# The gravitational cat on the Cauchy horizon

Christiane Klein<EMAIL_ADDRESS>Institut für
Theoretische Physik, Universität Leipzig, Brüderstraße 16, 04103 Leipzig,
Germany  Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France  AGM, CY Cergy
Paris Université, 2 av. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France
Jochen Zahn<EMAIL_ADDRESS>Institut für Theoretische Physik,
Universität Leipzig, Brüderstraße 16, 04103 Leipzig, Germany

###### Abstract

We show that the stress tensor of a real scalar quantum field on Reissner-
Nordström-deSitter spacetime exhibits correlations over macroscopic distances
near the Cauchy horizon. These diverge as the Cauchy horizon is approached and
are universal, i.e., state-independent. This signals a breakdown of the semi-
classical approximation near the Cauchy horizon. We also investigate the
effect of turning on a charge of the scalar field and consider the correlation
of the stress tensor between the two poles of the Cauchy horizon of Kerr-de
Sitter spacetime.

##  I Introduction

On Minkowski space and in the vacuum state, the correlations
${\langle\hat{T}_{\alpha\beta}(x)\hat{T}_{\gamma\delta}(x^{\prime})\rangle}$
of stress tensor components of a massless field at space-like separated points
$x$, $x^{\prime}$ fall off as $d^{-8}$ with the distance $d$. For massive
fields, the fall-off is even exponential. Hence, correlations of the stress
tensor over macroscopic distances are negligible. That correlations of the
stress tensor are negligible is one of the (not always outspoken) assumptions
underlying the semi-classical Einstein equation (here $\Lambda$ is the
cosmological constant, $T^{\mathrm{class}}_{\mu\nu}$ the stress tensor of
classical matter and $\Psi$ the state of the quantum matter)

$G_{\mu\nu}+\Lambda
g_{\mu\nu}=8\pi\left(T^{\mathrm{class}}_{\mu\nu}+{\langle\hat{T}_{\mu\nu}\rangle}_{\Psi}\right).$
(1)

An example of a situation in which correlations of the stress tensor over
macroscopic distances are not negligible would be a quantum superposition of
two macroscopically different spatial distributions of quantum matter, also
called a _gravitational cat_ state. Such superpositions played a crucial role
in Feynman’s famous Gedanken experiment Feynman (2011) which was used to argue
that in a consistent theory comprising gravity and quantum matter also the
gravitational field must be quantized. The analysis of this and similar
Gedanken experiments has been a major activity in quantum gravity research in
recent years Anastopoulos and Hu (2015); Bose  _et al._ (2017); Marletto and
Vedral (2017); Belenchia  _et al._ (2018); Danielson  _et al._ (2022) and the
actual realization and study of gravitational cat states is a key goal of
experimental studies of quantum gravity Carlesso  _et al._ (2019); Carney  _et
al._ (2022).

In quantum field theory on curved spacetimes, correlations of quantum fields
over macroscopic distances are crucial in inflationary cosmology Linde (1990),
where they provide the seeds of cosmological structure formation.

In the following, we show that also near the Cauchy horizon inside black
holes, there are correlations of the stress tensor over macroscopic distances.
These are generically of the same order as the square of the expectation value
of the stress tensor, and in particular they diverge as the Cauchy horizon is
approached. Moreover this behaviour is universal, i.e., the leading divergence
of the correlations is independent of the quantum state. Hence, the occurrence
of gravitational cat states (defined as states with non-negligible
correlations of the stress tensor over macroscopic distances) on the Cauchy
horizon of a black hole is a robust prediction of quantum field theory on
curved spacetime.

All stationary black hole solutions (with the exception of non-rotating,
uncharged black holes) possess a Cauchy horizon in their interior. While the
metric can be smoothly extended beyond this Cauchy horizon, the extension is
non-unique, as is the extension of any other field subject to hyperbolic field
equations. Hence, the occurrence of a Cauchy horizon signals the breakdown of
predictivity. It was conjectured by Penrose Penrose (1974) that this breakdown
of predictivity is not generic, i.e., under generic perturbations of the
gravitational and/or matter fields, the Cauchy horizon should become singular.
This _strong cosmic censorship_ (sCC) conjecture motivates the study of
classical McNamara (1978); Poisson and Israel (1990); Ori (1991); Mellor and
Moss (1990, 1992); Brady  _et al._ (1998); Luk and Oh (2017); Hintz and Vasy
(2017); Dafermos and Shlapentokh-Rothman (2017); Dafermos and Luk (2017);
Cardoso  _et al._ (2018a); Dafermos and Shlapentokh-Rothman (2018); Dias  _et
al._ (2018, 2019a); Cardoso  _et al._ (2018b) and quantum Birrell and Davies
(1978); Dias  _et al._ (2019b); Zilberman  _et al._ (2020); Hollands  _et al._
(2020a, b); Klein  _et al._ (2021); Zilberman and Ori (2021); Zilberman  _et
al._ (2022) fields near a Cauchy horizon.

