# Renormalized charged scalar current in the Reissner-Nordström-de Sitter
spacetime

Christiane Klein<EMAIL_ADDRESS>Jochen Zahn<EMAIL_ADDRESS>leipzig.de Institut für Theoretische Physik, Universität Leipzig,
Brüderstraße 16, 04103 Leipzig, Germany

###### Abstract

We perform a Hadamard point-split renormalization of the current density
$j_{\nu}$ of a charged scalar field in a Reissner-Nordström-de Sitter
spacetime. We compute numerically the expectation values of the components
$j_{r}$ and $j_{t}$ in the Unruh state in the exterior region and study their
dependence on the field parameters and the position.

##  I Introduction

In astrophysical contexts, it is expected that any charged black hole (BH)
will quickly lose its charge by accreting matter of the opposite charge. But
even an isolated charged BH loses its charge by quantum effects. The discharge
of charged BHs by Hawking radiation or pair creation due to the
electromagnetic background field has already been anticipated by Hawking
himself in his seminal work Hawking (1975). This effect has been studied
further, for example, by Zaumen (1974); Carter (1974); Damour and Ruffini
(1975); Gibbons (1975); Hiscock and Weems (1990); Herman and Hiscock (1994,
1996); Khriplovich (1999); Gabriel (2000); Chen  _et al._ (2012); Kim (2014);
Johnson (2020); Ong (2019, 2021); Balakumar  _et al._ (2020). These works
typically focus on the discharge in the exterior region and, as far as
quantitative results are concerned, on large field masses and charges. While
this is certainly the most physically interesting phenomenon and parameter
range, we think that there are still interesting open questions. Here, we want
to answer a few of them.

The study of discharge in the exterior region typically proceeds either by
considering pair creation (for example using Schwinger’s pair creation rate
Schwinger (1951)), or by computing the expectation value of the current
$j_{r}$. By charge conservation, this quantity is determined by its value at
an arbitrary position in stationary and rotationally symmetric states. We
compute it numerically at the event horizon in the Unruh state, albeit only
for small mass $\mu$ and charge $q$ of the field and a large cosmological
constant $\Lambda$. We find an exponential decay in the mass and a
proportionality $\sim q^{2}$ for small $q$.

Another potentially interesting observable, which to the best of our knowledge
has not yet been considered in the literature, is the current component
$j_{t}$, representing, up to a negative factor, the (vacuum polarization)
charge density, as measured by a stationary observer. Certainly, for
astrophysical BHs and realistic field masses, one expects this polarization
charge density to decay rapidly, as for QED in an external Coulomb potential
(exponentially according to Uehling’s result linear in the external potential
Uehling (1935) or subexponentially when treating the external potential
nonperturbatively Mohr  _et al._ (1998)). It would be nice to verify this
intuition. We make first steps in this direction, as we calculate a finite
counterterm from a Hadamard point-split renormalization. We also compute the
vacuum polarization numerically in the Unruh state, again in the regime of
small mass $\mu$ and charge $q$ of the field and a large cosmological constant
$\Lambda$.

The inclusion of a cosmological constant, i.e., working on a Reissner-
Nordström-de Sitter (RNdS) instead of a Reissner-Nordström (RN) spacetime,
goes beyond previous work (focusing on the asymptotically flat case) and has
further advantages: numerical calculations are easier, as the corresponding
mode equation has only regular singular points (instead of an irregular
singular point at $\infty$ for RN). This allows for an efficient numerical
scheme Hollands  _et al._ (2020a), which works best if the cosmological and
event horizons are of the same order of magnitude (corresponding to a large
cosmological constant). Furthermore, (quantum) fields on RNdS have attracted
considerable attention in recent years, due to the possibility of violations
of strong cosmic censorship (sCC) Penrose (1974) on such spacetimes Cardoso
_et al._ (2018a); Dias  _et al._ (2019); Cardoso  _et al._ (2018b).

Our results are also relevant regarding the ongoing debate about sCC. The sCC
conjecture roughly states that at the inner horizon of a charged or rotating
black hole, like $\mathcal{CH}^{R}$ in Fig. 1, local observables such as the
current of a charged scalar field will diverge. This prevents any observer
from crossing the inner horizon and entering the region beyond it, where
determinism breaks down. It was suspected long ago Birrell and Davies (1978)
and recently shown Hollands  _et al._ (2020b) that the stress tensor of a real
scalar field diverges in a universal way at the inner horizon. The divergence
is stronger than that of classical fields and strong enough to ensure the
validity of sCC.

In the context of charged BHs, the consideration of charged fields is natural.
In view of the sCC conjecture, it is then interesting to study the behavior of
the expectation value of the current density $j_{\nu}$ near the inner horizon.
Our results on the point-split renormalization of the current, which are also
valid in the interior region, are used in Klein  _et al._ (2021) to do this.

The study of charged scalars on RNdS is made slightly more complicated by the
fact that an instability is present in the regime of small mass $\mu$ and
charge $q$ of the field and small cosmological constant $\Lambda$ Zhu  _et
al._ (2014); Konoplya and Zhidenko (2014); Dias  _et al._ (2019); Hod (2018).
As far as our general results on the point-split renormalization of the
current density in the Unruh state are concerned, we assume that we are in a
parameter range in which the instability is absent. For our concrete
computations, we choose parameters such that the instability is avoided (this
is one of the reasons for choosing a large cosmological constant $\Lambda$).

Let us comment on the relation to previous similar work. In Herman and Hiscock
(1996), an expression for the Hadamard point-split renormalization of the
current in general spacetimes and external background on-shell electromagnetic
fields is derived (though not evaluated). A divergent counterterm from
Hadamard point splitting is obtained, in contrast to our result of a finite
counterterm. As explained below, this seems to be due to a sign mistake in
Herman and Hiscock (1996). Reference Balakumar and Winstanley (2020) has
derived a prescription for the calculation of the renormalized current density
of a charged scalar field in a curved spacetime with a background
electromagnetic field via Hadamard point splitting, but only applies it to
show that the renormalized current satisfies current conservation. Recently,
the expectation value of $j_{r}$ was computed in the massless case for the
Boulware state in a RN spacetime Balakumar  _et al._ (2020). We go beyond
these results in that we allow for a nonzero mass and cosmological constant,
use the physically more relevant Unruh state, and also compute the expectation
value of the component $j_{t}$, related to the charge density.

The rest of the paper is structured as follows. In Sec. II, we introduce the
dynamics of the scalar field and the Unruh vacuum. These are used in Sec. III
to derive a point-splitting regularized expression for the current in the
Unruh state. The parametrix for the current is determined in Sec. IV. We
present numerical results in Sec. V and conclude in Sec. VI.

Throughout, we work in units $\hbar=c=G=4\pi\epsilon_{0}=1$.

##  II Setup

We consider a charged scalar field $\Phi$ on a RNdS spacetime.

Figure 1: Penrose diagram for Reissner-Nordström-de Sitter spacetime.
${\mathrm{I}}$ and ${\mathrm{III}}$ constitute the exterior region, with
${\mathrm{III}}$ being the region beyond the cosmological horizon
$\mathcal{H}_{c}$, i.e., out of causal contact with the BH interior. The
latter is represented by regions ${\mathrm{II}}$ and ${\mathrm{IV}}$, which
are separated from the exterior by the event horizon $\mathcal{H}^{R}$ and
from each other by the Cauchy horizon $\mathcal{CH}^{R}$. The curvature
singularity is represented by the wiggled line. The green line indicates a
Cauchy surface $\Sigma$ for the region
${\mathrm{I}}\cup{\mathrm{II}}\cup{\mathrm{III}}$. Here we focus on region
${\mathrm{I}}$ and the event horizon $\mathcal{H}^{R}$.

This spacetime describes a spherically symmetric charged BH in the presence of
a positive cosmological constant. Its conformal diagram is depicted in Fig. 1.
We will focus mostly on the region ${\mathrm{I}}$, the exterior of the black
hole, and the event horizon $\mathcal{H}^{R}$. A consideration of region
${\mathrm{II}}$, the inner horizon, and the relation to strong cosmic
censorship can be found in Klein  _et al._ (2021). The dynamics of the field
$\Phi$ on this spacetime are described by the Klein-Gordon equation

$\displaystyle\left[D_{\nu}D^{\nu}-\mu^{2}\right]\Phi$ $\displaystyle=0$ (1a)
$\displaystyle D_{\nu}$ $\displaystyle=\nabla_{\nu}-iqA_{\nu}\,.$ (1b)

Here,

$A=-\frac{Q}{r}\text{d}t$ (2)

is the background electromagnetic field, and the covariant derivative $\nabla$
is taken with respect to the metric

$\displaystyle g$
$\displaystyle=-f(r)\text{d}t^{2}+f(r)^{-1}\text{d}r^{2}+r^{2}\text{d}\Omega^{2}$
(3a) $\displaystyle f(r)$
$\displaystyle=-\frac{\Lambda}{3}r^{2}+1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$
(3b)

of our RNdS spacetime, where $M$, $Q$ and $\Lambda$ describe the mass and
charge of the BH and the cosmological constant. The three positive real roots
of $f$, $r_{-}<r_{+}<r_{c}$, mark the inner ($r_{-}$), event ($r_{+}$), and
cosmological ($r_{c}$) horizon.

