$ is equivalent to
fixing the brane location in the bulk according to the relation

$z_{c}=R\sqrt{1-\tanh^{2}\rho_{c}}=\frac{R}{\cosh\rho_{c}},$ (22)

such that $z_{c}\ll R$ is equivalent to $\rho_{c}\gg 1$.

Since the bulk RT diamond is anchored to the boundary diamond, a light beam
traveling from tip-to-tip on the boundary diamond will arrive at the same time
as a light beam traveling from tip-to-tip on the bulk RT diamond. This time is
given by

$T=\frac{L}{z_{c}}(R-z_{c})=L(\cosh\rho_{c}-1)\approx\frac{e^{\rho_{c}}}{2}L.$
(23)

Then, combining Eq. 20 with Eq. 14, we have

$\frac{A(\Sigma_{d-2})}{4G_{d}}\simeq\frac{L^{D-2}\Omega_{D-2}}{4G_{D}}\left(\frac{e^{\rho_{c}}}{2}\right)^{D-2},$
(24)

where we are taking the $\rho_{c}\gg 1$ limit. We now identify

$A({\cal B})=L^{D-2}\Omega_{D-2}\left(\frac{e^{\rho_{c}}}{2}\right)^{D-2},$
(25)

as the area of $D-2$ dimensional surface ${\cal B}$. Therefore, utilizing the
$\alpha=1$ result for bulk RT diamonds of Ref. Verlinde and Zurek (2020), we
have

$\langle\Delta K^{2}\rangle=\frac{A(\Sigma_{d-2})}{4G_{d}}=\frac{A({\cal
B})}{4G_{D}}.$ (26)

This suggests that modular Hamiltonian fluctuations in Minkowski space are the
same as in AdS, i.e. $\alpha=1$.

Note that very similar reasoning, applied now to $\langle K\rangle$ rather
than $\langle\Delta K^{2}\rangle$, gives

$\langle K\rangle=\frac{A(\Sigma_{d-2})}{4G_{d}}=\frac{A({\cal B})}{4G_{D}}.$
(27)

The vacuum entanglement entropy of a causal diamond on an RS-II brane was also
calculated by independent methods in Ref. Myers  _et al._ (2013), and the same
result was obtained, supporting the results obtained here.

At first sight $\langle\Delta K^{2}\rangle=\langle K\rangle$ for Minkowski
diamonds may seem surprising, so we explore next why this result could have
been expected from the dynamics of conformal field theories in the near-
horizon limit.

##  III Near-Horizon Dynamics

We first consider a general coordinate system that covers a causal diamond
characterized by a geodesic proper time $T_{prop}$. There is a different near-
horizon behavior for large finite diamonds than for boundary-anchored and
small diamonds in AdS, or the cosmological horizon of dS space. The essential
difference is simple, and determined by whether the inequality

$T_{prop}^{d-2}\gtrsim A(\Sigma_{d-2})$ (28)

is satisfied. If it is, our claim is that there is sufficient time for the
near-horizon dynamics to equilibrate the state of the system, and $\alpha=1$.

  1. 1.

For diamonds smaller than the curvature radius $L$, area scales like
$T_{prop}^{d-2}$ and this scaling relation is valid for the proper time along
any trajectory of finite acceleration.

  2. 2.

For the cosmological horizon of dS the proper time along any timelike
trajectory connecting the tips of the causal patch can be arbitrarily long,
while the area is bounded by $L^{d-2}$. Thus, localized probes of the horizon
measure the equilibrium behavior of the entire diamond.

  3. 3.

For boundary-anchored diamonds in AdS, the same is true, though in this case
the system has infinite entropy.

In all three of these cases, Eq. 28 is satisfied. By contrast, for finite area
diamonds in AdS, the proper time along any trajectory is bounded by
$\frac{L\pi}{2}$, while the area can be arbitrarily large. Although the local
near-horizon dynamics is identical to that of the other kinds of diamonds, no
diamond observer can follow the system long enough to probe the equilibrium
state. The calculations of this section are valid locally on all horizons with
the near-horizon Rindler factorization exhibited below.

The authors of Hayden and Preskill (2007); Sekino and Susskind (2008) have
argued that the behavior of the quasi-normal modes on Minkowksi diamonds
indicates that the quantum systems responsible for the equilibration are in
fact fast scramblers. The same arguments apply to the cosmological horizon of
dS space, and indeed to the local behavior on any diamond horizon. Fast
scrambling systems have a characteristic time scale $t_{0}$ and become
completely equilibrated in a time $t_{0}{\rm ln}S$ where $S$ is the total
number of q-bits in the system. For small black holes $t_{0}$ is the
Schwarzschild radius, while for the dS horizon it is the Hubble time. For AdS-
Rindler $t_{0}$ is the AdS radius $L$. In all three of the cases enumerated
above, $t_{0}{\rm ln}\ S$ is of the same order or smaller than the proper time
along trajectories in the diamond and one can explore the equilibration of
perturbations over the entire horizon within the proper time available to a
detector on such a trajectory. For AdS-Rindler we must cut off the $z$
coordinate on the hyperboloid, similarly to the RS-II model, to render the
entropy finite in order to make this statement unambiguous.

In the case of large AdS diamonds/black holes, however, the local quasi-normal
modes dissipate after having spread to only a distance $\sim L$ on the
horizon, which is much smaller than their Schwarzschild radius. Furthermore,
as we will recall below, the quasi-normal spectrum of large AdS diamonds
includes sound modes with wavelengths $>L$, which indicate ballistic
propagation of energy and entropy densities, rather than fast scrambling. By
contrast, the quasi-normal spectrum of all other horizons, including AdS-
Rindler space does not have sound modes. We believe that this is an indication
that our local calculation of the modular fluctuations is valid for any causal
diamond with a Rindler horizon, but that the thermal equilibrium ensemble dual
to a large radius AdS causal diamond can have a different value of $\alpha$
because its entropy is dominated by modes that propagate ballistically on the
large horizon. This is of course the message of the AdS/CFT correspondence.
The equilibrium dynamics is that of a local field theory on the sphere.

