Path Integral Factorization and the Gravitational Effective Action

Patrick<EMAIL_ADDRESS>Szilard
<EMAIL_ADDRESS>and Manthos<EMAIL_ADDRESS>

(a)Illinois Center for Advanced Studies of the Universe &

Department of Physics, University of Illinois, Urbana, IL 61801

We discuss the factorization and continuity properties of fields in the
Euclidean gravitational path integral with higher dimension operators
constructed from powers of the Riemann tensor. We construct the boundary terms
corresponding to the microcanonical ensemble and show that the saddle point
approximation to the path integral with a quasilocal energy constraint
generally yields a saddle point with discontinuous temperature. This extends a
previous result for the Euclidean Schwarzschild-de Sitter geometry in Einstein
gravity and shows that it is robust against at least some types of quantum
corrections from heavy fields. As an application, we compute the entropy of
SdS in $\text{D}=4$ using the BTZ method. Our result matches the entropy
calculated using Wald’s formula.

###### Contents

  1. 1 Introduction
  2. 2 Conventions
  3. 3 Factorization Methodology
  4. 4 Canonical action of $f$(Riemann) gravity
    1. 4.1 Local Symmetries
  5. 5 $f$(Riemann) Path Integral Factorization
    1. 5.1 Purely bulk form of $I_{\text{ADM}}$
    2. 5.2 Microcanonical ensemble for $f$(Riemann)
  6. 6 Constrained states in EFT
  7. 7 Conclusions
  8. A Reduction to General Relativity
  9. B  Derivation of purely bulk form of $I_{\text{ADM}}$
  10. C SdS entropy/microcanonical data

##  1 Introduction

A desirable property of functional integrals is “factorizability”: it should
be possible to split the path integral defined on a spacetime manifold ${\cal
M}$ into integrals on subregions of ${\cal M}$. For factorization to hold,
generally the fields need to respect some continuity properties, so that the
classical action functional over subregions is suitably additive.

As a result of these continuity conditions, the factorized form of the path
integral on ${\cal M}$ is not usually a simple product of subregion path
integrals, but rather it is given by the integral of such a product over field
data defined on the codimension-1 boundaries between the subregions. Loosely
we may think of this decomposition analogously to the insertion of a complete
set of states in canonical quantization. Restricting the lower-dimensional
path integral so that some data on the interface are fixed corresponds to the
insertion of a constraint.

The continuity properties of the fields across a factorization surface follow
from the form of the classical action. Consequently there is a close
relationship between the factorization properties of the functional integral
and the variational problem associated with the semiclassical approximation.
Fields that are fixed on the boundary in the classical variational problem are
generally required to be continuous across a factorization surface, and thus
appear in the lower-dimensional integral over the interface data. Fields that
are not included in the lower-dimensional integral over the interface data are
not generally continuous. This can have interesting consequences for the
semiclassical approximation to factorized path integrals, for fixed
factorization surface data or equivalently in the presence of a constraint:
the fields at the stationary point may not be continuous.

An example of this phenomenon arises in the Euclidean path integral in
4-dimensional pure Einstein gravity with positive cosmological constant. In
the absence of a constraint, there is a completely smooth solution given by
the 4-sphere. However, imposing a spherically symmetric constraint on the
Brown-York (BY) quasilocal energy [1], the relevant solution is given by the
Euclidean Schwarzschild de-Sitter (SdS) metric [2, 3]. The black hole and
cosmological “horizons” can both be made smooth by a suitable choice of
Euclidean time periodicity on either side of the surface where the constraint
is imposed, but the choices are different, so the metric is not continuous
across the surface. The discontinuity may be encoded as a jump in the lapse
function in static slicing. The lapse is not part of the field data that needs
to be provided in the variational problem with fixed BY energy, and it is not
integrated over in the factorized form of the path integral, so it is
perfectly acceptable for it to appear with a discontinuity in the
semiclassical solution. At the leading semiclassical order, at least, the on-
shell action is independent of the radius at which the constraint is imposed
[2]. This resolution of the semiclassical interpretation of SdS also provides
support to the idea that localized matter should correspond to a constrained
state of the underlying quantum theory of de Sitter space [4].111See also [5]
and refs therein.

One should ask whether discontinuous saddle point geometries obtained in
Einstein gravity still make sense beyond the leading semiclassical order. The
approach to this question taken here will be to determine whether the
factorization properties of more general gravitational Wilsonian effective
actions are, in a suitable sense, consistent with the factorization properties
of Einstein gravity. Higher-dimension operators can account for fluctuations
of short-wavelength degrees of freedom, and so provide a test of whether the
continuity conditions inferred from the Einstein-Hilbert action continue to
hold beyond the leading order.

At a technical level, we work with “$f$(Riemann)” gravity, where the action
density is an arbitrary function of the Riemann tensor, in an auxiliary field
formulation [6] where the bulk Lagrangian is second order in derivatives, and
we consider “canonical ensemble” and “microcanonical ensemble” versions, where
boundary terms are chosen so that the semiclassical variational problem
corresponds to fixing the induced boundary metric or fixing the boundary
energy and momentum fluxes, respectively. The choice of $f$(Riemann) gravity
could be viewed as one of technical convenience, since we can borrow the
auxiliary field formulation of [6]. It is not a completely general effective
action, as it omits some terms of higher order in covariant derivatives as
well as parity odd operators. However, up to dimension four parity-even
operators it is general, and in pure gravity through dimension six it is
general up to field redefinitions and total derivatives [7].222The dimension
four terms $R^{2}$ and $R_{ab}R^{ab}$ may be removed by a field redefinition
$g_{ab}\rightarrow g_{ab}+bR_{ab}+c\,Rg_{ab}$, modulo a shift in $1/G_{N}$
proportional to the cosmological constant. The third independent dimension
four term may be taken to be the Gauss-Bonnet term, a total derivative. The
dimension six terms may be categorized as of $f$(Riemann) type, total
derivatives, and terms proportional to the vacuum Einstein equation. The
latter can again be removed by field redefinitions, up to total derivatives
and terms of higher dimension. To our knowledge there has been limited
investigation in the literature of how total derivatives, generated either by
integrating out massive fields or by field redefinitions, modify the
variational problem needed to implement the semiclassical approximation in
effective field theories with boundaries (see, however, [8] and refs therein.)
We will not take up this problem here, and in any case it is not important for
path integral factorization, where any consistent set of boundary conditions
can be considered on a factorization surface. (This is analogous to how we may
insert either a complete set of position states or a complete set of momentum
states in any expectation value in ordinary quantum mechanics.) Instead, we
rewrite the general effective Lagrangian of $f$(Riemann) type in the auxiliary
field formulation of [6], where it is particularly transparent how to
implement a consistent Dirichlet-type, or “canonical ensemble,” boundary
condition. The auxiliary field formulation has been useful for defining a
Brown-York quasi-local stress tensor for curvature squared Lagrangian [9],
with sensible results. The auxiliary field formulation is also essential to
pass to a Hamiltonian formulation, and so cleanly facilitates the construction
of a second set of consistent “microcanonical ensemble” boundary conditions.
In any event, the class of theories we examine is illustrative of a general
procedure which can be applied to other effective actions; for example, higher
derivatives may be accommodated by expanding the set of auxiliary fields.

Although we do not know the precise generalization of the Euclidean SdS
solution in an arbitrary $f$(Riemann) theory, the presence of two Euclidean
horizons at different temperatures implies that a lapse discontinuity is
unavoidable. To that end, one of the main results we will obtain is that the
microcanonical boundary condition (and associated microcanonical factorization
and constraints) can be naturally generalized to theories with higher
dimension operators, in such a way that the lapse function is still not part
of the boundary data in boundary-value problems or part of the continuous
field data in factorization problems. Thus we may infer that a similar “saddle
point of a constrained path integral” interpretation will apply to whatever
geometry corrects Euclidean SdS in the semiclassical treatment of more general
gravitational effective actions, at least ones that may be cast by field
redefinitions to the $f$(Riemann) class. In the simplest case of Einstein
gravity plus curvature-squared corrections, SdS remains an exact solution, and
we show that the Wald entropy [10, 11] matches the thermodynamic entropy
computed using the BTZ method which is well-adapted to the Hamiltonian
formalism [12].

This work is organized as follows. In Section 2 we set up the notation and
conventions for this paper. In Sec. 3 we give an overview of our approach to
path integral factorization. For a given action we explain the importance of
the set ${\cal B}_{i}$, which is the set of continuous fields required for the
action to satisfy the additivity property. This is most naturally identified
in the purely bulk form of the action. We also point out the close connection
between the set ${\cal B}_{i}$ and the boundary conditions needed for a well-
posed variational problem. In Sec. 4 we review the derivation of the canonical
action $I_{\text{ADM}}$ for $f$(Riemann) effective Lagrangians, the path
integral factorization of which we study in Sec. 5. We define the canonical
path integral in Eq. (39) and in Appendix A we exhibit the reduction of the
canonical path integral to that of General Relativity for a specific function
$f$. We end Sec. 4 with remarks regarding local symmetries in the canonical
form of the action. Section 5 is split into subsections 5.1 and 5.2 focused on
the canonical and microcanonical ensemble respectively. In subsection 5.1 we
find the set ${\cal B}_{i}$ for the canonical ensemble by expressing the
action in purely bulk form. Details of the derivation as well as the full form
of the purely bulk form are given in Appendix B. The purely bulk form of the
canonical action facilitates defining the microcanonical ensemble, following
[13], in subsection 5.2. We also identify the set ${\cal B}_{i}$ for the
microcanonical factorization, which, among other fields, contains the surface
energy density $\epsilon_{\text{BY}}$ and surface momentum density $J_{a}$,
for which explicit expressions are found in Eq. (51). In Section 6 we turn to
applications. We show that SdS in 4 dimensions is a constrained state for the
general parity-even effective action with operators up to dimension 4. We
calculate the constraints and show that SdS satisfies the equation of motion
on either side of the constrained surface. We also compute the entropy from
the Euclidean on-shell action and verify that it matches Wald’s formula. The
details of these computations are presented in Appendix C. Finally, we end
with some discussion and outlook in Sec. 7.

##  2 Conventions

We consider ambient D-dimensional spacetime ${\cal M}$ with metric $g_{MN}$,
cf. Fig. 1. The time coordinate is labeled by $\tau$ and is timelike/spacelike
for $\sigma=\mp 1$. All our expressions up to Section 6 will hold for general
signature, except the explicit path integral expressions in Eqs (10), (39),
and (56); it is straightforward to modify them to cover Lorentzian signature.
In Section 6 we will work with Euclidean signature only, i.e. $\sigma=+1$.
Capital Greek letters $\\{M,N,K,\Lambda,\dots\\}$ label the coordinates
$x^{M}$ of ${\cal M}$ and $\nabla_{M}$ is the metric-compatible covariant
derivative on ${\cal M}$.

$\Sigma_{\tau}$ denotes codimension-1 hypersurfaces of constant time $\tau$
with induced metric $h_{\mu\nu}$. Lowercase Greek letters
$\\{\mu,\nu,\kappa,\lambda,\dots\\}$ (except $\rho$) label coordinates
$x^{\mu}$ of $\Sigma_{\tau}$, and $D_{\mu}$ denotes the induced covariant
derivative on $\Sigma_{\tau}$, defined as

$D_{K}T\indices{{}^{M_{1}\cdot\cdot M_{p}}_{N_{1}\cdot\cdot
N_{q}}}:=\gamma\indices{{}^{M_{1}}_{M^{\prime}_{1}}}\cdot\cdot\gamma\indices{{}^{M_{p}}_{M^{\prime}_{p}}}\gamma\indices{{}^{N^{\prime}_{1}}_{N_{1}}}\cdot\cdot\gamma\indices{{}^{N^{\prime}_{q}}_{N_{q}}}\gamma\indices{{}^{\Sigma}_{K}}\nabla_{\Sigma}\left(\gamma\indices{{}^{M^{\prime}_{1}}_{M^{\prime\prime}_{1}}}\cdot\cdot\gamma\indices{{}^{N^{\prime\prime}_{1}}_{N^{\prime}_{1}}}\cdot\cdot
T\indices{{}^{M^{\prime\prime}_{1}\cdot\cdot
M^{\prime\prime}_{p}}_{N^{\prime\prime}_{1}\cdot\cdot
N^{\prime}_{q}}}\right),$ (1)

where $\gamma_{MN}=g_{MN}-\sigma n_{M}n_{N}$ is the projector on
$\Sigma_{\tau}$.

${\cal T}$ denotes the codimension-1 factorization surface. It is defined as a
surface of constant spatial coordinate $\rho(x^{\mu})$. Throughout this work
we assume ${\cal T}$ intersects ${\Sigma_{\tau}}$ normally. When the
factorization surface is fixed to a particular $\rho=r$ we denote it as ${\cal
T}_{r}$.

${\cal S}_{\tau}=\Sigma_{\tau}\cap{\cal T}$ denotes codimension-2 surfaces
defined as the intersection of $\Sigma_{\tau}$ and ${\cal T}$ with induced
metric $s_{ab}$. Lowercase Latin letters $\\{a,b,c,d,\dots\\}$ label
coordinates $x^{a}$ of ${\cal S}_{\tau}$ and ${{}^{(2)}}D_{\mu}$ denotes the
codimension-2 induced covariant derivative on ${\cal S}_{\tau}$. In our
analysis we will assume ${{\cal S}_{\tau}}$ surfaces are compact and thus drop
boundary terms of the form $\int d^{D-2}x\sqrt{s}\,{{}^{(2)}D_{a}}T^{a}$. For
SdS in $\text{D}=4$ the surfaces ${\cal S}_{\tau}$ are 2-spheres.

Figure 1: The manifold ${\cal M}$ with boundary $\partial{\cal
M}=\Sigma_{\tau_{1}}\cup{\cal B}\cup\Sigma_{\tau_{2}}$. The unit normal
$\underline{n}$ is future pointing. The unit normal $\underline{r}$ at
$\partial{\cal M}$ is defined as outwards pointing. We assume the
$\Sigma_{\tau}$ surfaces intersect ${\cal B}$ normally, i.e.
$g_{MN}n^{M}r^{N}=0$. The boundary $\cal B$ does not have to be connected, but
is drawn that way for simplicity.

For the divergence theorem in pseudo-Riemannian manifolds we assume a volume
${\cal M}$ with boundary $\partial{\cal M}={\cup_{i}}\Sigma_{i}$ comprised
from surfaces $\Sigma_{i}$ that are solely spacelike/timelike. We define the
outward unit normal $r_{M}$ to the boundary $\Sigma_{i}$ with
$r_{M}r^{M}=\epsilon_{i}$ and projector
$r_{MN}=g_{MN}-\epsilon_{i}r_{M}r_{N}$. The divergence theorem is then

$\int_{{\cal M}}d^{D}x\sqrt{|g|}\nabla_{M}A^{M}=\int_{\partial{{\cal
M}}}d^{D-1}y\,\epsilon_{i}\sqrt{|r|}r_{M}A^{M}\,,$ (2)

where $|r|$ is the determinant of the induced metric on $\partial{{\cal M}}$.
Note that in the case where $\sigma=-1$, the surfaces
$\Sigma_{1,2}:=\Sigma_{\tau_{1},\tau_{2}}$ of Figure 1 are spacelike with
$\epsilon_{1,2}=-1$. For the timelike boundary $\Sigma_{3}:={\cal B}$ we have
$\epsilon_{3}=+1$. We will assume ${\cal B}$ and $\Sigma_{\tau}$ intersect
normally. We will also drop corner terms, namely boundary total derivatives.

##  3 Factorization Methodology

In this section we review some aspects of the factorization properties of
gravitational path integrals at leading order in the semiclassical expansion.
Restricting to the leading order mainly allows us to be relaxed regarding
local functional measures. (For subleading semiclassics there are interesting
and subtle issues and ambiguities regarding the path integral measure that
will not be addressed here.)

On a manifold ${\cal M}$ of fixed topology the canonical gravitational action
takes the form

$I_{\text{ADM}}({\cal
M})=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left({\cal
P}_{i}\partial_{\tau}{\cal Q}^{i}\right)-H\left({\cal Q}^{i},{\cal
P}_{i},\lambda_{m}\right)\,,$ (3)

where $\lambda_{m}$ are fields that act as Lagrange multipliers. The
corresponding path integral is

$Z=\int_{{\cal M}}{{\cal D}{\cal Q}^{i}}\,{\cal D}{\cal P}_{i}\,{\cal
D}{\lambda}_{m}\,e^{-I_{\text{ADM}}({\cal M})}\,.$ (4)

Suppose that $\cal M$ is partitioned into two regions ${\cal M}_{1}$, ${\cal
M}_{2}$ separated by a surface ${\cal T}$ with ${\cal M}={\cal M}_{1}\cup{\cal
M}_{2}$. Then it is natural to try to write a factorized version of the path
integral,

$Z=\int_{{\cal T}}D{\cal B}_{i}\,Z({\cal M}_{1}|{\cal B}_{i})Z({\cal
M}_{2}|{\cal B}_{i})\,.$ (5)

Here the integrand is written as a product of two path integrals over ${\cal
M}_{1,2}$ with fixed boundary data ${\cal B}_{i}$ at the interface ${\cal T}$.
In the subscript of the integral in (5), ${\cal T}$ is a mnemonic indicating
that the integration variables are fields defined on the boundary ${\cal T}$.
The question is which field variables on ${\cal T}$ are included in ${\cal
B}_{i}$?

To answer this question, it is convenient to write the action in a “purely
bulk” form, where any boundary terms (like the GHY term, or the Hamiltonian
boundary terms) are rewritten as bulk total derivatives. Suppose that the
action has been written in such a form. Then Eq. (5) can hold if the path
integrals are over fields satisfying the additivity property

$I_{\text{ADM}}({\cal M})=I^{(1)}_{\text{ADM}}({\cal
M}_{1})+I^{(2)}_{\text{ADM}}({\cal M}_{2})\,.$ (6)

The fields ${\cal B}_{i}$ on ${\cal T}$ are given by the restriction to ${\cal
T}$ of all fields on ${\cal M}$ whose continuity across ${\cal T}$ is required
by the additivity property (6). (We use “fields” as a general term that
encompasses fundamental fields, their derivatives and linear combinations,
etc.) In the purely bulk form of the action, additivity will hold if there are
no Dirac delta functions in the action density at ${\cal T}$. Products of
discontinuous functions are permissable and such configurations can play an
important physical role.333The simplicity of these conditions motivate but do
not require the use of the purely bulk form of the action. If we prefer a form
of the action with explicit boundary terms, then the right-hand side of (6)
includes these terms on either side of ${\cal T}$. In this case, the
continuity properties required for factorization permit integrable Dirac delta
function singularities on ${\cal T}$ in the action density on the left-hand
side of (6), so long as they are captured by the boundary terms on the right-
hand side of (6).