To simplify (1a), one can introduce the tortoise radial coordinate $r_{*}$,
defined by $\text{d}r_{*}=f(r)^{-1}\text{d}r$. This coordinate approaches
$-\infty$ near the event horizon and $+\infty$ near the inner and cosmological
horizon. In addition to the change of variable, we employ the mode ansatz

$\displaystyle\Phi_{\ell m}(t,r_{*},\theta,\phi)=(4\pi)^{-1/2}r^{-1}Y_{\ell
m}(\theta,\phi)h_{\ell}(r_{*},t)\,$ (4)

for the scalar field. Here, the $Y_{\ell m}(\theta,\phi)$ are the spherical
harmonics. If we assume

$\displaystyle h_{\ell}(r_{*},t)=e^{-i\omega t}H_{\omega\ell}(r_{*})\,,$ (5)

(1a) is reduced to an ordinary differential equation for
$H_{\omega\ell}(r_{*})$. It takes the form of a one-dimensional scattering
problem

$\displaystyle\left[\partial_{r_{*}}^{2}+\left(\omega-\frac{qQ}{r}\right)^{2}-W\right]H_{\omega\ell}(r_{*})=0$
(6a) $\displaystyle
W=f(r)\left(\frac{\partial_{r}f(r)}{r}+\frac{\ell(\ell+1)}{r^{2}}+\mu^{2}\right)\,.$
(6b)

In terms of the tortoise coordinate $r_{*}$, the potential $W$ vanishes faster
than any power as $r_{*}\to\pm\infty$.

In contrast to the real scalar field, the charged scalar field allows for
gauge transformations of the form

$\displaystyle A_{\nu}$ $\displaystyle\to A_{\nu}+\partial_{\nu}\chi(x)$
$\displaystyle\Phi$ $\displaystyle\to e^{iq\chi(x)}\Phi\,.$ (7)

We will fix our gauge partially and restrict to transformations of the form

$\chi(t,r_{*},\theta,\phi)=\tfrac{Q}{r_{0}}t\,.$ (8)

For $r_{0}=r_{i}$, $i\in\\{+,-,c\\}$, we denote the corresponding gauge by an
$(i)$-superscript. The benefit of these gauges is that $A=0$ at the respective
horizon, so that the functions $h_{\ell}(r_{*},t)$ asymptotically behave as
free waves for $r_{*}\to\pm\infty$ if the gauge is chosen accordingly.

To extend the metric onto the horizons, we introduce lightlike coordinates
$v=t+r_{*}$, and $u=t-r_{*}$, and construct the Kruskal coordinates

$\displaystyle U$ $\displaystyle=\mp e^{-\kappa_{+}u}\,,$ $\displaystyle
V_{c}$ $\displaystyle=-e^{-\kappa_{c}v}\,,$ $\displaystyle V$
$\displaystyle=-e^{-\kappa_{-}v}\,,$ (9)

with $\kappa_{i}=\tfrac{1}{2}\left|\partial_{r}f(r)\right|_{r=r_{i}}$,
$i\in\\{c,+,-\\}$, the surface gravity of the corresponding horizon. In the
definition of $U$, the upper sign is used in ${\mathrm{I}}$ and the lower sign
is used in ${\mathrm{II}}$. $V_{c}$ and $V$ are defined in ${\mathrm{I}}$ and
${\mathrm{II}}$, respectively. Using $U$, we can smoothly extend the metric
across $\mathcal{H}^{R}$, and using $V$ ($V_{c}$) we can also extend it across
$\mathcal{CH}^{R}$ ($\mathcal{H}_{c}^{L}$). In these coordinates, the horizons
$\mathcal{H}^{R}$, $\mathcal{CH}^{R}$, and $\mathcal{H}_{c}^{L}$ are situated
at $U=0$, $V=0$, and $V_{c}=0$, respectively.

For the construction of the Unruh vacuum, we define a set of mode solutions
$h^{\mathrm{in}}_{\omega\ell}$, $h^{\mathrm{up}}_{\omega\ell}$, called Unruh
mode solutions, by initial conditions on
$\mathcal{H}^{L}\cup\mathcal{H}^{-}\cup\mathcal{H}_{c}^{-}\cup\mathcal{H}_{c}^{R}$.
The Unruh mode solutions are of the form (4), and the boundary conditions are

$\displaystyle h^{(c)\mathrm{in}}_{\omega\ell}$
$\displaystyle=\begin{cases}|\omega|^{-1/2}e^{-i\omega V_{c}}&\text{on
}\mathcal{H}_{c}^{-}\cup\mathcal{H}_{c}^{R}\\\ 0&\text{on
}\mathcal{H}^{-}\cup\mathcal{H}^{L}\\\ \end{cases}$ (10a) $\displaystyle
h^{(+)\mathrm{up}}_{\omega\ell}$ $\displaystyle=\begin{cases}0&\text{on
}\mathcal{H}_{c}^{-}\cup\mathcal{H}_{c}^{R}\\\ |\omega|^{-1/2}e^{-i\omega
U}&\text{on }\mathcal{H}^{-}\cup\mathcal{H}^{L}\,.\end{cases}$ (10b)

As mentioned in the Introduction, we assume the absence of classical
instabilities or bound states in region ${\mathrm{I}}$. This means that
solutions of the form (4), (5) with

$\displaystyle h^{(+)}(r_{*})$ $\displaystyle\sim e^{-i\omega v}$
$\displaystyle\text{ for }r_{*}\to-\infty,$ (11a) $\displaystyle
h^{(c)}(r_{*})$ $\displaystyle\sim e^{-i(\omega+\omega_{{\mathrm{I}}})u}$
$\displaystyle\text{ for }r_{*}\to+\infty,$ (11b)

where $\omega_{{\mathrm{I}}}=qQ(r_{+}^{-1}-r_{c}^{-1})$, are only possible for
frequencies $\omega$ with a negative imaginary part. In other words, the
spectral gap of quasi-normal frequencies is strictly positive. As discussed
above, this is the case except for a small parameter range of small
$q,\mu,\Lambda$. If this condition is fulfilled, we can quantize the scalar
field by an expansion in terms of the positive frequency Unruh modes as

$\displaystyle\Phi(x)$
$\displaystyle=\sum_{\lambda,\ell,m}\int\limits_{0}^{\infty}\text{d}\omega\left(\Phi_{\omega\ell
m}^{\lambda}(x)a_{\omega\ell m}^{\lambda}+\Phi_{-\omega\ell
m}^{\lambda}(x)b_{\omega\ell m}^{\lambda\dagger}\right)\,,$ (12)

with $\lambda$ running over $\mathrm{in}$ and $\mathrm{up}$. $a_{\omega\ell
m}^{\lambda}$ and $b_{\omega\ell m}^{\lambda}$ are taken to be the usual Fock
space annihilation operators, satisfying the canonical commutation relations.
The Unruh vacuum $|0\rangle_{\mathrm{U}}$ is the ground state of this Fock
space, for which $a^{\lambda}_{\omega\ell
m}|0\rangle_{\mathrm{U}}=0=b^{\lambda}_{\omega\ell m}|0\rangle_{\mathrm{U}}$.
As in Hollands  _et al._ (2020b) and using results for decay of the field
towards $i^{-}$ (c.f. Fig. 1) Hintz and Vasy (2017); Cardoso  _et al._
(2018b), one can show that this defines a proper quantum field/state (in the
absence of classical instabilities or bound states). Furthermore, this state
is Hadamard (see, for example, Hollands and Wald (2015) for a definition) in
${\mathrm{I}}\cup{\mathrm{II}}\cup{\mathrm{III}}$, i.e., across the
cosmological and the event horizon. This is in contrast to the Boulware state
considered in Balakumar  _et al._ (2020), which can not be extended as a
Hadamard state across the event horizon.