Our claim, following the work of Carlip Carlip (2012) and Solodukhin
Solodukhin (1999), is that locally on the bifurcation surface of a diamond
with a Rindler near-horizon expansion, the quantum dynamics is governed by a
Virasoro algebra in a representation satisfying Cardy’s theorem. The argument
begins with a rigorous classical theorem that the algebra of near-horizon
symmetries consists of conformal transformations of the flat two dimensional
Rindler space,555The authors of Ref. Carlip (2012); Solodukhin (1999)
emphasize that there are two such Virasoro algebras for a time symmetric
situation, which are time reflections of each other. Only one is apparent at a
given point on the horizon. The CFT on the past/future portion of the diamond
boundary has only incoming/outgoing modes. which acts on the horizons of all
of these diamonds in space-time. It is important to note that the $L_{0}$
generator of this algebra is precisely the CKV of the background geometry
which switches from time-like to space-like on the diamond boundary. The
quantum generator associated with the action of this CKV on the bifurcation
surface of the diamond was proposed Casini  _et al._ (2011); Jacobson and
Visser (2019) to be the modular Hamiltonian of the diamond. It has been shown
Carlip (2012); Solodukhin (1999) that, using the symplectic structure derived
from the EH Lagrangian, the Poisson brackets of these generators have a
central charge and there is a classical value for the $L_{0}$ generator,
coming from the boundary terms. If one assumes that the quantum model of
horizons is a representation of the Virasoro algebra satisfying the
constraints giving rise to the Cardy formula for the asymptotic density of
$L_{0}$ eigenvalues, then that formula reproduces the Bekenstein-Hawking area
law for black holes666In fact, the argument works for rotating black holes as
well as the spherically symmetric geometries considered here..

We will summarize and generalize these arguments using the formalism of
Solodukhin Solodukhin (1999), who dimensionally reduced the Einstein-Hilbert
action on a class of metrics of the form

$ds^{2}=g_{ab}dy^{a}dy^{b}+\rho^{2}(y)d\Sigma_{d-2}^{2}=\rho^{2}(y)[\rho^{-2}g_{ab}dy^{a}dy^{b}+d\Sigma_{d-2}^{2}].$
(29)

Solodukhin assumed that $d\Sigma_{d-2}^{2}$ was the round metric on a sphere.
We will argue that in fact $d\Sigma_{d-2}^{2}$ can be any fixed $d-2$
dimensional Riemannian manifold, and $g_{ab}$ a two dimensional Lorentzian
metric, which we will soon restrict to be Rindler space. Evaluated on such a
configuration, the EH action reduces to a metric for dilaton gravity, with the
dilaton a function of $\rho$. Such metrics, in the near-horizon limit, can be
expanded, following Solodukhin, in the form of Eq. 12. For our purposes, the
transverse metric $d\Sigma_{d-2}^{2}$ can be any Riemannian manifold, for
which this reduction produces a valid solution of the $d$ dimensional Einstein
equations. To see the reduction, note that the metric $\tilde{g}$ in square
brackets is a product manifold so that

$R_{d}(\tilde{g}_{\mu\nu})=R_{2}(\rho^{-2}g_{ab})+R_{d-2}.$ (30)

Our metric is conformal to this one so that Carroll (2019)

$R(g_{\mu\nu})=\rho^{-2}R_{d}(\tilde{g}_{\mu\nu})+2\rho^{-1}(d-1)\tilde{g}^{ab}\tilde{\nabla}_{a}\tilde{\nabla}_{b}(\rho^{-1})-d(d-1)\tilde{g}^{ab}\partial_{a}\rho^{-1}\partial_{b}\rho^{-1}.$
(31)

Here we have used the fact that the conformal factor depends only on the two
dimensional coordinates, to simplify the general relation between scalar
curvatures of conformally related metrics.

Combining the last two equations, we see that the geometry of the $d-2$
dimensional bifurcation surface enters into the two dimensional Lagrangian
only via overall factors of the area of the surface and its integrated scalar
curvature. Furthermore, the $d-2$ dimensional curvature term multiplies
something that is a pure potential for the dilaton, with no derivatives of the
two dimensional fields. As we will see below, Solodukhin argues that such
potential terms become irrelevant near the horizon, where the theory thus
becomes conformal.

The upshot of these remarks is that we can translate Solodukhin’s analysis of
spherically symmetric black hole horizons to the bifurcate horizon of a causal
diamond in a Lorentzian space-time solving the Einstein equations. Moreover,
the argument that the curvature of the bifurcation surface is negligible near
the horizon implies that the calculation is ultra-local on the horizon. This
is quite analogous to the replica calculation of $\langle K\rangle$ and
$\langle\Delta K^{2}\rangle$.

We now recapitulate Solodukhin’s analysis of the spherically symmetric case,
where he has carried out the algebra of the general discussion of the previous
paragraphs. Solodukhin writes the dimensionally reduced action as777This is
Eqn. 5.3 of Solodukhin (1999), with the substitution $r\rightarrow\rho$ since
we use $r$ to denote a coordinate rather than a field.

$I=-\frac{\Sigma_{d-2}}{16\pi G_{d}}\int d^{2}y\
\sqrt{-g_{2}}[\rho^{d-2}R_{2}(g_{ab})+(d-3)(d-2)\rho^{d-4}(\nabla\rho)^{2}+(d-3)(d-2)\rho^{d-4}].$
(32)

Note that a $d$ dimensional cosmological constant would similarly add a
potential term that is a power of $\rho$ . The field $\rho$ is the radius of
the $\Sigma^{d-2}$ entangling surface. When we solve its equation of motion it
will have the value $\rho_{0}$ at the horizon.

Now let us suppose that there is a solution of the $d$ dimensional equations,
which has a causal diamond whose bifurcation surface has fixed area
$\Sigma_{d-2}$, in Planck units. Near the bifurcation surface, generically,
the two-dimensional metric will be two dimensional Rindler space as in Eq. 12.
Following Solodukhin, define

$\rho^{d-2}=\rho_{0}^{d-2}\sqrt{\frac{2\pi}{S}\frac{(d-3)}{(d-2)}}q\phi.$ (33)

where $S=\Sigma_{d-2}\rho_{0}^{d-2}/4G_{d}$ and $q$ is an arbitrary parameter.
We immediately see that the dilaton $\phi$ is related to the area.

