aainstitutetext: Department of Physics, McGill University, Montreal, QC, H3A
2T8, Canadabbinstitutetext: Institute of Physics, University of Amsterdam,
Science Park 904, PO Box 94485, 1090 GL Amsterdam, the
Netherlandsccinstitutetext: Delta Institute for Theoretical Physics, Science
Park 904, PO Box 94485, 1090 GL Amsterdam, the Netherlands

# Static sphere observers and geodesics in Schwarzschild-de Sitter spacetime

Mir Mehedi Faruk b,c Edward Morvan b,c Jan Pieter van der Schaar
<EMAIL_ADDRESS>edward.morvan<EMAIL_ADDRESS>
###### Abstract

We analyze null- and spacelike radial geodesics in Schwarzschild-de Sitter
spacetime connecting two conjugate static sphere observers, i.e. free-falling
observers at a fixed radius in between the two horizons. We explicitly
determine the changes in the causal structure with respect to these natural
observers as a result of the inward bending of the black hole singularity, as
well as the outward bending of asymptotic infinity. Notably, the inward and
outward bending changes as a function of the black hole mass, first increasing
towards a maximum and then decreasing to vanish in the extreme Nariai limit.
For a generic mass of the black hole this implies the existence of finite size
(temporal) windows for the presence of symmetric radial geodesics between the
static sphere observers probing the interior region of the black hole, as well
as the exterior de Sitter region. We determine the size of the interior (black
hole) and exterior (de Sitter) temporal windows in $4$, $5$ and $6$ spacetime
dimensions, finding that they are equal in $D=5$, and compute the proper
lengths of the symmetric radial geodesics. We comment on the implications for
information exchange and the potential role of the symmetric radial geodesics
in a geodesic approximation of static sphere correlators in Schwarzschild-de
Sitter spacetime.

##  1 Introduction

Recent studies of quantum gravity in de Sitter Banks:2000fe ; Witten:2001kn ;
Spradlin:2001pw ; Goheer:2002vf ; Klemm:2004mb ; Anninos:2012qw ;
Susskind:2021dfc ; Susskind:2021omt ; Morvan:2022ybp ; Galante:2023uyf ;
Svesko:2022txo ; Balasubramanian:2002zh suggest that, inspired by recent
progress in our understanding of quantum gravity in Anti-de Sitter space
Maldacena:2013xja ; Penington:2019npb ; Almheiri:2019psf ;
Balasubramanian:1999zv , non-perturbative contributions in the (Euclidean)
path integral could play an important role Susskind:2021dfc ; Anninos:2022ujl
; Anninos:2023epi . The simplest, most elementary, solutions of relevance in
this context are black holes in de Sitter space, whose contribution the the
Euclidean path integral can be (partially) understood using the formalism of
constrained instantons Cotler:2020lxj ; Morvan:2022ybp . Very concretely, it
would be interesting to understand how such non-perturbative contributions
affect specific correlators in de Sitter space. Notably, if one assumes the
existence of a discrete Hilbert space of states in de Sitter, as suggested by
the finite entropy of a de Sitter static patch causal region, one would expect
on general grounds that the semi-classical late-time behaviour of large mass
correlators between conjugate static patch observers is modified from
exponential decay to a growing ’ramp’ type behavior that ultimately lands on a
constant ’plateau’ average Saad:2018ramp . In the absence of a holographic
description, and inspired by the important role of non-perturbative effects in
Anti-de Sitter quantum gravity, our ultimate goal is to understand the
contributions of de Sitter black holes to these late-time, large mass,
conjugate correlators.

As a first step in this direction, in this paper we present results on the
detailed causal structure of the Schwarzschild-de Sitter solution, from the
perspective of a pair of natural conjugate observers. We call these special
free-falling observers ’static sphere’ observers, as they are located at a
unique radius in between the black hole and cosmological horizons. We are
particularly interested in the associated (timelike) geodesics, as they
correspond to the natural (radial) locations for two conjugate large mass
(scalar) field operators Bousso:1996au ; Aalsma:2020aib ; Morvan:2022ybp ;
Svesko:2022txo . By analyzing the causal structure from the perspective of
these natural observers we compute the finite temporal windows for the
appearance of spacelike geodesics going through the black hole interior, as
well as the de Sitter exterior. These windows are the result of the inward
bending of the black hole singularity and the outward bending of asymptotic de
Sitter infinity, resulting in a smaller interior, and at the same time a
larger exterior, causal diamond as compared to empty de Sitter space. These
windows depend on the black hole’s mass and the specific dimension under
consideration, and we compute and compare their interior and exterior size in
spacetime dimensions $4$, $5$ and $6$. Notably, in $5$ spacetime dimensions
the interior inward and exterior outward bending, and consequently their
temporal windows, are the same. And in every dimension the windows show an
initial increase towards a maximum, but then decrease to vanish again in the
extreme Nariai limit. This detailed shape of the causal Penrose diagram is
also of interest for information recovery and exchange protocols involving
gravitational shockwaves Cornalba:2006xk ; Freivogel:2019lej ;
Freivogel:2019whb , which in de Sitter, necessarily imply disposing of
positive energy Aalsma:2021kle ; Morvan:2023tele , and therefore resulting in
Schwarzschild-de Sitter-like solution.

The opening of temporal windows implies the presence of spacelike geodesics
going through the interior black hole, as well as the exterior de Sitter,
region connecting the two static sphere observers. We compute the lengths of
these spacelike geodesics and consider their limit as we approach the
boundaries of the temporal windows. In general, assuming a holographic
description exists, understanding the appearance and structure of (spacelike)
geodesics can be associated to features of holographic entanglement. And as is
well known Fidkowski:2003nf , using a geodesic approximation for (large mass)
correlators the limits towards the boundaries of the temporal windows signals
a transition in the correlator, where one expects complex correlators to
become important. As such the specific behavior of geodesics between these
static sphere observers probe the black hole singularity, as well as the
structure of (exterior) de Sitter asymptotic infinity. We study these limits
and show that they behave differently for the interior and exterior spacelike
geodesics.

The organisation of this paper is as follows. We start with a short reminder
of computing geodesics in spherically symmetric spacetimes, uysing (complex)
static coordinates, ending with the Anti-de Sitter black hole and empty de
Sitter space as useful examples. We then introduce the Schwarzschild-de Sitter
geometry, explain the special role of ’static sphere’ observers, and then
study interior and exterior null geodesics to determine the temporal window
for the existence of spacelike geodesics in $4$, $5$ and $6$ spacetime
dimensions. Having done that we then look at symmetric spacelike geodesics
connecting the static sphere observers, determining their length and analyzing
the limits to the boundaries of the temporal windows. We end with some
conclusions and suggestions for further research.

##  2 Geodesics in static spherically symmetric spacetimes

In this introductory section we will set up our notation and conventions for
studying radial geodesics. We will remind the reader how to probe the details
of the causal structure, in particular the (inward) bending of the black hole
singularity, using null geodesics and the corresponding existence of symmetric
spacelike radial geodesics probing the interior of the black hole. After
setting up the general structure of the geodesic analysis, we end with a
reminder of two well-known basic examples of interest, namely Schwarzschild-
Anti-de Sitter and pure de Sitter spacetime, before we move on and study the
particular spacetime of interest, Schwarzschild-de Sitter.

###  2.1 Preliminaries

To start, let us briefly review the analysis of geodesics in a general curved
spherically symmetric background. Geodesics can be understood as stationary
points for the action of a point particle

$I=\int\mathrm{d}\lambda\,g_{\mu\nu}\mathrm{d}\dot{x}^{\mu}\mathrm{d}\dot{x}^{\nu}~{},$
(1)

here the $\dot{x}^{\mu}$ represents the derivative with respect to the affine
parameter $\lambda$. This naturally results in the well known geodesic
equation

$\ddot{x}^{\mu}+\Gamma_{\nu\sigma}^{\mu}\dot{x}^{\nu}\dot{x}^{\sigma}=0\,,$
(2)

where $\Gamma_{\nu\sigma}^{\mu}$ is the usual Levi-Civita connection symbol
(torsion-free). As $\lambda$ is an affine parameter, defining $\epsilon\equiv
g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$, we can classify the different types of
geodesics as

$\epsilon=\begin{cases}~{}~{}0\qquad\text{null-like}\\\ +1\qquad\text{space-
like}\\\ -1\qquad\text{time-like}\end{cases}$ (3)

Let us now focus on spherically symmetric line elements of the following form

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}\,d\Omega^{2}_{D-2}\,,$ (4)

where $d\Omega_{D-2}$ is the volume element of an unit sphere in $D-2$
dimensions. For radial geodesics we then obtain the following equation

$-f(r)\dot{t}^{2}+f(r)^{-1}\dot{r}^{2}=\epsilon\,.$ (5)

These geometries feature a timelike Killing vector $\partial_{t}$, one can
therefore infer the existence of a conserved quantity $\mathcal{E}$ along a
geodesic given by

$\mathcal{E}=\dot{t}\,f(r)\,.$ (6)

Using this in equation (5), one then obtains

$\dot{r}^{2}-\mathcal{E}^{2}=\epsilon f(r)\,,$ (7)

Let us now concentrate on null and spacelike geodesics. Studying null
geodesics ($\epsilon=0$) allows one to map out the causal structure of the
spacetime under consideration, which can be transformed into a Penrose
diagram. For radial null geodesics the two relevant equations (6) and (7) turn
into

$\displaystyle\dot{r}=\pm\mathcal{E}\quad\Longleftrightarrow\quad\frac{\dot{r}}{\dot{t}}=\frac{dr}{dt}=\pm
f(r)\,,$ (8)

where the sign determines the direction of the radial null geodesic (incoming
or outgoing). This equation can be solved for $t(r)$, for an outgoing radial
geodesic ($r^{\prime}>r_{\cal O}$), giving

$t(r^{\prime})-\tau=\int_{\tau}^{t(r^{\prime})}\mathrm{d}t=\int_{r_{\mathcal{O}}}^{r^{\prime}}\frac{\dot{t}}{\dot{r}}\mathrm{d}r=\int_{r_{\mathcal{O}}}^{r^{\prime}}\frac{\mathrm{d}r}{f(r)}\,,$
(9)

where $\tau$ labels the observer time at which the null trajectory intersects
$r^{\prime}=r_{\mathcal{O}}$ (see figure 1). Inserting the value of
$r^{\prime}$ in the above integral fixes the endpoint of the null radial
geodesic under consideration, so for an incoming radial geodesic
($r^{\prime}<r_{\cal O}$) one needs to add a minus sign on the right. For the
Schwarzschild-de Sitter geometry, we will be interested in radial null
geodesics passing through the black hole horizon and reaching the singularity
at $r^{\prime}=0$, as well as radial null geodesics crossing the cosmological
horizon and reaching asymptotic future infinity $\mathcal{I}^{+}$ where
$r^{\prime}\rightarrow\infty$.

Space-like separated points ($\epsilon=+1$) satisfy the following (radial)
geodesic relation,

$\dot{r}^{2}-\mathcal{E}^{2}=f(r)\quad\Longleftrightarrow\quad\frac{\dot{r}}{\dot{t}}=\frac{dr}{dt}=\pm\frac{f(r)\sqrt{f(r)+\mathcal{E}^{2}}}{\mathcal{E}}\,,$
(10)

where once again we made use of equation (6). For outgoing radial spacelike
geodesics (selecting the positive sign and assuming $\mathcal{E}$ is positive)
this then results in the following (coordinate) time difference

$t(r^{\prime})-\tau=\int_{r_{\mathcal{O}}}^{r^{\prime}}\frac{\dot{t}}{\dot{r}}\,dr=\int_{r_{\mathcal{O}}}^{r^{\prime}}\frac{\mathcal{E}\,dr}{f(r)\sqrt{f(r)+\mathcal{E}^{2}}}\,.$
(11)

As before $\tau$ refers to the initial starting time of the space-like
geodesic at the position of the observer $r=r_{\mathcal{O}}$. Again, for
incoming space-like radial geodesics one should add a minus sign. Also note
that for space-like geodesics a sign flip of the conserved quantity
$\mathcal{E}$ changes the sign of the (coordinate) time difference. Clearly,
the initial slope $\frac{dr}{dt}$ at $r=r_{\mathcal{O}}$ of a radial spacelike
geodesic is given by the expression on the right-hand side of equation (10)
evaluated at $r=r_{\mathcal{O}}$, which will depend on (the sign of)
$\mathcal{E}$, asymptoting to the slope of a null geodesic in the limit
$\mathcal{E}\rightarrow\pm\infty$.