##  III The regularized current in the Unruh state

The observable that we focus on in this work is the current density $j_{\nu}$,
which is classically given by

$\displaystyle
j_{\nu}(x)=iq\left(\Phi(x)D_{\nu}^{*}\Phi^{*}(x)-\Phi^{*}(x)D_{\nu}\Phi(x)\right)\,$
(13)

with $D^{*}_{\nu}={\partial}_{\nu}+iqA_{\nu}$. In the quantum theory, this
needs to be renormalized, as it is quadratic and local in the field. We do
this using Hadamard point-split renormalization, which is local and (gauge)
covariant Hollands and Wald (2001, 2015); Zahn (2014). The expectation value
of a renormalized field monomial $D_{\alpha}\Phi(x)D^{*}_{\beta}\Phi^{*}(x)$,
with multi-indices $\alpha$ and $\beta$ in a Hadamard state $\Psi$, is defined
as

$\langle
D_{\alpha}\Phi(x)D^{*}_{\beta}\Phi^{*}(x)\rangle_{\Psi}^{\mathrm{ren}}=\lim\limits_{x^{\prime}\to
x}D_{\alpha}(D^{\prime}_{\beta})^{*}\left(\langle\Phi(x)\Phi^{*}(x^{\prime})\rangle_{\Psi}-H(x,x^{\prime})\right)\,.$
(14)

Here, $H(x,x^{\prime})$ is the Hadamard parametrix of the wave operator
$D_{\nu}D^{\nu}-\mu^{2}$, which is discussed in detail in the following
section. For a Hadamard state $\Psi$, the expression in brackets on the right-
hand side (r.h.s.) is smooth, so the limit of coinciding points is well
defined and gauge invariant (before the limit of coinciding points is taken,
the expression on the r.h.s. is of course not gauge invariant).

Hadamard point-split renormalization is not the unique local and gauge
covariant renormalization scheme. There are in general finite renormalization
ambiguities. However, for the case of the current, any two locally gauge
covariant renormalization schemes yielding a conserved current $j_{\mu}$
differ, at most, by multiple of the current $J_{\mu}$ responsible for the
external electromagnetic field Zahn (2014); Balakumar and Winstanley (2020),
corresponding to a renormalization of the charge of the field. In the present
case, the external current $J_{\mu}$ vanishes, so that Hadamard point-split
renormalization yields a unique result. For some applications of this
renormalization procedure in the context of external electromagnetic
potentials or curved spacetimes, we refer to Schlemmer and Zahn (2015);
Wernersson and Zahn (2021) and Hollands and Wald (2015); Zilberman  _et al._
(2020); Hollands  _et al._ (2020b).

In this section, we focus on the evaluation of the first term on the r.h.s. of
(14), in the linear combination of monomials relevant for the current density
and evaluated in the Unruh state. Concretely, we first derive an expression
for

$\langle
j_{\nu}(x_{1},x_{2})\rangle_{\mathrm{U}}=iq\left\langle\left\\{\Phi(x_{1}),\partial_{\nu}\Phi^{*}(x_{2})\right\\}-\left\\{\Phi^{*}(x_{1}),\partial_{\nu}\Phi(x_{2})\right\\}\right\rangle_{\mathrm{U}}.$
(15)

For convenience, we performed a symmetrization here, as
$\\{A,B\\}=\tfrac{1}{2}(AB+BA)$. In addition, we reduced the gauge-covariant
derivative to a simple partial derivative by choosing a gauge of the form (8),
with $r_{0}=r(x_{2})$. As one can think of the point splitting as a
regularization, we henceforth refer to this as the regularized current. We
will further simplify it by choosing a particular prescription for the point
splitting.

As usual, it is convenient to re-express the Unruh modes
$h^{\mathrm{in}/\mathrm{up}}_{\omega\ell}$ in terms of so-called Boulware
modes $\tilde{h}^{\lambda\mathrm{N}}_{\omega\ell}$. Here $\lambda$ runs over
$\\{\mathrm{in},\mathrm{up}\\}$, as before, while $\mathrm{N}$ runs over
$\\{{\mathrm{I}},{\mathrm{II}}\\}$, labeling the region on which the modes are
defined (as we will not be interested in evaluating the current density in
region ${\mathrm{III}}$, we refrain from introducing Boulware modes for that
region). The boundary conditions for the modes in ${\mathrm{I}}$ are defined
on $\mathcal{H}^{-}\cup\mathcal{H}_{c}^{-}$, and are given by

$\displaystyle\tilde{h}^{(c)\mathrm{in}{\mathrm{I}}}_{\omega\ell}$
$\displaystyle\sim e^{-i\omega v}\quad\text{on }\mathcal{H}_{c}^{-}\,,$
$\displaystyle\tilde{h}^{(+)\mathrm{up}{\mathrm{I}}}_{\omega\ell}$
$\displaystyle\sim e^{-i\omega u}\quad\text{on }\mathcal{H}^{-}\,,$ (16)

with vanishing boundary conditions on the other horizon, respectively. The
modes in ${\mathrm{II}}$ are defined on $\mathcal{H}^{L}\cup\mathcal{H}^{R}$,
by the boundary conditions

$\displaystyle\tilde{h}^{(+)\mathrm{in}{\mathrm{II}}}_{\omega\ell}$
$\displaystyle\sim e^{-i\omega v}\quad\text{on }\mathcal{H}^{R}\,,$
$\displaystyle\tilde{h}^{(+)\mathrm{up}{\mathrm{II}}}_{\omega\ell}$
$\displaystyle\sim e^{-i\omega u}\quad\text{on }\mathcal{H}^{L}\,,$ (17)

again with vanishing boundary conditions understood on the other part of the
initial data surface. These boundary conditions are easier to use for
computations than the ones for the Unruh modes. Note also that the modes
defined in region ${\mathrm{I}}$ can be continued to region ${\mathrm{II}}$ by
comparing their asymptotic behavior near $\mathcal{H}^{R}$ with that of the
modes defined in ${\mathrm{II}}$.

In ${\mathrm{I}}\cup{\mathrm{II}}$, we can now express the Unruh modes in
terms of the Boulware modes as

$|\omega|^{1/2}h^{\lambda}_{\omega\ell}(r_{*},t)=\sum\limits_{\mathrm{N\in\\{{\mathrm{I}},{\mathrm{II}}\\}}}\int\limits_{-\infty}^{\infty}\frac{\text{d}\omega^{\prime}}{2\pi}\alpha^{\lambda\mathrm{N}}_{\omega\omega^{\prime}}\tilde{h}^{\lambda\mathrm{N}}_{\omega^{\prime}\ell}(r_{*},t)\,,$
(18)

where $\lambda\in\\{\mathrm{in},\mathrm{up}\\}$. The coefficients
$\alpha^{\lambda\mathrm{N}}_{\omega\omega^{\prime}}$ can be computed by
applying the Fourier transform twice Lanir  _et al._ (2018). In the
calculation, it is necessary to introduce a small imaginary part for
$\omega^{\prime}$ as a regularization of a linear divergence for
$\omega^{\prime}\to 0$. The regularized coefficients can be computed to be

$\displaystyle\alpha_{\omega\omega^{\prime}}^{\lambda{\mathrm{I}}}$
$\displaystyle=\lim\limits_{\epsilon\to
0}\frac{1}{\kappa_{j}}|\omega|^{i\tfrac{\omega^{\prime}}{\kappa_{j}}}e^{\mathrm{sgn}(\omega)\tfrac{\pi\omega^{\prime}}{2\kappa_{j}}}\Gamma\left(-i\frac{\omega^{\prime}+i\epsilon}{\kappa_{j}}\right)$
(19a) $\displaystyle\alpha_{\omega\omega^{\prime}}^{\mathrm{up}{\mathrm{II}}}$
$\displaystyle=\overline{\alpha_{\omega\omega^{\prime}}^{\mathrm{up}{\mathrm{I}}}}$
(19b) $\displaystyle\alpha_{\omega\omega^{\prime}}^{\mathrm{in}{\mathrm{II}}}$
$\displaystyle=0$ (19c)

where $j=c$ for $\lambda=\mathrm{in}$ and $j=+$ for $\lambda=\mathrm{up}$.

The expressions (18) for the Unruh mode functions can now be inserted into the
expression for $\Phi(x)$. The regularized current (15) is then expressed in
terms of the mode expansions of $\Phi(x)$, $\Phi^{*}(x)$, and their first
partial derivatives. Since we are most interested in the $v$, $t$, and
$r_{*}$-component of the current, we will assume that the partial derivative
does not act on the $\theta$\- and $\phi$\- variables (the corresponding
current components vanish by the rotational symmetry of the Unruh state). For
$x_{1}=(r_{1},t_{1},\theta_{1},\phi_{1})$, and
$x_{2}=(r_{2},t_{2},\theta_{2},\phi_{2})$, the expression for the regularized
current (15) takes the form

$\displaystyle\langle j_{\nu}(x_{1},x_{2})\rangle_{\mathrm{U}}$
$\displaystyle=\frac{q}{16\pi^{3}}\sum\limits_{\ell,m,\lambda,\mathrm{N},\mathrm{N}^{\prime}}\int\limits_{0}^{\infty}\frac{\text{d}\omega}{\omega}\int\limits_{-\infty}^{\infty}\text{d}\omega^{\prime}\int\limits_{-\infty}^{\infty}\text{d}\omega^{\prime\prime}\
\operatorname{Im}\bigg{[}\left(\overline{\alpha^{\lambda\mathrm{N}}_{\omega\omega^{\prime}}}\alpha^{\lambda\mathrm{N}^{\prime}}_{\omega\omega^{\prime\prime}}+\overline{\alpha^{\lambda\mathrm{N}}_{-\omega\omega^{\prime}}}\alpha^{\lambda\mathrm{N}^{\prime}}_{-\omega\omega^{\prime\prime}}\right)\times$
(20) $\displaystyle\times\frac{1}{r_{1}}\overline{Y_{\ell
m}}(\theta_{1},\phi_{1})Y_{\ell
m}(\theta_{2},\phi_{2})\overline{\tilde{h}_{\omega^{\prime}\ell}^{\lambda\mathrm{N}}}(r_{1},t_{1})\partial_{\nu}\left(\frac{1}{r_{2}}\tilde{h}^{\lambda\mathrm{N}^{\prime}}_{\omega^{\prime\prime}\ell}(r_{2},t_{2})\right)\bigg{]}\,,$

where the Boulware modes are evaluated in the gauge described below (15).

