# Interior Quasinormal Modes and Strong Cosmic Censorship

Filipe S. Miguel<EMAIL_ADDRESS>Department of Applied Mathematics and
Theoretical Physics, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, United Kingdom

###### Abstract

Recent work has argued that the strong cosmic censorship (SCC) conjecture is
violated by near-extremal Reissner–Nordström de Sitter (RNdS) black holes but
respected by Kerr-de Sitter black holes. It has also been shown that the
conjecture is violated by near-extremal BTZ black holes. The latter result
relies on a coincidence between “exterior” and “interior” quasinormal
frequencies. If this coincidence were to occur also for RNdS or Kerr-dS then
it would significantly modify the conclusions of earlier work. In this paper,
it is demonstrated that this coincidence does not occur for RNdS or Kerr-dS
and so the conclusions of the earlier work remain valid.

###### Contents

  1. I Introduction
  2. II Background Material
    1. II.1 The RNdS black hole solution
    2. II.2 The Kerr-dS black hole solution
  3. III Quasinormal modes and Strong Cosmic Censorship
    1. III.1 Quasinormal modes definition
    2. III.2 Relation with Strong Cosmic Censorship
  4. IV Strong Cosmic Censorship in the RNdS black hole
    1. IV.1 Master Equation
    2. IV.2 Analytical approximation for confluent horizons
    3. IV.3 Numerical computation
    4. IV.4 Comparison between interior and exterior modes for gravitoelectromagnetic perturbations
  5. V Strong cosmic censorship in the Kerr-dS black hole
    1. V.1 Wave equation and QNM
    2. V.2 Solving the wave equation in the BH interior
    3. V.3 Behaviour near the horizons
  6. VI Discussion
  7. VII Acknowledgements
  8. A Positivity of $\lambda_{-2}$

##  I Introduction

General relativity is a deterministic theory that describes our universe at
large scales. Nevertheless, there are solutions of Einstein equations, that
are not fully specified by the initial conditions. In fact, for some
spacetimes, we find Cauchy horizons. These are regions beyond which there are
infinitely many possible solutions to Einstein equations that are consistent
with the initial data. This is a violation of determinism, and an undesirable
feature of the theory. To attenuate this issue, in the 1970’s, Penrose put
forward the Strong Cosmic Censorship (SCC) conjecture [1]. In broad terms SCC
is the statement that spacetimes with Cauchy horizons, come from fine tuned
initial conditions. Generic spacetimes, should be deterministic.

Several solutions of Einstein equations are known to admit Cauchy horizons.
Famous examples in asymptotically flat spacetimes $(\Lambda=0)$ are Kerr and
Reissner–Nordström black holes. Fortunately, there is compelling evidence [2,
3, 4, 5, 6, 7, 8] that these horizons are not stable. In fact signals emitted
by an observer outside the black hole (BH), suffer an infinite blueshift when
approaching the Cauchy horizon, prompting the instability. On the other hand,
for asymptotically de-Sitter spacetimes $(\Lambda>0)$, there is a competing
redshift effect due to the existence of a cosmological horizon, which may
weaken the strength of the Cauchy horizon instability and make a violation of
SCC more likely [9].

Recently, Cardoso et al. [10], renewed the interest on SCC in asymptotically
de-Sitter spacetimes. The authors showed that if we couple the Einstein field
equations with a massless scalar field, then for a non-trivial set of RNdS
black holes it is possible to extend the scalar field across the Cauchy
horizon as a weak solution of the Einstein Klein-Gordon equations. More
formally, they found that for a generic class of RNdS black holes, the
massless scalar field will have finite energy at the Cauchy horizon. This work
was based in recent mathematical results [11, 12] about the boundedness of
linear fields near the Cauchy Horizon. In [13], the authors extended this
result for a charged scalar field. In these papers, the authors use [14, 15,
16], to argue that the results should remain valid in the non-linear case.
Further work from Dias et al. [17, 18] showed that in the case of
gravitoelectromagnetic perturbations, the violation is even more drastic. For
a non-trivial region of parameter space gravitoelectromagnetic perturbations
can be extended across the Cauchy horizon with arbitrary regularity.

RNdS BHs are not very relevant as astrophysical objects. It is not very likely
that a near extremal amount of electrical charge falls into a black hole, to
reproduce the SCC violation found in [17]. It would be much more interesting,
if we found SCC violations on the more physical Kerr-dS background. However,
in [19], the authors showed that this is not the case. In fact, scalar field
and gravitational perturbations are sufficiently irregular at the Cauchy
Horizon to preserve SCC.

In a recent paper, [20] Dias et al. studied the behaviour of several fields
propagating around the 3-dimensional, asymptotically AdS, BTZ black hole [21].
They found that for BHs sufficiently close to extremality, these fields can be
continued across the inner event horizon with arbitrary regularity. This
conclusion relies fundamentally on an unexpected coincidence found between
quasinormal mode (QNM) frequencies of waves propagating on the exterior of the
BH and waves propagating on the interior of the BH. This coincidence increases
the regularity of fields on the Cauchy horizon, leading to a violation of SCC.

If this coincidence also occurred for Kerr-dS or RNdS black holes, we could
find new violations of SCC in 4 dimensional gravity. In [17] the authors found
a large class of near extremal RNdS BHs that violate SCC, so this equality
would simply increase the region of parameter space that violates SCC. Much
more interesting is perhaps the Kerr-dS case. As discussed above, there is no
evidence for SCC violations in this spacetime. However, if this coincidence
was to occur, the conclusion of [19] would have to be revised. In fact, for a
set of near extremal (NE) black holes, we could expect SCC to be violated.

In this paper we prove that this is not the case. We study the spectrum of
interior and exterior QNMs for Kerr-dS and RNdS BHs. Using numerical and
analytical arguments, we prove that the frequencies are distinct, maintaining
the conclusions of [17] and [19].

It is also important to mention recent efforts in the direction of restoring
the faith in SCC. At the classical level, we should mention [22]. In this
paper, Dafermos and Shlapentokh-Rothman showed that even though some notions
of SCC are not respected for a class of RNdS black holes, if we allow non-
smooth initial data, then SCC is recovered. More recently, a couple of papers
by Hollands et al. , [23, 24] restored the faith in SCC, when considering
quantum fields on a RNdS / Kerr-dS BH. The authors argue that the energy
momentum tensor of quantum fields is sufficiently irregular at the Cauchy
horizon, recovering SCC. The BTZ case is a bit more complicated. In [20], the
authors prove that the stress energy of quantum fields is regular at the
Cauchy horizon. Hence, the backreaction onto the gravitational sector, should
also remain regular. However, in a recent paper [25] Emparan et al. , argued
that second order backreaction diverges at the Cauchy Horizon, preserving SCC.

The paper is organized as follows. In section II, we review the RNdS and Kerr-
dS black hole solutions. Then, we review the notion of exterior QNMs, define
interior QNMs and finally outline the argument that relates QNMs with SCC.
Then in section IV we study the families of interior and exterior QNMs that
are relevant for SCC violation in the RNdS black hole. Using analytical
approximations, we show that in the near extremal limit exterior and interior
QNM frequencies coincide to leading order. Then, using high precision
numerical methods, we find the values of these frequencies, and show that they
are different. We conclude, that in the RNdS case, there is no additional
violation of SCC. In section V we use a WKB argument to prove that exterior
QNM frequencies are not frequencies of interior QNMs. We conclude reinforcing
confidence of SCC in this spacetime.

##  II Background Material

###  II.1 The RNdS black hole solution

Let’s review the 4 dimensional RNdS black hole (see e.g. [26]). This solution
of Einstein’s equations describes an electrically charged black hole in a de-
Sitter background. The BH is completely specified by 3 parameters,
$(M,Q,\Lambda)$, (mass, charge and cosmological constant). In static
coordinates $(t,r,\theta,\phi)$, the line element of the metric can be written
as:

$\differential s^{2}\,=\,-\,f\,\differential t^{2}\,+\,\frac{\differential
r^{2}}{f}\,+\,r^{2}\,\differential\Omega_{2}^{2},$ (1)

where $\differential\Omega^{2}$ is the line element of a unit radius $S^{2}$,
and

$f(r)\,=\,1\,-\,\frac{2M}{r}\,+\,\frac{Q^{2}}{r^{2}}\,-\,\frac{\Lambda}{3}\,r^{2}.\\\
$ (2)

This spacetime is pathological unless $f(r)$ has 3 positive roots. $r_{-}\leq
r_{+}\leq r_{c}$, corresponding to the Cauchy horizon $\mathcal{CH}$, event
horizon $\mathcal{H}_{R}$ and cosmological horizon $\mathcal{H}_{C}$. These
will bound the different causal regions of the black hole. (see figure 1).

Figure 1: Penrose diagram for the RNdS and Kerr-dS black holes (truncated to
the physical regions I and II). Each point denotes an $S_{2}$ ($S_{1}$) in the
case of the RNdS (Kerr-dS black hole).

Region I (the black hole exterior), is bounded by
$r_{+}\,\leq\,r\,\leq\,r_{c}$. In here $r\,=\,r_{+}$ corresponds to the white
hole horizon ($\mathcal{H}^{-}$) at $t=-\infty$ and the right black hole
horizon at $t=+\infty$ ($\mathcal{H}_{R}^{+}$). The radial coordinate
$r\,=\,r_{c}$ will correspond to the past ($t=-\infty$) and future
($t=+\infty$) cosmological horizons: ($\mathcal{H}_{C}^{-}$ and
$\mathcal{H}_{C}^{+}$).

Similarly, region II is bounded by $r_{-}\,\leq\,r\,\leq\,r_{+}$. At $r=r_{-}$
we find the Left ($t=+\infty$) and Right ($t=-\infty$) Cauchy horizons
($\mathcal{CH}_{L}^{+}$ and $\mathcal{CH}_{R}^{+}$ respectively). On the other
hand, we have in the causal past $r=r_{+}$, corresponding to the left
($t=-\infty$) and right ($t=+\infty$) future event horizons
($\mathcal{H}_{L}^{+}$ and $\mathcal{H}_{R}^{+}$).

In the calculations that follow, we will work in units such that
$\Lambda\,=\,3$. Furthermore, it is instructive to use the roots of f to
characterize the black hole (instead of M and Q). To do so, we rewrite (2):

$f(r)\,=\,-\,\frac{\left(r\,-\,r_{1}\right)\,\left(r\,-\,r_{2}\right)\,\left(r\,-\,r_{3}\right)\,\left(r\,-\,r_{4}\right)}{r^{2}}\quad.$
(3)

where $r_{i}$ may specify any of the roots of $f$.