Let us consider a concrete example. We adopt a coordinate system where ${\cal
T}={\cal T}_{r}$ lies at a fixed “radial” coordinate $\rho=r$. (Here and
elsewhere “radial” will be a general term referring to some spatial
coordinate, a level surface of which specifies a factorization surface.)
Suppose that in a purely bulk form of the action all the terms with radial
derivatives have the following form

$I({{\cal M}})=\int_{{\cal M}}A\partial_{\rho}B\,,$ (7)

where $A,B$ are fields whose exact identity is not needed for now. It is clear
that the additivity property holds as long as $B$ is continuous in $\rho$ at
${\cal T}_{r}$. The field $A$ can be discontinuous in $\rho$. Thus the field
$B$ belongs to the set ${\cal B}_{i}$ mentioned before and must be integrated
on ${\cal T}_{r}$ in the factorized form of the path integral (5), while the
field $A$ is integrated over separately in each path integral appearing in the
integrand of (5). A standard example of this type is a canonical path integral
in quantum mechanics, where $\rho$ is a time coordinate, $B$ is a particle
coordinate, and $A$ is its conjugate momentum.

There is also a close relationship between the continuity properties required
for the factorization in (5) and the boundary conditions required for a well-
defined variational problem in each of the subregions ${\cal M}_{1,2}$. The
functional derivatives of the actions $I^{(1,2)}({\cal M}_{1,2})$ are well
defined for variations in which Dirichlet conditions are imposed on the ${\cal
B}_{i}$. If we restrict attention to fields for which the additivity property
holds, then we can implement the saddle point approximation on each path
integral $Z({\cal M}_{1,2}|{\cal B}_{i})$ separately, with consistent boundary
data on ${\cal T}_{r}$.

If the variational problem is well-defined with Dirichlet conditions on the
${\cal B}_{i}$, then we can also apply the saddle approximation to a
constrained path integral, namely a path integral where the values of the
fields ${\cal B}_{i}$ are held fixed on ${\cal T}_{r}$ and _not_ integrated
over. This is similar to inserting a projection into a matrix element, or a
delta functional into a path integral. The Euclidean SdS solution is precisely
a saddle point solution of such constrained path integral when the
$I_{\text{ADM}}$ is the Einstein-Hilbert (EH) action with microcanonical
boundary conditions [2].

In the next sections, we will determine the continuity properties of fields
required for factorization of the gravitational path integral, with action
given by a general functional of the Riemann tensor, and with microcanonical
boundary terms. We focus on the phase space path integral which is a natural
formulation for discussing the microcanonical ensemble. As an example
application of our results, in Sec. 6 we will show that Euclidean SdS is again
a saddle point solution of $\text{D}=4$ $f$(Riemann) gravity theories with
microcanonical boundary conditions and operators up to dimension 4, and the
on-shell action computes the leading EFT corrections to the entropy.

##  4 Canonical action of $f$(Riemann) gravity

In this section we review the derivation of the canonical action
$I_{\text{ADM}}$ for $f$(Riemann) theories of gravity, originally presented in
[6]. The main result is given in Eq. (36), which is input to the canonical
path integral factorization analysis in Section 5. In this section we keep
track both Euclidean/Lorentzian $\sigma=\pm 1$ signature of the time
coordinate.

$f$(Riemann) gravity theories have action of the following form:444For now we
ignore boundary terms in the action. We will introduce them in Eq. (10) after
we have defined the auxiliary fields needed for the Hamiltonian description.

$S[g_{MN}]=\frac{1}{2}\int_{{\cal M}}d^{D}x\sqrt{|g|}f({\cal
R}\indices{{}^{MN}_{K\Lambda}})\,,$ (8)

where ${\cal R}\indices{{}^{MN}_{K\Lambda}}$ is the Riemann tensor associated
with $g_{MN}$. The Euler-Lagrange equations are

${\cal R}\indices{{}^{(M}_{K\Lambda P}}\frac{\partial f}{\partial{\cal
R}_{N)K\Lambda P}}-2\nabla_{K}\nabla_{\Lambda}\frac{\partial f}{\partial{\cal
R}_{K(MN)\Lambda}}-\frac{1}{2}fg^{MN}=0\,.$ (9)

The action (8) generically contains second-order time derivatives which is not
suitable for Hamiltonian analysis. To convert the equations of motion (9) into
equivalent first order differential equations, one can introduce auxiliary
fields $\varrho_{MNK\Lambda},\varphi^{MNK\Lambda}$ that act as Lagrange
multipliers. The action and path integral proposed in [6] is

$\displaystyle I_{{\cal L}}$ $\displaystyle=\frac{1}{2}\int_{{\cal
M}}d^{D}x\sqrt{|g|}\left[f(\varrho\indices{{}^{MN}_{K\Lambda}})+\varphi^{MNK\Lambda}\left({\cal
R}_{MNK\Lambda}-{\varrho}_{MNK\Lambda}\right)\right]-{\cal S}_{\text{GHY}}\,,$
(10) $\displaystyle Z$ $\displaystyle=\int{\cal D}g_{MN}\,{\cal
D}\varrho\indices{{}^{MN}_{K\Lambda}}\,{\cal
D}\varphi\indices{{}^{MN}_{K\Lambda}}\,e^{-I_{{\cal
L}}[g_{MN},\varrho\indices{{}^{MN}_{K\Lambda}},\varphi^{MNK\Lambda}]}\,.$ (11)

The fields $\varrho_{MNKL},\varphi^{MNKL}$ are assumed to have the same
symmetries as the Riemann tensor. The generalized Gibbons-Hawking-York (GHY)
term ${\cal S}_{\text{GHY}}$, whose role we will see shortly, is defined as
[14] (see also [15])

${\cal S}_{\text{GHY}}=\int_{\partial{\cal M}}d^{D-1}x\sqrt{|r|}{\cal
K}_{MN}{\cal\Psi}^{MN}\,,$ (12)

where $|r|$ is the determinant of the induced metric on $\partial{\cal M}$ and

${\cal
K}_{MN}:=r\indices{{}_{M}^{K}}r\indices{{}_{N}^{\Sigma}}\nabla_{K}r_{\Sigma}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
{\cal\Psi}^{MN}:=-2r^{MA}r^{NB}\varphi_{A\Gamma
B\Delta}r^{\Gamma}r^{\Delta}\,.$ (13)

Note that ${\cal K}_{MN}$ is the extrinsic curvature of $\partial{\cal M}$,
and by definition $r_{M}$ is the outward pointing unit normal to
$\partial{\cal M}$ with $r_{M}r^{M}=\epsilon$. The tensor
$r_{MN}=g_{MN}-\epsilon r_{M}r_{N}$ is the projector (see Figure 1) on ${\cal
M}$. For timelike outward normal unit vector $\underline{r}$ we have
$\epsilon=-1$ and for spacelike $\epsilon=+1$.

At the classical level the auxiliary action (10) and $f$(Riemann) action in
(8) are equivalent, resulting in the same equations of motion for $g_{MN}$. To
see this we take the functional variation of the auxiliary action (10)

$\displaystyle\delta I_{{\cal L}}=$ $\displaystyle\frac{1}{2}\int_{{\cal
M}}d^{D}x\sqrt{|g|}$ $\displaystyle\bigg{[}E^{MN}\delta
g_{MN}+\left(\frac{\partial
f}{\partial\varrho\indices{{}^{MN}_{P\Sigma}}}-\varphi^{ABP\Sigma}g_{AM}g_{BN}\right)\delta\varrho\indices{{}^{MN}_{P\Sigma}}$
(14) $\displaystyle+\delta\varphi^{MNP\Sigma}\left({\cal
R}_{MNP\Sigma}-{\varrho}_{MNP\Sigma}\right)\bigg{]}$
$\displaystyle+\epsilon\int_{{\partial\cal M}}d^{D-1}x$
$\displaystyle\sqrt{|r|}\left(r_{\Sigma}(\nabla_{P}\varphi^{MPN\Sigma})\delta
g_{MN}-r_{\Sigma}\varphi^{MPN\Sigma}\nabla_{P}\delta g_{MN}\right)-\delta
S_{GHY}\,,$

where

$\begin{split}E^{MN}&=\frac{1}{2}g^{MN}f(\varrho\indices{{}^{AB}_{\Gamma\Delta}})+\frac{1}{2}g^{MN}\varphi^{ABP\Sigma}\left({\cal
R}_{ABP\Sigma}-{\varrho}_{ABP\Sigma}\right)\\\
&-2\varphi^{(M|ABP}\varrho\indices{{}^{N)}_{ABP}}+\varphi^{ABP(M}{\cal
R}\indices{{}_{ABP}^{N)}}+2\nabla_{A}\nabla_{B}\varphi^{A(MN)B}\,.\end{split}$
(15)

The equations of motion are

$E^{MN}=0\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\varphi\indices{{}_{MN}^{P\Sigma}}=\frac{\partial
f}{\partial\varrho\indices{{}^{MN}_{P\Sigma}}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \varrho_{MNK\Lambda}={\cal
R}_{MNK\Lambda}\,.$ (16)

If we substitute the second and third equations above into $E^{MN}=0$ we find
the $f$(Riemann) metric equations of motion given in (9). The role of
generalized GHY term (12) is to make the variational problem of the action
(10) well posed [14] with the following Dirichlet boundary conditions on
$\partial{\cal M}$555$\delta r^{MN}|_{{\partial\cal M}}$ implies only the
induced metric on $\partial{\cal M}$ needs to be fixed. Also note that
${\partial{\cal M}}=\Sigma_{\tau_{1}}\cup{\cal B}\cup\Sigma_{\tau_{2}}$ as
depicted in Figure 1.

$\delta r^{MN}|_{{\partial\cal
M}}=\delta\left(r_{K}r_{\Lambda}\varphi^{MKN\Lambda}\right)|_{{\partial\cal
M}}=0\,.$ (17)

The utility of the auxiliary action (10) is that it contains only first-order
time derivatives, suitable for canonical analysis. We assume a “time” function
$\tau:{\cal M}\to\mathbb{R}$ that defines a foliation of spacetime ${\cal M}$.
The surfaces of constant $\tau$ could be timelike ($\sigma=-1$) or spacelike
($\sigma=+1$). The metric is decomposed into an Arnowitt-Deser-Misner (ADM)
[16] form adapted to the foliation:

$ds^{2}=\sigma
N^{2}d\tau^{2}+h_{\mu\nu}\left(dx^{\mu}+N^{\mu}d\tau\right)\left(dx^{\nu}+N^{\nu}d\tau\right)\,.$
(18)

We define the unit normal form to constant time slices666Raising the index we
get
$\underline{n}=N^{-1}\underline{\partial}_{\tau}-N^{-1}N^{\mu}\underline{\partial}_{\mu}$.

$n_{M}=\sigma N\partial_{M}\tau\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ N>0,\leavevmode\nobreak\
\leavevmode\nobreak\ n_{N}n^{N}=\sigma\,.$ (19)

The sign convention for (19) is such that the dual vector $\underline{n}$
points to increasing values of $\tau$ i.e. $n^{M}\partial_{M}\tau\geq 0$. The
time evolution vector $\underline{\tau}=\frac{\partial}{\partial\tau}$, the
normal vector $\underline{n}$ and the shift vector $\underline{N}$ are related
by

$\underline{\tau}=N\underline{n}+\underline{N}\,.$ (20)

The codimension-1 projection tensor $\gamma$ on constant time surfaces is

$\gamma_{MN}=g_{MN}-\sigma n_{M}n_{N}\,.$ (21)

A very useful property of the projection tensor that we will repeatedly use is
that $\gamma^{\mu\nu}=h^{\mu\nu}$ and $\gamma^{MN}=0$ for
$(M,N)\neq(\mu,\nu)$. Here $h^{\mu\nu}$ is the inverse of the induced metric
on each time slice $\Sigma_{\tau}$. The Gauss-Codazzi and Ricci equations are

$\begin{split}\gamma\indices{{}^{\Sigma}_{A}}\gamma\indices{{}^{K}_{B}}\gamma\indices{{}^{P}_{\Gamma}}\gamma\indices{{}^{T}_{\Delta}}{\cal
R}_{\Sigma KPT}&=\sigma\left(K_{A\Delta}K_{\Gamma
B}-K_{A\Gamma}K_{B\Delta}\right)+R_{AB\Gamma\Delta}\,,\\\
\gamma\indices{{}^{\Sigma}_{A}}\gamma\indices{{}^{K}_{B}}\gamma\indices{{}^{P}_{\Gamma}}{\cal
R}\indices{{}_{\Sigma
KPT}}n^{T}&=D_{A}K\indices{{}_{B\Gamma}}-D_{B}K\indices{{}_{A\Gamma}}\,,\\\
\gamma\indices{{}^{\Lambda}_{M}}\gamma\indices{{}^{K}_{N}}{\cal
R}\indices{{}_{\Lambda
PK\Sigma}}n^{P}n^{\Sigma}&=K\indices{{}_{M}^{\Sigma}}K_{\Sigma N}-\sigma
N^{-1}D_{M}D_{N}N-N^{-1}{\cal L}_{N\underline{n}}K_{MN}\,,\end{split}$ (22)

where $D_{M}$ is the induced covariant derivative on each time slice (see
Section 2) and $K_{MN}$ is the extrinsic curvature tensor of the
$\Sigma_{\tau}$ hypersurface,

$K_{MN}:=\gamma\indices{{}^{K}_{M}}\gamma\indices{{}^{\Lambda}_{N}}\nabla_{K}n_{\Lambda}\,.$
(23)

Performing a $(D-1)+1$ decomposition of the fields $\varphi^{MNK\Lambda},{\cal
R}_{MNK\Lambda},\varrho_{MNK\Lambda}$, using the projection tensor
$\gamma\indices{{}^{M}_{N}}$ and Eqs. (22), the action (10) becomes

$I_{{\cal L}}=\int_{{\cal M}}d^{D}x\,{\cal L}-\sigma\int
d^{D}x\sqrt{|g|}\nabla_{M}\left(n^{M}K\cdot\Psi\right)-{\cal
S}_{\text{GHY}}\,,$ (24)

where777From (20) we have ${\cal
L}_{N\underline{n}}\Psi^{\mu\nu}=\dot{\Psi}^{\mu\nu}-{\cal
L}_{\underline{N}}\Psi^{\mu\nu}$.

$\begin{split}{\cal
L}=N\sqrt{h}\bigg{[}&\frac{1}{2}f(\varrho_{MNK\Lambda},g_{MN})+\frac{1}{2}\phi^{\mu\nu\kappa\lambda}\left(\sigma
K_{\mu\lambda}K_{\kappa\nu}-\sigma
K_{\mu\kappa}K_{\nu\lambda}+R_{\mu\nu\kappa\lambda}-\rho_{\mu\nu\kappa\lambda}\right)\\\
&+2\sigma\phi^{\mu\nu\kappa}\left(D_{\mu}K_{\nu\kappa}-D_{\nu}K_{\mu\kappa}-\rho_{\mu\nu\kappa}\right)\\\
&+\sigma\Psi^{\mu\nu}\left((K\cdot K)_{\mu\nu}-\sigma
N^{-1}D_{\mu}D_{\nu}N+KK_{\mu\nu}-\Omega_{\mu\nu}\right)+\sigma
N^{-1}K_{\mu\nu}{\cal L}_{N\underline{n}}\Psi^{\mu\nu}\bigg{]}\,,\end{split}$
(25)

and the “spatial” fields are

$\begin{split}\rho_{\mu\nu\kappa\lambda}&:=\varrho_{\mu\nu\kappa\lambda},\leavevmode\nobreak\
\leavevmode\nobreak\ \rho_{\mu\nu\kappa}:=\varrho_{\mu\nu\kappa
P}n^{P},\leavevmode\nobreak\ \leavevmode\nobreak\
\Omega_{\mu\nu}:=\varrho_{\mu K\nu\Lambda}\,n^{K}n^{\Lambda},\\\
\phi^{\mu\nu\kappa\lambda}&:=h^{\mu\mu^{\prime}}h^{\nu\nu^{\prime}}h^{\kappa\kappa^{\prime}}h^{\lambda\lambda^{\prime}}\varphi_{\mu^{\prime}\nu^{\prime}\kappa^{\prime}\lambda^{\prime}}\,,\leavevmode\nobreak\
\leavevmode\nobreak\
\phi^{\mu\nu\kappa}:=h^{\mu\mu^{\prime}}h^{\nu\nu^{\prime}}h^{\kappa\kappa^{\prime}}\varphi_{\mu^{\prime}\nu^{\prime}\kappa^{\prime}P}n^{P},\\\
\Psi^{\mu\nu}&:=2\sigma
h^{\mu\mu^{\prime}}h^{\nu\nu^{\prime}}n^{K}n^{\Lambda}\varphi_{\mu^{\prime}K\nu^{\prime}\Lambda}\,,\end{split}$
(26)

with $A\cdot B:=A_{\mu\nu}B^{\mu\nu}$, $\left(A\cdot
B\right)_{\mu\nu}:=A_{\mu\lambda}B\indices{{}^{\lambda}_{\nu}}$. Note that the
Riemannian symmetries of $\varphi^{MNK\Lambda}$, $\varrho_{MNK\Lambda}$ imply
$\phi^{(\mu\nu)\lambda}=\Omega_{[\mu\nu]}=\Psi^{[\mu\nu]}=0$ and
$\phi^{\mu\nu\kappa\lambda}$ has the same symmetries as the Riemann tensor. In
equation (25) we assume $f(\varrho_{MNK\Lambda})$ has been written in terms of
the fields
$\rho_{\mu\nu\kappa\lambda},\rho_{\mu\nu\kappa},\Omega_{\mu\nu},h_{\mu\nu}$.

