causal diamonds in general. The consistency with the positive Gibbons-Hawking
temperature hinges on the fact that, in gravitational thermodynamics, the
matter contribution to the first law enters either as an energy contribution,
or as an entropy contribution, unlike in ordinary thermodynamics. In the
latter form, the matter entropy combines with the Bekenstein-Hawking area
entropy to comprise the generalized entropy. We showed that it works this way
also for quantum corrections. In establishing this for nonconformal matter we
invoked a conjecture, previously postulated in [21] and checked in [37, 38],
to relate the conformal boost energy to the entanglement entropy.

We next reformulated the entanglement equilibrium proposal of [21], replacing
the role of generalized entropy maximization by a free energy extremization,
and showed that in this way, the need to fix the volume is no longer present.
We find that extremization of the free conformal energy
$H_{\zeta}-TS_{\text{BH}}$ at fixed cosmological constant is equivalent to the
stationarity of the generalized entropy at fixed cosmological constant and
fixed volume. Therefore, the Einstein equation is implied by stationarity of
the free energy at fixed cosmological constant, without fixing the volume.

In the final section we considered limiting cases of causal diamonds, such as
small diamonds, the de Sitter static patch, Rindler space, and AdS-Rindler
space, and showed how our general result for the first law appears and reduces
to known results in these different settings. In particular, we gained some
interesting perspective on the first law in Rindler space, viewed as the
infinite size limit of a causal diamond.

In one of the appendices we established a link between conformal Killing time
and York time in a maximally symmetric causal diamond. The latter time is
defined by the mean curvature on a foliation by constant mean curvature
slices. We found that these slices coincide with the foliation orthogonal to
the conformal Killing vector, and that the conformal Killing time parameter
also labels these surfaces.

We end with some questions and future research directions:

  * (i)

Can the first law for causal diamonds be generalized to non-spherical regions
and/or non-maximally symmetric spacetimes? The particular form (3.45) applies
only to maximally symmetric spaces, but the general form (3.28) holds in
general. On the other hand, the conformal Killing Hamiltonian variation may
have no particular geometric significance. Since the connection between the
York time Hamiltonian and proper volume holds for any solution to Einstein
gravity, however, perhaps this would produce a geometrically meaningful form
of the first law in the absence of a conformal Killing vector. Note also that
the diamond itself doesn’t play an essential role in the first law, since all
the quantities are evaluated at the ball. Therefore, the thermodynamics might
be a property of the ball rather than the full diamond.

  * (ii)

In the case of a large diamond in AdS, could there be a dual CFT
interpretation of the first law? Proposals exist for holographic CFT duals of
the maximal volume and boundary area in the bulk. The area of the cut-off
boundary surface in Planck units may be dual to the number of degrees of
freedom of the cut-off CFT [83]. The proposed dual quantities for the maximal
volume are computational complexity [84], fidelity susceptibility [85], and
(for variations) the symplectic form on the space of sources in the Euclidean
path integral [86, 87]. One can also consider varying the cosmological
constant, which is equivalent to varying the number of degrees of freedom $N$
in the CFT. The thermodynamic volume is the conjugate to the cosmological
constant, and in the AdS-Rindler case it is dual to the chemical potential for
$N$ in a ball in the CFT [58]. It would be interesting to find a variational
relation in the CFT between these quantities dual to the bulk area, maximal
volume, and thermodynamic volume of the maximal slice.

  * (iii)

Black hole thermodynamics, as well as de Sitter thermodynamics, has been well
studied from the viewpoint of the partition function, formulated as a path
integral over Riemannian geometries beginning with the work of Gibbons and
Hawking [5]. Can the thermodynamics of causal diamonds similarly be formulated
in terms of the Euclidean path integral in the canonical and/or microcanonical
ensemble? If so this should provide a foundation for determining the stability
of different ensembles, and for a systematic treatment of the quantum
corrections.

### Acknowledgements

It is a pleasure to thank Batoul Banihashemi, Pablo Bueno, Juan Maldacena,
Vincent Min, Maulik Parikh, Antony Speranza, and Erik Verlinde for insightful
comments and discussions about this work. TJ was supported in part by NSF
grants PHY-1407744 and PHY-1708139. MV acknowledges support from the Spinoza
Grant of the Dutch Science Organisation (NWO), and from the Delta ITP
consortium, a special program of the NWO. Finally, we thank the organizers of
the workshop “Holography for black holes and cosmology” in Brussels and of the
“Amsterdam String Workshop”, both held in 2016, where this work was initiated.

##  Appendix A Conformal isometry of causal diamonds in (A)dS

We seek the conformal isometry that preserves a causal diamond in de Sitter
space. Since the property of being a conformal isometry is invariant under a
Weyl rescaling of the metric, we will use the form of the line element (2.4),
leaving off the conformal factor $\text{sech}^{2}(r_{*}/L)$. Note that any
vector field of the form

$\zeta=A(u)\partial_{u}+B(v)\partial_{v}$ (A.1)

is then a conformal isometry of the $dudv$ factor of the metric:
$\mathcal{L}_{\zeta}dudv=[A^{\prime}(u)+B^{\prime}(v)]dudv$, where
$\mathcal{L}_{\zeta}$ is the Lie derivative along $\zeta$. It will be a
conformal isometry of the full metric provided
$\mathcal{L}_{\zeta}\sinh^{2}(r_{*}/L)=[A^{\prime}(u)+B^{\prime}(v)]\sinh^{2}(r_{*}/L)$.
Using $r_{*}=(v-u)/2$ we find that in fact
$\mathcal{L}_{\zeta}\sinh^{2}(r_{*}/L)=\frac{1}{L}(B-A)\sinh(r_{*}/L)\cosh(r_{*}/L)$,
so $\zeta$ is a conformal Killing field if

$[A^{\prime}(u)+B^{\prime}(v)]L\tanh(r_{*}/L)=B(v)-A(u)\,.$ (A.2)

For $u=v$, $r_{*}=0$ and thus $\tanh(r_{*}/L)=0$, so this implies $B(v)=A(v)$.
Then evaluating at $v=0$, we have $r_{*}=-u/2$, so (A.2) becomes

$[A^{\prime}(u)+A^{\prime}(0)]L\tanh(u/(2L))=A(u)-A(0)\,.$ (A.3)

The general solution to this equation is

$A(u)=B(u)=a+b\sinh(u/L)+c\cosh(u/L)\,.$ (A.4)