As we will soon discover, for symmetric radial space-like geodesics the time
$\tau$ is bounded, for real values of the parameter $\mathcal{E}$, where the
boundaries are given by special null geodesics
($\mathcal{E}\rightarrow\pm\infty$). Given a starting time $\tau$ and the
conserved quantity $\mathcal{E}$, the length of the associated symmetric,
radial, space-like geodesic is given by

$\mathcal{D}=2\int_{r_{\mathcal{O}}}^{r_{t}}\frac{dr}{\sqrt{f(r)+\mathcal{E}^{2}}}\,,$
(12)

where we introduced the turning point radius $r_{t}$, which is obtained by
solving

$f(r)+\mathcal{E}^{2}=0\,.$ (13)

As will become clear soon, the turning points correspond to the minimal
(interior) or maximal (exterior) radii a symmetric spacelike geodesic can
probe as a function of the conserved quantity $\mathcal{E}$. For vanishing
$\mathcal{E}$, the turning points actually coincide with the horizon radii,
implying that the radial geodesics never cross the interior or exterior
region. In principle, combining equations (11) and (12), we can compute the
length of these symmetric radial geodesics for allowed $\tau$ and (related)
conserved quantity $\mathcal{E}$.

###  2.2 Radial geodesics in Schwarzschild-Anti-de Sitter spacetime

In this section we want remind the reader of the techniques and results
described in the paper by Fidkowski et. al. Fidkowski:2003nf , as they will
play a key role in our study of Schwarzschild-de Sitter. In Fidkowski:2003nf
it was shown explicitly that the causal structure of Schwarzschild Anti-de
Sitter (SAdS) spacetimes in dimensions bigger than $3$ cannot be drawn as a
perfect square with respect to the two timelike boundaries, i.e. the black
hole singularity at $r=0$ is bending inwards. This then meant that the
boundary CFT correlators probe the singularity from a particular (boundary)
time forward. To briefly review these results, we start with the five
dimensional AdS blackening factor

$\displaystyle
f(r)=1+\frac{r^{2}}{\ell^{2}}-\frac{r_{b}^{2}}{r^{2}}\left(1+\frac{r_{b}^{2}}{\ell^{2}}\right)$
(14)

where, $\ell$ is the AdS radius and $r_{b}$ is the horizon radius. To further
simplify, let us consider the infinite mass limit of the black hole, i.e.
$\frac{r_{b}}{\ell}\rightarrow\infty$, for which the blackening factor reduces
to

$\displaystyle f(r)=\frac{r^{2}}{\ell^{2}}-\frac{\ell^{2}}{r^{2}}.$ (15)

Putting the relevant observer at the AdS boundary $r_{\mathcal{O}}=\infty$
equation (9) dictates that

$\displaystyle
t_{SAdS}=\tau+\frac{\ell}{2}\left[(1-i)\frac{\pi}{2}+\operatorname{arctanh}\left(\frac{r}{\ell}\right)-\arctan\left(\frac{r}{\ell}\right)\right]\,,$
(16)

From this expression we learn that a null geodesic starting from the AdS
boundary reaches the black hole singularity at $r=0$ at a coordinate time that
now contains an imaginary contribution

$\displaystyle t_{SAdS}(r=0)=\tau+\frac{\pi}{4}(1-i)\,\ell\,.$ (17)

The origin of the imaginary contribution can be understood by noting that by
introducing a complex time coordinate one can cover all the four different
patches of this global SAdS geometry using the same static coordinates, with
the imaginary part identifying the appropriate region. As one crosses the
event horizon, the imaginary part shifts by $-i\beta/4$, where $\beta$ is the
inverse temperature of the geometry, and the (real) time and space coordinates
swap roles. Going around all four patches (counter-clockwise, starting in the
left causal patch), each time crossing an event horizon, the imaginary part
changes from $0$ to $-i\beta$, consistent with the periodicity of the
imaginary time coordinate (i.e. Euclidean time). The inverse temperature
$\beta=\pi\ell$ in the infinite mass limit of the SAdS geometry. This
structure applies more generally of course: one encounters the same type of
behavior for the (complex) time coordinate in all (global) static metrics that
feature event horizons.

This behavior of the complex time coordinate implies that a null geodesic
starting from the boundary at $\tau=0$ reaches the singularity at $r=0$ off-
center in the Kruskal or Penrose diagram, as the real part of $t_{SAdS}(r=0)$
is nonzero. This actually happens for SAdS geometries in $D>3$
Fidkowski:2003nf ; Festuccia:2005pi . In fact, in general $d$ dimensions
Fidkowski:2003nf , a null radial geodesic reaches the the singularity exactly
at the center of the diagram only when emitted at a boundary AdS time
$\tau=-{\pi\ell}((D-1){\text{tan}(\frac{\pi}{D-1})})^{-1}$, denoted as
$-\mathcal{T}$ in figure 1(b). So apart from the $3$-dimensional BTZ black
hole, the black hole singularity in the SAdS geometry in all higher dimensions
is bend inwards, with respect to asymptotic AdS observers (drawn as straight
AdS boundary lines), see figures 1(a) and 1(b).

$x$$x^{\prime}$$\tau_{R}=0$$\tau_{L}=0$$t$$t$ (a) $D=3$

$x$$x^{\prime}$$\tau_{R}=\mathcal{T}$$\tau_{R}=-\mathcal{T}$ (b) $D>3$

Figure 1: Penrose diagrams of the SAdS geometry. On the left is the $D=3$
case, whereas for any dimensions $D>3$ the situation is described by the right
panel, with respect to straight line AdS boundary observers.

After studying the null geodesics and the associated causal structure, we can
now also look for radial space-like geodesics connecting the symmetric points
$x$ and $x^{\prime}$ on the opposite boundaries (see figure 1(a)). For a
specific boundary time, one then finds a unique symmetric radial spacelike
geodesic, labeled by $\mathcal{E}$. Using (11) and assuming real
$\mathcal{E}$, the turning points of these symmetric spacelike radial
geodesics are given by

$r_{t}=\sqrt{\sqrt{1+\mathcal{E}^{4}/4}-\mathcal{E}^{2}/2}\,.$ (18)

Not only do the turning points have a vanishing derivative, but for real
$\mathcal{E}$ they always lie exactly in the middle of the Penrose diagram,
i.e. they correspond to a coordinate time whose real part vanishes, ensuring
that the complete spacelike geodesic is a continuous differentiable function,
connecting the two opposite static patches of the SAdS geometry. Plugging in
the blackening factor (15), one finds that the integral (11) identifies the
following initial boundary time of the symmetric radial spacelike geodesics
Fidkowski:2003nf ,

$\tau=\text{Re}\left[\frac{1}{4}\ln{\left(\frac{1-\mathcal{E}+\mathcal{E}^{2}/2}{\sqrt{1+\mathcal{E}^{4}/4}}\right)}-\frac{i}{4}\ln{\left(\frac{1+i\mathcal{E}-\mathcal{E}^{2}/2}{\sqrt{1+\mathcal{E}^{4}/4}}\right)}\right]\,.$
(19)

From this relation between the boundary time and $\mathcal{E}$, one then
concludes that (for real $\mathcal{E}$) symmetric radial geodesics only exist
within a finite-size temporal window given by $-\mathcal{T}<\tau<\mathcal{T}$.
For $\mathcal{E}=0$ the endpoints of the symmetric spacelike radial geodesic
is $\tau_{R}=\tau_{L}=0$, however, when $\mathcal{E}$ approaches $\pm\infty$
the initial boundary time equals $\tau\rightarrow\pm\mathcal{T}$ and the
turning points get arbitrarily close to the singularity. In this limit the
radial geodesic becomes null and bounces off the black hole singularity at a
finite boundary time and its (appropriately renormalized) length diminishes to
zero.

For boundary times outside of this window the only geodesics connecting
symmetric points on the boundary of AdS are labeled by complex values of
$\mathcal{E}$, necessarily implying that their length has real and imaginary
parts as well Fidkowski:2003nf ; Fischetti:2014uxa . Although we considered an
infinite mass limit to simplify the equations, the same conclusions can be
derived more generally Fidkowski:2003nf , but it is not as straightforward to
derive analytic results using (17) and (19). Specifically, for the
Schwarzschild-de Sitter black hole we will not need to rely on special limits
to derive analytical results.

###  2.3 Radial geodesics in de Sitter spacetime

Another basic example of interest, one that also serves as a limiting case of
the full Schwarschild-de Sitter geometry that we will soon address, is pure de
Sitter spacetime. The de Sitter metric, written in terms of static patch
coordinates, the blackening factor in eq. (4) is given by

$\displaystyle f(r)=1-\frac{r^{2}}{\ell^{2}},$ (20)

with the coordinate singularity at $r=\ell$ corresponding to the cosmological
horizon. In this metric the only free-falling observer at fixed radius,
determined by

$f^{\prime}(r)\big{|}_{r=r_{\mathcal{O}}}=0\,.$ (21)

This of course gives the expected result $r_{\mathcal{O}}=0$, i.e. the
observer situated at the center of the static patch that is surrounded by the
cosmological horizon. In terms of a global slicing, one can identify this
position as one of the poles on the global $3$-sphere (for $4$-dimensional de
Sitter). The global geometry features another, causally disconnected, static
patch observer, for which time runs in the opposite direction, and which can
then be identified with the other pole of the spatial $3$-sphere slicing the
(global) de Sitter geometry. Let us now introduce radial null geodesics
originating from, or arriving at, the two static patch centers, depicted as
the red curves in the de Sitter Penrose diagram 2. For de Sitter space
equation (9) turns into

$t_{dS}(r)=\tau+\int_{0}^{r}\frac{\ell^{2}\,\mathrm{d}r^{\prime}}{(r^{\prime}-\ell)(r^{\prime}+\ell)}=\tau+\ell\operatorname{arctanh}{(r/\ell)}\,.$
(22)

Taking the limit $r\rightarrow\infty$, probing the structure of asymptotic
future infinity $\mathcal{I}^{+}$, allows one to infer that

$t_{dS}=\tau+i\frac{\pi\ell}{2}\,.$ (23)

As before, the imaginary part is generated by crossing the cosmological
horizon, however as opposed to black holes, it is now equal to a quarter of
its Euclidean periodicity $\beta_{dS}=1/T_{dS}=2\pi\ell$. This sign difference
is related to the fact when energy $M$ is added to de Sitter space, its
entropy $S$ decreases Gibbons:1976ue ; Bousso:1996au ; Morvan:2022ybp ;
morvan:2022rn ; Morvan:2023tele

$\mathrm{d}S_{b}=\beta_{b}\,\mathrm{d}M\quad,\quad\mathrm{d}S_{dS}=-\beta_{dS}\,\mathrm{d}M\,,$
(24)

where $\beta$ is kept positive definite. Equation 23 ensures that a null
radial geodesic moving outward from the symmetric center of the north or south
pole, i.e. $(\tau=0,r_{\mathcal{O}}=0)$, ends up exactly at the center of
asymptotic future infinity $\mathcal{I}^{+}$, as the real part of the complex
time coordinate vanishes. Similarly, inward moving radial null geodesics
arrive at the observer in the center exactly at the symmetric $\tau=0$ point
when originating from the center of past infinity $\mathcal{I}^{-}$. This
means that, with respect to the two free-falling observers at the origin of
the two conjugate static patches, past infinity $\mathcal{I}^{-}$ and future
infinity $\mathcal{I}^{+}$ do not bend inward or outward. Combined with the de
Sitter isometries, which guarantees the equivalence of all free-falling
observers, this proves that the Penrose diagram is a perfect square, with past
and future infinity represented by straight horizontal orange lines in figure
2.