One then redefines the background metric by a $\phi$ dependent Weyl
transformation

$g_{ab}=(\frac{\phi_{0}}{\phi})^{\frac{d-3}{d-2}}e^{\frac{2\phi}{\Phi_{0}}}\bar{g}_{ab},$
(34)

where
$\Phi_{0}=\sqrt{\frac{2S}{\pi}\frac{(d-3)}{(d-2)}}=2q(d-3)/(d-2)\phi_{0}$,
with $\phi_{0}$ is the value of $\phi$ near the horizon according to the
definition in Eq. 33. The action takes the form

$I=-\int
d^{2}y\sqrt{-\bar{g_{2}}}\left[\frac{(\nabla\phi)^{2}}{2}+q\phi\sqrt{\frac{(d-3)}{(d-2)}\frac{S}{8\pi}}~{}R_{2}(\bar{g}_{ab})+U(\phi)\right].$
(35)

Notice that nothing in this derivation requires that the metric is a black
hole. The calculation is valid, locally on the transverse space, for any
metric that is approximately the product of two-dimensional Rindler space and
a transverse manifold, as in Eq. 11. The action above, with $U=0$, is called
the (free) Liouville action.

Differentiating the free Liouville action with respect to the metric and then
setting it equal to the Rindler metric we find that the stress tensor of the
$\phi$ field is Solodukhin (1999)

$T_{ab}=\frac{1}{4}\left[\partial_{a}\phi\partial_{b}\phi-\bar{g}_{ab}(\nabla\phi)^{2}\right]-q\sqrt{\frac{(d-3)}{(d-2)}\frac{S}{8\pi}}\left[\partial_{a}\partial_{b}\phi-\bar{g}_{ab}\nabla^{2}\phi\right].$
(36)

The trace vanishes when the flat space equation of motion $\nabla^{2}\phi=0$
is satisfied, verifying that this is a classical conformal field theory.
Importantly, this criterion is met in the near-horizon limit. This is most
easily seen by defining

$z=-\frac{1}{4\pi T_{0}}{\rm ln}\ (r-r_{0})+z_{0},$ (37)

where $r$ is the spatial Rindler coordinate in Eq. 12, and $z_{0}$ is a
constant defined so that the horizon maps to $z=I/2$, with $I$ taken to
infinity at the end of the calculation. In this coordinate system, the
blackening factor becomes $f(z)=f_{0}e^{\frac{2z}{\beta_{0}}}$. Then the trace
of Eq. 36 gives an equation of motion for $\phi$ Solodukhin (1999):

$-\partial^{2}_{t}\phi+\partial_{z}^{2}\phi=\frac{1}{4}q\Phi_{0}f(z)R_{2}.$
(38)

The right-hand-side vanishes exponentially as $r\rightarrow r_{0}$, and in
this limit the theory becomes conformal. The non-vanishing trace can be viewed
as the (classical) conformal anomaly of the flat space space Liouville theory
in the background two dimensional metric, and it vanishes near the horizon
because the metric becomes flat.

More importantly, the classical limit of the representation theory for the
conformal spectrum of the free Liouville theory shows that all of the terms in
$U(\phi)$ are relevant perturbations of the free Liouville Lagrangian. The
near-horizon limit is the analog of a UV fixed point. We can zoom in on the
horizon by rescaling the coordinate $r-r_{0}$, which rescales the Rindler
metric by a Weyl transformation. The field $\phi$, which plays the role of an
invariant entropy, has weight zero under this transformation. Powers of
$r-r_{0}$ higher than linear, as well as potential terms for $\phi$, scale
away like relevant operators at an ultraviolet fixed point. We emphasize that
this is a purely classical argument, but we believe the analogy with the
renormalization group makes it more transparent. Details of the conformal
representation theory for the Liouville theory, with particular emphasis on
the classical limit, can be found in Seiberg (1990) and references therein.

We conclude that at the level of classical geometric symmetries, the generator
$L_{0}$, which, according to Refs. Casini  _et al._ (2011); Jacobson and
Visser (2019) becomes the modular Hamiltonian of a diamond in the quantum
theory, is embedded in a Virasoro algebra. Our conjecture, following Refs.
Carlip (2012); Solodukhin (1999) is that the full Virasoro algebra is realized
in the quantum theory, and that its representation satisfies the criteria for
Cardy’s theorem.888Knowledgeable readers will object that the Virasoro algebra
(even its finite $sl(2,R)$ sub-algebra) has no finite dimensional unitary
representations, contradicting the CEB for the diamond. However, the
considerations of Carlip (2012); Solodukhin (1999) and the present paper need
only the subspace with $L_{0}$ in the vicinity of a finite value. The true
dynamics of quantum gravity will impose a cutoff on the Virasoro algebra of a
diamond. Nothing in this paper is sensitive to that cutoff. For example, our
calculation of $\alpha=1$ for a system with square root behavior of the log
density of states is insensitive to cutoffs on the spectrum.

Utilizing the $z-t$ coordinate system, which make the conformal symmetry
manifest, the Poisson brackets of the dilaton

$\\{\phi(z),\partial_{t}\phi(z^{\prime})\\}_{PB}=\delta(z-z^{\prime}),$ (39)

allow one to write down the Poisson algebra of the charges
$T_{++}=T_{00}+T_{0z}$ by identifying

$L_{n}=\frac{I}{2\pi}\int_{-I/2}^{I/2}dze^{2\pi in\frac{z}{I}}T_{++}.$ (40)

Here $I$ is the length of the finite interval on which the Liouville theory is
defined. The $L_{n}$ are, as a result, found to have a Virasoro algebra with
central charge

$c=6q^{2}\frac{d-3}{d-2}S.$ (41)

Note that the free scalar stress tensor has vanishing classical central
charge, but the canonical Poisson bracket relations have induced a large
central charge proportional to the square of the coefficient of the term
linear in $\phi$. This classical central charge is much larger than the
quantum correction to the central charge $\Delta c=1$ of the free scalar, and
the quantized $\phi$ field is not the correct description of the fluctuating
degrees of freedom.

To be able to use Cardy’s formula and count the number of states $S_{\rm
CFT}=2\pi\sqrt{\frac{cL_{0}}{6}}$, one must have not only the central charge
but also the value of $L_{0}$. Applying Cardy’s formula will allow us to
determine $\langle K\rangle$ and $\langle\Delta K^{2}\rangle$. Since
$L_{0}=I^{2}P^{2}/8\pi$, with $P=\partial_{z}\phi$, we need the field
configuration. The classical solutions of the Liouville model are right moving
waves (the left moving waves don’t contribute to $T_{++}$) and a zero mode
solution which is a linear function of $z$. The equilibrium configuration of
the diamond is described by the zero mode. Solodukhin posits that the correct
zero mode is

$\phi=2\phi_{0}\frac{z-\beta_{0}}{I-2\beta_{0}}.$ (42)

This obviously has the right value at the horizon, while the behavior at $z=0$
is gotten by imposing that the boundary term in the EH action vanish
there.999Since $z$ denotes the position of the horizon, which scales as $1/n$
in a replica calculation, one immediately sees that $S$ must scale inversely
with the replica index $n$. In a replica calculation, this would imply
$\alpha=1$. $L_{0}$ for the classical solution is then

$L_{0}=\frac{\phi_{0}^{2}}{2\pi},$ (43)

for which Cardy’s formula is

$S_{CFT}=2\pi\sqrt{\frac{cL_{0}}{6}}=S.$ (44)

Carlip derived the same result, including the case of rotating black holes, by
purely $d$ dimensional methods.