To make the structure in (3) explicit, we take $r^{2}f(r)$ in (2) and
explicitly evaluate the polynomial quotient with respect to $(r\,-\,r_{1})$.
Then, we divide the result by $(r\,-\,r_{2})$. Equating the remainders of this
operations to zero, we obtain:

$\displaystyle
M=\,\frac{1}{2}\,\left(r_{1}\,+\,r_{2}\,\right)\,\left(1\,-\,r_{1}^{2}\,-\,r_{2}^{2}\right)$
(4) $\displaystyle
Q^{2}=\,r_{1}\,r_{2}\,\left(1\,-\,r_{1}\,r_{2}\,-\,r_{1}^{2}\,-\,r_{2}^{2}\right)$
(5)
$\displaystyle\\!\begin{multlined}\frac{r^{2}f(r)}{\left(r\,-\,r_{1}\right)\,\left(r\,-r_{2}\right)}=\\\
r^{2}\,+\left(r_{1}\,+\,r_{2}\right)\,r\,+\,r_{1}^{2}\,+\,r_{2}^{2}\,+\,r_{1}\,r_{2}\,\,-\,1\quad.\end{multlined}\frac{r^{2}f(r)}{\left(r\,-\,r_{1}\right)\,\left(r\,-r_{2}\right)}=\\\
r^{2}\,+\left(r_{1}\,+\,r_{2}\right)\,r\,+\,r_{1}^{2}\,+\,r_{2}^{2}\,+\,r_{1}\,r_{2}\,\,-\,1\quad.$
(8)

Finally, equating (8) to 0, we find that

$r_{3/4}=-\frac{1}{2}\left(r_{1}\,+\,r_{2}\pm\,\sqrt{4\,-\,2\,r_{1}\,r_{2}\,-\,3\,\left(r_{1}^{2}\,+\,r_{2}^{2}\right)}\right)\\\
$ (9)

Lets now define, $r_{1}\,=\,r_{+}$, $r_{2}\,=\,r_{c}$, $r_{3}\,=\,r_{-}$ and
$r_{4}\,=\,r_{\tilde{c}}$. Seeking physical solutions, we must impose that the
roots $r_{i}$ are real valued, and obey $0\leq r_{-}\leq r_{+}\leq r_{c}$. As
we can see in figure 2, this bounds the allowed RNdS black holes by three
limiting cases: $r_{-}\,\rightarrow\,0$ (Schwarzschild de Sitter limit),
$r_{-}\,\rightarrow\,r_{+}$ (extremal limit) and $r_{+}\,\rightarrow\,r_{c}$
(Nariai limit).

Figure 2: Parameter space of the RNdS black hole.

Although $r_{+},\,r_{-}$ and $r_{c}$ are quite useful to discuss properties of
the black hole, they are not the best when scanning the BH parameter space
(see 2). Instead, it is more intuitive to consider $Q^{2}/Q^{2}_{\text{max}}$
and $y_{+}=r_{+}/r_{c}$. Here, $Q^{2}_{\text{max}}$ is the maximal electric
charge for a given value of $y_{+}$. It corresponds to the charge of an
extremal RNdS black hole. To obtain this, we take $r_{1}\,=\,r_{3}$ in
equation (9), and solve with respect to $r_{1}/r_{2}$. We get:

$Q^{2}_{\text{max}}=\frac{y_{+}^{2}\,\left(2y_{+}\,+\,1\right)}{\left(3y_{+}^{2}\,+\,2y_{+}\,+\,1\right)^{2}}$
(10)

These parameters have the advantage of being defined in the interval
$(0,\,1)$, transforming the parameter space into a square. Given equations (4)
and (5), we may invert the definitions of $y_{+}$ and
$Q^{2}/Q_{\text{max}}^{2}$ to obtain $r_{+/-/c}$ as a function of $y_{+}$ and
$Q^{2}/Q_{\text{max}}^{2}$.

In the calculations below, we will also make use of the tortoise coordinate,
defined through:

$\differential r_{*}=\frac{\differential r}{f(r)}.$ (11)

$r_{*}$ goes from $+\infty$ at $r\,=\,r_{c}$, to $-\infty$ at $r\,=\,r_{+}$
and back to $+\infty$ at $r\,=\,r_{-}$. It is then important to specify the
$r$ region, when working with this coordinate. Finally, using ingoing
Eddington-Finkelstein coordinates, (see e.g. [26]) we obtain the surface
gravity $\kappa$ of the Horizons:

$\kappa_{i}=\left|\frac{f^{\prime}(r_{i})}{2}\right|$ (12)

###  II.2 The Kerr-dS black hole solution

The Kerr-dS black hole has very similar properties to the RNdS BH. This
solution describes a rotating black hole in a de Sitter background, see e.g.
[26]. In Boyer-Lindquist coordinates, the line element reads:

$\differential s^{2}=-\frac{\Delta_{r}}{\Sigma^{2}\rho^{2}}\left(\differential
t\,-\,a\sin^{2}\theta\,\differential\phi\right)^{2}\,+\,\frac{\rho^{2}}{\Delta_{r}}\differential
r^{2}\\\
+\frac{\Delta_{\theta}\sin^{2}\theta}{\Sigma^{2}\rho^{2}}\left(a\differential
t-(r^{2}+a^{2})\differential\phi\right)^{2}+\,\frac{\rho^{2}}{\Delta_{\theta}}\differential\theta^{2}\quad,$
(13)

where,

$\begin{split}\Delta_{r}\,&=\left(r^{2}+a^{2}\right)\left(1-\frac{\Lambda}{3}r^{2}\right)\,-\,2\,M\,r,\\\
\Delta_{\theta}\,&=\,1+\alpha\cos^{2}\theta,\\\ \Sigma&=1+\alpha,\\\
\rho^{2}&=r^{2}+a^{2}\cos^{2}\theta,\\\
\alpha&=\frac{\Lambda}{3}a^{2}\quad.\end{split}$ (14)

The black hole is completely specified by the parameters $(M,\,a,\,\Lambda)$,
denoting the mass, spin and cosmological constant respectively. As before, the
solution is pathological, unless $\Delta_{r}(r)$ possesses 3 positive roots,
$r_{-}\leq r_{+}\leq r_{c}$, corresponding to the Cauchy, event and
cosmological horizon. This condition imposes bounds on the allowed values of
the BH parameters, as in the RNdS case. As before, these roots indicate the
locations of $\mathcal{CH}$, $\mathcal{H}^{+}$ and $\mathcal{H}_{c}$.

The causal structure of a Kerr-dS black hole is very similar to the one for a
RNdS spacetime. Region I is the exterior of the black hole, bounded by
$\mathcal{H}^{+}$ and $\mathcal{H}_{C}$, while region II is the BH interior,
bounded by $\mathcal{H}^{+}$ and $\mathcal{CH}$ (see figure 1).

As before, we may define the tortoise coordinate, by integrating:

$\differential
r_{*}=\frac{\Sigma\left(r^{2}+a^{2}\right)}{\Delta_{r}(r)}\differential
r\quad.$ (15)

Finally, we may obtain the surface gravity of the different horizons:

$\kappa_{i}=\frac{1}{2\,a}\frac{\Delta_{r}^{\prime}(r_{i})}{\Sigma}\,\Omega\left(r_{i}\right)\quad,$
(16)

where:

$\Omega(r)=\frac{a}{a^{2}+r^{2}}\quad.$ (17)

##  III Quasinormal modes and Strong Cosmic Censorship

###  III.1 Quasinormal modes definition

Quasinormal modes have been studied extensively in the literature, see [27,
28] for a comprehensive review. In asymptotically de-Sitter spacetimes, QNMs
are responsible for governing the late time behaviour of linear fields. We
define QNMs as solutions of a given wave equation with specific boundary
conditions. Specifically, we require that QNMs are purely outgoing at
$\mathcal{H}_{C}$, and ingoing into $\mathcal{H}_{R}$.

As a toy example, let’s study QNMs arising from the Klein Gordon (KG) equation
in these spacetimes. A massless KG field will obey:

$\Box\,\Phi=0\quad.$ (18)

Quasinormal mode solutions of this equation, can be obtained by performing the
Ansatz:

$\Phi_{nlm}(t,r,\theta,\phi)=e^{-i\omega
t}e^{-im\phi}R_{nlm}(r)\,\Theta_{nlm}(\theta)\quad,$ (19)

where $\omega$ is the Quasinormal frequency, $n,l\,\text{and}\,m$ are integers
that label each mode and $R_{nlm}(r)$ and $\Theta_{nlm}(\theta)$ are the
radial and angular part of the solution. In the RNdS black hole, spherical
symmetry implies $e^{im\phi}\Theta_{nlm}(\theta)$ is given by the usual
spherical harmonic functions ($Y_{lm}$). In the Kerr-dS background there is no
closed form solution known for $\Theta_{nlm}(\theta)$. Plugging equation (19)
into (18), we separate the equation into two ODEs describing the radial and
angular part respectively. For convenience, we will henceforth drop the $nlm$
indexes. The radial equation reads:

$\left(\derivative[2]{r_{*}}\,+\left(\omega\,-\,m\,\Omega_{BH}(r)\right)^{2}-V_{BH}(r)\right)R(r)\,=\,0,$
(20)

where $V_{BH}$ is the scattering potential, $\Omega_{BH}(r)\,=\,0$ in the RNdS
case and $\Omega_{BH}(r)$ is given by equation (17) in the Kerr-dS case. For
both BHs, we have $V_{BH}(r_{c})\,=\,V_{BH}(r_{+})\,=\,V_{BH}(r_{-})\,=\,0$.