For our study, we mostly consider a scalar (charged or uncharged) field on
Reissner-Nordström-de Sitter (RNdS) spacetime, describing a static charged
black hole in a spacetime with positive cosmological constant. This choice is
motivated both by physical and practical considerations: From a practical
point of view, RNdS has the advantage of spherical symmetry as well as having
a further (cosmological) horizon at a finite radius $r$, which simplifies the
computation of the required scattering coefficients.

But RNdS is also interesting from a conceptual point of view, as (the
Christodoulou formulation Christodoulou (2009) of) sCC can be violated in that
case Cardoso  _et al._ (2018a); Dias  _et al._ (2019a); Cardoso  _et al._
(2018b), i.e., the classical stress tensor diverges weaker than $V^{-1}$ at
the Cauchy horizon. Here $V$ is a (Kruskal) coordinate with which one can
smoothly extend the RNdS metric beyond the Cauchy horizon, situated at $V=0$.
In contrast, for quantum fields, one finds the leading divergence

${\langle\hat{T}_{VV}\rangle}\sim CV^{-2}$ (2)

for the expectation value of the renormalized stress tensor component
$\hat{T}_{VV}$ near the Cauchy horizon Zilberman  _et al._ (2020); Hollands
_et al._ (2020a), with a universal (state-independent) coefficient $C$
Hollands  _et al._ (2020a); Hintz and Klein (2023).

In this work, we will consider the correlations

$\Delta\hat{T}_{VV}(\delta\theta):={\langle\hat{T}_{VV}(\theta)\hat{T}_{VV}(\theta+\delta\theta)\rangle}_{{\mathrm{U}}}-{\langle\hat{T}_{VV}\rangle}_{{\mathrm{U}}}^{2}$
(3)

of $\hat{T}_{VV}$ in the Unruh state at angular separation $\delta\theta$ near
the Cauchy horizon (note that we here used the spherical symmetry to simplify
the last term on the r.h.s.). We find that for the uncharged scalar field on
near-extremal RNdS

$\Delta\hat{T}_{VV}(\delta\theta)\sim D(\delta\theta)V^{-4},$ (4)

with a non-negative coefficient $D(\delta\theta)$ which is i) universal
(state-independent), ii) essentially flat (independent of $\delta\theta$)
except near spacetime parameters where the coefficient $C$ of (2) vanishes,
and iii) related to $C$ as $\lim_{\delta\theta\to 0}D(\delta\theta)=2C^{2}$.

This implies that the correlations of $\hat{T}_{VV}$ are of the same order as
(the square of) its expectation value and that these strong correlations exist
over macroscopic distances (the whole Cauchy horizon). This puts the
applicability of the semi-classical Einstein equation into question, as the
fluctuations of the stress tensor can not be considered as small compared to
the expectation value (of course also the divergence (2) of the expectation
value indicates a breakdown of the semi-classical Einstein equation for $V\to
0$, but for the present argument we do not need to consider this limit).

We also see that for spacetime parameters such that the coefficient $C$ of (2)
vanishes we nevertheless still have (4) with $D(\delta\theta)\neq 0$ (except
in the limit $\delta\theta\to 0$) and taking values which are of the same
order as $C^{2}$ for “nearby” spacetime parameters. Hence, even if $C=0$, in a
“typical realization” a measurement of $\hat{T}_{VV}$ will yield a result of
order $V^{-2}$, only that for repeated measurements positive and negative
results cancel on average.

