# Gravitational Thermodynamics of
Causal Diamonds in (A)dS

Ted Jacobson<EMAIL_ADDRESS>Maryland Center for Fundamental Physics,
University of Maryland, College Park, MD 20742, USA Manus Visser
<EMAIL_ADDRESS>Institute for Theoretical Physics, University of Amsterdam,
1090 GL Amsterdam, The Netherlands

###### Abstract

The static patch of de Sitter spacetime and the Rindler wedge of Minkowski
spacetime are causal diamonds admitting a true Killing field, and they behave
as thermodynamic equilibrium states under gravitational perturbations. We
explore the extension of this gravitational thermodynamics to all causal
diamonds in maximally symmetric spacetimes. Although such diamonds generally
admit only a conformal Killing vector, that seems in all respects to be
sufficient. We establish a Smarr formula for such diamonds and a “first law”
for variations to nearby solutions. The latter relates the variations of the
bounding area, spatial volume of the maximal slice, cosmological constant, and
matter Hamiltonian. The total Hamiltonian is the generator of evolution along
the conformal Killing vector that preserves the diamond. To interpret the
first law as a thermodynamic relation, it appears necessary to attribute a
negative temperature to the diamond, as has been previously suggested for the
special case of the static patch of de Sitter spacetime. With quantum
corrections included, for small diamonds we recover the “entanglement
equilibrium” result that the generalized entropy is stationary at the
maximally symmetric vacuum at fixed volume, and we reformulate this as the
stationarity of free conformal energy with the volume _not_ fixed.

###### Contents

  1. 1 Introduction
  2. 2 Causal diamonds in maximally symmetric spaces
  3. 3 Mechanics of causal diamonds in (A)dS
    1. 3.1 Smarr formula for causal diamonds
      1. 3.1.1 Smarr formula for AdS black holes
    2. 3.2 First law of causal diamonds
    3. 3.3 Further remarks on the first law
      1. 3.3.1 Role of maximal volume
      2. 3.3.2 Fixed volume and fixed area variations
      3. 3.3.3 Varying the cosmological constant
      4. 3.3.4 Gravitational field Hamiltonian and York time
  4. 4 Thermodynamics of causal diamonds in (A)dS
    1. 4.1 Negative temperature
    2. 4.2 Quantum corrections
      1. 4.2.1 Conformal matter
      2. 4.2.2 Non-conformal matter
    3. 4.3 Entanglement equilibrium
    4. 4.4 Free conformal energy
  5. 5 Limiting cases: small and large causal diamonds
    1. 5.1 De Sitter static patch
    2. 5.2 Small diamonds and Minkowski space
    3. 5.3 Rindler space
    4. 5.4 AdS-Rindler space
    5. 5.5 Wheeler-DeWitt patch in AdS
  6. 6 Discussion
  7. A Conformal isometry of causal diamonds in (A)dS
  8. B Conformal Killing time and mean curvature
  9. C Zeroth law for bifurcate conformal Killing horizons
  10. D Conformal group from two-time embedding formalism
    1. D.1 Conformal Killing vectors of de Sitter space
    2. D.2 Conformal Killing vectors of Anti-de Sitter space
  11. E Conformal transformation from Rindler space to causal diamond
  12. F Acceleration of the conformal Killing flow

##  1 Introduction

Horizon thermodynamics was first discovered in the context of black holes [1,
2, 3], but the principles are far more universal than that. The case of
cosmological horizons was quickly understood [4, 5] and, rather less quickly
insofar as entropy is concerned, that of acceleration horizons as well [6, 7,
8, 9, 10, 11]. Most recently, in the setting of AdS/CFT duality, the
gravitational thermodynamics of “entanglement wedges” has been discovered [12,
13]. An entanglement wedge is the domain of dependence of a partial Cauchy
surface of the bulk, whose boundary consists of a subregion ${\cal R}$ of a
conformal boundary Cauchy slice together with the minimal area bulk surface
that meets the conformal boundary at $\partial{\cal R}$. The area $A$ of the
minimal surface corresponds to the CFT entanglement entropy of the subregion
${\cal R}$ to leading order in Newton’s constant, via the Ryu-Takayanagi
formula, which is nothing but the Bekenstein-Hawking entropy $A/4\hbar G$ [14,
15]. In particular, a link has been established between fundamental properties
of CFT entanglement entropy and the bulk Einstein equation, drawing a
connection between the behavior of quantum information and gravitational
dynamics in this holographic setting [16, 17, 18, 19].

In the examples just described, the thermodynamic system extends to a boundary
of spacetime. Quasilocal relations analogous to the laws of thermodynamics
have been found for various sorts of ‘apparent’ horizons, but of course these
find application only when such horizons are present [20]. If the lesson from
all we have learned is that something fundamentally statistical underlies
gravitational dynamics, then that something should be at play everywhere in
spacetime. This viewpoint was the motivation for introducing the notion of
“local causal horizon,” with which it was possible to derive the Einstein
equation from the Clausius relation applied to the area-entropy changes of all
such horizons in spacetime [8]. Essential to that argument was the fact that
the near vicinity of any spacetime point looks like part of a slightly
deformed Minkowski spacetime, so that one can identify an approximate local
boost Killing field with respect to which the relevant notion of energy can be
defined. While that derivation was reasonably plausible on physical grounds,
the “system” under consideration was not sharply defined. It was taken to be a
very small, near horizon subsystem, but no precise definition was given, and
possible effects of entanglement with neighboring regions were not addressed.

To localize and isolate it, the thermodynamic system was defined in [21] as
the spacetime inside a causal diamond, i.e. the domain of dependence of a
spatial ball.111The thermal behavior of conformal quantum fields – without
gravity – inside a causal diamond in flat space was studied earlier, for
example in Refs. [22, 23, 24, 25, 12]. For such a system, rather than
describing a time dependent physical process, one can just consider
variations, comparing the equilibrium state to nearby states. A link was
established between the semiclassical Einstein equation, and the stationarity
of the total entanglement entropy of the diamond with respect to variations of
the geometry and quantum fields away from the vacuum state at fixed volume,
taking the diamond much smaller than the local curvature length scale of the
background spacetime.

A classical ingredient of this argument is a “first law of causal diamonds,”
which relates variations, away from flat spacetime, of the area, volume, and
matter energy-momentum tensor inside. This relation is similar to the first
law of black hole mechanics, and was derived in an Appendix of [21] using the
same methods as employed by Wald for black holes [26]. A diamond in Minkowski
spacetime plays the role of equilibrium state in this first law. Unlike a
black hole, however, a causal diamond is only conformally stationary. That is,
it does not admit a timelike Killing vector, but it admits a timelike
conformal Killing vector, for which the null boundary is a conformal Killing
horizon [27, 28], with a well-defined surface gravity.222It was shown in [29]
that the surface gravity $\kappa$ of a conformal Killing horizon has a
conformal invariant definition (see Appendix C for an explanation, and for a
proof of the zeroth law). This fact can be viewed as a hint that conformal
Killing horizons have thermodynamic properties, since $\kappa$ is identical to
the surface gravity of a conformally related true Killing horizon (for which
$\hbar\kappa/2\pi$ is the Hawking temperature). Remarkably, this is sufficient
to derive the first law, but an additional contribution arises, namely, the
gravitational Hamiltonian, which is proportional to the spatial volume of the
ball.

In the present paper we generalize the first law of causal diamonds to
(Anti-)de Sitter spacetime – i.e. it applies to any maximally symmetric space
– and we include variations of the cosmological constant and matter stress-
energy, both using a fluid description as done for black holes by Iyer [30].
Treated this way, the matter can contribute to a first order variation away
from a maximally symmetric spacetime.333Matter described by a typical field
theory action could not contribute, since the matter fields would vanish in
the maximally symmetric background spacetime, and the action is at least
quadratic in the fields. Motivation for varying the cosmological constant is
discussed in Sec. 3.3.3. The first law we obtain for Einstein gravity
minimally coupled to fluid matter takes the form

$\delta H_{\zeta}=-\frac{\kappa}{8\pi G}\,\delta A\,,$ (1.1)

where $\zeta$ is the conformal Killing vector of the unperturbed, maximally
symmetric diamond (computed explicitly in Appendix A), $H_{\zeta}$ is the
total Hamiltonian for gravity and matter along the flow of $\zeta$, and
$\kappa$ is the surface gravity (defined as positive) with respect to $\zeta$.
Much as for black holes, if we multiply and divide by Planck’s constant
$\hbar$, the quantity on the right hand side of (1.1) takes the form $-T_{\rm
H}\delta S_{\rm BH}$, where $T_{\text{H}}=\hbar\kappa/2\pi$ is the Hawking
temperature and $S_{\rm BH}=A/4\hbar G$ is the Bekenstein-Hawking entropy.
Unlike for black holes, however, there is a minus sign in front, indicating
that the temperature is negative. This minus sign is familiar from the
limiting case in which the diamond consists of the entire static patch of de
Sitter (dS) spacetime, bounded by the de Sitter horizon [5]. It was argued in
the dS case that the temperature of the static patch is therefore negative
[31], and further arguments in favor of this interpretation were given
recently in [32].

The left hand side of (1.1) consists of minimally coupled matter, cosmological
constant, and gravitational terms, i.e. $\delta H_{\zeta}=\delta
H_{\zeta}^{\text{\~{m}}}+\delta H_{\zeta}^{\Lambda}+\delta
H_{\zeta}^{\text{g}}$. The contribution of the fluid matter with arbitrary
equation of state is

$\delta
H^{\text{\~{m}}}_{\zeta}=\int_{\Sigma}\delta({T_{a}}^{b})^{\text{\~{m}}}\zeta^{a}u_{b}\,dV\,,$
(1.2)

where $({T_{a}}^{b})^{\text{\~{m}}}=g^{bc}T_{ac}^{\text{\~{m}}}$ is the
Hilbert stress-energy tensor with one index raised by the inverse metric and
$\Sigma$ is the maximal slice of the unperturbed diamond, with future pointing
unit normal vector $u^{a}$ and proper volume element $dV$. The cosmological
constant can be thought of as a perfect fluid with stress-energy tensor
$T^{\Lambda}_{ab}=-\Lambda/(8\pi G)g_{ab}$. Since it is maximally symmetric,
it may be nonzero in the background. Its Hamiltonian variation is given by

$\delta H^{\Lambda}_{\zeta}=\frac{V_{\zeta}}{8\pi
G}\,\delta\Lambda\,,\qquad\text{with}\qquad
V_{\zeta}:=\int_{\Sigma}|\zeta|dV,$ (1.3)

where $|\zeta|:=\sqrt{-\zeta\cdot\zeta}$ is the norm of the conformal Killing
vector. The quantity $V_{\zeta}$ is called the “thermodynamic volume” [33, 34,
35], and is well known in the context of the first law for black holes, when
extended to include variations of $\Lambda$. Finally, the gravitational term
takes the form

$\delta H_{\zeta}^{\text{g}}=-\frac{\kappa k}{8\pi G}\delta V,$ (1.4)

where $k$ is the trace of the (outward) extrinsic curvature of
$\partial\Sigma$ as embedded in $\Sigma$, and $V$ is the proper volume of the
ball-shaped spacelike region $\Sigma$. This term arises because $\zeta$ fails
to be a true Killing vector. In the special case of the static patch of dS
space, $\zeta$ is a true Killing vector, and indeed (1.4) vanishes, since we
have $k=0$. That this gravitational Hamiltonian variation is proportional to
the maximal volume variation suggests a connection with the York time
Hamiltonian [36], which generates evolution along a constant mean curvature
foliation and is proportional to the proper volume of slices of this foliation
(see Sec. 3.3.4). In fact, we show in Appendix B that slices of the constant
mean curvature foliation of a maximally symmetric diamond coincide with slices
of constant conformal Killing time.

Moreover, we extend the first law of causal diamonds to the semiclassical
regime, i.e. by considering quantum matter fields on a fixed classical
background. For any type of quantum matter in a small diamond we show that the
semiclassical first law can be written in terms of Bekenstein’s generalized
entropy444For conformal matter this expression for the first law actually
holds for any sized diamond, whereas for non-conformal matter the derivation
depends on an assumption (4.13) about the modular Hamiltonian variation for
small diamonds which was conjectured in [21] and tested in [37, 38].

$-\frac{\kappa k}{8\pi G}\delta V+\frac{V_{\zeta}}{8\pi
G}\delta\Lambda=T\delta S_{\rm{gen}}\,,$ (1.5)

where the temperature is minus the Hawking temperature, i.e.
$T=-\hbar\kappa/2\pi$, and the generalized entropy is defined as the sum of
the Bekenstein-Hawking entropy and the matter entanglement entropy, i.e.
$S_{\rm{gen}}:=S_{\rm{BH}}+S_{\rm\tilde{m}}$. At fixed volume $V$ and
cosmological constant $\Lambda$ this implies that the generalized entropy is
stationary in a maximally symmetric vacuum. This coincides with the
_entanglement equilibrium_ condition of [21], which was shown in that paper to
be equivalent to the semiclassical Einstein equation. We also argue in Sec.
4.4 that the entanglement equilibrium condition is equivalent to the
stationarity of a free energy at fixed cosmological constant, but _without_
fixing the volume. We thus find that the semiclassical Einstein equation is
also equivalent to the stationarity of free energy.

The present paper complements other investigations of causal diamonds in flat
space [39, 40, 41]. In particular, the first law of causal diamonds was
generalized to higher derivative gravity in [42], in which case the
Bekenstein-Hawking entropy should be replaced by the Wald entropy and the
proper volume by the “generalized volume” $W$. That first law is also valid in
any maximally symmetric space, but for (A)dS space the derivation was based on
certain identities which we prove below, in particular (2.8) and (2.13).
Further, in [43] the second order area variation of a small geodesic ball was
computed in the absence of matter and compared with the gravitational energy.
This is a step towards an extension of the first law of causal diamonds to
second order variations. Ref. [44] derives a Clausius relation for the
reversible part of the entropy change between time slices of causal diamonds
in flat space, and compares that to the first law for causal diamonds in
higher curvature gravity. Finally, in [45] the four laws of thermodynamics
were established for the causal complement of a diamond in flat space, in
particular a physical process version of the first law was derived. This
differs from our first law, in the sense that the latter is an equilibrium
state version.

This paper is organized as follows. In Sec. 2 we describe our setup of causal
diamonds in maximally symmetric spaces in more detail. The Smarr formula and
first law for causal diamonds are derived in Sec. 3 using Wald’s Noether
charge formalism. In Sec. 4 we give a thermodynamic interpretation to the
first law, and we derive the entanglement equilibrium condition from the
semiclassical first law. In Sec. 3.3 we comment on various aspects of the
first law, and in Sec. 5 we describe a number of limiting cases of the first
law for maximally symmetric causal diamonds: de Sitter static patch, flat
space, Rindler space, AdS-Rindler space and the Wheeler-DeWitt patch of AdS.
We end with a summary of results and a discussion of possible future research
directions in Sec. 6. The appendices are devoted to establishing several
properties of conformal Killing fields in (A)dS and in flat space, and of
bifurcate conformal Killing horizons in general.

##  2 Causal diamonds in maximally symmetric spaces

In this section we discuss causal diamonds and their conformal Killing vectors
in maximally symmetric spaces. It suffices to write the equations for the case
of positive curvature, i.e. de Sitter space, since the negative curvature
(Anti-de Sitter) and flat cases can be obtained from these by sending the
curvature length scale $L$ to $iL$, or to infinity, respectively. The line
element for a static patch of de Sitter space in $d$ spacetime dimensions is

$ds^{2}=-[1-(r/L)^{2}]dt^{2}+[1-(r/L)^{2}]^{-1}dr^{2}+r^{2}d\Omega_{d-2}^{2}\,,$
(2.1)

where we use units with $c=1$. In terms of retarded and advanced time
coordinates,

$u=t-r_{*}\,\qquad\text{and}\qquad v=t+r_{*}\,,$ (2.2)

with $r_{*}$ the “tortoise coordinate” defined by

$dr_{*}=\frac{dr}{1-(r/L)^{2}}\,,\qquad r=L\tanh(r_{*}/L)\,,$ (2.3)

the line element (2.1) takes the form

$\displaystyle ds^{2}$
$\displaystyle=-[1-(r/L)^{2}]\,dudv+r^{2}d\Omega_{d-2}^{2}$ (2.4)
$\displaystyle=\text{sech}^{2}(r_{*}/L)\left[-dudv+L^{2}\sinh^{2}(r_{*}/L)\,d\Omega_{d-2}^{2}\right].$

Note that $r_{*}=r+O(r^{3})$, so in particular $r=0=r_{*}$ at the origin. For
dS the cosmological horizon, $r=L$, corresponds to $r_{*}=\infty$. For AdS, we
have $r=L\tan(r_{*}/L)$, so $r=\infty$ corresponds to $r_{*}=L\pi/2$. In the
flat space limit, $L\to\infty$, the tortoise and radial coordinates coincide.