To map the diamond onto itself, the flow of $\zeta$ must leave invariant the
boundaries $u=-R_{*}$ and $v=R_{*}$. This implies $A(\pm R_{*})=0$, and hence
$A(u)=a_{1}(\cosh(u/L)-\cosh(R_{*}/L)).$ The requirement that the surface
gravity $\kappa$ of $\zeta$ be unity at the future conformal Killing horizon
implies $\kappa=-B^{\prime}(R_{*})=-\frac{1}{L}a_{1}\sinh(R_{*}/L)=1$.
Therefore, the conformal Killing vector that preserves a diamond in dS and has
unit surface gravity at the horizon is given by

$\zeta=\frac{L}{\sinh(R_{*}/L)}\Big{[}\left(\cosh(R_{*}/L)-\cosh(u/L)\right)\partial_{u}+\left(\cosh(R_{*}/L)-\cosh(v/L)\right)\partial_{v}\Big{]}\,.$
(A.5)

Expressed in terms of the standard $t$ and $r$ coordinates, $\zeta$ reads

$\zeta=\frac{L^{2}}{R}\left[\Big{(}1-\frac{\sqrt{1-(R/L)^{2}}}{\sqrt{1-(r/L)^{2}}}\cosh(t/L)\Big{)}\partial_{t}-\frac{r}{L}\sqrt{\left(1-(R/L)^{2}\right)\left(1-(r/L)^{2}\right)}\sinh(t/L)\partial_{r}\right].$
(A.6)

A similar expression exists for the conformal Killing vector which preserves a
causal diamond in AdS, which can be obtained by sending $L\rightarrow iL$.
Moreover, in the flat space limit $L\to\infty$, $\zeta$ reduces to the well-
known expressions [17, 21]

$\displaystyle\zeta^{\text{flat}}$
$\displaystyle=\frac{1}{2R}\left[\left(R^{2}-u^{2}\right)\partial_{u}+\left(R^{2}-v^{2}\right)\partial_{v}\right]$
(A.7)
$\displaystyle=\frac{1}{2R}\left[\left(R^{2}-t^{2}-r^{2}\right)\partial_{t}-2rt\partial_{r}\right]\,.$

##  Appendix B Conformal Killing time and mean curvature

In this appendix we show that on slices of constant conformal Killing time $s$
in a maximally symmetric causal diamond, the trace $K$ of the extrinsic
curvature is constant, and we establish the relation (3.50) between $K$ and
$s$. To this end, we first construct a coordinate system adapted to the flow
of $\zeta$, and then we compute $K$ on the constant $s$ slices using this
coordinate system.

Conformal Killing time is a function $s$ satisfying $\zeta\cdot ds=1$, but
there are many such functions. Given one choice of a constant $s$
hypersurface, $s$ is determined by integration along the flow lines of
$\zeta$. We choose the $t=0$ slice of the diamond to be the $s=0$ slice. This
slice is everywhere orthogonal to $\zeta$, hence $\zeta_{a}$ and $\nabla_{a}s$
are parallel on it. The Lie derivative ${\cal L}_{\zeta}\nabla_{a}s$ vanishes
by definition of $s$, and ${\cal L}_{\zeta}\zeta_{a}=2\alpha\zeta_{a}$, since
$\zeta$ is a conformal Killing vector satisfying ${\cal
L}_{\zeta}g_{ab}=2\alpha g_{ab}$. The flow therefore preserves the
proportionality of these two covectors, hence all of the constant $s$ slices
are orthogonal to $\zeta$.

Now choose a spherically symmetric coordinate $x$ on the $s=0$ slice, such
that $|dx|=|ds|$ and $x=0$ at $r=0$, and extend it to the diamond by the flow
of $\zeta$, i.e. so that $\zeta\cdot dx=0$. Both $ds$ and $dx$ are invariant
under the conformal Killing flow, and they are orthogonal and have equal norms
at $s=0$, hence these conditions hold for all values of $s$. The line element
(2.4) therefore takes the form

$ds^{2}=C^{2}(s,x)(-ds^{2}+dx^{2})+r^{2}d\Omega_{d-2}^{2}\,.$ (B.1)

The conformal Killing equation then implies that $r=C\rho$, where
$\rho=\rho(x)$ is a function of $x$ alone, so we have

$ds^{2}=C^{2}(s,x)[-ds^{2}+dx^{2}+\rho^{2}(x)d\Omega_{d-2}^{2}]\,.$ (B.2)

In this coordinate system the future horizon of the diamond is located at
$s=\infty$, and the past horizon is at $s=-\infty$ (see Fig. 2 for an
illustration of constant $s$ and $x$ surfaces). Further, we have
$\zeta=\partial_{s}$, the surface gravity is
$\kappa=-C^{-1}\partial_{s}C\big{|}_{s\to\infty}$, and the unit timelike
vector normal to the constant $s$ slices is $u=C^{-1}\partial_{s}$. The
divergence $\nabla_{a}u^{a}$ is the trace $K$ of the extrinsic curvature of
these slices,

$K=(d-1)C^{-2}\partial_{s}C=(1-d)\partial_{s}C^{-1}\,.$ (B.3)

It remains to show that $\partial_{s}C^{-1}$ is independent of $x$, and to
evaluate it explicitly.414141We note that the mean curvature can also be
expressed as $K=(d-1)\alpha/|\zeta|$, since
$\alpha=\nabla\cdot\zeta/d=C^{-1}\partial_{s}C$ and $|\zeta|=C$.

We proceed by finding the relation between the null coordinates $(u,v)$
defined in (2.2) and the null coordinates $\bar{u}=s-x$ and $\bar{v}=s+x$. The
form of the metric indicates that $\bar{u}=\bar{u}(u)$ and
$\bar{v}=\bar{v}(v)$, and the conditions $\zeta\cdot ds=1$ and $\zeta\cdot
dx=0$ imply $\zeta\cdot d\bar{u}=1=\zeta\cdot d\bar{v}$. Using the expression
(A.5) for $\zeta$ (which is the conformal Killing vector which has unit
surface gravity), we find the coordinate relations

$e^{\bar{u}}=\frac{\sinh[(R_{*}+u)/2L]}{\sinh[(R_{*}-u)/2L]}\,,\quad
e^{u/L}=\frac{\cosh[(R_{*}/L+\bar{u})/2]}{\cosh[(R_{*}/L-\bar{u})/2]}\,,\quad\text{and}\quad(u\rightarrow
v)\,.$ (B.4)

With the help of Mathematica we then find

$C^{-1}=\frac{\cosh
s+\cosh(x)\cosh(R_{*}/L)}{L\sinh(R_{*}/L)}\,,\qquad\rho=\sinh x\,.$ (B.5)