Before continuing, we want to argue that the $i\epsilon$-prescription
introduced in (19a), which shifts the poles of $\omega^{\prime}$ and
$\omega^{\prime\prime}$ in (20), can be safely dropped. This is due to the
fact that the Boulware modes $\tilde{h}^{\lambda}_{\omega^{\prime}\ell}$
vanish as $\mathcal{O}(\omega^{\prime})$ for $\omega^{\prime}\to 0$. To see
this, let us first focus on the $\mathrm{up}{\mathrm{I}}$-modes. We find that
they can be written in the form (5). The $\mathrm{up}{\mathrm{I}}$-modes enter
region ${\mathrm{I}}$ from $\mathcal{H}^{-}$. From there, they can be
scattered back to $\mathcal{H}^{R}$, or travel all the way to
$\mathcal{H}_{c}^{L}$. As a result, the asymptotic behavior of the radial part
$H_{\omega\ell}(r_{*})$ is

$\displaystyle H_{\omega\ell}\sim\begin{cases}e^{i\omega
r_{*}}+\mathcal{R}_{\omega\ell}e^{-i\omega r_{*}}&r_{*}\to-\infty\\\
\mathcal{T}_{\omega\ell}e^{i(\omega+\omega_{{\mathrm{I}}})r_{*}}&r_{*}\to\infty\,,\end{cases}$
(21)

with $\omega_{{\mathrm{I}}}$ as defined below (11). There is no analytic
expression for the scattering coefficients $\mathcal{R}_{\omega\ell}$ and
$\mathcal{T}_{\omega\ell}$, but the behavior of $\mathcal{R}_{\omega\ell}$ for
small $\omega$ can be estimated by a first-order approximation of the radial
equation (6a) in $r-r_{+}$ near the event horizon Sela (2018). One finds
$\mathcal{R}_{\omega\ell}\sim-1+\mathcal{O}(\omega)$ for small $\omega$.
Combining this result with the Wronskian relation

$\displaystyle|\mathcal{R}_{\omega\ell}|^{2}+\frac{\omega+\omega_{{\mathrm{I}}}}{\omega}|\mathcal{T}_{\omega\ell}|^{2}=1\,,$
(22)

of the scattering coefficients, one obtains that
$\mathcal{T}_{\omega\ell}\sim\mathcal{O}(\omega)$ for small $\omega$. As a
consequence, near the boundaries of region ${\mathrm{I}}$, $H_{\omega\ell}$
vanishes point-wise as $\mathcal{O}(\omega)$ for $\omega\to 0$. The same is
also true for $\partial_{r_{*}}H_{\omega\ell}(r_{*})$. Since $H_{\omega\ell}$
satisfy (6a), this implies $H_{\omega\ell}\to 0$ point-wise for all $r_{*}$ as
$\omega\to 0$. A similar estimate can be found for the $\mathrm{in}$-type
modes on ${\mathrm{I}}\cup{\mathrm{II}}$, as well as for the combination of
the reflected $\mathrm{up}{\mathrm{I}}$-modes and the
$\mathrm{up}{\mathrm{II}}$-modes in ${\mathrm{II}}$. As a result, the
regularization of $\alpha_{\omega\omega^{\prime}}^{\lambda\mathrm{N}}$ can be
dropped.

To proceed, we now compute the integral over $\omega$ in (20), which involves
only the coefficients in the first line. One obtains results of the form

$\displaystyle\int\limits_{0}^{\infty}\frac{d\omega}{\omega}(\overline{\alpha_{-\omega\omega^{\prime}}^{\lambda\mathrm{N}}}\alpha_{-\omega\omega^{\prime\prime}}^{\lambda\mathrm{N}^{\prime}}+\overline{\alpha_{\omega\omega^{\prime}}^{\lambda\mathrm{N}}}\alpha_{\omega\omega^{\prime\prime}}^{\lambda\mathrm{N}^{\prime}})=4\pi^{2}C_{\lambda\mathrm{N}\mathrm{N}^{\prime}}(\omega^{\prime})\delta(\omega^{\prime}-\omega^{\prime\prime})\,,$
(23)

where the $C_{\lambda\mathrm{N}\mathrm{N}^{\prime}}(\omega^{\prime})$ are real
functions given by

$\displaystyle C_{\mathrm{in},{\mathrm{I}},{\mathrm{I}}}(\omega)$
$\displaystyle=\coth\left(\pi\tfrac{\omega}{\kappa_{c}}\right)\omega^{-1}$
(24a) $\displaystyle
C_{\mathrm{up},{\mathrm{I}},{\mathrm{I}}}(\omega)=C_{\mathrm{up},{\mathrm{II}},{\mathrm{II}}}(\omega)$
$\displaystyle=\coth\left(\pi\tfrac{\omega}{\kappa_{+}}\right)\omega^{-1}$
(24b) $\displaystyle
C_{\mathrm{up},{\mathrm{I}},{\mathrm{II}}}(\omega)=C_{\mathrm{up},{\mathrm{II}},{\mathrm{I}}}(\omega)$
$\displaystyle=\left[\omega\sinh\left(\pi\tfrac{\omega}{\kappa_{+}}\right)\right]^{-1}\,,$
(24c)

and all other $C_{\lambda\mathrm{N}\mathrm{N}^{\prime}}(\omega)$ vanish.
Inserting this back in (20), we find

$\displaystyle\langle j_{\nu}(x_{1},x_{2})\rangle_{\mathrm{U}}=$
$\displaystyle\frac{q}{4\pi
r_{1}r_{2}}\sum\limits_{\ell,m,\lambda,\mathrm{N}\mathrm{N}^{\prime}}\int\limits_{-\infty}^{\infty}\text{d}\omega\
\operatorname{Im}\bigg{[}C_{\lambda\mathrm{N}\mathrm{N}^{\prime}}(\omega)\overline{Y_{\ell
m}}(\theta_{1},\phi_{1})Y_{\ell m}(\theta_{2},\phi_{2})\times$ (25)
$\displaystyle\times\overline{\tilde{h}_{\omega\ell}^{\lambda\mathrm{N}}}(r_{1},t_{1})\left(\partial_{\nu}\tilde{h}^{\lambda\mathrm{N}^{\prime}}_{\omega\ell}(r_{2},t_{2})-\frac{1}{r_{2}}\tilde{h}^{\lambda\mathrm{N}^{\prime}}_{\omega\ell}(r_{2},t_{2})\partial_{\nu}r_{2}\right)\bigg{]}\,.$

So far, we have not specified the point splitting we apply. We use a
$\theta$\- splitting as introduced in Levi and Ori (2016): We identify
$r_{2}=r_{1}$, $\phi_{2}=\phi_{1}$, but set
$\theta_{2}=\theta_{1}+\epsilon=\theta+\epsilon$ and in addition
$t_{2}=t_{1}+\delta=t+\delta$ for the regularization of the $\omega$-integral.
Note that we first take the limit $\delta\to 0$, before we take the limit
$\epsilon\to 0$. This allows us to write

$\sum\limits_{m=-\ell}^{\ell}\overline{Y_{\ell m}}(\theta,\phi)Y_{\ell
m}(\theta+\epsilon,\phi)=\frac{2\ell+1}{4\pi}P_{\ell}(\cos\epsilon)\,$ (26)

for the $\omega$-independent part of the regularized current in (25).