A spherical causal diamond in a maximally symmetric space can be defined as
the domain of dependence of a spherical spacelike region $\Sigma$ with
vanishing extrinsic curvature. Equivalently, it can be described as the
intersection of the future of some point $p$, with the past of another point
$p^{\prime}$ (see Fig. 1). All such diamonds are equivalent, once the geodesic
proper time between the vertices $p$ to $p^{\prime}$ has been fixed. The
intersection of the future light cone of $p$ and the past light cone of
$p^{\prime}$ is called the edge of the diamond. The edge is the boundary
$\partial\Sigma$ of a $(d-1)$-dimensional ball-shaped region $\Sigma$. The
symmetries of such diamonds are rotations about the $pp^{\prime}$ line,
reflection across $\Sigma$, and a conformal isometry to be discussed shortly.

Figure 1: A causal diamond in a maximally symmetric spacetime for a ball-
shaped spacelike region $\Sigma$. The past and future vertices of the diamond
are denoted by $p$ and $p^{\prime}$, respectively, and $\mathcal{H}$ is the
null boundary. The dashed arrows are the flow lines of the conformal Killing
vector $\zeta$, whose flow sends the boundary of the diamond into itself, and
vanishes at $\partial\Sigma$, $p$ and $p^{\prime}$.

Fixing a causal diamond, we place the origin of the above coordinate system at
the center, and choose the $t$ coordinate so that $\Sigma$ lies in the $t=0$
surface. The geodesic joining the vertices is then the line $r=0$, and the
diamond is the intersection of the two regions $u>-R_{*}$ and $v<R_{*}$, for
some $R_{*}$. The vertices are located at the points $p=\\{u=v=-R_{*}\\}$ and
$p^{\prime}=\\{u=v=R_{*}\\}$, and the edge of the diamond is the
$(d-2)$-sphere $\partial\Sigma=\\{v=-u=R_{*}\\}$, with coordinate radius
$r_{*}=R_{*}$ and area radius $r=R=L\tanh(R_{*}/L)$.

The unique conformal isometry that preserves the causal diamond is generated
by the conformal Killing vector

$\zeta=\frac{L}{\sinh(R_{*}/L)}\Big{[}\big{(}\cosh(R_{*}/L)-\cosh(u/L)\big{)}\partial_{u}+\big{(}\cosh(R_{*}/L)-\cosh(v/L)\big{)}\partial_{v}\Big{]}\,.$
(2.5)

A derivation of this fact is given in Appendix A. The flow generated by
$\zeta$ sends the boundary of the diamond into itself, and leaves fixed the
vertices and the edge. In the interior $\zeta$ is the sum of two future null
vectors, so it is timelike and future directed. It is the null tangent to the
past and future null boundaries, $\mathcal{H}=\\{v=R_{*},u=-R_{*}\\}$, so
those boundaries are conformal Killing horizons. The intersection of these
null boundaries, i.e. the edge $\partial\Sigma$, is therefore referred to as
the bifurcation surface of $\mathcal{H}$.555The conformal Killing vector also
acts outside the diamond, and remains null on the continuation of the null
boundaries of the diamond. In total there are four conformal Killing horizons
$\\{u=\pm R_{*},v=\pm R_{*}\\}$, which divide the maximally symmetric
spacetime up into six regions (see [45] for an analysis of the flow of the
conformal Killing field in Minkowski spacetime).

In terms of the $t$ and $r$ coordinates introduced above, the conformal
Killing vector reads

$\zeta=\frac{L^{2}}{R}\left[\Big{(}1-\frac{\sqrt{1-(R/L)^{2}}}{\sqrt{1-(r/L)^{2}}}\cosh(t/L)\Big{)}\partial_{t}-\frac{r}{L}\sqrt{\left(1-(R/L)^{2}\right)\left(1-(r/L)^{2}\right)}\sinh(t/L)\,\partial_{r}\right].$
(2.6)

Note that if the boundary of the diamond coincides with the cosmological
horizon, i.e. if $R=L$, then $\zeta=L\partial_{t}$. That is, the conformal
Killing symmetry becomes the usual time translation of the entire static patch
of dS, which is a causal diamond with infinite time duration but finite
spatial width. In Appendix D we use the two-time embedding formalism of dS and
AdS to derive an expression for $\zeta$ in terms of the generators of the
conformal group.666Note that maximally symmetric spaces are conformally flat,
so they admit the $O(2,d)$ group of conformal isometries. We show that $\zeta$
can be written as a linear combination of a time translation of the
surrounding static patch and a conformal transformation (which is a special
conformal transformation in the case of flat space) – see equation (D.8).

The surface gravity $\kappa$ of any conformal Killing vector with a
bifurcation surface ${\cal B}$ can be defined exactly as for a true Killing
vector: If we contract the conformal Killing equation,
$\nabla_{(a}\zeta_{b)}=\alpha g_{ab}$ with $m^{a}m^{b}$, for $m^{a}$ any
tangent vector to ${\cal B}$, the left hand side vanishes, since $\zeta^{a}=0$
on ${\cal B}$. Thus we learn that (as for the particular example of
$\zeta^{a}$ under study) $\alpha=0$ on ${\cal B}$. It follows that
$\partial_{a}\zeta^{b}$ is a generator of Lorentz transformations in the two-
dimensional normal plane at each point of ${\cal B}$. Like for true Killing
vectors, we may therefore define the surface gravity $\kappa$ at $\cal{B}$ (or
rather its absolute value) by
$2\kappa^{2}=(\partial_{a}\zeta^{b})(\partial_{b}\zeta^{a})$. Other
definitions for $\kappa$, which are equivalent for a true Killing vector, are
not equivalent for a conformal Killing vector. The definition that is
invariant under conformal rescalings of the metric [29],
$\nabla_{a}\zeta^{2}=-2\kappa\zeta_{a}$, is also constant along the generators
of the conformal Killing horizon, and coincides with the definition just given
at ${\cal B}$. We establish these general properties in Appendix C, along with
the zeroth law, i.e. the fact that $\kappa$ is constant on $\cal{H}$. The
normalization of $\zeta^{a}$ in (2.5) or (2.6) has been chosen so that
$\kappa=1$ at the future horizon ($v=R_{*}$) and $\kappa=-1$ at the past
horizon ($u=-R_{*}$). In the rest of the paper we take $\kappa$ to be positive
and keep it explicit, to indicate where it appears if a different
normalization is chosen.777If the conformal Killing vector were normalized
such that $\zeta^{2}=-1$ at $t=r=0$, then the surface gravity would be
$\kappa=(R/L^{2})(1-\sqrt{1-(R/L)^{2}})^{-1}$. This is the choice usually made
in the case of the static patch of de Sitter space, where $R=L$ and
$\kappa_{\text{dS}}=1/L$. In the flat space limit ($L\to\infty$) this surface
gravity becomes $\kappa_{\rm{flat}}=2/R$, and for an infinite diamond in AdS
($R/L\to\infty$, the “Wheeler-DeWitt patch”) it reduces to
$\kappa_{\rm{WdW}}=1/L,$ where $L$ is the AdS radius.

The Lie derivative of the metric with respect to a conformal Killing vector in
general has the form

$\mathcal{L}_{\zeta}g_{ab}=\nabla_{a}\zeta_{b}+\nabla_{b}\zeta_{a}=2\alpha
g_{ab}\,,\qquad\alpha=(\nabla_{c}\zeta^{c})/d\,.$ (2.7)

In addition, $\zeta$ has some special properties that will be important for
us. Under the reflection symmetry of the diamond, $\zeta\rightarrow-\zeta$, so
$\alpha\rightarrow-\alpha$. It follows that $\alpha$ vanishes on $\Sigma$, so
$\zeta$ acts “instantaneously” as a true Killing vector on $\Sigma$:

$\mathcal{L}_{\zeta}g_{ab}\big{|}_{\Sigma}=0\,.$ (2.8)

It also follows that $\nabla_{a}\alpha$ is normal to $\Sigma$, so we have

$\nabla_{c}\left(\mathcal{L}_{\zeta}g_{ab}\right)\big{|}_{\Sigma}=-2\dot{\alpha}\,u_{c}g_{ab}\,,\qquad\dot{\alpha}=u^{c}\nabla_{c}\alpha\,,$
(2.9)

where $u^{c}$ is the future pointing unit normal to $\Sigma$, given by
$u^{c}\partial_{c}=\left(1-(r/L)^{2}\right)^{-1/2}\\!\partial_{t}$.
Computation of $\alpha$ using the expression (2.6) easily yields

$\displaystyle\alpha$
$\displaystyle=-\frac{L}{R}\sqrt{\left(1-(R/L)^{2}\right)\left(1-(r/L)^{2}\right)}\sinh(t/L)\,,$
(2.10) $\displaystyle\dot{\alpha}\big{|}_{t=0}$
$\displaystyle=-\frac{1}{R}\sqrt{1-(R/L)^{2}}=\frac{-1}{L\sinh(R_{*}/L)}\,.$
(2.11)

Notice that $\dot{\alpha}$ is constant on $\Sigma$. This property will be
crucial for the existence of a geometric form of the first law of causal
diamonds. Further, since

$k:=\frac{d-2}{R}\sqrt{1-(R/L)^{2}}$ (2.12)

is the trace of the extrinsic curvature of the surface $\partial\Sigma$ as
embedded in $\Sigma$, we may write

$\dot{\alpha}\big{|}_{\Sigma}=-\frac{\kappa k}{d-2}\,,$ (2.13)

allowing for a normalization of $\zeta$ with surface gravity $\kappa$ rather
than unity.

While we established (2.13) by explicit computation, it can also be derived by
examining derivatives along the horizon at $\partial\Sigma$, and using the
properties that (i) the trace of the extrinsic curvature of $\Sigma$ vanishes,
and (ii) $\dot{\alpha}$ is constant on $\Sigma$. However, we have not found an
underlying geometric reason for the constancy of $\dot{\alpha}$ on $\Sigma$.
It probably requires maximal symmetry of the spacetime, since we checked that
$\dot{\alpha}$ is not constant for the case of a causal diamond in time cross
hyperbolic space, $\mathbb{R}\times\mathbb{H}^{d-1}$, which also admits a
diamond preserving conformal Killing vector. In Appendix B the constancy of
$\dot{\alpha}$ on $\Sigma$ is established in a different way.

##  3 Mechanics of causal diamonds in (A)dS

In this section we first derive a Smarr formula for causal diamonds in (A)dS,
by equating the Noether charge to the integral of the Noether current. This
method of obtaining the Smarr formula is quite general, and it illustrates the
origin of the “thermodynamic volume” term. As an aside, we also show how a
finite Smarr formula for AdS black holes can be obtained by subtracting the
(divergent) empty AdS Smarr formula from the black hole one. We then move on
to our main objective, which is to derive the first law of causal diamonds in
(A)dS. First we employ the usual dimensional scaling argument to deduce from
the Smarr formula a first law for variations between maximally symmetric
diamonds. Next we employ the Noether current method, used by Wald for the case
of black holes[26, 46], as was done for variations of Minkowski space diamonds
in Appendix D of [21]. Here we extend that derivation to (Anti-)de Sitter
space, and include fluid matter (allowing in particular for a variable
cosmological constant) as in [30], obtaining a first law that applies to
arbitrary variations to nearby solutions.

###  3.1 Smarr formula for causal diamonds

We start with deriving a Smarr-like formula for causal diamonds in (A)dS
space. We will obtain this relation using a slightly unusual but very general
method: equating the Noether current associated with diffeomorphism symmetry
to the exterior derivative of the Noether charge and integrating over the ball
$\Sigma$. Liberati and Pacilio [47] used the same Noether method to derive a
Smarr formula for Lovelock black holes. Throughout this section we will employ
Wald’s Noether charge formalism [26]. See e.g. [46, 48] for further details
about this formalism.

To every Lagrangian $d$-form depending on the dynamical fields $\phi$ there is
an associated symplectic potential $(d-1)$-form $\theta$, defined through

$\delta L=E\,\delta\phi+d\theta(\phi,\delta\phi)\,,$ (3.1)

where $E$ is the equation of motion $d$-form, and tensor indices are
suppressed. For a variation $\delta\phi=\mathcal{L}_{\chi}\phi$ induced by the
flow of a vector field $\chi$, there is an associated Noether current
$(d-1)$-form,

$j_{\chi}:=\theta(\phi,\mathcal{L}_{\chi}\phi)-\chi\cdot L\,.$ (3.2)

When $L$ is diffeomorphism covariant, the variation $\delta_{\chi}L$ produced
by $\delta\phi=\mathcal{L}_{\chi}\phi$ is equal to $\mathcal{L}_{\chi}L$,
which implies that the Noether current is closed for all $\chi$ when the
equation of motion $E=0$ holds, and so is an exact form,

$j_{\chi}=dQ_{\chi}.$ (3.3)

The $(d-2)$-form $Q_{\chi}$ is constructed from the dynamical fields together
with $\chi$ and its first derivative, and is called the Noether charge form.

The Smarr formula comes from the integral version of the identity (3.3),

$\oint_{\partial{\cal R}}Q_{\chi}=\int_{\cal R}j_{\chi},$ (3.4)

where ${\cal R}$ is a $(d-1)$-dimensional submanifold with boundary
$\partial{\cal R}$. For a black hole with bifurcate Killing horizon, $\chi$
can be taken as the horizon generating Killing vector, and ${\cal R}$ can be
taken as a hypersurface extending from the bifurcation surface to spatial
infinity. Since $\mathcal{L}_{\chi}\phi=0$ when $\chi$ is a Killing symmetry
of all the dynamical fields, the first term of (3.2) vanishes. For vacuum
Einstein gravity, without a cosmological constant, the second term of (3.2)
vanishes on shell, so (3.4) reduces to the statement that the Noether charge
of the horizon is equal to that of the sphere at spatial infinity, both
orientations being taken outward (toward larger radius). This yields the Smarr
formula [49],

$\frac{d-3}{d-2}M-\Omega_{\mathcal{H}}J=\frac{\kappa}{8\pi G}A,$ (3.5)

where $M$ is the mass, $J$ is the angular momentum, and $\Omega_{\mathcal{H}}$
is the angular velocity of the horizon, and $A$ is the horizon area.

For a maximally symmetric causal diamond in Einstein gravity, with or without
a cosmological constant, we can instead choose the region ${\cal R}$ to be the
ball $\Sigma$, and choose the vector field $\chi$ to be the conformal Killing
vector $\zeta$ of the diamond. The left hand side of (3.4) is then just a
single integral,888The orientation is chosen to be outward, toward larger
radius, according to Stokes’ theorem. The minus sign is unfamiliar, because in
the black hole case the orientation of the Noether charge integral on the
horizon is typically chosen, as in [26], so as to be towards spatial infinity.