For the charged scalar field, one still finds (4), but for increasing charge
$q$ of the field $D(\delta\theta)$ is more and more localized near
$\delta\theta=0$, i.e., correlations over macroscopic distances are suppressed
for large enough values of $q$. This result calls into question the genericity
of the result obtained for the uncharged scalar field (i.e., correlations over
macroscopic distances). As realistic black holes are (essentially) not charged
but rotating, this motivates us to finally consider the case of Kerr-de Sitter
(KdS) spacetime. Since all fields “couple” to the angular momentum (in a
fashion quite analogous to the “coupling” of a charged field to the charge of
the black hole), it is conceivable (by analogy to the charged case) that
correlations over macroscopic distances are suppressed in that case. For
simplicity, we consider the scalar field with “conformal mass”, which can be
seen as a proxy for the electromagnetic field, and calculate the correlation
between the north and the south pole. We again find correlations which are of
the same order as the (square of the) expectation value. Hence, also in this
case, there are correlations over macroscopic distances near the Cauchy
horizon, so that one can consider the occurrence of these as a generic feature
of (essentially massless) quantum fields in black hole spacetimes with a
Cauchy horizon.

The paper is structured as follows: In section II, we introduce the Reissner-
Nordström-de Sitter spacetime. The scalar quantum field will be introduced in
section III. In this section, we will also derive a formula for the
correlation of the energy flux at the Cauchy horizon based on the results of
Hollands  _et al._ (2020a, b). We will present and discuss the numerical
results in section IV. Finally, we conclude in section V.

##  II Reissner-Nordström-de Sitter spacetime

Reissner-Nordström-de Sitter (RNdS) spacetime describes a charged, spherically
symmetric black hole in the presence of a cosmological constant. It is
characterized by the metric

$\displaystyle g$ $\displaystyle=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\Omega^{2}$
(5) $\displaystyle f(r)$
$\displaystyle=-\frac{\Lambda}{3}r^{2}+1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\,.$
(6)

Here, $d\Omega^{2}$ is the area element of the unit 2-sphere. The metric is
determined by three constants: the cosmological constant $\Lambda$, the mass
$M$ and the charge $Q$ of the black hole. For simplicity, one can rescale to
$M=1$, reducing the free spacetime parameters to $Q/M$ and $\Lambda M^{2}$. In
the subextremal regime, the constants are chosen such that $f(r)$ has three
distinct positive roots $r_{-}<r_{+}<r_{c}$. These roots determine the
location of the Cauchy $(\mathcal{CH})$, event $(\mathcal{H})$ and
cosmological horizon $(\mathcal{H}_{c})$. The black hole exterior consists of
region ${\mathrm{III}}=\mathbb{R}_{t}\times(r_{c},\infty)\times\mathbb{S}^{2}$
beyond the cosmological horizon and region
${\mathrm{I}}=\mathbb{R}_{t}\times(r_{+},r_{c})\times\mathbb{S}^{2}$ causally
connected to the black hole, as well as the horizon $\mathcal{H}_{c}^{L}$
connecting them. The black hole interior up to the Cauchy horizon, region
${\mathrm{II}}=\mathbb{R}_{t}\times(r_{-},r_{+})\times\mathbb{S}^{2}$, is
connected to region ${\mathrm{I}}$ along the event horizon $\mathcal{H}^{R}$.
It is connected to the region ${\mathrm{IV}}$ containing the singularity by
the Cauchy horizon $\mathcal{CH}^{R}$. A Penrose diagram of the spacetime
under consideration is shown in Fig. 1.

Figure 1: The Penrose diagram of the subextremal RNdS spacetime. The regions
${\mathrm{I}}$ and ${\mathrm{III}}$ make up the black hole exterior. The black
hole interior consists of ${\mathrm{II}}$ and ${\mathrm{IV}}$. The green
surface $\Sigma$ indicates a Cauchy surface for
${\mathrm{I}}\cup{\mathrm{II}}\cup{\mathrm{III}}$. The blue arrows show the
directions of increasing $u$ and $v$ in ${\mathrm{I}}$ and ${\mathrm{II}}$,
while the red arrow points towards increasing $V$.

For the computation of the energy flux, it is convenient to introduce a double
null coordinate system akin to Eddington-Finkelstein coordinates. As a first
step, we introduce the tortoise coordinate $r_{*}$, defined by $fdr_{*}=dr$.
One can then define $u=t-r_{*}$ and $v=t+r_{*}$. These coordinates will run
over $\mathbb{R}$ in each of the regions ${\mathrm{I}}$, ${\mathrm{II}}$, and
${\mathrm{III}}$ separately. To cover the horizons, one can introduce Kruskal-
type coordinates. The null coordinate that is regular across the Cauchy
horizon $\mathcal{CH}^{R}$ is defined in region ${\mathrm{II}}$ by

$\displaystyle
V=-e^{-\kappa_{-}v}\,,\quad\kappa_{i}=\frac{1}{2}\left|f^{\prime}(r_{i})\right|\,,$
(7)

where the prime denotes a derivative with respect to $r$. Then $V=0$ on
$\mathcal{CH}^{R}$. These coordinates are also indicated in Fig. 1.