With this equation for $\rho$ the metric between brackets in (B.2) becomes
that of conformal Killing time cross hyperbolic space,
$\mathbb{R}\times\mathbb{H}^{d-1}$.424242The limit $L\rightarrow\infty$,
$R_{*}\rightarrow R$ of (B.4) and (B.5) yields the result for a diamond in
flat spacetime: $\bar{u}=\ln\\!\left[(R+u)/(R-u)\right]$,
$u=R\tanh(\bar{u}/2)$, and $C=R/({\cosh s+\cosh x})$. The limit $R\rightarrow
L$, $R_{*}\rightarrow\infty$ leads to the result for the static patch of de
Sitter spacetime: $\bar{u}=u/L$, and $C=L/\\!\cosh x$. And by replacing $L\to
iL$ and setting $R_{*}=L\pi/2$ we obtain for the Wheeler-DeWitt patch of AdS:
$C=L/\cosh s$, where $L$ is the AdS radius. From this expression for $C^{-1}$
it follows that

$K=\frac{1-d}{L\sinh(R_{*}/L)}\,\sinh s=(d-1)\dot{\alpha}|_{s=0}\,\sinh s\,,$
(B.6)

where we used (2.11) in the last equality. This establishes in particular that
$K$ is constant on the constant $s$ slices.434343A degenerate case is the dS
static patch ($R_{*}\to\infty$), since in that case $K=0$ for all $s$, i.e.
all the CMC slices have zero mean curvature, so there is no York time flow for
the dS static patch. Instead of using (2.11) we could use
$\alpha=\nabla\cdot\zeta/d=-C\partial_{s}C^{-1}$ to write
$\dot{\alpha}=u^{a}\nabla_{a}\alpha=-C^{-1}\partial_{s}(C\partial_{s}C^{-1})$.
Inserting (B.5) then yields $\dot{\alpha}|_{s=0}=-1/{[L\sinh(R_{*}/L)]}$,
which again establishes the fact that $\dot{\alpha}|_{s=0}$ is independent of
$x$.

Finally, as a side comment, let us explain how the conformal Killing flow is
related to the “new York” transformation of [86, 87]. In terms of the induced
metric and the extrinsic curvature of a Cauchy surface the latter
transformations reads: $\delta_{Y}h_{ab}=0$ and
$\delta_{Y}K_{ab}=\tilde{\alpha}\,h_{ab}$. This transformation is not a
diffeomorphism in general, so it is not expected to be the same as the
variation induced by the conformal Killing vector:
$\delta_{\zeta}\phi=\mathcal{L}_{\zeta}\phi$. In our setup the extrinsic
curvature is given by
$K_{ab}:=2\nabla_{(a}u_{b)}=(C^{-2}\partial_{s}C)h_{ab}=\alpha
h_{ab}/|\zeta|$. The Lie derivative of the induced metric
$h_{ab}:=u_{a}u_{b}+g_{ab}$ on $\Sigma$ and of the extrinsic curvature are
given by

$\mathcal{L}_{\zeta}h_{ab}=2\alpha
h_{ab}\,,\qquad\mathcal{L}_{\zeta}K_{ab}=(C^{-2}\partial_{s}^{2}C)h_{ab}=(\dot{\alpha}+\alpha^{2}/|\zeta|)h_{ab}\,.$
(B.7)

The first equation follows from the definition (2.7) of a conformal Killing
vector, and from $\mathcal{L}_{\zeta}u_{a}=\alpha u_{a}$, where
$u_{a}=\zeta_{a}/|\zeta|$. The second equation follows from the expression
above for $K_{ab}$ and from the fact that
$h_{ab}(s,x)=C^{2}(s,x)\sigma_{ab}(x)$, where $\sigma_{ab}(x)$ is the metric
on hyperbolic space $\mathbb{H}^{d-1}$. Since $\alpha=0$ on $\Sigma$, we find
that on the extremal slice of the diamond

$\mathcal{L}_{\zeta}h_{ab}\big{|}_{\Sigma}=0\,,\qquad\mathcal{L}_{\zeta}K_{ab}\big{|}_{\Sigma}=\dot{\alpha}|_{s=0}h_{ab}\,.$
(B.8)

This is of the same form as the new York transformation, if we identify
$\tilde{\alpha}=\dot{\alpha}\big{|}_{s=0}$. In other words, the new York
transformation and the conformal Killing transformation only coincide _at_ the
extremal surface $\Sigma$.

##  Appendix C Zeroth law for bifurcate conformal Killing horizons

A vector field $\zeta^{a}$ is a conformal Killing vector of the metric
$g_{ab}$ if $\mathcal{L}_{\zeta}g_{ab}=2\alpha g_{ab}$ for some function
$\alpha$. A conformal Killing horizon $\mathcal{H}$ is a null hypersurface
whose null generators are orbits of a conformal Killing field. The notion of
surface gravity for Killing horizons can be extended to conformal Killing
horizons [29, 27, 28], however definitions that are equivalent in the former
case (see e.g. Sec. 12.5 in [88]) are not generally equivalent in the latter
case. One of these definitions has two properties not shared by the other
definitions: it is (i) Weyl invariant, and (ii) constant on the horizon. In
this appendix we establish these properties, the second one assuming
$\mathcal{H}$ has a bifurcation surface ${\cal B}$ where $\zeta^{a}$ vanishes.
We also show that $\nabla_{a}\zeta_{b}\overset{\mathcal{B}}{=}\kappa n_{ab}$,
where $\kappa$ is the surface gravity and $n_{ab}$ is the binormal to ${\cal
B}$. In particular, the Killing equation holds at ${\cal B}$.

The conformal Killing horizon is a hypersurface defined by the equation
$\zeta^{2}=0$. The gradient $\nabla_{a}\zeta^{2}$ is normal to this surface;
and, since it is a null surface, the normal is proportional to $\zeta_{a}$, so

$\nabla_{a}\\!\left(\zeta^{b}\zeta_{b}\right)\overset{\mathcal{H}}{=}-2\kappa\zeta_{a}\,$
(C.1)

for some function $\kappa$. Under a Weyl transformation,
$g_{ab}\rightarrow\Omega^{2}g_{ab}$, $\zeta^{a}$ remains a conformal Killing
field, and $\mathcal{H}$ a conformal Killing horizon. Moreover both sides of
(C.1) transform homogeneously, acquiring a factor $\Omega^{2}$. This
definition of the surface gravity $\kappa$ is therefore Weyl invariant [29].

