aainstitutetext: Institute of Physics, University of Amsterdam, Science Park
904, PO Box 94485, 1090 GL Amsterdam, the Netherlandsbbinstitutetext: Delta
Institute for Theoretical Physics, Science Park 904, PO Box 94485, 1090 GL
Amsterdam, the Netherlandsccinstitutetext: Department of Applied Mathematics
and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WA, UK

# Action, entropy and pair creation rate of charged black holes in de Sitter
space

E. Morvan a,b J.P. van der Schaar c M.R. Visser<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute and clarify the interpretation of the on-shell Euclidean action for
Reissner-Nordström black holes in de Sitter space. We show the on-shell action
is minus the sum of the black hole and cosmological horizon entropy for
arbitrary mass and charge in any number of dimensions. This unifying
expression helps to clear up a confusion about the Euclidean actions of
extremal and ultracold black holes in de Sitter, as they can be understood as
special cases of the general expression. We then use this result to estimate
the probability for the pair creation of black holes with arbitrary mass and
charge in an empty de Sitter background, by employing the formalism of
constrained instantons. Finally, we comment that the decay of charged de
Sitter black holes is expected to be governed by the gradient flow of the
entropy function, and that as a consequence, as long as the regime of light,
superradiant, rapid charge emission is excluded, the effective decay may be
described by the tunneling formalism.

##  1 Introduction

In this work we continue our study of the Euclidean action of black holes in
de Sitter space and their potential role in the partition function. In
previous work SdSaction  we provided a general and covariant derivation of the
Euclidean on-shell action for neutral, nonrotating black holes in de Sitter
space, generalizing classic results for the Nariai instanton Ginsparg:1982rs ;
Bousso:1996au , and showed it is equal to minus the sum of the two horizon
entropies for arbitrary mass and in any number of dimensions (as was
previously done for four dimensions in Chao:1997em ; Chao:1997osu ;
Bousso:1998na ; Gregory:2014 ). We emphasized the special role of the conical
singularities, as well as a proper interpretation in terms of constrained
instantons (also explained in Draper:2022xzl  and first suggested in the older
works Chao:1997em ; Chao:1997osu ), and computed the pair production rate of
an arbitrary mass black hole in de Sitter space from first principles.

Our main motivation for studying these objects is that they might play an
important role in a Euclidean path integral description of quantum gravity in
de Sitter space. This is suggested by recent developments in string theory and
AdS/CFT addressing black hole unitarity from a bulk perspective
Penington:2019npb ; Almheiri:2019psf , which led to the discovery of (new)
non-perturbative saddle solutions of the Euclidean action of Einstein gravity
with negative cosmological constant and how they restore unitarity of black
hole evaporation Penington:2019kki ; Almheiri:2019qdq . Here, we will consider
non-perturbative, non-regular saddles of the Euclidean action of Einstein
gravity with positive cosmological constant and we revisit, extend and
reinterpret existing results on charged black hole instantons.

What makes de Sitter space especially interesting is the fact that the
effective dynamics appears to be completely governed by the (finite) entropy
of the cosmological horizon Gibbons:1976ue ; Gibbons:1977mu , as first
emphasized in Fischler-talk ; Banks:2000fe ; Banks:2006rx , but also nicely
spelled out in more recent work PeningtonWitten:2022cip . Vacuum de Sitter
corresponds to the maximum entropy state, and (perturbative or non-
perturbative) excited states in the effective Einstein gravity theory
correspond to configurations with lower entropy, which can be described in
terms of constraints on the energy (and other charges) defining the state. The
probability to create such a state in the static patch of de Sitter is
governed by the entropy difference and the contribution of non-perturbative
states, such as black holes, to the partition function can then be elegantly
accounted for using the constrained instanton formalism Cotler:2019nbi ;
Cotler:2020lxj . The hope is that these results might contribute to resolving
a cosmological version of the information paradox Goheer:2002vf ;
Aalsma:2020aib ; Aalsma:2021bit ; Aalsma:2022eru , or equivalently, provide
insights into the underlying microscopic holographic description of the de
Sitter static patch Anninos:2011af ; Nomura:2017fyh ; Leuven:2018ejp ;
Susskind:2021dfc ; Susskind:2021omt  (see also the excellent reviews
Witten:2001kn ; Klemm:2004mb ; Anninos:2012qw ).

As realized long ago, compared to neutral black holes in de Sitter space, the
configuration space of charged black holes is quite rich, see for instance
MELLOR1989361 ; Mellor:1989wc ; Romans:1991nq ; Mann:1995vb ; Cardoso:2004uz .
Next to extremal black holes, bounding the maximal charge of a given mass
black hole, there also exists an upper bound on the mass in a given charge
sector, extending the neutral Nariai black hole to its charged analogue.
Except for the special extremal, Nariai and so-called lukewarm solutions, the
absence of thermal equilibrium implies that the Euclidean solution will be
singular in general, a priori obscuring the meaning of the corresponding on-
shell action. We will nevertheless proceed with computing the on-shell action,
since we argue the singular Euclidean solutions are meaningful as instantons
of a constrained path integral. Although (partial) results for the on-shell
Euclidean action of charged de Sitter black holes have been reported before
MELLOR1989361 ; Mellor:1989wc ; Mann:1995vb ; Chao:1997osu ; Dias:2004px , see
also the recent paper Wang:2022sbp , in this work we will present a completely
general derivation, for any charge and mass and in any number of dimensions.
Moreover, we will correct the on-shell action in the extremal and so-called
ultracold limits.

We will start with some preliminaries in section 2 on charged black holes in
de Sitter, setting up our conventions. We will then confirm in section 3 that
the action of a general Euclidean Reissner-Nordström-de Sitter (RNdS) solution
with conical singularities at the black hole and cosmological horizons, in
arbitrary dimensions $D>3$, after imposing the (nonlinear) Smarr relation
between the two horizons and including a boundary term for the electromagnetic
field, is independent of the Euclidean time period and is equal to

$I=-\frac{\mathcal{A}_{+}+\mathcal{A}_{++}}{4G}=-S_{RNdS}\,.$ (1)

Here $\mathcal{A}_{+}$ is the area of the (outer) black hole horizon and
$\mathcal{A}_{++}$ is the cosmological horizon area. This formula suggests
that the total entropy of the RNdS static patch between the two horizons is
given by the sum of the horizon entropies. Applying this result we then
compute the pair production rate of any static black hole in de Sitter space
using the formalism of constrained instantons in section 4. We end with a
comment on the Festina Lente bound FestinaLente:2019  in section 5, and a
short summary and some open questions in section 6.

##  2 Charged black holes in de Sitter space

In this section we review some geometric and thermodynamic properties of
nonrotating, charged black holes in a de Sitter background (see also
Tangherlini:1963bw ; Mellor:1989wc ; Romans:1991nq ; Mann:1995vb ;
Astefanesei:2003gw ; Cardoso:2004uz ; Dolan:2013ft ; Kubiznak:2015bya ;
Montero:2019ekk ). We mostly keep the number of spacetime dimensions $D$
arbitrary, but the plots are for the case $D=4.$ Specific formulas in four
dimensions that are useful for making plots can be found in Appendix A.

###  2.1 Reissner-Nordström-de Sitter geometry and phase diagram

Charged black holes in asymptotically de Sitter spacetime are described by the
Reissner-Nordström-de Sitter solution to the Einstein-Maxwell field equation
with a positive cosmological constant $\Lambda$. In static coordinates the
RNdS line element reads

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\Omega_{D-2}^{2}\,,$ (2)

where $d\Omega_{D-2}^{2}$ is the metric on the round unit ($D-2$)-sphere, and
the blackening factor $f(r)$ is given by

$f(r)=1-\frac{r^{2}}{l^{2}}-\frac{m}{r^{D-3}}+\frac{q^{2}}{r^{2(D-3)}}\,.$ (3)

The length scale $l$ is the de Sitter curvature radius, which is related to
the cosmological constant by $\Lambda=(D-1)(D-2)/2l^{2}$. The mass and charge
parameters, respectively $m$ and $q$, are related to the mass $M$ and electric
charge $Q$ as

$M=\frac{(D-2)\,\Omega_{D-2}}{16\pi G}\,m\,,\qquad
Q=\sqrt{\frac{(D-3)(D-2)}{8\pi G}}\,q\,,$ (4)

where $\Omega_{D-2}$ is the volume of the unit ($D-2$)-sphere. Here $M$ is
defined in a similar way as the ADM mass in asymptotically flat space
Myers:1986un . However, unlike in asymptotically flat and AdS space, the
asymptotic charges in de Sitter are “conserved in space” rather than in time,
since the Killing vector field $\partial_{t}$ is spacelike near future and
past infinity Ghezelbash:2001vs . We will ignore this subtlety and refer to
$M$ simply as the “mass” of the black hole. Further, we assume throughout the
paper that $m,q>0$.