There are two boundary terms appearing in the Lagrangian (24). The first
boundary term comes from the Gauss-Codazzi decomposition, and we will refer it
as the Gauss-Codazzi boundary term, while the second is the generalized GHY
term (12) introduced at the beginning of this section. Assuming the boundary
is of the form $\partial{{\cal M}}=\Sigma_{\tau_{1}}\cup{\cal
B}\cup\Sigma_{\tau_{2}}$ as shown in Figure 1, we can use the divergence
theorem (2) on the Gauss-Codazzi boundary term and check that it exactly
cancels the GHY term on $\Sigma_{\tau_{1},\tau_{2}}$. Meanwhile it vanishes on
the boundary ${\cal B}$, since we assume the orthogonality relation
$g^{MN}n_{M}r_{N}=0$. Thus the only boundary term surviving in the form (24)
is the GHY term (12) on the boundary ${\cal B}$, labeled as ${\cal S}^{{\cal
B}}_{\text{GHY}}$

$I_{{\cal L}}=\int_{{\cal M}}d^{D}x\,{\cal L}-{\cal S}^{{\cal
B}}_{\text{GHY}}\,,$ (27)

where ${\cal L}$ is given in (25). The GHY term expressed in terms of the
spatial fields is

$\begin{split}{\cal S}^{\cal B}_{\text{GHY}}&=\int_{{\cal
B}}d^{D-1}x\sqrt{|r|}\bigg{[}-2k_{\mu\nu}\phi^{\mu\gamma\nu\delta}r_{\gamma}r_{\delta}-4\sigma
r^{\beta}K_{\beta\alpha}\phi^{\alpha\gamma\delta}r_{\gamma}r_{\delta}-(D_{\underline{r}}\ln
N)(r_{\alpha}r_{\beta}\Psi^{\alpha\beta})\bigg{]}\,,\end{split}$ (28)

where $\sqrt{|r|}=N\sqrt{s}$, $D_{\underline{r}}:=r^{\mu}D_{\mu}$ and
$r_{\mu}$ is the unit normal one form at ${\cal B}$ which is also tangent to
$\Sigma_{\tau}$.

The Lagrangian (25) is first order in time derivatives since the constant time
extrinsic curvature tensor is888For all ambient components $K_{MN}$ we find
from (23) that $K_{MN}=\frac{1}{2N}{\cal
L}_{N\underline{n}}\gamma_{MN}=\frac{1}{2N}\dot{\gamma}_{MN}-\frac{1}{2N}{\cal
L}_{\underline{N}}\gamma_{MN}$.

$K_{\mu\nu}=\frac{1}{2N}(\partial_{\tau}h_{\mu\nu})-\frac{1}{2N}{\cal
L}_{\underline{N}}h_{\mu\nu}\,,$ (29)

The non-vanishing canonical momentum densities are

$p^{\mu\nu}:=\frac{\delta{\cal
L}}{\delta(\partial_{\tau}h_{\mu\nu})}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Pi_{\mu\nu}:=\frac{\delta{\cal L}}{\delta(\partial_{\tau}\Psi^{\mu\nu})}\,,$
(30)

and can be calculated from the action (27). We find

$\begin{split}p^{\mu\nu}=\sigma\sqrt{h}\bigg{[}&\phi^{\mu\alpha\beta\nu}K_{\alpha\beta}-2N^{-1}D_{\beta}\left(N\phi^{\beta(\mu\nu)}\right)+\left(\Psi\cdot
K\right)^{(\mu\nu)}+\frac{1}{2}\left(K\cdot\Psi\right)h^{\mu\nu}\\\
&+\frac{1}{2}K\Psi^{\mu\nu}+\frac{1}{2}N^{-1}{\cal
L}_{N\underline{n}}\Psi^{\mu\nu}\bigg{]}\,,\\\
\Pi_{\mu\nu}=\sqrt{h}\sigma&K_{\mu\nu}\,.\end{split}$ (31)

The Hamiltonian integral is

$\int_{\tau_{1}}^{\tau_{2}}d\tau
H\,=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(p^{\mu\nu}\partial_{\tau}h_{\mu\nu}+\Pi_{\mu\nu}\partial_{\tau}\Psi^{\mu\nu}{\cal}\right)-I_{\cal
L}\,,$ (32)

where $I_{{\cal L}}$ is the action given in (27). We find

$\int_{\tau_{1}}^{\tau_{2}}d\tau
H=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\bigg{[}N^{\mu}{\cal
C}_{\mu}+N{\cal C}+\sqrt{h}D_{\mu}V^{\mu}\bigg{]}+{\cal S}^{{\cal
B}}_{\text{GHY}}\,,$ (33)

where ${\cal S}_{\text{GHY}}$ is given in (28) and

$V^{\mu}=2\frac{p^{\mu\nu}}{\sqrt{h}}N_{\nu}-2\frac{\Pi_{\alpha\beta}}{\sqrt{h}}\Psi^{\beta\mu}N^{\alpha}+\Psi^{\mu\beta}D_{\beta}N-ND_{\beta}\Psi^{\beta\mu}\,,$
(34) $\begin{split}{\cal
C}_{\mu}&=2\sqrt{h}D_{\nu}\left(\frac{\Pi\indices{{}_{\mu}{}_{\lambda}}}{\sqrt{h}}\Psi^{\lambda\nu}\right)-2\sqrt{h}h_{\mu\sigma}D_{\nu}\left(\frac{p^{\sigma\nu}}{\sqrt{h}}\right)+\sqrt{h}\left(\frac{\Pi_{\alpha\beta}}{\sqrt{h}}D_{\mu}\Psi^{\alpha\beta}\right)\,,\\\
{\cal
C}&=\sqrt{h}\left(D_{\mu}D_{\nu}\Psi^{\mu\nu}-\frac{1}{2}f(\varrho\indices{{}^{MN}_{K\Lambda}})+\sigma\Psi\cdot\Omega\right)+\frac{1}{\sqrt{h}}\left(2\sigma
p\cdot\Pi-\sigma\Psi\cdot\Pi\cdot\Pi-\sigma\Pi(\Psi\cdot\Pi)\right)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
-\frac{\sqrt{h}}{2}\phi^{\mu\nu\kappa\lambda}\left(\frac{2\sigma}{h}\Pi_{\mu\lambda}\Pi_{\nu\kappa}+R_{\mu\nu\kappa\lambda}-\rho_{\mu\nu\kappa\lambda}\right)+2\sigma\sqrt{h}\phi^{\mu\nu\kappa}\left(\rho_{\mu\nu\kappa}-2\sigma
D_{\mu}\left(\frac{\Pi_{\nu\kappa}}{\sqrt{h}}\right)\right)\,.\end{split}$
(35)

From equations (27), (32) and (33) we can express the action (10) in
_canonical form_

$I_{\text{ADM}}=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(p^{\mu\nu}\partial_{\tau}h_{\mu\nu}+\Pi_{\mu\nu}\partial_{\tau}\Psi^{\mu\nu}-N{\cal
C}-N^{\mu}{\cal C}_{\mu}\right)+B_{\underline{N}}+B_{N}\,,$ (36)

where $B_{\underline{N}}$, $B_{N}$ are 999We use the notation $B_{N}$,
$B_{\underline{N}}$ since they reduce to the General Relativity analogs [17]
(see Appendix A).

$\displaystyle B_{\underline{N}}$
$\displaystyle=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\,N^{\mu}\left(2\frac{\Pi\indices{{}_{\mu}{}_{\lambda}}}{\sqrt{h}}\Psi^{\lambda\nu}-2h_{\mu\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}\right)r_{\nu}\,,$
(37) $\displaystyle B_{N}$
$\displaystyle=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\,N\left(D_{\lambda}\Psi^{\lambda\nu}-\Psi^{\nu\lambda}D_{\lambda}\ln
N\right)r_{\nu}-{\cal S}^{{\cal B}}_{\text{GHY}}\,,$ (38)

and ${\cal C},{\cal C}_{\mu}$ are given in (35) and ${\cal S}^{{\cal
B}}_{\text{GHY}}$ in (28). The corresponding canonical path integral is

$Z=\int{\cal D}N{\cal D}N^{\mu}{\cal D}h_{\mu\nu}{\cal D}p^{\mu\nu}{\cal
D}\Psi^{\mu\nu}{\cal D}\Pi_{\mu\nu}{\cal
D}[\Omega_{\mu\nu},\rho_{\mu\nu\kappa\lambda},\rho_{\mu\nu\kappa}\phi^{\mu\nu\kappa\lambda}\phi^{\mu\nu\kappa}]\,e^{-I_{\text{ADM}}}\,,$
(39)

where $I_{\text{ADM}}$ is given in (36). The path integral (39) corresponds to
the canonical ensemble [1] of $f$(Riemann) gravity, since part of the boundary
data for a well-defined variational problem requires fixing the induced metric
on $\partial{\cal M}$, as can be verified from Eq. (17). We will see this
explicitly in the next section.

In Appendix A we reduce the path integral (39) to the familiar one of General
Relativity when
$f^{\text{GR}}(\varrho\indices{{}^{MN}_{K\Lambda}})=-\frac{\sigma}{8\pi
G_{N}}\left(g^{MP}g^{N\Sigma}\varrho_{MNP\Sigma}-2\Lambda\right)$.

###  4.1 Local Symmetries

Our analysis is suitable for the evaluation of path integrals, with fixed data
on a factorization surface, in the semiclassical approximation. In going
beyond the leading order, one must identify the gauge symmetries of the action
and fix them. We will not do so here, but we make some brief remarks about
this procedure.

In the canonical formalism of GR, the constraints are first class and depend
on the canonical variables only. They generate the symmetries of the canonical
variables and their Poisson algebra also determines the transformation rules
of the lapse and the shifts [18]. We have written the $f$(Riemann) path
integral in the Hamiltonian formalism retaining auxiliary fields, which are
extra variables that are not canonical. The status of local symmetries of the
off-shell action is more obscure in this formulation of $f$(Riemann) than it
is in ordinary GR.

By solving the equation of motion of the canonical momenta, one can fall back
on the Lagrangian formulation (10) and write down the path integral with a
manifestly diffeomorphism-invariant action. Another option can be the
elimination of the auxiliary fields by solving some of their equations of
motion. This way one gets an action that depends only on the lapse and the
shifts in addition to the canonical variables. The gauge symmetries of this
action can be identified as in canonical GR.

In either case, the auxiliary variables are of great use for reading off
continuity properties in the path integral factorization, and the Hamiltonian
formalism is useful – although not essential – for working out the
microcanonical boundary terms. The possibility of evaluating the path integral
in the auxiliary field formulation beyond the leading saddle point
approximation will be investigated in future work.

##  5 $f$(Riemann) Path Integral Factorization

In this section we find the continuity conditions for the path integral
factorization of $f$(Riemann) gravity theories in both canonical and
microcanonical ensemble. For this purpose we need to isolate all radial
derivatives appearing in the purely bulk form of the canonical action (36),
which corresponds to the canonical ensemble. (Again we stress that “radial” is
used as a general term describing a spatial coordinate, constant values of
which will label factorization surfaces. By a change of coordinatization we
can work with any factorization surface orthogonal to the temporal foliation.)
To achieve this we employ another ADM type decomposition of the induced metric
and the rest of spatial fields on $\Sigma_{\tau}$ along a foliation of
constant $\rho(x^{\mu})$ surfaces. This allows us to implement a Gauss-Codazzi
type decomposition of the canonical action and convert it to purely bulk form
such that the $\rho$-derivatives appear explicitly. The purely bulk form is
also important from another perspective; it will allow us to define the
microcanonical ensemble by adding the necessary boundary terms to the
canonical action.

###  5.1 Purely bulk form of $I_{\text{ADM}}$

In Section 4 we derived the canonical action $I_{\text{ADM}}$ given in Eq.
(36) for the $f$(Riemann) Lagrangian (10). The canonical action
$I_{\text{ADM}}$ is obviously not in purely bulk form due to the existence of
the boundary terms $B_{N}$ and $B_{\underline{N}}$ given in Eqs. (37)-(38).
Both these boundary terms can be cancelled by surface terms produced from some
of the bulk terms after integration by parts.

To isolate these terms we consider a foliation of the time slices
$\Sigma_{\tau}$ by surfaces of constant radial coordinate $\rho(x^{\mu})$. We
then employ another ADM decomposition of the induced metric $h_{\mu\nu}$ as
follows

$\begin{split}ds^{2}|_{\Sigma_{\tau}}:&=h_{\mu\nu}dx^{\mu}dx^{\nu}=P^{2}d\rho^{2}+s_{ab}(dx^{a}+\beta^{a}d\rho)(dx^{b}+\beta^{b}d\rho)\,,\end{split}$
(40)

where $P:=\sqrt{h_{\rho\rho}}$ is the radial lapse, $\underline{\beta}$ is the
radial shift vector and $s_{ab}$ is the induced metric on the codimension-2
surface ${\cal S}_{\tau}=\Sigma_{\tau}\cap{\cal T}$. The Latin indices label
the coordinates $x^{a}$ on ${\cal S}_{\tau}$. The unit one form $r_{\mu}$ and
dual vector $\underline{r}$ normal to $\rho$\- surfaces on $\Sigma_{\tau}$ are

$r=Pd\rho\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\underline{r}=P^{-1}\underline{\partial}_{\rho}-P^{-1}\underline{\beta}\,,$
(41)

where $\underline{\beta}=\beta^{a}\underline{\partial}_{a}$ and
$r_{\mu}r^{\mu}=1$. The ${\cal S}_{\tau}$ \- projector
$r_{\mu\nu}=h_{\mu\nu}-r_{\mu}r_{\nu}$ allows us to define the _radial
extrinsic curvature tensor_

$k_{\mu\nu}=r\indices{{}^{\sigma}_{\mu}}r\indices{{}^{\lambda}_{\nu}}D_{\sigma}r_{\lambda}\,.$
(42)

One can show that $k_{\mu\nu}$ contains radial derivatives of $s_{ab}$ only.
The Riemann tensor on $\Sigma_{\tau}$ has the familiar Gauss-Codazzi
decomposition (analogous to Eqs. (22))

$\begin{split}r\indices{{}^{\sigma}_{\alpha}}r\indices{{}^{\kappa}_{\beta}}r\indices{{}^{\rho}_{\gamma}}r\indices{{}^{\tau}_{\delta}}{R}_{\sigma\kappa\rho\tau}&=\left(k_{\alpha\delta}k_{\gamma\beta}-k_{\alpha\gamma}k_{\beta\delta}\right)+{{}^{(2)}}R_{\alpha\beta\gamma\delta}\\\
r\indices{{}^{\sigma}_{\alpha}}r\indices{{}^{\kappa}_{\beta}}r\indices{{}^{\rho}_{\gamma}}{R}\indices{{}_{\sigma\kappa\rho\tau}}r^{\tau}&={{}^{(2)}}D_{\alpha}k\indices{{}_{\beta\gamma}}-{{}^{(2)}}D_{\beta}k\indices{{}_{\alpha\gamma}}\\\
r\indices{{}^{\lambda}_{\mu}}r\indices{{}^{\kappa}_{\nu}}{R}\indices{{}_{\lambda\rho\kappa\sigma}}r^{\rho}r^{\sigma}&=-k\indices{{}_{\mu}^{\sigma}}k_{\sigma\nu}+{{}^{(2)}}D_{\mu}a_{\nu}-a_{\mu}a_{\nu}-D_{\underline{r}}k_{\mu\nu}-2r_{(\mu}k_{\nu)\lambda}a^{\lambda}\,,\end{split}$
(43)

where ${{}^{(2)}}D_{\mu}$ and ${{}^{(2)}}R_{\alpha\beta\gamma\delta}$ denote
the induced covariant derivative and Riemann tensor on ${\cal S}_{\tau}$ and
$a_{\mu}:=r^{\sigma}D_{\sigma}r_{\mu}$ is the _radial acceleration_. One can
check that $a$ has the general expression

$a_{\mu}=-{{}^{(2)}}D_{\mu}\ln P,$ (44)

and thus it does not contain any radial derivatives. From the Gauss-Codazzi
equations in Eqs. (43) we can spot where the radial derivatives are hiding in
the Riemann tensor components $R_{\mu\nu\kappa\lambda}$. Most of them are
contained inside the radial extrinsic curvature tensor $k_{\mu\nu}$ which
contain radial derivatives of $s_{ab}$. There is also a radial derivative of
the extrinsic curvature tensor itself that appears as the fourth term in the
third equation of (43). To make other radial derivatives in the canonical
action visible we first use the radial projector $r_{\mu\nu}$ to decompose
(some of) the canonical fields into tensors tangent to ${\cal S}_{\tau}$
similar to the ADM decomposition of the metric field $h_{\mu\nu}$ in (40). We
find