$\oint_{\partial\Sigma}Q_{\zeta}=-\frac{\kappa}{8\pi G}\,A.$ (3.6)

On the other hand, the contribution from the integral of the Noether current
on the right hand side of (3.4) no longer vanishes: since $\zeta$ is not a
Killing vector, the symplectic potential term in the Noether current (3.2) is
nonzero and, if the cosmological constant is nonvanishing, the Lagrangian no
longer vanishes on shell so the second term in the Noether current is also
nonzero. To evaluate the contribution from the first term in the Noether
current, we note that the symplectic potential for Einstein gravity is given
by [50, 46]

$\theta(g,\delta g)=\frac{1}{16\pi
G}\epsilon_{a}(g^{ab}g^{ce}-g^{ae}g^{bc})\nabla_{e}\delta g_{bc}\,,$ (3.7)

where $\epsilon_{a}=\epsilon_{aa_{2}\cdots a_{d}}$ is the volume form with the
first index displayed and the remaining $(d-1)$ indices suppressed. Setting
$\delta g_{bc}={\cal L}_{\zeta}g_{bc}=2\alpha g_{bc}$, evaluating on $\Sigma$,
and using (2.9), we obtain

$\theta(g,{\cal L}_{\zeta}g)|_{\Sigma}=\frac{(d-1)\dot{\alpha}}{8\pi
G}\,u\cdot\epsilon,$ (3.8)

and together with (2.13) this yields

$\int_{\Sigma}\theta(g,{\cal L}_{\zeta}g)=-\frac{d-1}{d-2}\frac{\kappa k}{8\pi
G}\,V,$ (3.9)

where $V=\int_{\Sigma}u\cdot\epsilon$ is the proper volume of the ball. To
evaluate the contribution from the second term in the Noether current, we note
that the off-shell Lagrangian is

$L=\frac{R-2\Lambda}{16\pi G}\,\epsilon\,.$ (3.10)

On shell we have $R-2\Lambda=4\Lambda/(d-2)$, so the on-shell Lagrangian is

$L^{\rm on-shell}=\frac{\Lambda}{(d-2)4\pi G}\,\epsilon,$ (3.11)

and the second term in the integral of the Noether current is thus

$-\int_{\Sigma}\zeta\cdot L=-\frac{\Lambda}{(d-2)4\pi G}\,V_{\zeta},$ (3.12)

where

$V_{\zeta}:=\int_{\Sigma}\zeta\cdot\epsilon\,.$ (3.13)

Since $\zeta$ is orthogonal to $\Sigma$, $V_{\zeta}$ is just the proper volume
of $\Sigma$ weighted locally by the norm of the conformal Killing vector,
given in (1.3). For the case of a black hole in asymptotically Anti-de Sitter
spacetime, a quantity close to $V_{\zeta}$ was first identified in [33] as the
variable thermodynamically conjugate to $\Lambda$. (See subsection 3.1.1 for a
discussion of that case.) For a true Killing vector it is commonly called the
thermodynamic volume [34, 35], and we will use that term also in the conformal
Killing case. For the conformal Killing vector (2.6) in dS space $V_{\zeta}$
is easily found to be given by

$V_{\zeta}=\frac{\kappa
L^{2}}{R}\left(V_{R}^{\text{flat}}-\sqrt{1-(R/L)^{2}}\,\,V_{R}\right)\,.$
(3.14)

where $V_{R}^{\text{flat}}=R^{d-1}\Omega_{d-2}/(d-1)$ is the volume of a
sphere of radius $R$ in Euclidean space and $V_{R}$ is the proper volume of a
ball of radius $R$ in dS space. It follows from the definition (3.13) that
$V_{\zeta}$ is positive in both dS and AdS space, although that is not obvious
from the expression in (3.14). In the flat space limit $R/L\to 0$ it becomes
$V_{\zeta}^{\text{flat}}=\kappa RV_{R}^{\text{flat}}/(d+1)$.

Combining the two terms (3.9) and (3.12), the integral of the Noether current
is thus given by

$\int_{\Sigma}j_{\zeta}=-\frac{{(d-1)}{\kappa k}\,V+2\Lambda
V_{\zeta}}{(d-2)8\pi G}\,,$ (3.15)

so (3.4) yields the Smarr formula,

$(d-2){\kappa}A={{(d-1)}{\kappa k}\,V+2{V_{\zeta}}\Lambda}\,.$ (3.16)

At the cosmological horizon of de Sitter space the extrinsic curvature trace
$k$ vanishes, hence the Smarr formula reduces to a relation between the
horizon area and the cosmological constant. In flat space the cosmological
constant is zero, so that the formula turns into the trivial connection
between the area and the volume. For generic sizes of the causal diamond, and
for a positive cosmological constant, equation (3.16) can be checked
explicitly by using the formulas (2.12) and (3.14) for $k$ and $V_{\zeta}$,
respectively, and the expression for the cosmological constant of dS space:
$\Lambda=+(d-1)(d-2)/2L^{2}$.

####  3.1.1 Smarr formula for AdS black holes

As an aside from the main topic of our paper, in this subsection we discuss
briefly how the Smarr formula for asymptotically AdS black holes [33] can be
derived using (3.4). In that setting $\chi$ is replaced by the horizon
generating Killing field $\xi$, and the domain of integration $\Sigma$ is from
the black hole horizon to infinity. This does not yet yield a meaningful Smarr
formula, since both $\oint_{\infty}Q_{\xi}$ and $\int_{\Sigma}j_{\xi}$
diverge. However, these divergences are the same as those that arise for pure
AdS, so by subtracting the pure AdS Smarr formula from the AdS black hole
Smarr formula, one obtains a finite relation:

$\oint_{\infty}(Q_{\xi}-Q_{\xi}^{\text{AdS}})-\oint_{\mathcal{H}}Q_{\xi}=\int_{\Sigma}j_{\xi}-\int_{\Sigma^{\prime}}j_{\xi}^{\text{AdS}}\,,$
(3.17)

where the Noether charge integrals are both outward oriented. The domain of
integration $\Sigma$ in the black hole integral on the right extends from the
horizon to infinity, while in the pure AdS integral the domain
$\Sigma^{\prime}$ extends across the entire spacetime.

The first integral on the left hand side of (3.17) is proportional to the (AdS
background subtracted) Komar mass and angular momentum, and for Einstein
gravity the horizon integral is proportional to the surface gravity times the
horizon area. Using (3.11) and $\theta(g,\mathcal{L}_{\xi}g)=0$ for Killing
vectors, we find that the Noether current is
$-\Lambda\xi\cdot\epsilon/((d-2)4\pi G)$. Thus the Smarr formula for AdS black
holes is [33]

$\frac{d-3}{d-2}M-\Omega_{\mathcal{H}}J=\frac{\kappa A}{8\pi
G}-\frac{2\bar{V}_{\xi}\Lambda}{(d-2)8\pi G}\,,$ (3.18)

where

$\bar{V}_{\xi}:=\int_{\Sigma}\xi\cdot\epsilon-\int_{\Sigma^{\prime}}\xi^{\text{AdS}}\cdot\epsilon^{\text{AdS}}$
(3.19)

is the background subtracted thermodynamic volume.999 In the literature
$\bar{V}_{\xi}$ is usually denoted by $\Theta$. Moreover, the background
subtracted thermodynamic volume is expressed in [33] in terms of surface
integrals of the Killing potential $(d-2)$-form $\omega_{\xi}$, defined
through $\xi\cdot\epsilon=d\omega_{\xi}$, which can be solved at least locally
for $\omega_{\xi}$ because $\xi\cdot\epsilon$ is closed for Killing vectors.
Thus,
$\bar{V}_{\xi}=\oint_{\infty}(\omega_{\xi}-\omega_{\xi}^{\text{AdS}})-\oint_{\mathcal{H}}\omega_{\xi}$,
where the orientation is outward (toward larger radius) at both $\infty$ and
${\cal H}$. This agrees with the expression (22) in [33], up to a minus sign
in the definition of $\omega_{\xi}$.  The relative sign between the area term
and cosmological constant term is the same as in the Smarr formula for causal
diamonds (3.16). For AdS-Schwarzschild, however, the quantity $\bar{V}_{\xi}$
is negative (it is minus the ‘flat’ volume excluded by the black hole, i.e.
$\bar{V}_{\xi}=-V^{\text{flat}}_{r_{\mathcal{H}}}$), whereas for causal
diamonds $V_{\zeta}:=\int_{\Sigma}|\zeta|dV$ is positive.

###  3.2 First law of causal diamonds

From the Smarr formula one can derive a variational identity analogous to the
first law for black holes using a simple scaling argument (see e.g. [33]). In
fact, Smarr [49] originally derived the relation (3.5) for stationary black
holes from the first law of black hole mechanics by using Euler’s theorem for
homogeneous functions, applied to the black hole mass considered as a function
of the horizon area, angular momentum, and charge. In the context of causal
diamonds, the area is a function of the volume and the cosmological constant
alone, $A(V,\Lambda)$, since $V$ and $\Lambda$ determine a unique diamond up
to isometries. It follows from dimensional analysis that
$\lambda^{d-2}A(V,\Lambda)=A(\lambda^{d-1}V,\lambda^{-2}\Lambda)$, where
$\lambda$ is a dimensionless scaling parameter. For a function with this
property Euler’s theorem implies

$(d-2)A=(d-1)\left(\frac{\partial A}{\partial
V}\right)_{\\!\\!\Lambda}\\!V-2\left(\frac{\partial
A}{\partial\Lambda}\right)_{\\!\\!V}\\!\Lambda\,.$ (3.20)

Comparing this with the Smarr formula (3.16) we find that

$\left(\frac{\partial A}{\partial
V}\right)_{\\!\\!\Lambda}=k\,,\qquad\left(\frac{\partial
A}{\partial\Lambda}\right)_{\\!\\!V}=-\frac{V_{\zeta}}{\kappa}\,,$ (3.21)

which yields the first law for causal diamonds in (A)dS,

$\kappa\,\delta A=\kappa k\,\delta V-V_{\zeta}\,\delta\Lambda\,.$ (3.22)

Notice that an increase of the cosmological constant at fixed volume leads to
a decrease of the area. This is because the spatial curvature is increased
inside the ball. The fact that the coefficient of $\delta V$ is the extrinsic
curvature $k$ can be understood by considering a variation of the radius of a
ball in a fixed, maximally symmetric space. If the proper radius increase is
$\delta\ell$, the volume increase is $\delta V=A\delta\ell$, while the area
increase is $\delta A=kA\delta\ell$, hence $\delta A=k\delta V$.

The first law (3.21) involves only variations of the parameters that
characterize the maximally symmetric causal diamond, and matter fields are not
included in this approach because there are no maximally symmetric solutions
with matter (except the cosmological constant). The first law can be extended
to allow for variations away from maximal symmetry, thereby permitting
variations of the matter stress tensor, as has been done both for black holes
[2] and for de Sitter space [5]. We next derive such an extended first law by
varying the identity (3.4), as was done for vacuum black holes in [26], but
including matter stress-tensor variations as in Refs. [30, 51]. The variations
we consider are arbitrary variations of the dynamical fields $\phi$ to nearby
solutions, while keeping the manifold, the vector field $\zeta^{a}$ and the
surface $\Sigma$ of the unperturbed diamond fixed.

The variation of the Noether current (3.2) is given on shell by

$\delta
j_{\chi}=\omega(\phi,\delta\phi,\mathcal{L}_{\chi}\phi)+d(\chi\cdot\theta(\phi,\delta\phi)),$
(3.23)

where
$\omega(\phi,\delta_{1}\phi,\delta_{2}\phi):=\delta_{1}\theta(\phi,\delta_{2}\phi)-\delta_{2}\theta(\phi,\delta_{1}\phi)$
is the _symplectic current_ $(d-1)$-form. The variation of the integral
identity (3.4) thus yields

$\int_{\Sigma}\omega(\phi,\delta\phi,\mathcal{L}_{\chi}\phi)=\oint_{\partial\Sigma}\left[\delta
Q_{\chi}-\chi\cdot\theta(\phi,\delta\phi)\right].$ (3.24)

This relation holds provided the background equations for all the fields and
the linearized constraint equations associated with the diffeomorphism
generated by $\chi$ are satisfied on the hypersurface $\Sigma$.101010The fact
that (3.24) invokes only the (linearized) initial value constraint equations
(as opposed to linearized dynamical field equations), is explained in the
Appendix of [52] and Appendix B of [17]. The left hand side of (3.24) is the
symplectic form on the (covariant) phase space of solutions which, by
Hamilton’s equations,111111The variation of a Hamiltonian $H$ for a general
dynamical system is related to the symplectic form $\omega$ on phase space and
the flow vector field $X_{H}$ of the background solution via Hamilton’s
equations, $dH(v)=\omega(v,X_{H})$, where $v$ is any tangent vector on phase
space. In the present case $v$ corresponds to $\delta\phi$, $X_{H}$ to
$\mathcal{L}_{\chi}\phi$, and $dH(v)$ is written as $\delta H$ [53].  is equal
to the variation of the Hamiltonian,

$\delta
H_{\chi}=\int_{\Sigma}\omega(\phi,\delta\phi,\mathcal{L}_{\chi}\phi)\,.$
(3.25)

Equation (3.24) thus yields the on-shell identity relating the Hamiltonian
variation to the Noether charge variation and the symplectic potential,

$\delta H_{\chi}=\oint_{\partial\Sigma}\left[\delta
Q_{\chi}-\chi\cdot\theta(\phi,\delta\phi)\right]\,.$ (3.26)

If $\chi$ is a true Killing vector of the background metric and matter fields,
then (3.25) implies $\delta H_{\chi}=0$, so the variational identity reduces
to a relation between the boundary integrals. This is how the first law of
black hole mechanics arises [26, 46]: taking $\Sigma$ to be a hypersurface
bounded by the black hole horizon and spatial infinity, the identity relates
the variation of the horizon Noether charge to the variations of total energy
and angular momentum.

A special case for the first law of causal diamonds is the first law of a
static patch of dS space [5], which in vacuum is just the statement that the
variation of the area of the de Sitter horizon vanishes. The first law derived
by Gibbons and Hawking allowed for variations in the Killing energy of matter,
but matter contributions do not appear in $\delta H_{\chi}$ if the matter is
described by fields that appear quadratically in the Lagrangian and vanish in
the de Sitter background. However, for matter described by a diffeomorphism
invariant fluid theory first order variations of the matter stress tensor can
arise, and because the fields are potentials which do not share the background
Killing symmetry enjoyed by the stress tensor, a volume contribution
containing the matter Killing energy appears in the variational relation [2,
30]. In the derivation of the first law for causal diamonds below we also
allow for variations of fluid matter fields.

We consider the case where the gravitational theory is general relativity, the
matter sector consists of minimally coupled fluids with arbitrary equation of
state, the background metric is pure dS, and the vector field $\chi$ is the
conformal Killing vector $\zeta$ of a causal diamond.121212Many steps in the
derivation below remain valid for other conformally flat solutions. First of
all, causal diamonds in conformally flat spacetimes still allow for a unique
conformal Killing field whose flow preserves the diamond. Moreover, the
diamonds still have a reflection symmetry around $\Sigma$, so that
$\mathcal{L}_{\zeta}g_{ab}=0$ on $\Sigma$. However, $\dot{\alpha}$ might not
be constant in other spacetimes, so all the equations up to (3.34) hold, but
not (3.35), since the relation (2.13) for $\dot{\alpha}$ might be specific to
maximally symmetric spacetimes. One of the fluids describes the cosmological
constant, with equation of state $p=-\rho=-\Lambda/(8\pi G)$. Since $\zeta$ is
zero at the edge $\partial\Sigma$ of the diamond, the second term on the right
hand side in (3.26) vanishes. The surface integral of $\delta Q_{\zeta}$ in
this case is

$\oint_{\partial\Sigma}\delta Q_{\zeta}=-\frac{\kappa}{8\pi G}\delta A\,,$
(3.27)

where $\kappa$ is the surface gravity and $A$ is the area of the bifurcation
surface $\partial\Sigma$.131313The minus sign appears for the same reason as
in (3.6), which is explained in footnote 8. With this result, the identity
(3.26) takes the form

$\delta H_{\zeta}=-\frac{\kappa}{8\pi G}\delta A\,.$ (3.28)

For the present field content the variation of the total Hamiltonian splits
into a (nonvanishing) term associated to the background metric and one
associated to the matter fields

$\delta H_{\zeta}=\delta H_{\zeta}^{\text{g}}+\delta H_{\zeta}^{\text{m}}\,.$
(3.29)

In the following we will first evaluate the gravitational term and then the
matter term. The result will be that the gravitational term is proportional to
minus the variation of the volume of $\Sigma$, the matter term contains a term
proportional to the thermodynamic volume times the variation of the
cosmological constant as well as the variation of the canonical Killing energy
for the other fluids.

We evaluate $\delta H^{\text{g}}$ through its relation to the symplectic form
(3.25). For general relativity the symplectic current takes the form [54, 50]

$\omega(g,\delta_{1}g,\delta_{2}g)=\frac{1}{16\pi
G}\epsilon_{a}P^{abcdef}\left(\delta_{2}g_{bc}\nabla_{d}\delta_{1}g_{ef}-\delta_{1}g_{bc}\nabla_{d}\delta_{2}g_{ef}\right),$
(3.30)

with

$P^{abcdef}=g^{ae}g^{bf}g^{cd}-\frac{1}{2}g^{ad}g^{be}g^{cf}-\frac{1}{2}g^{ab}g^{cd}g^{ef}-\frac{1}{2}g^{bc}g^{ae}g^{fd}+\frac{1}{2}g^{bc}g^{ad}g^{ef}.$
(3.31)

Note that in (3.25) the symplectic current is evaluated on the Lie derivative
of the fields along $\zeta$. If $\zeta$ were a Killing vector, the metric
contribution $\delta H_{\zeta}^{\text{g}}$ would hence vanish, as it does when
deriving the first law of black hole mechanics [26]. However, since for a
diamond $\zeta$ is only a conformal Killing vector, $\delta
H_{\zeta}^{\text{g}}$ makes a nonzero contribution to the first law. When
$\delta_{1}g=\delta g$ and $\delta_{2}g=\mathcal{L}_{\zeta}g$, the first term
in (3.30) is zero at $\Sigma$, since $\mathcal{L}_{\zeta}g|_{\Sigma}=0$ (2.8).
Using (2.9) the second term yields

$\omega(g,\delta
g,\mathcal{L}_{\zeta}g)\big{|}_{\Sigma}=-\frac{(d-2)\dot{\alpha}}{16\pi
G}\epsilon_{a}\\!\left(h^{ab}u^{c}-h^{bc}u^{a}\right)\delta g_{bc}\,,$ (3.32)

where $h_{ab}:=g_{ab}+u_{a}u_{b}$ is the induced metric on $\Sigma$ and
$u_{a}$ is the unit normal to $\Sigma$. Only the pullback of $\omega$ to
$\Sigma$ is relevant in the integral in (3.25). Using

$\epsilon_{a}\big{|}_{\Sigma}=-u_{a}(u\cdot\epsilon),$ (3.33)

this pullback can be simplified as

$\omega(g,\delta
g,\mathcal{L}_{\zeta}g)\big{|}_{\Sigma}=\frac{(d-2)\dot{\alpha}}{16\pi
G}(u\cdot\epsilon)\,h^{bc}\delta h_{bc}=\frac{(d-2)\dot{\alpha}}{8\pi
G}\delta(u\cdot\epsilon)\,,$ (3.34)

where pullback of all forms to $\Sigma$ is implicit. The metric contribution
to $\delta H_{\zeta}$ is therefore equal to

$\delta H_{\zeta}^{\text{g}}=\frac{d-2}{8\pi
G}\int_{\Sigma}\dot{\alpha}\,\delta(u\cdot\epsilon)=-\frac{\kappa k}{8\pi
G}\delta V\,,$ (3.35)

where $V=\int_{\Sigma}u\cdot\epsilon$ is the proper volume of $\Sigma$, and in
the last equality we used (2.13) and the fact that $\dot{\alpha}$ is constant
over $\Sigma$. The constancy of $\dot{\alpha}$ is hence crucial for arriving
at an intrinsic geometric quantity, the variation of the proper volume.