The RNdS solution could be electrically or magnetically charged, but in this
paper we only consider the solution that is supported by an electric gauge
field

$F=-\frac{Q}{r^{D-2}}dt\wedge dr.$ (5)

For a fixed number of dimensions, the RNdS geometry is fully characterized by
three independent parameters $m,q$ and $l$ (or, equivalently, $M,Q$ and
$\Lambda$). This implies the blackening factor $f(r)$ has four real roots,
denoted by $r_{--}$, $r_{-}$, $r_{+}$ and $r_{++}$ in increasing order (for
$D\geq 5$ there are additional complex roots). The negative root at $r=r_{--}$
is deemed unphysical. The other three real roots are positive and they
correspond to horizon radii: $r=r_{-}$ is the inner black hole horizon,
$r_{+}$ is the outer black hole horizon, and $r_{++}$ is the cosmological
horizon (see Figure 1).

a) Arbitrary $m$ and $q$

b) Charged Nariai black hole

c) Extremal or cold black hole

d) Ultracold black hole

Figure 1: Plots of the blackening factor $f$ as a function of the radius
$r>0$, for a) arbitrary $m$ and $q$, where $r_{-}\neq r_{+}\neq r_{++}$; b)
the Nariai black hole, where $r_{+}=r_{++}\equiv r_{N}$; c) the extremal black
hole, where $r_{-}=r_{+}\equiv r_{E}$; d) the ultracold black hole, where
$r_{-}=r_{+}=r_{++}\equiv r_{U}$.

In order to avoid naked singularities $m$ and $q$ should satisfy the following
bounds

$m_{E}\leq m\leq m_{N}\,,\qquad q_{N}\leq q\leq q_{E}\,.$ (6)

Outside this domain the roots $r_{-},r_{+}$ and $r_{++}$ can become complex.
The left of Figure 2 shows the phase diagram of the RNdS solution in terms of
$m$ and $q$, where we see that the domain (6) takes the form of a “shark fin”
Montero:2019ekk . Configurations inside the shark fin (orange region) have
three distinct horizons, whereas the left and right edges of the shark fin are
associated with the extreme cases where two of the horizon radii coincide,
respectively:

  * •

Extremal solution: The lower bound on $m$ and the upper bound on $q$
correspond to the case where the inner and outer horizon radii coincide,
$r_{E}\equiv r_{-}=r_{+}$. The near-horizon topology is $AdS_{2}\times
S^{D-2}$ and the temperature of the extremal black hole vanishes. In the de
Sitter context this is also known as the cold solution Romans:1991nq .

  * •

Charged Nariai solution: The upper bound on $m$ and the lower bound on $q$
arise from the case where the outer horizon radius and cosmological horizon
radius are the same, $r_{N}\equiv r_{+}=r_{++}$. The near-horizon topology is
$dS_{2}\times S^{D-2}$, and the outer black hole horizon and cosmological
horizon are in thermal equilibrium.

The mass and charge of the cold and Nariai black holes can be found by solving
$f(r_{E,N})=f^{\prime}(r_{E,N})=0$, which yields the following expressions in
terms of the horizon radii

$m_{E,N}=2\left(1-\frac{D-2}{D-3}\frac{r_{E,N}^{2}}{l^{2}}\right)\,r_{E,N}^{D-3}\,,\qquad
q_{E,N}^{2}=\left(1-\frac{D-1}{D-3}\frac{r_{E,N}^{2}}{l^{2}}\right)\,r_{E,N}^{2(D-3)}\,.$
(7)

a) “Shark fin” phase diagram for $m/m_{U}$ vs. $q/q_{U}$

b) Phase diagram for $\alpha$ vs. $\rho$.

Figure 2: Phase diagram of RNdS parameterized by $(m/m_{U},q/q_{U})$ (left)
and $(\alpha,\rho)$ (right). In the orange region the bounds (6) are
satisfied, and in the turquoise region (left) the metric has a naked
singularity. The blue boundary corresponds to the Nariai limit, and the green
edge to the extremal (cold) limit. Those boundaries meet at the magenta
tip/line (left/right), associated to the ultracold limit. The red line
corresponds to the lukewarm solution, for which the black hole and
cosmological horizon temperatures are identical, and the yellow dot is the
Nariai-lukewarm limit. In the light orange region black holes are colder than
the cosmological horizon, whereas in the dark orange region they are hotter.

Since for any fixed mass parameter $m$, the charge parameter $q$ is bounded
from above (by the cold limit), and similarly for any fixed $q$ there is an
upper bound on $m$ (due to Nariai limit), there exists a de Sitter black hole
with maximal mass and charge, known as:

  * •

Ultracold solution: In this case all three horizon radii coalesce,
$r_{U}\equiv r_{-}=r_{+}=r_{++}$. This corresponds to the tip in the shark fin
diagram where the extremal and Nariai boundaries meet. All three horizons have
vanishing temperature in this case, and the near-horizon geometry is
$M_{2}\times S^{D-2},$ where $M_{2}$ is two-dimensional Minkowski space.

The horizon radius, mass and charge for the ultracold black hole are obtained
by solving $f(r_{U})=f^{\prime}(r_{U})=f^{\prime\prime}(r_{U})=0$, leading to

$r_{U}=\frac{D-3}{\sqrt{(D-1)(D-2)}}\,l\,,\qquad
m_{U}=4\frac{r_{U}^{D-3}}{D-1}\,,\qquad
q_{U}=\frac{r_{U}^{D-3}}{\sqrt{D-2}}\,.$ (8)

Even though the ultracold black hole has the largest mass and charge, its
(outer) horizon radius is not maximal. The extremal horizon radius $r_{E}$ has
the range $[0,r_{U}]$, whereas the Nariai horizon radius $r_{N}$ ranges from
$r_{U}$ to $r_{N}(q=0)=l\sqrt{\frac{D-3}{D-1}}$. Hence, the neutral Nariai
black hole has the largest outer horizon radius.

Finally, on the right of Figure 2 we rescaled the shark fin diagram in order
to accommodate for the fact that the interval $[m_{E},m_{N}]$ shrinks with
increasing $q$. This opens up the region near the ultracold solution and makes
the so-called lukewarm solution (red line) more visible, for which the black
hole and cosmological horizon have the same temperature. We introduced a
dimensionless mass parameter $\alpha\in[0,1]$, defined via

$m=\alpha\,m_{N}+(1-\alpha)\,m_{E}\,.$ (9)

This allows us to better compare and visualize thermodynamic quantities, such
as the horizon temperatures and entropies. Note that $\alpha$ is defined such
that low values correspond to small (allowed) masses, and high values to large
(allowed) masses. Following SdSaction , we also introduce the ratios

$\mu\equiv m/m_{N}\,,\qquad\mu_{E}\equiv m_{E}/m_{N}\,,$ (10)

in terms of which the relation for $\alpha$ becomes

$\mu=\alpha+(1-\alpha)\,\mu_{E}.$ (11)

Finally, for the charge we often use the dimensionless ratio

$\rho\equiv q/q_{U}\,,$ (12)

which ranges from $\rho=0$ (neutral black hole) to $\rho=1$ (ultracold black
hole).

###  2.2 Thermodynamics: horizon temperatures, first law and Smarr formula

Next we briefly discuss the thermodynamics of the Reissner-Nordström-de Sitter
spacetime. Since RNdS has three event horizons, there are three associated
temperatures and entropies

$T_{h}=\frac{\kappa_{h}}{2\pi}\,,\qquad
S_{h}=\frac{\mathcal{A}_{h}}{4G}\,,\qquad\text{with}\quad h=\\{-,+,++\\}\,,$
(13)

where $\kappa$ denotes the surface gravity and $\mathcal{A}$ the horizon area.
For general values of the mass and charge, the surface gravities of the three
horizons are different, hence the horizons are not in thermodynamic
equilibrium. However, as we will discuss in more detail below, for specific
values of $m$ and $q$ (two or three of) the surface gravities coincide, hence
the associated horizon temperatures are the same. We will focus on the
thermodynamics of the static patch between the outer black hole horizon and
the cosmological horizon $r\in[r_{+},r_{++}]$, since that is the region
captured by the Euclidean RNdS geometry.

#### Horizon temperatures

a) Black hole temperature $T_{+}$

b) Cosmological horizon temperature $T_{++}$

Figure 3: Heat maps in terms of $(\alpha,\rho)$ for the black hole temperature
(left) and the cosmological horizon temperature (right). For low mass and
charge (in the white region at the bottom left corner) the black hole is
extremely hot.

The horizon temperature depends on the normalization of the horizon generating
Killing vector $\xi$, which cannot be fixed at the asymptotic boundary in de
Sitter space, as is usually done for Schwarzschild spacetime. Bousso and
Hawking Bousso:1996au  normalized the timelike Killing vector of
Schwarzschild-de Sitter spacetime instead on the geodesic at constant angular
variables between the black hole and the cosmological horizon. The geodesic
lies at a particular radius $r_{\mathcal{O}}$ where an observer can stay in
place without accelerating, i.e., the gravitational pull from the black hole
and cosmological acceleration balance out. The blackening factor attains a
maximum at this radius, $f^{\prime}(r_{\mathcal{O}})=0$, and the Killing
vector is normalized such that $\xi^{2}=-1$ at $r=r_{\mathcal{O}}$. In the
limit to ($\Lambda=0$) Schwarzschild spacetime the radius goes to infinity,
and in the limit to empty ($M=0$) de Sitter spacetime the radius goes to zero.
Hence, the radius $r_{\mathcal{O}}$ agrees in these limits with the standard
location where the Killing vector is normalized.