The other Kruskal coordinates relevant to this work are defined in region
${\mathrm{I}}$ by

$\displaystyle U=-e^{-\kappa_{+}u}\quad\text{and}\quad
V_{c}=-e^{-\kappa_{c}v}\,.$ (8)

##  III Correlations of the energy flux

We will mostly consider a real, minimally coupled massive scalar field of
“conformal” mass $\mu^{2}=2\Lambda/3$, i.e., satisfying the Klein-Gordon
equation

$\displaystyle(\nabla^{\nu}\nabla_{\nu}-\mu^{2})\phi=0.$ (9)

In terms of the differential equation, this is the same as the one for a
massless, conformally coupled scalar field on RNdS (or KdS). This ambiguity
also leads to an ambiguity in the classical stress-energy tensor. We will
stick to the stress-energy tensor obtained for the minimally coupled scalar
field, even though we will sometimes refer to this particular choice of mass
as a ”conformally coupled” scalar.

For such a scalar field, the classical stress-energy tensor is given by

$\displaystyle T_{\nu\rho}(x)$
$\displaystyle={\partial}_{\nu}\phi(x){\partial}_{\rho}\phi(x)$ (10)
$\displaystyle-\frac{1}{2}g_{\nu\rho}\left({\partial}_{\alpha}\phi(x){\partial}^{\alpha}\phi(x)-\mu^{2}\phi^{2}(x)\right)\,.$

In the quantum theory, this needs to be renormalized, since it is quadratic
and local in the quantum field. According to the principles of quantum field
theory on curved spacetime Hollands and Wald (2015), this must be done in a
local and covariant way, i.e., by Hadamard point split renormalization (up to
possible finite renormalizations, which are irrelevant for the divergent
behaviour near the Cauchy horizon). In this scheme, the renormalized
expectation value of a Wick square with any number of derivatives acting on it
is given by

$\displaystyle{\langle{\partial}^{\alpha}\phi(x){\partial}^{\beta}\phi(x)\rangle}^{\mathrm{ren}}_{\Psi}$
(11) $\displaystyle=\lim\limits_{x^{\prime}\to
x}{\partial}^{\alpha}_{x}{\partial}^{\beta}_{x^{\prime}}\left({\langle\phi(x)\phi(x^{\prime})\rangle}_{\Psi}-H(x,x^{\prime})\right),$

with $\alpha,\beta$ multi-indices. Here, $H(x,x^{\prime})$ is the Hadamard
parametrix for the Klein-Gordon operator $\nabla^{\nu}\nabla_{\nu}-\mu^{2}$,
which is determined by the local geometry of the spacetime. If the state
$\Psi$ is Hadamard, i.e., with a two-point function whose singularities for
$x^{\prime}\to x$ agree with that of $H(x,x^{\prime})$, then the resulting
expectation value will be finite. Thus, one may write the operator-valued
distribution for the $vv$-component of the renormalized stress-energy tensor
(henceforth referred to as the _energy flux_) on RNdS (somewhat formally) as

$\displaystyle\hat{T}_{vv}(x)=\lim\limits_{x^{\prime}\to
x}\left({\partial}_{v}\phi(x){\partial}_{v}\phi(x^{\prime})-{\partial}_{v}{\partial}_{v}^{\prime}H(x,x^{\prime})\mathbf{1}\right).$
(12)

We are interested in the correlations of this operator, defined by

$\Delta\hat{T}_{vv}(x,y)_{\Psi}={\langle\hat{T}_{vv}(x)\hat{T}_{vv}(y)\rangle}_{\Psi}-{\langle\hat{T}_{vv}(x)\rangle}_{\Psi}{\langle\hat{T}_{vv}(y)\rangle}_{\Psi}.$
(13)

If the state $\Psi$ is quasi-free (Gaussian), as is the case for the Unruh
state considered below, then one finds, using Wick’s formula,

$\Delta\hat{T}_{vv}(x,y)_{\Psi}=2\left({\langle{\partial}_{v}\phi(x){\partial}_{v}\phi(y)\rangle}_{\Psi}\right)^{2}.$
(14)

We would like to compute this correlation for the Unruh state on the Cauchy
horizon $\mathcal{CH}^{R}$ of RNdS.