Using a Frobenius analysis, we may define solutions of equation (20) according
to their asymptotic behaviour near each horizon. We will denote these
solutions according to their ingoing/outgoing character (in/out) at the
horizon. For $r\,<\,r_{c}$, taking the limit $r\rightarrow r_{c}$ we have:

$\begin{array}[]{c c}&R_{\text{out},\,c}(r_{*})\sim
e^{i\,\left(\omega\,-\,m\Omega_{BH}(r_{c})\right)\,r_{*}},\\\
&R_{\text{in},\,c}(r_{*})\sim
e^{-i\,\left(\omega\,-\,m\Omega_{BH}(r_{c})\right)\,r_{*}},\end{array}\qquad\text{as}\qquad
r_{*}\rightarrow+\infty,$ (21)

Similarly, for $r\,>\,r_{+}$, we define ingoing and outgoing modes such that:

$\begin{array}[]{c c}&R_{\text{out},\,+}(r_{*})\sim
e^{i\,\left(\omega\,-\,m\Omega_{BH}(r_{+})\right)\,r_{*}},\\\
&R_{\text{in},\,+}(r_{*})\sim
e^{-i\,\left(\omega\,-\,m\Omega_{BH}(r_{+})\right)\,r_{*}}\end{array}\qquad\text{as}\qquad
r_{*}\rightarrow-\infty,$ (22)

The same approach is valid for $r\,<\,r_{+}$ in the $r\rightarrow r_{+}$
limit, by taking $r_{*}$ as a function of $r\in(r_{-},\,r_{+})$. The
asymptotic behaviour of such solutions is identical to the one in (22).
Finally, for $r>r_{-}$, in the limit $r\rightarrow r_{-}$, we have:

$\begin{array}[]{c c}&R_{\text{out},\,-}\sim
e^{i\left(\omega\,-\,m\Omega_{BH}(r_{-})\right)\,r_{*}}\\\
&R_{\text{in},\,-}\,\sim\,e^{-i\left(\omega\,-\,m\Omega_{BH}(r_{-})\right)\,r_{*}}\end{array}\qquad\text{as}\qquad
r_{*}\rightarrow+\infty\quad.$ (23)

We can now define quasinormal modes as solutions of (20) that are proportional
to $R_{\text{in},\,+}$ and $R_{\text{out},\,c}$:

$R(r)\sim R_{\text{in},\,+}\sim R_{\text{out},\,c}\quad,$ (24)

The condition of proportionality between $R_{\text{in},\,+}$ and
$R_{\text{out},\,c}$ is very stringent and quantizes the spectrum of
frequencies that lead to QNMs. We call these values of $\omega$ quasinormal
frequencies.

This condition can be rewritten using the Wronskian W. Given two solutions
$f,\,g$ of a linear ODE, $W\left[f,g\right]$ is defined as

$W\left[f,g\right]=f^{\prime}\,g\,-\,f\,g^{\prime}\quad.$ (25)

If f is proportional to g, we have $W[f,g]\,=\,0$. We can thus define pairs
(QNM, quasinormal frequencies) as solutions of

$W\left[R_{\text{in},\,+},\,R_{\text{out},\,c}\right]\,=\,0\quad.$ (26)

where ′ denotes differentiation wrt. $r^{*}$.

###  III.2 Relation with Strong Cosmic Censorship

The late time behaviour of linear fields in a RNdS or Kerr-dS background can
be obtained by taking a linear combination of QNMs. Hence, we may expect these
modes to control the regularity of fields at the Cauchy horizon. In fact, SCC
violations are fully dependent on the behaviour of some families of QNMs. A
proper justification of this is a bit lengthy, and can be found in refs. [17,
20, 18, 20]. Nevertheless, we will outline the general idea below.

Given a RNdS or Kerr-dS black hole, there will be a violation of SCC, if we
may extend generic small metric perturbations across the Cauchy Horizon, as a
solution of the equations of motion. QNMs describe the late time behaviour of
perturbations, so we expect the slowest decaying QNM (lowest negative
imaginary part) to control the regularity of generic perturbations at
$\mathcal{CH}_{R}$.

Black holes created by gravitational collapse do not have a left Cauchy
horizon, hence, we are interested in characterizing the behaviour of QNMs at
$\mathcal{CH}_{R}$. For simplicity, we will restrict to the
$\Omega_{BH}\,=\,0$ case. The same analysis is valid in the general case by
performing an adequate coordinate transformation on $\phi$ (see [19]).

We start by defining ingoing and outgoing EF coordinates, $u\,=\,t\,-\,r_{*}$
and $v\,=\,t\,+\,r_{*}$. Then, we define Kruskal coordinates in the BH
interior:

$\begin{split}U_{-}&=-e^{\kappa_{-}\,u}\quad,\\\
V_{-}&=-e^{\kappa_{-}\,v}\quad.\end{split}$ (27)

Reinstating time-dependence in the radial equation, we obtain:

$\begin{array}[]{c c}e^{-i\omega\,t}\,R_{\text{in},\,-}\sim
V_{-}^{\frac{i\omega}{\kappa_{-}}}\\\ e^{-i\omega\,t}\,R_{\text{out},\,-}\sim
U_{-}^{-\frac{i\omega}{\kappa_{-}}}\end{array}\qquad\text{as}\qquad\begin{array}[]{c
c}r_{*}&\rightarrow+\infty\\\ t&\rightarrow+\infty\quad.\end{array}$ (28)

The right Cauchy horizon is obtained by taking the $V\rightarrow 0$ limit,
with $U_{-}>0$. The outgoing solution is smooth at the horizon, whereas the
regularity of the ingoing part is dictated by the value of

$\beta\,=\,-\text{Im}\left(\frac{\omega}{\kappa_{-}}\right)\quad.$ (29)

In fact, if $\beta\,>\,\frac{1}{2}$, $R_{\text{in},-}$ can be continued across
$\mathcal{CH}_{R}$ with finite energy. Similarly, for some positive integer k,
if $\beta\geq k$ then $e^{-i\omega t}R_{\text{in},-}(r)$ has $C^{k}$
regularity at the Cauchy Horizon.

At $\mathcal{H}^{+}$, QNMs are proportional to $R_{in,+}$. Extending this to
the interior of the Black Hole, we may decompose each mode as

$R_{\text{in},+}(r)=\mathcal{A}(\omega)\,R_{\text{in},-}(r)\,+\mathcal{B}(\omega)\,R_{\text{out},-}(r).$
(30)

Given the smoothness of $e^{-i\omega t}R_{\text{out},-}$, the regularity of a
QNM at $\mathcal{CH}_{R}$ should be dictated by the value of $\beta$. We may
then argue that the irregular behaviour of our generic perturbation is
controlled by the frequency of the slowest decaying QNM. Nevertheless, there
is a subtlety we must be wary of. If the slowest decaying QNM happens to be
purely ingoing at the Cauchy Horizon, i.e. $\mathcal{A}(\omega)\,=\,0$, then
the mode will be smooth there. The regularity of waves will be dictated by the
slowest decaying mode, with non-vanishing $\mathcal{A}(\omega)$.

The condition $\mathcal{A}(\omega)\,=\,0$ can be rewritten as

$W[R_{\text{in},+},\,R_{\text{out},-}]\,=\,0\quad.$ (31)

This condition is akin to equation (26). In fact, it quantizes the spectrum of
solutions to (18). We define the modes in this spectrum as interior QNMs [20].
These solutions will be purely ingoing at $\mathcal{H}_{+}$ and outgoing at
$\mathcal{CH}_{R}$. We conclude that if the lowest lying exterior QNM has the
same frequency as onw of the interior modes, the regularity of linear fields
at $\mathcal{CH}_{R}$ will be increased.

In [17, 18, 19], studies of SCC in the RNdS and Kerr-dS BHs were performed
assuming that $\mathcal{A}(\omega)\,\neq\,0$ generically. On the other hand,
in [20], Dias et al. show that $\mathcal{A}(\omega)\,=\,0$, for a family of
exterior modes. This coincidence leads to the subsequent SCC violation found
in the paper. It would be very interesting if this coincidence was present in
the Kerr-dS black hole. The increased regularity of modes at the Cauchy
horizon would lead to a SCC violation in the vacuum Einstein Equations, with
$\Lambda>0$. Nevertheless, the authors of [20] argue that this coincidence is
most likely a special case of the 3-dimensional BTZ BH. In fact, the wave
equation is hypergeometric in this background, containing only 3 singular
points. Contrarily, for the 4 dimensional RNdS / Kerr-dS BHs, the equations of
motion are of the Huen type, with 4 or more singular points, with more degrees
of freedom. Hence, it is less likely we find a coincidence between interior
and exterior QNM frequencies. In this paper we will explicitly compare the
frequency spectrum of interior and exterior QNMs, checking the validity of
this argument.

##  IV Strong Cosmic Censorship in the RNdS black hole

###  IV.1 Master Equation

(a) Nariai limite spectrum

(b) Interior near extremal spectrum

Figure 3: Comparison between the analytical approximations obtained for the
Nariai limit (left panel) and the near extremal interior spectrum (right
panel) with results obtained numerically. In both cases we studied quasinormal
modes of $\Phi_{-}$ with $l=2$. On the left panel we have
$Q^{2}/Q^{2}_{\text{max}}\,=\,0.5$, $y_{+}\,=\,0.999$ and in the right panel
we have $y_{+}\,=\,0.5$, $Q^{2}/Q^{2}_{\text{max}}\,=\,0.999$. Notice that in
the right panel, the frequencies are purely imaginary, the horizontal
displacement is artificial for readability purposes.

For most cases, the equations of motion of linear fields in a RNdS black hole,
can be written in the form of the following master equation:

$\left(\derivative[2]{r_{*}}\,+\,\omega^{2}-V(r)\right)\,R(r)\,=\,0,$ (32)

where $V(r)$ is smooth for $r>0$. In most relevant cases, we can further
decompose

$V(r)=f(r)\,A(r),$ (33)

where $V(r)$ is smooth for $r>0$. The expression for $V(r)$ in (32) depends on
the specifics of the field we are considering. In this work we will focus in
gravito-electromagnetic perturbations, as studied in [17]. These modes may be
constructed from the combination of two scalar modes
($\Phi^{\pm}_{\text{s}}(r)$) and two vector modes
($\Phi^{\pm}_{\text{v}}(r)$), obeying equation (32), with
$V_{\text{v/s},\,\pm}(r)$ given by,

$\displaystyle V_{\text{s},\pm}$
$\displaystyle=\beta_{\pm}f(r)\derivative{\check{F}_{\pm}}{r}\,+\,\beta^{2}_{\pm}\check{F}_{\pm}(r)^{2}\,+\,\tilde{\kappa}\check{F}_{\pm}(r)\quad,$
(34) $\displaystyle V_{\text{v},\pm}$
$\displaystyle=-\beta_{\pm}f(r)\derivative{\check{F}_{\pm}}{r}\,+\,\beta^{2}_{\pm}\check{F}_{\pm}(r)^{2}\,+\,\tilde{\kappa}\check{F}_{\pm}(r)\quad,$
(35)

where,

$\displaystyle\beta_{\pm}$
$\displaystyle=3M\,\mp\,\sqrt{9M^{2}+4\,Q^{2}(l-1)(l+2)},$ (36)
$\displaystyle\tilde{\kappa}$ $\displaystyle=(l-1)(l+2)[(l-1)(l+2)+2],$ (37)
$\displaystyle\check{F}_{\pm}$
$\displaystyle=\frac{f(r)}{r[(l-1)(l+2)r+\beta_{\pm}]}.$ (38)

Here, $l$ is the integer azimuthal quantum number, coming from the expansion
in spherical harmonics $Y_{lm}(\theta,\phi)$. For $\Phi^{+}_{\text{v/s}}$,
$l\geq 1$ whereas for $\Phi^{-}_{\text{v/s}}$, $l\geq 2$.