It is a mistake to think that the quantized Liouville theory is the $1+1$
dimensional CFT to which the Cardy formula is being applied. In that model the
lowest value of $L_{0}$ is not the conformally invariant state, and Cardy’s
formula for the spectral density is not valid. The density of states of the
quantized Liouville theory is much too small to account for the entropy of the
horizon. Moreover, the Liouville model is integrable, while the entanglement
spectra of horizons are expected to be chaotic. Instead, one should think of
the results of Carlip and Solodukhin as a classical calculation of black hole
entropy, analogous to that following from saddle point approximations to
functional integrals over Euclidean metrics. We expect, however, that a
stochastic scalar field theory of $\phi$, with entropy given by Cardy’s
formula, and fluctuations satisfying $\alpha=1$, could give an adequate
effective description of the full theory. We leave the precise formulation of
such an effective description for future work.

It is useful to recall the calculation of Brown and Henneaux Brown and
Henneaux (1986) which motivated the results of Carlip and Solodukhin.
Strominger Strominger (1998) argued that the Brown-Henneaux calculation should
be viewed as an avatar of the AdS/CFT correspondence. The entropy of black
holes in $AdS_{3}$ had been, in many cases, shown to be calculable in terms of
$1+1$ CFT. Earlier work of Brown and Henneaux showed that the asymptotic
symmetry group of classical $AdS_{3}$ gravity contained a Virasoro algebra (in
this case two sided) with a classical central charge. Indeed, the boundary
modes of $AdS_{3}$ gravity obey the equations of motion of the Liouville
theory. The classical central charge derived from $AdS_{3}$ gravity coincides
with the actual central charge of the CFT describing AdS space according to
the AdS/CFT correspondence. So the results of Carlip and Solodukhin should be
viewed as a conjecture, analogous to the AdS/CFT conjecture, about the near-
horizon quantum dynamics of black holes. The conjecture is that the modular
Hamiltonian of a small black hole is the $L_{0}$ generator of a Virasoro
algebra in a representation satisfying Cardy’s theorem. We have extended that
conjecture to the spectrum of the modular Hamiltonian of any causal diamond
with Rindler behavior near its horizon.

While we think this conjecture is extremely plausible, and have argued that
Solodukhin’s method generalizes to the near-horizon dynamics of any causal
diamond with geometry $R_{2}\times\Sigma_{d-2}$, for the purposes of
calculating $\alpha$, we only need a more modest conjecture. The classical
calculations of Carlip (2012); Solodukhin (1999) tell us that that the $L_{0}$
generator which preserves the diamond is embedded in a Virasoro algebra with a
large central charge. Since the modular Hamiltonian $K$ was also identified in
Refs. Casini  _et al._ (2011); Jacobson and Visser (2019) as the generator of
boosts that preserves the diamond, this suggests that $L_{0}$ should be
identified with $K$. We denote the spectrum of eigenvalues of $L_{0}$ (or
equivalently $K$) as $E$, in which case the density of states, assuming that
Cardy’s theorem Eq. 44 is valid for the entropy of the quantum theory, is
given by $e^{B\sqrt{E}}$. We will assume a large constant $B$ such that we can
carry out a saddle point approximation (this corresponds to large entropy). We
then identify

$\beta F_{\beta}=-\left({\rm ln}\ \int_{E_{1}}^{E_{2}}dE\ e^{B\sqrt{E}-\beta
E}\right)$ (45)

in Eq. 2, from which one can calculate the modular fluctuations
thermodynamically:

$\langle\Delta K^{2}\rangle=\beta^{2}\partial^{2}_{\beta}\left({\rm ln}\
\int_{E_{1}}^{E_{2}}dE\ e^{B\sqrt{E}-\beta E}\right)_{\beta=1}.$ (46)

For large $B$, the integral is dominated by a stationary point on the contour
of integration

$\frac{B}{\sqrt{E_{*}}}=2\beta.$ (47)

Taking just the saddle point contribution we have

$\langle\Delta
K^{2}\rangle=\beta^{2}\partial_{\beta}^{2}\left(\frac{B^{2}}{4\beta}\right)_{\beta=1}=\frac{B^{2}}{2\beta}.$
(48)

At the same time, the entropy is the microcanonical entropy at the saddle
point,

$\langle K\rangle=B\sqrt{E_{*}}=\frac{B^{2}}{2\beta},$ (49)

so that $\alpha=1$. For a $1+1$ dimensional CFT of course, the results
$\alpha=1$ follows from the fact that the entropy is linear in the temperature
as a consequence of scale invariance and extensivity. This calculation shows
that the result $\alpha=1$ is more general than $1+1$ dimensional CFTs. In
particular, it shows that SYK models Sachdev and Ye (1993); Kitaev (2015) (by
which we mean Hamiltonians with fixed couplings belonging to one of the SYK
ensembles) also have $\alpha=1$ since they all have a density of states
$\rho(E)\sim\sinh(B\sqrt{E})$101010 SYK Hamiltonians with fixed couplings have
a chaotic spectrum with no degeneracies. The average density of states emerges
after coarse-graining this discrete spectrum over appropriate energy
intervals. Similar remarks are valid for Cardy’s formula, which applies to
chaotic CFTs with no degeneracies..

We recall that for RT diamonds in AdS/CFT with an Einstein-Hilbert dual, the
result $\alpha=1$ was already obtained by multiple independent calculations
Verlinde and Zurek (2020); Perlmutter (2014b); Nakaguchi and Nishioka (2016);
De Boer  _et al._ (2019). If one accepts the RS II conjecture relating
gravitational physics on a stabilized brane in AdS space to CFT dynamics, then
our calculation in the RS II model extends this result to causal diamonds in
Minkowski space. Our generalization of the conjecture of Carlip (2012);
Solodukhin (1999) to arbitrary Rindler horizons implies that we should have
$\alpha=1$ for local fluctuations on any such horizon, including the
cosmological horizon of dS space. Below we will explain why this conjecture
does not contradict the result $\alpha=d-2$, for large diamonds in AdS space,
which follows from the thermal equilibrium fluctuations of large black holes.