Let us now take a closer look at the integral over $\omega$. Consider first
the part with the derivative acting on $r$. In the coincidence limit, only the
imaginary part of $\partial_{\nu}r$ will contribute to the integral. However,
$\partial_{\nu}r$ is real. Hence this part does not contribute to the integral
in the coincidence limit, and we can drop it. The remaining integrals over
$\omega$ are of the form

$\displaystyle\sum\limits_{\lambda,\mathrm{N},\mathrm{N}^{\prime}}\int\limits_{-\infty}^{\infty}\text{d}\omega\
C_{\lambda\mathrm{N}\mathrm{N}^{\prime}}(\omega)\operatorname{Im}\left[\overline{\tilde{h}_{\omega\ell}^{\lambda\mathrm{N}}}(r,t)\partial_{\nu}\tilde{h}^{\lambda\mathrm{N}^{\prime}}_{\omega\ell}(r,t+\delta)\right]\,.$
(27)

The splitting in $t$ will lead to an oscillatory term of the form
$e^{-i\delta(\omega-\omega_{g})}$, where $\omega_{g}$ is the frequency shift
between the gauge in which the modes are defined and the one in which they are
evaluated. To be able to take the limit $\delta\to 0$ later, we will shift the
integrals by $\omega_{g}$. This means that the integral involving the
$\mathrm{in}{\mathrm{I}}$-type Boulware mode will be shifted by
$\omega_{r,\mathrm{in}}=qQ(r^{-1}-r_{c}^{-1})$ and the one involving the
$\mathrm{up}{\mathrm{I}}$\- or $\mathrm{up}{\mathrm{II}}$-type mode will be
shifted by $\omega_{r,\mathrm{up}}=qQ(r^{-1}-r_{+}^{-1})$.

Next, consider the leading order in the large $|\omega|$-limit. There, (6a)
becomes

$\displaystyle\left[\partial_{r_{*}}^{2}+\omega^{2}\right]H_{\omega\ell}(r_{*})=0\,$
(28)

so that the Boulware modes become plane waves. Then, the leading order
contribution to the integrand of (27), evaluated in region ${\mathrm{I}}$, is
given by

$\displaystyle\coth\left(\pi\tfrac{\omega}{\kappa_{c}}\right)\left(\partial_{\nu}t+\partial_{\nu}r_{*}\right)\cos(\delta\omega)+\coth\left(\pi\tfrac{\omega}{\kappa_{+}}\right)\left(\partial_{\nu}t-\partial_{\nu}r_{*}\right)\cos(\delta\omega)\,.$
(29)

This integrand is antisymmetric in $\omega$, and the contributions from
$\omega\to\infty$ and $\omega\to-\infty$ cancel. Similar results also hold for
the integrand in ${\mathrm{II}}$. The results for situations close to the
present one Yafaev (2010), as well as the numerical results obtained with this
formula, indicate that this cancellation is indeed strong enough to make the
integral converge, even when the coincidence limit is taken. Assuming this to
be true, we can take the limit $\delta\to 0$, and hence

$\displaystyle\langle j_{\nu}(x)\rangle_{\mathrm{U}}$
$\displaystyle=\lim\limits_{\epsilon\to
0}\sum\limits_{\ell}\frac{q(2\ell+1)}{16\pi^{2}r^{2}}P_{\ell}(\cos\epsilon)\int\limits_{0}^{\infty}\text{d}\omega\sum\limits_{\lambda,\mathrm{N},\mathrm{N}^{\prime}}\left(C_{\lambda\mathrm{N}\mathrm{N}^{\prime}}(\omega+\omega_{r,\lambda})\operatorname{Im}\left[\overline{\tilde{h}_{(\omega+\omega_{r,\lambda})\ell}^{\lambda\mathrm{N}}}(r,t)\partial_{\nu}\tilde{h}^{\lambda\mathrm{N}^{\prime}}_{(\omega+\omega_{r,\lambda})\ell}(r,t)\right]+\omega\leftrightarrow-\omega\right).$
(30)

From numerical computations, it turns out that this expression is already
finite in the limit $\epsilon\to 0$. This is further confirmed by the
calculation of the second term in (14), in the combination relevant for the
current, which must also be, and indeed is, finite.

##  IV The calculation of the parametrix

We now turn to the discussion of the second term on the r.h.s. of (14). In
four dimensions, the Hadamard parametrix can be written as

$\displaystyle H(x,x^{\prime})$
$\displaystyle=\frac{1}{8\pi^{2}}\left[\frac{U(x,x^{\prime})}{\sigma_{\epsilon}(x,x^{\prime})}+\sum\limits_{n}V_{n}(x,x^{\prime})\sigma^{n}(x,x^{\prime})\ln\left(\frac{\sigma_{\epsilon}(x,x^{\prime})}{K^{2}}\right)\right]\,.$
(31)

Here $K$ is an arbitrary scale, and $\sigma_{\epsilon}(x,x^{\prime})$ is
Synge’s world function Poisson  _et al._ (2011) equipped with an
$i\epsilon$-prescription, which is only relevant for causally related $x$,
$x^{\prime}$ (and is thus irrelevant in the $\theta$-splitting that we are
employing). The so-called Hadamard coefficients, $U(x,x^{\prime})$ and
$V_{n}(x,x^{\prime})$, are smooth functions, which are determined in a local
and (gauge) covariant way and fulfill

$\displaystyle\overline{U(x,x^{\prime})}$ $\displaystyle=U(x^{\prime},x),$
$\displaystyle\overline{V_{n}(x,x^{\prime})}$
$\displaystyle=V_{n}(x^{\prime},x).$ (32)

The sum in (31) does not converge in general, but for the application in the
Hadamard point split (14), only a finite number of terms is relevant (whose
number depends on the number of derivatives in the field monomial, i.e.,
$|\alpha|$ and $|\beta|$). From the requirement that the application of
$D_{\nu}D^{\nu}-\mu^{2}$ yields a smooth function, one derives the transport
equations

$\displaystyle\left[\sigma^{\nu}D_{\nu}+\frac{1}{2}\Box\sigma-2\right]U$
$\displaystyle=0,$ (33a) $\displaystyle
2\left[\sigma^{\nu}D_{\nu}+\frac{1}{2}\Box\sigma-1\right]V_{0}$
$\displaystyle=-\left[D_{\nu}D^{\nu}-\mu^{2}\right]U,$ (33b) $\displaystyle
2(n+1)\left[\sigma^{\nu}D_{\nu}+\frac{1}{2}\Box\sigma+n\right]V_{n+1}$
$\displaystyle=-[D_{\nu}D^{\nu}-\mu^{2}]V_{n}.$ (33c)

Here we used the notation $\sigma^{\nu}=\nabla^{\nu}\sigma$. That the leading
short-distance singularity is as in the flat case is ensured by the initial
condition $U(x,x)=1$. This determines $U(x,x^{\prime})$ uniquely as

$U(x,x^{\prime})=\Delta^{\tfrac{1}{2}}(x,x^{\prime})P(x,x^{\prime})\,,$ (34)

where $\Delta(x,x^{\prime})$ is the Van Vleck-Morette determinant Poisson  _et
al._ (2011), and $P(x,x^{\prime})$ is the parallel transport with respect to
$D_{\nu}$ along the geodesic from $x^{\prime}$ to $x$ (which can be assumed to
be unique by restricting $x^{\prime}$ to a suitably small neighborhood of
$x$), i.e., solving $\sigma^{\nu}D_{\nu}P=0$ with the initial condition
$U(x,x)=1$. The higher order Hadamard coefficients are then determined
recursively.

For the application in Hadamard point splitting (14), only the coinciding
point limit of a suitable number of derivatives of the Hadamard coefficients
is relevant. More concretely, for the point-split renormalization of the
current, we need to determine all the divergent and finite contributions to
the coinciding point limit $x^{\prime}\to x$ of
$\partial^{\prime}_{\nu}\operatorname{Im}[H(x,x^{\prime})]$, where we used
(32). For this, we perform a Taylor expansion of the imaginary part of the
Hadamard coefficients, in the form

$\displaystyle F(x,x^{\prime})$
$\displaystyle=F^{(0)}(x)+F^{(1)}_{\alpha}(x)\sigma^{\alpha}(x,x^{\prime})+F^{(2)}_{\alpha\beta}(x)\sigma^{\alpha}(x,x^{\prime})\sigma^{\beta}(x,x^{\prime})+\dots$
(35)

Evaluating successively higher covariant derivatives of $U$, $V_{n}$ in the
limit of coinciding points, one obtains from the transport equations (33) the
following results Balakumar and Winstanley (2020):

$\displaystyle\operatorname{Im}\left(U^{(1)}_{\alpha}\right)$
$\displaystyle=qA_{\alpha}$ (36a)
$\displaystyle\operatorname{Im}\left(U^{(2)}_{\alpha\beta}\right)$
$\displaystyle=-\frac{q}{2}\nabla_{(\alpha}A_{\beta)}$ (36b)
$\displaystyle\operatorname{Im}\left(U^{(3)}_{\alpha\beta\gamma}\right)$
$\displaystyle=\frac{q}{6}\operatorname{Re}\left(D_{(\alpha}D_{\beta}A_{\gamma)}\right)+\frac{q}{12}A_{(\alpha}R_{\beta\gamma)}$
(36c) $\displaystyle\operatorname{Im}\left(V_{0\alpha}^{(1)}\right)$
$\displaystyle=\frac{q}{2}\left[\mu^{2}-\frac{1}{6}R\right]A_{\alpha}-\frac{q}{12}\nabla^{\nu}F_{\nu\alpha}$
(36d) $\displaystyle\operatorname{Im}\left(V_{0\alpha\beta}^{(2)}\right)$
$\displaystyle=-\frac{q}{4}\left[\mu^{2}-\frac{1}{6}R\right]\operatorname{Re}\left(D_{(\alpha}A_{\beta)}\right)+\frac{q}{24}A_{(\alpha}\nabla_{\beta)}R-\frac{q}{24}\nabla_{(\alpha}\nabla^{\nu}F_{\beta)\nu}$
(36e) $\displaystyle\operatorname{Im}\left(V^{(0)}_{1}\right)$
$\displaystyle=0\,,$ (36f)

where symmetrization with respect to the indices in round brackets is
understood. In addition, we will need an expansion of $\sigma^{\mu}$ in terms
of the angular separation $\epsilon$. For this, we use the expansion given in
Ottewill and Wardell (2009), and find