Combining (3.28), (3.29) and (3.35), we find the extended first law for causal
diamonds, which includes a variation of the matter Hamiltonian,

$\delta H^{\text{m}}_{\zeta}=\frac{\kappa}{8\pi G}\left(-\delta A+k\delta
V\right)\,.$ (3.36)

Next, we compute the matter Hamiltonian variation explicitly through its
relation with the symplectic form.

The precise on-shell relation between the symplectic current
$\omega^{\text{m}}$ and the Noether current $j^{\text{m}}_{\chi}$ for matter
fields is [30]

$\omega^{\text{m}}(\phi,\delta\phi,\mathcal{L}_{\chi}\phi)=\delta
j^{\text{m}}_{\chi}+\frac{1}{2}\chi\cdot\epsilon\,T^{ab}\delta
g_{ab}-d(\chi\cdot\theta^{\text{m}}(\phi,\delta\phi))\,.$ (3.37)

Here, $T^{ab}$ is the Hilbert stress-energy tensor defined through the matter
Lagrangian.141414In particular, the variation of the matter Lagrangian
$d$-form with respect to the matter fields $\psi$ and the metric $g_{ab}$ is
given by: $\delta
L^{\text{m}}=E^{\text{m}}\delta\psi+\epsilon\frac{1}{2}T^{ab}\delta
g_{ab}+d\theta^{\text{m}}(\phi,\delta\phi)$, where $E^{\text{m}}=0$ are the
matter equations of motion, $T^{ab}$ is the stress-energy tensor, and
$\theta^{\text{m}}$ is the symplectic potential associated to $L^{\text{m}}$
[30]. Compared to the equivalent identity (3.23) for all the dynamical fields,
we see that in the identity above for the matter sector the term involving the
stress-energy tensor is new. This term arises from the metric variation of the
matter Lagrangian. A similar identity exists for the pure metric sector, with
the extra term being $-(1/16\pi G)\chi\cdot\epsilon\,G^{ab}\delta g_{ab}$, so
that (3.23) holds when the pure metric and matter sectors are combined and the
metric equation of motion is imposed.

Further, the Noether current for matter fields is on shell given by [30]

$j_{\chi}^{\text{m}}=dQ_{\chi}^{\text{m}}-T_{a}{}^{b}\chi^{a}\epsilon_{b}\,.$
(3.38)

The stress-energy term appears on the right hand side because only the full
Noether current $j_{\chi}=j^{\text{g}}_{\chi}+j^{\text{m}}_{\chi}$ is an exact
form on shell. Inserting (3.38) into the variational identity (3.37) and using
Hamilton’s equations (3.25), we find that the matter Hamiltonian variation is

$\delta H^{\text{m}}_{\chi}=\oint_{\partial\Sigma}\left[\delta
Q_{\chi}^{\text{m}}-\chi\cdot\theta^{\text{m}}(\phi,\delta\phi)\right]+\int_{\Sigma}\left[-\delta(T_{a}{}^{b}\chi^{a}\epsilon_{b})+\frac{1}{2}\chi\cdot\epsilon\,T^{ab}\delta
g_{ab}\right].$ (3.39)

This equality is true for an arbitrary smooth vector field $\chi$ on
spacetime, and holds provided the field equations and the linearized equations
of motion are satisfied for the matter fields, i.e. $E^{\text{m}}=\delta
E^{\text{m}}=0$.

We now specialize to the conformal Killing vector $\zeta$ that preserves a
diamond in (A)dS (the analysis below is actually valid in any conformally flat
spacetime). Since $\zeta=0$ at $\partial\Sigma$, the second term in the
boundary integral in (3.39) vanishes. In addition, the Noether charge
variation also does not contribute at the bifurcation surface
$\partial\Sigma$. This is because for a generic diffeomorphism invariant
Lagrangian the Noether charge ($d-2$)-form can be expressed as
$Q_{\zeta}=W_{c}(\phi)\zeta^{c}+X^{cd}(\phi)\nabla_{[c}\zeta_{d]}$
[46].151515Here we have fixed the ambiguity in the definition of the Noether
charge, coming from the freedom to shift the symplectic potential $\theta$ by
an exact form $dY(\phi,\delta\phi)$, such that
$Y(\phi,\mathcal{L}_{\zeta}\phi)=0$. If one were to allow for a nonzero $Y$
form, then the first law would not be modified, since the Noether charge
variation (together with symplectic potential) associated to matter fields in
(3.39) cancels anyway in the variational identity (3.26) against an identical
term on right hand side. The cancellation of $\oint_{\partial\Sigma}[\delta
Q^{\text{m}}_{\chi}-\chi\cdot\theta^{\text{m}}(\phi,\delta\phi)]$ in the first
law was pointed out by Iyer in [30]. The first term vanishes at
$\partial\Sigma$, and the second term involves a form $X^{cd}$, which is
purely constructed from derivatives of the Lagrangian with respect to the
Riemann tensor (and its covariant derivatives). For minimally coupled matter
fields, this form does not receive contributions from the matter sector, so
$Q^{\text{m}}_{\zeta}=0$ at $\partial\Sigma$ for the present field content
(and also $\delta Q^{\text{m}}_{\zeta}$ vanishes at $\partial\Sigma$).

The matter Hamiltonian variation on the maximal slice $\Sigma$ in a diamond is
therefore given by

$\delta
H^{\text{m}}_{\zeta}=\int_{\Sigma}\left[-\delta(T_{a}{}^{b}\zeta^{a}\epsilon_{b})+\frac{1}{2}\zeta\cdot\epsilon\,T^{ab}\delta
g_{ab}\right]\,,$ (3.40)

which can be rewritten, using
$\delta\epsilon_{b}=\frac{1}{2}\epsilon_{b}g^{cd}\delta g_{cd}$, as a sum of
stress tensor and metric variation terms,

$\delta
H^{\text{m}}_{\zeta}=\int_{\Sigma}\left[-\delta{T_{a}}^{b}+\frac{1}{2}({\delta_{a}}^{b}T^{cd}-{T_{a}}^{b}g^{cd})\delta
g_{cd}\right]\\!\zeta^{a}\epsilon_{b}\,.$ (3.41)

Notice that the trace part drops out of the second term.161616We should have
been able to anticipate this feature, but have not yet found a way to do so.
In a maximally symmetric background the tracefree part of the stress tensor
must vanish, hence the matter Hamiltonian variation takes the form

$\delta
H^{\text{m}}_{\zeta}=-\int_{\Sigma}\delta{T_{a}}^{b}\zeta^{a}\epsilon_{b}\,,$
(3.42)

which receives contributions from all types of matter.171717Using (3.33), the
integrand becomes $\delta{T_{a}}^{b}\zeta^{a}u_{b}\,u\cdot\epsilon$, and the
unfamiliar minus sign in (3.42) disappears.

Since the cosmological constant can be obtained from a field or fields
covariantly coupled to the metric, its variation falls within the class of
“matter” variations to which the first law (3.36) applies, and so may be
included in (3.42). To separate out this contribution, we split the matter
stress tensor as

$T_{ab}=T_{ab}^{\text{\~{m}}}+T_{ab}^{\Lambda}\,,$ (3.43)

where $T_{ab}^{\text{\~{m}}}$ is the stress-energy tensor of matter other than
the cosmological constant, and $T^{\Lambda}_{ab}=-(\Lambda/8\pi G)g_{ab}$ is
the “vacuum” energy-momentum tensor corresponding to the cosmological
constant. The contribution of the $\Lambda$ term to the variation of the
Hamiltonian is thus

$\delta H^{\Lambda}_{\zeta}=\frac{V_{\zeta}\,\delta\Lambda}{8\pi G}\,,$ (3.44)

where $V_{\zeta}$ is the thermodynamic volume defined in (3.13). Note that
$V_{\zeta}$ is not varied, since the metric variation was already separated
out in (3.41).

In conclusion, by inserting $\delta H^{\text{m}}_{\zeta}=\delta
H^{\text{\~{m}}}_{\zeta}+\delta H^{\Lambda}_{\zeta}$ and (3.44) into (3.36),
we arrive at the final form of the first law of causal diamonds,

$\delta H^{\text{\~{m}}}_{\zeta}=\frac{1}{8\pi G}\left(-\kappa\,\delta
A+\kappa k\,\delta V-V_{\zeta}\,\delta\Lambda\right).$ (3.45)

We remind the reader of what all these symbols represent:
$H^{\text{\~{m}}}_{\zeta}$ is the conformal Killing energy of matter other
than the cosmological constant $\Lambda$, $\kappa$ is the surface gravity, $A$
is the area of the edge $\partial\Sigma$, $k$ is the trace of the (outward)
extrinsic curvature of $\partial\Sigma$ as embedded in the maximal slice
$\Sigma$, $V$ is the proper volume of the maximal slice, and $V_{\zeta}$ is
the proper volume weighted locally by the norm of the conformal Killing vector
$\zeta$.

The derivation above also goes through for causal diamonds in AdS. Note that
the form of the first law is the same for (A)dS as for Minkowski space, except
that all the quantities should now be evaluated in (A)dS. Hence, we have
established a variational identity in general relativity which holds for
spherical regions of any size in maximally symmetric spacetimes.

###  3.3 Further remarks on the first law

Below we collect several comments on aspects of the first law of causal
diamonds.

####  3.3.1 Role of maximal volume

When we evaluated the variation of the gravitational Hamiltonian $\delta
H_{\zeta}^{\text{g}}$ in (3.35), we chose to carry out the integral over the
maximal slice of the unperturbed diamond. Because the symplectic current is
conserved, the value would have been the same had we chosen any other slice
bounded by $\partial\Sigma$, although it would not have been given in the same
way by the volume variation. The slice $\Sigma$ therefore has a somewhat
preferred status. Furthermore, although we described this as the variation of
the volume of “the slice that was the maximal slice in the unperturbed
diamond,” we could just as well describe it as the variation of the volume of
the maximal slice itself. This is because the volume change due to the
variation of the location of the maximal slice itself vanishes, precisely
because that slice is maximal to begin with. This is satisfying, since it
allows the first law to be stated in a manifestly “gauge-invariant” fashion —
i.e. independently of how the spacetime interior of the varied diamond is
identified with that of the original diamond — and the second and higher order
variations of the maximal slice are unambiguously defined.

####  3.3.2 Fixed volume and fixed area variations

For variations that fix the volume, (3.36) becomes a relation between the area
variation at fixed volume and the variation of the matter Hamiltonian,

$\frac{\kappa}{8\pi G}\delta A\big{|}_{V}=-\delta H_{\zeta}^{\text{m}}.$
(3.46)

That is, the presence of positive conformal Killing energy matter produces an
area deficit at fixed volume. Similarly, for variations in which the area is
fixed we obtain the relation

$\frac{\kappa k}{8\pi G}\delta V\big{|}_{A}=\delta H_{\zeta}^{\text{m}}.$
(3.47)

Hence, the presence of positive conformal Killing energy matter produces a
volume excess at fixed area.

Importantly, the combination of variations $\delta A-k\delta V$ that appears
in the first law (3.36) is _equivalent_ to the area variation at fixed volume,
i.e.

$\delta A-k\delta V\sim\delta A\big{|}_{V},$ (3.48)

where the equivalence means modulo diffeo-induced variations. This is because
(i) it is always possible to compose any variation with a variation
$\delta_{\xi}$ induced by a diffeomorphism $\xi$, such that the volume $V$ is
unchanged under the complete variation; and (ii) for _all_ diffeo-induced
variations the combination $\delta_{\xi}A-k\delta_{\xi}V$ vanishes (as shown
below). The combination of variations $\delta A-k\delta V$ is thus equal to
the area change that would remain if one were to compose a generic variation
with a diffeo-variation restoring the volume to its original value.

Further, the matter Hamiltonian variation $\delta H_{\zeta}^{\text{m}}=\delta
H_{\zeta}^{\text{\~{m}}}+\delta H_{\zeta}^{\Lambda}$ in the first law (3.36)
is also unaffected by composing the variation with a diffeo-induced variation:
since $\Lambda$ is constant in the background,
$\delta_{\xi}\Lambda:=\mathcal{L}_{\xi}\Lambda=0$, and since matter (other
than the $\Lambda$ contribution) can be present only after the field variation
away from maximal symmetry, $\hat{\delta}H_{\zeta}^{\text{\~{m}}}$ is non-
vanishing only at the next variational order. Thus, both sides of the first
law vanish for variations that are induced by diffeomorphisms. Hence, we are
free to add to the first law (3.36) a diffeo-induced variation that restores
the volume $V$ to its original value, so that the first law takes the form
(3.46). Similarly, one can also freely add a diffeo-induced variation that
restores the area $A$ to its original value, such that to the first law
becomes (3.47). This means that the first law at fixed volume (3.46) and the
first law at fixed area (3.47) are equivalent to the one (3.36) without
holding the volume and area fixed.

Finally, we prove statement (ii) above. That is, if $\mathcal{R}$ is a
codimension-one submanifold with vanishing mean extrinsic curvature, and if
the boundary $\partial\mathcal{R}$ has mean extrinsic curvature (normal to
$\partial\mathcal{R}$) that vanishes in the direction normal to $\mathcal{R}$
and is constant in the direction tangent to $\mathcal{R}$, then under an
infinitesimal diffeo-variation the variations of the area $A$ of
$\partial\mathcal{R}$ and the volume $V$ of $\mathcal{R}$ are related
by181818If the assumptions are relaxed, then the diffeo-variation of the
volume is
$\delta_{\xi}V=\int_{\mathcal{R}}K\sigma_{a}u^{a}u\cdot\epsilon+\int_{\partial\mathcal{R}}\\!\\!\left(\frac{1}{k_{n}}\delta_{\tau}\mu+\frac{k_{u}}{k_{n}}\sigma_{a}u^{a}\mu\right)\,.$
Here, $\sigma$ and $\tau$ are the components of $\xi$ normal and tangent to
$\mathcal{R}$, respectively, $\mu$ is the volume form on the codimension-two
surface $\partial\mathcal{R}$, and $k_{u}$ and $k_{n}$ are the mean curvatures
in the two normal directions to $\partial R$. Under the assumptions
$K=0=k_{u}$ and $k_{n}$ is constant, we recover (3.49) with $k:=k_{n}$. A
similar expression for the volume variation is given by equation (4.32) in
[55]. We thank Antony Speranza for clarifying the required assumptions for the
validity of (3.49).

$\delta_{\xi}A-k\,\delta_{\xi}V=0\,,$ (3.49)

where $k$ is the mean curvature of $\partial\mathcal{R}$ in $\mathcal{R}$. To
see this, consider the diffeomorphism as an active transformation of the
spacetime points. Under the stated assumptions, the normal component of the
diffeomorphism will do nothing to $V$ and $A$, while the piece tangential to
$\mathcal{R}$ will deform each surface element $dA$ by
$\delta_{\xi}dA=k\,dA\,\delta_{\xi}s=k\,\delta_{\xi}dV$, where $\delta_{\xi}s$
is the normal deformation distance. If $k$ is constant on
$\partial\mathcal{R}$, then when integrated over $\partial\mathcal{R}$ this
yields $\delta_{\xi}A=k\delta_{\xi}V$. The maximal slice $\Sigma$ of a
maximally symmetric diamond possesses the assumed properties, so the result
applies in particular to that surface.