In [17] it was shown that scalar and vector perturbations are isospectral, so
we will henceforth drop the v/s subscript and concern ourselves only with
scalar perturbations. It was shown that the lowest lying QNM is either a
solution for $\Phi_{+}$ with $l=1$ or $\Phi_{-}$, with $l=2$. Below, we will
compare the interior QNM spectrum with the frequencies of these two modes.

Defining $r_{*}$ in the appropriate domain, the boundary conditions that
define interior and exterior QNMs can be written as:

$\begin{split}&R(r)\,\sim\,e^{i\omega r_{*}}\quad\text{as}\quad
r_{*}\rightarrow-\infty\,,\\\ &R(r)\,\sim\,e^{-i\omega
r_{*}}\quad\text{as}\quad r_{*}\rightarrow+\infty\,.\end{split}$ (39)

For interior (exterior) QNM, we denote $r_{-}\,\left(r_{c}\right)$ as $r_{2}$
and $r_{+}$ as $r_{1}$. Restoring $r$ dependence in equation (39), we get:

$\begin{split}&R(r)\,\sim\,\left|r\,-\,r_{1}\right|^{-\frac{i\omega}{2\,k_{1}}}\quad\text{as}\quad
r\,\rightarrow\,r_{1}\quad,\\\
&R(r)\,\sim\,\left|r\,-\,r_{2}\right|^{-\frac{i\omega}{2\,k_{2}}}\quad\text{as}\quad
r\,\rightarrow\,r_{2}\quad.\\\ \end{split}$ (40)

It is useful to incorporate the boundary conditions in the equations of
motion. Thus, we redefine $R(r)$:

$R(r)\,=\,\left|r\,-\,r_{1}\right|^{-\frac{i\omega}{2\,k_{1}}}\,\left|r\,-\,r_{2}\right|^{-\frac{i\omega}{2\,k_{2}}}\,\tilde{R}(r)\quad,$
(41)

where $\tilde{R}(r)$ is smooth in $r_{1}$ and $r_{2}$. We now define the
dimensionless variable,

$y\,=\,\frac{r\,-\,r_{1}}{r_{2}\,-\,r_{1}}\quad,$ (42)

with $y=0$ at $r_{1}$ and $y=1$ at $r_{2}$. Substituting (42) and (41) in
(32), we get

$\tilde{R}^{\prime\prime}(y)\,+\,\frac{P(y,\omega)}{y\,(y-1)}\,\tilde{R}^{\prime}(r)\,+\,\frac{Q(y,\omega)}{y(y-1)}\,\tilde{R}(y)\,=\,0\quad,$
(43)

with $P(y,\omega)$ and $Q(y,\omega)$ analytic functions of $y$ and $\omega$ in
$[0,1]\times\mathbb{C}$. For values of $y>1$ or $y<0$ we might find
singularities in P and Q. We now define QNMs as solutions of (43) that are
smooth at $y\,=\,0$ and $y\,=\,1$, with $\tilde{R}(0),\,\tilde{R}(1)\neq 0$.

###  IV.2 Analytical approximation for confluent horizons

In [17], violations of SCC were found for BHs that were close to extremality.
These BHs have $Q^{2}\rightarrow Q^{2}_{max}$ (see equation (10)),
corresponding to $r_{-}\rightarrow r_{+}$. The authors found that the slowest
decaying QNMs in these BHs are QNMs inherited from the limiting extremal RNdS
BHs. These solutions are sharply peaked near $\mathcal{H}^{+}$, and vanish
quickly for $r>r_{+}$. We denote them as near extremal QNMs. As argued in
III.2 we want to explicitly check if the frequencies of these modes coincide
with a subset of the interior QNM spectrum. If frequencies were to coincide,
fields would have higher regularity at the Cauchy Horizon.

Lets first focus in the more general case of modes propagating between two
horizons that are arbitrarily close. This situation describes interior QNMs in
the near extremal limit, but also QNMs of near Nariai black holes
($r_{+}\rightarrow r_{c}$ limit). We will use the second case as a control on
our calculations.

Analysing the equation (12), in the $r_{1}\rightarrow r_{2}$ limit, we have
$\kappa_{i}\rightarrow 0$. Given that in (41) $\omega$ shows up always divided
by $k_{i}$, and the same happens in (43) if expanded around $y\,=\,0$, we
expect that QNM frequencies will vanish proportionally to $k_{i}$. Thus, we
will work with the dimensionless parameter $\rho\,=\,\frac{\omega}{k_{1}}$.
Due to $r_{1}$ and $r_{2}$ being very close, we expect that $A$ (defined in
(33)) will be approximately constant in $r\,\in\,(r_{1},\,r_{2})$. We expect
the same from $F$ defined through

$f(r)=(r\,-\,r_{1})\,(r\,-\,r_{2})\,F(r)\quad.$ (44)

Hence, a Taylor expansion of (43) about $r=r_{1}$ should yield accurate
results. We now define the perturbation parameter
$\varepsilon=\frac{r_{2}-r_{1}}{r_{1}}$ , and expand $\tilde{R}$ and $\rho$:

$\begin{split}\tilde{R}(r)&=\tilde{R}_{0}(r)\,+\,\varepsilon\,\tilde{R}_{1}(r)\,+\,...\\\
\rho&=\rho_{0}\,+\,\varepsilon\,\rho_{1}\,+\,...\end{split}$ (45)

Plugging this into equation (43), to zero order in $\varepsilon$, we get:

$y(1\,-\,y)\,R_{0}^{\prime\prime}(y)\,+\,\left(1\,-\,i\,\rho_{0}\,-\,2\left(1\,-\,i\,\rho_{0}\right)y\right)\,R_{0}^{\prime}(y)\\\
+\left(\frac{A\left(r_{1}\right)}{F\left(r_{1}\right)}\,+\,\rho_{0}^{2}\,+\,i\,\rho_{0}\right)\,R_{0}(y)\,=\,0\quad.$
(46)

This is the hyper-geometric equation. In the canonical form, we write this
equation as:

$y(1-y)w^{\prime\prime}(y)+\left(c-\left(a+b+1\right)y\right)w^{\prime}(y)-abw(y)=0\quad.$
(47)

This result is analogous to the one obtained in [29], as the near horizon
geometry of Black Holes is approximately $AdS_{2}$. From (46) we read off the
coefficients a, b and c:

$\begin{split}a&=-i\,\rho_{0}\,+\,\frac{1-\Delta}{2}\quad,\\\
b&=-i\,\rho_{0}\,+\,\frac{1+\Delta}{2}\quad,\\\
c&=1\,-\,i\,\rho_{0}\quad.\end{split}$ (48)

where,

$\Delta=\sqrt{1\,+\,\frac{4\,A(r_{1})}{F(r_{1})}}\quad.\\\ $ (49)

This result is still valid for $1+4\,A(r_{1})/F(r_{1})<0$. Here, $\Delta$ will
be the root of a negative number, and we must choose a branch to define it.
Notice that swapping the branch choice is equivalent to swapping the
definition of a and b, so this choice can be arbitrary.

The hypergeometric equation is solved by hypergeometric functions. These have
been well documented in the literature (see e.g. [30, 31] ). Following the
procedure in [20], we prove that requiring smoothness at $y\,=\,0$ and
$y\,=\,1$ implies a quantization of a or b. We have that either $a\,=\,-k$ or
$b\,=\,-k$ with $k\,=\,0,\,1,\,2,\,...$. Solving this with respect to
$\rho_{0}$ we obtain two families of QNMs, defined as type-a and type-b, with
frequencies:

$\displaystyle\frac{\omega_{a}}{k_{+}}\,$
$\displaystyle=\,-i\left(\frac{1-\Delta}{2}+\,k\right)\,+\,\mathcal{O}\left(\varepsilon\right)\quad,$
(50) $\displaystyle\frac{\omega_{b}}{k_{+}}\,$
$\displaystyle=\,-i\left(\frac{1+\Delta}{2}+\,k\right)\,+\,\mathcal{O}\left(\varepsilon\right)\quad.$
(51)

Taking $r_{1}=r_{+}$ and $r_{2}=r_{c}$, we can estimate the spectrum of
exterior modes in the Nariai limit ($r_{+}\rightarrow r_{c}$). Here $\Delta$
will be purely imaginary as discussed above. $\omega_{a}$ and $\omega_{b}$
will have the same imaginary part and symmetric real parts. In figure 3(a), we
compare the spectrum of Nariai modes obtained numerically (see section IV.3),
with this analytic prediction. These are consistent with the WKB approximation
in [17] and the supplementary material of [10].

Similarly, taking $r_{1}=r_{+}$ and $r_{2}=r_{-}$, we obtain an estimate for
the interior frequency spectrum of near extremal black holes. Here, $\Delta$
will be real and positive, with no a-priori upper bound. In fact, there are
cases where it is greater than 1, leading to modes with positive imaginary
frequencies. This is not cause for concern, as there is no relation between
interior QNMs and spacetime stability (see [20]). In figure 3(b) we compare
the analytic prediction with the numerical results obtained in IV.3.

This approach is entirely analogous to the analytical approach used in [17] to
estimate exterior QNM frequencies. In fact, we may extend y to negative
values, to explore the behaviour of $R_{0}(y)$ in the BH exterior. Taking
$R_{0}(y)$, to be smooth in $y\,=\,0$ and exponentially decaying at
$y\rightarrow-\infty$, we get a mode that is smooth in $r_{+}$ and
exponentially decaying for $r>r_{+}$. These are precisely the conditions
required in [17] to estimate NE modes. It is important to mention that the
decaying condition is purely empirical, motivated by the behaviour of
solutions observed in numerical calculations. Analysing the solutions of
equation (47) in terms of hypergeometric functions, we prove that exponential
decay at $y\rightarrow-\infty$ implies smoothness at $y=1$. This means that
extending the NE QNMs of [17] to the interior of the BH, we get interior QNMs.
Hence, up to 0th order in $\varepsilon$ the spectrum of exterior NE QNM
frequencies is a subset of the spectrum of interior QNM frequencies. In fact,
NE QNMs coincide with the type-b family of modes defined in (51).

This result might seem a bit worrying, as it hints against the conclusion of
our paper. In fact, we must compute higher order corrections to distinguish
the interior and exterior spectrum. Doing this analytically is hard, and it is
not clear how to proceed. Hence, we will resort to numerics, to pinpoint
whether there is a difference between interior and exterior QNM frequencies.