In RNdS, the equation $f^{\prime}(r_{\mathcal{O}})=0$ has two positive roots:
a local minimum between the inner and outer horizon, and a local maximum
between the outer horizon and cosmological horizon. We normalize the Killing
vector on the local maximum to mimic the normalization for the timelike
Killing vector of Schwarzschild-de Sitter. The blackening factor evaluated at
$r_{\mathcal{O}}$ is given by

$f(r_{\mathcal{O}})=1-\frac{D-1}{D-3}\frac{r_{\mathcal{O}}^{2}}{l^{2}}-\frac{q^{2}}{r_{\mathcal{O}}^{2(D-3)}}\,.$
(14)

The normalized horizon generating Killing vector is
$\xi=\partial_{t}/\sqrt{f(r_{\mathcal{O}})}$ and the surface gravity defined
with respect to this Killing vector is given by
$\kappa_{h}=f^{\prime}(r_{h})/(2\sqrt{f(r_{\mathcal{O}})})$. Hence, for the
RNdS solution the temperatures of the different horizons takes the following
form

$\displaystyle
T_{h}=\frac{D-3}{4\pi\,r_{h}\sqrt{f(r_{\mathcal{O}})}}\left|1-\frac{D-1}{D-3}\frac{r_{h}^{2}}{l^{2}}-\frac{q^{2}}{r_{h}^{2(D-3)}}\right|\,.$
(15)

We will now discuss several special cases for which the horizon temperatures
are the same. The normalization with respect to the geodesic observer between
$r_{+}$ and $r_{++}$ is particularly appropriate for the Nariai black hole,
since the outer horizon and cosmological horizon temperatures remain finite in
this limit, which is expected since the near-horizon Nariai geometry is two-
dimensional pure de Sitter space times a sphere. (The temperatures $T_{+}$ and
$T_{++}$ defined with respect to the Killing vector $\partial_{t}$ seem
inappropriate for the Nariai solution since they vanishes in that limit.) In
the Nariai case the black hole and cosmological horizons have the same radius
as the geodesic, i.e., $r_{+}=r_{++}=r_{\mathcal{O}}=r_{N}$. Moreover, the
temperatures of the black hole and cosmological horizon are the same, and
equal to

$T_{N}=\frac{\sqrt{D-3}}{2\pi\,r_{N}}\sqrt{1-(D-2)\,\frac{q^{2}}{r_{N}^{2(D-3)}}}\,.$
(16)

For the critical values $q=0$ and $q=q_{U}$ the Nariai temperature becomes,
respectively,

$T_{N}(q=0)=\frac{\sqrt{D-1}}{2\pi\,l}\,,\qquad T_{U}\equiv
T_{N}(q=q_{U})=0\,.$ (17)

The former is the temperature of the neutral Nariai black hole, which agrees
with previous findings Bousso:1996au ; Svesko:2022txo ; SdSaction , and the
latter is the temperature of the ultracold solution which vanishes for all
three horizons.

Furthermore, in the extremal case the temperatures of the outer and inner
horizons vanish,111This is not the case if the temperatures is normalized with
respect to the other positive root $\tilde{r}_{\mathcal{O}}$ of
$f^{\prime}(r)=0$, where the blackening factor has a local minimum, located
between $r_{-}$ and $r_{+}$. For this other normalization the black hole
temperature does not go to zero in the extremal limit,
$\tilde{T}_{E,-}=\tilde{T}_{E,+}\neq 0$, whereas the Nariai black hole
temperature does vanish, $\tilde{T}_{N,+}=\tilde{T}_{N,++}=0$, and the
temperature of the ultracold black hole as well, $\tilde{T}_{U}=0$.
$T_{E,+}=0\,,$ and the temperature of the cosmological horizon is finite and
given by

$T_{E,++}=\frac{3-D}{4\pi\,r_{E,++}\sqrt{f(r_{\mathcal{O}})}}\left(1-\frac{D-1}{D-3}\frac{r_{E,++}^{2}}{l^{2}}-\frac{q^{2}}{r_{E,++}^{2(D-3)}}\right)\,,$
(18)

where $r_{E,++}$ stands for the cosmological horizon radius of the extremal
solution. For zero charge $q=0$ ($r_{E,++}=l$) the temperature reduces to the
empty de Sitter temperature $T_{dS}$ (20) and for $q=q_{U}$ ($r_{E,++}=r_{U}$)
the temperature vanishes, cf. (17).

Besides the Nariai solution there is another thermal equilibrium solution for
which the (outer) black hole horizon and cosmological horizon temperature are
the same. This is the so-called _lukewarm_ solution Romans:1991nq . In Figure
4 we show that the black hole temperature $T_{+}$ and the cosmological horizon
temperature $T_{++}$ cross for specific combinations of $\alpha$ and $\rho$,
or equivalently $m$ and $q$, which corresponds to the lukewarm case.

a) $\rho=0,1/2$.

b) $\alpha=0,1/2$

Figure 4: Horizon temperatures (a) as a function of the mass parameter
$\alpha$ for fixed $\rho=0,1/2$ and (b) as a function of the charge parameter
$\rho$ for $\alpha=0,1/2$. The black hole temperature $T_{+}$ is shown in blue
and the cosmological horizon temperature $T_{++}$ in orange. Dark colors
correspond to $\alpha,\rho=0$, and light colors to $\alpha,\rho=1/2.$

When solving for $T_{++}=T_{+}$, one finds that the mass and charge parameter
are related by $m_{L}=2q_{L}$ in $D=4$. In higher dimensions the relation
between the (lukewarm) mass and charge is more complicated, since it depends
non-trivially on the horizon radii, cf. equation (81). Further, the
temperature of the lukewarm black hole, $T_{+}=T_{++}(\equiv T_{L})$, can be
expressed in general dimensions as a function of the outer horizon radius
$r_{L,+}$ and the cosmological horizon radius $r_{L,++}$

$T_{L}=\frac{1}{2\pi\sqrt{f(r_{\mathcal{O}})}\,l^{2}}\left[r_{L,++}+r_{L,+}^{D-2}(r_{L,+}+r_{L,++})\left(\frac{D-3}{r_{L,+}^{D-2}-r_{L,+}r_{L,++}^{D-3}}+\frac{D-2}{r_{L,+}^{D-2}-r_{L,++}^{D-2}}\right)\right]\,.$
(19)

For $q=0$ and $q=q_{N}$ this reduces to, respectively,

$T_{dS}\equiv T_{L}(q=0)=\frac{1}{2\pi l}\,,\qquad T_{LN}\equiv
T_{L}(q=q_{N})=\frac{\sqrt{D-1}}{\sqrt{6}\pi l}\,.$ (20)

The former is the de Sitter temperature and the latter the lukewarm
temperature of the Nariai black hole.

#### First law and generalized Smarr formula

The horizon temperatures, entropies, and charge are related by the first law
for the static patch between the black hole and cosmological horizon (see,
e.g., Dolan:2013ft )

$T_{+}\delta S_{+}+T_{++}\delta S_{++}+(\Phi_{+}-\Phi_{++})\delta Q=0\,.$ (21)

Here $\Phi_{+}$ and $\Phi_{++}$ are the static electric potentials evaluated
at the black hole and cosmological horizon, respectively,

$\Phi_{+,++}=\frac{Q}{(D-3)r_{+,++}^{D-3}}\,.$ (22)

Without variations the thermodynamic quantities satisfy the (generalized)
Smarr formula

$T_{+}S_{+}+T_{++}S_{++}+\frac{D-3}{D-2}(\Phi_{+}-\Phi_{++})Q-\frac{\Theta\Lambda}{(D-2)4\pi
G}=0\,.$ (23)

The quantity $\Theta$ is the “thermodynamic volume” or “Killing volume”,
defined as Jacobson:2018ahi

$\Theta\equiv\int_{\Sigma}dV\sqrt{-\xi^{2}}\,,$ (24)

where $\Sigma$ is the spatial slice between the outer horizon and cosmic
horizon, and $\xi$ is the time translation generating Killing vector.