The zeroth law for stationary black holes asserts that the surface gravity is
constant over the entire event horizon. This was originally proven in [2] by
assuming the Einstein field equations and the dominant energy condition for
matter.444444There exists another proof of the zeroth law by Carter [89] and
Rácz and Wald [90] that does not depend on the Einstein equation or an energy
condition. The proof assumes the black hole horizon is a Killing horizon and
the black hole is either (i) static or (ii) stationary-axisymmetric with
“$t-\phi$” reflection isometry. It would be interesting to see whether this
proof can be generalized to conformal Killing horizons. A simpler, and in a
sense more general, proof of the zeroth law for black hole horizons was given
by Kay and Wald in [91].454545Gibbons and Geroch are acknowledged,
respectively, in [91] for this version of the zeroth law and for the proof.
This proof applies to Killing horizons that contain a bifurcation surface
where the Killing vector vanishes — also called _bifurcate Killing horizons_ —
and it holds independently of any gravitational field equation or energy
condition. Here we extend this proof to bifurcate conformal Killing horizons:

Zeroth Law. Let $\mathcal{H}$ be a (connected) conformal Killing horizon with
bifurcation surface $\mathcal{B}$. Then the surface gravity $\kappa$, defined
in (C.1), is constant on $\mathcal{H}$.

Proof: We show first that $\kappa$ is constant along the generators of
$\mathcal{H}$, and then that its value also does not vary from generator to
generator.

That $\kappa$ is constant on each generator can be expected, since the flow of
the conformal Killing vector leaves the metric unchanged up to a Weyl
transformation, and $\kappa$ is Weyl invariant. To make this into a
computational proof, we take the Lie derivative of both sides of (C.1) along
$\zeta$. On the lhs we have

$\mathcal{L}_{\zeta}\nabla_{a}\zeta^{2}=\nabla_{a}{\cal
L}_{\zeta}\zeta^{2}=\nabla_{a}{\cal
L}_{\zeta}(g_{bc}\zeta^{b}\zeta^{c})=\nabla_{a}(2\alpha\zeta^{2})\overset{\mathcal{H}}{=}2\alpha\nabla_{a}\zeta^{2}\overset{\mathcal{H}}{=}-4\alpha\kappa\zeta_{a}.$
(C.2)

On the rhs we have

$\mathcal{L}_{\zeta}(-2\kappa\zeta_{a})=(-2\mathcal{L}_{\zeta}\kappa-4\alpha\kappa)\zeta_{a}\,.$
(C.3)

Since (C.2) and (C.3) must be equal, we conclude that

$\mathcal{L}_{\zeta}\kappa\overset{\mathcal{H}}{=}0,$ (C.4)

i.e. $\kappa$ is constant along the flow of $\zeta^{a}$ and hence along each
null generator of $\mathcal{H}$.

Next, to prove that $\kappa$ does not vary from generator to generator, we
will show that it is constant on the bifurcation surface ${\cal B}$. We begin
by noting that, since $\zeta^{a}$ is a conformal Killing vector, we have

$\nabla_{a}\zeta_{b}=\alpha g_{ab}+\omega_{ab},$ (C.5)

where $\omega_{ab}=-\omega_{ba}$ is an antisymmetric tensor. If $m^{a}$ is
tangent to $\mathcal{B}$, then the contraction of this equation with
$m^{a}m^{b}$ yields zero on the lhs (since $\zeta_{b}=0$ on ${\cal B}$), and
yields $\alpha m^{2}$ on the rhs. It follows that the conformal factor
$\alpha$ vanishes on $\mathcal{B}$, and thus the Killing equation
$\nabla_{(a}\zeta_{b)}\overset{\mathcal{B}}{=}0$ holds there. In addition, the
contraction of the lhs of (C.5) with a single vector $m^{a}$ vanishes on
${\cal B}$, so $m^{a}\omega_{ab}\overset{\mathcal{B}}{=}0$ for all $m^{a}$
tangent to ${\cal B}$, which implies that $\omega_{ab}$ is proportional to the
binormal $n_{ab}$ to ${\cal B}$, i.e.
$\nabla_{a}\zeta_{b}\overset{\mathcal{B}}{=}\beta n_{ab}$ for some function
$\beta$. To evaluate $\beta$, contract both sides of this equation with a null
vector $k^{b}$ that is tangent to ${\cal H}$. On the lhs we have
$k^{b}\nabla_{a}\zeta_{b}$, which according to (C.1) off of ${\cal B}$ is
equal to $-\kappa k_{a}$ (since $k_{a}$ is proportional to $\zeta_{a}$).
Taking the limit as ${\cal B}$ is approached along the horizon thus yields
$k^{b}\nabla_{a}\zeta_{b}\overset{\mathcal{B}}{=}-\kappa k_{a}$. On the rhs we
have $\beta n_{ab}k^{b}=-\beta k_{a}$, so it follows that $\beta=\kappa$, if
the sign of the binormal is chosen so that $n_{ab}k^{b}=-k_{a}$.464646Note
that this means that the sign of $\kappa$ is opposite for the two sheets of a
bifurcate Killing horizon. This establishes the useful relation474747This
relation is also needed to derive the expressions (3.6) and (3.27) for the
Noether charge (see section 7 in [43]).

$\nabla_{a}\zeta_{b}\overset{\mathcal{B}}{=}\kappa\,n_{ab}.$ (C.6)

To demonstrate that $\kappa$ is constant on ${\cal B}$, we act on both sides
of (C.6) with $n^{ab}m^{c}\nabla_{c}$. Since $n^{ab}n_{ab}=-2$ is constant on
${\cal B}$, we obtain

$n^{ab}m^{c}\nabla_{c}\nabla_{a}\zeta_{b}\overset{\mathcal{B}}{=}-2m^{a}\nabla_{a}\kappa.$
(C.7)

Finally, a conformal Killing vector satisfies a generalization of the usual
Killing identity,

$\nabla_{c}\nabla_{a}\zeta_{b}=\zeta^{d}R_{dcab}+g_{ab}\nabla_{c}\alpha+g_{bc}\nabla_{a}\alpha-
g_{ac}\nabla_{b}\alpha\,.$ (C.8)

The contraction of the rhs of this identity with $n^{ab}m^{c}$ vanishes on
${\cal B}$, since $\zeta^{d}$ and $\alpha$ vanish on ${\cal B}$, and
$n^{ab}m_{b}=0$. It follows that the lhs of (C.7) vanishes, which establishes
that $\kappa$ is constant on ${\cal B}$. $\Box$

##  Appendix D Conformal group from two-time embedding formalism

It is well-known that $d$-dimensional Minkowski spacetime can be embedded in
$\mathbb{R}^{2,d}$ as a section of the light cone through the origin,
described by the equation (see e.g. [92])