####  3.3.3 Varying the cosmological constant

In a thermodynamic interpretation, causal diamonds in maximally symmetric
spacetimes are all “equilibrium states” from which variations can be made. The
diamonds differ only in size, and in the cosmological constant of the
background. It is natural to allow also $\Lambda$ to vary, since it is
evidently an equilibrium state variable, and there are circumstances under
which it might vary. For instance, there may be mechanisms by which it can
decay. Also, in the context of the AdS/CFT correspondence, the negative
cosmological constant is controlled by the number of stacked D-branes, which
could in principle change [33]. Another reason to consider variable $\Lambda$
arises in formulating the principle of vacuum entanglement equilbrium for non-
conformal matter fields, see Sec. 4.2.2 and Ref. [21]. Consistency with the
Bianchi identity made it necessary to allow for an initially undetermined
local cosmological constant in small causal diamonds, which ended up being
related to the part of the entanglement entropy variation not captured by the
energy-momentum tensor. It is thus of interest to include variations of
$\Lambda$ in the first law. Ref. [56] provides an extensive review of black
hole thermodynamics extended to include variable $\Lambda$, a.k.a. “black hole
chemistry”.

There are many ways to accommodate a cosmological constant variation in the
first law. In the literature this has been done for the first law for black
holes and for holographic entanglement entropy by employing various methods,
see e.g. [33, 57, 58, 59, 60, 56]. In Sec. 3.2 we treated the cosmological
constant as a perfect fluid, and made use of Iyer’s generalized derivation of
the first law to allow for matter fields which are non-stationary yet have a
stationary stress-energy tensor [30]. In this approach the cosmological
constant term in the first law comes from the variation of the stress-energy
tensor of the fluid. Yet another way of introducing a cosmological constant is
to promote it to a dynamical scalar field, and to add it to the Lagrangian
together with a $(d-1)$-form field $B$ as: $\Lambda(dB-\epsilon)/(8\pi G)$
[61]. The $B$ field equation implies that $\Lambda$ is constant, while the
$\Lambda$ field equation implies $dB=\epsilon$. The addition to the symplectic
potential due to this Lagrangian is $\theta(\phi,\delta\phi)=\Lambda\delta
B/(8\pi G)$, where $\phi=(\Lambda,B)$. Moreover, the additional term in the
symplectic current is given on shell by
$\omega(\phi,\delta\phi,\mathcal{L}_{\zeta}\phi)=[\delta\Lambda\,\zeta\cdot\epsilon+d(\delta\Lambda\,\zeta\cdot
B)]/(8\pi G)$. When integrated over $\Sigma$ this gives precisely
$V_{\zeta}\delta\Lambda/(8\pi G)$, as in (3.44), since $\zeta$ vanishes at the
edge $\partial\Sigma$.

####  3.3.4 Gravitational field Hamiltonian and York time

Figure 2: The $(x,s)$ coordinate chart of a maximally symmetric causal
diamond. The coordinate $s\in(-\infty,\infty)$ is the conformal Killing time,
defined as the function that vanishes on the maximal slice $\Sigma$ and
satisfies $\zeta\cdot ds=1$. The coordinate $x\in[0,\infty)$ is spherically
symmetric and satisfies $\zeta\cdot dx=0$ and $|dx|=|ds|$. Constant $s$ and
$x$ lines are plotted at equal coordinate intervals of $0.5$. See Appendix B
for a demonstration that $ds$ and $dx$ are everywhere orthogonal, and for the
line element in these coordinates.

We have seen that the gravitational contribution to the variation of the
Hamiltonian $H_{\zeta}$ generating evolution along the conformal Killing flow
of the background maximally symmetric diamond is proportional to the volume
variation. This “volume as Hamiltonian” is reminiscent of a “York time”
Hamiltonian for general relativity [36], which generates evolution along a
foliation by spacelike hypersurfaces with constant mean curvature $K$ (i.e.
along a “CMC” foliation), using $K$ as the time parameter, and with an
arbitrary shift vector field. (Mean curvature can be defined as
$K:=\nabla_{a}u^{a}$, where $u^{a}$ is the future pointing unit normal to a
spacelike hypersurface.) Such a Hamiltonian is proportional to the spatial
volume of the CMC slices.

The similarity is not accidental. It arises from the fact that (i) the
conformal Killing vector $\zeta^{a}$ is orthogonal to $\Sigma$, which is a CMC
surface, and (ii) $\zeta^{a}\nabla_{a}K$ is constant on $\Sigma$. Actually,
these two properties hold on all leaves of the CMC foliation: as shown in
Appendix B, surfaces of constant conformal Killing parameter $s$ — defined by
$\zeta^{a}\nabla_{a}s=1$ with the initial condition $s=0$ on $\Sigma$ —
coincide with surfaces of constant $K$ everywhere in the diamond, and
$\zeta^{a}$ is everywhere orthogonal to these surfaces (see Fig. 2 for an
illustration). More specifically, $K$ and $s$ are related by

$K=(d-1)\dot{\alpha}|_{s=0}\sinh s\,,$ (3.50)

where $\alpha=\nabla_{a}\zeta^{a}/d$ and $\dot{\alpha}=u^{a}\nabla_{a}\alpha$.
In particular, $K$ vanishes at the extremal surface $s=0$, and its first
derivative with respect to $s$ at $s=0$ is given by

$\frac{dK}{ds}\Big{|}_{s=0}=(d-1)\dot{\alpha}|_{s=0}=-\frac{d-1}{d-2}\kappa
k\,,$ (3.51)

where (2.13) is used in the last equality.191919Note that $K$ decreases as $s$
increases, and is hence negative to the future of the slice $\Sigma$ (and
positive to the past of $\Sigma$). Equation (3.51) establishes that York time
and conformal Killing time are proportional, to first order about the maximal
slice, for a maximally symmetric diamond. This indicates, as we will now
argue, that the variation $\delta H_{\zeta}^{\rm g}$ agrees, up to the
constant (3.51), with the York time Hamiltonian variation $\delta H_{\rm Y}$.

In the context of the first law, the perturbed spacetime is not the maximally
symmetric one. When the metric is varied, the definition of York time varies,
so the surface on which we should be computing the volume varies, as does the
rate of time flow. Nevertheless, since the $t=0$ surface has vanishing $K$,
the volume variation induced by varying the surface vanishes. Also, the field
variation is already first order, so the change of flow rate of $K$ makes a
higher order contribution to $\delta H_{\rm Y}$. It follows that

$\delta H_{\zeta}^{\rm g}=\frac{dK}{ds}\Big{|}_{s=0}\delta H_{\rm
Y}\,,\qquad\text{with}\qquad\delta H_{\rm Y}=\frac{d-2}{d-1}\frac{\delta
V}{8\pi G}\,.$ (3.52)

Therefore, the gravitational Hamiltonian variation (3.35) is indeed equal to a
constant times the York time Hamiltonian variation. The York time Hamiltonian
variation would be equal to the negative of the proper volume variation, i.e.
$\delta H_{\rm Y}=-\delta V$, if one used the time variable
$t_{\rm{Y}}=-\frac{d-2}{d-1}\frac{K}{8\pi G}$ instead of $K$. This is
precisely the time variable that York originally introduced for general
relativity in $d=4$ [36]. In the literature the minus sign in the time
variable is often omitted, in which case the Hamiltonian is equal to the
volume (see e.g. [62]).

##  4 Thermodynamics of causal diamonds in (A)dS

As is well known black holes admit a true thermodynamic interpretation. In
this section we will explore to what extent the same is true for causal
diamonds in (A)dS. We will also relate the first law of causal diamonds to the
entanglement equilibrium proposal in [21].

###  4.1 Negative temperature

Like the first law of black hole mechanics, and its generalizations mentioned
in the introduction, the first law of causal diamonds (3.28),

$\delta H_{\zeta}=-\frac{\kappa}{8\pi G}\delta A,$ (4.1)

admits a thermodynamic interpretation. What is unusual, however, is the minus
sign on the right hand side. The $\kappa\,\delta A$ term is usually identified
with a $T_{\text{H}}\delta S_{\text{BH}}$ term, where $S_{\text{BH}}=A/4\hbar
G$ is the Bekenstein-Hawking entropy and $T_{\text{H}}=\kappa\hbar/2\pi$ is
the Hawking temperature. However, in the present context this identification
calls for a negative temperature202020The conceptual possibility of negative
absolute temperature was discussed for the first time by Afanassjewa in 1925
[63]. In 1951 Purcell and Pound [64] prepared and measured a nuclear spin
system at negative temperature in an external magnetic field. Subsequently,
the thermodynamic and statistical mechanical implications of negative
temperature were studied in detail by Ramsey [65]. We thank Jos Uffink for
bringing the work of Afanassjewa to our attention [66]. See [67] for a recent
review of negative temperature.

$T=-T_{\rm H},$ (4.2)

because an increase of conformal Killing energy in the diamond is associated
with a decrease of horizon entropy.212121It was suggested in Ref. [68] that in
global de Sitter space this minus sign (which was there attached to the
entropy rather than to the temperature) results from imposing the first law
with the energy inside the horizon, rather than the energy outside the horizon
which is the negative of the former. The sensibility of this proposal is
debatable, since the opposite sign of the energy results from the fact that
the Killing vector is past-oriented outside the horizon. Moreover, in flat
spacetime or AdS the energy outside the horizon is not the negative of that
inside, and yet we still encounter the same minus sign. This negative
temperature interpretation has previously been suggested by Klemm and Vanzo
[31] in the special case of the static patch of de Sitter space, and was
recently advocated in the context of multiple Killing horizons in [32], where
the cosmological event horizon was also assigned a negative (Gibbsian)
temperature.222222The arguments in [32] appealed to the fact that the surface
gravity of the cosmological horizon is negative. Although not stated in [32]
(nor elsewhere in the literature that we are aware of), this holds only on the
past cosmological horizon (if we take the Killing vector to be future pointing
on the past and future horizons). The surface gravity of the future
cosmological horizon is positive. (The surface gravity is positive (negative)
if the Killing vector is stretched (shrunk) with respect to affine parameter
along the Killing flow on the Killing horizon.) A varied diamond can be viewed
as the result of a physical process in which a perturbation has passed through
the past horizon, and entered what would otherwise have been a maximally
symmetric diamond. The surface gravity of the past horizon should thus be
expected to play the role of temperature in the first law. Negative
temperature typically requires of a system that i) its energy spectrum is
bounded above and ii) the Hilbert space is finite-dimensional. Klemm and Vanzo
have argued that these requirements are indeed satisfied for the de Sitter
space static patch.232323The proposal that the Hilbert space of an observer’s
patch in asymptotic de Sitter space is finite-dimensional is due to Banks and
Fischler [69, 70]. Their arguments can actually be applied to all causal
diamonds: i) the mass inside is bounded above by the mass of the largest black
hole that fits inside a diamond with a given boundary area and ii) the entropy
associated to the horizon is finite due to the holographic principle or
covariant entropy bound [71, 72]. It therefore seems feasible that causal
diamonds have a negative temperature in quantum gravity.

Using the negative temperature (4.2), the first law (4.1) can be written as

$\delta H_{\zeta}=T\delta S_{\text{BH}},$ (4.3)

which is a standard thermodynamic relation between energy, temperature and
entropy. As a special case, we find that the static patch of de Sitter space
has a negative temperature (see Sec. 5.1). This is in apparent conflict with
the positive Gibbons-Hawking temperature for dS space, computed using quantum
field theory on a fixed background [5]. In Sec. 4.2 below we shall propose a
resolution to this apparent conflict involving the quantum corrections to the
first law, but first we want to further discuss the thermodynamic
interpretation of the leading order classical quantities in this relation.

Instead of writing the cosmological constant term (3.44) in $\delta H_{\zeta}$
as an energy variation, we can also take it to the right hand side of the
first law and write it as the thermodynamic volume times the variation of the
pressure, i.e. $V_{\zeta}\,\delta p$. This is because the cosmological
constant can be interpreted both as an energy density, $\rho=\Lambda/8\pi G$,
and as a pressure $p=-\Lambda/8\pi G$. In this way (4.3) can be expressed as

$\delta H^{\rm g+\rm\tilde{m}}_{\zeta}=T\delta S_{\text{BH}}+V_{\zeta}\,\delta
p\,,$ (4.4)

where $\rm g$ labels the gravitational contribution (1.4) to the Hamiltonian
variation, and $\rm\tilde{m}$ labels the matter contribution (1.2) other than
the cosmological constant. This form of the first law suggests that $H^{\rm
g+\rm\tilde{m}}_{\zeta}$ is an enthalpy, rather than an energy, just like the
ADM mass for black holes [33]. The matter Hamiltonian vanishes in the
background, so the enthalpy of causal diamonds in Minkowski and (A)dS space is
$H_{\zeta}^{\rm g}$, which is defined above only through its variation. We
leave it to future work to evaluate $H_{\zeta}^{\rm g}$ itself.

Through a Legendre transformation, $U=H^{\rm
g+\rm\tilde{m}}_{\zeta}-p\,V_{\zeta}$, the first law can be written in the
standard form

$\delta U=T\delta S_{\rm BH}-p\,\delta V_{\zeta}\,,$ (4.5)

where $U$ plays the role of the internal energy associated to causal diamonds.
Using the equation of state $p=-\rho$, the contribution of the $\Lambda$ term
to $U$ may be expressed as $\rho\,V_{\zeta}$, which is the redshifted vacuum
energy associated to the cosmological constant. (See Sec. 5.1 for a similar
discussion for the special case of the de Sitter static patch.)

###  4.2 Quantum corrections

The first law can be extended into the semiclassical regime by considering
quantum matter fields (instead of classical fields) on a classical background
spacetime. The “quantum corrected” first law of causal diamonds reads

$\delta\langle H^{\rm\tilde{m}}_{\zeta}\rangle+\delta H_{\zeta}^{\rm
g+\Lambda}=T\delta S_{\rm BH}\,,$ (4.6)

which can be derived (along the lines of Sec. 3.2) from the semiclassical
Einstein equation, where the stress-energy tensor is replaced by its quantum
expectation value but the metric is kept classical. Our aim in this section is
to show how this first law can be written in terms of the variation of
Bekenstein’s generalized entropy [1] — defined in (4.11) — which at the same
time explains why the negative temperature $T$ is consistent with the positive
Gibbons-Hawking temperature $T_{\rm H}$. We will first restrict the discussion
to conformal matter and then generalize it to any quantum matter.

####  4.2.1 Conformal matter

In the particular case of the vacuum state of a conformal quantum field
theory, the matter Hamiltonian $H^{\rm\tilde{m}}_{\zeta}$ associated to a
spherical region $\Sigma$ in flat space or (A)dS is equal to the so-called
modular Hamiltonian $K$, i.e.

$H^{\rm\tilde{m}}_{\zeta}:=\int_{\Sigma}({T_{a}}^{b})^{\rm\tilde{m}}\zeta^{a}u_{b}dV=K\,,$
(4.7)

where $K$ is the operator implicitly defined via the reduced density matrix of
the vacuum restricted to the region $\Sigma$, $\rho_{\rm
vac}=e^{-K/T_{H}}/Z$.242424It is well known that the reduced density matrix of
the vacuum in Rindler space is thermal with respect to the Lorentz boost
Hamiltonian [23]. The thermal behavior of conformal quantum fields in a global
vacuum state inside maximally symmetric causal diamonds can be derived from
the Weyl equivalence between these diamonds and Rindler space. In Appendix B
we show explicitly that all maximally symmetric diamonds are Weyl equivalent
to conformal Killing time cross hyperbolic space,
$\mathbb{R}\times\mathbb{H}^{d-1}$, and therefore to each other. It is also
known that a diamond in flat space is Weyl equivalent to Rindler space, see
Appendix E. Therefore, the reduced density matrix of the conformal vacuum on a
diamond in (A)dS and flat space is thermal. For the special case of diamonds
in flat space this was discussed before in [22, 23, 24, 25, 12]. For
infinitesimal variations of the reduced density matrix, the variation of the
expectation value of the modular Hamiltonian is equal to the variation of the
matter entropy $S_{\rm\tilde{m}}$, by

$\delta\langle K\rangle=T_{\rm H}\delta S_{\rm\tilde{m}},$ (4.8)

with the positive sign for the temperature $T_{\rm H}$. If the variation is to
a global pure state $\delta S_{\rm\tilde{m}}$ is purely entanglement entropy,
which is why this is known as the “first law of entanglement”.