$\mathcal{I}^{+}$$\mathcal{I}^{-}$$\tau_{S}=0$$\tau_{N}=0$$t$$t$ Figure 2:
Penrose diagram of the de Sitter spacetime geometry. Null rays sent from the
north or south pole at $\tau=0$ meet at future infinity $\mathcal{I}^{+}$.
Similarly, null rays emanating from the center of past infinity
$\mathcal{I}^{-}$ reach the north or south pole at $\tau=0$.

Let us now consider symmetric radial spacelike geodesics in pure de Sitter
spacetime. With symmetric we main connecting two points at the separate poles
at opposite static times ($\tau_{S}=-\tau_{N}$). As is well known, in pure de
Sitter, the unique time for which a family of radial spacelike geodesics
exists is at the symmetric center, i.e. $\tau_{S}=-\tau_{N}=0$, for any value
of the conserved quantity $\mathcal{E}$. The turning points of these symmetric
radial geodesics are found at $r_{t}=l\sqrt{1+\mathcal{E}^{2}}$. This is easy
to show starting from (11) Aalsma:2022swk ; job ; Chapman:2021eyy ;
Galante:2022nhj ,

$\displaystyle\tau-t(r_{t})=\int_{0}^{r_{t}}\frac{\mathcal{E}}{(1-\frac{r^{2}}{\ell^{2}})(1-\frac{r^{2}}{\ell^{2}}+\mathcal{E}^{2})}dr=i\frac{\beta_{dS}}{4}$
(25)

implying that symmetric radial geodesics, assuming the conserved quantity
$\mathcal{E}$ is real, only exist when $\tau=0$. Computing the geodesic length
we then find Chapman:2021eyy ,

$\mathcal{D}_{dS}(\mathcal{E})=2\int_{0}^{r_{t}}\frac{\mathrm{d}r}{\sqrt{1-\frac{r^{2}}{\ell^{2}}+\mathcal{E}^{2}}}=2\ell\left(\lim_{r\rightarrow
r_{t}}\arctan\frac{r}{r_{t}-r}\right)=\pi\ell\,.$ (26)

Notably, we see that all these symmetric geodesics have the same geodesic
length, in other words the length is independent of $\mathcal{E}$. As also
pointed out in Chapman:2021eyy ; Galante:2022nhj , even in the limit
$\mathcal{E}\rightarrow\infty$, the geodesic length remains $\pi\ell$. This
could be considered surprising as one typically expects spacelike geodesics to
become null in this limit. However, one should realize that this limit, for
outgoing radial spacelike geodesics in de Sitter space, is expected to be
singular, since the null geodesics bounce off asymptotic (spacelike) infinity
($\mathcal{I}^{+}$ or $\mathcal{I}^{-}$) in this particular case (instead of
the black hole singularity in the SAdS geometry)111Analogous to comparing
outgoing null and spacelike geodesics in AdS, where the AdS boundary is at a
finite distance for null geodesics but at an infinite distance for spacelike
geodesics.. We stress once more that symmetric radial spacelike geodesics do
not exist for any other points (i.e. times) on the two poles. Instead, as
shown in Aalsma:2022swk ; Galante:2022nhj , one needs to introduce complex
geodesics in order to connect symmetric points on the north and south pole
lying that are not located at $\tau=0$.

This ends our introduction. In the rest of the paper, we will study how the
pure de Sitter causal structure changes as one introduces a black hole. We
will find that both the (interior) black hole singularity, as well as
(exterior) asymptotic infinity, are bend inwards and outwards respectively,
with respect to special, static sphere, observers. As a consequence, two
conjugate static sphere observers can be connected by interior and exterior
symmetric radial spacelike geodesics in a finite temporal window, whose
duration is defined by the magnitude of the inward and outward bending of the
interior black hole singularity and exterior asymptotic infinity respectively.

##  3 Geodesics in Schwarzschild-de Sitter spacetimes

We are now ready to explore the geodesic structure of the Schwarzschild-de
Sitter (SdS) geometry. We will determine both the size of inward bending of
the interior black hole singularity, as well as the exterior outward bending
of asymptotic infinity, with respect to a special pair of conjugate static
sphere observers. This implies corresponding changes in the size of the
interior and exterior causally connected regions connected to these special
observers and, related to that, a finite-size temporal window for symmetric
radial spacelike geodesics going either through the interior black hole
region, or the exterior de Sitter region.

Before we delve into the detailed computations, let us first briefly sketch
the SdS global geometry and how it compares to pure de Sitter. The presence of
the black hole in the Sitter requires an embedding of the black hole
singularity into an asymptotic de Sitter like structure, changing the topology
of the global spatial slices from $S^{3}$ to $S^{2}\otimes S^{1}$ for the
four-dimensional SdS spacetime Bousso:1996au . The black hole obviously breaks
the dS isometries, which reduce to just time translations (connecting the
north and south pole observers) and rotations. This also implies the existence
of a special pair of free-falling conjugate observers, namely those that stay
at rest with respect to the black hole and cosmological horizon in the two
conjugate static patches of global SdS. We will study the SdS geometry in $4$,
$5$ and $6$ spacetime dimensions, for which we can clearly distinguish
qualitative differences in the behavior.

In arbitrary dimensions $D$ the blackening factor that describes a black hole
in de Sitter space, a.k.a. SdS spacetime, is given Morvan:2022ybp ;
morvan:2022rn by

$f(r)=1-\frac{r^{2}}{\ell^{2}}-\alpha\frac{M}{r^{D-3}}\,.$ (27)

Here, $\alpha$ is a constant defined as

$\alpha=\frac{16\pi G}{(D-2)\Omega_{D-2}}.$ (28)

Where $M$ represents the mass parameter associated with the black hole
geometry, and $G$ stands for Newton’s constant. This metric features two
(spherical) horizons: a black hole and a cosmological horizon, with radii
$r_{b}$ and $r_{c}$ respectively. The existence of these horizons is due to
the presence of two positive real roots of the blackening factor, i.e. where
$f(r)=0$. In contrast to Schwarzschild black holes embedded in flat space,
which admit solutions for all values of the mass, in de Sitter space the range
of masses is finite. There exists an upper limit for the mass of a black hole
in de Sitter space, $M\leq M_{N}\equiv\frac{2}{\alpha(D-1)}{r_{N}^{D-3}}$, the
Nariai mass Nariai ; Morvan:2022ybp ; morvan:2022rn . The corresponding Nariai
radius $r_{N}$ can be derived from the condition
$f(r_{N})=f^{\prime}(r_{N})=0$ and equals

$r_{N}=\ell\sqrt{\frac{D-3}{D-1}}\,.$ (29)

It follows that for all smaller values of the mass, the black hole horizon is
smaller than the cosmological horizon. In the limit of vanishing mass $M=0$,
one of course ends up in empty pure de Sitter spacetime. Due to the presence
of a black hole, SdS not only has a cosmological temperature but also an
additional one, emanating from the black hole. For masses distinct from
Nariai, the two temperatures are different (as well as gravitational
entropies) and as such the system is not equilibrium, which will not play an
important role in our (classical) analysis. As usual, the temperatures of the
black hole and cosmological horizon are related to their respective surface
gravities $\kappa_{b,c}$,

$T_{b,c}=\frac{\kappa_{b,c}}{2\pi}\,.$ (30)

It is worth noting that only in the Nariai limit do these two temperatures
coincide, while for any other value of $M$, $T_{b}>T_{c}$. As the black hole
temperature is larger, ultimately this will lead to the decay to empty de
Sitter space, which indeed also corresponds to the maximal entropy state. To
identify the physical temperatures, corresponding to the relevant Euclidean
periodicities, let us discuss the proper normalization of the Killing vectors
associated with the surface gravity. The relationship between surface gravity
and the Killing vector $\zeta=\gamma\,\partial_{t}$, which generates time
translations, is expressed as follows

$\zeta^{\mu}\nabla_{\mu}\zeta^{\nu}=\kappa\zeta^{\nu}\,,$ (31)

where the value of $\gamma$ depends on the position of the observer under
consideration. In Anti-de Sitter and Minkowski spacetime, one usually
considers observers infinitely far from the black hole, rendering $\gamma=1$.
However, since that is not an opton in a SdS spacetime, we will have to make a
choice at what radius to evaluate the temperature, normalizing
$\gamma=1/\sqrt{f(r_{\mathcal{O}})}$, giving the Tolman temperature Toleman .
The natural candidate for this particular radius is the static sphere at
$r_{\mathcal{O}}$, defined by the free-falling observer at a fixed radius in
between the black hole and cosmological horizon. This unique value of the
radius in a given SdS metric also reduces nicely to the special observers in
the pure de Sitter ($r_{\mathcal{O}}=0$) and flat space limit
($r_{\mathcal{O}}\rightarrow\infty$). With the natural normalization fixed to
observers at $r=r_{\mathcal{O}}$, we can now compute the surface gravities of
the black hole and cosmological horizons

$\kappa_{b,c}=\frac{\gamma}{2}f^{\prime}(r)\Big{|}_{r=r_{b,c}}.$ (32)

Here, $r_{b,c}$ denote the radii of the black hole and cosmological horizons,
respectively. This relation can be evaluated and expressed in arbitrary
dimensions as Morvan:2022ybp ,

$\displaystyle\kappa_{b}$
$\displaystyle=\frac{\gamma(D-1)}{2}\frac{r_{N}^{2}-r_{b}^{2}}{r_{b}\ell^{2}}$
(33) $\displaystyle\kappa_{c}$
$\displaystyle=\frac{\gamma(D-1)}{2}\frac{r_{c}^{2}-r_{N}^{2}}{r_{c}\ell^{2}}$
(34)

Let us also stress here that using the standard value of the normalization
($\gamma=1$) actually leads to confusing results, as the temperature
associated to both horizons would vanish in the Nariai limit Gibbons:1977mu
(see figure 3), whereas the entropy does not Gibbons:1976ue . That the more
natural normalization is related to a static sphere observer was only realized
relatively recently Bousso:1996au ; Bousso:1997wi ; Morvan:2022ybp ;
Svesko:2022txo . As already emphasized, for the remainder of this paper, we
will consider the natural static sphere observers, which are free-falling at a
fixed radius. This radius is determined by the stationary point of the
blackening factor $f^{\prime}(r_{\mathcal{O}})=0$ and in arbitrary dimensions
equals Morvan:2022ybp

$\displaystyle
r_{\mathcal{O}}^{D-1}=\frac{(D-3)}{2}r_{b,c}^{D-3}(\ell^{2}-r_{b,c}^{2}).$
(35)

The blackening factor at this radius is
$f(r_{\mathcal{O}})=1-(r_{\mathcal{O}}/r_{N})^{2}$, implying that the
normalization of the temperature becomes
$\gamma=\sqrt{1-(r_{\mathcal{O}}/r_{N})^{2}}$. As a consequence, the physical
temperature in the Nariai limit, where $r_{b}=r_{c}=r_{N}$, equals
Morvan:2022ybp ; Svesko:2022txo

$\displaystyle\kappa_{N}=\frac{\sqrt{D-1}}{\ell}$ (36)

This nicely reproduces the temperature of the $dS_{2}\times S^{D-2}$ near-
horizon geometry that appears in the Nariai limit Bousso:2000md , i.e.
$T_{N}=\frac{1}{2\pi\ell}$. Interestingly, as briefly reviewed in section 2.3,
we would then also expect the causal structure to be represented by a perfect
square in the maixmal mass Narai limit. And indeed, that is what we will find.
In Figure 3 we plotted the appropriately normalized temperatures, as measured
by static sphere observers, as a function of the black hole mass in $4$ and
$5$ dimensions.

a) Normalized SdS temperatures in $D=4$

b) UN-Normalized SdS temperatures in $D=4$

Figure 3: Left: Normalized temperatures $T_{b,c}$ as a function of the black
hole mass in $D=4$. The orange and blue curves correspond to the cosmological
and black hole temperature respectively. Right: Un-normalized ($\gamma=1$)
temperatures.