The Unruh state for the real scalar field on RNdS is Hadamard in
${\mathrm{I}}\cup{\mathrm{II}}\cup{\mathrm{III}}$, i.e., up to the Cauchy
horizon Hollands  _et al._ (2020a). It can be defined by expanding the quantum
scalar field in terms of mode solutions to the Klein-Gordon equation (9),

$\displaystyle\phi(x)=\sum\limits_{\lambda,\ell,m}\int\limits_{0}^{\infty}\phi_{\omega\ell
m}^{\lambda}(x)a^{\lambda}_{\omega\ell m}+\overline{\phi}_{\omega\ell
m}^{\lambda}(x)a^{\lambda\dagger}_{\omega\ell m}\,\text{d}\omega,$ (15)

where the $a^{\lambda}_{\omega\ell m}$ and $a^{\lambda\dagger}_{\omega\ell m}$
are creation and annihilation operators. The mode solutions
$\phi^{\lambda}_{\omega\ell m}$ are chosen such that they together with their
complex conjugates form a basis for the space of smooth, space-like compact
solutions to (9). The index $\lambda$ runs over two families of such modes,
called ”in”- and ”up”- modes, while $\ell$ and $m$ are the usual angular
quantum numbers. The modes are also required to satisfy the usual symplectic
normalization condition.

For the Unruh state, the modes are characterized by their asymptotic behaviour
near
$\mathcal{H}^{L}\cup\mathcal{H}^{-}\cup\mathcal{H}_{c}^{-}\cup\mathcal{H}_{c}^{R}$
(see Fig. 1). The ”up”- modes vanish near
$\mathcal{H}_{c}^{-}\cup\mathcal{H}_{c}^{R}$, and behave like

$\displaystyle\phi_{\omega\ell
m}^{\mathrm{up}}\sim(4\pi|\omega|)^{-\tfrac{1}{2}}r^{-1}Y_{\ell
m}(\theta,\varphi)e^{-i\omega U}$ (16)

near $\mathcal{H}^{L}\cup\mathcal{H}^{-}$, while the ”in”-modes vanish near
$\mathcal{H}^{L}\cup\mathcal{H}^{-}$ and behave like

$\displaystyle\phi_{\omega\ell
m}^{\mathrm{in}}\sim(4\pi|\omega|)^{-\tfrac{1}{2}}r^{-1}Y_{\ell
m}(\theta,\varphi)e^{-i\omega V_{c}}$ (17)

near $\mathcal{H}_{c}^{-}\cup\mathcal{H}_{c}^{R}$. Here, $Y_{\ell
m}(\theta,\varphi)$ are spherical harmonics. The Unruh state is a stationary
state modelling the final state of a quantum field in a gravitational collapse
spacetime.

For the evaluation of the correlations at the Cauchy horizon, we will pick $x$
as any point on $\mathcal{CH}^{R}$, and $y=x+\delta\theta$ separated from $x$
in the $\theta$-direction. Thanks to the spherical symmetry of the Unruh
state, the correlations will be independent of the point $x$ and will only
depend on $\delta\theta$.

As the Unruh state is stationary, we can also compute the correlations on
$\mathcal{CH}^{L}$, which turns out to be advantageous. To explain this, we
first recall the calculation of the expectation value
${\langle\hat{T}_{vv}\rangle}_{{\mathrm{U}}}$ on $\mathcal{CH}^{R}$ as
performed in Hollands  _et al._ (2020a). There, a further “comparison” state
${\langle\cdot\rangle}_{{\mathrm{C}}}$ is introduced, which is defined by
final data on $\mathcal{CH}^{L}\cup\mathcal{CH}^{+}$ (see Fig. 1), and which
is Hadamard in ${\mathrm{II}}\cup{\mathrm{IV}}$, i.e., across the Cauchy
horizon $\mathcal{CH}^{R}$. Due to the latter property, the renormalized
expectation value of $\hat{T}_{vv}$ in this state must vanish at
$\mathcal{CH}^{R}$, so that instead of
${\langle\hat{T}_{vv}\rangle}_{\mathrm{U}}$ one can consider
${\langle\hat{T}_{vv}\rangle}_{\mathrm{U}}-{\langle\hat{T}_{vv}\rangle}_{\mathrm{C}}$.
The advantage is that the Hadamard parametrix drops out in this difference and
the result can be computed (on $\mathcal{CH}^{L}$) in terms of a mode integral
as