$\begin{split}N^{\mu}&=N^{\mu}_{|}+r^{\mu}N_{0}\\\
\Pi_{\mu\nu}&=\Pi^{|}_{\mu\nu}+2r_{(\mu}\Pi^{|}_{\nu)}+r_{\mu}r_{\nu}\Pi_{0}\\\
\Psi^{\mu\nu}&=\Psi^{\mu\nu}_{|}+2r^{(\mu}\Psi^{\nu)}_{|}+r^{\mu}r^{\nu}\Psi_{0}\\\
\phi^{\mu\nu\kappa}&=\phi_{|}^{\mu\nu\kappa}+2r^{[\nu}\phi^{\mu]\kappa}_{2,|}+\left(\phi^{\mu\nu}_{1,|}+2r^{[\nu}\phi^{\mu]}_{|}\right)r^{\kappa}\\\
\phi^{\mu\nu\kappa\lambda}&=\theta^{\mu\nu\kappa\lambda}_{|}+2\theta_{|}^{\mu\nu[\kappa}r^{\lambda]}+2\theta_{|}^{\kappa\lambda[\mu}r^{\nu]}+2r^{\nu}\theta_{|}^{\mu[\kappa}r^{\lambda]}+2r^{\mu}\theta^{\nu[\lambda}_{|}r^{\kappa]}\,,\end{split}$
(45)

where we defined the fields

$\begin{split}N_{|}^{\mu}&:=r\indices{{}^{\mu}_{\mu^{\prime}}}N^{\mu^{\prime}}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
N_{0}:=r_{\lambda}N^{\lambda}\,,\\\
\Pi^{|}_{\mu\nu}&:=r\indices{{}^{\mu^{\prime}}_{\mu}}r\indices{{}^{\nu^{\prime}}_{\nu}}\Pi_{\mu^{\prime}\nu^{\prime}}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Pi^{|}_{\mu}:=r\indices{{}^{\mu^{\prime}}_{\mu}}\Pi_{\mu^{\prime}\lambda}r^{\lambda}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Pi_{0}:=\Pi_{\alpha\beta}r^{\alpha}r^{\beta}\,,\\\
\Psi_{|}^{\mu\nu}&:=r\indices{{}^{\mu}_{\mu^{\prime}}}r\indices{{}^{\nu}_{\nu^{\prime}}}\Psi^{\mu^{\prime}\nu^{\prime}}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Psi_{|}^{\mu}:=r\indices{{}^{\mu}_{\mu^{\prime}}}\Psi^{\mu^{\prime}\lambda}r_{\lambda}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Psi_{0}:=\Psi^{\alpha\beta}r_{\alpha}r_{\beta}\,,\\\
\theta^{\mu\nu\kappa\lambda}_{|}&:=r\indices{{}^{\mu}_{\mu^{\prime}}}r\indices{{}^{\nu}_{\nu^{\prime}}}r\indices{{}^{\kappa}_{\kappa^{\prime}}}r\indices{{}^{\lambda}_{\lambda^{\prime}}}\phi^{\mu^{\prime}\nu^{\prime}\kappa^{\prime}\lambda^{\prime}}\leavevmode\nobreak\
,\leavevmode\nobreak\
\theta_{|}^{\mu\nu\kappa}:=r\indices{{}^{\mu}_{\mu^{\prime}}}r\indices{{}^{\nu}_{\nu^{\prime}}}r\indices{{}^{\kappa}_{\kappa^{\prime}}}\phi^{\mu^{\prime}\nu^{\prime}\kappa^{\prime}\lambda}r_{\lambda}\leavevmode\nobreak\
,\leavevmode\nobreak\
\theta^{\mu\nu}_{|}:=r\indices{{}^{\mu}_{\mu^{\prime}}}r\indices{{}^{\nu}_{\nu^{\prime}}}\phi^{\mu^{\prime}\alpha\nu^{\prime}\beta}r_{\alpha}r_{\beta}\,,\\\
\phi^{\mu\nu\kappa}_{|}&:=r\indices{{}^{\mu}_{\mu^{\prime}}}r\indices{{}^{\nu}_{\nu^{\prime}}}r\indices{{}^{\kappa}_{\kappa^{\prime}}}\phi^{\mu^{\prime}\nu^{\prime}\kappa^{\prime}},\phi^{\mu\nu}_{1,|}:=r\indices{{}^{\mu}_{\mu^{\prime}}}r\indices{{}^{\nu}_{\nu^{\prime}}}\phi^{\mu^{\prime}\nu^{\prime}\lambda}r_{\lambda},\,\phi^{\mu\nu}_{2,|}:=r\indices{{}^{\mu}_{\mu^{\prime}}}r\indices{{}^{\nu}_{\nu^{\prime}}}\phi^{\mu^{\prime}\lambda\nu^{\prime}}r_{\lambda},\leavevmode\nobreak\
\phi_{|}^{\mu}:=r\indices{{}^{\mu}_{\mu^{\prime}}}\phi^{\mu^{\prime}\alpha\beta}r_{\alpha}r_{\beta}\,.\end{split}$
(46)

The symbol $|$ above serves as a reminder that the field is tangent to ${\cal
S}_{\tau}$, e.g.
$r_{\lambda}\Psi_{|}^{\lambda\nu}=r_{\lambda}\Psi_{|}^{\mu\lambda}=0$ and
similarly for the other fields. The symmetries of the spatial fields are
inherited to the projected fields above, e.g. $\phi^{(\mu\nu)}_{1,|}=0$ since
$\phi^{(\mu\nu)\kappa}=0$. All other spatial fields that do not appear above
need not be decomposed since they do not participate with terms that have
radial derivatives in the canonical action.

The Gauss-Codazzi equations in Eqs. (43) together with the projected fields in
Eqs. (46) can be used to express the canonical action (36) in purely bulk
form. For economic purposes we present a shortened version of the purely bulk
form up to a function ${\cal G}(\text{fields})$, namely

$\begin{split}&I_{\text{ADM}}=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(p^{\mu\nu}\partial_{\tau}h_{\mu\nu}+\Pi_{\mu\nu}\partial_{\tau}\Psi^{\mu\nu}\right)\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}{\color[rgb]{0,0,1}(D_{\underline{r}}N)}\bigg{[}2\,{{}^{(2)}}D_{\lambda}\Psi^{\lambda}_{|}-{\color[rgb]{0,0,1}k_{\alpha\beta}}\Psi^{\alpha\beta}_{|}+{\color[rgb]{0,0,1}k}\Psi_{0}-2a_{\lambda}\Psi^{\lambda}_{|}+{\color[rgb]{0,0,1}D_{\underline{r}}\Psi_{0}}+2{\color[rgb]{0,0,1}k_{\alpha\beta}}\theta^{\alpha\beta}_{|}+4\phi^{\lambda}_{|}\frac{\Pi^{|}_{\lambda}}{\sqrt{h}}\bigg{]}\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}\bigg{[}({\color[rgb]{0,0,1}D_{\underline{r}}N^{\mu}_{|}})\bigg{(}2\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-2\frac{h_{\mu\sigma}p^{\sigma\nu}r_{\nu}}{\sqrt{h}}\bigg{)}+{\color[rgb]{0,0,1}D_{\underline{r}}N_{0}}\bigg{(}\frac{2r^{\mu}\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-\frac{2r_{\mu}p^{\mu\nu}r_{\nu}}{\sqrt{h}}\bigg{)}\bigg{]}\\\
&-4\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N\phi^{\nu\kappa}_{2,|}{\color[rgb]{0,0,1}D_{\underline{r}}\left(\frac{\Pi^{|}_{\nu\kappa}}{\sqrt{h}}\right)}+2\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N{\color[rgb]{0,0,1}k_{\mu\nu}D_{\underline{r}}\theta^{\mu\nu}_{|}}\\\
&+4\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N({\color[rgb]{0,0,1}D_{\underline{r}}\phi^{\mu}_{|}})\frac{\Pi^{|}_{\mu}}{\sqrt{h}}\\\
&-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N_{0}\left(\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi^{\alpha\beta}_{|}}+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi^{\beta}_{|}}+\frac{\Pi_{0}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi_{0}}\right)\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}\,{\cal
G}\left({\color[rgb]{0,0,1}k_{\mu\nu}},a_{\mu},\text{other field combinations
with no radial derivatives}\right)\,.\end{split}$ (47)

The detailed derivation of (47) is presented in Appendix B where the exact
form of the function ${\cal G}(\text{fields})$ is also given (see Eq. (90)).
The blue terms in (47) are all the terms that contain radial derivatives.

Let us make some important remarks. First, note that the blue terms in (47)
contain radial (spatial) covariant derivatives and not partial derivative of
the form $A\partial_{\rho}B$. One could wonder whether there are more radial
derivatives in the Christoffel symbols $\gamma\indices{{}^{\lambda}_{\mu\nu}}$
of the spatial metric $h_{\mu\nu}$. From the radial ADM decomposition in Eq.
(40) one can show that the Christoffel symbols needed contain only (partial)
radial derivatives of the induced metric $\partial_{\rho}s_{ab}$. The reason
is that the radial covariant derivatives appearing in (47) act only on tensors
tangential to ${\cal S}_{\tau}$, which the subscript symbol $|$ signifies, as
explained below Eqs. (46).101010For $D_{\underline{r}}$ acting on tangential
tensors on ${\cal S}_{\tau}$ the Christoffels
$\gamma\indices{{}^{\rho}_{\rho\rho}}$, $\gamma\indices{{}^{a}_{\rho\rho}}$
are not used. These are the only components that contain radial derivatives of
$h_{\rho\rho}$.

Second, from the purely bulk form of the canonical action we can easily read
off the continuity conditions required for factorization as described in
Section 3. Since our analysis assumes the factorization surface ${\cal T}$
intersects the time slices $\Sigma_{\tau}$ orthogonally, we have
$N_{0}|_{{\cal T}}=0$, and the purely bulk from in Eq. (47) implies that the
following fields111111Without loss of generality in Eq. (48) we replaced
lowercase greek $\mu,\nu,\dots$ indices with latin $a,b,\dots$ on ${\cal
S}_{\tau}$. need to be continuous in $\rho$ across ${\cal T}$

${\cal B}_{i}^{\text{canonical}}=\\{N,\leavevmode\nobreak\
N^{a},\leavevmode\nobreak\ s_{ab},\leavevmode\nobreak\
\theta^{ab}_{|},\leavevmode\nobreak\ \phi^{a}_{|},\leavevmode\nobreak\
\Psi_{0}\,,\leavevmode\nobreak\ \text{and }\leavevmode\nobreak\
\Pi^{|}_{ab}/\sqrt{h}\,\\}\quad\mbox{at}\leavevmode\nobreak\ {\cal T}\,.$ (48)

For the variational problem of the canonical action $I_{\text{ADM}}$ the set
${\cal B}_{i}^{\text{canonical}}$ is indeed the set of fields that need to be
fixed at the boundary. As we have mentioned, there is a one-to-one
relationship between the continuous fields ${\cal B}_{i}$ required for
factorization and the boundary conditions in the variational problem.

In a classical boundary value problem, the fields in Eq. (48) represent a
minimal set of data that must be fixed in order to have a well-defined
functional derivative of the canonical action. However, the classical
equations of motion will only admit a solution if the boundary data for
$\Pi_{ab}^{|}$ is consistent with Hamilton’s first equation (30). Relatedly,
we have to impose Hamilton’s first equation to achieve equivalence with the
Lagrangian form of the boundary conditions in Eq. (17) (using Eq. (46) to
express (17) in terms of projected fields.) In terms of the projected fields
the relevant component reads

$\frac{\Pi^{|}_{ab}}{\sqrt{h}}=\frac{\sigma}{N}\left(\frac{1}{2}(\partial_{\tau}s_{ab})-h_{c(a}{{}^{(2)}}D_{b)}N^{c}_{|}-k_{ab}N_{0}\right)\,.$
(49)

The above equation together with the orthogonality condition $N_{0}|_{{\cal
T}}=0$ show that the momenta $\Pi^{|}_{ab}/\sqrt{h}$ depend only on the
boundary metric (and tangential derivatives thereof) that is already fixed in
the Lagrangian variational problem. The canonical and Lagrangian problems
variational problems are consistent with each other when Hamilton’s first
equation holds.

Since the boundary metric is fixed in the variational problem, the canonical
action in Eq. (36) together with the boundary terms in Eqs. (37)-(38)
correspond to the canonical ensemble [1].121212This justifies the notation
${\cal B}_{i}^{\text{canonical}}$ in Eq. (48). In the next section we utilize
the purely bulk form in Eq. (47) to define the microcanonical ensemble for
$f$(Riemann) gravity theories and find the set ${\cal
B}_{i}^{{\text{micro}}}$.

###  5.2 Microcanonical ensemble for $f$(Riemann)

The canonical ensemble for $f$(Riemann) corresponds to the canonical action
(36). The purely bulk form for the canonical ensemble was found in Eq. (47)
and the continuity conditions across the factorization surface were found in
Eq. (48). In this sections we derive the microcanonical ensemble [13].

The prescription basically amounts to adding appropriate boundary terms to the
action that change the boundary conditions on ${\cal B}$ from fixed $N$ and
shift $N^{a}$ to fixed _energy-surface density_ (Brown-York quasilocal energy)
$\epsilon_{\text{BY}}$ and _momentum surface-density_ $J_{a}$, while all the
other boundary conditions remain the same. From the purely bulk form in Eq.
(47) we find that the variation among classical solutions of $I_{\text{ADM}}$
is

$\begin{split}\delta\left(I_{\text{ADM}}\big{|}_{\text{on-
shell}}\right)&=\int_{\Sigma_{\tau_{1}},\Sigma_{\tau_{2}}}d^{D-1}x\left(p^{\mu\nu}\delta
h_{\mu\nu}+\Pi_{\mu\nu}\delta\Psi^{\mu\nu}\right)\\\
&-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\left[\epsilon_{\text{BY}}\delta N-J_{a}\delta
N^{a}-J^{ab}\delta s_{ab}\right]\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\bigg{[}2Nk_{ab}\,\delta\theta^{ab}_{|}+4N\left(\frac{\Pi^{|}_{a}}{\sqrt{h}}\right)\delta\phi^{a}_{|}+(D_{\underline{r}}N)\delta\Psi_{0}\bigg{]}\,,\end{split}$
(50)

where

$\begin{split}\epsilon_{\text{BY}}&=-2\,{{}^{(2)}}D_{\lambda}\Psi^{\lambda}_{|}+k_{\alpha\beta}\Psi^{\alpha\beta}_{|}-k\Psi_{0}+2a_{\lambda}\Psi^{\lambda}_{|}-D_{\underline{r}}\Psi_{0}-2k_{\alpha\beta}\theta^{\alpha\beta}_{|}-4\phi^{\lambda}_{|}\frac{\Pi^{|}_{\lambda}}{\sqrt{h}}-4\phi^{\mu\nu}_{2,|}\left(\Pi^{|}_{\mu\nu}/\sqrt{h}\right)\,,\\\
J_{a}&=2\frac{\Pi_{a\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-2h_{a\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}r_{\nu}-4\sigma\,s_{af}\,{{}^{(2)}}D_{b}\left(\phi^{(bf)}_{2,|}\right)\,.\end{split}$
(51)

In the derivation of Eq. (50) we used Hamilton’s equation (49) for the term
$\phi_{2,|}^{ab}\delta(\Pi^{|}_{ab}/\sqrt{h})$, which is responsible for the
final terms in the expressions for $\epsilon_{\text{BY}}$, $J_{a}$ in Eq.
(51). The precise form of $J^{ab}$ is quite complicated but is not needed for
our purposes. Note that in the derivation of (50) we have assumed the
orthogonality condition $N_{0}|_{{\cal T}}=0$ and as usual dropped total
derivatives of the form ${{}^{(2)}}D_{a}v^{a}$. It is now clear from Eq. (50)
that in order to define the microcanonical action one needs to add the
following boundary terms in the canonical action (36)

$I_{\text{micro}}=I_{\text{ADM}}+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\left[N\,\epsilon_{\text{BY}}-J_{a}N^{a}\right]\,.$
(52)

From Eq (36) and (37), (38) the microcanonical action (52) becomes

$I_{\text{micro}}=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(p^{\mu\nu}\partial_{\tau}h_{\mu\nu}+\Pi_{\mu\nu}\partial_{\tau}\Psi^{\mu\nu}-N{\cal
C}-N^{\mu}{\cal C}_{\mu}\right)+B_{\text{micro}}\,,$ (53)

where the boundary term $B_{\text{micro}}$ is

$B_{\text{micro}}=-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\bigg{[}4N\phi^{ab}_{2,|}\frac{\Pi^{|}_{ab}}{\sqrt{h}}-4\sigma
s_{af}N^{a}\,{{}^{(2)}}D_{b}\left(\phi^{(bf)}_{2,|}\right)\bigg{]}\,.$ (54)

It is interesting to note the appearance of the boundary term above vanishes
in GR [13] because in that case we have $\phi^{ab}_{2,|}=0$.

The variation of $I_{\text{micro}}\big{|}_{\text{on-shell}}$ given in Eq. (53)
is

$\begin{split}\delta\left(I_{\text{micro}}\big{|}_{\text{on-
shell}}\right)&=\int_{\Sigma_{\tau_{1}},\Sigma_{\tau_{2}}}d^{D-1}x\left(p^{\mu\nu}\delta
h_{\mu\nu}+\Pi_{\mu\nu}\delta\Psi^{\mu\nu}\right)\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\left[N\delta\epsilon_{\text{BY}}-N^{a}\delta
J_{a}-J^{ab}\delta s_{ab}\right]\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\bigg{[}2Nk_{ab}\,\delta\theta^{ab}_{|}+4N\left(\frac{\Pi^{|}_{a}}{\sqrt{h}}\right)\delta\phi^{a}_{|}+(D_{\underline{r}}N)\delta\Psi_{0}\bigg{]}\,,\end{split}$
(55)

where we used Eq. (50).