As emphasized, the temperatures obtained from (33) and (34) are associated
with the timelike Killing vector $\partial_{\tilde{t}}$ instead of
$\partial_{t}$, related to the unique static sphere radius in SdS. Before we
move on to discuss null geodesics in SdS, it is important to realize that the
relevant (proper) time for a static sphere observer is of course determined by
the same normalization factor

$t\longrightarrow\tilde{t}\equiv\gamma{t}=t\,\sqrt{f(r_{\mathcal{O}})}=t\,\sqrt{1-(r_{\mathcal{O}}/r_{N})^{2}}\,.$
(37)

As we will be interested in computing time differences for static sphere
observers, in order to understand the causal structure with respect to those
observers, it will be important to use the appropriately normalized time
parameter $\tilde{t}$, instead of the SdS coordinate time $t$.

###  3.1 Radial null geodesics from static sphere observers in SdS

As in the basic examples, we will make use of (9) to understand radial null
geodesics in SdS. Importantly, we locate an observer at the static sphere
radius (35), from which an emitted radial null geodesic either goes inwards
towards the black hole horizon or outwards towards the cosmological horizon.
As already emphasized, the radius $r_{\mathcal{O}}$ corresponds to the unique
radius for which free-falling observers remain at a fixed distance from the
black hole and cosmological horizon respectively. In the Penrose diagrams
these locations will be identified with bold (vertical) straight lines.

a) $\text{Re}[\tilde{\mathcal{T}}_{b,c}]$, temporal windows.

b) $\text{Im}[\tilde{\mathcal{T}}_{b,c}]$, (-1/4) euclidean periodicities.

Figure 4:  The complex time $\tilde{\mathcal{T}}_{b,c}$ for $\tilde{\tau}=0$,
as a function of the black hole radius $r_{b}$ in $5$ dimensions. The blue
curves stand for the black hole and the orange curves for the cosmological
horizon.

As we fix the observer at $r=r_{\mathcal{O}}$, implying we should use the
appropriately rescaled (proper) time coordinate $\tilde{t}$ (37), the equation
for the time parameter along the null geodesic (9) becomes

$\displaystyle\tilde{t}_{b,c}(r^{\prime})=\tilde{\tau}_{b,c}\pm\gamma\int_{r^{\prime}}^{r_{\mathcal{O}}}\frac{\mathrm{d}r}{f(r)}\,,$
(38)

where $\tilde{\tau}_{b,c}$ stands for the initial time of the null radial
geodesic originating from the static sphere $r=r_{\mathcal{O}}$. The
subscripts $b$ and $c$ are added to distinguish between the two different
directions of the radial null geodesic: one moving inward heading towards the
black hole horizon and the other moving outward towards cosmological future
infinity. Imposing that the inward moving radial null geodesic reaches the
black hole singularity we set $r^{\prime}$ to zero, while instead setting
$r^{\prime}$ to infinity for an outward moving radial null geodesic. In
summary, we will be interested in the following two limiting (complex)
quantities

$\displaystyle\tilde{\mathcal{T}}_{c}=\lim_{r\rightarrow\infty}\tilde{t}_{c}(r)$
(39) $\displaystyle\tilde{\mathcal{T}}_{b}=\lim_{r\rightarrow
0}\tilde{t}_{b}(r)$ (40)

We will evaluate these limits in $4$, $5$ and $6$ dimensions, to identify and
distinguish two different types of characteristic behavior. We will start in
$5$ dimensions, which is the easiest to analyse analytically and for which the
results show a striking symmetry between interior black hole and exterior de
Sitter behavior. In five dimensions the blackening factor takes the following
form

$f(r)=\frac{\left(r^{2}-r_{b}^{2}\right)\left(r_{c}^{2}-r^{2}\right)}{\ell^{2}r^{2}}\,,$
(41)

and the horizon radii can be expressed as Morvan:2022ybp ,

$r_{b,c}=r_{N}\sqrt{1\mp\sqrt{1-M/M_{N}}}\,.$ (42)

The static sphere observers are located at
$r_{\mathcal{O}}=\sqrt{r_{b}\,r_{c}}$, such that the respective Euclidean
periodicities, i.e. the temperatures, are determined by

$\gamma=\sqrt{f(r_{\mathcal{O}})}\,.$ (43)

We can now readily compute eq. (39) and eq. (40). For $D=5$ SdS we find

$\displaystyle\tilde{\mathcal{T}}_{b,c}$
$\displaystyle=\tilde{\tau}_{b,c}\pm\mathcal{T}_{b,c}\mp
i\frac{\beta_{b,c}}{4}\,,$ (44)

where $+\mathcal{T}_{b}$ identifies the real contribution for inward moving
null shells and $-\mathcal{T}_{c}$ for outward moving shells respectively. The
imaginary part in (44) is connected to the Euclidean periodicity, or the
inverse temperature, generated by each respective horizon. The real
contributions $\mathcal{T}_{b,c}$ are in fact the same in $D=5$, and equal to

$\displaystyle\mathcal{T}_{b}=\mathcal{T}_{c}=\ell\frac{\left(r_{b}-r_{c}\right)}{\left(r_{b}+r_{c}\right)}\operatorname{arctanh}\left(\frac{r_{b}}{r_{\mathcal{O}}}\right)\,.$
(45)

The result for the real contribution is nonzero and in this case the same, up
to a sign flip, for both the inward moving radial null geodesic reaching the
black hole singularity, as well as the outward moving radial null geodesics
reaching future infinity, which is distinctively different compared to empty,
pure, de Sitter spacetime.

$\tilde{\tau}=-\mathcal{T}_{b}$$\tilde{\tau}=\mathcal{T}_{c}$$r=r_{\mathcal{O}}$$\tilde{\tau}=0$$r=0$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$r=0$$x$$x^{\prime}$$x^{\prime}$

Figure 5: Causal Penrose diagram for $5$-dimensional SdS spacetime, with
respect to static sphere observers represented by straight vertical lines. The
black hole singularity (blue wavy lines) and de Sitter future/past infinity
(solid orange) bend inward and outward respectively. The dashed diagonal lines
on the left stand for the black hole and the ones on the the right for the
cosmological horizon respectively. Also indicated are the times for which
inward or outward moving radial null shells reach the center of the
future/past black hole singularity or future/past asymptotic infinity, and a
pair of symmetric conjugate events $x$ and $x^{\prime}$.

For radial null geodesics moving inward from $r=r_{\mathcal{O}}$ and reaching
the black hole singularity the real contribution $+\mathcal{T}_{b}$ is
positive unless $M=0$ or $M=M_{N}$, where it exactly vanishes as anticipated.
The same happens for $D>3$ Schwarzschild black holes in AdS spacetime
Fidkowski:2003nf , where the real part of the integral eq. (9) is also
positive. This means that radial null geodesics moving inwards, emitted at
$\tilde{\tau}=0$ from $r=r_{\mathcal{O}}$, reach the singularity $r=0$ "off-
center", implying that the black hole singularity is bend inwards with respect
to (straight vertical line) static sphere observers. Alternatively, for a null
shell to reach the singularity’s center, i.e.,
$\text{Re}[\tilde{\mathcal{T}}_{b}]=0$, it should be emitted not at
$\tilde{\tau}=0$, but at $\tilde{\tau}=-\mathcal{T}_{b}$. Clearly, as a
consequence of (global) time reversal symmetry, outward moving radial null
geodesics emitted from the center of the white hole singularity $r=0$ will
reach the static sphere observer at a time $\tilde{\tau}=+\mathcal{T}_{b}$.
Also, the same conclusions hold for the conjugate static sphere observer,
implying that two inward moving radial null geodesics emitted at the symmetric
events $\tilde{\tau}=-\mathcal{T}_{b}$ on the right, and
$\tilde{\tau}=+\mathcal{T}_{b}$ on the left, will meet exactly in the center
of the (future) black hole singularity $r=0$.

Let us now shift our attention to radial null geodesics that move outwards
towards future de Sitter infinity, i.e. towards the exterior region of the SdS
spacetime, emitted from the static sphere observer. In that case the real
contribution $-\mathcal{T}_{c}$ is negative, vanishing only for pure dS or in
the maximal mass, Nariai limit. As a result, when the radial null geodesic is
emitted from $r=r_{\mathcal{O}}$ at $\tilde{\tau}=0$ it will pass the center
of future infinity, ending off-center to the right. Instead, when two radial
null shells are emitted outwards from two conjugate static sphere observers at
(symmetric) times $\tilde{\tau}=\mathcal{T}_{c}$ on the right and
$\tilde{\tau}=\mathcal{T}_{c}$ on the left, will they meet each other, in the
future infinity limit, in the center. Similarly, considering inward moving
null shells, the sign of the real contribution changes, and null shells
starting at the center of past infinity will reach the static sphere at
$\tilde{\tau}=-\mathcal{T}_{c}$.

In Figure 4 the real and imaginary part of $\tilde{\mathcal{T}}_{b,c}$ are
plotted with respect to the black hole radius. Remarkably, as the real
contributions $\mathcal{T}_{b,c}$ are the same, up to a sign, for
$5$-dimensional SdS, we conclude that in five dimensions the magnitude of the
black hole singularity bending inwards matches the magnitude of future
infinity bending outwards. Note that choosing the appropriate (proper) time
normalization is crucial for obtaining these results. If one would have
instead picked $\gamma=1$, both the real and imaginary part of
$\tilde{\mathcal{T}}_{b,c}$ would have diverged in the Nariai limit. Having
determined the details of the causal structure, with respect to static sphere
observers, we can already anticipate the presence of finite size temporal
window for the existence of radial spacelike geodesics connecting symmetric
points between two conjugate static sphere observers in SdS spacetime. These
symmetric points can either be connected through the interior of the black
hole, or through the exterior asymptotic de Sitter region, and in five
dimensions both temporal windows have the same size.

As we move on to $4$ and $6$ dimensions we will encounter the same type of
qualitative behavior, except for the fact that the magnitude of the respective
interior black hole (inward) or exterior future infinity (outward) bending
will no longer be the same.

In four dimensions the horizon radii $r_{b,c}$ can be written as
Morvan:2022ybp

$\displaystyle
r_{b,c}=r_{N}\left(\cos{\eta}\mp\sqrt{3}\sin{\eta}\right),\quad\text{where}\quad\eta\equiv\frac{1}{3}\arccos{M/M_{N}}\,.$
(46)

In terms of the event horizon radii the blackening factor equals

$\displaystyle f(r)=\frac{(r-r_{b})(r-r_{c})(r+r_{b}+r_{c})}{\ell^{2}r}\,,$
(47)

which implies that the sphere static observers are located at the special
radius

$r_{\mathcal{O}}^{3}=\frac{r_{b}r_{c}}{2}(r_{b}+r_{c})=\frac{r_{b,c}}{2}(\ell^{2}-r^{2}_{b,c})\,,\
$ (48)

a) $\text{Re}[\tilde{\mathcal{T}}_{b,c}]$, temporal windows.

b) $\text{Im}[\tilde{\mathcal{T}}_{b,c}]$, (-1/4) euclidean periodicities.

Figure 6: The complex time $\tilde{\mathcal{T}}_{b,c}$ for $\tilde{\tau}=0$,
as a function of the black hole radius $r_{b}$ in $4$ dimensions. The blue
curve represents $\tilde{\mathcal{T}}_{b}$, and the orange curve represents
$\tilde{\mathcal{T}}_{c}$.