Initially it appears that this opposite sign for the temperature indicates an
inconsistency: if $\delta\langle H^{\rm\tilde{m}}_{\zeta}\rangle$ were added
to the rest of the conformal energy variation $\delta H_{\zeta}$, and at the
same time $\delta S_{\rm\tilde{m}}$ were added to $\delta S_{\rm BH}$ in the
classical first law (4.3), then — because of the mismatch in the signs of the
temperatures — the first law would no longer be valid for any temperature.
However, in gravitational thermodynamics, it would be incorrect to add both
the energy term and the entropy term. The derivation of the gravitational
first law follows from diffeomorphism invariance and the gravitational field
equation, and the matter entropy does not enter in the derivation [26, 30]. On
the other hand, this first law must be consistent with the thermodynamic first
law, so it must also be possible to take into account the matter entropy.

The way this works is perhaps easiest to understand in the setting of a
stationary, asymptotically flat black hole surrounded by a fluid [2, 30]. The
gravitational first law indicates that a matter Killing energy variation
$\delta H^{\rm\tilde{m}}_{\xi}$ increases the variation of the total ADM mass
$M$,

$\delta M=\delta H^{\rm\tilde{m}}_{\xi}+\kappa\delta A/8\pi G,$ (4.9)

where $\xi$ is the time translation Killing vector, and variation of angular
momentum of the black hole is suppressed for simplicity.252525Equation (4.9)
is equivalent to the first law as given in Ref. [2, 30], although that is not
immediately apparent since we write the fluid term as the variation of its
contribution to the Killing Hamiltonian. The equivalence can be established
using (3.39) for this variation. The $\Sigma$ integral in (3.39) is the same
as the fluid contribution in Eqn. (50) of [30], and it can be shown that the
$\partial\Sigma$ boundary term vanishes for the case of a prefect fluid (using
the variational formulation given by Schutz and reviewed in [30]). On the
other hand, the energy variation of a thermal fluid can be expressed in terms
of its entropy and particle number density variations, $\delta
dS_{\rm\tilde{m}}$ and $\delta dN$, and redshifted comoving temperature and
chemical potential, $\bar{T}$ and $\bar{\mu}$, together with a possible
angular momentum variation that is also suppressed. If that is done, the
entropy of the fluid registers explicitly in the thermodynamical first law,

$\delta M=\int_{\Sigma}(\bar{T}\delta dS_{\rm\tilde{m}}+\bar{\mu}\delta
dN)+\kappa\delta A/8\pi G,$ (4.10)

while the energy of the fluid registers only implicitly in the $\delta M$
term, to which it contributes via the constraints. It is quite peculiar to
gravitational thermodynamics that the first law has simultaneously these two
different meanings, one gravitational and one thermodynamical [73], and that
the fluid does not contribute explicitly to both the entropy and energy
variations, unlike in ordinary thermodynamics.

It thus seems that the correct procedure is to add only the matter energy
variation $\delta\langle H^{\rm\tilde{m}}_{\zeta}\rangle$ to the classical
first law, as we anticipated in (4.6), where the classical matter Hamilonian
variation $\delta H^{\rm\tilde{m}}$ is _replaced_ by its expectation value. If
desired, one can use (4.7) and (4.8) to express $\delta\langle
H^{\rm\tilde{m}}_{\zeta}\rangle$ in terms of the matter entropy variation.
Thanks to the opposite sign of the temperature in (4.8), this can then be
combined with $\delta S_{\rm BH}$ in (4.6) to form the variation of the
generalized entropy,

$S_{\rm gen}:=S_{\rm BH}+S_{\rm\tilde{m}}\,,$ (4.11)

in terms of which the quantum corrected first law of causal diamonds becomes

$\delta H^{\rm g+\Lambda}_{\zeta}=T\delta S_{\rm gen}\,.$ (4.12)

In fact, this appears to be a more satisfactory formulation of the first law
because, unlike the matter entropy and Bekenstein-Hawking entropy separately,
the generalized entropy is plausibly invariant under a change of the UV cutoff
(see the appendix of [74] for a discussion of this idea).

In this way, we see that the opposite sign of the temperature in (4.3) and
(4.8) is precisely what is needed in order for the Bekenstein-Hawking entropy
and matter entropy to combine as $S_{\rm gen}$. At least for conformal fields,
this resolves the apparent conflict alluded to before between the negative
temperature in the first law for causal diamonds and the positive temperature
in the first law of entanglement. The negative and positive temperature seem
compatible with each other, since the first law of entanglement (4.8) and the
quantum corrected first law of causal diamonds (4.12) are valid
simultaneously.

####  4.2.2 Non-conformal matter

We now show how the generalized entropy variation can be obtained in the first
law for generic quantum fields. For non-conformal fields the matter
Hamiltonian is not equal to the modular Hamiltonian, and hence the term
$\delta\langle H_{\zeta}^{\rm\tilde{m}}\rangle$ cannot be directly related to
the entanglement entropy variation. However, in [21] it was postulated that,
for causal diamonds that are small compared to the local curvature scale, the
length scale of the quantum state, and any length scale in the quantum field
theory defined by a relevant deformation of a conformal field theory, this
term is in fact related to the variation of the expectation value of the
modular Hamiltonian via

$\delta\langle H_{\zeta}^{\rm\tilde{m}}\rangle=\delta\langle K\rangle-
V_{\zeta}\,\delta X\,.$ (4.13)

Here $X$ is a spacetime scalar that can depend on the size of the diamond but
is invariant under Lorentz boosts that leave the center of the diamond fixed.
The thermodynamic volume $V_{\zeta}$ has been factored out for later
convenience.262626For small diamonds the conformal Killing energy variation
can be approximated by $\delta\langle
H^{\rm\tilde{m}}_{\zeta}\rangle=V_{\zeta}\delta\langle
T_{00}^{\rm\tilde{m}}\rangle$ and the thermodynamic volume is to first order
given by $V_{\zeta}=\kappa\,\Omega_{d-2}R^{d}/(d^{2}-1)$ (see Sec. 5.2).
Inserting these approximations into (4.13) yields the actual conjecture (21)
in [21]. The modular Hamiltonian $K$ is here defined for the vacuum of a
quantum field theory restricted to a ball-shaped region, and the variation
denotes a perturbation of the vacuum state. The assumption (4.13) was checked
in [37, 38], and it was found in particular that $\delta X$ may depend on the
radius $R$ of the ball, and can dominate at small $R$ (depending on the
conformal dimension of the operator that deforms the CFT).

With the postulate (4.13) and the first law of entanglement (4.8), the quantum
first law (4.6) can be written in terms of the generalized entropy variation
as

$\delta H^{\rm g+\lambda}_{\zeta}-V_{\zeta}\,\delta X=T\delta S_{\rm gen}\,.$
(4.14)

Here, we have denoted the local cosmological constant in a small maximally
symmetric diamond by $\lambda$, in order to distinguish it from the total
cosmological constant variation to be introduced below. The variations $\delta
H^{\rm g}$ and $\delta H^{\lambda}$ are explicitly given by (1.4) and (1.3),
respectively, so we can further rewrite this first law as

$-\frac{\kappa k}{8\pi G}\delta V+\frac{V_{\zeta}}{8\pi G}(\delta\lambda-8\pi
G\delta X)=T\delta S_{\rm gen}\,.$ (4.15)

Now, since the $\delta X$ contribution from non-conformal matter appears
together with the local cosmological constant variation $\delta\lambda$ in
this way, we may combine them into one net variation,

$\delta\Lambda:=\delta\lambda-8\pi G\delta X,$ (4.16)

in terms of which (4.15) is expressed as

$-\frac{\kappa k}{8\pi G}\delta V+\frac{V_{\zeta}}{8\pi
G}\delta\Lambda=T\delta S_{\rm gen}\,.$ (4.17)

The first law including non-conformal quantum fields is thus also expressed in
terms of the generalized entropy variation, and the temperature in this first
law is still negative. We conclude that the assignment of a negative
temperature to the diamond remains consistent when extended to the
semiclassical realm.

###  4.3 Entanglement equilibrium

If the proper volume $V$ and cosmological constant $\Lambda$ are held fixed in
(4.17), then the generalized entropy is stationary in a maximally symmetric
vacuum,

$\delta S_{\rm{gen}}\big{|}_{V,\Lambda}=\delta S_{\rm
BH}\big{|}_{V,\Lambda}+\delta S_{\rm\tilde{m}}=0\,,$ (4.18)

There is no need to fix $V$ and $\Lambda$ in the matter entropy variation,
because there is no first order metric variation of the matter entropy, since
the zeroth order matter state is the vacuum. The condition $\delta\Lambda=0$
means that $\delta\lambda$ is chosen to cancel the change in the effective
local cosmological constant, $-8\pi G\delta X$.

In [21], it was shown, assuming the conjecture (4.13), that the semiclassical
Einstein equation holds if and only if the generalized entropy is stationary
at fixed volume in small local diamonds everywhere in spacetime.272727In that
context, the area term in the generalized entropy was assumed to have the form
$\eta A$, and the gravitational constant $G$ was found to be given by
$1/4\hbar\eta$. The validity of the latter property was called the “maximal
vacuum entanglement hypothesis”, but we shall refer to it as the _entanglement
equilibrium_ hypothesis. Here we have deduced the entanglement equilibrium
statement (4.18) from the conjecture (4.13) together with the quantum
corrected first law of causal diamonds (4.6), which itself was derived from
the semiclassical gravitational equations of motion. In this sense the
semiclassical Einstein equation is equivalent to the quantum corrected first
law for small, and therefore maximally symmetric diamonds. However, the
variations in the entanglement equilibrium setting [21] and in the present
paper are viewed somewhat differently, so the precise relation between the two
results is not immediately clear. We shall now explain how they may be brought
into alignment.

In the setting of [21], an arbitrary spacetime and matter state,
$(g,|\psi\rangle)$, are considered, in every small causal diamond, as a
variation of a maximally symmetric spacetime (MSS) and vacuum
$(g_{\lambda},|0\rangle)$, with an initially arbitrary $\lambda$. The idea is
that any spacetime is locally close to an “equilibrium” state, and that all
maximally symmetric states qualify as equilibria. The generalized entropy
$S_{\rm{gen}}$ in the diamond is then compared to that of the diamond with the
same volume in the MSS, the difference being

$\delta
S_{\rm{gen}}|_{V,\lambda}=S_{\rm{gen}}|_{V(g,|\psi\rangle)}-S_{\rm{gen}}|_{V(g_{\lambda},|0\rangle)}.$
(4.19)

The notation suggests “fixed $\lambda$”, but at this stage $\lambda$ is just
an arbitrary background value for the comparison. The entanglement equilibrium
assumption amounts to the postulate that there exists some $\lambda$, in each
diamond, for which the stationary condition $\delta S_{\rm
gen}|_{V,\lambda}=0$ holds. When applied to all diamonds this condition,
together with energy-momentum conservation and the Bianchi identity, implies
that $\lambda$ for each diamond is determined, up to one overall spacetime
constant $\Lambda$, by the $\delta X$ of the state, and it implies that the
Einstein equation holds for that $\Lambda$.

To bring this more in line with the variational relations of the present
paper, instead of setting the difference (4.19) to zero, we may first reckon
the diamond entropy of $(g,|\psi\rangle)$ relative to that of a diamond in
flat spacetime. In the notation of [21] this yields

$\delta S_{\rm{gen}}|_{V}=\eta\delta A|_{V}+\frac{2\pi}{\hbar}\delta\langle
K\rangle=\frac{\Omega_{d-2}R^{d}}{d^{2}-1}\left[-\eta
G_{00}+\frac{2\pi}{\hbar}(\delta\langle T_{00}\rangle+\delta X)\right].$
(4.20)

Now, rather than postulating that this variation vanishes, we postulate that
it is the same as would be obtained by varying from the Minkowski vacuum to a
MSS vacuum with some cosmological constant $\lambda$ (3.22),

$\delta_{\rm MSS}S_{\rm{gen}}|_{V}=\eta\,\delta_{\rm
MSS}A|_{V}=-\frac{\Omega_{d-2}R^{d}}{d^{2}-1}\eta\,\lambda.$ (4.21)

The equality of (4.20) and (4.21) implies the relation

$G_{00}+\lambda\,g_{00}=\frac{2\pi}{\hbar\eta}(\delta\langle
T_{00}\rangle-\delta X\,g_{00}),$ (4.22)

(since $g_{00}=-1$ in Riemann normal coordinates at the center of the diamond)
and the validity of this relation for all small diamonds in spacetime implies
the tensor equation

$G_{ab}+\Lambda\,g_{ab}=\frac{2\pi}{\hbar\eta}\delta\langle T_{ab}\rangle,$
(4.23)

where282828Eq. (4.24) agrees with the result in Ref. [21] after correcting the
sign error there of the $\delta X$ term in (25), which appears due to an error
in equation (24).

$\Lambda:=\lambda+\frac{2\pi}{\hbar\eta}\delta X.$ (4.24)

With the identification $\eta=1/4\hbar G$, the relation (4.24) matches that
found in (4.16), when it is recognized that $\lambda$ and $\Lambda$ here refer
to a maximally symmetric comparison spacetime, whereas in (4.16) the
background spacetime is implicit and $\delta\lambda$ and $\delta\Lambda$ are
part of the one overall variation that is made. Adding $\lambda$ to the
comparison spacetime yields the same change of entropy as including a
variation $\delta\lambda=-\lambda$ in the variation being considered, and
similarly for $\Lambda$, so that the appropriate identification is
$\lambda=-\delta\lambda$ and $\Lambda=-\delta\Lambda$. This establishes how
the first law derived in the present paper from the Einstein equation is
related to the entanglement equilibrium postulate used in [21] to derive the
Einstein equation.

###  4.4 Free conformal energy

In this section we address two questions concerning the entanglement
equilibrium proposal. First, in standard thermodynamics the stationarity of
entropy at fixed energy follows from the stationarity of free energy at fixed
temperature. Hence, it is natural to ask whether we can identify a
thermodynamic potential (e.g. free energy) in our setting whose stationarity
corresponds to an equilibrium condition. The stationarity is a condition on
the first order variation of the free energy. Whether the free energy is
minimized or maximized can only be determined from the second order variation
of the free energy, which we leave for future work (see also [43]).292929We
expect that maximally symmetric causal diamonds have _maximum_ free energy.
This is because local thermodynamic stability requires that entropy be
maximized at fixed energy, which for systems with negative temperature (such
as our diamonds) implies that free energy is maximized (see e.g. [75]). We
thank Batoul Banihashemi for pointing this out. Second, an essential
ingredient in the derivation of the Einstein equation in [21] was the fixed
volume requirement. One advantage of characterizing the equilibrium state in
terms of free energy stationarity, instead of entropy stationarity, is that
the fixed volume constraint is relaxed. It is desirable to understand the
fixed volume requirement better, and to see whether the free energy
stationarity can be used as an input assumption in deriving the Einstein
equation.

Let us start with the free energy for classical matter configurations. The
classical first law (4.3) implies that the _free conformal energy_ 303030The
term “conformal” here refers to the fact that the Hamiltonian $H_{\zeta}$
generates evolution along the conformal Killing vector $\zeta$, and not to a
conformal symmetry of the matter fields (as for CFTs).

$F=H_{\zeta}-TS_{\rm BH}$ (4.25)

is stationary at fixed temperature, i.e. $\delta F=-S_{\rm BH}\delta
T=0$.313131We remind the reader that the temperature of a vacuum causal
diamond is always $T=-\hbar/2\pi$. This means that causal diamonds in flat
space and (A)dS are equilibrium states.

Next, in the semiclassical regime the quantum corrected free conformal energy
can be defined by replacing the matter Hamiltonian by its quantum expectation
value

$F_{\rm quan}=\langle H_{\zeta}^{\rm\tilde{m}}\rangle+H_{\zeta}^{\rm
g+\Lambda}-TS_{\rm BH}\,.$ (4.26)

The stationarity of the quantum free energy at fixed temperature in the
vacuum, i.e. $\delta F_{\rm quan}=0$, follows from the quantum first law of
causal diamonds (4.6). We woud like to show that the free energy stationarity
is equivalent to the entanglement equilibrium hypothesis (4.18), i.e. the
stationarity of generalized entropy at fixed $V$ and $\Lambda$. In
establishing the equivalence we will treat conformal quantum fields and non-
conformal quantum matter separately. Using the expressions (1.3) and (1.4) for
$H_{\zeta}^{\Lambda}$ and $H_{\zeta}^{\rm g}$, respectively, in terms of the
variation of the volume and the cosmological constant, we see that the
variation of the quantum free energy (4.26) at fixed $V$ and $\Lambda$ is
equal to

$\delta F_{\rm quan}\big{|}_{V,\Lambda}=\delta\langle
H_{\zeta}^{\rm\tilde{m}}\rangle-T\delta S_{\rm BH}\big{|}_{V,\Lambda}\,.$
(4.27)

For conformal matter, the first law of entanglement (4.8) can be used to trade
the conformal Killing energy variation $\delta\langle
H_{\zeta}^{\rm\tilde{m}}\rangle$ for a matter entropy variation, which then
combines with $\delta S_{\rm BH}$ to form the generalized entropy variation,
i.e. $\delta F_{\rm quan}|_{V,\Lambda}=T_{\rm H}\delta S_{\rm
gen}|_{V,\Lambda}.$ The free energy for conformal quantum fields is thus
stationary at fixed $V$ and $\Lambda$ if and only if the generalized entropy
is stationary at fixed $V$ and $\Lambda$.