Readers uncomfortable with the idea of attributing entropy to empty causal
diamonds, should remember the fact that effective field theory implies the
existence of a state independent111111State independent, means independent of
the state in the Hilbert space constructed by completion of the space of
states obtained by acting on the vacuum with a finite product of local
operators. Thermal states of the field theory do not lie in this Hilbert
space. entanglement entropy of any such diamond, whose leading term is
proportional to $A(\Sigma_{d-2})\Lambda^{d-2}$, with $\Lambda$ a UV cutoff.
The map in Ref. Casini  _et al._ (2011) of this UV divergence into a volume
divergence on hyperbolic space shows that for an EFT flowing from a UV fixed
point, the coefficient of this term is finite and calculable. It depends on
the UV fixed point, but in effective field theory the whole effect is a
renormalization of $G_{d}$, if we accept the Covariant Entropy Bound, which
attributes a classical gravitational entropy to the diamond boundary. Thus we
expect the universal entropy $A(\Sigma_{d-2})/4G_{d}$, for any small diamond
in any finite model of quantum gravity121212A version of this argument can be
constructed using the AdS/CFT correspondence. Polchinski Polchinski (1999) and
Susskind Susskind (1999) showed that for a CFT with an EH dual, one can
construct states localized in a diamond much smaller than the AdS radius, by
acting on the CFT with special combinations of local boundary operators $A_{i\
\diamond}$. The small diamond is described by a code subspace, which is
maximally entangled with the much larger Hilbert space of the CFT. Its density
matrix, in generic states of the CFT, has high entropy, even though the
diamond is empty. .

Our conjecture thus unifies effective field theory methods for calculating
gravitational entropy Susskind and Uglum (1994); Carlip and Teitelboim (1995);
Callan and Wilczek (1994); Cooperman and Luty (2014); Banks  _et al._ (2021)
with the arguments of Carlip (2012); Solodukhin (1999), and with the explicit
calculations of Verlinde and Zurek (2020) for RT diamonds. Like effective
field theory replica methods, it is ultra-local on the $d-2$ dimensional
bifurcation surface. Physically, one might think of all of these techniques as
giving one the snapshot of horizon dynamics seen by a measuring device with
Planck scale acceleration localized at a particular point on the bifurcation
surface of the diamond boundary. Ultra-locality on the bifurcation surface
will be the key to understanding the discrepancy between the universal result
$\alpha=1$ and thermal entropy fluctuations for large AdS black holes.

To conclude this section we note that our conjecture is also suggested by the
results of Bredberg  _et al._ (2011). These authors showed that the viscosity
to entropy density ratio for the hydrodynamic equations arising from
Einstein’s equations on any weakly curved diamond boundary, is universal, and
equal to $\frac{1}{4\pi}$. The Fluctuation-Dissipation theorem relates that
ratio to the fluctuations of a hypothetical quantum operator, dual to the
$T^{\tau i}$ components of the Brown-York stress tensor on a time-like near-
horizon surface, in any model of quantum gravity in which Einstein’s equations
are approximately valid. Our conjecture is the analogous statement for entropy
fluctuations. The hydrodynamics of entropy is compressible fluid dynamics,
because of the thermodynamic relation for entropy density

$\sigma=(p+\rho)/T.$ (50)

We conjecture that the hydrodynamic equations equivalent to Einstein’s
equation are incompressible because of the fast scrambling Hayden and Preskill
(2007); Sekino and Susskind (2008) of information on certain null surfaces.
The dynamics that equilibrates entropy operates on a time scale $t_{0}{\rm
ln}\ S$ shorter than the diffusive time scale, $t_{0}$ times a power of $S$,
which determines the flow of transverse momentum. As a consequence, the
entropy density in the Einstein/Navier-Stokes equations is constant. This
insight will be important in Section IV, where we seek to understand why the
equilibrium fluctuations of large AdS black holes do not satisfy our universal
rule.

In order to relate the capacity of entanglement to a transport coefficient,
and demonstrate the analog of the Fluctuation-Dissipation Theorem for entropy
fluctuations, one would have to work out the analog of the Navier-Stokes
equations that describes the fast scrambling of entropy on a small enough area
of any stretched horizon. This is the subject of future work.

##  IV Large Diamonds/Black Holes in AdS Space

We now turn to the exception that violates the rule, large diamonds or black
holes in AdS space. For large black holes, the rules of the AdS/CFT
correspondence say that they are dual to the thermal ensemble of a boundary
CFT on a sphere. The entropy and temperature computed by black hole
thermodynamics are those of the density matrix Eq. 1, and as a consequence we
can calculate the modular fluctuations via the thermodynamic formulae Eqs. 3,
4, by taking a temperature derivative $\langle\Delta
K^{2}\rangle=T\frac{dS}{dT}.$ Since $S\propto r_{S}^{d-2}$, it follows that we
must find the temperature dependence of $r_{S}$. For large mass, the
Schwarzschild radius is given by

$r_{S}^{d-1}=C_{d}ML^{2},$ (51)

with $C_{d}$ determined from Eq. 19. The black hole temperature, given by
$T=\frac{f^{\prime}(r_{S})}{4\pi}$, is related to $L$ as

$T\propto(3-d)C_{d}\frac{M}{r_{S}(T)^{d-2}}+2\frac{r_{S}(T)}{L^{2}}=(5-d)\frac{r_{S}(T)}{L^{2}}.$
(52)

The entropy is thus $S\propto T^{d-2},$ and hence,

$\alpha=d-2.$ (53)

As we discussed already in Sec. III, we can understand the failure of the
$1+1$ dimensional CFT description of the horizon variables by noting that for
large AdS black holes, the range of validity of the near-horizon approximation
is small in comparison to the time required to span the horizon, since
$r_{S}\gg L\pi/2$, with the latter bounding the proper time of a generic time-
like trajectory.

There is a feature of the dynamics that gives a clue to what is going on. The
spectrum of quasi-normal modes of a large AdS black hole includes sound modes,
which indicate ballistic propagation of energy, momentum and information over
the horizon Crossley  _et al._ (2016); Rangamani (2009). Note that there are
no sound modes in the quasi-normal spectra of small diamonds, the dS horizon,
or the boundary of an RT diamond. This is of course expected from the AdS/CFT
correspondence: dynamics on the $d-2$ sphere is described by a local field
theory. Thus the temperature scaling of the entropy follows from scale
invariance and extensivity of the entropy of a CFT on a boundary sphere
$S^{d-2}$ of radius $R$. So, $\alpha=d-2$ follows both from black hole
thermodynamics, and from the same kind of scaling argument that we used for
$1+1$ dimensional CFT, applied to $d-1$ dimensional CFT.