###  IV.3 Numerical computation

(a)

(b)

Figure 4: In these images we plot $\gamma$ as a function of
$Q^{2}/Q^{2}_{max}$ for $y_{+}=0.47$. On the left panel, we study quasinormal
modes corresponding to $\Phi_{-}$, with l=2, whereas on the right panel we
have $\Phi_{+}$, with $l=1$. In both plots, we zoomed the circled region into
a subplot on the top right corner. As we can see, the value of $\gamma$ takes
very small values, oscillating around 0. We can see that $\gamma$ tends to 0
in the $Q^{2}\rightarrow Q^{2}_{max}$ limit, as predicted in section IV.2.

(a)

(b)

Figure 5: As in figure 4, we plot the value of $\gamma$ in a non-trivial
region of parameter space. On the left panel we study $\Phi_{-}$, with l=2,
whereas on the right panel we have $\Phi_{+}$, with $l=1$. To produce these
plots, we used the Newton-Raphson algorithm to find the frequency of
$\omega_{\text{int}}$ and $\omega_{\text{ext}}$ discretely varying the
parameter space in a 200x200 grid. As described in section IV.3, we repeated
the calculation twice for each black hole with different grid sizes, to
estimate the numerical precision of our method. We checked that in the whole
plot, the numerical precision was always above the difference between interior
and exterior frequencies. We remark, that as seen in the one dimensional plot,
$\gamma$ goes to 0 as $Q^{2}\rightarrow Q^{2}_{max}$.

To obtain the spectrum of QNMs numerically we approximate the continuous ODE
problem by a discrete system of equations, as seen in [32, 33, 34, 35]. More
specifically, we sample solutions of equation (43) in a discrete Chebyshev
grid, mapping the differential operator into a matrix operator. We then solve
the resulting matrix equation using standard linear algebra methods.

Figure 6: In this figure we show the exponential convergence of the Newton-
Raphson method. We plot the normalized difference between
$\beta=\omega/\kappa_{-}$ computed using a given Chebyshev grid size $(N_{c})$
and $\beta$ computed using very high resolution $(N_{c}=601)$. We see that the
relative numerical error decays exponentially with $N_{c}$ as expected. Here,
$\omega$ is the lowest lying NE mode, of $\Phi_{-}$ with $l=1$, for a BH with
$y_{+}\,=\,0.35$ and $Q^{2}/Q^{2}_{max}=0.99$.

For a given interval $[a,\,b]$ we may define a Chebyshev grid with size
$N_{c}$ as the set of points $x_{i}$ such that

$x_{i}=\frac{(b-a)}{2}\cos\left(\frac{i\,\pi}{N_{c}}\right)\,+\,\frac{a+b}{2},\quad\quad
i=0,\,1,\,2,\,...,\,N_{c}\quad.$ (52)

Constraining the domain to the discrete Chebyshev grid, we may rewrite the
action of the derivative using matrix multiplication. We denote this matrix
$D_{c}$, (see [34] for an explicit definition). Multiplying equation (43) by
$y(y-1)$ and sampling it in the Chebyshev grid above, we obtain a quadratic
matrix eigenvalue problem. The eigenvectors will be discrete approximations of
solutions to equation (43), smooth at $y\,=\,0$ and $y\,=\,1$. The eigenvalues
will correspond to approximations of the Quasinormal Frequencies.

To find the frequencies we use a combination of two methods. We start by
obtaining a direct eigenvector decomposition of the system using Mathematica’s
native Eigensystem function. This method yields an approximation of the full
frequency spectrum. Bear in mind it is very important to use a software that
supports arbitrary precision to obtain good results (we used up to 500 digits
of precision in intermediate calculations). This procedure, is not the best to
approximate the frequency of individual modes. It is too computational
intensive, and does not allow arbitrary high precision of a given frequency.
Bearing this in mind, we apply the version of the Newton-Raphson method
described in [33, 32, 36] using the direct method as a seed. Although faster
and more precise, this method requires a very accurate initial frequency
estimate to converge. In figure 6, we plot the numerical error on the
calculation of a given quasinormal frequency as a function of the Chebyshev
grid size $(N_{c})$. As expected, we see that the usage of a Chebyshev grid
actually guarantees that this method has exponential convergence.

In the analysis below, we computed the lowest lying NE exterior and type-b
interior QNMs (see equation (51) for a definition) in a non-trivial region of
the parameter space. To achieve this, we used the direct method to obtain
these modes for a given RNdS BH, and then used them as seed for a Newton-
Raphson method applied to BHs with similar mass and charge. We then used the
results as estimates on other similar BHs and henceforth. For each frequency
calculated, we repeated this procedure twice, once with a Chebyshev grid size
of 500, and then 550. The difference between the obtained frequencies was used
to estimate the precision of our calculations. We always made sure this was at
least one order of magnitude below the difference between interior and
exterior QNM frequencies. This limit would only saturate close to extremality,
otherwise, the precision was usually much higher than the difference in
frequencies.

###  IV.4 Comparison between interior and exterior modes for
gravitoelectromagnetic perturbations

As seen in section IV.2, in the NE limit of RNdS BHs, there is a
correspondence between the spectrum of NE exterior modes and type-b interior
modes. Using numerics, we verified whether this correspondence extends to the
remainder of parameter space. To do so, we explicitly computed the lowest
lying frequency of the NE family of exterior modes ($\omega_{\text{ext}}$) and
the lowest lying frequency of type-b interior modes ($\omega_{\text{int}}$).
To make sure we chose the appropriate families, we compared with the analytic
prediction in the near extremal limit (equation (51)). We performed this
calculation for $\Phi_{+}\,(l=1)$ and $\Phi_{-}\,(l=2)$ gravitoeletric QNMs in
a large class of RNdS black holes.

In our plots, we characterize the difference using the quantity

$\gamma=\left(\frac{\omega_{\text{in}}\,-\,\omega_{\text{ext}}}{\omega_{\text{in}}\,+\,\omega_{\text{ext}}}\right).$
(53)

This will always be a real number, as the modes we are concerned with have
purely imaginary frequency.

In figure 4 we plot $\gamma$ as a function of $Q^{2}/Q^{2}_{\text{max}}$ for a
fixed value of $y_{+}$. Similarly, in figure 5, we depict $\gamma$ in a non-
trivial region of parameter space using a density plot. We see that $\gamma$
oscillates around 0, attaining the value in some simple roots and occasional
double roots (as seen in the beginning of figure 4(a)).

Figure 7: In this figure we study the double roots of $\gamma$ found in
figures 4 and 5. As an example we establish that ($y_{+}\approx 0.47$,
$Q^{2}/Q^{2}_{\text{max}}\approx 0.908$) is indeed a double root of $\gamma$,
from $\Phi_{+}$ QNMs. We fixed $y_{+}$ and varied $Q^{2}/Q^{2}_{\text{max}}$
in a grid of 20 entries, with values in $\overline{r}_{0}\pm\delta_{0}/2$,
($\overline{r}_{0}=0.0075$, $\delta_{0}=0.005$). We sampled $\gamma$ there and
found the grid point with smallest $\gamma$. Setting this as
$\overline{r}_{1}$ we created a new grid centred at this point with width
$\delta_{1}=10^{-1}\delta_{0}$. We sampled values of $\gamma$ in this new grid
and iterated the process 10 times. In the plot we see the value of the minimum
as a function of the sampling density $\rho=20/\delta$. The value decreases
exponentially with sampling density, hinting that $y_{+}\approx 0.47$,
$Q^{2}/Q^{2}_{\text{max}}\approx 0.908$ is indeed a double root of $\gamma$.

Analysing figure 5, we see $\gamma$ is always very small, reaching a maximum
of $\mathcal{O}(10^{-6})$. Although the frequencies are very similar as
expected, they are not equal. It is interesting to analyse the dark regions in
5. When the colour changes from purple to orange, we have single 0’s of
$\gamma$, whereas when there is no colour change, we have double zeros, as
seen above. In figure 7, we prove that this features are actually present, by
zooming in around one of these roots. These features are unexpected as both
the frequencies and their gradient coincide. Close to these lines, there is a
wider region where $\gamma$ is close to 0. The roots of $\gamma$ are
1-dimensional regions of parameter space. For these black holes, fields will
have increased regularity at the Cauchy horizon. Nevertheless, these regions
are not generic in the 2-dimensional parameter space of RNdS BHs, so they are
not pointers to SCC violations. Finally, it is worth remarking that $\gamma$
approaches 0 in the extremal limit, validating the discussion in section IV.2.

We can conclude that lowest lying exterior frequencies, generically, are not
the frequencies of interior modes. Hence the regularity of
gravitoelectromagnetic perturbations on a generic RNdS black hole, agree with
the ones found in [17]. Hence, there are no new violations of SCC.

##  V Strong cosmic censorship in the Kerr-dS black hole

###  V.1 Wave equation and QNM

As a toy example, we will focus in the behaviour of massless scalar fields in
Kerr-dS spacetime. Our argument should be extendible to other fields without
much difficulty. In the following, we will show that generic solutions of

$\Box\Phi\,=\,0$ (54)

do not have finite energy at the Cauchy horizon. We will take purely ingoing
modes at the event horizon, with the frequency of an exterior mode, and
scatter it through the BH interior, studying their regularity at the Cauchy
horizon. In a Kerr-dS background, the radial and angular parts of the KG
equation are given by the Teukolsky equation. Plugging (19) into (18), we
obtain the following equations:

$\displaystyle\begin{split}&{}\derivative{\theta}\left(\Delta_{\theta}\sin\theta\derivative{\Theta_{\omega
lm}}{\theta}\right)\\\
&{}-(m-a\omega\sin^{2}\theta)^{2}\frac{\Sigma^{2}}{\Delta_{\theta}\sin\theta}\Theta_{\omega
lm}(\theta)\\\ &{}+K_{lm}\sin(\theta)\Theta_{\omega
lm}(\theta)\,=\,0\quad,\end{split}$ (55)
$\displaystyle\begin{split}&\derivative{r}\left(\Delta_{r}\derivative{R_{\omega
lm}(r)}{r}\right)\\\
&+\left[\frac{\Sigma^{2}}{\Delta_{r}}(\omega(r^{2}+a^{2})-am)^{2}\,-\,K_{lm}\right]R_{\omega
lm}(r)=0\quad,\end{split}$ (56)

here $m\in\mathbb{Z},\,|m|\leq l$ and $K_{lm}$ is the constant arising from
separation of variables. For ease of notation we will henceforth drop the
$\omega lm$ subscript.