The corresponding microcanonical path integral is

$Z^{\text{micro}}=\int{\cal D}N{\cal D}N^{\mu}{\cal D}h_{\mu\nu}{\cal
D}p^{\mu\nu}{\cal D}\Psi^{\mu\nu}{\cal D}\Pi_{\mu\nu}{\cal
D}[\Omega_{\mu\nu},\rho_{\mu\nu\kappa\lambda},\rho_{\mu\nu\kappa}\phi^{\mu\nu\kappa\lambda}\phi^{\mu\nu\kappa}]\,e^{-I_{\text{micro}}}\,.$
(56)

The purely bulk form of the microcanonical action in Eq. (53) is

$\begin{split}I_{\text{micro}}&=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(p^{\mu\nu}\partial_{\tau}h_{\mu\nu}+\Pi_{\mu\nu}\partial_{\tau}\Psi^{\mu\nu}\right)\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}\sqrt{h}N{\color[rgb]{0,0,1}D_{\underline{r}}\epsilon_{\text{BY}}}-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N^{a}_{|}{\color[rgb]{0,0,1}D_{\underline{r}}J_{a}}+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N\,{\color[rgb]{0,0,1}k\,\epsilon_{\text{BY}}}\\\
&-4\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}\phi^{\nu\kappa}_{2,|}{\color[rgb]{0,0,1}D_{\underline{r}}\left(N(\Pi^{|}_{\nu\kappa}/\sqrt{h})+\sigma{{}^{(2)}}D_{(\nu}N^{|}_{\kappa)}\right)}\\\
&+2\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N{\color[rgb]{0,0,1}k_{\mu\nu}D_{\underline{r}}\theta^{\mu\nu}_{|}}+4\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N({\color[rgb]{0,0,1}D_{\underline{r}}\phi^{\mu}_{|}})\frac{\Pi^{|}_{\mu}}{\sqrt{h}}\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}({\color[rgb]{0,0,1}D_{\underline{r}}N_{0}})\left(2r^{\mu}\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-2r_{\mu}\frac{p^{\mu\nu}}{\sqrt{h}}r_{\nu}\right)\\\
&-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N_{0}\bigg{(}\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi^{\alpha\beta}_{|}}+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi^{\beta}_{|}}+\frac{\Pi_{0}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi_{0}}\bigg{)}\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}\,{\cal
G^{\prime}}\left({\color[rgb]{0,0,1}k_{\mu\nu}},a_{\mu},\text{other field
combinations with no radial derivatives}\right)\,,\end{split}$ (57)

where $\epsilon_{\text{BY}}$, $J_{a}$ were defined in (51). The precise form
of ${\cal G^{\prime}}$ is modified compared to the function ${\cal G}$ in Eq.
(47) and its precise form is not needed. Similarly to the canonical ensemble
the blue terms in Eq. (57) are all the terms that contain radial derivatives.
From the purely bulk form of the microcanonical action we find the continuous
fields across ${\cal T}$ are

${\cal B}_{i}^{\text{micro}}=\\{\epsilon_{\text{BY}},\leavevmode\nobreak\
J_{a},\leavevmode\nobreak\ s_{ab},\leavevmode\nobreak\
\theta^{ab}_{|},\leavevmode\nobreak\ \phi^{a}_{|},\leavevmode\nobreak\
\Psi_{0}\,,\leavevmode\nobreak\ \text{and }\leavevmode\nobreak\
N(\Pi^{|}_{ab}/\sqrt{h})+\sigma{{}^{(2)}}D_{(a}N_{b)}\,\\}\quad\mbox{at}\leavevmode\nobreak\
{\cal T}\,,$ (58)

while the lapse $N$ and shift $N^{a}$ can be discontinuous, just as in
Einstein gravity. One way to impose continuity of
$N(\Pi^{|}_{ab}/\sqrt{h})+\sigma{{}^{(2)}}D_{(a}N_{b)}$ would be to impose
Hamilton’s first equation (49) as a surface condition for (17), upon which
continuity of $N(\Pi^{|}_{ab}/\sqrt{h})+\sigma{{}^{(2)}}D_{(a}N_{b)}$ becomes
continuity of $\frac{1}{2}\sigma\partial_{\tau}s_{ab}$, which is already
continuous according to (58). Since we are discussing factorization, however,
which can take place on any surface, this would essentially mean imposing
Hamilton’s first equation for $\Pi^{|}_{ab}$ everywhere. The precise
continuity condition is somewhat more general.131313A practical approach to
imposing it would be to change path integral variables from $\Pi^{|}_{ab}$ to
$Z_{ab}$ that are related by
$\Pi^{|}_{ab}/\sqrt{h}=N^{-1}Z_{ab}-\sigma{{}^{(2)}}D_{(a}N_{b)}$, and then
enforce continuity of $Z_{ab}$.

The fields in Eq. (58) were defined in Eqs. (46), (26) and the currents
$\epsilon_{\text{BY}},J_{a}$ in Eq. (51). With these continuity properties on
the fields the microcanonical action in (52) satisfies the additivity property
(6) and the corresponding path integral (56) factorizes

$Z^{\text{micro}}=\int_{{\cal T}}D{\cal B}_{i}^{\text{micro}}\,Z({{\cal
M}}_{1}|{\cal B}_{i}^{\text{micro}})Z({\cal B}_{i}^{\text{micro}}|{{\cal
M}}_{2})\,.$ (59)

Let us emphasize the role of the discontinuous lapse allowed by the
microcanonical path integral. Such a lapse discontinuity is actually realized
semiclassically in a number of simple settings involving horizons out of
equilibrium. For example, it allows us to make sense of Euclidean
$\text{SdS}_{4}$ as a saddle point of a constrained path integral, exhibiting
two smooth horizons but a finite lapse discontinuity. The constraint amounts
to fixing the fields ${\cal B}_{i}^{\text{micro}}$ on the surface ${\cal T}$.
We will see this in detail in the next section. Finally, as we mentioned above
the microcanonical ensemble also allows for discontinuous shift $N^{a}$.
Although this is not needed for the SdS geometry it is probably important for
other geometries, like the (complex Euclidean continuation of) the Kerr-de
Sitter spacetime [19].

##  6 Constrained states in EFT

Now we consider a constrained path integral, obtained by inserting a delta
functional to $Z^{\text{micro}}$ that fixes the microcanonical data (58) to
the values $\underline{{\cal B}}_{i}^{\text{micro}}$ on a codimension-1
surface ${{\cal T}}$. The constrained path integral then becomes

$Z^{\text{constrained}}=Z({{\cal M}}_{1}|\underline{{\cal
B}}_{i}^{\text{micro}})Z(\underline{{\cal B}}_{i}^{\text{micro}}|{{\cal
M}}_{2})\,,$ (60)

i.e., Eq. (59) with the integral omitted. Similar expressions hold for other
types of surface constraints.

The saddle point approximation may then by applied to each of the two factors
separately. Since the boundary data ${\cal B}_{i}$ does not generally include
all the field data, the semiclassical solution may be discontinuous on ${\cal
M}$. However, the continuity requirements are inferred from a classical
gravitational action, which is scale dependent when coupled to quantum fields.
For this reason it is important to establish the continuity conditions implied
by general Wilsonian effective actions.

The canonical and microcanonical factorization properties of the leading
truncation of the effective action, Einstein gravity, were discussed in [2].
Here we have shown that some of the most physically interesting properties –
namely the lack of continuity restrictions on $N$ and $N^{a}$ – are robust
against ultraviolet quantum corrections, in the sense that the more general
effective actions studied in the previous section have the same properties. In
particular, a low-curvature, discontinuous-$N$ or -$N^{a}$ saddle point
solution to a constrained path integral in Einstein gravity, or to a
constrained path integral with a more general Wilsonian gravitational
effective action, should provide a good approximation to a saddle point
solution with similar discontinuities arising from a new gravitational
effective action obtained by a small change in renormalization scale.

(Since we have only worked with the $f$(Riemann) class of effective actions,
which includes Einstein gravity but not all possible higher dimension
operators, and moreover since the local effective action does not incorporate
all infrared quantum effects, we cannot draw a completely general conclusion
about quantum corrections to solutions of Einstein gravity. However, as
discussed in the introduction, $f$(Riemann) is sufficiently general to discuss
factorization provided one performs suitable field and coupling redefinitions,
at least through the first few orders in the derivative expansion.)

Furthermore, we may use the Euclidean effective action to compute quantum
corrections, parameterized by the effective couplings, to the thermodynamic
potentials of constrained states. Examples include the free energies of out-
of-equilibrium black holes in asymptotically flat or AdS space at finite
temperature, and the Schwarzschild de Sitter black hole, which is always a
constrained state and always out of equilibrium. Let us work out the latter in
$D=4$, including general curvature-squared terms in the effective action.

We have a path integral over metrics on a manifold of $S^{4}$ topology, and a
microcanonical constraint applied on a surface corresponding to a fixed
“radial” coordinate. The path integral then factorizes into the form (60).

We can now apply the saddle point approximation on each factor separately.
What are values $\underline{{\cal B}}_{i}^{\text{micro}}$ should we fix on
${{\cal T}}$? With malice aforethought we will fix them to be the SdS boundary
data. To find their precise form, first note that including operators up to
dimension 4, the function $f$(Riemann) in Eq. (8) takes the form

$f({\cal R}\indices{{}^{MN}_{K\Lambda}})=-\frac{1}{8\pi G_{N}}\left({\cal
R}-2\Lambda\right)+b_{1}\;{\cal R}_{MNK\Lambda}{\cal
R}^{MNK\Lambda}+b_{2}\;{\cal R}_{MN}{\cal R}^{MN}+b_{3}\;{\cal R}^{2}\,,$ (61)

with $\Lambda>0$. The Einstein-Hilbert case $b_{1}=b_{2}=b_{3}=0$ was analyzed
in [2]. The auxiliary field Lagrangian $I_{\cal L}$ corresponding to the above
$f$(Riemann) action is given in Eq. (10) with

$f(\varrho\indices{{}^{MN}_{K\Lambda}})=-\frac{1}{8\pi
G_{N}}\left(\varrho-2\Lambda\right)+b_{1}\;\varrho_{MNK\Lambda}\varrho^{MNK\Lambda}+b_{2}\;\varrho_{MN}\varrho^{MN}+b_{3}\;\varrho^{2}\,,$
(62)

where $\varrho=g^{MP}g^{N\Sigma}\varrho_{MNP\Sigma}$.

The last two equations of motion in Eq. (16) become

$\begin{split}\varphi^{MNK\Lambda}&=-\frac{1}{16\pi
G_{N}}\left(g^{MK}g^{N\Lambda}-g^{NK}g^{M\Lambda}\right)+2b_{1}{\cal
R}^{MNK\Lambda}+b_{3}{\cal
R}\left(g^{MK}g^{N\Lambda}-g^{NK}g^{M\Lambda}\right)\\\ &\leavevmode\nobreak\
\leavevmode\nobreak\ +\frac{1}{2}b_{2}\left({\cal R}^{N\Lambda}g^{MK}-{\cal
R}^{M\Lambda}g^{NK}-{\cal R}^{NK}g^{M\Lambda}+{\cal
R}^{MK}g^{N\Lambda}\right)\,,\\\ \varrho_{MNK\Lambda}&={\cal
R}_{MNK\Lambda}\,.\end{split}$ (63)

Substituting the above into the first equation of motion in (16), $E^{MN}=0$,
we find the following metric equation of motion for the EFT Lagrangian in
(61),

$\begin{split}&-\frac{1}{8\pi G_{N}}\left(G^{MN}+\Lambda
g^{MN}\right)-\frac{1}{2}g^{MN}\bigg{(}(b_{2}+4b_{1}){\cal R}_{AB}{\cal
R}^{AB}+(b_{3}-b_{1}){\cal R}^{2}-(b_{2}+4b_{3})\nabla_{K}\nabla^{K}{\cal
R}\bigg{)}\\\ &-2(b_{2}+4b_{1}){\cal R}_{AB}{\cal
R}^{MABN}+2(b_{3}-b_{1}){\cal R}{\cal
R}^{MN}-(b_{2}+2b_{1}+2b_{3})\nabla^{M}\nabla^{N}{\cal R}\\\
&+(4b_{1}+b_{2})\nabla^{K}\nabla_{K}{\cal R}^{MN}=0\,.\end{split}$ (64)

It is now easy to check that general Einstein manifolds satisfy the equation
of motion if we substitute $R_{MN}=\Lambda g_{MN}$ into Eq. (64). Thus SdS is
a solution of the gravitational EFT up to curvature-squared operators. To find
the constrained values of $\underline{{\cal B}}_{i}^{\text{micro}}$ we need to
compute them for the SdS geometry. Before we present the result let us
describe the SdS geometry in more detail.

The Euclidean SdS geometry is

$ds^{2}={\cal
N}^{2}f(\rho)d\tau^{2}+f(\rho)^{-1}d\rho^{2}+\rho^{2}d\Omega^{2}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ f(\rho)=1-2M/\rho-\rho^{2}/L^{2}\leavevmode\nobreak\ \,,$
(65)

where $L=\sqrt{3/\Lambda}$ is the de Sitter radius, $M$ the black hole mass
parameter and $d\Omega^{2}$ the $S_{2}$ metric element. We impose Euclidean
time periodicity $\tau\sim\tau+\beta$ with $\beta=1/T_{b}$. We also define the
lapse function ${\cal N}=1+(T_{b}/T_{c}-1)\theta(\rho-r)$ that keeps track of
the lapse discontinuity. $T_{b}=\frac{L^{2}-4r_{b}^{2}}{4\pi L^{2}r_{b}}$ is
the temperature of the black hole horizon located at $\rho=r_{b}$, and
$T_{c}=\frac{3r_{c}^{2}-L^{2}}{4\pi L^{2}r_{c}}$ is the temperature of the
cosmological horizon at $\rho=r_{c}$. The factorization surface ${\cal T}_{r}$
is located at radial coordinate $\rho=r$. The Heaviside step function that
appears in lapse function ${\cal N}$ indicates that for the black hole side
$\rho<r$, we have ${\cal N}=1$, while for the cosmological side $\rho>r$, we
have ${\cal N}=\frac{T_{b}}{T_{c}}$. Note that although both sides have the
same Euclidean time periodicity $\beta=1/T_{b}$, both horizons are smooth due
to the different lapse functions on each side.

The continuation of the discontinuous Euclidean solution (65) to Lorentzian
singnature can be done as follows [2]. First we rescale the Euclidean time
periodicity of the cosmological side such that the lapse is continuous but the
time periodicity is not. We then continue $\tau\to t$ on both sides which
erases the discontinuity in the periodicity by decompactifying $t$ in both
regions. The end result is the Lorentzian SdS solution with mass M.

Let us compute the factorization surface values $\underline{{\cal
B}}_{i}^{\text{micro}}$ (58) of the continuous fields for the SdS geometry
given in Eq. (65). We find

$\begin{split}\epsilon_{\text{BY}}\big{|}_{\text{SdS}}&=-2r^{-1}f(r)^{1/2}\left(-\frac{1}{8\pi
G_{N}}+2\Lambda(4b_{3}+b_{2})\right)+2b_{1}f(r)^{1/2}\left(2r^{-1}f^{\prime\prime}(r)+f^{\prime\prime\prime}(r)\right)\\\
J_{a}\big{|}_{\text{SdS}}&=0\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
s_{ab}\big{|}_{\text{SdS}}=r^{2}t_{ab}\leavevmode\nobreak\
,\leavevmode\nobreak\ \Psi_{0}\big{|}_{\text{SdS}}=-\frac{1}{8\pi
G_{N}}+2\Lambda\left(4b_{3}+b_{2}\right)-2b_{1}f^{\prime\prime}(r)\\\
\theta^{ab}_{|}|_{\text{SdS}}&=r^{-2}\bigg{[}\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)-b_{1}r^{-1}f^{\prime}(r)\bigg{]}t^{ab}\leavevmode\nobreak\
,\leavevmode\nobreak\ \phi^{a}_{|}\big{|}_{\text{SdS}}=0\\\
N(\Pi^{|}_{ab}&/\sqrt{h})+{{}^{(2)}}D_{(a}N_{b)}\big{|}_{\text{SdS}}=0\,,\end{split}$
(66)

where $t_{ab}$ are the unit round metric components in $(\theta,\varphi)$
coordinates and $f(\rho)$ is given in Eq. (65). The details of the proof of
Eqs. (66) are presented in Appendix C. The constrained path integral (60) with
$\underline{{\cal B}}_{i}^{\text{micro}}$ fixed to (66) on ${\cal T}$ will
have the SdS geometry (65) on both sides of ${\cal T}$ as a saddle point
solution. Then in the leading saddle approximation, if there are no saddles of
lower action, $Z^{\text{constrained}}\approx
Z^{\text{constrained}}_{\text{SdS}}$. The geometry is a genuine saddle because
the geometry solves the equations of motion everywhere off of the constraint
surface – in particular, both horizons are smooth.

Thus SdS can be interpreted as a constrained state of the gravitational path
integral with an effective action up through arbitrary curvature-squared
operators. Beyond this order in the derivative expansion of the effective
action, the saddle point geometry will receive corrections from the effective
couplings. Our results from the previous section show that if the corrections
are of $f$(Riemann) type then there is no obstruction to maintaining a
discontinuous lapse function, so we expect that the change in the saddle point
geometry will be a small effect, well controlled in the derivative and
semiclassical expansions.

Finally, let us compute the curvature squared corrections to the entropy of
the constrained SdS state. We need to compute the on-shell value of the
microcanonical action (53). Although the momenta $p^{\mu\nu},\Pi_{\mu\nu}$,
time derivatives $\partial_{\tau}h_{\mu\nu},\,\partial_{\tau}\Psi^{\mu\nu}$
and the boundary terms in (53) vanish for SdS, one cannot conclude that the
on-shell action vanishes. The reason is that the time foliation breaks down at
the horizons $r_{b,c}$. To calculate the on-shell action we will use the
method of [12] which we refer to as the _BTZ prescription_.

The BTZ prescription is as follows: first we draw infinitesimal boundaries
${\cal T}_{b}$ and ${\cal T}_{c}$ around the black hole and cosmological
horizons. To compute the on-shell action we use the ADM form of the
microcanonical action for the region between ${\cal T}_{b}$ and ${\cal
T}_{c}$, while for the regions inside the boundaries ${\cal T}_{b}$ and ${\cal
T}_{c}$ we use the Lagrangian form of the microcanonical action. This means
that we use the microcanonical action (52) in these infinitesimal regions, but
instead of $I_{\text{ADM}}$, we use the Lagrangian form (27). For the large
bulk region in between ${\cal T}_{b}$ and ${\cal T}_{c}$, the microcanonical
action vanishes on-shell for SdS. To see this note that on-shell the lapse and
shift constraints should hold ${\cal C}={\cal C}_{\mu}=0$ and since
$\partial_{\tau}h_{\mu\nu},\partial_{\tau}\Psi^{\mu\nu}$ vanish for SdS, the
bulk terms in Eq. (53) also vanish. The boundary terms $B_{\text{micro}}$ in
Eq. (54) also vanish because
$\Pi^{|}_{ab}|_{\text{SdS}}=N^{a}|_{\text{SdS}}=0$. For the infinitesimal
horizon regions “inside” ${\cal T}_{b}$ and ${\cal T}_{c}$, one can show that
the boundary terms in Eq. (52) vanish for the infinitesimal surfaces ${\cal
T}_{b,c}$. The bulk terms in (27) also vanish because the curvature is finite
but the integration volume goes to zero. The entirety of the classical action
is thus due to generalized GHY terms ${\cal S}^{{\cal B}}_{\text{GHY}}$ on
${\cal T}_{b,c}$. From Eq. (12) and the SdS metric (65), we find

$\begin{split}I_{\text{micro}}^{\text{SdS}}&=-\frac{\pi r_{b}^{2}+\pi
r_{c}^{2}}{G_{N}}+64\pi^{2}b_{1}+48\pi^{2}\frac{r_{b}^{2}+r_{c}^{2}}{L^{2}}(4b_{3}+b_{2})\\\
&=-\frac{A_{b}+A_{c}}{4G_{N}}+64\pi^{2}b_{1}+12\pi(4b_{3}+b_{2})\frac{A_{b}+A_{c}}{L^{2}}\,,\end{split}$
(67)

where the horizons $r_{b},r_{c}$ satisfy $f(r_{b,c})=0$ with $f(\rho)$ given
in (65) and $A_{b,c}=4\pi r_{b,c}^{2}$. We see that the result is independent
of the location $r$ where the constrained was imposed. The entropy is minus
the total on-shell action.