Dealing with non-conformal matter is more subtle since, to relate the
variation of the conformal Killing energy to the matter entropy variation,
apart from the first law of entanglement we need the additional assumption
(4.13), which holds only for small diamonds. For small maximally symmetric
diamonds we denote the local cosmological constant by $\lambda$, rather than
$\Lambda$ in (4.27), and the variation of the associated Hamiltonian is
$(V_{\zeta}/8\pi G)\delta\lambda$. Inserting the assumption (4.13) for
$\delta\langle H_{\zeta}^{\rm\tilde{m}}\rangle$ and the expression above for
$\delta H^{\lambda}_{\zeta}$ into the variation of the quantum free energy at
fixed volume yields

$\delta F_{\rm quan}\big{|}_{V}=T_{\rm H}\delta S_{\rm
gen}\big{|}_{V}+\frac{V_{\zeta}}{8\pi G}(\delta\lambda-8\pi G\delta X)\,.$
(4.28)

The variations in parenthesis can be combined into a single net variation
$\delta\Lambda$, cf. (4.16). Thus, restricting the variations such that
$\delta\Lambda=0$ and holding the volume $V$ fixed, we establish that the
stationarity of quantum free energy is equivalent to the stationarity of
generalized entropy

$\delta F_{\rm quan}\big{|}_{V,\Lambda}=0\qquad\Longleftrightarrow\qquad\delta
S_{\rm gen}\big{|}_{V,\Lambda}=0\,.$ (4.29)

Holding the volume and (effective) cosmological constant fixed is therefore
analogous to holding the internal energy fixed in an ordinary thermodynamic
system. In fact, it corresponds here to holding fixed the metric and $\Lambda$
contributions to the conformal Hamiltonian $H_{\zeta}$ (assuming, for non-
conformal matter, that there exists a Hamiltonian such that $\delta
H^{\Lambda}_{\zeta}=(V_{\zeta}/8\pi G)\,\delta\Lambda$, where $\Lambda$ is the
effective cosmological constant (4.16) in a small diamond.323232It is not
clear to us whether this Hamiltonian exists, since $\Lambda$ includes both a
local cosmological constant and a piece from the non-conformal matter. The
latter contribution seems to spoil the derivation of equation (3.44). In this
sense, the entanglement equilibrium hypothesis would correspond to the
statement that generalized entropy is stationary at fixed “internal energy”
(as it should be for an equilibrium state).

In standard thermodynamical equilibrium, not only is entropy stationary at
fixed energy, but also energy is stationary at fixed entropy. Let us now see
to what extent the latter is true for causal diamonds. For conformal matter,
and for any sized diamond, if the generalized entropy is kept fixed then Eq.
(4.26) implies that the Hamiltonian associated to $\rm{g}$ and $\Lambda$ is
stationary in the vacuum

$\delta F_{\rm
quan}\big{|}_{S_{\rm{gen}}}=0\qquad\Longleftrightarrow\qquad\delta
H_{\zeta}^{\rm g+\Lambda}\big{|}_{S_{\rm{gen}}}=0\,.$ (4.30)

For non-conformal matter, we must restrict to small diamonds if we want to
hold the generalized entropy fixed, and as above assume the existence of
$H^{\Lambda}_{\zeta}$. But if instead we keep the Bekenstein-Hawking entropy
fixed, rather than the full generalized entropy, then we need not restrict to
small diamonds and the _total_ Hamiltonian is stationary in the vacuum, i.e.

$\delta F_{\rm
quan}\big{|}_{S_{\rm{BH}}}=0\qquad\Longleftrightarrow\qquad\delta\big{(}\langle
H_{\zeta}^{\rm\tilde{m}}\rangle+H_{\zeta}^{\rm
g+\Lambda}\big{)}\big{|}_{S_{\rm{BH}}}=0\,.$ (4.31)

This equilibrium condition states that the total conformal Killing energy is
stationary in the vacuum if the dimension of the Hilbert space is fixed (if we
interpret the Bekenstein-Hawking entropy as the logarithm of that dimension).
It would be interesting to explore this energy relation further.

Next, we return to the role of the fixed volume constraint in (4.29). For
small diamonds the free energy variation at fixed $\Lambda$ can be expressed
as

$\frac{1}{T_{H}}\delta F_{\rm quan}\big{|}_{\Lambda}=\frac{1}{4G\hbar}(\delta
A-k\,\delta V)\big{|}_{\Lambda}+\delta S_{\rm\tilde{m}}\,,$ (4.32)

(where the volume variation comes from the variation of $H^{\rm{g}}_{\zeta}$
in (4.26)) whereas the variation of the free energy at fixed $V$ and $\Lambda$
is given by

$\frac{1}{T_{H}}\delta F_{\rm quan}\big{|}_{V,\Lambda}=\frac{1}{4G\hbar}\delta
A\big{|}_{V,\Lambda}+\delta S_{\rm\tilde{m}}\,.$ (4.33)

Now any given variation at fixed $\Lambda$ can be expressed as a variation at
fixed $V$ and $\Lambda$, together with a variation induced by a diffeomorphism
that changes the volume. Under such a diffeo-induced variation, $\delta
A-k\delta V$ vanishes, as explained in Sec. 3.3.2, and $\delta
S_{\rm\tilde{m}}$ vanishes because the renormalized matter entropy is zero in
the vacuum. The two free energy variations are therefore _equivalent_ , i.e.

$\delta F_{\rm quan}\big{|}_{\Lambda}\sim\delta F_{\rm
quan}\big{|}_{V,\Lambda}\,,$ (4.34)

where the equivalence is modulo diffeo-induced variations. This implies that
the stationarity of the free energy at fixed $\Lambda$ is equivalent to the
stationarity of the free energy at fixed $V$ and $\Lambda$, and hence — using
the equivalence (4.29) — equivalent to the entanglement equilibrium postulate.
Therefore, the derivation of the Einstein equation in [21] can be rephrased
without the requirement to fix the volume. One can thus take as the input
assumption that the free energy is stationary at fixed $\Lambda$, rather than
that the generalized entropy is stationary at fixed $V$ and $\Lambda$ (see
also Ref. [76]).

##  5 Limiting cases: small and large causal diamonds

In this section we comment on various limiting cases of the first law of
causal diamond mechanics (3.45). Since the law applies to arbitrary sized
diamonds in (A)dS, it has a wide domain of applicability. Here we apply it to
the static patch of de Sitter spacetime, small diamonds in any maximally
symmetric spacetime, flat and AdS Rindler spacetimes, and to the so-called
“Wheeler-DeWitt patch” in AdS, tying these limiting cases together into one
framework.

###  5.1 De Sitter static patch

Figure 3: A causal diamond in the de Sitter space static patch. The conformal
Killing vector $\zeta$ turns into the timelike Killing vector $\xi$ if the
boundary of the diamond coincides with the cosmological horizon
$\mathcal{H}_{C}$.

If the boundary of a causal diamond in dS space coincides with the
cosmological horizon, i.e. if $R=L$, then the conformal Killing vector (2.6)
becomes the time translation Killing vector of the static patch,333333In this
section we use the letter $\xi$ for Killing vectors and retain $\zeta$ for
conformal Killing vectors.

$\xi^{\text{dS}}=L\partial_{t},$ (5.1)

normalized so that the surface gravity is unity (see Fig. 3). Since this is a
true Killing vector, the variation of the gravitational part of the
Hamiltonian (3.25) vanishes. This is consistent with (3.35), because the
Sitter horizon has extremal area on the $(d-1)$-sphere, so $k=0$. The first
law (3.45) thus reduces to

$\delta H_{\xi}^{\text{\~{m}}}=-\frac{1}{8\pi G}\left(\kappa\,\delta
A+V_{\xi}\,\delta\Lambda\right)\,.$ (5.2)

The thermodynamic volume (3.14) in this case reduces to $V_{\xi}^{\text{dS-
static-patch}}=\kappa LV^{\text{flat}}_{L}$. The relation (5.2) with
$\delta\Lambda=0$ was established long ago by Gibbons and Hawking [5], and was
generalized to include a variation of the cosmological constant in [77, 57].

If one assigns a negative temperature $T=-\kappa\hbar/2\pi$ and pressure
$p=-\Lambda/8\pi G$ to the dS static patch, the first law can be turned into a
proper thermodynamic relation343434For $\delta H_{\xi}^{\text{\~{m}}}=0$, it
might look like one could assign a positive temperature to the dS static
patch, i.e. $T=T_{\rm H}>0$; however, according to the first law, the
“pressure” would then have to be defined as $p=+\Lambda/8\pi G$, in
contradiction to the sign of the pressure associated to the cosmological
constant when viewed as a stress-energy contribution.

$\delta H_{\xi}^{\text{\~{m}}}=T\delta S_{\text{BH}}+V_{\xi}\,\delta p\,.$
(5.3)

Since this thermodynamic relation is of the form $dH=TdS+V^{\text{th}}dp$,
where $H$ is the enthalpy of the system and $V^{\text{th}}$ is the
thermodynamic volume, $H_{\xi}^{\text{\~{m}}}$ coincides with the enthalpy
instead of simply the internal energy. The matter Hamiltonian vanishes in the
background, hence we observe that the enthalpy of the static patch of dS space
is zero.

Through a Legendre transformation, $U=H-pV^{\text{th}}$, the first law can be
rewritten in the standard form $dU=TdS-p\,dV^{\text{th}}$. But what is the
internal energy $U$ of de Sitter space? The common lore, cf. e.g. [5], is that
the energy of dS space is zero, because it has no asymptotic infinity.
However, we find that its (vacuum) internal energy is nonzero and given by
$U_{\text{vac}}=\rho\,V_{\xi}$, the redshifted vacuum energy, where we used
the equation of state $p=-\rho$ and the fact that $H=0$ for dS. The first law
can thus be expressed as

$\delta U=T\delta S_{\text{BH}}-p\,\delta V_{\xi}\,,$ (5.4)

where $U=U_{\text{vac}}+H_{\xi}^{\text{\~{m}}}$ is the internal energy.
Finally, we note that the Gibbs free energy of dS space is $G=H-TS=-TS$, and
the Helmholtz free energy is $F=U_{\text{vac}}-TS$. As usual, the former is
extremized in a fixed $(T,p)$ ensemble, whereas the latter is extremized in a
fixed $(T,V_{\xi})$ ensemble. The free energy computed in [5] from the on-
shell Euclidean action agrees with the Gibbs free energy, and not the
Helmholtz free energy, because the Euclidean action was extremized there at
fixed period (i.e. fixed temperature) and fixed cosmological constant.353535If
one takes the timelike Killing vector to be $\xi^{\text{dS}}=\partial_{t}$, so
that $\xi^{2}=-1$ at the center of the diamond, then temperature and pressure
are not independent in the dS static patch. That is because the surface
gravity is set by the dS radius in this case, i.e. $\kappa_{\text{dS}}=1/L$,
and the pressure is determined by the cosmological constant
$\Lambda=(d-1)(d-1)/2L^{2}$. Hence, by fixing the temperature one also fixes
the pressure, and vice versa, when using this normalization of the Killing
field to define the temperature.

###  5.2 Small diamonds and Minkowski space

(a)

(b)

Figure 4:  (a) A causal diamond associated to a ball at the center of
Minkowski space. If one the normalizes the conformal Killing vector that
preserves the diamond such that $\zeta^{2}=-1$ at the center of the ball, then
it becomes identical to the timelike Killing vector $\xi=\partial_{t}$ in the
infinite-size limit. (b) A causal diamond whose edge touches the bifurcation
surface of two Rindler horizons in Minkowski space. In the infinite-size limit
the diamond coincides with the right Rindler wedge, and the conformal Killing
vector $\zeta$ becomes the boost Killing vector $\xi$.

In the small radius limit $R\ll L$ the mean curvature (2.12) and the
thermodynamic volume (3.14) are given up to second order in $R/L$ by

$\displaystyle k$
$\displaystyle=\frac{d-2}{R}\left(1-\frac{1}{2}\frac{R^{2}}{L^{2}}+\dots\right)\,,$
(5.5) $\displaystyle V_{\zeta}$
$\displaystyle=\frac{\kappa\,\Omega_{d-2}R^{d}}{d^{2}-1}\left(1+\frac{d}{d+3}\frac{R^{2}}{L^{2}}+\dots\right)\,.$
(5.6)

To first order in $R/L$, the first law (3.45) thus reduces to the one that
would be found in flat spacetime,

$\delta H^{\text{\~{m}}}_{\zeta}=-\frac{\kappa}{8\pi G}\left[\delta
A-\frac{d-2}{R}\,\delta
V^{\text{flat}}+\frac{\Omega_{d-2}R^{d}}{d^{2}-1}\delta\Lambda\right]\,.$
(5.7)

This identity, without the cosmological constant term, is the one derived in
[21], both by Riemann normal coordinate expansion, and by varying the Noether
current for the conformal Killing vector that preserves a causal diamond in
flat space. That conformal Killing vector can be recovered from (2.6) in the
limit $L\rightarrow\infty$, and is given by

$\zeta^{\text{flat}}=\frac{1}{2R}\left[\left(R^{2}-t^{2}-r^{2}\right)\partial_{t}-2tr\partial_{r}\right]\,.$
(5.8)

(See Fig. 4(a) for a conformal diagram of a diamond in Minkowski space.) As a
side remark, if the variation $\delta T_{ab}^{\text{\~{m}}}u^{a}u^{b}$ is
constant on $\Sigma$,363636Note that since $\Sigma$ has vanishing extrinsic
curvature, $m^{c}\nabla_{c}u^{a}=0$ for any vector $m^{c}$ tangent to
$\Sigma$. Thus constancy on $\Sigma$ of the scalar $\delta
T_{ab}^{\text{\~{m}}}u^{a}u^{b}$ is equivalent to the condition
$(m^{c}\nabla_{c}\delta T_{ab}^{\text{\~{m}}})u^{a}u^{b}=0$. then the
conformal Killing energy variation becomes proportional to the thermodynamic
volume

$\delta H_{\zeta}^{\text{\~{m}}}=\delta
T_{ab}^{\text{\~{m}}}u^{a}u^{b}\,V_{\zeta}\,,$ (5.9)

where $\delta T_{ab}^{\text{\~{m}}}u^{a}u^{b}$ is evaluated at the center of
the ball.

###  5.3 Rindler space

Figure 5: Sequence of causal diamond edges anchored at the origin and with
growing radius, shown on a time slice of compactified 2+1 dimensional
Minkowski spacetime. In the infinite radius limit, the entire Rindler horizon
subtends an infinitesimal angle of the diamond’s edge, and the remainder of
the edge lies at spatial infinity.

A Rindler wedge is an infinite diamond in flat space, as can be seen from the
Penrose diagram in Fig. 4(b). More precisely, the right Rindler wedge can be
obtained by inflating the causal diamond whose edge touches the origin of flat
space and whose center is located at
$\\{t=0,x^{1}=R,x^{2}=0,\dots,x^{d-1}=0\\}$.373737Instead of increasing the
size of the diamond to infinity, one could also directly relate the diamond
nestled in the corner of the right Rindler wedge to the entire wedge itself
through a conformal map. See Appendix E for further details. In the infinite
$R$ limit, only an infinitesimal solid angle of the edge coincides with the
entire Rindler horizon, and the rest of the edge covers half the sphere at
spatial infinity (see Fig. 5). Moreover, by replacing the coordinate $x^{1}$
by $x^{1}-R$ in $\zeta^{\text{flat}}$ and then taking the limit $R\to\infty$,
the conformal Killing vector (5.8) becomes the boost Killing vector of Rindler
space,

$\xi^{\text{Rindler}}=x^{1}\partial_{t}+t\partial_{x^{1}}\,.$ (5.10)

Given that the regions agree it is interesting to understand the relation
between the infinite-size limit of the first law of this causal diamond and
the first law for the Rindler wedge [10, 11, 78, 79].