##  3 Euclidean action of Reissner-Nordström-de Sitter space

###  3.1 Hartle-Hawking wavefunction and instantons

We are interested in computing the pair creation rate of charged black holes
in a de Sitter background for any mass and charge. The pair creation of such
black holes is described by the propagation from nothing to a spatial surface
$\Sigma$ with topology $S^{D-2}\times S^{1}$ for generic mass and charge (and
$\Sigma$ has topology $S^{D-2}\times R$ for extreme and ultracold black
holes). Given certain boundary conditions on $\Sigma$ the probability
amplitude for this process is given by the no-boundary Hartle-Hawking
wavefunction, which can be represented formally as a Euclidean path integral
Hawking:1995ap ; Mann:1995vb

$\Psi=\int[dg][dA]e^{-I_{1/2}[g,A]}\,.$ (25)

The integral is over all metrics and gauge fields that agree with the boundary
data on $\Sigma$, and $I_{1/2}$ is the action of a Euclidean manifold with
boundary $\Sigma$. If there exists a Euclidean saddle solution that
interpolates within the boundary, then in the saddle point approximation the
wavefunction is estimated by $\Psi\approx e^{-I_{1/2}}$, where $I_{1/2}$ is
the on-shell action of the instanton. Moreover, the pair creation rate of
fixed charge and mass black holes is approximated by the square of the
amplitude, i.e.,

$\Gamma=\Psi^{*}\Psi\approx e^{-2I_{1/2}}.$ (26)

We are interested, however, in the probability to produce a pair of
_arbitrary_ mass and charge black holes. We will argue in the next section
that this is computed by the full gravitational path integral (without any
boundary conditions) which can be written as an integral of $\Psi^{*}\Psi$
over the mass and charge parameters

$\int[dg][dA]e^{-I[g,A]}=\int d\alpha d\rho\Psi^{*}\Psi$ (27)

Here $I[g,A]$ is the Euclidean action of the entire manifold. In the saddle
point approximation the partition function is given by $\int d\alpha
d\rho\exp[-I]$, where $I=2I_{1/2}$ is the action of a “bounce” Euclidean
solution. In this section we compute the on-shell action of these “bounce”
Euclidean solutions, which is twice the action of a standard instanton with
boundary $\Sigma$, and in the next section we will calculate the pair creation
rate.

The Euclidean RNdS solutions provide the instantons for the pair creation of
charged de Sitter black holes. Their metric can be obtained from the
Lorentzian metric by analytically continuing $t\to it=\tau$,

$ds^{2}=f(r)d\tau^{2}+f^{-1}(r)dr^{2}+r^{2}d\Omega_{D-2}^{2},$ (28)

where $f(r)$ is given by (3) and the Euclidean time $\tau$ is periodically
identified with inverse temperature $\beta$. For arbitrary $\beta$ the
Euclidean RNdS spacetime is regular everywhere except at the black hole and
cosmological horizon, where there are conical singularities. One of the
conical singularities can be removed by choosing $\beta$ appropriately.
However, it is not possible to remove both conical singularities at the same
time, except in the extreme cases discussed in the previous section. Indeed,
the extremal, Nariai, cold and lukewarm Euclidean spacetimes are smooth
everywhere, whereas for a generic mass and charge the Euclidean RNdS geometry
is singular at (at least one of) the Euclidean horizons. The extreme RNdS
solutions are, therefore, proper instantons of the Euclidean path integral,
but we will argue in section 4 that generic Euclidean RNdS geometries are
instantons of a constrained path integral where the mass and charge are kept
fixed.

###  3.2 On-shell Euclidean action

In order to compute the on-shell action of the entire Euclidean RNdS section
$\mathcal{M}$ we cut the compact geometry in half. We first calculate the
action of the two “caps” separately, denoted by $\mathcal{M}_{1/2}$, and then
sum the results to obtain the action of the entire Euclidean section. This
cutting and pasting procedure is consistent with the path integral
representation in (27). Moreover, it seems necessary due to a subtlety related
to the boundary term of the Maxwell action, which localizes on the conical
singularities, and does not vanish when the two caps are glued together. We
cut the Euclidean section along the spatial slice that has zero extrinsic
curvature, which corresponds to the standard choice of boundary $\Sigma$ for
the Hartle-Hawking wavefunction.

The appropriate off-shell action of half the Euclidean section
$\mathcal{M}_{1/2}$, for boundary data that fix the induced metric and
electric charge on $\Sigma$, is given by Braden:1990hw ; Hawking:1995ap

$I_{1/2}=-\int_{\mathcal{M}_{1/2}}\\!\\!d^{D}x\sqrt{g}\left[\frac{1}{16\pi
G}(R-2\Lambda)-\frac{1}{4}F^{2}\right]-\int_{\Sigma}d^{D-1}x\sqrt{h}F^{\mu\nu}n_{\mu}A_{\nu}\,.$
(29)

Here $n_{\mu}$ is the unit normal to the boundary $\Sigma$ and $h$ is the
determinant of the induced metric on $\Sigma$. We did not include the Gibbons-
Hawking-York boundary term, since it vanishes on a surface $\Sigma$ that has
zero extrinsic curvature. We want to evaluate the action on half of the
Euclidean RNdS geometry where the Euclidean time has the range
$\tau\in[0,\beta/2]$, for an arbitrary inverse temperature $\beta$. The Ricci
scalar of the RNdS geometry has a regular part and a divergent part

$R=R_{{bulk}}+R_{{con}}\,,$ (30)

where the second part is due to the conical singularities at the two Euclidean
horizons and is given by

$R_{con}=4\pi\delta(r=r_{+})+4\pi\delta(r=r_{++})\,.$ (31)

To begin with, we compute the contribution from the regular part of the Ricci
scalar to the on-shell Euclidean action. Away from the conical singularities
the RNdS spacetime satisfies the Einstein equation

$R_{\mu\nu}-2\Lambda g_{\mu\nu}=8\pi
G\left(T_{\mu\nu}-\frac{T}{D-2}g_{\mu\nu}\right)\,.$ (32)

For the action (29) the Maxwell energy-momentum tensor is given by

$T_{\mu\nu}=F_{\mu\alpha}{F_{\nu}}^{\alpha}-\frac{1}{4}g_{\mu\nu}F^{2}\,.$
(33)

Taking the trace of the Einstein equation yields an on-shell expression for
the regular part of the Ricci scalar

$R_{bulk}=\frac{2D}{D-2}\Lambda+\frac{D-4}{D-2}4\pi GF^{2}\,.$ (34)

Hence, the bulk contribution to the on-shell action is given by

$I_{1/2,bulk}=-\frac{2}{D-2}\int_{\mathcal{M}_{1/2}}d^{D}x\sqrt{g}\left(\frac{1}{8\pi
G}\Lambda-\frac{1}{4}F^{2}\right)\,.$ (35)

Further, it can be shown that the square of the electromagnetic field tensor
(5) in Euclidean static coordinates is

$F_{\mu\nu}F^{\mu\nu}=-2\left(\frac{Q}{r^{D-2}}\right)^{2}\,.$ (36)

By inserting this into (35) we find the bulk action becomes

$I_{1/2,bulk}=-\frac{\beta\Theta\Lambda}{8\pi G(D-2)}-\frac{\beta
Q}{2(D-2)}(\Phi_{+}-\Phi_{++})\,.$ (37)

Here, we used equation (22) for the electric potentials, and we noted that the
Euclidean spacetime volume $\int_{\mathcal{M}_{1/2}}d^{D}x\sqrt{g}$ is equal
to $\beta\Theta/2$, where $\Theta$ is the Killing volume (24).

Second, integrating the conical part of the Ricci scalar yields

$I_{1/2,con}=-\frac{\mathcal{A}_{+}\epsilon_{+}}{8\pi
G}-\frac{\mathcal{A}_{++}\epsilon_{++}}{8\pi
G}=-\frac{\mathcal{A}_{+}}{8G}-\frac{\mathcal{A}_{++}}{8G}+\frac{\beta}{2}\left(\frac{\mathcal{A}_{+}\kappa_{+}}{8\pi
G}+\frac{\mathcal{A}_{++}\kappa_{++}}{8\pi G}\right)\,,$ (38)

where $\epsilon_{+,++}=\pi(1-n_{+,++})$ is the deficit angle associated to the
conical singularities. The range of the polar angle of one cap is $\pi
n_{+,++}$, which is related to the Euclidean time period by $\pi
n_{+,++}=\beta\kappa_{+,++}/2$. By adding the conical deficit and bulk terms
and using the Smarr formula (23), the on-shell action becomes

$I_{1/2,bulk}+I_{1/2,con}=-\frac{\mathcal{A}_{+}}{8G}-\frac{\mathcal{A}_{++}}{8G}-\frac{\beta
Q}{2}(\Phi_{+}-\Phi_{++})\,.$ (39)

Next let us add the boundary term of the Maxwell action, $I_{bdy}$, which
originates from the fact that we are fixing the charge on the boundary, which
is equivalent to fixing $n_{\mu}F^{\mu i}$ on the boundary Hawking:1995ap . In
order to compute the Maxwell boundary term we need to choose a gauge for the
gauge potential. A suitable gauge choice that is regular everywhere on the
Euclidean geometry is Mann:1995vb

$A_{1}=-i\frac{Q\tau}{r^{D-2}}dr\,.$ (40)

Further, we choose coordinates on the boundary $\Sigma$ such that it is
located at the surface $\tau=0,\beta/2$ and the coordinate $r$ runs from the
black hole horizon to the cosmological horizon. Then, the boundary term can be
computed to be

$I_{1/2,bdy}=-\int_{\Sigma}d^{D-1}x\sqrt{h}F^{\mu\nu}n_{\mu}A_{\nu}=\frac{\beta
Q}{2}(\Phi_{+}-\Phi_{++})\,.$ (41)