We have verified by explicit computation that the same result for the entropy
is obtained using Wald’s formula [10, 11]. Namely, we find

$\begin{split}S^{\text{SdS}}_{\text{W}}&:=-2\pi\oint_{\text{horzn.}}d\theta
d\varphi\sqrt{s}\,\hat{\epsilon}^{MN}\hat{\epsilon}_{K\Lambda}\frac{\partial{\cal
L}}{\partial{\cal R}\indices{{}^{MN}_{K\Lambda}}}\\\
&\,\,=-I_{\text{micro}}^{\text{SdS}}\end{split}$ (68)

where ${\cal L}=-\frac{1}{2}f({\cal R}\indices{{}^{MN}_{K\Lambda}})$ 141414The
minus sign in ${\cal L}=-\frac{1}{2}f({\cal R}\indices{{}_{MNK\Lambda}})$ is
due to rotation from Euclidean to Lorentzian signature. and $f({\cal
R}\indices{{}^{MN}_{K\Lambda}})$ was given in Eq. (61), $s_{ab}=r^{2}t_{ab}$
is the cod-2 induced metric at the horizon,
$\hat{\epsilon}_{MN}=\nabla_{M}\tilde{\xi}_{N}$ and $\tilde{\xi}^{M}$ is the
Killing field that is null at the horizon with normalized surface gravity
$\kappa:=-\frac{1}{2}\nabla_{A}\tilde{\xi}_{B}\nabla^{A}\tilde{\xi}^{B}\big{|}_{\text{horzn.}}=1$.
Some details of the computation are presented at the end of Appendix C.

Let us make some remarks on the relation between BTZ and Wald’s method of
calculating the entropy. The relation with the ordinary Gibbons-Hawking
Euclidean approach was already pointed by Wald [10]. The equivalence of the
former to the BTZ method then implies that we have two methods, one Lorentzian
and one Euclidean, for computing horizon entropies just from data localized
near the horizons. Thus there should be a geometric demonstration that the two
computations are equivalent for geometries with Killing horizons. Let us
assume spherical topology for the Euclidean horizon (or bifurcate horizon in
Lorentzian signature). Then the BTZ surface ${\cal T}_{r}$ enclosing the
Euclidean horizon $r_{h}$ is $S_{\beta_{r}}^{1}\times S^{D-2}$, where
$\beta_{r}$ denotes the size of $S_{\beta_{r}}^{1}$. In the horizon limit
$\beta_{r}$ shrinks to zero and the horizon has finite area. For $f$(Riemann)
Lagrangians, we can compare Wald’s formula in Eq. (68) with the generalized
GHY term in Eqs. (12),(13), and using the on-shell value for
$\varphi_{MNK\Lambda}$ given in Eq. (16) we find that for the two formulas to
match we must have

$\lim_{r\to r_{h}}\leavevmode\nobreak\ \beta_{r}\,r_{[M}{\cal
K}_{A][B}\,r_{N]}=\lim_{r\to
r_{h}}\left(-\frac{\pi}{2}\nabla_{M}\tilde{\xi}_{A}\nabla_{B}\tilde{\xi}_{N}\right)\,,$
(69)

where the brackets on the left hand side mean anti-symmetrization and the
right hand side is computed using the appropriate Euclidean continuation of
the Killing vector $\underline{\tilde{\xi}}$. We note that both sides of Eq.
(69) should be computed in a coordinate system that is well-defined at the
Euclidean horizon, e.g. Kruskal-Szekeres coordinates, where the Riemann
components are finite. It is easy to check directly that Eq. (69) holds for
the Schwarzschild horizon.

Returning to SdS, finally we can estimate that the nucleation probability of
black hole of mass M in the de Sitter background is

$P=Z^{\text{constrained}}/Z\sim
e^{S_{\text{SdS}}-S_{\text{dS}}}=e^{-I_{\text{micro}}^{\text{SdS}}-\frac{\pi
L^{2}}{G_{N}}+16\pi^{2}\left(4b_{1}+12b_{3}+3b_{2}\right)}\,,$ (70)

where $I_{\text{micro}}^{\text{SdS}}$ was given in (67).

##  7 Conclusions

Let us summarize our main results and sketch some possible directions for
future progress.

The primary general result is the derivation of the continuity properties of
the gravitational fields in factorizations of the path integral, corresponding
to Dirichlet and microcanonical factorizations, in a large class of
gravitational effective Lagrangians. These continuity properties can be used
to implement the semiclassical approximation with fixed data on a
factorization surface, including the insertion of constraints.

Our main application of the general result was to the Euclidean Schwarzschild-
de Sitter geometry, where the detailed continuity properties of the factorized
path integral are already necessary to make semiclassical sense of the
geometry in Einstein gravity. By showing that the continuity properties of
Einstein gravity extend in a simple way to a large class of more general
effective actions, we conclude that the discontinuity present in Euclidean
Schwarzschild-de Sitter is not an accident of working in the leading
truncation of the effective Lagrangian, and something SdS-like can be expected
to persist after taking into account the quantum fluctuations of massive
matter fields. At the next-to-leading truncation, SdS itself remains a
solution, and the BTZ method of computing the on-shell action – which reduces
to a horizon-localized integral in the Euclidean signature – matches the
Lorentzian signature result obtained by Wald’s formula.

Let us now discuss some possible extensions of this work.

As discussed above, we consider effective actions of the $f$(Riemann) type.
This restricts the EFT operators to the parity-even sector with no explicit
covariant derivatives. (In some cases, or at least through low orders, it is
possible to perform field redefinitions and integrations-by-parts to rewrite
the action in the $f$(Riemann) form, but we are not aware of a general proof
that this can be extended to arbitrary order.) At a fixed order in the
derivative expansion, we believe it should be possible to incorporate other
classes of terms by introducing a finite number of additional auxiliary
fields, but it would be interesting to exhibit this construction explicitly.

In addition, as briefly mentioned at the end of Section 5, one could
investigate other types of factorizations apart from fixing canonical or
microcanonical data. For example, one could fix the surface angular momentum
density $J_{a}$ along with $\epsilon_{\text{BY}}$ to non-zero values. It would
be interesting to examine whether the Kerr-de Sitter metrics, which are
complex metrics in Euclidean signature, provide a consistent semiclassical
approximation of the contribution to the partition function.

Finally, we have confined our analysis to the leading order in the
semiclassical expansion. Going beyond the leading order introduces many
interesting technical and conceptual complications. Some have been discussed
in Sec. 4.1. Issues of ellipticity that affect ordinary one-sided boundaries
in gravity [20] are not a problem for factorization surfaces (essentially
because they are two-sided, and “boundary moving diffs” should be treated as
genuine diffs), but they do highlight the technical issue of gauge fixing in
the presence of a factorization surface. Another interesting related question
concerns the renormalization of boundary terms, and thus of semiclassical
boundary conditions: are all relevant boundary conditions radiatively stable?

We hope to address these matters in future work. Nearly a half century after
the discovery of the Gibbons-Hawking entropy, the gravitational path integral
continues to pose as many questions as it answers.

## Acknowledgements

This work was supported in part by the U.S. Department of Energy, Office of
Science, Office of High Energy Physics under award number DE-SC0015655.

##  Appendix A Reduction to General Relativity

In this appendix we reduce the path integral with the auxiliary fields to the
gravitational path integral of pure General Relativity (GR). There are two
different approaches for showing this, based on the Lagrangian or canonical
path integral. We will present both.151515We note that our manipulations do
not keep track of determinants that should be present when integrating out
fields but do not affect leading semiclassics.

In the first approach we use the Lagrangian formalism defined by the action
(10) and path integral (11) and reduce to the GR Lagrangian by integrating out
the auxiliary fields. We find the effective Lagrangian (75) is indeed the EH
Lagrangian with the GHY boundary term. Similarly we repeat the procedure
starting from the canonical form of the action (36) and path integral (39). In
this second approach we find the system indeed reduces to the usual GR
canonical action (86) with GHY boundary term. We will present both approaches.
For GR the function $f(\varrho_{MNP\Sigma})$ appearing in (10) is

$f^{\text{GR}}(\varrho\indices{{}^{MN}_{K\Lambda}})=-\frac{\sigma}{8\pi
G_{N}}\left(g^{MP}g^{N\Sigma}\varrho_{MNP\Sigma}-2\Lambda\right)\,.$ (71)

Note that in GR given the function
$f^{GR}(\varrho\indices{{}^{MN}_{K\Lambda}})$ defined in (71) above, the
fields $\rho_{MNK\Lambda}$ appear linearly in the action (10) and thus act as
Lagrange multipliers. Integrating them out along the imaginary contour in
field space inserts the Dirac delta functional in the path integral (11).
After integrating out the fields $\varphi^{MNK\Lambda}$, the Dirac delta
functional imposes the following constraint

$\varphi_{\text{GR}}^{MNK\Lambda}=-\frac{\sigma}{16\pi
G_{N}}\left(g^{MK}g^{N\Lambda}-g^{NK}g^{M\Lambda}\right)\,.$ (72)

Enforcing the constraint (72) to the generalized GHY term (12) we obtain the
usual GHY boundary term of GR,

${\cal S}^{\text{GR}}_{\text{GHY}}=\frac{\sigma}{8\pi
G_{N}}\epsilon\int_{\partial{{\cal M}}}d^{D-1}y\sqrt{|r|}{\cal K}\,,$ (73)

where ${\cal K}:=g^{MN}{\cal K}_{MN}$ is the extrinsic curvature trace. In
deriving (73) we used ${\cal\Psi}_{\text{GR}}^{MN}=\frac{\sigma}{8\pi
G_{N}}\epsilon\,r^{MN}$ which can be proven from the definition (13) if we use
the constraint (72). The effective action for the metric degrees of freedom
can be found by integrating out $\varrho_{MNP\Sigma}$ and then
$\varphi^{MNK\Lambda}$ in the partition function (10)

$Z=\int{\cal D}g_{MN}\,e^{-I^{\text{eff}}_{{\cal L}}[g_{MN}]}\,.$ (74)

Enforcing the constraint (72), we find the effective action
$I^{\text{eff}}_{{\cal L}}[g_{MN}]$ to be the Einstein-Hilbert action with GHY
boundary term

$I^{\text{eff}}_{{\cal L}}[g_{MN}]=-\frac{\sigma}{16\pi G_{N}}\int_{{{\cal
M}}}d^{D}x\sqrt{|g|}\left({\cal R}-2\Lambda\right)-\frac{\sigma}{8\pi
G_{N}}\epsilon\int_{\partial{{\cal M}}}\sqrt{|r|}d^{D-1}y{\cal K}\,.$ (75)

For Euclidean metrics we have $\sigma=\epsilon=1$ which matches the action in
equation (2) of [2].161616The GR path integral (74) can be expressed for both
Lorentzian/ Euclidean signature $\sigma=\mp 1$ as $Z=\int\left[{\cal
D}g_{MN}\right]e^{-\sigma\sqrt{\sigma}I^{\text{eff}}_{{\cal L}}[g_{MN}]}$,
where $i=\sqrt{-1}$ and $I^{\text{eff}}_{{\cal L}}[g_{MN}]$ is given in (75).

We now repeat the same procedure, starting from the ADM form of the action
(36) with corresponding path integral (39). The function
$f^{\text{GR}}(\varrho_{MNK\Lambda})$ in (71) can be expressed in terms of the
spatial fields

$f^{\text{GR}}(\varrho\indices{{}^{MN}_{K\Lambda}})=-\frac{\sigma}{8\pi
G_{N}}\left(h^{\mu\lambda}h^{\nu\sigma}\rho_{\mu\nu\lambda\sigma}+2\sigma
h^{\mu\nu}\Omega_{\mu\nu}-2\Lambda\right)\,,$ (76)

where $h_{\mu\nu}$ is the induced metric on $\Sigma_{\tau}$, and the fields
$\rho_{\mu\nu\lambda\sigma}$, $\Omega_{\mu\nu}$ are defined in (26). Note that
from (76) and the lapse constraint ${\cal C}$ given in (35) the fields
$\Omega_{\mu\nu}$, $\rho_{\mu\nu\kappa\lambda}$, $\rho_{\mu\nu\kappa}$ appear
linearly in the ADM action (36). Integrating them out along the imaginary
contour inserts the following delta functionals to the path integral171717We
inserted appropriate factors $1/\sqrt{h}$ in the delta functionals using the
inner product convention $(A,B)=\int d^{D-1}x\sqrt{h}\,A\cdot B$ for tensors
$A$,$B$.

$\begin{split}&\delta\left(\sigma N\left(\Psi^{\mu\nu}+\frac{\sigma}{8\pi
G_{N}}h^{\mu\nu}\right)\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\
\delta\left(\frac{N}{2}\left(\frac{\sigma}{16\pi
G_{N}}(h^{\mu\kappa}h^{\nu\lambda}-h^{\nu\kappa}h^{\mu\lambda})+\phi^{\mu\nu\kappa\lambda}\right)\right)\,,\\\
&\delta\left(2\sigma N\,\phi^{\mu\nu\kappa}\right)\,.\end{split}$ (77)

If we integrate out $\Psi^{\mu\nu}$, $\phi^{\mu\nu\kappa\lambda}$,
$\phi^{\mu\nu\kappa}$ the delta functional arguments are imposed as
constraints and reduce the path integral (39) to

$Z=\int{\cal D}N{\cal D}N^{\mu}{\cal D}h_{\mu\nu}{\cal D}p^{\mu\nu}{\cal
D}\Pi_{\mu\nu}\,e^{-\tilde{I}_{\text{ADM}}}\,,$ (78)

where

$\tilde{I}_{\text{ADM}}=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left((p^{\mu\nu}+\frac{\sigma}{8\pi
G_{N}}\Pi^{\mu\nu})\partial_{\tau}h_{\mu\nu}-N\tilde{\cal
C}-N^{\mu}\tilde{\cal
C}_{\mu}\right)+\tilde{B}_{N}+\tilde{B}_{\underline{N}}\,,$ (79)

and $\tilde{\cal C}$, $\tilde{\cal C}_{\mu}$, $\tilde{B}_{N}$,
$\tilde{B}_{\underline{N}}$ are

$\begin{split}\tilde{\cal C}&=\frac{1}{\sqrt{h}}\frac{1}{16\pi
G_{N}}\left(3\Pi\cdot\Pi+\Pi^{2}\right)+\frac{\sigma}{16\pi
G_{N}}\sqrt{h}\left(R-2\Lambda\right)+\frac{1}{\sqrt{h}}2\sigma
p\cdot\Pi\,,\\\ \tilde{\cal C}_{\mu}&=-\frac{\sigma}{8\pi
G_{N}}\sqrt{h}h_{\mu\lambda}D_{\nu}\left(2\frac{\Pi\indices{{}^{\lambda\nu}}}{\sqrt{h}}\right)-2\sqrt{h}h_{\mu\lambda}D_{\nu}\left(\frac{p^{\lambda\nu}}{\sqrt{h}}\right)\,,\\\
\tilde{B}_{N}&=-\frac{\sigma}{8\pi
G_{N}}\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{|s|}Nk\,,\\\
\tilde{B}_{\underline{N}}&=-2\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}N^{\mu}h_{\mu\sigma}\left(\frac{p^{\sigma\nu}}{\sqrt{h}}+\frac{\sigma}{8\pi
G_{N}}\frac{\Pi\indices{{}^{\sigma\nu}}}{\sqrt{h}}\right)r_{\nu}\,.\end{split}$
(80)

Note that there is one extra degree of freedom left in the path integral (78)
compared to the GR canonical path integral. One way to proceed is to integrate
out $p^{\mu\nu}$, which yields the GR path integral in Lagrangian form. Let us
take a different route and obtain it in the phase space form. We define the
new momenta $\tilde{p}^{\mu\nu},\tilde{\Pi}^{\mu\nu}$

$\tilde{p}^{\mu\nu}=p^{\mu\nu}+\frac{\sigma}{8\pi
G_{N}}\Pi^{\mu\nu}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{\Pi}^{\mu\nu}=p^{\mu\nu}-\frac{\sigma}{8\pi G_{N}}\Pi^{\mu\nu}\,,$ (81)

after which the $\tilde{I}_{\text{ADM}}$ takes the more familiar form

$\begin{split}\tilde{I}_{\text{ADM}}&=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(\tilde{p}^{\mu\nu}\partial_{\tau}h_{\mu\nu}-N\tilde{\cal
C}-N^{\mu}\tilde{\cal
C}_{\mu}\right)+\tilde{B}_{N}+\tilde{B}_{\underline{N}}\,,\end{split}$ (82)

where $\tilde{\cal C},\tilde{\cal C}_{\mu}$ and the boundary terms
$\tilde{B}_{\underline{N}}$,$\tilde{B}_{N}$ of Eq. (80) expressed in terms of
$\tilde{p}^{\mu\nu}$ and $\tilde{\Pi}^{\mu\nu}$ are

$\begin{split}\tilde{\cal C}&=\frac{\pi
G_{N}}{\sqrt{h}}\left(7\tilde{p}\cdot\tilde{p}-6\tilde{p}\cdot\tilde{\Pi}-2\tilde{p}\tilde{\Pi}-\tilde{\Pi}\cdot\tilde{\Pi}+\tilde{p}^{2}+\tilde{\Pi}^{2}\right)+\frac{\sigma}{16\pi
G_{N}}\sqrt{h}\left(R-2\Lambda\right)\,,\\\ \tilde{\cal
C}_{\mu}&=-2\sqrt{h}h_{\mu\lambda}D_{\nu}\left(\frac{\tilde{p}^{\lambda\nu}}{\sqrt{h}}\right)\,,\\\
\tilde{B}_{N}&=-\frac{\sigma}{8\pi
G_{N}}\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}Nk\,,\\\
\tilde{B}_{\underline{N}}&=-2\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}N^{\mu}h_{\mu\sigma}\left(\frac{\tilde{p}^{\sigma\nu}}{\sqrt{h}}\right)r_{\nu}\,.\end{split}$
(83)

Note that $\tilde{\Pi}^{\mu\nu}$ appears quadratically in (82). Integrating
out $\tilde{\Pi}^{\mu\nu}$ is equivalent to substituting the
$\tilde{\Pi}^{\mu\nu}$ variation equations of motion

$\tilde{\Pi}^{\mu\nu}=-3\tilde{p}^{\mu\nu}+\frac{4}{D-2}\tilde{p}h^{\mu\nu}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{\Pi}=\frac{D+2}{D-2}\tilde{p}\,,$ (84)

into (82). The path integral (78) then reduces to the GR canonical path
integral

$Z=\int{\cal D}N\,{\cal D}N^{\mu}\,{\cal D}h_{\mu\nu}\,{\cal
D}\tilde{p}^{\mu\nu}\,e^{-I^{\text{eff}}_{\text{ADM}}}\,,$ (85)

where

$\begin{split}I^{\text{eff}}_{\text{ADM}}=&\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(\tilde{p}^{\mu\nu}\partial_{\tau}h_{\mu\nu}-N\tilde{\cal
C}_{\text{eff}}-N^{\mu}\tilde{\cal
C}^{\text{eff}}_{\mu}\right)+\tilde{B}_{N}+\tilde{B}_{\underline{N}}\,,\end{split}$
(86)

and

$\begin{split}\tilde{\cal C}_{\text{eff}}&=\frac{16\pi
G_{N}}{\sqrt{h}}\left(\tilde{p}^{\mu\nu}\tilde{p}_{\mu\nu}-\frac{1}{D-2}\tilde{p}^{2}\right)+\frac{\sigma}{16\pi
G_{N}}\sqrt{h}\left(R-2\Lambda\right)\,,\\\ \tilde{\cal
C}^{\text{eff}}_{\mu}&=-2\sqrt{h}h_{\mu\lambda}D_{\nu}\left(\frac{\tilde{p}^{\lambda\nu}}{\sqrt{h}}\right)\,,\\\
\tilde{B}_{N}&=-\frac{\sigma}{8\pi
G_{N}}\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}Nk\,,\\\
\tilde{B}_{\underline{N}}&=-2\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}N^{\mu}h_{\mu\lambda}\left(\frac{\tilde{p}^{\lambda\nu}}{\sqrt{h}}\right)r_{\nu}\,.\end{split}$
(87)

The action (86) together with the lapse and shift constraints (87) is the well
known canonical form of the EH action with the GHY boundary term in D
dimensions.