$\displaystyle{\langle\hat{T}_{vv}\rangle}_{\mathrm{U}}$
$\displaystyle=\sum_{\ell=0}^{\infty}T^{(\ell)}_{vv},$ (18) $\displaystyle
T_{vv}^{(\ell)}$
$\displaystyle=\frac{2\ell+1}{16\pi^{2}r_{-}^{2}}\int\limits_{0}^{\infty}\,\text{d}\omega\,\omega
n_{\ell}(\omega),$ (19)

with $n_{\ell}(\omega)$ a function of the scattering coefficients of the so-
called Boulware modes on the RNdS spacetime, whose explicit form is given in
(Hollands  _et al._ , 2020a, eq. (123)).

Returning to the correlations (14), we note that in principle these do not
require any renormalization for spacelike separated $x$, $y$ (as is the case
for $\delta\theta\neq 0$ on $\mathcal{CH}^{R}$). However, the necessary mode
integral (and sum over $\ell$) is only weakly convergent. To improve
convergence, we again use stationarity to evaluate the expression on
$\mathcal{CH}^{L}$ and we subtract the correlation of $\hat{T}_{vv}$ in the
comparison state. This does not alter the result for $\delta\theta\neq 0$, as,
on $\mathcal{CH}^{L}$,
${\langle{\partial}_{v}\phi(x){\partial}_{v}\phi(x+\delta\theta)\rangle}_{\mathrm{C}}=0$
for $\delta\theta\neq 0$ (a “blind spot” in the terminology of Levi and Ori
(2016)), but it greatly enhances the convergence of the mode integral. In
fact, one finds

$\Delta\hat{T}_{vv}(\delta\theta)_{\mathrm{U}}=2\left(\sum_{\ell=0}^{\infty}P_{\ell}(\cos\delta\theta)T^{(\ell)}_{vv}\right)^{2},$
(20)

with $P_{\ell}(x)$ the Legendre polynomials and $T^{(\ell)}_{vv}$ as defined
in (19). In particular, we see that the angular dependence comes from the
terms with $\ell>0$.

From the result for $\Delta\hat{T}_{vv}(\delta\theta)_{\mathrm{U}}$, one
straightforwardly obtains (with the tensor transformation law) the correlation

$\Delta\hat{T}_{VV}(\delta\theta)\sim\kappa_{-}^{-4}V^{-4}\Delta\hat{T}_{vv}(\delta\theta)_{\mathrm{U}}$
(21)

near the Cauchy horizon in adapted (Kruskal) coordinates, which determines the
coefficient $D(\delta\theta)$ of (4). In particular, these diverge strongly as
the Cauchy horizon is approached. Furthermore, using the arguments of Hollands
_et al._ (2020a); Hintz and Klein (2023), one sees that this result is
universal, i.e., the coefficient $D(\delta\theta)$ of the leading divergence
is the same for all states which are Hadamard in
${\mathrm{I}}\cup{\mathrm{II}}\cup{\mathrm{III}}$.

##  IV Numerical results

We now present numerical results for
$\Delta\hat{T}_{vv}(\delta\theta)_{\mathrm{U}}$. A method for the numerical
computation of the integrand $n_{\ell}(\omega)$ with Mathematica has been
developed in Hollands  _et al._ (2020b). When restricting to a conformally
coupled scalar field, the computation simplifies slightly, since one can
employ built-in functions of Mathematica, see Hollands  _et al._ (2020b);
Klein  _et al._ (2021), and the discussion in Hatsuda (2020). We will focus on
the regime of large $Q$, since this is where sCC is violated classically. We
will also set $\Lambda M^{2}=0.02$, which allows us to compare our results
with Hollands  _et al._ (2020b).

First, we investigate the behaviour of $T_{vv}^{(\ell)}$, recall (19), for
different values of $\ell$ and $Q/M$. The results are shown in Fig. 2.

Figure 2: The contribution $T^{(\ell)}_{vv}$ of different $\ell$-modes to the
expectation value of the energy flux ${\langle T_{vv}\rangle}_{U-C}$ of a
real, minimally coupled scalar field with mass $\mu^{2}=2\Lambda/3$. The
cosmological constant is set to $\Lambda M^{2}=0.02$.