$X\cdot
X=-\left(X^{-1}\right)^{2}-\left(X^{0}\right)^{2}+\left(X^{1}\right)^{2}+\dots+\left(X^{d}\right)^{2}=0\,.$
(D.1)

Here $X^{A}$ are the standard flat coordinates on $\mathbb{R}^{2,d}$. The
embedding space naturally induces a metric on a hyper-lightcone section, for
example the section $X^{-1}+X^{d}=1$ realizes the standard Minkowski metric on
$\mathbb{R}^{1,d-1}$. In fact, any conformally flat manifold can be embedded
in $\mathbb{R}^{2,d}$ as a section of the light cone, because under the
coordinate transformation $\widetilde{X}^{A}=\Omega(x)X^{A}$ the induced
metric becomes

$d\widetilde{s}^{2}=(Xd\Omega+\Omega dX)^{2}=\Omega^{2}dX\cdot
dX=\Omega^{2}(x)ds^{2}\,,$ (D.2)

where we used the light cone properties $X\cdot dX=0$ and $X\cdot X=0$. This
means that the induced metrics on two different hyper-lightcone sections are
related by a Weyl transformation.

This embedding construction is particularly useful for characterizing the
conformal group $O(2,d)$ of Minkowski space, since it corresponds to the
symmetry group that preserves a light cone in $\mathbb{R}^{2,d}$. Moreover, it
follows from the observation above that the conformal group generators of
_any_ conformally flat spacetime can be obtained from the embedding space. In
this appendix we will use the two-time embedding formalism to derive all
conformal Killing vectors of dS and AdS space – which are both conformally
flat – and, in particular, we will write the conformal Killing vector (A.6) in
terms of embedding coordinates.

###  D.1 Conformal Killing vectors of de Sitter space

The embedding coordinates for the static line element (2.1) of de Sitter space
are

$\displaystyle X^{0}$ $\displaystyle=\sqrt{L^{2}-r^{2}}\sinh(t/L)\,,\qquad
X^{-1}=L\,,$ (D.3) $\displaystyle X^{d}$
$\displaystyle=\sqrt{L^{2}-r^{2}}\cosh(t/L)\,,\qquad
X^{i}=r\,\Omega^{i}\,,\quad i=1,\dots,d-1\,,$

with
$\Omega^{1}=\cos\theta_{1},\Omega^{2}=\sin\theta_{1}\cos\theta_{2},\,...,\,\Omega^{d-1}=\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{d-2}$
and the condition $\sum_{i=1}^{d-1}(\Omega^{i})^{2}=1$. Note that we have
promoted the de Sitter length scale to a coordinate, which is a convenient
trick to obtain the conformal group. If the relation $X^{-1}=L$ is inserted
back into (D.1), then the embedding constraint turns into the equation for a
hyperboloid in $\mathbb{R}^{1,d}$, which is the standard embedding of de
Sitter space.

The conformal group of $d$-dimensional Lorentzian de Sitter space is $O(2,d)$,
whereas the isometry group is $O(1,d)$. Hence, de Sitter space admits
$\frac{1}{2}d(d+1)$ true Killing vectors and $d+1$ conformal Killing vectors
which do not generate isometries (in total there are $\frac{1}{2}(d+1)(d+2)$
conformal generators). The generators of $O(2,d)$ are boosts and rotations in
embedding space, and hence take the form

$J_{AB}=i\left(X_{A}\partial_{X^{B}}-X_{B}\partial_{X^{A}}\right)\,.$ (D.4)

Coordinates with lower indices are defined as $X_{A}=\eta_{AB}X^{B}$, such
that for instance the generator
$J_{01}=-i\left(X^{0}\partial_{X^{1}}+X^{1}\partial_{X^{0}}\right)$ is a
proper boost. The generators satisfy the usual Lorentz algebra commutation
relations

$\left[J_{AB},J_{CD}\right]=i\left(\eta_{AD}J_{BC}+\eta_{BC}J_{AD}-\eta_{AC}J_{BD}-\eta_{BD}J_{AC}\right)\,.$
(D.5)

In embedding space the true Killing vectors correspond to boosts and rotations
in the $X^{0},\dots,X^{d}$ directions, since these generators preserve the
light cone section, whereas the extra $d+1$ conformal Killing vectors are
boosts and rotations in the $X^{-1}$ direction, which do not preserve the
section.

We are now ready to compute the conformal generators explicitly. In terms of
static coordinates the true Killing vectors of de Sitter space are given by

$\displaystyle iJ_{0d}$ $\displaystyle=L\partial_{t}$ (D.6) $\displaystyle
iJ_{0i}$
$\displaystyle=\frac{Lr\Omega_{i}\cosh(t/L)}{\sqrt{L^{2}-r^{2}}}\partial_{t}+\sqrt{L^{2}-r^{2}}\Omega_{i}\sinh(t/L)\partial_{r}+\frac{\sqrt{L^{2}-r^{2}}}{r}\sinh(t/L)\nabla_{i}$
$\displaystyle iJ_{id}$
$\displaystyle=\frac{Lr\Omega_{i}\sinh(t/L)}{\sqrt{L^{2}-r^{2}}}\partial_{t}+\sqrt{L^{2}-r^{2}}\Omega_{i}\cosh(t/L)\partial_{r}+\frac{\sqrt{L^{2}-r^{2}}}{r}\cosh(t/L)\nabla_{i}$
$\displaystyle iJ_{ij}$
$\displaystyle=\Omega_{j}\partial_{\Omega^{i}}-\Omega_{i}\partial_{\Omega^{j}}\,,$

where
$\nabla_{i}=\partial_{\Omega^{i}}-\Omega_{i}\Omega^{j}\partial_{\Omega^{j}}$
is the covariant operator484848 In terms of the angular coordinates
$\theta_{1},\dots,\theta_{d-2}$ the covariant operator on the unit sphere is
given by  $\displaystyle\nabla_{1}$
$\displaystyle=-\sin\theta_{1}\partial_{\theta_{1}},\quad\nabla_{2}=\cos\theta_{1}\cos\theta_{2}\partial_{\theta_{1}}-\frac{\sin\theta_{2}}{\sin\theta_{1}}\partial_{\theta_{2}},\,\,\dots\,,$
$\displaystyle\nabla_{d-2}$
$\displaystyle=\sum_{j=1}^{d-3}\frac{\cos\theta_{j}\sin\theta_{j}\cdots\sin\theta_{d-3}\cos\theta_{d-2}}{\sin\theta_{1}\cdots\sin\theta_{j}}\partial_{\theta_{j}}-\frac{\sin\theta_{d-2}}{\sin\theta_{1}\cdots\sin\theta_{d-3}}\partial_{\theta_{d-2}}\,,$
$\displaystyle\nabla_{d-1}$
$\displaystyle=\sum_{j=1}^{d-3}\frac{\cos\theta_{j}\sin\theta_{j}\cdots\sin\theta_{d-3}\sin\theta_{d-2}}{\sin\theta_{1}\cdots\sin\theta_{j}}\partial_{\theta_{j}}+\frac{\cos\theta_{d-2}}{\sin\theta_{1}\cdots\sin\theta_{d-3}}\partial_{\theta_{d-2}}\,.$
on the unit sphere $S^{d-2}$, and the other $d+1$ conformal generators take
the form