$\displaystyle\sigma$
$\displaystyle=\frac{r^{2}}{2}\epsilon^{2}-\frac{fr^{2}}{24}\epsilon^{4}+\mathcal{O}\left(\epsilon^{6}\right)$
(37a) $\displaystyle\sigma_{\theta}$
$\displaystyle=-r^{2}\epsilon+\frac{fr^{2}}{6}\epsilon^{3}+\mathcal{O}\left(\epsilon^{4}\right)$
(37b) $\displaystyle\sigma_{r}$
$\displaystyle=\frac{r}{2}\epsilon^{2}+\mathcal{O}\left(\epsilon^{4}\right)$
(37c) $\displaystyle\sigma_{t}$
$\displaystyle\approx\sigma_{\varphi}=0+\mathcal{O}\left(\epsilon^{4}\right)\,.$
(37d)

Taking into account that we are in a gauge such that $A(x)=A(x^{\prime})=0$,
we can now put these ingredients together and compute the coinciding point
limit of $\partial^{\prime}_{\nu}\operatorname{Im}[H(x,x^{\prime})]$ in the
$\theta$-splitting, which is the relevant quantity for our purpose of
renormalizing the current density. We find that the divergent parts cancel,
consistent with our finding in the previous section, and for the finite part
we obtain

$-2q\partial^{\prime}_{\nu}\operatorname{Im}\left[H(x,x^{\prime})\right]=-\frac{1}{4\pi^{2}}\frac{q^{2}Qf(r)}{6r^{3}}\delta^{t}_{\nu}+\mathcal{O}(\epsilon)\,.$
(38)

We note that this term is only relevant for the $t$-component of the current
and vanishes on the horizons. Similar finite contributions from Hadamard point
splitting frequently occur in the renormalization of the current density (see
the discussion and examples in Schlemmer and Zahn (2015); Wernersson and Zahn
(2021)). Comparing this expression to the result of Herman and Hiscock (1996),
where a divergence in the coinciding point limit is found, we note that there
is an error in the calculation of Herman and Hiscock (1996): in their formula
for the current, Eq. (8), the difference rather than the sum of contributions
with derivatives acting on the first and second variable should be taken. If
this is taken into account, the results do agree.

Note also that the parametrix satisfies the condition for the conservation of
the current, as can be shown by an explicit calculation in the
$(t,r_{*},\theta,\phi)$ coordinates. Employing that by spherical symmetry and
stationarity the angular components of the current must vanish,
$j_{\theta}=j_{\phi}=0$, and that it must be time-independent, the current
conservation $\nabla_{\nu}j^{\nu}=0$ reduces to
$\partial_{r_{*}}(r^{2}j_{r_{*}})=0$ Balakumar  _et al._ (2020). Since the
parametrix only affects the $t$-component of the current and is time-
independent, the current is conserved if the expression without the parametrix
yields a conserved current.

Finally, our expression for the renormalized current is given by subtracting
the Hadamard parametrix from the result of the previous section. Since this
parametrix is finite, and supported by the numerical results, we take the
limit $\epsilon\to 0$. We find

$\displaystyle\langle j_{\nu}(x)\rangle_{\mathrm{U}}$
$\displaystyle=\sum\limits_{\ell}\frac{q(2\ell+1)}{16\pi^{2}r^{2}}\int\limits_{0}^{\infty}\text{d}\omega\sum\limits_{\lambda,\mathrm{N},\mathrm{N}^{\prime}}\left(C_{\lambda\mathrm{N}\mathrm{N}^{\prime}}(\omega+\omega_{r,\lambda})\operatorname{Im}\left[\overline{\tilde{h}_{(\omega+\omega_{r,\lambda})\ell}^{\lambda\mathrm{N}}}(r,t)\partial_{\nu}\tilde{h}^{\lambda\mathrm{N}^{\prime}}_{(\omega+\omega_{r,\lambda})\ell}(r,t)\right]+\omega\leftrightarrow-\omega\right)$
$\displaystyle\quad+\frac{1}{4\pi^{2}}\frac{q^{2}Qf(r)}{6r^{3}}\delta^{t}_{\nu}.$
(39)

##  V Numerical results

While there is no analytic solution for the functions $\tilde{h}^{\lambda
N}_{\omega\ell}(r_{*},t)$, they can be approximated numerically. This allows
us, for example, to calculate the $v$-component of the current on the event
horizon. Here, we can use the asymptotic behavior of the Boulware modes (21)
to get an expression in terms of scattering coefficients. We find

$\displaystyle\langle j_{v}\rangle_{\mathrm{U}}$
$\displaystyle=\sum\limits_{\ell}\frac{q(2\ell+1)}{16\pi^{2}r^{2}}\int\limits_{0}^{\infty}\text{d}\omega\
\left(F(\omega)+F(-\omega)\right)$ (40a) $\displaystyle F(\omega)$
$\displaystyle=\coth\left(\pi\tfrac{\omega+\omega_{{\mathrm{I}}}}{\kappa_{c}}\right)\left(1-|\mathcal{R}_{\omega\ell}|^{2}\right)+\coth\left(\pi\tfrac{\omega}{\kappa_{+}}\right)|\mathcal{R}_{\omega\ell}|^{2}\,.$
(40b)

Due to the fact that the radial part of the Klein-Gordon equation can be
brought into the form of a Heun equation in the conformally coupled case,
$\mu^{2}=2\Lambda/3$, Suzuki  _et al._ (1999); Hollands  _et al._ (2020a),
this case can be dealt with most efficiently. However, more general masses can
be handled as well Hollands  _et al._ (2020a): one can obtain solutions to
(6a) which are defined in the neighborhood of one of the three horizons.
Comparing these solutions on the overlap of their domains, one can then
compute the scattering coefficients. In this way, the scattering coefficients
and the integrand in (40a) or (IV) are computed for different values of
$\omega$ and $\ell$ up to some maximal values, beyond which the contributions
become negligibly small. The integral is finally estimated using a Riemann
sum. The sum is also used to estimate the error of this procedure. For more
details, see Hollands  _et al._ (2020a).

Note that in order to achieve a good performance of the code, we restrict
ourselves to small charges $qQ$ and masses $\mu^{2}$ of the scalar field. As
discussed above, we thus need to choose the cosmological constant $\Lambda$
large enough in order to avoid the instability parameter region. Equation (46)
in Dias  _et al._ (2019) gives a condition to determine whether the
instability appears for a chosen set of spacetime parameters. We checked this
condition and find no instability in the parameter range we consider.

Figure 2: The $v$-component of the current of a conformally coupled scalar
field on the event horizon as a function of the scalar field charge in the
Unruh vacuum for $Q=0.95M$ and $\Lambda=0.14M^{-2}$. The smaller graph shows
the results for $Q=0.95M$ for small $qQ$, the dashed line indicates a
$q^{2}$-fit. Figure 3: The $v$-component of the current of a charged scalar
field of charge $qQ=0.1$ at the event horizon as a function of the scalar
field mass in the Unruh vacuum for $Q=0.95M$ and $\Lambda=0.14M^{-2}$.

Figures 2 and 3 show the $v$-component of the current on the event horizon. In
both plots, the cosmological constant is chosen as $\Lambda=0.14M^{-2}$, for
comparability with Klein  _et al._ (2021). In Fig. 2 the current is plotted as
a function of the scalar charge $qQ$ for a conformally coupled scalar field
and for different values of $Q/M$. We observe that $\langle
j_{v}\rangle_{\mathrm{U}}$ increases with $Q/M$. This is in line with the
results of Chen  _et al._ (2012), who found that the pair-production rate of
charged scalars by an extremal BH is larger than that of a near-extremal one.
In addition, we find that for small charges $q$ of the scalar field the
current behaves as $q^{2}$. This can be seen in the smaller graph in Fig. 2,
where the results for $Q=0.95$ and $qQ\leq 0.1$ have been fitted with
$a\times(qQ)^{2}$, shown by the dashed line. This is seemingly in contraction
with Schwinger’s pair creation formula Schwinger (1951) widely used in the
literature on black hole discharge, Zaumen (1974); Carter (1974); Damour and
Ruffini (1975), according to which the pair creation rate $\Gamma$ should be
nonperturbative in $q$ near $q=0$, involving a factor

$\Gamma\sim\exp\left[-\frac{\pi\mu^{2}r_{+}^{2}}{qQ}\right]\,.$ (41)

For the parameters used for Fig. 2, $\pi\mu^{2}r_{+}^{2}\sim 0.84$. Thus, the
effect of such an exponential suppression should be clearly visible in the
embedded figure. That this is not the case indicates that (41) is not
applicable for small (conformal) mass $\mu$ and small charge $q$. This is not
very surprising, as for conformal mass the Compton length coincides with the
Hubble length. Intuitively, the proportionality $\sim q^{2}$ can be understood
as follows: a charge $q$ created near the horizon is accelerated away from the
BH with acceleration proportional to $q$. Its contribution to the current is
thus $\sim q^{2}$.