We see that the $\ell=0$ term is dominant for most values of $Q/M$, except for
a small range around $Q_{0}/M\simeq 0.966$, at which $T_{vv}^{(0)}=0$.
Therefore, apart from this small region, the behaviour of the expectation
value ${\langle\hat{T}_{vv}\rangle}_{{\mathrm{U}}}$ is essentially determined
by the $\ell=0$ mode. Furthermore, the neighbourhood of $Q_{0}$ also contains
$Q_{*}/M\simeq 0.9666$, at which ${\langle\hat{T}_{vv}\rangle}_{{\mathrm{U}}}$
vanishes due to a cancellation of $T_{vv}^{(0)}$ and the higher $\ell$-modes,
mostly $T_{vv}^{(1)}$.

Due to the dominance of the $\ell=0$ mode, one expects that the dependence of
$\Delta\hat{T}_{vv}(\delta\theta)_{\mathrm{U}}$ on $\delta\theta$ is quite
weak except for the regime near $Q_{0}$.

Figure 3: Correlations of the energy flux at the inner horizon at angular
separation $\delta\theta$ for different values of $Q/M$ and $\Lambda
M^{2}=0.02$. The red line marks $Q_{*}$ at which the energy flux vanishes. The
blue line represents twice the square of the flux as a function of $Q/M$.

This is verified in Fig. 3, where we see that the spectrum of fluctuations is
essentially flat, i.e., independent of $\delta\theta$, and coincides
approximately with $2{\langle T_{vv}\rangle}_{{\mathrm{U}}}^{2}$. From (21) it
follows that the correlation of $\hat{T}_{VV}$ in the Unruh state will diverge
as $V^{-4}$ as the Cauchy horizon is approached, with a coefficient of the
same order of magnitude as that of the leading divergence of
${\langle\hat{T}_{VV}\rangle}_{{\mathrm{U}}}^{2}$. We thus see strong
fluctuations of the stress tensor which are correlated over macroscopic
distances.

The only parameter region where a stronger dependence of the correlation on
$\delta\theta$ is expected is near $Q_{*}$. Therefore, we focus on the regime
around $Q_{*}$ in Fig. 4.

Figure 4: Correlations of the energy flux at the inner horizon at angular
separation $\delta\theta$ for $\Lambda M^{2}=0.02$ around $Q_{*}$. The red
line marks $Q_{*}$ at which the energy flux vanishes. The blue line represents
twice the square of the flux as a function of $Q/M$.

There, $Q_{*}$, where ${\langle\hat{T}_{vv}\rangle}_{{\mathrm{U}}}$ vanishes,
is indicated by a red line. We see that even at this parameter value, the
correlations of $\hat{T}_{vv}$ at different angular separations are non-zero
and of a size similar to ${\langle\hat{T}_{vv}\rangle}_{{\mathrm{U}}}^{2}$ at
other (nearby) values of $Q$. This means that even at $Q_{*}$ the typical
realization of $\hat{T}_{vv}$ in a single measurement must be of the same
order of magnitude as for other nearby values of $Q$. The positive and
negative measurement results only cancel out on average. Thus, even if the
leading divergence of the expectation value of the energy flux vanishes for
this particular choice of parameters, one would expect that in a typical
realisation the quantum effects will still lead to a quadratic divergence of
the stress tensor and thereby restore sCC.

It is also interesting to note that at $Q_{*}$, the correlations at large
angular separation are much larger than at small angular separation. This is
counter-intuitive, since one would usually expect the correlations to decay
with the separation.

To see how generic our finding of strong correlations of the energy flux over
macroscopic distances near the Cauchy horizon is, we study two variations of
the above. The first one is to turn on a charge of the scalar field. This
indeed alters the picture substantially. First of all, if the charge of the
scalar field is sufficiently large, the sign change of $T_{vv}^{(0)}$ as a
function of $Q/M$ that we observed for the uncharged field in Fig. 2 is
absent, and therefore also the sign change in
${\langle\hat{T}_{vv}\rangle}_{\mathrm{U}}$. Second, as the charge of the
scalar field is increased, the relative size of the higher-$\ell$ modes,
$T_{vv}^{(\ell)}$ with $\ell\geq 1$, compared to $T_{vv}^{(0)}$, increases.
This leads to a stronger dependence of the correlations on $\delta\theta$.
This can be observed in Fig. 5, where
$\Delta\hat{T}_{vv}(\delta\theta)_{\mathrm{U}}$ (normalized with
${\langle\hat{T}_{vv}\rangle}_{\mathrm{U}}^{2}$) is shown as a function of
$\delta\theta$ for different values of the field charge $q$ (for a fixed
choice of spacetime parameters $\Lambda$ and $Q$). We see that as $qQ$
increases, the correlations start to localize stronger around
$\delta\theta=0$.