$\displaystyle iJ_{-10}$
$\displaystyle=\frac{L^{2}\cosh(t/L)}{\sqrt{L^{2}-r^{2}}}\partial_{t}+\frac{r}{L}\sqrt{L^{2}-r^{2}}\sinh(t/L)\partial_{r}$
(D.7) $\displaystyle iJ_{-1i}$
$\displaystyle=\frac{L^{2}-r^{2}}{L}\Omega_{i}\partial_{r}+\frac{L}{r}\nabla_{i}$
$\displaystyle iJ_{-1d}$
$\displaystyle=-\frac{L^{2}\sinh(t/L)}{\sqrt{L^{2}-r^{2}}}\partial_{t}-\frac{r}{L}\sqrt{L^{2}-r^{2}}\cosh(t/L)\partial_{r}\,.$

Finally, by comparing expression (A.6) for the conformal Killing vector that
preserves a causal diamond in dS with the previous conformal generators, we
see that $\zeta$ can be written as a linear combination of $J_{0d}$ and
$J_{-10}$

$\zeta^{\text{dS}}=\frac{iL}{R}\left(J_{0d}-\sqrt{1-(R/L)^{2}}J_{-10}\right)\,.$
(D.8)

###  D.2 Conformal Killing vectors of Anti-de Sitter space

Global coordinates for $d$-dimensional Anti-de Sitter space are defined by

$\displaystyle X^{-1}$ $\displaystyle=\sqrt{L^{2}+r^{2}}\cos(t/L)\,,\qquad
X^{d}=L\,,$ (D.9) $\displaystyle X^{0}$
$\displaystyle=\sqrt{L^{2}+r^{2}}\sin(t/L)\,,\qquad
X^{i}=r\,\Omega^{i}\,,\quad i=1,\dots,d-1\,.$

Note that here we have set a space coordinate $X^{d}$, instead of a time
coordinate $X^{-1}$, equal to the curvature scale $L$. In this way the
embedding constraint (D.1) turns into the familiar embedding equation for AdS,
which is that of a hyperboloid in $\mathbb{R}^{2,d-1}$. It is manifest from
this embedding that the isometry group of Lorentzian AdSd is $O(2,d-1)$,
whereas the full conformal group is the same as that of dSd, i.e. $O(2,d)$.

In terms of these coordinates, the induced metric on the light cone reads

$ds^{2}=-[1+(r/L)^{2}]dt^{2}+[1+(r/L)^{2}]^{-1}dr^{2}+r^{2}d\Omega_{d-2}^{2}\,.$
(D.10)

For $r\geq 0$ and $0\leq t<2\pi L$ this solution covers the entire
hyperboloid. However, to avoid closed timelike curves AdS is usually defined
as the universal cover of the hyperboloid. This means that the timelike cycle
is unwrapped, i.e. the range of the coordinate $t$ is extended:
$-\infty<t<\infty$.

The generators of the conformal group take the following form in global AdS
coordinates

$\displaystyle iJ_{-10}$ $\displaystyle=L\partial_{t}$ (D.11) $\displaystyle
iJ_{-1i}$
$\displaystyle=-\frac{Lr\Omega_{i}\sin(t/L)}{\sqrt{L^{2}+r^{2}}}\partial_{t}+\sqrt{L^{2}+r^{2}}\Omega_{i}\cos(t/L)\partial_{r}+\frac{\sqrt{L^{2}+r^{2}}\cos(t/L)}{r}\nabla_{i}$
$\displaystyle iJ_{0i}$
$\displaystyle=\frac{Lr\Omega_{i}\cos(t/L)}{\sqrt{L^{2}+r^{2}}}\partial_{t}+\sqrt{L^{2}+r^{2}}\Omega_{i}\sin(t/L)\partial_{r}+\frac{\sqrt{L^{2}+r^{2}}\sin(t/L)}{r}\nabla_{i}$
$\displaystyle iJ_{ij}$
$\displaystyle=\Omega_{j}\partial_{\Omega^{i}}-\Omega_{i}\partial_{\Omega^{j}}$
$\displaystyle iJ_{-1d}$
$\displaystyle=-\frac{L^{2}\sin(t/L)}{\sqrt{L^{2}+r^{2}}}\partial_{t}-\frac{r}{L}\sqrt{L^{2}+r^{2}}\cos(t/L)\partial_{r}$
$\displaystyle iJ_{0d}$
$\displaystyle=\frac{L^{2}\cos(t/L)}{\sqrt{L^{2}+r^{2}}}\partial_{t}-\frac{r}{L}\sqrt{L^{2}+r^{2}}\sin(t/L)\partial_{r}$
$\displaystyle iJ_{id}$
$\displaystyle=\frac{L^{2}+r^{2}}{L}\Omega_{i}\partial_{r}+\frac{L}{r}\nabla_{i}\,.$

The first four generators are true Killing vector fields, whereas the latter
three are conformal Killing fields. The conformal Killing vector that
preserves a causal diamond centered at the origin $r=0$ of AdS can be obtained
from (A.6) by analytically continuing the curvature scale $L\rightarrow iL$.
One then finds the following expression in terms of the conformal generators

$\zeta^{\text{AdS}}=\frac{iL}{R}\left(\sqrt{1+(R/L)^{2}}J_{0d}-J_{-10}\right)\,.$
(D.12)

Notice that for a causal diamond of infinite size, i.e.
$R/L\rightarrow\infty$, the conformal Killing field $\zeta$ reduces to the
conformal generator $iJ_{0d}$. The $t=0$ time slice of this diamond covers the
entire AdS time slice, and at the center of this diamond $\zeta$ is equal to
the timelike Killing vector $L\partial_{t}$.