In Fig. 3, the current is displayed as a function of the mass $\mu$ in units
of the conformal mass $\sqrt{2\Lambda/3}$, for $qQ=0.1$ and $Q=0.95M$. We find
that $\langle j_{v}\rangle_{\mathrm{U}}$ decays exponentially with $\mu$, and
the corresponding fit is represented by the dashed line (the data point of the
massless case is excluded from the fit). This also seems to be in
contradiction to (41), which would yield an exponential decay in $\mu^{2}$
rather than $\mu$. This matches our expectation that the Schwinger formula is
not applicable for masses of the order of $\sqrt{2\Lambda/3}$.

As the Unruh state is Hadamard across $\mathcal{H}^{R}$, the renormalized
expectation value $\langle j_{U}\rangle_{\mathrm{U}}$ must be finite on
$\mathcal{H}^{R}$. By the tensor transformation law, it follows that $\langle
j_{u}\rangle_{\mathrm{U}}$ vanishes on $\mathcal{H}^{R}$. This implies that on
the event horizon $\langle j_{v}\rangle_{\mathrm{U}}=\langle
j_{r_{*}}\rangle_{\mathrm{U}}=\langle j_{t}\rangle_{\mathrm{U}}$. Hence the
results presented above also apply to $\langle j_{t}\rangle_{\mathrm{U}}$. In
particular, we find that near the horizon the charge density is negative and
its magnitude decreases exponentially as the Compton-wavelength $1/\mu$
decreases.

With our general formula (IV), we can also calculate the current in the
interior of region ${\mathrm{I}}$. This can for example be used to check the
conservation of the current by calculating $r^{2}\langle
j_{r_{*}}\rangle_{\mathrm{U}}$ at different values of the radius $r$. If the
current is conserved, the result must be a constant that agrees with
$r_{+}^{2}\langle j_{v}\rangle_{\mathrm{U}}|_{\mathcal{H}}$. The results for
$qQ=0.1$, $\Lambda=0.14M^{-2}$ and $Q=0.95M$ are shown in Fig. 4. We find that
our results satisfy the condition for current conservation very well.

Figure 4: $r^{2}\langle j_{r_{*}}\rangle_{\mathrm{U}}$ for the conformally
coupled scalar at different radii $r$. This quantity must be constant for the
current conservation. The vertical lines mark $r_{+}$ and $r_{c}$,
respectively, where the first and last point are computed. $Q=0.95M$,
$\Lambda=0.14M^{-2}$, and $qQ=0.1$.

Furthermore, we can compute $\langle j_{t}\rangle_{\mathrm{U}}$, related to
the charge density, in the exterior region. This result is the hardest to
obtain numerically. In particular, estimating the errors of our calculation
via the Riemann sum is insufficient in this case, since it strongly
overestimates the inaccuracy. Instead, we carefully consider all error sources
separately. First of all, we calculate the integral in (IV) only up to some
$\omega_{\mathrm{max}}$. The rest of the integral can, however, be
approximated by fitting the exponential decay we expect for large $\omega$ to
the last few $\omega\leq\omega_{\mathrm{max}}$, and adding the integral of the
fitted function from $\omega_{\mathrm{max}}$ to $\infty$ to our result. For
our chosen parameters, the correction term is smaller than $3\times 10^{-11}$
for all $\ell$ considered. Second, we approximate the integral by a Riemann
sum. To estimate the error made by this discretization, we compare the result
of the Riemann sum with the full number of points to the result obtained by
using only every second point. We find that the change in the result is less
than $10^{-10}$ for all values of $\ell$ considered, becoming slightly larger
as $\ell$ increases. Finally, we only consider $\ell$ up to some finite
$\ell_{\mathrm{max}}$. To estimate the error due to this cutoff, we observe
that for all $\ell$ taken into account, the contribution to the current
decreases by a factor $k<0.9$ as $\ell$ increases, and the contribution from
$\ell_{\mathrm{max}}$ is of the order $10^{-10}$. Due to this analysis, we
estimate that the uncertainty of our results is about $10^{-8}-10^{-9}$, which
corresponds to $1-10\%$.

The results are shown in Fig. 5. Note that the first and last point shown in
Fig. 5 are results obtained on $\mathcal{H}^{R}$ and $\mathcal{H}_{c}^{L}$
respectively, including conservative errors estimated as detailed in Hollands
_et al._ (2020a). For the results in the interior, we omitted the error bars.
However the data points seem to interpolate reasonably between the two
horizons.

Figure 5: $\langle j_{t}\rangle_{\mathrm{U}}$ for the conformally coupled
scalar in region ${\mathrm{I}}$ as a function of $r$ for $qQ=0.1$, $Q=0.95M$,
and $\Lambda=0.14M^{-2}$. The first and last point are results obtained at
$r_{+}$ and $r_{c}$ respectively, including realistic errors. $r_{+}$ and
$r_{c}$ are also marked by the vertical lines.

We observe that $\langle j_{t}\rangle_{\mathrm{U}}$ starts positive near the
event horizon and becomes negative as one approaches the cosmological horizon.
This is to be expected in the Unruh state, which is Hadamard across the
cosmological horizon: Using the same argument as for the $u$ component on
$\mathcal{H}^{R}$, we find that $\langle j_{v}\rangle_{\mathrm{U}}$ vanishes
on $\mathcal{H}_{c}^{L}$. Hence, we must have $\langle
j_{t}\rangle_{\mathrm{U}}=-\langle j_{r_{*}}\rangle_{\mathrm{U}}$ on the
cosmological horizon. However, by current conservation and our previous
results, $\langle j_{r_{*}}\rangle_{\mathrm{U}}=Kr_{c}^{-2}$ on
$\mathcal{H}_{c}$, where $K=\langle r^{2}j_{r_{*}}\rangle_{\mathrm{U}}$ is a
positive constant. Hence $\langle j_{t}\rangle_{\mathrm{U}}<0$ on the
cosmological horizon. This is also to be expected on physical grounds: in
region ${\mathrm{III}}$, ${\partial}_{t}$ is spacelike and directed inwards,
so that $-j_{t}$ is the outward current. For a positively charged BH, we
indeed expect this to be positive.

##  VI Conclusion

We derived a formula for the current of a charged scalar field in the RNdS
spacetime, using point splitting and Hadamard renormalization. We found that
the Hadamard parametrix only contributes a finite term, in contrast to results
obtained previously Herman and Hiscock (1996). In addition, this term vanishes
on all horizons. We obtained an explicit expression for the current of the
charged scalar in the RNdS spacetime, which can also be used in the interior
of the BH, as was done in Klein  _et al._ (2021). We tested the result by
evaluating it numerically. We evaluated the $v$-component of the current on
$\mathcal{H}^{R}$ in the Unruh state and found an exponential decay in the
mass $\mu$, as well as a quadratic dependence on the charge $q$ of the field
for small $q$. We also calculated the $r_{*}$-component of the current in the
exterior region of the spacetime to test current conservation, and found it to
hold. Finally, $\langle j_{t}\rangle_{\mathrm{U}}$, related to vacuum
polarization, was computed inside region ${\mathrm{I}}$, with results
compatible with those obtained on the horizons bounding region ${\mathrm{I}}$.

## Acknowledgements

C.K. wants to thank Stefan Hollands for suggesting this topic. 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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We take $K_{j}:=\log\circ L_{j}$ which are singular kernel functions
satisfying (PM0), and $J:=\log\circ w$ which is an $n$-field function subject
to ($\infty_{+}$ ‣ 7) and ($\infty_{-}$ ‣ 7). The Homeomorphism Theorem 7.1
can be applied to conclude the proof as for Theorem 9.9. ∎

If we specialize $L_{j}(t)=L(t):=|\sin(\pi t)|$ and carry out the usual sign-
tracking, we obtain the following version of Videnskii’s Theorem 9.1 for
trigonometric polynomials.