Notice that the boundary term localizes on the two Euclidean horizons, and
exactly cancels against the term proportional to $\beta$ in (39). Summing the
three different contributions to the action yields the total Euclidean action
of $\mathcal{M}_{1/2}$

$I_{1/2,RNdS}=I_{1/2,bulk}+I_{1/2,con}+I_{1/2,bdy}=-\frac{\mathcal{A}_{+}+\mathcal{A}_{++}}{8G}\,.$
(42)

The action is the same for both halves of the Euclidean section, and hence the
on-shell action of the entire Euclidean RNdS section $\mathcal{M}$ is given by
minus the sum of the black hole entropy and the cosmological horizon entropy

$I_{RNdS}=-\frac{\mathcal{A}_{+}+\mathcal{A}_{++}}{4G}=-S_{RNdS}\,.$ (43)

We emphasize that this expression is independent of the Euclidean time period
$\beta$, and that it holds in any number of dimensions for arbitrary mass and
charge. This is the action in the canonical ensemble where the temperature and
charge are kept fixed.222In the grand canonical ensemble that fixes the
electric potential instead of the charge, there is no boundary term for the
gauge field and the on-shell action of $\mathcal{M}_{1/2}$ is given by (39)
instead. The same result for the on-shell action $I_{1/2,RNdS}$ was obtained
in four dimensions in Chao:1997osu ; Wang:2022sbp , which was used to compute
the pair creation of fixed mass and charge black holes in de Sitter. For zero
charge the result (43) is consistent with earlier findings for the action of
Euclidean Schwarzschild-de Sitter space Chao:1997em ; Bousso:1998na ;
Gregory:2014 ; SdSaction . Further, the on-shell action of the extreme
Euclidean solutions (proper instantons) was computed before in Mellor:1989wc ;
Hawking:1995ap ; Mann:1995vb ; Cardoso:2004uz , which we discuss in more
detail below.

###  3.3 Extreme cases: cold, lukewarm, Nariai and ultracold instantons

In the literature the action is usually only evaluated for the extreme
Euclidean RNdS solutions: cold, lukewarm, Nariai and ultracold instantons.
This is because they are proper stationary points of the Euclidean Einstein
action, i.e., they are regular and hence satisfy the Einstein equation
everywhere. Below we specialize to these true instanton cases and compare our
results for the on-shell action to those in the literature, especially to the
results in the comprehensive paper by Mann and Ross Mann:1995vb . From our
general expression (43) it follows that the on-shell actions of the extremal,
lukewarm, Nariai and ultracold instantons are, respectively,

$\displaystyle I_{E}$
$\displaystyle=-\frac{\mathcal{A}_{E,+}+\mathcal{A}_{E,++}}{4G}=-\frac{\Omega_{D-2}}{4G}\,\left(r_{E}^{D-2}+r_{E,++}^{D-2}\right)\,,$
(44) $\displaystyle I_{L}$
$\displaystyle=-\frac{\mathcal{A}_{L,+}+\mathcal{A}_{L,{++}}}{4G}=-\frac{\Omega_{D-2}}{4G}\,\left(r_{L,+}^{D-2}+r_{L,{++}}^{D-2}\right)\,,$
(45) $\displaystyle I_{N}$
$\displaystyle=-2\frac{\mathcal{A}_{N}}{4G}=-\frac{\Omega_{D-2}}{2G}\,r_{N}^{D-2}\,,$
(46) $\displaystyle I_{U}$
$\displaystyle=-2\frac{\mathcal{A}_{U}}{4G}=-\frac{\Omega_{D-2}}{2G}\,r_{U}^{D-2}\,.$
(47)

Obviously, since the expression (43) for the Euclidean action is a continuous
function of the mass and charge, the action of the proper instantons is also
equal to minus the sum of the black hole and cosmological horizon entropy. On
the other hand, Mann and Ross computed the action of each proper instanton
separately by using the metric of each instanton. Their results for the
Euclidean action of the lukewarm and charged Nariai instantons agree with
(half) the expressions (46) and (45) above. Furthermore, following earlier
work by Hawking, Horowitz and Ross Hawking:1994ii , they found for the
extremal instanton that the black hole horizon makes no contribution to the
action and hence to the entropy. This is because only the cosmological horizon
is part of the instanton, and the black hole horizon is infinitely far away
from any other point. Thus, the extremal black hole entropy vanishes according
to them, even though its horizon area is nonzero (see also Carroll:2009maa ).
This is in disagreement, however, with the modern view on the entropy of
extremal black holes (see e.g. Nayak:2018qej ), and with the microstate
counting of extremal black holes in string theory Strominger:1996sh .
Moreover, a zero extremal black hole entropy would imply that the RNdS action
is discontinuous as a function of the mass and charge, which contradicts our
result for the action. The same confusion about extremal black hole entropy
seems to arise in the more recent papers Dias:2004px ; Wang:2022sbp  on the
Euclidean RNdS action.

Furthermore, for the ultracold instanton Mann and Ross obtained two different
results for the on-shell action depending on the near-horizon metric. The
action of one of the ultracold metrics, called $UC1,$ agrees with the limit of
their action of the extremal solution as it approaches the ultracold solution,
but it is not the same as the ultracold limit of the action of the Nariai
solution. The action of the other ultracold metric $UC2$ vanishes identically,
since the metric describes $2D$ Minkowski spacetime (times a sphere) in
standard coordinates without horizons. Our result does not agree with either
of these actions. In fact, our action for the ultracold solution can be
obtained as a special case of the action for both the extremal solution and
the charged Nariai solution. In Figure 5 we plotted the fraction of the action
of the proper instantons and the action for de Sitter space against the
dimensionless mass $m/m_{U}.$ One can see that the ultracold point (magenta
point) is connected to both the Nariai branch (blue line) and the extremal
branch (green line). This plot should be contrasted with Figure 3 in the paper
by Mann and Ross Mann:1995vb , in which the Nariai branch and extremal branch
are not connected.

Figure 5: Fraction of the Euclidean action of proper instantons over the
action of pure de Sitter $I/I_{dS}$ vs. the mass parameter $m/m_{U}$. The blue
line corresponds to charged Nariai $I_{N}$, the green line to the extremal
solution $I_{E}$, and the red line to the lukewarm solution $I_{L}$. The blue
dashed curve corresponds to the action of the Schwarzschild-de Sitter black
hole. $I_{LN}$ and $I_{U}$ are shown in yellow and in magenta, respectively.

As a function of the mass the action of the proper instantons is cumbersome.
But in terms of the charge the Euclidean on-shell action takes a relatively
simple form in $D=4$

$\displaystyle I_{E}$ $\displaystyle=-\frac{\pi}{\Lambda
G}\left(3-\sqrt{2+2\rho_{E}^{2}-2\sqrt{1-\rho_{E}^{2}}}\right)\,,$ (48)
$\displaystyle I_{L}$ $\displaystyle=-\frac{\pi}{\Lambda
G}\left(3-\sqrt{3}\rho_{L}\right)\,,$ (49) $\displaystyle I_{N}$
$\displaystyle=-\frac{\pi}{\Lambda G}\left(1+\sqrt{1-\rho_{N}^{2}}\right)\,,$
(50) $\displaystyle I_{dS}$ $\displaystyle=-\frac{3\pi}{\Lambda G}\,,\quad
I_{NN}=-\frac{2\pi}{\Lambda G}\,,\quad I_{LN}=-\frac{3\pi}{2\Lambda G}\,,\quad
I_{U}=-\frac{\pi}{\Lambda G}\,,$ (51)

where $I_{NN}$ denotes the action of the neutral Nariai solution
($\rho_{NN}=0$) and $I_{LN}$ the action of the lukewarm-Nariai solution
($\rho_{LN}=\frac{\sqrt{3}}{2}$). We plotted the fraction of the proper
instanton actions and the de Sitter action against the charge parameter
$\rho=q/q_{U}$ in Figure 6. We note the actions of the lukewarm (red line) and
cold (green line) solutions coincide at $\rho_{dS}=0$ and are equal to the de
Sitter action $I_{dS}$ in that case. Further, the actions of the lukewarm and
Nariai (blue line) solutions coincide for the lukewarm-Nariai solution (yellow
point). Moreover, we can see from the figure that the actions of the extremal
and Nariai solutions are the same at $\rho=1$, which corresponds to the
ultracold solution (magenta point). The extremal and Nariai solutions also
have the same action at $\rho=\frac{\sqrt{5}}{3}$, but this is not a special
point in the shark fin diagram. Figure 6 can be contrasted with Figure 3 in a
paper by Bousso and Hawking Bousso:1996au  where the fraction $I/I_{dS}$ is
plotted against the square of the charge parameter (in fact, the dimensionless
quantity $q^{2}\Lambda).$ In that figure made by Bousso and Hawking (which is
based on the computations of the action by Mann and Ross Mann:1995vb ) the
Nariai branch and the extremal branch do not meet at the ultracold point,
which seems at odds with the shark fin diagram.

Figure 6: Fraction of the Euclidean action of proper instantons over the
action of pure de Sitter $I/I_{dS}$ vs. the charge parameter $\rho$. The blue
line corresponds to charged Nariai $I_{N}$, the green line to the extremal-
cold solution $I_{E}$, and the red line to the lukewarm solution $I_{L}$.
$I_{LN}$ and $I_{U}$ are shown in yellow and in magenta, respectively.