A further piece of evidence is the fact that the sound modes on a large AdS
black hole have a shortest wavelength, of order the AdS curvature radius $L$.
To see this, note that scale invariance of the boundary CFT, and the fact that
the UV cutoff on sound modes is independent of the boundary volume, implies
that the cutoff wavelength on the boundary is $\lambda_{c}\sim
T^{-1}\sim\frac{L^{2}}{r_{S}}$. The large black holes in AdS have a blackening
factor

$f(r)=\frac{r^{2}}{L^{2}}-1-\frac{r_{S}^{d-1}}{r^{d-3}L^{2}}.$ (54)

The metric Eq. 11 with this blackening factor has a scaling symmetry Horowitz
and Hubeny (2000), which tells us that ratios of length scales on the black
hole horizon are the same as those on the boundary. So the cutoff length scale
$l_{c}$ on the horizon at radius $r_{S}$ satisfies

$l_{c}\sim\lambda_{c}\frac{r_{S}}{L}\sim L\ll r_{S}.$ (55)

On the other hand the spectrum of quasi-normal modes of shorter wavelength
behave exactly like those on the horizon of a flat space black hole, because
the geometries are identical on scales less than $L$. The authors of Hayden
and Preskill (2007); Sekino and Susskind (2008) have argued that this behavior
indicates fast scrambling of the information in a localized probe of some
point on the horizon over a region of radius $L$, in a time $\sim L~{}{\rm
ln}(L/l_{p})$.

The picture that one gets by combining these arguments is of a lattice of fast
scrambling systems with $\sim(L/l_{p})^{d-2}$ q-bits, and asymptotic densities
of states like those of a $1+1$ dimensional CFT, coupled together in a manner
local on the sphere. The black hole entropy formula tells us that the modular
fluctuations of large AdS diamonds are dominated by those of the $d-1$
dimensional “sound modes”. This is to be expected. The inequality $r_{S}\gg L$
tells us that the entropy of the full system scales like
$(L/l_{p})^{d-2}(r_{S}/L)^{d-2}$, which is much larger than the entropy of
“individual lattice points”.

This picture is very natural in the context of tensor network regularizations
of CFTs, which tile a spatial slice of AdS with a sequence of lattices. The
large coefficient of $T^{d-2}$ in the thermal entropy of CFTs with an EH dual
implies that each of the nodes of the tensor network must be a Hilbert space
of large dimension. In this context, our conjecture states that the dynamics
internal to each node are described by a model with the spectral density of a
$1+1$ dimensional CFT. Sound modes arise because of the couplings between
different nodes. The black hole entropy formula tells us that the entropy of
the sound modes dominates the localized entropy at fixed temperature. The
authors of Bredberg  _et al._ (2011) showed that the incompressible
hydrodynamics of large black holes is identical to that of Minkowski diamonds
in the near-horizon limit. A localized perturbation of the large black hole,
rapidly spreads its information over a node of the tensor network, on a time
scale much shorter than the hydrodynamic scale, so the local entropy density
appears constant. It is only on length scales greater than the AdS radius that
sound propagation becomes important.

To summarize, the quadratic thermal fluctuations of large black holes, which
we claim are the same as the modular fluctuations of large empty diamonds in
AdS space, are dominated by the behavior of the system on length scales much
larger than the AdS radius. They are proportional to the entropy, but with
$\alpha=d-2$ rather than $1$. On length scales below the AdS radius large AdS
black hole horizons behave like other horizons studied in this paper. It is
important to realize that this reasoning does not apply to the boundary
anchored RT diamonds studied in Verlinde and Zurek (2020). There, the proper
time along time-like trajectories in the diamond are much larger than the AdS
curvature radius; this allows ample time for information to propagate through
the entire diamond. Furthermore, there are no sound modes appearing at length
scales above the AdS radius.

##  V Nested Causal Diamonds and the Random Walk

The results we have presented so far are mathematical formulae/conjectures
applying to general causal diamonds. In order to apply them to an actual
interferometer experiment we must take into account the fact that the light
beam in the apparatus passes through many causal diamonds. More precisely, as
depicted in Fig. 3, we can cover the maximal causal diamond of the experiment
by a nested sequence of diamonds (somewhat akin to Matryoshka dolls)
corresponding to proper time intervals. We must identify the interval $\delta
x_{0}$ over which subsequent nested causal diamonds of size $x_{0}$ become
statistically uncorrelated. For simplicity we restrict ourselves to a
spherically symmetric diamond in Minkowski space (relevant for an experiment),
though the result can be generalized to RT and dS diamonds. We calculate the
entropy associated to a diamond at each subsequent step

$S_{\rm ent}=\frac{\Omega_{d-2}(x_{0}+\delta x_{0})^{d-2}}{4l_{p}^{d-2}}\simeq
A_{0}+\frac{\Omega_{d-2}(d-2)}{4}\frac{x_{0}^{d-2}}{l_{p}^{d-2}}\frac{\delta
x_{0}}{x_{0}},$ (56)

where the $d$-dimensional Planck length is defined here by
$l_{p}^{d-2}=G_{d}$, and $A_{0}$ is the entangling surface area when $\delta
x_{0}=0$.

Figure 3: An interferometer of arm length $R$ traces out a series of nested
causal diamonds, with each subsequent diamond separated by the scrambling
length $\delta x_{0}$. In the left panel, the nested causal diamonds are shown
in position space, with each concentric sphere representing subsequent nested
diamonds, while the right panel depicts the nested diamonds in the
$x_{i}-x_{0}$ plane.