Equation (56) has four regular singular points at the roots $r_{i}$ of
$\Delta_{r}(r_{i})$. For $r>0$ this happens at the Cauchy horizon $(r_{-})$,
the event horizon $(r_{+})$ and the Cosmological horizon $(r_{c})$. Solutions
to this equation are defined in (21), (22) and (23).

The analysis of section III.2 tells us that for a QNM with

$\frac{|Im(\omega)|}{\kappa_{-}}<\frac{1}{2}\quad,$ (57)

$\Phi_{in,-}$ will have infinite energy at the Cauchy horizon. In [19], using
a WKB analysis, it was proved that taking $l$ large enough we find modes
respecting condition (57). We will now prove that for these modes the value of
$\mathcal{A}(\omega)$ in equation $\eqref{eq:RinpExpansion}$ is non-zero,
validating the conclusions of [19].

###  V.2 Solving the wave equation in the BH interior

Below, we will study the continuation of an exterior QNM solution, onto the
interior of the BH, and show that it has $\mathcal{A}(\omega)\neq 0$. To do
so, we fix $K_{lm}$ and $\omega$ to the values obtained for WKB QNMs defined
in [19]. Exterior QNMs are purely ingoing at the event horizon. We use this to
set the initial conditions for the wave function in the BH interior. Using a
WKB approximation, we solve the radial equation, decomposing it in the
$R_{in/out,-}$ basis, using equation (30).

Consider $r_{*}$ defined in (15), such that $r_{*}(r_{+})=+\infty$ and
$r_{*}(r_{-})=-\infty$. In terms of $r_{*}$, the radial equation is

$\derivative[2]{S}{r_{*}}\,+\,U(r,\omega,l,m)\,S(r)\,=\,0\quad,$ (58)

where $S(r)\,=\,\sqrt{r^{2}+a^{2}}\,R(r)$, and

$U=\left(\omega-\frac{am}{r^{2}+a^{2}}\right)^{2}-\frac{K_{lm}\,\Delta_{r}}{\Sigma^{2}(r^{2}+a^{2})^{2}}\\\
+\,(r^{2}+a^{2})^{-4}\Bigg{[}-\Delta_{r}^{2}(r^{2}+a^{2})+3\Delta_{r}^{2}r^{2}\Bigg{.}\\\
\Bigg{.}-r\Delta_{r}\,\derivative{\Delta_{r}}{r}\,(r^{2}+a^{2})\Bigg{]}\quad.$
(59)

The modes studied in [19], are such that $m=l$. Using a WKB approximation, the
authors show that in the large $l$ limit:

$\omega=\omega_{R}+i\,\omega_{I}=l\,\Omega_{c}+\mathcal{O}(1)\quad,$ (60)

here $\Omega_{c}$ is the rotation frequency of a photon propagating in the
inner photon sphere, outside the black hole. This corresponds to
$\Omega^{+}_{c}$ defined in equation (4.19) of [19]. We expand $K_{ll}$ in
powers of $l$:

$K_{ll}\,=\,l^{2}\,\lambda_{-2}\,+\,l\,\lambda_{-1}\,+\,...$ (61)

In appendix A, we prove that this expansion is valid and $\lambda_{-2}$ is
real and positive. We can now define:

$U_{0}=\left[\left(\Omega_{c}-\frac{a}{r^{2}+a^{2}}\right)^{2}-\frac{\,\lambda_{-2}\,\Delta_{r}}{\Sigma^{2}(r^{2}+a^{2})^{2}}\right]\quad,$
(62)

so that

$U=l^{2}U_{0}\,+\,\mathcal{O}(l)\quad\text{for}\quad l>>1\quad.$ (63)

Notice that $U_{0}(r_{*})$ is a real function of $r_{*}$, whereas $U(r_{*})$
has non vanishing imaginary part. We explicitly verified, that for Kerr-dS
BHs:

$\Omega(r_{-})>\Omega_{c}>\Omega(r_{+})\quad,$ (64)

where $\Omega(r)$ is defined in (17).

Using equations (64), (61) and the fact that $\Delta_{r}(r)<0$ in
$r\in[r_{-},r_{+}]$, we can establish that $U_{0}(r_{*})>0$ in
$r\in[r_{-},r_{+}]$. In the BH exterior $\Delta_{r}(r)>0$ and this is no
longer the case.

We now solve equation (58) using a standard WKB expansion (see [30]):

$S(r_{*})=\exp\left(\frac{1}{l}\sum_{n=0}^{\infty}\delta^{n}S_{n}(r_{*})\right).$
(65)

Truncating to $n=1$, we obtain that in the large $l$ limit:

$S(r_{*},\omega,l)=\sqrt{r^{2}+a^{2}}\,R_{in,+}\approx\\\
U_{0}(r_{*})^{-\frac{1}{4}}\Bigg{[}A(\omega,l)\exp\left(il\int_{0}^{r_{*}}U_{0}(s)^{\frac{1}{2}}\differential
s\right)\\\
+B(\omega,l)\exp\left(-il\int_{0}^{r_{*}}U_{0}(s)^{\frac{1}{2}}\differential
x\right)\Bigg{]}\quad,$ (66)

where $A(\omega,l)$ and $B(\omega,l)$ are integration constants.

Given that $U_{0}(r_{*})>0$ everywhere, equation (66) is valid everywhere.
Physically, this means that to leading order in $l$, there is no scattering of
WKB modes propagating in the BH interior. However, to get
$\mathcal{A}(\omega)=0$ we would need a purely ingoing mode at
$\mathcal{H}^{+}$ to be fully scattered into an outgoing mode at
$\mathcal{CH}_{R}$. Therefore we can already anticipate that no violations of
SCC in the Kerr-dS spacetime will be found. Nevertheless, lets explicitly
check this below.

###  V.3 Behaviour near the horizons

For ease of notation, define

$\omega_{+}=\omega-l\,\Omega_{+}\quad,\quad\omega_{-}=\omega-l\,\Omega_{-}\quad.$
(67)

For $r_{*}\to-\infty$ (near $\mathcal{H}_{+}$), using equations (59) and (60),
we obtain:

$\lim_{r_{*}\to-\infty}\int_{0}^{r_{*}}\left(l\sqrt{U_{0}}-\omega_{+}\right)\,=\,\alpha_{+}\,l\quad,$
(68)

where $\alpha_{+}=\mathcal{O}(1)$ as $l\to\infty$. Similarly, for
$r_{*}\to\infty$ (near $\mathcal{CH}_{+}$), we get

$\lim_{r_{*}\to\infty}\int_{0}^{r_{*}}\left(l\sqrt{U_{0}}-\omega_{-}\right)\,=\,\alpha_{-}\,l\quad.$
(69)

Dropping the finite term $\sqrt{r^{2}+a^{2}}$ in front of $R(r)$, we find that
near $\mathcal{H}_{+}$,

$R(r_{*})\sim\left(\frac{\omega_{+}}{l}\right)^{-\frac{1}{4}}\Bigg{[}A(\omega,l)e^{-i\left(\omega_{+}r_{*}\,+\,\alpha_{+}l\right)}\\\
+B(\omega,l)e^{i\left(\omega_{+}r_{*}\alpha\,+\,l\right)}\Bigg{]}\quad\text{as}\quad
r_{*}\rightarrow-\infty\quad.$ (70)

Similarly, at $\mathcal{CH}_{R}^{+}$, we get:

$R(r_{*})\sim\left(\frac{\omega_{-}}{l}\right)^{-\frac{1}{4}}\Bigg{[}A(\omega,l)e^{-i\left(\omega_{-}r_{*}\,+\,\alpha_{-}l\right)}\\\
+B(\omega,l)e^{i\left(\omega_{-}r_{*}\,+\,\alpha_{-}l\right)}\Bigg{]}\quad\text{as}\quad
r_{*}\rightarrow\infty\quad.$ (71)

Now, we take $R(r_{*})$ to be $R_{in,+}$. The ingoing condition (equation
(22)) at $\mathcal{H}_{+}$ yields $R_{in,+}(r_{*})\sim e^{-i\omega_{+}r_{*}}$
as $r_{*}\rightarrow\infty$. This gives:

$\displaystyle B(\omega,l)$ $\displaystyle\sim\mathcal{O}(l^{-1})\quad,$ (72)
$\displaystyle A(\omega,l)$ $\displaystyle\sim\mathcal{O}(1)\quad.$ (73)

Now, focusing in the behaviour near $\mathcal{CH}_{R}^{+}$, using equation
(30), we obtain:

$\displaystyle\mathcal{A}(\omega)$
$\displaystyle=\left(\frac{\omega_{-}}{l}\right)^{-\frac{1}{4}}A(\omega,l)e^{-i\,\alpha_{-}l}\quad,$
(74) $\displaystyle\mathcal{B}(\omega)$
$\displaystyle=\left(\frac{\omega_{-}}{l}\right)^{-\frac{1}{4}}B(\omega,l)e^{i\,\alpha_{-}l}\quad,$
(75)

where

$\frac{\omega_{-}}{l}=\mathcal{O}(1)\quad.$ (76)

In particular, in large $l$ limit, $\mathcal{B}$ vanishes, so
$\mathcal{A}(\omega)$ must be non zero, or we would have the trivial solution.
Hence, we prove that $\mathcal{A}(\omega)\neq 0$ for $\omega$ a QMN frequency.
The conclusion of [19] is maintained, preserving SCC for the Kerr-dS BH.

##  VI Discussion

In this paper, we proved that there is no coincidence between the interior and
exterior spectrum of QNMs for Kerr-dS and RNdS BHs. Hence, there is no new
violation of SCC for these spacetimes. As further work, it would be
interesting to relate the discussion in this paper with the quantum
instability of Cauchy horizons of Kerr-dS and RNdS black holes, found in [23].
In fact, this instability is proportional to a factor, that depends on the
reflection and transmition coefficients of waves around the BH (see equation
123 of that paper). We suspect that this function could vanish, if interior
and and exterior QNMs coincide, akin to $\mathcal{A}(\omega)$. In fact, in the
same paper, the authors study the case of the 3-dimensional BTZ black hole,
and they find that due to the hypergeometric nature of the wave equation, this
factor vanishes identically in there.

##  VII Acknowledgements

The author thanks the contribution of his supervisor Harvey S. Reall, in
motivating, discussing and reviewing the contents of this paper, Felicity C.
Eperon, for outlining the calculations used in the Kerr-dS case, Jorge E.
Santos, Oscar J.C. Dias, Aron Wall, Jonathan Crabbé, Rifath Khan, Bilyana
Tomova, Gonçalo Regado and Rita Costa for fruitful physics discussions. The
author also thanks Patrícia Jorge for help with producing figure 1. The author
is jointly funded by the IOA (University of Cambridge), the University of
Cambridge Trust and King’s College (University of Cambridge).