##  4 Constrained instantons and the pair creation rate

Having computed the on-shell Euclidean action for charged black holes in de
Sitter space in any number of dimensions, it is clear that, for general charge
and mass, they can only be understood as proper stationary instanton solutions
after introducing a constraint. After a very brief introduction of the general
method of constrained instantons Cotler:2020lxj ; Cotler:2021 , we will
proceed in the same way as in our previous work SdSaction  to compute the pair
creation rate of charged black holes in de Sitter using a constrained path
integral. Except that now, in addition to the introduction of a functional
that implements fixed mass, we also have to introduce a functional that fixes
the charge. In essence, as also emphasized by the construction of the Hartle-
Hawking wavefunction of RNdS, the presence of conical singularities implies
“conical boundary” contributions to the on-shell Euclidean action where the
fields are not allowed to vary, translating to fixed mass and charge, and
necessary in order to understand the solution as a stationary contribution to
the partition function.

To understand the contribution of the constrained stationary solutions to the
Euclidean gravitational partition function, we introduce the mass and charge
constraint in terms of a delta function and integrate over them to rewrite the
partition function as follows

$\int[dg][dA]\,e^{-I[g,A]}=\int d\alpha\int d\rho\int[dg][dA]\,\delta({\cal
C}[g,A]-\alpha)\,\delta({\cal D}[g,A]-\rho)e^{-I[g]}\,.$ (52)

Here $g$ and $A$ refer to the metric and gauge field degrees of freedom and
$I$ is the Euclidean off-shell action. Although the above expression is
completely general, what we have in mind is that the constraint functional
${\cal C}[g,A]$ fixes the mass to be $m$, whereas the constraint functional
${\cal D}[g,A]$ fixes the charge to be $q$. Here, as before, the dimensionless
ratio $\alpha$ runs from $0$ to $1$ for any charge $q$, and $\rho$ equals
$q/q_{U}$, where $q_{U}$ corresponds to the maximum ultracold charge. The next
step is to introduce the integral representation of the delta functions to
impose the constraints by means of two Lagrange multiplier terms added to the
action

$\int[dg][dA]\,e^{-I[g,A]}=\int d\alpha\int d\rho\int d\lambda_{m}\int
d\lambda_{q}\int[dg][dA]\,e^{-I[g,A]+\lambda_{m}({\cal
C}[g,A]-\alpha)+\lambda_{q}({\cal D}[g,A]-\rho))}\,,$ (53)

where it should be understood that the integral over the Lagrange multipliers
$\lambda_{m}$ and $\Lambda_{q}$ are parallel to the imaginary axis. With this
formal rewriting of the general partition function we can now identify the
contributions from the charged de Sitter black holes in terms of an actual
saddle point approximation. For any fixed $\alpha$ and $\rho$, only allowing
$\lambda_{m}$, $\lambda_{q}$ and the degrees of freedom $g$ and $A$ to vary,
the stationary points of the new action, including the Lagrange multiplier
terms, correspond to solutions of the following (schematic) equations

$\displaystyle\delta_{g}I_{E}[g,A]+\lambda_{m}\,\delta_{g}{\cal
C}[g,A]+\lambda_{q}\,\delta_{g}{\cal D}[g,A]=0\,,$
$\displaystyle\delta_{A}I_{E}[g,A]+\lambda_{m}\,\delta_{A}{\cal
C}[g,A]+\lambda_{q}\,\delta_{A}{\cal D}[g,A]=0\,,$ $\displaystyle{\cal
C}[g,A]=\alpha\,,\quad{\cal D}[g,A]=\rho\,.$ (54)

As in the neutral SdS case, the important point is that the charged black hole
is only a solution for non-vanishing $\lambda_{m}$ and $\lambda_{q}$, where
both Lagrange multipliers compensate for variations of the mass and charge
respectively, in such a way that the solution is now a true stationary point
of the modified action. That then allows us to use the saddle point
approximation, valid at weak gravitational coupling, to estimate the partition
function by integrating over all the saddles parameterized by the two
constraint parameters, the mass and the charge respectively. The constrained
instanton interpretation for black holes in de Sitter space was already
anticipated in Chao:1997em ; Chao:1997osu ; Bousso:1998vz , as well as more
recently in Draper:2022xzl , and after applying it to the neutral case in
SdSaction , here we are pointing out that it can straightforwardly be
generalized to charged black holes in de Sitter space as well. Note that the
evaluation of the new action, featuring the constraints, will simply reproduce
the on-shell Euclidean action of the charged black hole that we computed in
the previous section.

Importantly, any set of constraint functionals effectively fixing the mass and
charge should give the same result for the partition function. Since we will
only be interested in the exponential dependence on mass and charge as given
by the horizon entropies, and ignore any contributions coming from the one-
loop determinant, an explicit construction of the constraints will not be
necessary. We will assume that constraint functionals linear in the mass and
charge can be constructed, meaning that we can integrate over the
dimensionless variables $\mu$ and $\rho$ to determine the contributions to the
partition function without any additional dependence on the mass and charge
coming from the Gaussian path integral (the one-loop determinant).

We are now ready to use this saddle point approximation of the gravitational
partition function to derive an expression for the probability to
spontaneously nucleate charged black hole of arbitrary mass and charge in de
Sitter space. By integrating over the charge and mass, and comparing to the
on-shell action of empty de Sitter space, we get

$\Gamma=\int d\alpha\,d\rho\,e^{-(I_{RNdS}-I_{dS})}=\int
d\alpha\,d\rho\,e^{-(S_{dS}-S_{RNdS})}\,.$ (55)

Unfortunately, as in the neutral case SdSaction , it is not possible to
analytically perform this integral without making some approximations. From
Figure 7, as in the neutral case, one notices that the entropy is roughly
linear in $\alpha$, at a fixed value for the charge. So we will fit the total
entropy $S_{RNdS}$ as a linear function of $\alpha$, at fixed charge ratio
$\rho$, as follows

$\displaystyle S_{fit}$
$\displaystyle\equiv\alpha\,S_{N}+(1-\alpha)\,S_{E}\,,$ (56)

where the extremal entropies $S_{N}$ and $S_{E}$ are given by

$S_{E,N}(\alpha_{E,N},\rho)\equiv\frac{\mathcal{A}_{+}(\alpha_{E,N},\rho)+\mathcal{A}_{++}(\alpha_{E,N},\rho)}{4G}\,.$
(57)

This linear approximation, for fixed charge, gets better as the number of
dimensions increases (at worst, in $D=4$, the relative difference is $1.4\%$
at $\alpha=0.23$ and $\rho=0.81$).

a) $\rho=0,1/2$.

b) $\alpha=0,1/2$

Figure 7: Horizon entropies (a) as a function of the mass parameter $\alpha$
for fixed $\rho=0,1/2$, and (b) as a function of the charge parameter $\rho$
for $\alpha=0,1/2$. The black hole entropy $S_{+}$ is shown in blue, the
cosmological horizon entropy $S_{++}$ in orange, and the total entropy
$S_{RNdS}=S_{+}+S_{++}$ in red. Dark colors correspond to $\alpha,\rho=0$, and
light colors to $\alpha,\rho=1/2.$

Armed with this linear fit we can now straightforwardly perform the integral
over $\alpha$. The transition rate from empty de Sitter space to an arbitrary
mass black hole, in a fixed charge sector labeled by $\rho$, becomes

$\displaystyle\Gamma_{\rho}$
$\displaystyle\approx\int_{0}^{1}\,d\alpha\,\,e^{-(S_{dS}-S_{fit})}=e^{-(S_{dS}-S_{E})}\left(\frac{1-e^{S_{N}-S_{E}}}{S_{E}-S_{N}}\right)\,.$
(58)

Obviously, this reduces to our previous result SdSaction  for $\rho=0$ as
$S_{E}(\rho=0)=S_{dS}$. In the opposite limit $\rho=1$, the extremal entropies
$S_{E}$ and $S_{N}$ become equal to $S_{U}=\mathcal{A}_{U}/(2G)$ and the
result is proportional to the entropy deficit between the ultracold solution
and empty de Sitter space, as one might have expected. Now, to compute the
creation rate of arbitrary massive and charged black holes, one would be left
with computing the following integral

$\displaystyle\Gamma$ $\displaystyle=\int_{0}^{1}\,d\rho\,\Gamma_{\rho}\,,$
(59)

for which it is challenging to find an expression in closed form. Introducing,
however, another linear approximation, this time for the extremal entropies
$S_{E,N}$ as a function of the charge ratio $\rho$

$S_{E}\approx\rho\,S_{U}+(1-\rho)S_{dS}\,,\qquad
S_{N}\approx\rho\,S_{U}+(1-\rho)\,S_{N}^{0}\,,$ (60)

where $S_{U}=\mathcal{A}_{U}/(2G)$ is the ultracold entropy and
$S_{N}^{0}=S_{N}(\rho=0)$ the entropy of the neutral Nariai black hole, a two-
dimensional linear fit $\tilde{S}_{fit}$ for the entropy takes the following
form