Let us now again use (38) to evaluate the (complex) time parameter along a
radial null geodesic emitted from the static sphere observer to any point
$r^{\prime}$. As compared to $5$ dimensions, applying (46) and (48) and
evaluating the result is much more involved and we refer to the appendix for
the details, but in the end we can write

$\displaystyle\tilde{t}_{b,c}(r^{\prime})$
$\displaystyle=\tilde{\tau}\pm\gamma\ell^{2}\left(\frac{\left(r_{b}-r_{c}\right)\left(r_{b}+r_{c}\right)\ln\left(r_{b}+r_{c}+r\right)+r_{c}\left(2r_{b}+r_{c}\right)\ln\left(r-r_{c}\right)-r_{b}\left(r_{b}+2r_{c}\right)\ln\left(r-r_{b}\right)}{\left(r_{b}-r_{c}\right)\left(2r_{b}+r_{c}\right)\left(r_{b}+2r_{c}\right)}\bigg{|}_{r^{\prime}}^{r_{\mathcal{O}}}\right)\,.$
(49)

Here, as before, the plus sign is selected for the inward moving null shells
and the minus sign corresponds to outward moving null shells. Taking the
relevant limits we can evaluate the (real) times $\mathcal{T}_{b,c}$ for null
geodesics to reach the black hole singularity or dS future infinity. Not
surprisingly, we find expressions qualitatively similar to (44) for
$\mathcal{T}_{b,c}$, which we plotted in figure 6. Again, the real
contribution $+\mathcal{T}_{b}$ is positive, while it is negative for
$-\mathcal{T}_{c}$, except in the pure de Sitter or Nariai limit, where it
vanishes. So as in five dimensions, with respect to static sphere observers,
the black hole singularity is bending inwards, whereas asymptotic de Sitter
infinity is bending outwards. In four dimensions however, one finds
$|\mathcal{T}_{b}|\geq|\mathcal{T}_{c}|$, implying that the magnitude of the
inward bending of the black hole singularity exceeds the outward bending of
future infinity, for any value of the mass of the black hole.

For $D=4$ we can therefore draw similar conclusions as in five dimensions: an
inward null shell emitted from the static sphere reaches the center of the
black hole singularity only when emitted at $\tilde{\tau}=-\mathcal{T}_{b}$.
While to reach the center of de Sitter future infinity from the static sphere
the null shell should be emitted at a positive (later) time, i.e.
$\tilde{\tau}=\mathcal{T}_{c}$. Again, we can use the time reversal symmetry
of the SdS spacetime to identify the other null shells of interest (arriving
from the past white hole singularity or dS past infinity). The inward bending
of the interior black and white hole singularity imply a decrease of the size
of the interior causal diamond, whereas the outward bending of future and past
infinity define an increase of the exterior causal diamond, with a magnitude
measured by the two temporal windows $\mathcal{T}_{b}$ and $\mathcal{T}_{c}$.
Clearly, if exterior future infinity was ’straight’, null shells emitted
outward from symmetric (positive time for the right, negative time for the
left) static sphere observers would never intersect. In other words, for times
within the temporal window $-\mathcal{T}_{c}<\tilde{\tau}<\mathcal{T}_{c}$ the
two conjugate static sphere observers share the same causal region.

$r=0$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$r=0$$r=r_{\mathcal{O}}$$\tilde{\tau}=\mathcal{T}_{c}$$\tilde{\tau}=-\mathcal{T}_{c}$$\tilde{\tau}=\mathcal{T}_{b}$$\tilde{\tau}=-\mathcal{T}_{b}$$x$$x^{\prime}$$x^{\prime}$

a) Static sphere centered

$\tilde{\tau}=-\mathcal{T}_{c}$$\tilde{\tau}=\mathcal{T}_{c}$$r=0$$r=0$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$\mathcal{I}^{-}$$\mathcal{I}^{+}$$r=r_{\mathcal{O}}$$r=r_{\mathcal{O}}$$\tilde{\tau}=\mathcal{T}_{b}$$\tilde{\tau}=-\mathcal{T}_{b}$$x$$x^{\prime}$

b) Black hole centered

Figure 7: Causal Penrose diagram for $4$-dimensional SdS spacetime, with
respect to static sphere observers represented by straight vertical lines,
centered around a static sphere observer or the black hole. The black hole
singularity (blue wavy lines) and de Sitter future/past infinity (solid
orange) bend inward and outward respectively. The dashed blue and orange lines
are the black hole and cosmological horizon respectively. We also indicated
some inward and outward radial null shells, as well as a pair of symmetric
conjugate events.

The opposite behavior is observed in the four-dimensional black hole region.
The causally connected region between the black and white hole singularity
shrinks. This means that null shells emitted from conjugate static sphere
observers at a symmetric (global) time never intersect in the interior if
emitted after $\tilde{\tau}>-\mathcal{T}_{b}$. Equivalently, two null shells
cannot have a common origin in the interior of the black hole if arriving
before $\tilde{\tau}<\mathcal{T}_{b}$ by a conjugate pair of static sphere
observers. In summary, not surprisingly the qualitative behavior in four
dimensions is the same, but the size of the temporal window for inward radial
null geodesics, related to the magnitude of the inward bending of the black
and white hole singularity, exceeds the size of the temporal window for
outward radial null geodesics, related to the magnitude of the outward bending
of asymptotic de Sitter future and past infinity, for any value of the mass.
We have sketched the four-dimensional causal Penrose diagrams using two
different perspectives, either centered around the static sphere observer or
centered around the black hole, in Figure 7.

To finish this section on radial null geodesics in SdS spacetime, let us
discuss the results for the static sphere time parameter along inward and
outward radial null geodesics in $6$-dimensional SdS, which are plotted in fig
8. For this case we only plotted the real part of the (complex) static sphere
time parameter, and in addition sketched the corresponding $6$-dimensional
Penrose diagram. Although technically more involved (for the detailed
expression we refer to the appendix), in $6$ dimensions we conclude that
$|\mathcal{T}_{c}|\geq|\mathcal{T}_{b}|$ for any value of the black hole mass.
In other words, in $6$ dimensions the size of the outward bending of future
(or past) infinity, as measured in terms of the temporal window at the static
sphere, exceeds the size of the inward bending of the black (or white) hole
singularity, for any black hole mass. Qualitatively this is opposite from the
situation in $4$ dimensions, with $5$ dimensions sitting exactly in between
the two types of behavior, where the interior (inward) and exterior (outward)
temporal window are exactly the same. In dimensions more than $6$ we expect
the larger exterior (outward) bending of future (past) infinity to become more
pronounced as the number of dimensions grows.

a) $\text{Re}[{\mathcal{T}}_{b,c}]$, temporal windows

$r=0$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$r=0$$r=r_{\mathcal{O}}$$x$$x^{\prime}$$x^{\prime}$$\tilde{\tau}=\mathcal{T}_{b}$$\tilde{\tau}=-\mathcal{T}_{b}$$\tilde{\tau}=\mathcal{T}_{c}$$\tilde{\tau}=-\mathcal{T}_{c}$

b) SdS Penrose diagram

Figure 8: (a) $\mathcal{T}_{b}$ (blue curve) and $\mathcal{T}_{c}$ (orange
curve) as a function of the black hole radius $r_{b}$ in $6$ dimensions. (b)
Causal Penrose diagram of $6$-dimensional SdS, with respect to static sphere
observers, showing larger outward bending of future (past) infinity as
compared to the inward bending of the black (white) hole singularity.

As before we conclude that symmetric conjugate static sphere observers, on the
same global time slice, are part of the same exterior causal region as long as
$-\mathcal{T}_{c}<\tilde{\tau}<\mathcal{T}_{c}$. Correspondingly, the temporal
window $-\mathcal{T}_{b}<\tilde{\tau}<\mathcal{T}_{b}$ informs us that the
interior black hole region has decreased, i.e. for all times within that
window symmetric emitted or received null shells from the pair of conjugate
static sphere observers cannot intersect in the interior black hole region.
Finally, let us emphasize that our conclusions for the relative size of the
interior (inward) and exterior (outward) bending of the black hole singularity
and asymptotic de Sitter infinity hold for any value of the mass. Moreover,
for any number of dimensions, in the Nariai limit, the causal structure for
radial null geodesics reduces again to that of empty, pure, de Sitter, as
anticipated from the near-horizon de Sitter structure of the geometry. In the
Nariai limit the causal Penrose diagram therefore effectively reduces to that
of $2$-dimensional de Sitter spacetime, as portrayed in Figure 9.

$r=r_{\mathcal{O}}$$\tilde{\tau}=0$$r=0$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$r=0$$x$$x^{\prime}$$x^{\prime}$

Figure 9: Causal Penrose diagram of SdS in the Nariai limit.

This ends our discussion of null geodesics and the corresponding bending
inwards of the black (white) hole singularity, as well as the bending outwards
of future/past infinity. We can rephrase these results as follows: as compared
to pure de Sitter spacetime, where the causally connected regions (defined by
the center of future and past infinity) of two conjugate observers touch at
their boundaries, in the SdS geometry the interior black hole (defined by the
center of the black/white hole singularity) and the exterior de Sitter
(defined by the center of future/past infinity) causal regions are
disconnected in $D=4$, they touch (as in pure de Sitter) in $D=5$, and they
overlap in $D=6$.

This structure should have consequences for the symmetric correlations between
conjugate (large mass) operators. In Fidkowski:2003nf the modified causal
structure of the interior black hole region signalled a ’phase transition’ for
symmetric correlations between conjugate (large mass) operators exactly as one
crosses the boundary of the temporal window, when the correlator starts to
probe the singularity. Using a geodesic approximation for the symmetric
correlators of (large mass) conjugate operators this transition is signalled
by the presence (and disappearance) of spacelike geodesics between the
conjugate operators. The spatial compactness of the de Sitter and SdS geometry
implies that two conjugate operators can be connected by spacelike geodesics
either through the black hole interior or the de Sitter exterior. As is well
known, in pure de Sitter, except for the symmetric point at $\tau=0$, no
spacelike geodesics exist that connect symmetric pairs of conjugate operators,
but the appearance of (two) temporal windows suggests otherwise for SdS. In
the next section we will generalize our analysis of radial null geodesics in
SdS to study the presence (and length) of spacelike geodesics between
symmetric pairs of conjugate static sphere observers.

###  3.2 Spacelike geodesics between static sphere observers in SdS

In the previous section, we observed that the causal connectivity in the SdS
background exhibits some interesting features. In general, the interior and
exterior causal regions shrink and expand respectively, opening up two
temporal windows for conjugate static sphere observers. The details, depicted
in figures 5, 7 and 8, depend on the dimension of the SdS spacetime under
consideration. In this section, we will focus on the appearance of radial
spacelike geodesics connecting symmetric conjugate points $x$ and
$x^{\prime}$, located on opposite static spheres at the same global time
slice.

We start our discussion again in $D=5$, but will also briefly discuss the
$4$\- and $6$-dimensional case. For radial spacelike geodesics the conserved
quantity $\mathcal{E}$ plays an important role, as is evident from (11) and
(12). For example, in empty dS space, two (symmetric) points lying at the
north and south poles at $\tau_{N}=\tau_{S}=0$ are connected by a family of
real spacelike geodesics characterized by $\mathcal{E}$ (see Chapman:2021eyy
for a nice review with some specific examples). When the conserved quantity
$\mathcal{E}$ approaches infinity, the spacelike geodesics from the north pole
at $\tau_{N}=0$ get ever closer to future infinity before they turn around and
reach the south pole at $\tau_{S}=0$. For any other (symmetric) time, no
(real) spacelike geodesics exist that connect two conjugate points
Lars:2022geo ; Galante:2022nhj . As our analysis of the SdS causal structure
has shown, introducing a black hole in de Sitter significantly affects the
above situation for a pair of conjugate static sphere observers. For SdS one
should instead be able to construct spacelike geodesics connecting two
symmetric conjugate points for some (finite) temporal window, both through the
interior black hole region, as well as the exterior de Sitter region. Let us
now work out some of the details, keeping in mind that we have two horizons,
giving rise to an interior and exterior connection between conjugate points
and that we should use the appropriately normalized static sphere time (37) in
our analysis.

Figure 10: Turning points for spacelike geodesics in the SdS geometry, for
some value of $\mathcal{E}$. The green and magenta points correspond to the
turning points, while the blue and orange points are the black hole and
cosmological horizon respectively. The grey point denotes the position of the
static sphere.