##  Appendix B  Derivation of purely bulk form of $I_{\text{ADM}}$

In this appendix we derive the complete form of the purely bulk form given in
Eq. (47). We start with the canonical form of the action given in Eq. (36). By
definition the purely bulk form should have no boundary terms so our goal is
to cancel $B_{N}$, $B_{\underline{N}}$ given in Eqs. (37)-(38). As we will see
shortly, some of the bulk terms after integration by parts will generate
boundary terms that cancel $B_{N}$, $B_{\underline{N}}$.

The first step is to integrate by parts the first two terms in the shift
constraint ${\cal C}_{\mu}$ in Eq. (35). The covariant derivative for these
two terms will get shifted to $N^{\mu}$ and the boundary term generated will
cancel the $B_{\underline{N}}$ boundary term. The next step is to integrate by
parts the first term of the lapse constraint ${\cal C}$ in Eq. (35). The
boundary term generated will cancel the first term in $B_{N}$ given in Eq.
(37). After these two integrations by parts the canonical action (36) becomes

$\begin{split}I_{\text{ADM}}&=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\bigg{[}p^{\mu\nu}\partial_{\tau}h_{\mu\nu}+\Pi_{\mu\nu}\partial_{\tau}\Psi^{\mu\nu}\\\
&+\sqrt{h}D_{\nu}N^{\mu}\left(2\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}-2h_{\mu\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}\right)-\sqrt{h}\left(\frac{\Pi_{\alpha\beta}}{\sqrt{h}}N^{\lambda}D_{\lambda}\Psi^{\alpha\beta}\right)+\sqrt{h}(D_{\mu}N)D_{\lambda}\Psi^{\mu\lambda}\\\
&+\sqrt{h}N\left(\frac{1}{2}\phi^{\mu\nu\kappa\lambda}R_{\mu\nu\kappa\lambda}\right)+4\sqrt{h}N\phi^{\mu\nu\kappa}D_{\mu}\left(\frac{\Pi_{\nu\kappa}}{\sqrt{h}}\right)\\\
&-\sqrt{h}N\left(-\frac{1}{2}f(\varrho\indices{{}^{MN}_{K\Lambda}})+\sigma\Psi\cdot\Omega+2\sigma\sqrt{h}\phi^{\mu\nu\kappa}\rho_{\mu\nu\kappa}\right)\\\
&-N\left(\frac{1}{\sqrt{h}}\left(2\sigma\,p^{\mu\nu}\Pi_{\mu\nu}-\sigma\,\Psi\cdot\Pi\cdot\Pi-\sigma\,\Pi(\Psi\cdot\Pi)\right)-\frac{\sqrt{h}}{2}\phi^{\mu\nu\kappa\lambda}\left(\frac{2\sigma}{h}\Pi_{\mu\lambda}\Pi_{\nu\kappa}-\rho_{\mu\nu\kappa\lambda}\right)\right)\bigg{]}\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{{\cal
S}_{\tau}}d^{D-2}x\sqrt{s}\bigg{[}-Nr_{\nu}\Psi^{\nu\lambda}D_{\lambda}\ln
N+2Nk_{\mu\nu}\phi^{\mu\gamma\nu\delta}r_{\gamma}r_{\delta}+4Nr^{\beta}\left(\frac{\Pi_{\alpha\beta}}{\sqrt{h}}\right)\phi^{\alpha\gamma\delta}r_{\gamma}r_{\delta}\\\
&\qquad\qquad\qquad\qquad\qquad\leavevmode\nobreak\ +N(D_{\underline{r}}\ln
N)(r_{\alpha}r_{\beta}\Psi^{\alpha\beta})\bigg{]}\,.\end{split}$ (88)

Note there are boundary terms remaining in the above expression so the action
is not in purely bulk form yet. However the terms in the second and third line
of Eq. (88) above contain terms which after integration by parts will cancel
the remaining boundary terms. To see this explicitly we use Eq. (45) and find
the following expressions

$\begin{split}D_{\nu}N^{\mu}&\left(2\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}-2h_{\mu\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}\right)=D_{\underline{r}}N^{\mu}_{|}\left(2\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-2h_{\mu\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}r_{\nu}\right)\\\
&+D_{\underline{r}}N_{0}\left(2r^{\mu}\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-2r_{\mu}\frac{p^{\mu\nu}}{\sqrt{h}}r_{\nu}\right)\\\
&+\left({{}^{(2)}}D_{\nu}N^{\mu}_{|}+(k\indices{{}^{\mu}_{\nu}}+r_{\nu}a^{\mu})N_{0}+r^{\mu}({{}^{(2)}}D_{\nu}N_{0}-N_{|}^{\sigma}k_{\sigma\nu})\right)\left(2\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}-2h_{\mu\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}\right)\\\
\\\
\frac{\Pi_{\alpha\beta}}{\sqrt{h}}N^{\lambda}D_{\lambda}\Psi^{\alpha\beta}&=\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}N^{\lambda}\,{{}^{(2)}}D_{\lambda}\Psi^{\alpha\beta}_{|}+2\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}N^{\lambda}_{|}k\indices{{}_{\lambda}^{\alpha}}\Psi^{\beta}_{|}-2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N_{|}^{\lambda}k\indices{{}_{\lambda\alpha}}\Psi^{\alpha\beta}_{|}+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N^{\lambda}_{|}{{}^{(2)}}D_{\lambda}\Psi^{\beta}_{|}\\\
&+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N^{\lambda}_{|}k\indices{{}_{\lambda}^{\beta}}\Psi_{0}-2\frac{\Pi_{0}}{\sqrt{h}}N^{\lambda}_{|}k\indices{{}_{\lambda\beta}}\Psi^{\beta}_{|}+\frac{\Pi_{0}}{\sqrt{h}}N_{|}^{\lambda}{{}^{(2)}}D_{\lambda}\Psi_{0}-2\frac{\Pi_{0}}{\sqrt{h}}N_{0}a_{\beta}\Psi^{\beta}_{|}+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N_{0}a^{\beta}\Psi_{0}\\\
&+\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}N_{0}D_{\underline{r}}\Psi_{|}^{\alpha\beta}+2\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}N_{0}a^{\alpha}\Psi^{\beta}_{|}+2N_{0}\frac{\Pi^{|}_{\beta}}{\sqrt{h}}D_{\underline{r}}\Psi^{\beta}_{|}+\frac{\Pi_{0}}{\sqrt{h}}N_{0}D_{\underline{r}}\Psi_{0}-2N_{0}\frac{\Pi^{|}_{\beta}}{\sqrt{h}}a_{\lambda}\Psi^{\lambda\beta}_{|}\,\\\
\\\
(D_{\mu}N)D_{\lambda}\Psi^{\mu\lambda}&={{}^{(2)}}D_{\mu}N\left({{}^{(2)}}D_{\lambda}\Psi^{\mu\lambda}_{|}-\Psi_{|}^{\mu\lambda}a_{\lambda}+\Psi_{|}^{\sigma}k\indices{{}_{\sigma}^{\mu}}+a^{\mu}\Psi_{0}\right)\\\
&-(D_{\underline{r}}N)\left(k_{\alpha\beta}\Psi^{\alpha\beta}_{|}-2{{}^{(2)}}D_{\lambda}\Psi^{\lambda}_{|}-k\Psi_{0}+2\Psi^{\lambda}_{|}a_{\lambda}-D_{\underline{r}}\Psi_{0}\right)\\\
&+D_{\mu}\left(r^{\mu}\Psi_{|}^{\lambda}\leavevmode\nobreak\
{{}^{(2)}}D_{\lambda}N\right)-D_{\lambda}\left(\Psi^{\lambda}_{|}D_{\underline{r}}N\right)+\Psi^{\lambda}_{|}k\indices{{}_{\lambda}^{\sigma}}{{}^{(2)}}D_{\sigma}N\,,\\\
\\\
\phi^{\mu\nu\kappa\lambda}R_{\mu\nu\kappa\lambda}&=\theta_{|}^{\mu\nu\kappa\lambda}{{}^{(2)}}R_{\mu\nu\kappa\lambda}+\theta_{|}^{\mu\nu\kappa\lambda}\left(k_{\mu\lambda}k_{\nu\kappa}-k_{\mu\kappa}k_{\nu\lambda}\right)+4\theta_{|}^{\mu\nu\lambda}({{}^{(2)}}D_{\mu}k_{\lambda\nu}-{{}^{(2)}}D_{\nu}k_{\lambda\mu})\\\
&+4\theta_{|}^{\mu\nu}(-k\indices{{}_{\mu}^{\sigma}}k_{\sigma\nu}+{{}^{(2)}}D_{\mu}a_{\nu}-a_{\mu}a_{\nu})+4k\theta^{\mu\nu}_{|}k_{\mu\nu}+4k_{\mu\nu}D_{\underline{r}}\theta_{|}^{\mu\nu}-4D_{\mu}\left(r^{\mu}\theta_{|}^{\alpha\beta}k_{\alpha\beta}\right)\,,\\\
\\\
\phi^{\mu\nu\kappa}D_{\mu}\left(\frac{\Pi_{\nu\kappa}}{\sqrt{h}}\right)&=\phi^{\mu\nu\kappa}_{|}{{}^{(2)}}D_{\mu}\left(\frac{\Pi^{|}_{\nu\kappa}}{\sqrt{h}}\right)+\phi^{\mu\nu\kappa}_{|}k_{\mu\kappa}\frac{\Pi^{|}_{\nu}}{\sqrt{h}}-\phi^{\mu\kappa}_{2,|}k\indices{{}_{\mu}^{\sigma}}\frac{\Pi^{|}_{\sigma\kappa}}{\sqrt{h}}+\phi^{\mu\kappa}_{2,|}{{}^{(2)}}D_{\mu}\left(\frac{\Pi^{|}_{\kappa}}{\sqrt{h}}\right)\\\
&-2\phi_{2,|}^{\mu\nu}\frac{a_{(\mu}\Pi^{|}_{\nu)}}{\sqrt{h}}-\phi_{1,|}^{\mu\nu}k\indices{{}_{\mu}^{\kappa}}\frac{\Pi^{|}_{\nu\kappa}}{\sqrt{h}}+\phi_{1,|}^{\mu\nu}\,{{}^{(2)}}D_{\mu}\left(\frac{\Pi^{|}_{\nu}}{\sqrt{h}}\right)+\phi^{\nu}_{|}a^{\mu}\frac{\Pi^{|}_{\mu\nu}}{\sqrt{h}}\\\
&-2\phi^{\mu}_{|}k\indices{{}_{\mu}^{\sigma}}\frac{\Pi^{|}_{\sigma}}{\sqrt{h}}+\phi_{|}^{\mu}\,{{}^{(2)}}D_{\mu}\left(\frac{\Pi_{0}}{\sqrt{h}}\right)-\phi^{\mu}_{|}a_{\mu}\frac{\Pi_{0}}{\sqrt{h}}+k\phi^{\mu}_{|}\frac{\Pi^{|}_{\mu}}{\sqrt{h}}\\\
&-D_{\mu}\left(r^{\mu}\phi^{\lambda}_{|}\frac{\Pi^{|}_{\lambda}}{\sqrt{h}}\right)+\frac{\Pi^{|}_{\mu}}{\sqrt{h}}D_{\underline{r}}\phi_{|}^{\mu}-\phi^{\nu\kappa}_{2,|}D_{\underline{r}}\left(\frac{\Pi^{|}_{\nu\kappa}}{\sqrt{h}}\right)\,.\end{split}$
(89)

For the definition of the projected fields (with $|$ subscript) see Eq. (46).
If we substitute expressions (89) into Eq. (88) the boundary terms cancel from
the total derivative terms in the last three equations of Eq. (89), and we
finally find the purely bulk form of the canonical action

$\begin{split}&I_{\text{ADM}}=\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\left(p^{\mu\nu}\partial_{\tau}h_{\mu\nu}+\Pi_{\mu\nu}\partial_{\tau}\Psi^{\mu\nu}\right)\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}{\color[rgb]{0,0,1}(D_{\underline{r}}N)}\left[2\,{{}^{(2)}}D_{\lambda}\Psi^{\lambda}_{|}-{\color[rgb]{0,0,1}k_{\alpha\beta}}\Psi^{\alpha\beta}_{|}+{\color[rgb]{0,0,1}k}\Psi_{0}-2a_{\lambda}\Psi^{\lambda}_{|}+{\color[rgb]{0,0,1}D_{\underline{r}}\Psi_{0}}+2{\color[rgb]{0,0,1}k_{\alpha\beta}}\theta^{\alpha\beta}_{|}+4\phi^{\lambda}_{|}\frac{\Pi^{|}_{\lambda}}{\sqrt{h}}\right]\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N\left[({\color[rgb]{0,0,1}D_{\underline{r}}N^{\mu}_{|}})\left(2\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-2h_{\mu\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}r_{\nu}\right)+{\color[rgb]{0,0,1}D_{\underline{r}}N_{0}}\left(2r^{\mu}\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}r_{\nu}-2r_{\mu}\frac{p^{\mu\nu}}{\sqrt{h}}r_{\nu}\right)\right]\\\
&-4\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N\phi^{\nu\kappa}_{2,|}{\color[rgb]{0,0,1}D_{\underline{r}}\left(\frac{\Pi^{|}_{\nu\kappa}}{\sqrt{h}}\right)}+2\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N{\color[rgb]{0,0,1}k_{\mu\nu}D_{\underline{r}}\theta^{\mu\nu}_{|}}\\\
&+4\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N({\color[rgb]{0,0,1}D_{\underline{r}}\phi^{\mu}_{|}})\frac{\Pi^{|}_{\mu}}{\sqrt{h}}\\\
&-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N_{0}\left(\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi^{\alpha\beta}_{|}}+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi^{\beta}_{|}}+\frac{\Pi_{0}}{\sqrt{h}}{\color[rgb]{0,0,1}D_{\underline{r}}\Psi_{0}}\right)\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}\left({{}^{(2)}}D_{\nu}N^{\mu}_{|}+({\color[rgb]{0,0,1}k\indices{{}^{\mu}_{\nu}}}+r_{\nu}a^{\mu})N_{0}+r^{\mu}({{}^{(2)}}D_{\nu}N_{0}-N_{|}^{\sigma}{\color[rgb]{0,0,1}k_{\sigma\nu}})\right)\left(2\frac{\Pi_{\mu\lambda}}{\sqrt{h}}\Psi^{\lambda\nu}-2h_{\mu\sigma}\frac{p^{\sigma\nu}}{\sqrt{h}}\right)\\\
&-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}\bigg{(}\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}N^{\lambda}\,{{}^{(2)}}D_{\lambda}\Psi^{\alpha\beta}_{|}+2\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}N^{\lambda}_{|}{\color[rgb]{0,0,1}k\indices{{}_{\lambda}^{\alpha}}}\Psi^{\beta}_{|}-2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N_{|}^{\lambda}{\color[rgb]{0,0,1}k\indices{{}_{\lambda\alpha}}}\Psi^{\alpha\beta}_{|}+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N^{\lambda}_{|}{{}^{(2)}}D_{\lambda}\Psi^{\beta}_{|}\\\
&+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N^{\lambda}_{|}{\color[rgb]{0,0,1}k\indices{{}_{\lambda}^{\beta}}}\Psi_{0}-2\frac{\Pi_{0}}{\sqrt{h}}N^{\lambda}_{|}{\color[rgb]{0,0,1}k\indices{{}_{\lambda\beta}}}\Psi^{\beta}_{|}+\frac{\Pi_{0}}{\sqrt{h}}N_{|}^{\lambda}{{}^{(2)}}D_{\lambda}\Psi_{0}-2\frac{\Pi_{0}}{\sqrt{h}}N_{0}a_{\beta}\Psi^{\beta}_{|}+2\frac{\Pi^{|}_{\beta}}{\sqrt{h}}N_{0}a^{\beta}\Psi_{0}\\\
&+2\frac{\Pi^{|}_{\alpha\beta}}{\sqrt{h}}N_{0}a^{\alpha}\Psi^{\beta}_{|}-2N_{0}\frac{\Pi^{|}_{\beta}}{\sqrt{h}}a_{\lambda}\Psi^{\lambda\beta}_{|}\bigg{)}\\\
&+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}{{}^{(2)}}D_{\mu}N\left({{}^{(2)}}D_{\lambda}\Psi^{\mu\lambda}_{|}-\Psi_{|}^{\mu\lambda}a_{\lambda}+2\Psi_{|}^{\sigma}{\color[rgb]{0,0,1}k\indices{{}_{\sigma}^{\mu}}}+a^{\mu}\Psi_{0}\right)\\\
&+\frac{1}{2}\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}\bigg{(}\theta_{|}^{\mu\nu\kappa\lambda}{{}^{(2)}}R_{\mu\nu\kappa\lambda}+\theta_{|}^{\mu\nu\kappa\lambda}\left({\color[rgb]{0,0,1}k_{\mu\lambda}k_{\nu\kappa}}-{\color[rgb]{0,0,1}k_{\mu\kappa}k_{\nu\lambda}}\right)\\\
&+4\theta_{|}^{\mu\nu\lambda}({{}^{(2)}}D_{\mu}{\color[rgb]{0,0,1}k_{\lambda\nu}}-{{}^{(2)}}D_{\nu}{\color[rgb]{0,0,1}k_{\lambda\mu})}+4\theta_{|}^{\mu\nu}(-{\color[rgb]{0,0,1}k\indices{{}_{\mu}^{\sigma}}k_{\sigma\nu}}+{{}^{(2)}}D_{\mu}a_{\nu}-a_{\mu}a_{\nu})+4{\color[rgb]{0,0,1}k}\theta^{\mu\nu}_{|}{\color[rgb]{0,0,1}k_{\mu\nu}}\bigg{)}\\\
&+4\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N\bigg{[}\phi^{\mu\nu\kappa}_{|}{{}^{(2)}}D_{\mu}\left(\frac{\Pi^{|}_{\nu\kappa}}{\sqrt{h}}\right)+\phi^{\mu\nu\kappa}_{|}{\color[rgb]{0,0,1}k_{\mu\kappa}}\frac{\Pi^{|}_{\nu}}{\sqrt{h}}-\phi^{\mu\kappa}_{2,|}{\color[rgb]{0,0,1}k\indices{{}_{\mu}^{\sigma}}}\frac{\Pi^{|}_{\sigma\kappa}}{\sqrt{h}}+\phi^{\mu\kappa}_{2,|}{{}^{(2)}}D_{\mu}\left(\frac{\Pi^{|}_{\kappa}}{\sqrt{h}}\right)\\\
&-2\phi_{2,|}^{\mu\nu}\frac{a_{(\mu}\Pi^{|}_{\nu)}}{\sqrt{h}}-\phi_{1,|}^{\mu\nu}{\color[rgb]{0,0,1}k\indices{{}_{\mu}^{\kappa}}}\frac{\Pi^{|}_{\nu\kappa}}{\sqrt{h}}+\phi_{1,|}^{\mu\nu}\,{{}^{(2)}}D_{\mu}\left(\frac{\Pi^{|}_{\nu}}{\sqrt{h}}\right)+\phi^{\nu}_{|}a^{\mu}\frac{\Pi^{|}_{\mu\nu}}{\sqrt{h}}-2\phi^{\mu}_{|}{\color[rgb]{0,0,1}k\indices{{}_{\mu}^{\sigma}}}\frac{\Pi^{|}_{\sigma}}{\sqrt{h}}+\phi_{|}^{\mu}\,{{}^{(2)}}D_{\mu}\left(\frac{\Pi_{0}}{\sqrt{h}}\right)\\\
&-\phi^{\mu}_{|}a_{\mu}\frac{\Pi_{0}}{\sqrt{h}}+{\color[rgb]{0,0,1}k}\phi^{\mu}_{|}\frac{\Pi^{|}_{\mu}}{\sqrt{h}}\bigg{]}-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N\left(-\frac{1}{2}f(\varrho\indices{{}^{MN}_{K\Lambda}})+\sigma\Psi\cdot\Omega+2\sigma\sqrt{h}\phi^{\mu\nu\kappa}\rho_{\mu\nu\kappa}\right)\\\
&-\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\Sigma_{\tau}}d^{D-1}x\sqrt{h}N\left(\frac{1}{h}\left(2\sigma\,p\cdot\Pi-\sigma\,\Psi\cdot\Pi\cdot\Pi-\sigma\,\Pi(\Psi\cdot\Pi)\right)-\frac{1}{2}\phi^{\mu\nu\kappa\lambda}\left(\frac{2\sigma}{h}\Pi_{\mu\lambda}\Pi_{\nu\kappa}-\rho_{\mu\nu\kappa\lambda}\right)\right)\,.\end{split}$
(90)