$\tilde{S}_{fit}=\rho\,S_{U}+\alpha\,(1-\rho)\,S_{N}^{0}+(1-\alpha)(1-\rho)\,S_{dS}\,.$
(61)

Note that this linear fit is not a good approximation everywhere in the
parameter space of the charged de Sitter black hole. In fact, this fit for the
entropy is clearly worse as compared to the one for fixed charge (in $D=4$,
the maximal relative difference is now $13.6\%$ at $\alpha=1$ and
$\rho=0.77$). Even so, we believe it is useful to allow for an approximation
of the creation rate in closed form. Indeed, using this fit we can now
analytically compute the integral and obtain

$\displaystyle\Gamma\approx\frac{e^{-(S_{dS}-S_{U})}}{S_{dS}-S_{N}^{0}}\bigg{(}$
$\displaystyle\gamma(\sigma_{N})-\gamma(\sigma_{dS})+\log{(\sigma_{N}/\sigma_{dS})}\bigg{)}\,,$
(62)

where $\sigma_{N}\equiv S_{U}-S_{N}^{0}$ and $\sigma_{dS}\equiv S_{U}-S_{dS}$,
and $\gamma(\sigma)$ corresponds to the incomplete Euler function
$\gamma(a=0,\sigma)$, that we relabeled in order to not be confused with the
creation rate

$\gamma(\sigma)\equiv\int_{\sigma}^{\infty}t^{-1}e^{-t}\,dt\,.$

The expression (62) gives a reasonable first estimate in closed form for the
total creation rate. However, we should stress that these expressions for the
creation rates are merely indicative, in the sense that they could receive
several corrections. First of all, we expect corrections to the linear
approximations we used. Secondly, we assumed for simplicity the Gaussian
integral over the stationary solutions does not introduce additional mass or
charge dependence. However, we expect that the one-loop determinant does
depend on the mass and charge, which would alter our closed form for the pair
creation rate. The latter is clearly important to check in future work.

##  5 A comment on the Festina Lente bound

Figure 8: RNdS entropy $S_{RNdS}$ in terms of $\alpha$ and $\rho$, where
reddish colors correspond to high values, whereas blueish colors to low
values. The arrows stand for the gradient and indicate a tendency to escape
from the ultracold region at the top. Notice, that the lukewarm path (red
line) separates the arrows pointing left from those pointing right
(losing/gaining mass).

In flat space, as well as Anti-de Sitter space, the Weak Gravity Conjecture is
based on the idea that any black hole should be able to decay, even if the
black hole is extremal and has a vanishing temperature, without producing a
remnant naked singularity, in line with the Cosmic Censorship Conjecture (see
Wald:CC  and references therein). This puts a lower bound on the charge
$q_{e}$ to mass $m_{e}$ ratio of an elementary excitation, explaining why the
gravitational force is weak relative to the electromagnetic force.

In de Sitter space the configuration space of charged black holes has a more
elaborate structure (see Figure 2), implying that the Cosmic Censorship
Conjecture should forbid both decay by emitting too much charge $\delta q$, as
well as emitting too much mass $\delta m$, to avoid ending up in the region to
the right of the Nariai boundary or to the left of the extremal boundary,
respectively. Moreover, we expect the effective evolution to be described by
thermodynamics, i.e., the gradient of the entropy function (see Figure 8).
This implies that the probability of decay from a macroscopic black hole with
mass and charge parameters $(m,q)$ to one with parameters
$(m^{\prime},q^{\prime})=(m-\delta m,q-\delta q)$ is always proportional to
$\exp[\Delta S]=\exp[-(S(m,q)-S(m^{\prime},q^{\prime})]$.

As explained in Aalsma:2018 , using an s-wave tunneling formalism in flat and
AdS space, this is true as long as one avoids the so-called “superradiant”
regime at low energies, which describes a fast decay of all the charge of the
black hole. We expect an analogous line of reasoning should hold just as well
in de Sitter space. Indeed, in FestinaLente:2019  one concludes, supported by
explicit calculations of the Schwinger pair creation rate, that the adiabatic,
small mass regime should be avoided,333This should coincide with the
superradiant regime, as explained in Aalsma:2018 . otherwise the decay to
singular states to the right of the Nariai boundary would be possible,
violating the Cosmic Censorship Conjecture and prohibiting an effective
description of the decay in terms of thermodynamics. This sets a lower
“Festina Lente” bound on the mass of an elementary excitation, in contrast to
the Weak Gravity Conjecture, which sets an upper bound on the mass of an
elementary excitation.

This strongly suggests an s-wave tunneling calculation in de Sitter space
should be able to reproduce the effective thermodynamical evolution, as long
as one excludes the superradiant regime by imposing the absence of very light
charged particles. This would be interesting to check explicitly, especially
in the ultracold limit, where the effective evolution, described in terms of
$dq/dm$, is fixed uniquely and equals

$\displaystyle\left.\frac{dq}{dm}\right|_{U}=\frac{\sqrt{D-2}}{2}\,.$ (63)

It is important to realize that this unique limiting direction of the flow in
the top region of the shark fin should be reproduced for a range of elementary
charges and masses, as long as the superradiant (and potentially a sub-
extremal) regime in the mass and charge parameter space is excluded. In
general, a tunneling calculation should allow for a precise determination
(including backreaction) of the superradiant regime and therefore a more
precise Festina Lente type bound (in arbitrary dimensions).

##  6 Summary and conclusions

Let us summarize our results. First of all, using exactly the same general
techniques as for the neutral, static de Sitter black hole, we have shown that
the on-shell Euclidean action of charged, static de Sitter black holes in
arbitrary dimensions equals minus the sum of the black hole and the
cosmological horizon entropy. This expected result relies on an important
subtlety. Namely that the presence of conical singularities implies the
necessary inclusion of (gauge field) boundary terms providing a localized
contribution to the on-shell action on the conical singularities that
precisely cancels the contribution to the action from the bulk term depending
on the charge and the gauge potentials. As in the neutral case, this again
emphasizes the intriguing role of conical singularities in Euclidean quantum
gravity. This general expression, valid for any charge and mass, also helps to
clarify some confusion in the literature Mann:1995vb ; Dias:2004px ;
Wang:2022sbp  for the extremal and ultracold black hole Euclidean actions in
de Sitter, which can now be understood as a limit of the general expression,
with a clear and unambiguous answer.

Furthermore, equipped with the on-shell Euclidean action and an understanding
in terms of constrained instantons, we used a saddle point approximation to
compute an analytic expression for the probability to spontaneously nucleate
an arbitrary mass and charge black hole in de Sitter space. In order to be
able to compute this we used a linear approximation in the mass and charge of
the entropy deficit. Although this is certainly not a good approximation
everywhere in the full configuration space of charged de Sitter black holes,
it does provide a reasonable result in closed form. We should stress that we
expect corrections, in particular we ignored any potential contributions from
the one-loop determinants around the two-parameter family of saddle-points. We
hope to come back to this point in the future, starting with the neutral case.

Finally, with the result for the Euclidean action, in arbitrary dimensions,
and the expectation that thermodynamics should be governing the decay of
charged black holes in de Sitter, we also made the suggestion that as long as
the superradiant regime is avoided, a tunneling calculation similar to
Aalsma:2018  should be able to explicitly reproduce the correct behavior in
terms of entropy differences. This would be particularly interesting to check
in the ultracold limit, where the effective flow describing the decay is fixed
uniquely. We suggest that by using the tunneling formalism it should also be
possible to derive a more precise Festina Lente type bound in arbitrary
dimensions.

Last, but not least, this study has further explored the space of non-
perturbative corrections in de Sitter space, as described by saddle points in
a Euclidean (quantum) gravity description. Although AdS/CFT has provided
strong hints that saddle points in the partition function of Euclidean
Einstein gravity have an important role to play, it is not obvious whether
this can be generalized to de Sitter space. Nevertheless, we believe it is
important to further study the potential implications of these non-
perturbative contributions, in particular how they might contribute to
relevant observables, such as late-time correlators Aalsma:2022eru . The hope
is, of course, that these corrections are such that these observables behave
as they are expected from a unitary (holographic) microscopic perspective.

###### Acknowledgements.

We thank Miguel Montero, Thomas van Riet and Gerben Venken for useful
correspondence. This work is part of the Delta ITP consortium, a program of
the Netherlands Organisation for Scientific Research (NWO) that is funded by
the Dutch Ministry of Education, Culture and Science (OCW). EM is supported by
Ama Mundu Technologies (Adoro te Devote Grant 2019). MRV is supported by SNF
Postdoc Mobility grant P500PT-206877 “Semi-classical thermodynamics of black
holes and the information paradox”.

##  Appendix A Four dimensions

In this appendix we specialize to four dimensions and provide some potentially
useful expressions for the horizon radii, mass, charge and temperatures of the
RNdS solution.