Another useful set of coordinates for AdS are the Poincaré coordinates, which
are defined by

$\displaystyle X^{-1}$
$\displaystyle=\frac{1}{2z}\left(L^{2}-t^{2}+\vec{x}^{2}+z^{2}\right),\qquad
X^{1}=\frac{1}{2z}\left(L^{2}+t^{2}-\vec{x}^{2}-z^{2}\right)\,,$ (D.13)
$\displaystyle X^{0}$ $\displaystyle=Lt/z\,,\qquad X^{i}=Lx^{i}/z\,,\qquad
X^{d}=L\,,\qquad i=2,\dots,d-1\,.$

The induced metric in Poincaré coordinates is

$ds^{2}=\frac{L^{2}}{z^{2}}\left(-dt^{2}+d\vec{x}^{2}+dz^{2}\right)\,,$ (D.14)

where the coordinates $t$ and $x^{i}$ range from $-\infty$ to $\infty$. The
coordinate $z$ behaves as a radial coordinate and divides the hyperboloid into
two charts ($z\lessgtr 0$). The Poincaré patch of AdS is typically taken to be
the chart $z>0$, which covers one half of the hyperboloid. The location $z=0$
corresponds to the conformal boundary of AdS.

It is convenient to introduce the following generators of the conformal group

$\displaystyle D$ $\displaystyle=J_{-11}\,,\qquad P_{\mu}=J_{\mu-1}-J_{\mu
1}\,,$ (D.15) $\displaystyle M_{\mu\nu}$
$\displaystyle=J_{\mu\nu}\,,\qquad\,K_{\mu}=J_{\mu-1}+J_{\mu 1}\,,$

where $\mu=0,i,$ or $d$, corresponding to the coordinates $(t,x^{i},z)$
respectively. On the conformal boundary of AdS these generators turn into the
standard conformal generators of flat space (where $D$ denotes the generator
of dilatations, $P_{\mu}$ of translations, $M_{\mu\nu}$ of Lorentz
transformations, and $K_{\mu}$ of special conformal transformations). In
Poincaré coordinates the conformal generators are equal to

$\displaystyle iD$
$\displaystyle=-\left(t\partial_{t}+x^{i}\partial_{i}+z\partial_{z}\right)\,,$
(D.16) $\displaystyle iP_{t}$
$\displaystyle=-L\partial_{t}\,,\quad\,\,iP_{i}=-L\partial_{i}\,,\quad\,\,iP_{z}=-L\partial_{z}\,,$
$\displaystyle iK_{t}$
$\displaystyle=\frac{-1}{L}\left[(t^{2}+\vec{x}^{2}+z^{2})\partial_{t}+2t(x^{i}\partial_{i}+z\partial_{z})\right]\,,$
$\displaystyle iK_{i}$
$\displaystyle=\frac{1}{L}\left[(t^{2}-\vec{x}^{2}-z^{2})\partial_{i}+2x_{i}(t\partial_{t}+x^{j}\partial_{j}+z\partial_{z})\right]\,,$
$\displaystyle iK_{z}$
$\displaystyle=\frac{1}{L}\left[(t^{2}-\vec{x}^{2}+z^{2})\partial_{z}+2z(t\partial_{t}+x^{i}\partial_{i})\right]\,,$
$\displaystyle iM_{ti}$
$\displaystyle=x_{i}\partial_{t}+t\partial_{i}\,,\qquad
iM_{iz}=z\partial_{x}-x\partial_{z}\,,$ $\displaystyle iM_{tz}$
$\displaystyle=z\partial_{t}+t\partial_{z}\,,\qquad
iM_{ij}=x_{j}\partial_{i}-x_{i}\partial_{j}\,.$

The generators $P_{z},K_{z},M_{tz}$ and $M_{iz}$ are conformal Killing
vectors, and the other generators are true Killing vectors. With this list of
conformal generators at hand, we are able to write the conformal Killing field
(D.12) that preserves a diamond in AdS in terms of Poincaré coordinates

$\displaystyle\zeta^{\text{AdS}}$
$\displaystyle=\frac{iL}{R}\left[\sqrt{1+(R/L)^{2}}J_{0d}+J_{0-1}\right]=\frac{iL}{R}\left[\sqrt{1+(R/L)^{2}}M_{tz}+\frac{1}{2}\left(P_{t}+K_{t}\right)\right]$
(D.17)
$\displaystyle=\frac{1}{2R}\left[\sqrt{L^{2}+R^{2}}(2z\partial_{t}+2t\partial_{z})-(L^{2}+t^{2}+\vec{x}^{2}+z^{2})\partial_{t}-2tx^{i}\partial_{i}-2tz\partial_{z}\right]\,.$

This conformal Killing vector field generates a modular flow inside a causal
diamond $\mathcal{D}$ in the Poincaré patch of AdS. The center of the diamond
$\mathcal{D}$ is located at $\\{t=0,z=L,x^{i}=0\\}$, and its boundaries are
given by

$\displaystyle\text{past and future vertices:}\quad p,p^{\prime}=\left\\{t=\pm
R,\,z=\sqrt{L^{2}+R^{2}},\,x^{i}=0\right\\}\,,$
$\displaystyle\text{edge/bifurcation
surface:}\,\,\,\,\,\mathcal{B}=\left\\{t=0,\,z=\sqrt{L^{2}+R^{2}}\pm\sqrt{R^{2}-\vec{x}^{2}}\right\\}\,,$
$\displaystyle\text{past null
boundary:}\qquad\,\,\,\,\,\mathcal{H}_{\text{past}}=\left\\{z=\sqrt{L^{2}+R^{2}}\pm\sqrt{(R+t)^{2}-\vec{x}^{2}}\right\\}\,,$
$\displaystyle\text{future null
boundary:}\qquad\mathcal{H}_{\text{future}}=\left\\{z=\sqrt{L^{2}+R^{2}}\pm\sqrt{(R-t)^{2}-\vec{x}^{2}}\right\\}\,.$

One can readily check that the conformal Killing vector (D.17) vanishes at
$p,p^{\prime}$ and $\mathcal{B}$, and acts as a null flow on
$\mathcal{H}_{\text{past/future}}$. Hence this causal diamond is entirely
contained within the $z>0$ Poincaré patch.