All the terms in Eq. (90) that contain radial derivatives are colored with
blue.

##  Appendix C SdS entropy/microcanonical data

In this appendix we present details of the computation of the microcanonical
boundary data $\underline{{\cal B}}_{i}^{\text{micro}}$ in Eqs. (66) and SdS
entropy from Wald’s formula given in Eq. (68).

We are going to use some geometric facts about the SdS geometry given in Eq.
(65)

$ds^{2}={\cal
N}^{2}f(\rho)d\tau^{2}+f(\rho)^{-1}d\rho^{2}+\rho^{2}d\Omega^{2}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ f(\rho)=1-2M/\rho-\rho^{2}/L^{2}\leavevmode\nobreak\ \,,$
(91)

where $d\Omega^{2}=d\theta^{2}+\text{sin}^{2}\theta
d\varphi^{2}:=t_{ab}dx^{a}dx^{b}$ and $x^{a}=\\{\theta,\varphi\\}$ are the
usual polar coordinates on the sphere. We note that the quantities computed
hold for points except at $\rho=r$ where the lapse is discontinuous.181818For
the continuous quantities that we will compute, the value at $\rho=r$ can be
defined by taking the limit. The normal unit form $n$ and vector
$\underline{n}$ defined in Eq. (19) in $(\tau,\rho,\theta,\varphi)$
coordinates are

$n\big{|}_{\text{SdS}}={\cal N}f(\rho)^{1/2}d\rho\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\underline{n}\big{|}_{\text{SdS}}={\cal
N}^{-1}f(\rho)^{-1/2}\underline{\partial}_{\tau}\,.$ (92)

The induced metric (40) on $\Sigma_{\tau}$ time slices in
$x^{\mu}=\\{\rho,\theta,\varphi\\}$ is

$h_{\mu\nu}dx^{\mu}dx^{\nu}=f(\rho)^{-1}d\rho^{2}+\rho^{2}t_{ab}dx^{a}dx^{b}\,.$
(93)

The radial unit one form $r$ and vector $\underline{r}$ defined in Eq. (41)
are $r|_{\text{SdS}}=f(\rho)^{-1/2}d\rho\leavevmode\nobreak\
,\leavevmode\nobreak\
\underline{r}|_{\text{SdS}}=f(\rho)^{1/2}\underline{\partial}_{\rho}$, the
radial acceleration $a_{\mu}\big{|}_{\text{SdS}}=0$, and the radial extrinsic
curvature tensor (42) components $k_{\mu\nu}$ are

$\begin{split}k_{ab}\big{|}_{\text{SdS}}&=\rho
f(\rho)^{1/2}t_{ab}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ k_{\rho\rho}=k_{\rho a}=0\,,\\\
k\big{|}_{\text{SdS}}:&=h^{ab}k_{ab}=2\rho^{-1}f(\rho)^{1/2}\,.\end{split}$
(94)

It is convenient to use the Gauss Codazzi equations (22), (43)and the fact
that for SdS $K_{MN}=0$ to find the following Riemann tensor components for
SdS

$\begin{split}{\cal
R}_{\rho\tau\rho\tau}\big{|}_{\text{SdS}}&=-\frac{1}{2}{\cal
N}^{2}f^{\prime\prime}(\rho)\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\cal
R}_{a\tau b\tau}\big{|}_{\text{SdS}}=-\frac{1}{2}{\cal N}^{2}\rho
f(\rho)f^{\prime}(\rho)t_{ab}\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\cal
R}_{\mu\nu\lambda\tau}\big{|}_{\text{SdS}}=0\,,\\\ {\cal R}_{a\rho
b\rho}\big{|}_{\text{SdS}}&=-\frac{\rho}{2}\frac{f^{\prime}(\rho)}{f(\rho)}t_{ab}\leavevmode\nobreak\
,\leavevmode\nobreak\ {\cal
R}_{abc\rho}\big{|}_{\text{SdS}}=0\leavevmode\nobreak\ ,\leavevmode\nobreak\
{\cal
R}_{abcd}\big{|}_{\text{SdS}}=\rho^{2}(f(\rho)-1)(t_{ad}t_{cb}-t_{ac}t_{bd})\,.\end{split}$
(95)

The formulas above will be useful for computing the boundary data
$\underline{{\cal B}}_{i}^{\text{micro}}$. First we compute
$\phi^{MNK\Lambda}$ for SdS using the formula in Eq. (63). We find

$\varphi^{MNK\Lambda}\big{|}_{\text{SdS}_{4}}=\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)\left(g^{MK}g^{N\Lambda}-g^{NK}g^{M\Lambda}\right)\big{|}_{\text{SdS}}+2b_{1}{\cal
R}^{MNK\Lambda}\big{|}_{\text{SdS}}\,,$ (96)

where $\Lambda=3/L^{2}$. We use the above formula (96) to compute (some) of
the spatial fields defined in Eq. (26) for SdS using the formulas (95). We
find

$\begin{split}\phi^{a\rho
b\rho}\big{|}_{\text{SdS}}&=\rho^{-2}f(\rho)\bigg{[}\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)-b_{1}\rho^{-1}f^{\prime}(\rho)\bigg{]}t^{ab}\,,\\\
\phi^{abc\rho}\big{|}_{\text{SdS}}&=0\,,\\\
\phi^{abcd}\big{|}_{\text{SdS}}&=\rho^{-4}\bigg{[}\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)-2b_{1}\rho^{-2}(f(\rho)-1)\bigg{]}(t^{ac}t^{bd}-t^{bc}t^{ad})\,,\\\
\phi^{abcd}\big{|}_{\text{SdS}}&=\rho^{-4}\bigg{[}\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)-2b_{1}\rho^{-2}(f(\rho)-1)\bigg{]}(t^{ac}t^{bd}-t^{bc}t^{ad})\,,\\\
\phi^{\mu\nu\rho}\big{|}_{\text{SdS}}&=0\,,\\\
\Psi^{ab}\big{|}_{\text{SdS}}&=2\rho^{-2}\bigg{[}-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})-b_{1}\rho^{-1}f^{\prime}(\rho)\bigg{]}t^{ab}\,,\\\
\Psi^{\rho a}\big{|}_{\text{SdS}}&=0\,,\\\
\Psi^{\rho\rho}\big{|}_{\text{SdS}}&=2f(\rho)\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})-b_{1}f^{\prime\prime}(\rho)\right)\,.\end{split}$
(97)

Having found the spatial fields components above we can compute the radial
projected field defined in (46) using (97),

$\begin{split}N_{|}^{\mu}\bigg{|}_{\text{SdS}}&=0\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
N_{0}\bigg{|}_{\text{SdS}}=0\,,\\\
\Pi^{|}_{\mu\nu}\bigg{|}_{\text{SdS}}&=0\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Pi^{|}_{\mu}\bigg{|}_{\text{SdS}}=0\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\Pi_{0}\bigg{|}_{\text{SdS}}=0\,,\\\
\Psi_{|}^{ab}\bigg{|}_{\text{SdS}}&=2\rho^{-2}\bigg{[}-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})-b_{1}\rho^{-1}f^{\prime}(\rho)\bigg{]}t^{ab}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\Psi_{|}^{\mu}\bigg{|}_{\text{SdS}}=0\leavevmode\nobreak\ \leavevmode\nobreak\
,\\\ \Psi_{0}\bigg{|}_{\text{SdS}}&=-\frac{1}{8\pi
G_{N}}+2\Lambda\left(4b_{3}+b_{2}\right)-2b_{1}f^{\prime\prime}(\rho)\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
D_{\underline{r}}\Psi_{0}=-2b_{1}f(\rho)^{1/2}f^{\prime\prime\prime}(\rho)\,,\\\
\theta^{abcd}_{|}|_{\text{SdS}}&=\rho^{-4}\bigg{[}\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)-2b_{1}\rho^{-2}(f(\rho)-1)\bigg{]}(t^{ac}t^{bd}-t^{bc}t^{ad})\leavevmode\nobreak\
,\leavevmode\nobreak\ \theta_{|}^{abc}|_{\text{SdS}}=0\,,\\\
\theta^{ab}_{|}|_{\text{SdS}}&=\rho^{-2}\bigg{[}\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)-b_{1}\rho^{-1}f^{\prime}(\rho)\bigg{]}t^{ab}\,,\\\
\phi^{abc}_{|}|_{\text{SdS}}&=0,\leavevmode\nobreak\
\phi^{ab}_{1,|}|_{\text{SdS}}=0,\leavevmode\nobreak\
\phi^{ab}_{2,|}|_{\text{SdS}}=0,\leavevmode\nobreak\
\phi_{|}^{a}|_{\text{SdS}}=0\,.\end{split}$ (98)

We can now compute the $\underline{{\cal B}}_{i}^{\text{micro}}$, namely the
fields in Eq. (58) for $\text{SdS}_{4}$, namely

$\begin{split}\epsilon_{\text{BY}}\big{|}_{\text{SdS}}&=-2\rho^{-1}f(\rho)^{1/2}\left(-\frac{1}{8\pi
G_{N}}+2\Lambda(4b_{3}+b_{2})\right)+2b_{1}f(\rho)^{1/2}\left(2\rho^{-1}f^{\prime\prime}(\rho)+f^{\prime\prime\prime}(\rho)\right)\,,\\\
J_{a}\big{|}_{\text{SdS}}&=0\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
s_{ab}\big{|}_{\text{SdS}}=\rho^{2}t_{ab}\leavevmode\nobreak\
,\leavevmode\nobreak\ \Psi_{0}\big{|}_{\text{SdS}}=-\frac{1}{8\pi
G_{N}}+2\Lambda\left(4b_{3}+b_{2}\right)-2b_{1}f^{\prime\prime}(\rho)\,,\\\
\theta^{ab}_{|}|_{\text{SdS}}&=\rho^{-2}\bigg{[}\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)-b_{1}\rho^{-1}f^{\prime}(\rho)\bigg{]}t^{ab}\leavevmode\nobreak\
,\leavevmode\nobreak\ \phi^{a}_{|}\big{|}_{\text{SdS}}=0\,,\\\
N(\Pi^{|}_{ab}/&\sqrt{h})+{{}^{(2)}}D_{(a}N_{b)}\big{|}_{\text{SdS}}=0\,.\end{split}$
(99)

Evaluating the formulas in Eq. (99) at the location $\rho=r$ of the
factorization surface ${\cal T}_{r}$ we arrive at Eq. (66) given in Section 6.

We end this appendix by presenting some details of the calculation SdS entropy
from Wald’s formula given in Eq. (68). The first thing to note is that Wal’ds
formulas uses Lorentzian signature metrics. This amounts to rotating ${\cal
N}=i$ and decompactifying the time coordinate $\tau\to t$ in Eq. (65). For
$f(Riemann)$ type theories the formula reduces to Eq. (68). The minus sign in
${\cal L}=-(1/2)f({\cal R}\indices{{}^{MN}_{K\Lambda}})$ also comes from the
continuation to Lorentzian signature. There will be two contributions to the
entropy coming from the two killing horizons for the Lorentzian SdS geometry.
If we use Eq. (16) and its value for SdS (96) into the first line of (68) for
each horizon $\rho=r_{H}$ we get

$\begin{split}S^{\text{SdS}}_{\text{W}}\big{|}_{r_{H}}&=-2\pi\oint_{\text{horzn.}}d\theta
d\varphi\sqrt{s}\,\hat{\epsilon}^{MN}\hat{\epsilon}_{K\Lambda}\frac{\partial{\cal
L}}{\partial{\cal R}\indices{{}^{MN}_{K\Lambda}}}\\\
&=\pi\oint_{\text{horzn.}}d\theta
d\varphi\sqrt{s}\hat{\epsilon}^{MN}\hat{\epsilon}_{K\Lambda}\,\varphi\indices{{}_{MN}^{K\Lambda}}\\\
&=\pi\oint_{\text{horzn.}}d\theta d\varphi\sqrt{s}\left[\left(-\frac{1}{16\pi
G_{N}}+\Lambda(4b_{3}+b_{2})\right)2\hat{\epsilon}^{MN}\hat{\epsilon}_{MN}+2b_{1}\hat{\epsilon}^{MN}\hat{\epsilon}^{K\Lambda}{\cal
R}^{\text{SdS}}_{MNK\Lambda}\right]\\\ &=\pi
r_{H}^{2}-48\pi^{2}\frac{r_{H}^{2}}{L^{2}}(4b_{3}+b_{2})+8b_{1}\pi^{2}r_{H}^{2}\hat{\epsilon}^{MN}\hat{\epsilon}^{K\Lambda}{\cal
R}^{\text{SdS}}_{MNK\Lambda}\end{split}$ (100)

where we used $\Lambda=3/L^{2}$, $\hat{\epsilon}^{MN}\hat{\epsilon}_{MN}=-2$,
since $\underline{\tilde{\xi}}$ is normalized to unit surface gravity. The
quantity $\hat{\epsilon}^{MN}\hat{\epsilon}^{K\Lambda}{\cal
R}^{\text{SdS}}_{MNK\Lambda}$ can computed at the static coordinate system at
some $\rho=r$ and then take the horizon limit $r\to r_{H}$. Being a scalar
quantity we expect to get the correct value at horizon by taking the limit
even though the static coordinates break down at the horizon. We find using
(95) that $\lim_{\rho\to
r_{H}}\hat{\epsilon}^{MN}\hat{\epsilon}^{K\Lambda}{\cal
R}^{\text{SdS}}_{MNK\Lambda}=4{\cal
R}^{\text{SdS}}_{\tau\rho\tau\rho}|_{r_{H}}=-4/r_{H}^{2}$, and if we
substitute to the last line of Eq. (100) we finally find

$\begin{split}S^{\text{SdS}}_{\text{W}}\big{|}_{r_{H}}&=\pi
r^{2}_{H}-48\pi^{2}\frac{r_{H}^{2}}{L^{2}}(4b_{3}+b_{2})-32\pi^{2}b_{1}\,.\end{split}$
(101)

If we add the results for both black hole and cosmological horizons
$r_{H}=r_{b,c}$ we find the result presented in Eqs. (67)-(68).

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