We will start with the general expressions for the proper lengths of spacelike
radial geodesics connecting two conjugate symmetric points

$\displaystyle\mathcal{D}_{b}=2\int_{r_{1}}^{r_{\mathcal{O}}}\frac{\mathrm{d}r}{\sqrt{f(r)+\mathcal{E}^{2}}}$
(50)
$\displaystyle\mathcal{D}_{c}=2\int_{r_{\mathcal{O}}}^{r_{2}}\frac{\mathrm{d}r}{\sqrt{f(r)+\mathcal{E}^{2}}}.$
(51)

Here, $\mathcal{D}_{b}$ and $\mathcal{D}_{c}$ are the lengths of radial
spacelike geodesics connecting opposite symmetric points on the static sphere
through the interior black hole and exterior de Sitter regions. Note that
$\mathcal{E}$ will be a function of the initial (proper) time on the static
sphere. In other words, for the case at hand: a spacelike geodesic connecting
symmetric points labeled by some time $\tilde{t}$ only exists for a specific
value of $\mathcal{E}$, or does not exist at all. This relation is governed by
(11). The radii $r_{1}$ and $r_{2}$ are the turning points for the radial
spacelike geodesics inside the interior black hole and the exterior asymptotic
dS region, respectively. They correspond to the positive roots of
$f(r)+\mathcal{E}^{2}$, assuming $\mathcal{E}$ is real, where the two
conjugate parts of the radial spacelike geodesic are glued together in a
consistent, continuous, way222This means that the turning points always lie in
the ’center’ $\text{Re}[\tilde{t}]=0$ and the derivative $d\tilde{t}/dr$
vanishes there as well.. The presence of turning points can be nicely
understood graphically by plotting the blackening factor, with an opposite
sign, behaving as a potential barrier Fidkowski:2003nf , as shown in Figure
10, and where the conserved quantity $\mathcal{E}$ can then be interpreted as
the energy determining the location of the turning points. In $4$\- and
$5$-dimensional SdS spacetime they are given by

$r_{1,2}=\begin{cases}r_{N}\sqrt{(1+\mathcal{E}^{2})\mp\sqrt{(1+\mathcal{E}^{2})^{2}-M/M_{N}}}&,\,D=5\\\
r_{N}\sqrt{1+\mathcal{E}^{2}}\left(\cos{\eta^{\prime}}\mp\sqrt{3}\sin{\eta^{\prime}}\right)&,\,D=4\,,\end{cases}$
(52)

where

$\eta^{\prime}=\frac{1}{3}\arccos{\frac{M}{M_{N}(1+\mathcal{E}^{2})^{3/2}}}\,,$
(53)

Note that the initial derivative of the radial spacelike geodesic
$(d\tilde{t}/dr)|_{r_{\mathcal{O}}}$ also depends on $\mathcal{E}$, and that
it should approach the (inward or outward) radial null geodesic in the
$\mathcal{E}\rightarrow\pm\infty$ limit.

Let us rewrite (11) for the specific case of a spacelike radial geodesic
passing into the black hole interior from a particular time $\tilde{\tau}_{b}$
at the static sphere radius

$\tilde{t}_{\mkern
2.0mub}(r)-\tilde{\tau}_{b}=\int^{r_{\mathcal{O}}}_{r}\frac{\gamma\,\mathcal{E}\,\mathrm{d}r^{\prime}}{f(r^{\prime})\sqrt{f(r^{\prime})+\mathcal{E}^{2}}}\,.$
(54)

Introducing the interior, black hole, turning point $r=r_{1}$ (at which
$\text{Re}[\tilde{t}_{b}]=0$), to allow for a (symmetric) connection to the
other conjugate static sphere observer, one obtains

$-\tilde{\tau}_{b}=i\frac{\beta_{b}}{4}+\int_{r_{1}}^{r_{\mathcal{O}}}\frac{\gamma\,\mathcal{E}\,\mathrm{d}r}{f(r)\sqrt{f(r)+\mathcal{E}^{2}}}\,.$
(55)

The above equation relates the conserved quantity $\mathcal{E}$ to the initial
time $\tilde{\tau}_{b}$ on the static sphere for interior, black hole,
spacelike geodesics reaching an (interior) turning point. Similarly, we find a
relation between $\mathcal{E}$ and the initial time $\tilde{\tau}_{c}$ on the
static sphere for the other, exterior de Sitter, spacelike geodesic reaching
an (exterior) turning point

$-\tilde{\tau}_{c}=-i\frac{\beta_{c}}{4}+\int_{r_{\mathcal{O}}}^{r_{{2}}}\frac{\gamma\,\mathcal{E}\,\mathrm{d}r}{f(r)\sqrt{f(r)+\mathcal{E}^{2}}}\,,$
(56)

As we swapped the boundaries of the integral when switching from interior to
exterior spacelike radial geodesics, we will pick up a relative minus sign in
comparing interior and exterior geodesics (related to the distinction between
inward and outward radial geodesics) that can also be incorporated in the sign
of the conserved quantity $\mathcal{E}$. However, to more easily distinguish
the results for the interior and exterior geodesics in our plots, we will keep
this relative minus sign for the static sphere time comparing interior and
exterior spacelike geodesics, for positive values of $\mathcal{E}$. Fixing one
overall consistent sign for the two different integrals will remove this
relative sign difference. In other words, the negative values we will find for
the exterior static sphere time, for positive $\mathcal{E}$, are useful for
plotting them in the same graph, but actually correspond to negative
$\mathcal{E}$. As long as we are aware of this small subtlety, we can now go
ahead and evaluate the integrals (55) and (56) in $5$ dimensions, for which we
obtain the following (analytic) results for the relation between (real) static
sphere times and the conserved quantity $\mathcal{E}$

$\displaystyle\tilde{\tau}_{b}$
$\displaystyle=\text{Re}\left[\frac{\ell^{2}\,\mathcal{E}}{r_{b}+r_{c}}\left(\frac{r_{b}^{2}\operatorname{arctanh}\left(\frac{\sqrt{r_{\mathcal{O}}^{2}-r_{1}^{2}}}{\sqrt{r_{2}^{2}-r_{\mathcal{O}}^{2}}}\frac{\sqrt{r_{2}^{2}-r_{b}^{2}}}{\sqrt{r_{b}^{2}-r_{1}^{2}}}\right)}{\sqrt{r_{2}^{2}-r_{b}^{2}}\sqrt{r_{b}^{2}-r_{1}^{2}}}-\frac{r_{c}^{2}\arctan\left(\frac{\sqrt{r_{\mathcal{O}}^{2}-r_{1}^{2}}}{\sqrt{r_{2}^{2}-r_{\mathcal{O}}^{2}}}\frac{\sqrt{r_{c}^{2}-r_{2}^{2}}}{\sqrt{r_{c}^{2}-r_{1}^{2}}}\right)}{\sqrt{r_{c}^{2}-r_{1}^{2}}\sqrt{r_{c}^{2}-r_{2}^{2}}}\right)\right]\,,$
(57) $\displaystyle\tilde{\tau}_{c}$ $\displaystyle=-\tilde{\tau}_{b}\,.$ (58)

Figure 11:  Static sphere times as a function of $\mathcal{E}$ for interior
(green) and exterior (magenta) radial spacelike geodesics in $5$-dimensional
SdS at fixed $r_{b}/r_{N}=1/2$.

We refer to the appendix for the full expression. The final $5$-dimensional
result is conveniently expressed in terms of the turning point radii, whose
dependence on $\mathcal{E}$ is determined by (52). As for the radial null
geodesics, the absolute values are the same, meaning that the interior and
exterior static sphere time are the same for a fixed $\mathcal{E}$. We plotted
the results for the real part of the static sphere times as a function of the
conserved quantity $\mathcal{E}$ in figure 11.

For the spacelike radial geodesic passing through the interior black hole
region, as the conserved quantity $\mathcal{E}$ goes to infinity, the initial
time for the spacelike geodesic approaches
$\tilde{\tau}_{b}\rightarrow\mathcal{T}_{b}$. Similarly, in the opposite limit
$\mathcal{E}\rightarrow-\infty$, we find that
$\tilde{\tau}_{b}\rightarrow-\mathcal{T}_{b}$. For $\mathcal{E}=0$ the
spacelike radial geodesic connects two conjugate static sphere points at the
center $\tilde{t}=0$, never entering the black hole interior, connecting
through the bifurcation point (where the past and future event horizon
intersect).

(a) $D=4$

(b) $D=6$

Figure 12: Static sphere times as a function of $\mathcal{E}$ for interior
(green) and exterior (magenta) radial spacelike geodesics in a)
$4$-dimensional and b) $6$-dimensional SdS at fixed $r_{b}/r_{N}=1/2$.

A similar story holds for the exterior radial spacelike geodesic passing
through the asymptotic dS region, as is evident from figure 11. In the limit
$\mathcal{E}\rightarrow\pm\infty$, we find, as expected, that the radial
spacelike geodesics approach $\tilde{\tau}_{c}\rightarrow\mp\mathcal{T}_{c}$.
As we emphasized before, the $5$-dimensional SdS causal structure is special
in the sense that the inward interior (black/white hole) bending equals the
exterior (future/past infinity) bending, implying that the interior and
exterior temporal windows are the same. This also implies that the map between
static sphere times and $\mathcal{E}$ is the same for inward (interior) and
outward (exterior) spacelike geodesics, i.e. the inward and outward radial
spacelike geodesics can be smoothly connected (it is differentiable). In
Figure 11 this property is made evident through the symmetry around the
$\mathcal{E}$ axis.

In $4$ and $6$ dimensions this symmetry is absent, as can be seen in Figure
12. The integrals in eq. (55) and (56) are more complicated and harder to
solve in four and six dimensions. The analytic results in $4$ dimensions are
given in the Appendix (80), anf we will briefly allude to them below. In $6$
dimensions we were only able to derive approximate results that we have
plotted in Figure 12. We will briefly comment on qualitative differences as
compared to $D=4$ and $D=5$.

In four dimensions we concluded that the interior (black/white hole) temporal
window exceeds the exterior (asymptotically de Sitter) window
$|\mathcal{T}_{b}|>|\mathcal{T}_{c}|$, implying that the relation between
static sphere time and $\mathcal{E}$ is not the same for interior and exterior
radial spacelike geodesics. As a corollary, for a given static sphere time,
the interior and exterior (conserved) quantity $\mathcal{E}$ is different,
meaning that the interior and exterior radial spacelike geodesic do not
connect in a differentiable way (as the inward and outward initial slope are
not the same, as they are set by different $\mathcal{E}$). In four dimensions,
for any finite mass below the Nariai limit, and for a fixed $\mathcal{E}$, the
interior static sphere time is always bigger than the exterior static sphere
time, i.e. $|\tilde{\tau}_{b}|>|\tilde{\tau}_{c}|$. And no symmetric radial
spacelike geodesics exist between two conjugate static sphere observers beyond
the maximum of the two temporal windows. Presumably, as for the AdS black hole
Fidkowski:2003nf and in pure de Sitter Aalsma:2022swk ; Galante:2022nhj , one
should instead consider complex geodesics for static sphere times outside the
temporal windows (related to complex values of the conserved quantity
$\mathcal{E}$). In five dimensions this point coincides for interior (black
hole) and exterior (de Sitter) geodesics, but in four dimensions one first
crosses the exterior temporal window before reaching the interior temporal
boundary.

Since we now understand when and how to connect two symmetric static sphere
points by a radial spacelike geodesics either going through the interior black
hole region or the exterior asymptotic de Sitter region, in $4$ and $5$
dimensions for any real value of $\mathcal{E}$, we can proceed and compute the
proper lengths (50) and (51). In $5$ dimensions one then obtains relatively
nice analytic expressions for the interior and exterior geodesic lengths

$\displaystyle\mathcal{D}_{b}$
$\displaystyle=2\ell\arcsin{\sqrt{\frac{r_{1}^{2}-r_{\mathcal{O}}^{2}}{r_{1}^{2}-r_{2}^{2}}}}\,,$
(59) $\displaystyle\mathcal{D}_{c}$ $\displaystyle=\pi\ell-\mathcal{D}_{b}\,.$
(60)

Remarkably, we conclude that the sum of the (interior plus exterior) lengths
for $5$-dimensional SdS equals

$\displaystyle\mathcal{D}_{b}+\mathcal{D}_{c}=\pi\ell$ (61)

for any energy $\mathcal{E}$, which is the same as the length of any spacelike
geodesic connecting conjugate observers at $\tau_{N}=\tau_{S}=0$ in empty,
pure, de Sitter spacetime. Apparently the introduction of a black hole in
$5$-dimensional de Sitter does not change the total length of the spacelike
geodesics, it just splits it up in interior black hole and exterior de Sitter
parts, the division of the split depending on the mass of the black hole.