We posit statistical independence when $\delta S_{\rm ent}\sim\sqrt{S_{0}}$,
where $S_{0}$ in the entanglement entropy when $\delta x_{0}=0$. This implies
a scrambling length

$\delta
x_{0}\simeq\frac{x_{0}}{(d-2)}\sqrt{\frac{4l_{p}^{d-2}}{\Omega_{d-2}x_{0}^{d-2}}}.$
(57)

As the light beam in the interferometer travels to the mirror at a distance
$R$ from the beam splitter, it traverses a sequence of statistically
uncorrelated causal diamonds, the number of which is ${\cal N}=\frac{R}{\delta
x_{0}}$. The uncertainty in the length traversed, caused by fluctuations of
the near-horizon degrees of freedom–the pixellons of Ref. Zurek (2020)–in each
of these diamonds is of order $\delta x_{0}$, so the total length traversed
should be thought of as a one dimensional random walk. We thus have

$\delta R^{2}\simeq\delta x_{0}^{2}~{}{\cal
N}=\frac{R^{2}}{d-2}\frac{1}{\sqrt{S_{0}}}.$ (58)

This result is in agreement with the length fluctuations obtained in Verlinde
and Zurek (2020), computed via a topological black hold foliation of a
boundary-anchored diamond in AdS. These length fluctuations can be thought of
as the quantum width of the horizon of the causal diamond. In $d=4$, this
implies $\delta R^{2}\sim l_{p}R$, an observably large fluctuation as proposed
in Ref. Verlinde and Zurek (2019). In higher dimensions, the effect is
suppressed by higher powers of $l_{p}$ and would be unobservable.

Note that an important aspect of our picture is that the transverse directions
do not experience these length fluctuations, leaving the shape of the light
cone intact. This is important phenomenologically, because it implies that the
images of stars do not become blurred over cosmological distances, a
consequence of the entangling surface being minimal. It is also a consequence,
as we discussed above, of there being quasi-normal modes on Minkowski diamonds
that are fast scramblers. We expect, as described in Verlinde and Zurek
(2019), that transverse fluctuations are given by an approximately Gaussian
distribution with covariance $K(\Omega,\Omega^{\prime})$ that is rotationally
invariant; such a distribution is described in terms of spherical harmonics in
Ref. Verlinde and Zurek (2019). The calculations in this paper describe
fluctuations only along the light cone.

##  VI Conclusions

We have conjectured that the near-horizon dynamics of causal diamonds in flat
space, boundary-anchored AdS and dS, are described in terms of a conformal
field theory. We focused on three pieces of evidence. All three pieces of
evidence imply $\langle\Delta K^{2}\rangle=\langle
K\rangle=\frac{A(\Sigma_{d-2})}{4G_{d}}$, with the first two being explicit
calculations and the third a heuristic argument. First, we calculated
$\langle\Delta K^{2}\rangle=\frac{A(\Sigma_{d-2})}{4G_{d}}$ for a causal
diamond on a flat RS-II brane in the bulk of AdS. Second, we calculated the
same from the density of states derived from the dimensional reduction of the
Einstein-Hilbert action in flat space, following seminal work of Carlip and
Solodukhin. Third, we argued that the fluid-gravity correspondence combined
with the fluctuation-dissipation theorem, as well as the nature of fast-
scrambling horizons (in particular, the absence of sound modes for dS,
Minkowski and boundary-anchored diamonds), supports $\alpha=1$.

We also showed that such an entropic (or holographic) picture of gravity
implies that metric fluctuations in the causal development of a spherical ball
${\cal B}$ will accumulate, with each nested causal diamond (analogous to a
Matryoshka doll, each with radius $\delta R\sim R/\sqrt{S}$ larger than the
previous) uncorrelated with the previous causal diamond. In $d=4$ dimensions,
such trajectory-integrated fluctuations are observable in a future experiment,
utilizing the advances in spacetime measurements achieved with gravitational
wave observatories.

Taken together, these ideas imply that formal developments, often in the
context of AdS or conformal field theories, need not be far removed from the
realm of observation. Possible concrete theoretical tools that could be
utilized in the context of trajectory-integrated fluctuations are SYK
models131313We include SYK models because they also give rise to $\alpha=1$.
Strict adherence to the philosophy of Carlip (2012); Solodukhin (1999) would
require us to restrict attention to CFTs., tensor networks and OTOCs that
describe fast-scrambling systems. We look forward to further development along
these lines, including observational tests.

Acknowledgments

This paper is dedicated to the memory of M. Virasoro, whose algebra proved
pivotal. We thank Patrick Draper for collaboration at the early stages of this
work, Y. Chen, S. Gukov, V. Lee, D. Li for discussion, and S. Shenker, E.
Verlinde, and especially C. Keeler and J. Parra-Martinez for comments on the
manuscript. TB thanks M. Rangamani, R. Myers and S. Carlip for email
discussions of their work. The work of TB is partially supported by the
Department of Energy under grant DOE SC0010008. The work of KZ is supported by
the Heising-Simons Foundation “Observational Signatures of Quantum Gravity”
collaboration grant 2021-2817, by the DoE under contract DE-SC0011632, and by
a Simons Investigator award.