Figure 5: The correlation of the renormalized stress-energy tensor, normalized
to the square of the expectation value, as a function of $\delta\theta$ for
different values of the scalar field charge $qQ$ at $Q/M=1$ and $\Lambda
M^{2}=0.14$. For larger $qQ$, the range of the correlations begins to shrink.

The study of quantum fields on RNdS has practical advantages, but
astrophysical black holes are expected to have significant angular momentum,
but negligible charge. Hence, we finally consider the case of a real scalar
field (of conformal mass) on KdS, describing a rotating black hole in the
presence of a positive cosmological constant. Due to the lack of spherical
symmetry, this is considerably more involved than RNdS. Results for the
expectation values of $\hat{T}_{vv}$ (and $\hat{T}_{v\phi}$) on Kerr (KdS)
were recently obtained in Zilberman  _et al._ (2022) (Klein  _et al._ (2023)).
For simplicity, we here restrict to the correlation between the north and the
south pole. This has the practical advantage that only the azimuthal quantum
number $m=0$ needs to be considered. Using the results of Klein  _et al._
(2023), one obtains the correlations shown in Fig. 6. We see that the
correlation between the two poles is of the same order of magnitude as
${\langle\hat{T}_{vv}\rangle}_{{\mathrm{U}}}^{2}$ at the pole, except around
the point at which ${\langle\hat{T}_{vv}\rangle}_{{\mathrm{U}}}^{2}$ vanishes,
similar to the real scalar field on RNdS. Hence, there indeed are strong
correlations over macroscopic distances, also near the Cauchy horizon of KdS.

Figure 6: The correlation of the renormalized stress-energy tensor between the
two poles for a real conformally coupled scalar field on Kerr-de Sitter as a
function of the black-hole angular momentum $a/M$ for $\Lambda M^{2}=1/270$.
The correlations are of the same order as the square of the expectation value,
except around $a/M=0.7$, where ${\langle\hat{T}_{vv}\rangle}(\theta=0)$
changes its sign (for $a/M=0.75$, the error bar is omitted, due to the small
denominator).

An intuitive understanding for this finding can easily be given: At the poles,
only the $m=0$ mode is relevant, which does not couple to the angular momentum
of the black hole, analogously to the real scalar field not coupling to the
black hole charge. Hence the similarity to the behaviour of the real scalar on
RNdS. However, we also verified the existence of strong correlations (of the
order of the square of the expectation value) away from the poles (for pairs
of points related by reflection at the equatorial plane). Quite generally, we
expect strong correlations of the energy flux over macroscopic distances near
the Cauchy horizon whenever only a small number of $\ell$\- and $m$ modes
contribute to the expectation value of the energy flux. This is expected to no
longer hold for fields of a sizeable mass.

##  V Conclusion

We have seen that there are strong (divergent as $V^{-4}$) correlations of the
component $\hat{T}_{VV}$ of the stress tensor near the Cauchy horizon which,
in some cases, extend over macroscopic distances and are of the same order as
the square ${\langle\hat{T}_{VV}\rangle}^{2}$ of the expectation value. This
in particular applies for the real (or only weakly charged) scalar field (with
conformal mass $\mu^{2}=2\Lambda/3$) on RNdS. But also on KdS we found, for
the scalar field of conformal mass, correlations of $\hat{T}_{VV}$ between the
two poles which are of the same order as the square of the expectation value.
We expect such strong correlations to be generic for fields of vanishing or
small mass.

If the correlations of $\hat{T}_{VV}$ are of the same order as the square of
its expectation value over macroscopic distances, then correlations, i.e.,
fluctuations, can no longer be neglected, calling into question the validity
of the semi-classical Einstein equation. We refer to Hu and Verdaguer (2020);
Pinamonti and Siemssen (2015) for approaches to take correlations of the
stress tensor into account in the description of backreaction. Considering the
results obtained in this work, an ansatz incorporating also the fluctuations
of the stress energy tensor will be necessary to unravel the effect of quantum
fields on the formation of singularities at the Cauchy horizon.

###### Acknowledgements.

We would like to thank M. Soltani and M. Casals for providing the data for the
results on Kerr-de Sitter. We would also like to thank S. Hollands and A. Ori
for stimulating discussions. This work has been funded by the Deutsche
Forschungsgemeinschaft (DFG) under the Grant No. 406116891 within the Research
Training Group RTG 2522/1.

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