##  Appendix A Positivity of $\lambda_{-2}$

In this appendix, we will prove the positivity of $\lambda_{-2}$ defined in
(61). Taking the angular Teukolsky equation (55) and making the substitution
$x=\cos(\theta)$, we get:

$\derivative{x}\left(1+\alpha x^{2}\right)(1-x^{2})\derivative{x}\Theta\\\
-\frac{(m-a\omega(1-x^{2}))^{2}\Sigma^{2}}{(1+\alpha
x^{2})(1-x^{2})}\Theta+K_{lm}\Theta\,=\,0\quad.$ (77)

Multiplying both sides by $\Theta^{*}$ and integrating in $x\in(-1,1)$, we
get:

$\int_{-1}^{1}\Theta^{*}P_{x}\Theta\,\differential
x+K_{lm}||\Theta||^{2}=0\quad,$ (78)

where

$P_{x}=\derivative{x}\left(1+\alpha
x^{2}\right)(1-x^{2})\derivative{x}+\frac{(a\omega(1-x^{2})-m)^{2}\Sigma^{2}}{(1+\alpha
x^{2})(1-x^{2})}\quad,$ (79)

and

$||\Theta||^{2}=\int_{-1}^{1}\Theta^{*}\Theta\,\differential x>0\quad.$ (80)

Integrating by parts the first term of $P_{x}$ and noting that the boundary
terms (evaluation at $x=1$ and $x=-1$) vanish, we get:

$K_{lm}=\frac{\int_{-1}^{1}\frac{(a\omega(1-x^{2})-m)^{2}\Sigma^{2}}{(1+\alpha
x^{2})(1-x^{2})}|\Theta|^{2}\differential x}{||\Theta||^{2}}\\\
+\frac{\int_{-1}^{1}(1+\alpha
x^{2})(1-x^{2})|\derivative{x}\Theta|^{2}\differential
x}{||\Theta||^{2}}\quad.$ (81)

Here, the second term is real and positive, whereas the first term is complex.
We have:

$K_{lm}=K_{lmR}+iK_{lmI}\quad.$ (82)

However, taking $m=l$ and replacing equation (60) in (81), we obtain:

$\frac{K_{lmR}}{K_{lmI}}=\delta\,l+\mathcal{O}(1)\quad,$ (83)

where $\delta$ is some real number. This means that, to leading order in $l$,
the coefficients in (77), are real. Now, we multiply both sides of (77) by
$(1+\alpha x^{2})(1-x^{2})$, and define

$\differential x^{*}=\frac{\differential x}{(1+\alpha\,x^{2})(1-x^{2})}\quad,$
(84)

with $x^{*}\in(-\infty,+\infty)$. We rewrite (77) as:

$\derivative[2]{x_{*}}\,\Theta(x_{*})+Q(x^{*})\,\Theta(x^{*})\,=\,0\quad,$
(85)

with

$Q(x_{*}(x))=l^{2}\Bigg{(}(1+\alpha x^{2})(1-x^{2})\frac{K_{llR}}{l^{2}}\\\
-(1-a\,\Omega_{c}(1-x^{2}))^{2}\,\Sigma^{2}\Bigg{)}+\mathcal{O}(l)\\\
=:l^{2}\,Q_{-2}(x_{*})+\mathcal{O}(l)\quad.$ (86)

We may now study the large $l$ limit of $K_{ll}$, by solving:

$\derivative[2]{x_{*}}\,\Theta_{-2}(x_{*})+l^{2}\,Q_{-2}(x_{*})\,\Theta_{-2}(x_{*})\,=\,0\quad.$
(87)

Using the fact that $a\Omega_{c}<1$ (see [19] equation 4.19) and
$\alpha\,<\,1$, we obtain that $Q_{2}(x_{*})$ has 0, 1, or 2 roots. Now, the
angular equation must be regular at $\theta\,=0\,$ and $\theta\,=\,\pi$. This
is the statement that $\Theta(x_{*})$ must decay exponentially at both
infinities, i.e. an ingoing wave at $\theta\,=\,0$ must be fully scattered
into $\theta\,=\,\pi$. Using a standard WKB approximation (see [30]) we see
that this is the case if

$K_{llR}\,=\,K_{ll}+\mathcal{O}(l)\,=\,l^{2}\lambda_{-2}+\mathcal{O}(l).$ (88)

Hence, we prove equation (61). Now, the positivity of $\lambda_{-2}$ follows
simply by the substitution of equations (60) and (61) into equation (81). We
get:

$K_{ll}=l^{2}\lambda_{-2}=l^{2}\,\Bigg{(}\frac{\int_{-1}^{1}\frac{(a\Omega_{c}(1-x^{2})-1)^{2}\Sigma^{2}}{(1+\alpha
x^{2})(1-x^{2})}|\Theta|^{2}\differential x}{||\Theta||^{2}}\\\
+\,\frac{\int_{-1}^{1}(1+\alpha
x^{2})(1-x^{2})|\derivative{x}\Theta|^{2}\differential
x}{l^{2}||\Theta||^{2}}\Bigg{)}+\mathcal{O}(l)\quad,$ (89)

where both leading order terms are real and positive. Hence, we we deduce that
$\lambda_{-2}>0$.

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##  Appendix A Contribution of lines and AIBs in imaging bands

Figs. 14 to 20 show the three template spectra in Peeters et al. (2023),
illustrating the variation of the contribution of different lines into each
imaging band. More detailed analysis with a full set of lines will be
presented in the science enable product and the associated article (Chown et
al. 2023).

Figure 14: Three template spectra (from top to bottom) shown in Peeters et al.
(2023) in the range covered by F162M (left) and F164N (right). Response
functions of the filters are shown with thin orange lines. Expected positions
of HI, HeI, H2, and [FeII] lines are marked. Figure 15: Same as Fig. 14, but
for F182M (left) and F187N (right). Figure 16: Same as Fig. 14, but for F480M
(left) and F405N (right). Figure 17: Same as Fig. 14, but for F210M (left) and
F212N (right). Figure 18: Same as Fig. 14, but for F300M (left) and F323N
(right). Figure 19: Same as Fig. 14, but for F480 (left) and F470N (right).
Figure 20: Same as Fig. 20, but for F300M (left) and F335M (right).

##  Appendix B JWST and ground-based telescopes images of the Bar

Fig. 21 shows the composite MIRI images of the southwest part of Bar. This
region designated the ”SW Bright Bar” in O’Dell et al. (2017b) presents the
sharpest ionization front of the Bar. Its structure and physical conditions
near the MIF are studied in detail from HST, MUSE and velocity data in their
Section 4.3.

Figs. 22-24 present all NIRCam images of the Bar obtained in the filters
listed in Table 1, and some filters continuum subtracted. Fig. 25 shows for
the southwestern part of the Bar NIRCAM map of the 0-0 S(9) H2 line at 4.69
$\mu$m, Keck/NIRC2 map of the 1-0 S(1) H2 line at 2.12 $\mu$m, and ALMA maps
of the HCO+ J=4-3 and CO J=3-2 lines.

Figure 21: Southwestern part of Bar as seen by the JWST’s MIRI instrument with
north up and east left. Several images in different filters were combined to
produce an RGB composite image: F1500W (green), F1130W (blue) and F770W (red)
that traces respectively continuum emission from hot/warm dust and
hydrocarbons (AIBs). The individual images used to make the RGB composite one
are shown below. The size of the images is $\sim$42$\arcsec\times 42\arcsec$
and it is centered on RA=05h35m20$\aas@@fstack{s}$33,
DEC=-05$\degr$25$\arcmin$03.77$\arcsec$ very close the irradiated proto-
planetary disk 203-504.  Figure 22: NIRCam maps of the Bar in different
filters. Units are in MJy/sr. Figure 23: Same as Fig. 22, but for other
filters. Figure 24: Same as Fig. 22, but for other filters.  Figure 25:
NIRCam, Keck and ALMA maps of the southwestern part of Bar. The maps have been
rotated so that the ionizing radiation strikes the Bar from the left. Upper
left: Map in the F470N filter continuum subtracted (${\rm H_{2}}$ 0-0 S(9)
line at 4.7 $\mu$m). Units are in erg s-1 cm-2 sr-1. Upper right: Keck/NIRC2
map of the ${\rm H_{2}}$ 1-0 S(1) line at 2.12 $\mu m$ with a resolution of
0.085$\arcsec$ (Habart et al. 2023). Our NIRCam observations show a very good
agreement with Keck/NIRC2 observations in terms of line distribution and
intensity. Lower left and right: ALMA observation at 1$\arcsec$ resolution of
the HCO+ J=4–3 line integrated intensity (in K km s-1) and the CO J=3-2 line
peak temperatures ($T_{peak}$) (Goicoechea et al. 2016). There exists a
remarkably similar spatial distribution between the ${\rm H_{2}}$ and HCO+
emission from ALMA.

##  Appendix C NIRCam images of the north of the Dark Bay, north of M42 and
M43

In this Appendix, we present several prominent features arising in the images
within the fields centered on north of the Dark Bay, north of M42 and M43
obtained in the NIRCam parallel mode. Figs. 26, 27 and 29. show composite
images in three selected filters F187N, F335M and F470N in north of the Dark
Bay and F187N, F335M and F212N in north of M42 and M43. As in the field
centered on the Bar, the molecular cloud border appears quite structured.
Multiple patterns emerge such as ridges, waves, evaporating globules and bow-
shocks (Fig. 28). In Figs. 30-36, all the images obtained in the filters
listed in Table 1, and some filters continuum subtracted, are presented. Figs.
37, 38, 39, 40 show the proplyds detected in the north of the Dark Bay and
M43. New proplyd candidates identified in the NIRCam F187N images of the north
of the Dark Bay and M43 are given in Table 3.