## References

  * Ryu and Takayanagi (2006a) S. Ryu and T. Takayanagi, JHEP 08, 045 (2006a), arXiv:hep-th/0605073 .
  * Casini  _et al._ (2011) H. Casini, M. Huerta, and R. C. Myers, JHEP 05, 036 (2011), arXiv:1102.0440 [hep-th] .
  * Jacobson and Visser (2019) T. Jacobson and M. Visser, Int. J. Mod. Phys. D 28, 1944016 (2019), arXiv:1904.04843 [gr-qc] .
  * Susskind and Uglum (1994) L. Susskind and J. Uglum, Phys. Rev. D 50, 2700 (1994), arXiv:hep-th/9401070 .
  * Carlip and Teitelboim (1995) S. Carlip and C. Teitelboim, Class. Quant. Grav. 12, 1699 (1995), arXiv:gr-qc/9312002 .
  * Callan and Wilczek (1994) C. G. Callan, Jr. and F. Wilczek, Phys. Lett. B 333, 55 (1994), arXiv:hep-th/9401072 .
  * Cooperman and Luty (2014) J. H. Cooperman and M. A. Luty, JHEP 12, 045 (2014), arXiv:1302.1878 [hep-th] .
  * Banks  _et al._ (2021) T. Banks, P. Draper, and S. Farkas, Phys. Rev. D 103, 106022 (2021), arXiv:2008.03449 [hep-th] .
  * Srednicki (1993) M. Srednicki, Phys. Rev. Lett. 71, 666 (1993), arXiv:hep-th/9303048 .
  * Bousso (1999a) R. Bousso, JHEP 07, 004 (1999a), arXiv:hep-th/9905177 .
  * Bousso (1999b) R. Bousso, JHEP 06, 028 (1999b), arXiv:hep-th/9906022 .
  * Bousso (1999c) R. Bousso, JHEP 06, 028 (1999c), arXiv:hep-th/9906022 .
  * Lewkowycz and Maldacena (2013) A. Lewkowycz and J. Maldacena, JHEP 08, 090 (2013), arXiv:1304.4926 [hep-th] .
  * Dong (2016) X. Dong, Nature Commun. 7, 12472 (2016), arXiv:1601.06788 [hep-th] .
  * Ryu and Takayanagi (2006b) S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006b), arXiv:hep-th/0603001 .
  * Kwon and Hogan (2016) O. Kwon and C. J. Hogan, Class. Quant. Grav. 33, 105004 (2016), arXiv:1410.8197 [gr-qc] .
  * Hogan and Kwon (2017) C. J. Hogan and O. Kwon, Class. Quant. Grav. 34, 075006 (2017), arXiv:1506.06808 [gr-qc] .
  * Chou  _et al._ (2017) A. Chou _et al._ (Holometer), Class. Quant. Grav. 34, 165005 (2017), arXiv:1703.08503 [gr-qc] .
  * Verlinde and Zurek (2019) E. P. Verlinde and K. M. Zurek, (2019), arXiv:1902.08207 [gr-qc] .
  * Verlinde and Zurek (2020) E. Verlinde and K. M. Zurek, JHEP 04, 209 (2020), arXiv:1911.02018 [hep-th] .
  * Zurek (2020) K. M. Zurek, (2020), arXiv:2012.05870 [hep-th] .
  * De Boer  _et al._ (2019) J. De Boer, J. Järvelä, and E. Keski-Vakkuri, Phys. Rev. D 99, 066012 (2019), arXiv:1807.07357 [hep-th] .
  * Nakaguchi and Nishioka (2016) Y. Nakaguchi and T. Nishioka, JHEP 12, 129 (2016), arXiv:1606.08443 [hep-th] .
  * Perlmutter (2014a) E. Perlmutter, JHEP 03, 117 (2014a), arXiv:1308.1083 [hep-th] .
  * Hung  _et al._ (2011) L.-Y. Hung, R. C. Myers, M. Smolkin, and A. Yale, JHEP 12, 047 (2011), arXiv:1110.1084 [hep-th] .
  * Perlmutter (2019) E. Perlmutter, (2019).
  * Randall and Sundrum (1999) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999), arXiv:hep-th/9906064 .
  * Carlip (1995) S. Carlip, Phys. Rev. D 51, 632 (1995), arXiv:gr-qc/9409052 .
  * Carlip (1999) S. Carlip, Phys. Rev. Lett. 82, 2828 (1999), arXiv:hep-th/9812013 .
  * Carlip (1998) S. Carlip, Class. Quant. Grav. 15, 3609 (1998), arXiv:hep-th/9806026 .
  * Carlip (2012) S. Carlip, AIP Conf. Proc. 1483, 54 (2012), arXiv:1207.1488 [gr-qc] .
  * Solodukhin (1999) S. N. Solodukhin, Phys. Lett. B 454, 213 (1999), arXiv:hep-th/9812056 .
  * Damour (1978) T. Damour, Phys. Rev. D 18, 3598 (1978).
  * Damour (1979) T. Damour,  _Quelques proprietes mecaniques, electromagnet iques, thermodynamiques et quantiques des trous noir_ , Ph.D. thesis, Paris U., VI-VII (1979).
  * Bredberg and Strominger (2012) I. Bredberg and A. Strominger, JHEP 05, 043 (2012), arXiv:1106.3084 [hep-th] .
  * Bredberg  _et al._ (2012) I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, JHEP 07, 146 (2012), arXiv:1101.2451 [hep-th] .
  * Crossley  _et al._ (2016) M. Crossley, P. Glorioso, H. Liu, and Y. Wang, JHEP 02, 124 (2016), arXiv:1504.07611 [hep-th] .
  * Rangamani (2009) M. Rangamani, Class. Quant. Grav. 26, 224003 (2009), arXiv:0905.4352 [hep-th] .
  * Compere  _et al._ (2011) G. Compere, P. McFadden, K. Skenderis, and M. Taylor, JHEP 07, 050 (2011), arXiv:1103.3022 [hep-th] .
  * Bredberg  _et al._ (2011) I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, JHEP 03, 141 (2011), arXiv:1006.1902 [hep-th] .
  * Policastro  _et al._ (2002a) G. Policastro, D. T. Son, and A. O. Starinets, JHEP 09, 043 (2002a), arXiv:hep-th/0205052 .
  * Policastro  _et al._ (2002b) G. Policastro, D. T. Son, and A. O. Starinets, JHEP 12, 054 (2002b), arXiv:hep-th/0210220 .
  * Kovtun  _et al._ (2005) P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005), arXiv:hep-th/0405231 .
  * Myers  _et al._ (2013) R. C. Myers, R. Pourhasan, and M. Smolkin, JHEP 06, 013 (2013), arXiv:1304.2030 [hep-th] .
  * Hayden and Preskill (2007) P. Hayden and J. Preskill, JHEP 09, 120 (2007), arXiv:0708.4025 [hep-th] .
  * Sekino and Susskind (2008) Y. Sekino and L. Susskind, JHEP 10, 065 (2008), arXiv:0808.2096 [hep-th] .
  * Carroll (2019) S. M. Carroll,  _Spacetime and Geometry_ (Cambridge University Press, 2019).
  * Seiberg (1990) N. Seiberg, Prog. Theor. Phys. Suppl. 102, 319 (1990).
  * Brown and Henneaux (1986) J. D. Brown and M. Henneaux, Commun. Math. Phys. 104, 207 (1986).
  * Strominger (1998) A. Strominger, JHEP 02, 009 (1998), arXiv:hep-th/9712251 .
  * Sachdev and Ye (1993) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993), arXiv:cond-mat/9212030 .
  * Kitaev (2015) A. Kitaev, KITP strings seminar and Entanglement 2015 program (Feb 12, April 7 and May 27 2015).
  * Perlmutter (2014b) E. Perlmutter, JHEP 03, 117 (2014b), arXiv:1308.1083 [hep-th] .
  * Polchinski (1999) J. Polchinski, (1999), arXiv:hep-th/9901076 .
  * Susskind (1999) L. Susskind, AIP Conf. Proc. 493, 98 (1999), arXiv:hep-th/9901079 .
  * Horowitz and Hubeny (2000) G. T. Horowitz and V. E. Hubeny, Phys. Rev. D 62, 024027 (2000), arXiv:hep-th/9909056 .