In four dimensions the final expressions for the integrals eq. (50) and (51)
are much more involved, but can still be written elegantly in terms of special
functions of the horizon and turning point radii

$\displaystyle\mathcal{D}_{c}$
$\displaystyle=\frac{4i\ell}{\sqrt{r_{2}}\sqrt{2r_{1}+r_{2}}}\left\\{r_{1}F(A|B)+\left(r_{1}-r_{2}\right)\Pi\left(\frac{r_{2}}{r_{1}}B;A|B\right)\right\\}\,,$
(62) $\displaystyle\mathcal{D}_{b}$
$\displaystyle=\mathcal{D}_{c}(r_{1}\longleftrightarrow r_{2})\,,$ (63)

Figure 13: Left: different lengths of interior and exterior spacelike
geodesics in $D=5$ as a function of the black hole radius $r_{b}$, for
different $\mathcal{E}=0,1,2,3$, indicated by their respective opacity. Right:
Penrose diagram with spacelike geodesics connecting symmetric conjugate static
sphere points ((interior in green, exterior in magenta) for different
$\mathcal{E}$.

where $F$ is the elliptic function of the first kind and $\Pi$ is the elliptic
function of the second kind. Furthermore, we have identified,

$\displaystyle A$
$\displaystyle\equiv\arcsin{\left(\sqrt{\frac{\left(2r_{1}+r_{2}\right)\left(r_{\mathcal{O}}-r_{2}\right)}{\left(r_{1}+2r_{2}\right)\left(r_{\mathcal{O}}-r_{1}\right)}}\right)}\,,\quad
B\equiv\frac{r_{1}\left(r_{1}+2r_{2}\right)}{r_{2}\left(2r_{1}+r_{2}\right)}\,.$
(64)

Figure 14: Different lengths of interior and exterior spacelike geodesics in
$D=4$ (left) and $D=6$ (right) as a function of the black hole radius $r_{b}$.
Plotted for different $\mathcal{E}=0,1,2,3$, indicated by their respective
opacity.

In this case the sum of the two lengths, for arbitrary $\mathcal{E}$, no
longer satisfies (61). We do note however that the total geodesic length
connecting the symmetric points on the static sphere becomes less and less
sensitive to the presence of the black hole in the limit
$|\mathcal{E}|\rightarrow\infty$ and approaches the empty de Sitter result

$\mathcal{D}_{b}\xrightarrow{\mathcal{E}\rightarrow\infty}0\,,\quad\mathcal{D}_{c}\xrightarrow{\mathcal{E}\rightarrow\infty}\pi\ell=\mathcal{D}_{dS}\,.$
(65)

Figure 15: Penrose diagram with spacelike geodesics connecting symmetric
conjugate static sphere points ((interior in green, exterior in magenta) for
different $\mathcal{E}$, in $D=4$ (right) and $D=6$ (left).

The vanishing of $\mathcal{D}_{b}$ in that limit is not surprising, as the
radial spacelike geodesic crossing the black hole interior becomes null, but
the other finite limit (65) might at first be considered surprising. However,
as for empty de Sitter spacetime, this can be understood by realizing that the
limit to a radial null geodesic through the exterior cannot be regular, as
null geodesics reach (or ‘bounce off’) future infinity in de Sitter, which is
an infinite distance away for any spacelike geodesic.

Having understood the details of the symmetric radial spacelike geodesics
between a pair of conjugate static sphere observers, as well as the lengths of
these geodesics, we can sketch them in the Penrose diagram of SdS spacetime.
The results can be found in Figure 13, Figure 14, and Figure 15 for the five,
four, and six-dimensional examples, respectively.

This ends our (partially qualitative, especially in $D=6$) discussion on
radial spacelike geodesics. The most important conclusions are that in the
presence of a black hole, symmetric radial spacelike geodesics between
conjugate static sphere observers, probing the interior black hole, and the
exterior de Sitter region, exist within two generically different temporal
windows. Only in $D=5$ are the interior and exterior windows the same,
implying that the inward and outward radial spacelike geodesics can be
smoothly connected (have the same value of $\mathcal{E}$), and their combined
length is independent of $\mathcal{E}$ and equal to the length of spacelike
geodesics in empty de Sitter.

In $D=4$ the interior/exterior symmetry is absent, with the exterior spacelike
geodesics absent before the interior spacelike geodesics, meaning that the
inward and outward radial spacelike geodesics cannot be connected smoothly,
and that the interior black hole (centered) causal diamond and the exterior de
Sitter (centered) causal diamond are disconnected. In $D=6$ this situation is
reversed, implying that the interior black hole (centered) causal diamond is
overlapping with the exterior de Sitter (centered) causal diamond. We believe
that these modifications to the causal structure might have interesting
consequences for (symmetric) correlators between a pair of conjugate static
sphere observers.

##  4 Conclusions and discussion

In this paper we analyzed the (changes in the) causal structure of SdS
spacetime, with respect to special static sphere observers, in $D=4,5$ and
$6$. We traced radial null geodesics as they cross the interior black hole
horizon or the exterior cosmological horizon, introducing imaginary
contributions to the (static) time parameter along the radial null geodesic.
The generated imaginary parts are related to the Euclidean periodicity of the
corresponding horizon and as long as one separates the calculations for
geodesics going through the interior black hole from geodesics going through
the exterior de Sitter region, this procedure is completely well-defined and
regular. Using these standard techniques, we explicitly derived the inward
bending of the black hole singularity, as expected Fidkowski:2003nf , and the
outward bending of spacelike (asymptotic) de Sitter infinity, with respect to
free-falling, static sphere, observers. Related to this we then computed the
temporal windows at the static sphere observer marking the relevant interior
(black hole) and exterior (de Sitter) causal regions, in $D=4,5$ and $6$
dimensions. The $5$-dimensional SdS spacetime is special in the sense that the
relevant interior and exterior causal regions just connect to cover a complete
Cauchy slice of the SdS spacetime, whereas in $D=4$ the interior and exterior
causal regions are disconnected, and in $D=6$ they share a finite size region.
In all cases the inward and outward bending increases to a maximum size, as a
function of the black hole mass, and then decreases again to vanish in the
maximal black hole Nariai limit.

This modified structure could be important when considering protocols for
information recovery and exchange in de Sitter space times Aalsma:2021kle ;
Morvan:2023tele . In particular, the outward bending of de Sitter future
infinity implies that signals crossing the de Sitter exterior have more time
to be intersect a (positive energy) shockwave emitted Gao:2000ga ;
Galante:2023uyf from the conjugate static sphere observer, and be scattered
into the causal region of that same conjugate static sphere observer. Black
holes play a natural role of providing an energy reservoir for the positive
energy shockwaves in these scenarios and it would be interesting to study the
details of those information exchange protocols Freivogel:2019lej ;
Freivogel:2019whb , especially regarding the question of how much information
can be transmitted this way Morvan:2023tele .

The existence of temporal windows for static sphere observers also has
interesting consequences for the presence of radial spacelike geodesics
connecting conjugate static sphere observers at the same (global) time, which
should be of interest when applying a geodesic approximation of correlators
between two conjugate static sphere observers. By introducing a black hole
into de Sitter spacetime conjugate static sphere observers are connected by
spacelike geodesics for some finite amount of time, depending on the mass of
the black hole, and whether one probes the interior (crossing the black hole)
or the exterior (crossing the de Sitter region). In $D=5$ the interior and
exterior times and conserved quantities $\mathcal{E}$ match, implying the
connected geodesic is everywhere smooth, but in $D=4$ and $D=5$ the slope at
the static sphere time is discontinuous. We also computed their lenghts in
$D=4$ and $D=5$, finding the particularly interesting result that the combined
interior and exterior total length exactly equals the length of all radial
spacelike geodesics in pure (empty) de Sitter spacetime. In addition, beyond
some finite time, that we quantified analytically in $D=4$ and $D=5$, either
the interior or the exterior spacelike geodesic disappears and one expects a
phase transition that introduces complex geodesics (labeled by complex
$\mathcal{E}$) connecting static sphere observers. We expect this to play an
important role in a geodesic approximation to the correlator Fidkowski:2003nf
; Aalsma:2022swk ; Galante:2022nhj .

For example, consider the geodesic approximation to the (equal time)
correlation function of a scalar field operators $\phi$ of (large) mass $m\gg
1/l$

$\braket{\phi\phi}(t)=\sum_{i}^{N}e^{-m\,\mathcal{D}_{i}(t)}\,,$ (66)

which is a sum is over all lengths of the spacelike geodesics connecting the
two points in the correlator. Recently, the geodesic approximation was used to
establish a connection between $2$-dimensional de Sitter space-time and a
particular implementation of (an entangled pair of) Sachdev–Ye–Kitaev (SYK)
models VerlindeH:2023SYKdS , claiming to be a potential unitary holographic
description of de Sitter that is correctly able to reproduce the known bulk
correlators. Generalising to SdS spacetimes, which arguably should be
considered as a finite energy excitation in a holographic description of de
Sitter, it would be of interest to construct the relevant generalised SdS
correlators in a geodesic approximation. Generalising Fidkowski:2003nf , after
some particular time that we identified in this work, one requires complex
values for the parameter $\mathcal{E}$, rendering the lengths
$\mathcal{D}_{i}$ complex as well, suggesting the following structure of the
SdS (conjugate static sphere) correlator

$\sum_{i}^{N}\mathcal{D}_{i}=\begin{cases}\mathcal{D}_{b}+\mathcal{D}_{c}&,\,0\leq|t|\leq\mathcal{T}_{min}\\\
(\mathcal{D}_{c,b}+\mathcal{D}_{c,b}^{*})+\mathcal{D}_{b,c}&,\,\mathcal{T}_{min}<|t|\leq\mathcal{T}_{max}\\\
(\mathcal{D}_{b,c}+\mathcal{D}_{b,c}^{*})+(\mathcal{D}_{c,b}+\mathcal{D}_{c,b}^{*})&,\,\mathcal{T}_{max}<|t|\leq\infty\end{cases}\,,$
(67)

where we should use the $c$ index in the second line for $D=4$ and the $b$
index for $D>5$. For $D=5$, the black hole interior and exterior de Sitter
temporal windows are the same $\mathcal{T}_{min}=\mathcal{T}_{max}$, avoiding
a distinction between interior and exterior (real or complex) geodesics.
Although we have not studied the complex geodesic regime in this paper, with
the help of the detailed results for radial null and spacelike geodesics in
SdS spacetimes we hope to soon report on progress for complex geodesics in SdS
as well.

Finally, let us make a final comment on the appropriate SdS vacuum state that
should be selected by these static sphere correlators. As a consequence of the
semi-classical non-equilibrium nature of SdS the identification of an
appropriate vacuum state is problematic. However, our analysis clearly
suggests, in line with earlier observations Draper:2022ofa ; Morvan:2022ybp ,
that it should be consistent to split SdS spacetime, for instance at the
static sphere, into an interior black hole part and and exterior de Sitter
part that each correspond to smooth (complex) geometries for which a natural
Euclidean (Hartle-Hawking type) vacuum can be selected. Physically one can
presumably understand this in terms of a adding a reflective sphere at the
static sphere radius Morvan:2023tele , keeping both the black hole and
cosmological horizon in equilibrium. It would be interesting to work out the
details of such a SdS vacuum state construction Brum:2014nea ; Witten:2023xze
and corresponding correlation functions.

###### Acknowledgements.

We would like to thank Ben Freivogel, Lars Aalsma, Shradha Ramakrishnan and
Damian Galante for useful discussion. 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). MMF is partly supported by funds from NSERC.