###  C.1 North of the Dark Bay

The NIRCam images of the material North of the Dark Bay reveal that there are
linear bright features, such as the Bar (panels A and B in Figs. 26 and 28).
Some of these features were previously seen in extinction corrected HST images
(e.g., eastern and northern bright bar in Fig. 6 of O’Dell 2001b). The Dark
Bay (and similar structures) are foreground extinction of visible light (Fig.
1) but seen in emission in PDRs tracers, such as AIBs and H2 observed with
NIRCam, and [CII] cooling line emission from Herschel and SOFIA (Goicoechea et
al. 2015; Pabst 2021). The AIBs and H2 line emission show the edges of these
neutral warm regions with a very high level of detail. The bright PDR features
(panels A and B in Figs. 26 and 28), which are at a projected distance of the
order of 110$\arcsec$-120$\arcsec$ (0.22-0.24 pc) from the illuminating
O7-type star $\theta^{1}$ Ori C, are illuminated by a strong FUV radiation
field comparable to the one on the Bar. Goicoechea et al. (2015) studied with
Herschel/HIFI the [CII] velocity emission along with the CO J=2-1 line with
IRAM-30m and found that most of these PDRs lack detectable CO emission (CO-
dark molecular gas). In the northern-west part of the NIRCam map, there is a
zone devoid of nebula emission where background stars are well visible. This
is in agreement with the fact that no CO emission was previously detected and
that this region does not contain much dense molecular gas.

In the southwest corner of the continuum subtracted H2 0-0 S(9) 4.69 $\mu$m
line map (Fig. 32), several very bright features produced by shocks are
detected. No AIB emission or Pa $\alpha$ and [FeII] line emission are detected
on these features. The well studied outflow HH 513 lies in this region.
Furthermore, several arches (like a bent-over cylinder) which are most likely
bow-shocks are detected in the southest corner of field (see panel A in Figs.
26 and 28). Bow-shocks can also be the result of multiple outflows that are
close together in time like seen from the ground in clusters in various star-
forming regions (Plunkett et al. 2013; Nony et al. 2020).

###  C.2 North of M42

Here, we present images within the fields covered the north of M42 obtained in
the NIRCam parallel mode. As shown by the ionized gas line map in Fig. 27, the
north of M42 contains one main ionized gas region in the southwest part of the
map. This region is irradiated by the Trapezium cluster. At the edge of the
ionized region are the neutral zones, the PDRs. Particularly interesting PDRs
are the ones in panel C in Figs. 27 and 28 which are irradiated from the South
by star $\theta^{1}$ Ori C and from the north by the B0.5V star NU Ori in the
center of M43 (see Sect. C.3). As shown in Fig. 27, a remarkable spatial
correlation is observed between the F335M filter (AIBs) and the F212N filter
(vibrationally excited line H2 1-0 S(1)). Most likely, these PDRs are in a
lower $G_{0}/n_{H}$ regime than in the Bar. In fact, the incident FUV flux
from the south on that PDR can be estimated to be about 10 times lower than in
the Bar considering the projected distance from the star $\theta^{1}$ Ori C.
For lower $G_{0}/n_{H}$, the H0/H2 transition is not driven by dust opacity
but by H2 self shielding (e.g, Hollenbach & Tielens 1999), and the H2 emission
peak is expected to spatially coincide with the emission peak of the AIBs.
Spatial coincidence between H2 and AIB emission peaks has been observed in
several lower excited PDRs such as the $\rho$ Oph-W and Horsehead nebula PDR
(e.g., Habart et al. 2003, 2005). In high $G_{0}/n_{H}$ regime (as at the main
edge of the Bar), where the H0/H2 is driven by dust opacity and H2 is mostly
photo-dissociated in the outer PDR layers ($A_{V}<1$), H2 emission appears
after the emission peak of the AIBs. In the panel C of Fig. 28, a thin
structure (width of 1$\arcsec$), probably an Evaporating Gaseous Globule, is
detected at the PDR edge.

###  C.3 M43

As shown in Fig. 29, M43 contains one main ionized gas region. M43 exhibits an
expanding bubble structure, with an expansion velocity of 6 km s-1 Pabst et
al. (2020). Looking at the NIRCam images, the UV flux appears significantly
lower than in the inner Orion Nebula. The Pa $\alpha$ line is in fact much
less bright that in the inner region of the nebula. The emission in the F335M
filter is also significantly weaker. This is expected due to the much cooler
and less luminous exciting star NU Ori. Pabst et al. (2020, 2022) estimated
the local PDR physical conditions in M43, the incident radiation field $G_{0}$
($\sim 10^{3}$) and the gas density $n$ from the SED results and their [C II]
and CO data. In the rim of M43, Pabst et al. (2020) found $n=10^{4}$ cm-3 from
the [C II] and CO peak separation. Gas temperature estimated from [CII]
excitation temperature is about 100-120 K. Thus, the thermal pressure must be
around few $10^{6}$ K cm-3, 10 to 100 times lower than in the Bar. The PDRs
around M43 are bright in aromatic emission but quite faint in H2. This is
consistent with the fact that the density should not be too high ($n\leq
10^{4}$ cm-3). The conditions of the local PDR are compatible with the
conditions of pressure equilibrium with the ionized region using the results
of O’Dell & Harris (2010). From their M43 Samples A in Table 10, the averages
electronic temperature and density are $T_{e}=7950\pm 60$ K and $n_{e}=510\pm
40\leavevmode\nobreak\ $cm-3.

Figure 26: The North of the Dark Bay as seen by the JWST’s NIRCam instrument
with north up and east left. Several images in different filters were combined
to produce an RGB composite image as in Fig. 3: F335M (red), F470N (green) and
F187N (blue) that trace emission from hydrocarbons (AIBs), dust and molecular
gas (H2 0-0 S(9) line), and ionized gas (Pa $\alpha$ line), respectively. The
individual images used to make the RGB composite one are shown below. No
continuum was subtracted. The size of the images is $\sim$150$\arcsec\times
150\arcsec$ and it is centered on RA=05h35m21$\aas@@fstack{s}$09,
DEC=-05$\degr$21$\arcmin$40$\arcsec$01\. White boxes (labeled A, B) are
enlarged in Fig. 28.

Figure 27: North of M42 as seen by the JWST’s NIRCam instrument with north up
and east left. Several images in different filters were combined to produce an
RGB composite image : F335M (red), F212N (green) and F187N (blue) that traces
respectively emission from hydrocarbons (AIBs), molecular gas (H2 1-0 S(1)
line) and ionized gas (Pa $\alpha$ line), respectively. The individual images
used to make the RGB composite one are shown below. No continuum was
subtracted. The image field was chosen to have an overlap between the
different individual images. The size of the images is $\sim$135$\arcsec\times
145\arcsec$ and it is centered on RA=05h35m29$\aas@@fstack{s}$12,
DEC=-05$\degr$20$\arcmin$3.66$\arcsec$. White boxes (labeled C, D) are
enlarged in Fig. 28.

Figure 28: Zoom into the F335M image areas shown in Figs. 26 (for panels A, B)
and 27 (for panels B, C). A Irradiated molecular edges showing many patterns
such as ridges and arches (which are most likely bow-shocks) detected in the
southeast area of M42 North. B Irradiated molecular edges detected in the
north area of M42 North. C Irradiated molecular edges showing many patterns
such as ridges, waves and evaporating gaseous globule detected in the
northwest area of the M43 South field. D Features driven by a large outflow
detected in the south area of the M43 South field.

Figure 29: The M43 region as seen by the JWST’s NIRCam instrument with north
up and east left. Several images in different filters were combined to produce
an RGB composite image : F335M (red), F212N (green) and F187N (blue) that
traces respectively emission from hydrocarbons (AIBs), molecular gas (H2 1-0
S(1) line) and ionized gas (Pa $\alpha$ line), respectively. The individual
images used to make the RGB composite one are shown below. No continuum was
subtracted. The size of the images is $\sim$158$\arcsec\times 158\arcsec$ and
it is centered on RA=05h35m28$\aas@@fstack{s}$92,
DEC=-05$\degr$17$\arcmin$04.31$\arcsec$.  Figure 30: NIRCam maps of the North
region of the Orion Nebula in different filters. Units are in MJy/sr. Figure
31: Same as in Fig. 30, but for other filters. Figure 32: Same as in Fig. 30,
but for other filters. Figure 33: NIRCam maps of the M43 South region in
different filters. Units are in MJy/sr. Figure 34: Same as in Fig. 33, but for
other filters. Figure 35: NIRCam maps of the M43 center region in different
filters. Units are in MJy/sr. Figure 36: Same as in Fig. 35, but for other
filters. Figure 37: Zoom on the proplyds with extended structure identified in
the F187N NIRCam image of the M42 north-east region (module A). The tiles are
$5\arcsec\times 5\arcsec$, with exception of the object 208-122 for which
tiles are $8\arcsec\times 8\arcsec$, with north is up and east to the left.
Some images suffer from instrumental effects such as the diffraction pattern
from bright stars and uncorrected cosmic ray events, affecting particularly
the edge of the images because of lack of redundancy.  Figure 38: Zoom on the
3 new bright proplyds identified in the F187N NIRCam image north-east of the
Trapezium stars (module A): 171-212, 180-218 and 234-104. They have faint
cusps and only one shows a tail in Pa$\alpha$, that is proplyd 171-212. The
tiles are $5\arcsec\times 5\arcsec$ with north up and east to the left.
Figure 39: Zoom on the giant proplyd 332-1605 found in M43 (module A) and
pointing directly to the ionizing star NU Ori, just $27\arcsec$ to the west
and not visible in the images. This object shows a long tail in Pa$\alpha$
that is nearly 10 times larger than proplyd HST10. The H$\alpha$ HST/ACS image
of this object does not show any tail but only a faint cusp. The NIRCam F187N
(left) and HST F1658N (right) images have similar spatial resolution allowing
for a direct comparison of proplyd morphology in the optical and near-IR.
North is up and east to the left. Figure 40: Zoom on the new proplyd found in
the M43 north region (detector A) at $97\arcsec$ southwest of the ionizing
star NU Ori (top figure). This object also shows a prominent jet identified by
a chain of knots visible in both Pa$\alpha$ and H$\alpha$ images (bottom
figure). However, the central star is only visible in the near-IR, reflecting
the youth of this object. The JWST/NIRCam F187N images in the left and the
HST/ACS F658N images in the right have similar spatial resolution. North is up
and east to the left. Table 3: New proplyd candidates identified in the NIRCam
F187N images of M42 north and M43 (module A).

Proplyda | $\alpha_{J2000}$ | $\delta_{J2000}$ | Region | Comments | 2MASS
---|---|---|---|---|---
$171-212$ | 05 35 17.14 |  $-05$ 22 11.9 | M42 | bright cusp with tail | no
$180-218$ | 05 35 18.04 |  $-05$ 22 18.2 | M42 | near to filament, no tail | no
$234-104$ | 05 35 23.37 |  $-05$ 21 03.7 | M42 | faint cusp, no tail | no
$269-1713$ | 05 35 26.87 |  $-05$ 17 12.9 | M43 | bipolar jet, HH objects | yes, star

  * a

Proplyd name following O’Dell & Wen (1994) coordinate-based convention.