### Horizon radii

In four dimensions the blackening factor can be written as

$f(r)=\frac{(r-r_{+})(r-r_{++})(r-r_{-})(r-r_{--})}{l^{2}\,r^{2}}\,.$ (64)

The outer horizon radius $r_{+}$ and cosmological horizon radius $r_{++}$ can
be expressed as

$r_{+,++}=r_{U}\left(x\mp\sqrt{3-x^{2}-2m/(x\,m_{U})}\right)\,,$ (65)

where $x\equiv\sqrt{1+\sqrt{1-\rho^{2}}\cos{2\eta}}$ with
$\eta\equiv\frac{1}{3}\arccos{\sqrt{(m_{E}^{2}-m^{2})/(m_{E}^{2}-m_{N}^{2})}}$.

a) $\rho=0,1/2$.

b) $\alpha=0,1/2$.

Figure 9: Horizon radii (a) as a function of the mass parameter $\alpha$ and
(b) as a function of the charge parameter $\rho$. The inner horizon radius
$r_{-}$ is shown in green, the outer horizon radius $r_{+}$ in blue, and the
cosmological horizon radius $r_{++}$ in orange. Dark colors correspond to
$\alpha,\rho=0$, and light colors to $\alpha,\rho=1/2.$ Note that for
$\alpha=0$, the inner and outer horizons are identical (dashed on the right).

Moreover, the horizon radius, mass and charge of the ultracold black hole are
given by

$r_{U}=\frac{l}{\sqrt{6}}\,,\qquad
m_{U}=l\,\left(\frac{2}{3}\right)^{\frac{3}{2}}\,,\qquad
q_{U}=\frac{l}{2\sqrt{3}}\,.$ (66)

The mass for the extremal and Nariai cases as a function of the charge $\rho$
is given by

$\displaystyle m_{E,N}=2r_{U}\sqrt{1+3\rho^{2}\mp(1-\rho^{2})^{3/2}}/3\,,$
(67)

and the corresponding horizon radii can be found to be

$\displaystyle r_{E,N}$ $\displaystyle=r_{U}\sqrt{1\mp\sqrt{1-\rho^{2}}}\,.$
(68)

Finally, we can use the relations between the roots of $f(r)$

$\displaystyle\sum_{i=1}^{2(D-2)}r_{i}^{n}=\left\\{\begin{array}[]{ll}0\quad\,\,\,\,\,\text{for
$0<n\leq D-2$\, odd}\\\ 2l^{n}\,\,\,\,\,\,\text{for $0<n\leq D-2$\, even\,
}\end{array}\right.,\qquad\sum_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{2(D-2)}r_{i}\,r_{j}=-l^{2}\,,$ (71)

to find the following expressions for the other two roots

$\displaystyle r_{-}$
$\displaystyle=-\frac{1}{2}\left(r_{+}+r_{++}-\sqrt{2(2l^{2}-(r_{+}^{2}+r_{++}^{2}))-(r_{+}+r_{++})^{2}}\right)$
(72) $\displaystyle r_{--}$ $\displaystyle=-(r_{-}+r_{+}+r_{++})\,.$ (73)

### Horizon temperatures

The radius $r_{\mathcal{O}}$ of the stationary free falling observer in $D=4$
is given by

$r_{\mathcal{O}}=a\left(1+\sqrt{\frac{m}{m_{U}}\frac{r_{U}^{3}}{a^{3}}-1}\,\,\right)\quad\text{where}\quad
a\equiv
r_{U}\sqrt{\rho\cos{\left[\frac{1}{3}\arccos{\left(\frac{m^{2}}{m_{U}^{2}}\rho^{-3}\right)}\right]}}\,,$
(74)

and the black hole and cosmological horizon temperatures take the form

$T_{+,++}=\frac{r_{++}-r_{+}}{4\pi
l^{2}\sqrt{f(r_{\mathcal{O}})}}\left[2+\left(1+\frac{r_{+,++}}{r_{++,+}}\right)^{2}-\left(\frac{l}{r_{++,+}}\right)^{2}\right]\,.$
(75)

For the Nariai solution, where $r_{+}=r_{++}=r_{\mathcal{O}}=r_{N}$, these
temperatures are the same and equal to

$T_{N}=\frac{\sqrt[4]{1-\rho^{2}}}{2\pi\,r_{N}}=\frac{\sqrt{3}}{\pi\,l\sqrt{2+2/\sqrt{1-\rho^{2}}}}\,.$
(76)

The other equilibrium solution for which $T_{+}=T_{++}$, but $r_{+}\neq
r_{++}$, is the lukewarm solution, which satisfies the relations
$r_{L,+}+r_{L,++}=l$ and $m_{L}=2q_{L}$ in four dimensions. The horizon radii
and temperature of the lukewarm solution as a function of the charge are

$r_{{L,+,++}}=\frac{l}{2}\left(1\mp\sqrt{1-q/q_{LN}}\right),\qquad
T_{L}=\frac{\sqrt{1-q/q_{LN}}}{2\pi\,l\sqrt{1-q^{2}/r_{\mathcal{O}}^{2}-3r_{\mathcal{O}}^{2}/l^{2}}}\,.$
(77)

Finally, for the lukewarm-Nariai solution we find

$r_{LN}=\frac{l}{2},\qquad m_{LN}=\frac{l}{2},\qquad q_{LN}=\frac{l}{4},\qquad
T_{LN}=\frac{1}{\sqrt{2}\pi\,l}\,.$ (78)

##  Appendix B Arbitrary dimensions

In this appendix we collect some cumbersome relations for the mass, charge and
horizon radii of RNdS, which are valid in general dimensions. By solving
$f(r_{+})=f(r_{++})=0$ the mass and charge parameters can be expressions in
terms of the outer horizon radius $r_{+}$, the cosmological horizon radius
$r_{++}$ and de Sitter radius $l$

$m=\frac{r_{+}^{2(D-3)}(r_{+}^{2}-l^{2})-r_{++}^{2(D-3)}(r_{++}^{2}-l^{2})}{(r_{++}^{D-3}-r_{+}^{D-3})l^{2}}\,,\quad
q^{2}=\frac{r_{++}^{D-3}(r_{++}^{2}-l^{2})-r_{+}^{D-3}(r_{+}^{2}-l^{2})}{(r_{++}^{3-D}-r_{+}^{3-D})l^{2}}\,.$
(79)

Further, for the lukewarm solution, solving $T_{+}=T_{++}$ (with the Bousso-
Hawking normalization) yields the following expression for (the square of) the
de Sitter radius

$\displaystyle l^{2}=$
$\displaystyle\frac{1}{(D-3)\left(r_{L,++}^{D-2}-r_{L,+}^{D-2}\right)\left(r_{L,++}^{D-3}-r_{L,+}^{D-3}\right)^{2}}\Bigg{[}(D-3)\left(r_{L,++}^{3(D-2)}-r_{L,+}^{3(D-2)}\right)$
(80)
$\displaystyle-(D-2)\left(r_{L,++}^{D-4}-r_{L,+}^{D-4}\right)(r_{L,+}r_{L,++})^{D-1}-(D-1)\left(r_{L,++}^{D-2}-r_{L,+}^{D-2}\right)(r_{L,+}r_{L,++})^{D-2}\Bigg{]}\,.$

By plugging this into equation (79) one can obtain an equation for the ratio
of the mass $m_{L}$ and charge $q_{L}$ in the lukewarm case

$\displaystyle\left(\frac{m_{L}}{q_{L}}\right)^{2}$
$\displaystyle=\frac{4(r_{L,+}+r_{L,++})}{l^{2}(D-3)\left(r_{L,++}^{D-2}-r_{L,+}^{D-2}\right)\left(r_{L,++}^{D-3}-r_{L,+}^{D-3}\right)^{2}}\,\,\,\times$
(81)
$\displaystyle\times\frac{\left[(D-3)\left(r_{L,++}^{2(D-2)}-r_{L,+}^{2(D-2)}\right)-(D-2)r_{L,+}r_{L,++}\left(r_{L,++}^{2(D-3)}-r_{L,+}^{2(D-3)}\right)\right]^{2}}{(D-3)\left(r_{L,++}^{D-1}-r_{L,+}^{D-1}\right)-(D-1)r_{L,+}r_{L,++}\left(r_{L,++}^{D-3}-r_{L,+}^{D-3}\right)}\,.$

In four dimensions this relation simplifies to $m_{L}=2q_{L}.$ The maximal
values of $m_{L}$ and $q_{L}$ are set by the Nariai black hole. Solving
$m_{L}=m_{N}(\equiv m_{LN})$ and $q_{L}=q_{N}(\equiv q_{LN})$ yields in
general dimensions

$\left(\frac{m_{LN}}{m_{U}}\right)^{2}=3^{D-3}(2D-5)^{2}\frac{(D-2)^{D-3}}{(3D-8)^{D-1}}\,,\qquad\left(\frac{q_{LN}}{q_{U}}\right)^{2}=3^{D-3}\left(\frac{D-2}{3D-8}\right)^{D-2}\,.$
(82)

Finally, the horizon radius of the lukewarm-Nariai black hole is given by

$r_{LN}^{2}=\frac{3(D-3)^{2}}{(D-1)(3D-8)}\,l^{2}\,.$ (83)

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