##  Appendix A Appendix

The SDS metric is given by,

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}\,d\Omega^{2}_{D-2}\,,$ (68)

where,

$f(r)=1-\frac{r^{2}}{\ell^{2}}-\alpha\frac{M}{r^{D-3}}\,.$

The roots of blackening factors can be understood from the following relation,

$\displaystyle\sum_{i=1}^{D-1}r_{i}^{n}$
$\displaystyle=\left\\{\begin{array}[]{ll}0\quad\,\,\,\,\,\text{for $0<n\leq
D-2$\, odd,}\\\ 2\ell^{n}\,\,\,\,\text{for $0<n\leq D-2$\,
even,}\end{array}\right.\,,\quad\ell^{2}=\frac{r_{c}^{D-1}-r_{b}^{D-1}}{r_{c}^{D-3}-r_{b}^{D-3}}\,,$
(71)

Solving $f(r_{i})=0$ in $D$ dimensions provides $D-2$ relations between the
$D-1$ roots. This enable us to rewrite every root $r_{i}$ as a function of
$r_{b}$ and/or $r_{c}$. The $r_{b}$ and $r_{c}$ are the real positive roots in
this discussion.

In section 3.1 we have not explicitly provided the analytic formula for
integral (39) and (40), which determines the finite temporal windows for the
appearance of spacelike geodesics going through the black hole interior, as
well as the de Sitter exterior. The sizes of the windows can be explicitly
obtained in function of the black hole and cosmological horizon radius. We
give the explicit relations in four and six dimensions in this appendix. We
also provide some of long analytic expressions from 3.2.

###  A.1 4D windows

In four dimensions, we find that the temporal windows are given by

$\displaystyle\tilde{{\mathcal{T}}}_{c}=\frac{\gamma\ell^{2}}{\left(r_{b}-r_{c}\right)\left(2r_{b}+r_{c}\right)\left(r_{b}+2r_{c}\right)}$
$\displaystyle\Big{\\{}ir_{c}\left(2r_{b}+r_{c}\right)\Big{(}i\ln\left(r_{c}-r_{\mathcal{O}}\right)-\pi\Big{)}$
$\displaystyle+r_{b}\left(r_{b}+2r_{c}\right)$
$\displaystyle\ln\left(r_{\mathcal{O}}-r_{b}\right)+\left(r_{c}^{2}-r_{b}^{2}\right)\ln\left(r_{b}+r_{c}+r_{\mathcal{O}}\right)\Big{\\}}\,,$
(72)
$\displaystyle\tilde{\mathcal{T}}_{b}=\,\tilde{\mathcal{T}}_{c}(r_{\mathcal{O}}\longrightarrow
0)-\tilde{\mathcal{T}}_{c}\,.\quad\quad\,\quad\,\,\quad$ (73)

Evaluating the real part of those expressions leads to the left panel of
figure (6), whereas taking the imagery part yields the right panel.

###  A.2 6D windows

In six dimensions, the temporal windows can be expressed as

$\displaystyle\tilde{\mathcal{T}}_{c}=\gamma\ell^{2}$
$\displaystyle\bigg{\\{}-\frac{r_{a}^{3}\ln\left(r_{a}-r_{\mathcal{O}}\right)}{\left(r_{a}-r_{b}\right)\left(r_{a}-r_{c}\right)\left(r_{a}-r_{d}\right)\left(r_{a}-r_{e}\right)}+\frac{r_{b}^{3}\ln\left(r_{b}-r_{\mathcal{O}}\right)}{\left(r_{a}-r_{b}\right)\left(r_{b}-r_{c}\right)\left(r_{b}-r_{d}\right)\left(r_{b}-r_{e}\right)}$
$\displaystyle+$
$\displaystyle\frac{r_{c}^{3}\ln\left(r_{c}-r_{\mathcal{O}}\right)}{\left(r_{a}-r_{c}\right)\left(r_{c}-r_{b}\right)\left(r_{c}-r_{d}\right)\left(r_{c}-r_{e}\right)}+\frac{r_{d}^{3}\ln\left(r_{d}-r_{\mathcal{O}}\right)}{\left(r_{a}-r_{d}\right)\left(r_{d}-r_{b}\right)\left(r_{d}-r_{c}\right)\left(r_{d}-r_{e}\right)}$
$\displaystyle+$
$\displaystyle\frac{r_{e}^{3}\ln\left(r_{e}-r_{\mathcal{O}}\right)}{\left(r_{a}-r_{e}\right)\left(r_{e}-r_{b}\right)\left(r_{e}-r_{c}\right)\left(r_{e}-r_{d}\right)}\bigg{\\}}$
(74) $\displaystyle\tilde{\mathcal{T}}_{b}=\,\tilde{\mathcal{T}}_{c}($
$\displaystyle r_{\mathcal{O}}\rightarrow 0)-\tilde{\mathcal{T}}_{c}\,,$ (75)

where $r_{a},r_{d}$ and $r_{e}$ are the additional roots of the blackening
factor $f(r)$. $r_{a}$ is the negative root while $r_{d}$ and $r_{e}$ are the
complex roots. Taking the real part of equations (74) and (75) yields the left
panel of figure 8.

##  Appendix B 5D spatial geodesic lengths $\mathcal{D}_{b,c}$ and starting
times $\tilde{\tau}_{b,c}$

The five dimensional spatial lengths can be compactly written in terms of
$\mathcal{E}$, we find

$\displaystyle\mathcal{D}_{b,c}$
$\displaystyle=\pi\ell-2\ell\arcsin\sqrt{\frac{\sqrt{\left(1-2\frac{r_{b}^{2}}{\ell^{2}}\right)^{2}+\mathcal{E}^{2}(\mathcal{E}^{2}+2)}\pm\mathcal{E}^{2}\pm
1\mp
2\frac{r_{b}}{\ell}\sqrt{1-\frac{r_{b}^{2}}{\ell^{2}}}}{2\sqrt{\left(1-2\frac{r_{b}^{2}}{\ell^{2}}\right)^{2}+\mathcal{E}^{2}(\mathcal{E}^{2}+2)}}}\,.$
(76)

The dependence of starting time over energy $E$,

$\displaystyle\tilde{\tau}_{b}$
$\displaystyle=\frac{\ell^{2}\,E}{r_{b}+r_{c}}\left(\frac{r_{b}^{2}\operatorname{arctanh}\left(\frac{\sqrt{r_{\mathcal{O}}^{2}-r_{1}^{2}}}{\sqrt{r_{2}^{2}-r_{\mathcal{O}}^{2}}}\frac{\sqrt{r_{2}^{2}-r_{b}^{2}}}{\sqrt{r_{b}^{2}-r_{1}^{2}}}\right)}{\sqrt{r_{2}^{2}-r_{b}^{2}}\sqrt{r_{b}^{2}-r_{1}^{2}}}-\frac{r_{c}^{2}\arctan\left(\frac{\sqrt{r_{\mathcal{O}}^{2}-r_{1}^{2}}}{\sqrt{r_{2}^{2}-r_{\mathcal{O}}^{2}}}\frac{\sqrt{r_{c}^{2}-r_{2}^{2}}}{\sqrt{r_{c}^{2}-r_{1}^{2}}}\right)}{\sqrt{r_{c}^{2}-r_{1}^{2}}\sqrt{r_{c}^{2}-r_{2}^{2}}}\right)-i\frac{\beta_{b}}{4}\,,$
(77) $\displaystyle\tilde{\tau}_{c}$
$\displaystyle=\frac{i\pi\ell^{2}\,E}{2\left(r_{b}+r_{c}\right)}\left(\frac{r_{c}^{2}}{\sqrt{r_{2}^{2}-r_{c}^{2}}\sqrt{r_{c}^{2}-r_{1}^{2}}}-\frac{r_{b}^{2}}{\sqrt{r_{2}^{2}-r_{b}^{2}}\sqrt{r_{b}^{2}-r_{1}^{2}}}\right)-i\frac{\beta_{c}}{4}$
$\displaystyle-\frac{\ell^{2}\,E}{r_{b}+r_{c}}\left(\frac{r_{b}^{2}\operatorname{arctanh}\left(\frac{\sqrt{r_{\mathcal{O}}^{2}-r_{1}^{2}}}{\sqrt{r_{2}^{2}-r_{\mathcal{O}}^{2}}}\frac{\sqrt{r_{2}^{2}-r_{b}^{2}}}{\sqrt{r_{b}^{2}-r_{1}^{2}}}\right)}{\sqrt{r_{2}^{2}-r_{b}^{2}}\sqrt{r_{b}^{2}-r_{1}^{2}}}-\frac{r_{c}^{2}\arctan\left(\frac{\sqrt{r_{\mathcal{O}}^{2}-r_{1}^{2}}}{\sqrt{r_{2}^{2}-r_{\mathcal{O}}^{2}}}\frac{\sqrt{r_{c}^{2}-r_{2}^{2}}}{\sqrt{r_{c}^{2}-r_{1}^{2}}}\right)}{\sqrt{r_{c}^{2}-r_{1}^{2}}\sqrt{r_{c}^{2}-r_{2}^{2}}}\right)\,,$
(78)

##  Appendix C 4D starting times $\tilde{\tau}_{b,c}$

###  C.1 Starting times $\tilde{\tau}_{b,c}$

$\displaystyle\tilde{\tau}_{c}$
$\displaystyle=\frac{2i\mathcal{E}\ell^{3}\gamma\left(r_{1}-r_{2}\right)}{\sqrt{r_{2}}\sqrt{2r_{1}+r_{2}}}\left\\{\frac{r_{1}^{2}F(A|B)}{\left(r_{1}-r_{2}\right)\left(r_{1}-r_{b}\right)\left(r_{1}-r_{c}\right)\left(r_{b}+r_{c}+r_{1}\right)}\right.$
$\displaystyle+$
$\displaystyle\frac{r_{b}^{2}\Pi\left(\frac{\left(r_{b}-r_{1}\right)}{\left(r_{b}-r_{2}\right)}\frac{r_{2}}{r_{1}}B;A|B\right)}{\left(r_{b}-r_{1}\right)\left(r_{b}-r_{2}\right)\left(r_{b}-r_{c}\right)\left(2r_{b}+r_{c}\right)}-\frac{r_{c}^{2}\Pi\left(\frac{\left(r_{c}-r_{1}\right)}{\left(r_{c}-r_{2}\right)}\frac{r_{2}}{r_{1}}B;A|B\right)}{\left(r_{c}-r_{1}\right)\left(r_{c}-r_{2}\right)\left(r_{b}-r_{c}\right)\left(r_{b}+2r_{c}\right)}$
$\displaystyle+$
$\displaystyle\left.\frac{\left(r_{b}+r_{c}\right){}^{2}\Pi\left(\frac{\left(r_{1}+r_{b}+r_{c}\right)}{\left(r_{2}+r_{b}+r_{c}\right)}\frac{r_{2}}{r_{1}}B;A|B\right)}{\left(r_{b}+r_{c}+r_{1}\right)\left(r_{b}+r_{c}+r_{2}\right)\left(5r_{b}r_{c}+2r_{b}^{2}+2r_{c}^{2}\right)}\right\\}\,,$
(79) $\displaystyle\tilde{\tau}_{b}$
$\displaystyle=\tilde{\tau}_{c}(r_{1}\longleftrightarrow r_{2})\,,$ (80)

where $F$ is the elliptic function of the first kind and $\Pi$ is the elliptic
function of the second kind. Furthermore, we have defined

$\displaystyle A$
$\displaystyle\equiv\arcsin{\left(\sqrt{\frac{\left(2r_{1}+r_{2}\right)\left(r_{\mathcal{O}}-r_{2}\right)}{\left(r_{1}+2r_{2}\right)\left(r_{\mathcal{O}}-r_{1}\right)}}\right)}\,,\quad
B\equiv\frac{r_{1}\left(r_{1}+2r_{2}\right)}{r_{2}\left(2r_{1}+r_{2}\right)}\,.$
(81)

This was used to produce the left panel of figure 12.

##  Appendix D Approximating the starting times $\tilde{\tau}_{b,c}$

We find that the dependence of the connected times $\tilde{\tau}_{b,c}$ on
real values of the energy parameter $\mathcal{E}$ can be well approximated by

$\tilde{\tau}_{b,c}(\mathcal{E})\approx\pm\frac{\mathcal{T}_{b,c}}{\pi/2}\arctan{2\mathcal{E}}\,,$
(82)

especially for $|\mathcal{E}|\ll 1$ and $|\mathcal{E}|\gg 1$. This
approximation was used to produce the right panel of figure 12.

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