One can, of course, shift the causal diamond such that its center is located
at a different position. An interesting special case is the diamond which is
centered at the boundary of AdS, i.e. $z=0$, and whose past and future null
boundaries coincide with the Killing horizon of AdS-Rindler space. This can be
established by sending $z\rightarrow z+\sqrt{L^{2}+R^{2}}$ in (D.17). The
conformal Killing vector then turns into the boost Killing vector of AdS-
Rindler space, given in [17]

$\displaystyle\xi^{\text{AdS-Rindler}}$
$\displaystyle=\frac{iL}{2R}\left[\left(1-(R/L)^{2}\right)J_{0-1}+\left(1+(R/L)^{2}\right)J_{01}\right]=\frac{iL}{2R}\left[K_{t}-(R/L)^{2}P_{t}\right]$
(D.18)
$\displaystyle=\frac{1}{2R}\left[\left(R^{2}-t^{2}-\vec{x}^{2}-z^{2}\right)\partial_{t}-2tx^{i}\partial_{i}-2tz\partial_{z}\right]\,.$

Note that the normalization of the boost Killing vector is different than in
[17], since we have set the surface gravity of the Killing horizon equal to
one. The center of this diamond is located at $\\{t=0,z=0,x^{i}=0\\}$, and the
past and future vertices are at $\left\\{t=\pm R,z=0,x^{i}=0\right\\}$.
Moreover, the bifurcation surface corresponds to the hemisphere
$\mathcal{B}=\left\\{t=0,\,z^{2}+\vec{x}^{2}=R^{2}\right\\}$. There exist two
solutions to this quadratic equation, i.e. $z=\pm\sqrt{R^{2}-\vec{x}^{2}}$, so
there are in fact two equal-sized Rindler wedges in AdS that are preserved by
this boost Killing vector (see Fig. 6(a)).

##  Appendix E Conformal transformation from Rindler space to causal diamond

It is well known in the literature [22, 23, 12] that there exists a special
conformal transformation that maps Rindler space to a causal diamond in flat
space. In this appendix we describe this transformation and derive the
conformal Killing vector that preserves the diamond from the boost Killing
vector of Rindler space. In particular, we focus on the conformal
transformation from the right Rindler wedge to the diamond whose left edge is
fixed at the bifurcation surface of the Rindler wedge. (See Fig. 4(b) for a
depiction of this diamond inside the right Rindler wedge.) The special
conformal transformation and its inverse are given by

$x^{\mu}=\frac{X^{\mu}-X\cdot XC^{\mu}}{1-2X\cdot C+X\cdot XC\cdot C}\,,\qquad
X^{\mu}=\frac{x^{\mu}+x\cdot x\,C^{\mu}}{1+2\,x\cdot C+x\cdot x\,C\cdot C}\,,$
(E.1)

where $C^{\mu}=(0,-1/(2R),0,0)$, and $x^{\mu}$ and $X^{\mu}$ are Cartesian
coordinates in Minkowski space. This is just a shift of the map mentioned in
[12] (and we included a minus sign in $C^{\mu}$, correcting a typo in their
paper). The transformation $X^{\mu}\to x^{\mu}$ maps the Minkowski metric to a
metric on the diamond given by
$ds^{2}=\Omega^{2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}=\eta_{\mu\nu}dX^{\mu}dX^{\nu},$
where the conformal factor is given by

$\Omega=1-2X\cdot C+X\cdot XC\cdot C=\left[1+2x\cdot C+x\cdot x\,C\cdot
C\right]^{-1}.$ (E.2)

Note that the conformal transformation maps spatial infinity
$i^{0}=(T=0,X^{1}=\infty)$ to the right edge of the diamond $(t=0,x^{1}=2R)$,
and the origins are also mapped to each other.

Further, with the help of Mathematica, we find that under the mapping (E.1)
the boost Killing field

$\xi=X^{1}\partial_{T}+T\partial_{X^{1}}$ (E.3)

becomes the conformal Killing field that preserves the diamond

$\zeta=\frac{1}{2R}\left[\left(R^{2}-(x^{1}-R)^{2}-(x^{i})^{2}-t^{2}\right)\partial_{t}-2t(x^{1}-R)\partial_{x^{1}}-2tx^{i}\partial_{i}\right],$
(E.4)

where $i=2,\dots,d-1$. By replacing $x^{1}$ by $x^{1}+R$ one obtains the
standard conformal Killing vector (A.7) that preserves a diamond centered at
the origin of flat space. At $T=t=0$ the norm of the two vector fields (E.3)
and (E.4) is

$|\xi|\overset{(T=0)}{=}X^{1}\,,\qquad|\zeta|\overset{(t=0)}{=}\frac{X^{1}}{\Omega}=x^{1}-\frac{\vec{x}^{2}}{2R}\,.$
(E.5)

The former blows up at spatial infinity $i^{0}$, whereas the latter vanishes
at the edge of the diamond $(x^{1}-R)^{2}+(x^{i})^{2}=R^{2}$. This is because,
although both $X^{1}$ and $\Omega$ diverge at spatial infinity, their ratio
vanishes in this limit.

##  Appendix F Acceleration of the conformal Killing flow

In this appendix we show that the orbits of the conformal Killing field that
preserves a maximally symmetric causal diamond have uniform acceleration
inside the diamond. In the $(s,x)$ coordinate system, defined in Appendix B,
the conformal Killing vector that preserves the diamond is
$\zeta=\partial_{s}$. The flow lines of $\zeta$ coincide at the past and
future tip of the diamond, but never cross inside the diamond (see Fig. 2).
The velocity vector of the conformal Killing flow is
$u^{a}=\zeta^{a}/\sqrt{-\zeta\cdot\zeta}={\delta_{s}}^{a}C^{-1}$, and the
acceleration vector is defined as
$a^{b}=u^{a}\nabla_{a}u^{b}={\delta_{x}}^{b}C^{-3}\partial_{x}C$. Using the
expression (B.5) for $C=C(s,x)$, we find that the proper acceleration
$a:=\sqrt{a^{b}a_{b}}$ is equal to

$a(x)=C^{-2}\partial_{x}C=R^{-1}\sinh x\,.$ (F.1)

Note that the proper acceleration depends only on $x$, which is constant on
the conformal Killing orbits.494949We were unable to anticipate this
surprising fact. The central orbit at $x=0$ is unaccelerated, and at the edge
where $x\to\infty$ the acceleration diverges.

Finally, we note that the surface gravity associated to $\zeta$ is equal to
the redshifted proper acceleration evaluated at the bifurcation surface
$\mathcal{B}$, i.e.

$\kappa=\lim_{x\to\infty}Ca=1\,.$ (F.2)

This formula for the surface gravity is known to be equivalent to other
definitions of surface gravity for Killing horizons (see e.g. [88] p. 332).
For bifurcate conformal Killing horizons, following the same steps as [88],
the acceleration definition (F.2) can be shown to be equivalent to the
definition (C.1), but only in the limit as the bifurcation surface is
approached, since the conformal Killing field is instantaneously a true
Killing field at $\mathcal{B}$ (see Appendix C).

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