Although the first law of causal diamonds was originally derived for diamonds
which are centered at the origin of flat space, it actually applies to any
causal diamond associated to a spherical region whose center is located at
$\\{t=t_{0},x^{i}=x^{i}_{0}\\}$. Hence, in particular it applies to the causal
diamond described above which is centered at $x^{1}=R$. We evaluated the terms
in the first law (5.7) for the variation produced by a point mass $m$ at a
distance $\delta$ from the Rindler horizon at $t=0$, using the weak field
approximation and with $\delta\Lambda=0$. In the infinite $R$ limit the area
and volume variations both diverge as $mR$, and these divergences cancel each
other. The $k\delta V$ term contains a left-over finite part equal to
$m\delta$ which balances the boost Killing energy variation of the Rindler
wedge. Strangely, at least when defined by this limiting procedure, the first
law for the Rindler wedge involves not only an area variation but also a
volume variation, and the finite geometric variation comes entirely from the
volume term. Using the fact that the combination $\delta A-k\delta V$ is
diffeomorphism invariant — as shown in Section 3.3.2 — this finite
contribution could presumably be shifted entirely onto the area term by
composing the variation with an appropriate infinitesimal diffeomorphism.

The presence of the volume term in the infinite-size limit is unexpected,
since the conformal Killing vector of the diamond becomes the Rindler boost
Killing vector (5.10) in this limit, and the gravitational Hamiltonian
variation $\delta H^{\rm g}_{\xi}$ (3.35) vanishes for a Killing vector $\xi$.
Nevertheless, since the volume variation diverges, $k\delta V$ survives (and
diverges) in this limit. On the other hand, one can derive an equilibrium
variation version of the first law for the Rindler wedge by applying the
Noether identity (3.26) for the boost Killing vector (5.10) in a finite
region, and then taking the infinite region limit. In this way, no volume
variation term would be present. For example, for a finite half-spherical ball
with radius $R$ and center $o$ on the Rindler horizon the identity takes the
form

$\delta H_{\xi}^{\rm m}=-\frac{\kappa}{8\pi G}\delta A_{\rm disk}+\int_{\rm
hemisphere}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\delta
Q_{\xi}-\xi\cdot\Theta),$ (5.11)

where $\delta A_{\rm disk}$ is the area variation of the disk on the Rindler
horizon and the boundary integral on the right-hand side is over the
hemispherical boundary of the half ball. We evaluated the terms in this
identity for a weak field variation produced by a point mass at a
perpendicular distance $\delta$ from $o$, in $d=4$ dimensions. The disk
contribution is $-mR/2+m\delta/2$, while the hemisphere contribution is
$mR/2+m\delta/2$, up to $\mathcal{O}(1/R)$ corrections. In the infinite $R$
limit, the half-spherical ball fills the entire Rindler wedge at $t=0$, and
these two contributions add to $m\delta$, which equals the boost energy
variation $\delta H_{\xi}^{\rm m}$. This result too is peculiar. We would have
expected a first law with just the area variation of the Rindler horizon, as
in the physical process version of the first law for Rindler horizons, but
that is not the result of this limiting procedure.

A _physical process_ version of the first law for Rindler space was stated in
[10], and subsequently proven using the Einstein equation in [11, 78, 79].
This version of the first law describes how the horizon area changes if an
infinitesimal amount of Killing energy flows through the horizon. The physical
process first law is

$\delta E_{\xi}=\frac{\kappa}{8\pi G}\delta A\,,$ (5.12)

where $\delta A$ is the horizon area change and $\delta
E_{\xi}=\int_{\mathcal{H}}\delta T_{ab}\xi^{a}d\Sigma^{b}$ is the flux of
Killing energy across the horizon. This differs in two ways from the
equilibrium version of the first law (5.11): the hemispherical boundary
integral is absent, and the sign of the coefficient of the area variation term
is opposite. In the physical process version it is assumed that the energy
flux crosses the horizon rather than escapes to infinity, which perhaps
accounts for the fact that only the area change of the Rindler horizon enters.
As for the sign difference, this is due to a different definition of the area
change: the equilibrium version compares the areas of the bifurcation surface
in two infinitesimally nearby solutions (Rindler space and the spacetime
perturbed by the presence of matter), whereas the process version involves the
difference between the asymptotic area of the future horizon and the area of
the bifurcation surface in one solution involving a matter boost energy flux
across the horizon.

The positive coefficient in the process version (5.12) arises because the
generators of the horizon are parallel at future null infinity (due to the
teleological nature of the horizon), so that the sign of the area change is
the same as the same as the sign of the boost energy flux. For example, when
matter with positive Killing energy crosses the future horizon, the horizon
expansion decreases because of the attractive nature of gravity. In order to
satisfy the zero expansion boundary condition at infinity, the generators must
initially have positive expansion, and hence the area of the horizon cross-
section _increases_ towards the future.383838Ref. [79] also interpreted this
in terms of an equilibrium variation version of the first law, which refers to
the variation of the asymptotic horizon area (assuming the bifurcation surface
area is fixed) rather than the variation of the bifurcation surface as in our
version.

Finally, note that the area change in the process version can also be
expressed as a comparison of two solutions,

$(\delta A)_{\rm phys.\,\,proc.}=A_{\rm final}-A_{\rm initial}=\delta A_{\rm
final}-\delta A_{\rm initial},$ (5.13)

since the area change is zero in the background Minkowski solution. If we
consider variations to solutions with the final horizon area held fixed, then
$(\delta A)_{\rm phys.\,\,proc.}=-\delta A_{\rm initial}$. With this
replacement, the sign of the coefficient in the process version becomes the
same as that in the equilibrium version.393939For another viewpoint on the
negative sign, note that the process version for the _past_ horizon takes the
form: $\delta E_{\xi}=-\kappa\delta A/8\pi G$, where $\kappa$ is positive and
Killing energy flows through the past horizon into the right Rindler wedge.
For example, if the Killing energy flux is positive, then the horizon
generators converge, so that the area of the past horizon decreases, hence
$\delta A=A_{\rm final}-A_{\rm initial}$ is negative.

(a)

(b)

Figure 6:  (a) A causal diamond centered at the origin of AdS space can be
mapped to two equally large AdS-Rindler wedges in two different Poincaré
patches. The situation is depicted for a coordinate radius $R<L/\sqrt{3}$ of
the centered ball, since for larger values $R>L/\sqrt{3}$ the diamond overlaps
with the right Rindler wedge. (b) A small causal diamond and infinite size
diamond in AdS space. The infinite size diamond is called the Wheeler-DeWitt
patch of global AdS, and is also preserved by a conformal Killing vector.

###  5.4 AdS-Rindler space

In empty AdS space there also exists a Rindler wedge, which admits a boost
Killing vector. Unlike in Minkowski space, accelerating observers in AdS start
and end on the boundary at a finite global Killing time. In the Penrose
diagram the AdS-Rindler wedge therefore has the shape of a half diamond rather
than a full diamond (see Fig. 6(a)). The vertices of the diamond are located
at the AdS boundary, and are separated by an infinite proper time but a finite
global Killing time. Due to the presence of the conformal boundary, in
addition to the area variation one should take the boundary term
$\int_{\infty}[\delta Q_{\xi}-\xi\cdot\theta(g,\delta g)]$ at spatial infinity
into account in the variational identity (3.26). This boundary term has been
shown in [80, 81] to be equal to the gravitational energy variation $\delta
E_{\zeta}^{\text{g}}=\int_{\infty}\delta
T_{ab}^{\rm{g}}\zeta^{a}_{\text{flat}}d\Sigma^{b}$, where $T_{ab}^{\rm{g}}$ is
the holographic stress-energy tensor, and $\zeta^{\text{flat}}$ is the
boundary conformal Killing vector given by (5.8). The first law for AdS-
Rindler space can subsequently be derived along the lines of Sec. 3.2 using
Wald’s variational identity, but now applied to the boost Killing vector.

The variational identity was established in [13] for general relativity, and
later generalized in [17] to an arbitrary higher-derivative theory of gravity.
The bulk modular energy term $\delta H_{\xi}^{\tilde{\rm{m}}}$ was included in
the first law in [18, 82], and it has been extended to allow for variations of
the cosmological constant in [58, 59]. For general relativity the full first
law reads

$\delta\bar{E}_{\zeta}^{\text{g}}=\frac{1}{8\pi G}\left(\kappa\,\delta
A+\bar{V}_{\xi}\,\delta\Lambda\right)+\delta H_{\xi}^{\tilde{\rm{m}}}\,.$
(5.14)

The proper volume term is absent since the gravitational Hamiltonian variation
$\delta H^{\rm{g}}_{\xi}$ vanishes for a true Killing vector. In the energy
variation $\delta\bar{E}_{\zeta}^{\text{g}}:=\int_{\infty}[\delta
Q_{\xi}-\delta_{\Lambda}Q^{\text{AdS}}_{\xi}-\xi\cdot\theta(g,\delta g)]$ we
have subtracted the boundary term in the AdS background due to a variation of
$\Lambda$, such that it stays finite when the cosmological constant is
varied.404040We note that $\theta(g,\delta_{\Lambda}g)=0$ since the symplectic
potential for general relativity is linear in $\nabla_{c}\delta g_{ab}$, and
we have $\delta_{\Lambda}g_{ab}=-(\delta\Lambda/\Lambda)g_{ab}$ (see also
appendix C of [59]). From the RHS of the equation we have also subtracted the
outer boundary integral of the Noether charge variation
$\delta_{\Lambda}Q_{\xi}^{\text{AdS}}$, which for general relativity is given
by

$\delta_{\Lambda}Q_{\xi}^{\text{AdS}}=-\frac{d-2}{2\Lambda}Q_{\xi}^{\text{AdS}}\delta\Lambda=\frac{\delta\Lambda}{8\pi
G}\,\omega_{\xi}^{\text{AdS}}\,,$ (5.15)

where $\omega_{\xi}^{\text{AdS}}$ is the Killing potential form in AdS. This
first equality is due to the scaling $Q_{\xi}^{\text{AdS}}\sim L^{d-2}$, and
the second equality follows from the definition of the Noether charge
$j_{\xi}=dQ_{\xi}$ and the Killing potential form
$\xi\cdot\epsilon=d\omega_{\xi}$, together with equations (3.2) and (3.11).
This amounts to replacing the thermodynamic volume by the background
subtracted thermodynamic volume:
$\bar{V}_{\xi}=\int_{\infty}(\omega_{\xi}-\omega_{\xi}^{\text{AdS}})-\int_{\mathcal{H}}\omega_{\xi}$
(similar to the expression in footnote 9). Since
$\omega_{\xi}=\omega_{\xi}^{\text{AdS}}$ for the AdS-Rindler wedge, the
boundary term at infinity cancels and only the horizon surface integral
remains. More explicitly, in [58] the background subtracted thermodynamic
volume was found to be equal to: $\bar{V}_{\xi}=-\kappa\,A\,L^{2}/(d-1)$,
where $A$ is the area of the bifurcation surface of the AdS-Rindler horizon.

As in the Rindler case, although there exists a map from the finite causal
diamond to the AdS-Rindler wedge that takes the conformal Killing vector of
the former to the true Killing vector of the latter, the relation between the
first laws for these two regions is not straightforward. Here we will just
describe the nature of the map from a causal diamond in AdS space to an AdS-
Rindler wedge. This map consists of shifting the diamond towards the boundary
in the $z$ direction, where $z$ is the radial Poincaré coordinate. At the
level of the (conformal) isometries, the conformal Killing vector which
preserves the diamond transforms under this shift into the boost Killing
vector of AdS-Rindler space. In Appendix D we find the Poincaré coordinate
expression for the conformal Killing vector that preserves a diamond in AdS
with center located at $\\{t=0,z=L,x^{i}=0\\}$ (see equation (D.17))

$\zeta^{\text{AdS}}=\frac{1}{2R}\left[\sqrt{L^{2}+R^{2}}(2z\partial_{t}+2t\partial_{z})-(L^{2}+t^{2}+\vec{x}^{2}+z^{2})\partial_{t}-2tx^{i}\partial_{i}-2tz\partial_{z}\right].$
(5.16)

Now, by shifting the radial coordinate as $z\rightarrow z+\sqrt{L^{2}+R^{2}}$,
the conformal Killing vector above turns into the boost Killing vector of an
AdS-Rindler wedge [17]

$\xi^{\text{AdS-
Rindler}}=\frac{1}{2R}\left[\left(R^{2}-t^{2}-\vec{x}^{2}-z^{2}\right)\partial_{t}-2tx^{i}\partial_{i}-2tz\partial_{z}\right].$
(5.17)

At the boundary ($z=0$) this reduces to the conformal Killing vector
$\zeta^{\text{flat}}$ in (5.8). The boost Killing vector becomes null on the
Killing horizon $\mathcal{H}=\left\\{z^{2}=(R\pm t)^{2}-\vec{x}^{2}\right\\}$,
and vanishes at the vertices $\left\\{t=\pm R,\,z=x^{i}=0\right\\}$ and
hemisphere $\mathcal{B}=\left\\{t=0,\,z^{2}+\vec{x}^{2}=R^{2}\right\\}$.
Notice, though, that there are two solutions to these quadratic equations, one
for which $z$ takes positive values and the other for which $z$ takes negative
values. Thus, one part of the causal diamond is mapped to the AdS-Rindler
wedge in the $z>0$ Poincaré patch, and the other part is shifted to the
equally large Rindler wedge in the $z<0$ Poincaré patch (see Fig. 6(a)).

###  5.5 Wheeler-DeWitt patch in AdS

The first law applies also to causal diamonds whose spatial slice is an entire
slice of AdS. The spacetime region covered by such an infinite diamond is
commonly known as the “Wheeler-DeWitt patch” of AdS (see Fig. 6(b)). In this
limit, i.e. $R/L\rightarrow\infty$, the vector field $\zeta^{\text{AdS}}$
(5.16) takes the following simple form in Poincaré coordinates

$\zeta^{\text{WdW}}=z\partial_{t}+t\partial_{z}\,.$ (5.18)

This limiting value also follows directly from expression (D.17) for
$\zeta^{\text{AdS}}$ in terms of the generators of the conformal group. By
taking the limit $R/L\to\infty$ of (3.45), we obtain the first law for the
Wheeler-DeWitt patch

$\delta H^{\text{\~{m}}}_{\zeta}=-\frac{\kappa}{8\pi G}\left[\delta
A-\frac{d-2}{L}\delta V\right].$ (5.19)

In this limit the proper volume $V\approx A\,L/(d-2)$ of the $t=0$ timeslice
in AdS is divergent, but its variation can be finite. Here we have restricted
to fixed cosmological constant, i.e. $\delta\Lambda=0$. Note that the proper
volume variation is still present in the first law for the Wheeler-dDeWitt
patch, because the conformal Killing vector (5.18) is not a true Killing
vector.

##  6 Discussion

In this paper we explored aspects of the gravitational thermodynamics of
causal diamonds in maximally symmetric spacetimes and their first order
variations. Our starting point was the notion that the maximally symmetric
diamonds behave as thermodynamic equilibrium states. This is initially
motivated by the examples offered by the static patch of de Sitter spacetime
and the Rindler wedge of Minkowski spacetime, which are special cases of
causal diamonds admitting a true Killing field. Other maximally symmetric
causal diamonds, in particular finite ones, admit only a conformal Killing
vector. Since neither general relativity nor ordinary matter are conformally
invariant, it is not at all clear from the outset that the presence of a
conformal Killing symmetry should be adequate to support the interpretation of
a physical equilibrium state. However, it seems in all respects to be
sufficient. This can be traced to three important facts: (i) the conformal
Killing vector is an “instantaneous” Killing vector at the maximal volume
slice, which (ii) behaves at the edge as a boost-like Killing vector, with a
well-defined surface gravity; and (iii) in a maximally symmetric diamond the
vacuum of a conformal matter field restricted to the diamond is thermal with
respect to the Hamiltonian that generates the conformal Killing flow.

We first established a classical Smarr formula and a “first law” variational
identity for causal diamonds in maximally symmetric spacetime, i.e. in either
Minkowski, de Sitter, or Anti-de Sitter spacetime. Since we include a
cosmological constant and variations thereof, the “thermodynamic volume” [33,
34, 35] plays a role, generalized here to the case of a conformal Killing
vector. The name is appropriate for this quantity since it is
thermodynamically conjugate to the cosmological constant which is a type of
pressure. It is defined, given a (conformal) Killing vector $\zeta$, by
$V_{\zeta}=\int_{\Sigma}\zeta\cdot\epsilon$, which might well be called the
“redshifted volume”. For finite causal diamonds it appears as such, while for
infinite asymptotically Anti-de Sitter diamonds or black hole spacetimes it
diverges. We reviewed how finite relations have nevertheless been obtained by
subtracting the same divergence from both sides of an equation.

We then analyzed the thermodynamic interpretation of the first law, finding
that the gravitational temperature of a diamond is minus the Hawking
temperature associated with the horizon of the conformal Killing vector. The
idea that a negative temperature should be assigned to the static patch of de
Sitter spacetime has been floated before [31], but not followed up. We found
that the notion appears sound, and indeed is required by the thermodynamics of