######  Corollary 9.13.

Let $n\in\mathbb{N}$, and let $w:[0,1]\to[0,\infty)$ be a differentiable
function with at least $n+1$ non-zero values within $(0,1)$ and satisfying
$\lim_{t\downarrow 0}w(t)=0$ and $\lim_{t\uparrow 1}w(t)=0$. Further, let
$\nu_{1},\ldots,\nu_{n}>0$. For every
$\alpha_{0},\alpha_{1},\ldots,\alpha_{n}>0$ the following are true:

  1. (a)

There exist a unique system of points $0<y_{1}<y_{2}<\cdots<y_{n}<1$, and a
unique $C>0$ such that for the function

$S(t):=C\prod_{k=1}^{n}\sin^{\nu_{k}}(\pi|t-y_{k}|)$

there are $z_{0},\dots,z_{n}\in(0,1)$ with $0<z_{0}<y_{1}<z_{1}<y_{2}<\cdots
z_{n-1}<y_{n}<z_{n}<1$ and

$w(z_{j})S(z_{j})=\alpha_{j}\quad\text{and}\quad(wS)^{\prime}(z_{j})=0\quad\quad\text{for
$j=0,1,\ldots,n$}.$

  2. (b)

If $\nu_{1},\dots,\nu_{n}\in\mathbb{N}$, then for the previously chosen
$0<y_{1}<y_{2}<\cdots<y_{n}<1$ and $C$ the function

$T(t):=C\prod_{k=1}^{n}\sin^{\nu_{k}}(\pi(t-y_{k}))$

satisfies

$w(z_{j})T(z_{j})=(-1)^{\sum_{k=j}^{n}\nu_{k}}\alpha_{j}\quad\text{for
$j=0,1,\ldots,n$}.$

(Note that $T$ is in fact a trigonometric polynomial if
$\sum_{k=1}^{n}\nu_{k}$ is even.)

##  10\. Preview of further applications

Our original and real goal with working out the homeomorphism theorems of this
paper was, similarly to [11], its expected application in obtaining _minimax_
and _equioscillation_ type results for sums of translates on the interval.
Both of these have numerous applications, and the strength of our results,
which will be presented in the companion paper [12], allow us to derive new
information even in such thoroughly investigated classical problems as that of
Chebyshev of minimal maximum norm monic polynomials on the interval. Moreover,
these results will be obtained in quite a generality, even if the generality
of context will be somewhat different than here. A particular point is that we
need to assume some mild conditions about the field $J$, like upper
semicontinuity, but kernels may be not singular. The companion paper [12] will
restrict attention to the situation when all kernels are constant multiples of
each other, i.e. one base kernel $K$: $K_{j}=\nu_{j}K$ ($j=1,\ldots,n$). We
will term this situation as “kernels satisfying a proportionality condition”.

Bojanov proved in [2] the following.

######  Theorem 10.1 (Bojanov).

Let $\nu_{1},\ldots,\nu_{n}$ be fixed positive integers, and
$[a,b]\subset{\mathbb{R}}$ be a finite, non-degenerate, closed interval. Then
there exists a unique system of points $x_{1}\leq\ldots\leq x_{n}$ such that

$\|(x-x_{1})^{\nu_{1}}\ldots(x-x_{n})^{\nu_{n}}\|=\inf_{a\leq
y_{1}<\ldots<y_{n}\leq b}\|(x-y_{1})^{\nu_{1}}\ldots(x-y_{n})^{\nu_{n}}\|,$

where $\|\cdot\|$ denotes the sup-norm over $[a,b]$. Moreover,
$a<x_{1}<\ldots<x_{n}<b$ and the extremal polynomial

$P^{*}(x):=(x-x_{1})^{\nu_{1}}\ldots(x-x_{n})^{\nu_{n}}$

is uniquely characterized by the property that there exist
$a=s_{0}<s_{1}<\ldots<s_{n-1}<s_{n}=b$ such that

$|P^{*}(s_{j})|=\|P^{*}\|\quad\text{for $j=0,1,\ldots,n$.}$

Furthermore, in this situation

$P^{*}(s_{j+1})=(-1)^{\nu_{j+1}}P^{*}(s_{j})\quad\text{for
$j=0,1,\ldots,n-1$.}$

In [12] we prove the following generalization.

######  Theorem 10.2.

Let $\nu_{1},\nu_{2},\ldots,\nu_{n}$ be fixed positive numbers,
$[a,b]\subset{\mathbb{R}}$ a finite, non-degenerate, closed interval, and
$w:[a,b]\to[0,\infty)$ be an upper semicontinuous, non-negative weight
function, assuming non-zero values at more than $n$ points of the interval
$[a,b]$ (the endpoints $a,b$ are counted with weight $1/2$).

Then there exists a unique extremizer set of points $x_{1}^{*}\leq
x_{2}^{*}\leq\ldots\leq x_{n}^{*}$ such that

$\left\|w(x)|x-x_{1}^{*}|^{\nu_{1}}|x-x_{2}^{*}|^{\nu_{2}}\ldots|x-x_{n}^{*}|^{\nu_{n}}\right\|\\\
=\inf_{a\leq x_{1}\leq x_{2}\leq\ldots\leq x_{n}\leq
b}\left\|w(x)|x-x_{1}|^{\nu_{1}}|x-x_{2}|^{\nu_{2}}\ldots|x-x_{n}|^{\nu_{n}}\right\|,$

where $\|\cdot\|$ is the sup-norm over $[a,b]$. Moreover,
$a<x_{1}^{*}<x_{2}^{*}<\ldots<x_{n}^{*}<b$ and the extremal generalized
polynomial $T(x):=\prod_{k=1}^{n}|x-x_{k}^{*}|^{\nu_{k}}$ is uniquely
determined by the following equioscillation property: There exists an array of
$n+1$ points $a\leq t_{0}<t_{1}<t_{2}<\ldots<t_{n-1}<t_{n}\leq b$, interlacing
with the $x_{i}^{*}$, i.e., $a\leq
t_{0}<x_{1}^{*}<t_{1}<x_{2}^{*}<\ldots<x_{n}^{*}<t_{n}\leq b$ such that

(28) $w(t_{k})T(t_{k})=\|wT\|\qquad(k=0,1,\ldots,n).$

Furthermore, if $w$ is in addition log-concave, then the unique extremal
generalized polynomial $T$ is uniquely determined by the property that there
exists an array of $n+1$ points $a\leq
t_{0}<t_{1}<t_{2}<\ldots<t_{n-1}<t_{n}\leq b$ such that (28) holds.

In particular, this extends Bojanov’s result to the case of general weighted
norms, for essentially arbitrary upper semicontinuous weights—a result not
known before. As a matter of fact, a more general result for product systems,
replacing generalized polynomials, can be proved, see [12]. In fact, very
general minimax, equioscillation and maximin results are given for sum of
translates functions in that paper. The connection to polynomials and product
systems is that we arrive at the sum of translates setup after taking
logarithms of absolute values. For the concrete results we refer to [12].

Here we point out only one further result, which goes beyond existing
knowledge even in the centuries old classical case of the original Chebyshev
extremal problem for the interval. The following is Theorem 4.2 in [12].

######  Theorem 10.3 (Intertwining theorem).

Let $K$ be a singular ($\infty$ ‣ 2), strictly concave and (strictly) monotone
kernel function and let $J:[0,1]\to\underline{\mathbb{R}}$ be an upper
semicontinuous field function. Further, let $n\in\mathbb{N}$ and
$\nu_{j}>0\leavevmode\nobreak\ (j=1,\ldots,n)$ be arbitrary positive numbers.
Put $K_{j}:=\nu_{j}K\leavevmode\nobreak\ (j=1,\ldots,n)$.

Then for nodes $\mathbf{x},\mathbf{y}\in Y$ majorization cannot hold, i.e.,
the coordinatewise inequality
$\mathbf{m}(\mathbf{x})\leq\mathbf{m}(\mathbf{y})$ can only hold if
$\mathbf{x}=\mathbf{y}$.

######  Corollary 10.4.

Consider the (almost) two centuries old classical Chebyshev problem, where in
our terminology $K(t):=\log|t|$, $J(t)\equiv 0$,
$\nu_{j}=1\leavevmode\nobreak\ (j=1,\ldots,n)$, and so we have strict
concavity and monotonicity. Then for any two node systems
$\mathbf{x},\mathbf{y}\in S$ we necessarily have some indices $0\leq i\neq
j\leq n$ such that

$\displaystyle\max_{t\in
I_{i}(\mathbf{x})}\Bigl{|}\prod_{k=1}^{n}(t-x_{k})\Bigr{|}<\max_{t\in
I_{i}(\mathbf{y})}\Bigl{|}\prod_{k=1}^{n}(t-y_{k})\Bigr{|},$
$\displaystyle\max_{t\in
I_{j}(\mathbf{x})}\Bigl{|}\prod_{k=1}^{n}(t-x_{k})\Bigr{|}>\max_{t\in
I_{j}(\mathbf{y})}\Bigl{|}\prod_{k=1}^{n}(t-y_{k})\Bigr{|}.$

######  Remark 10.5.

It seems that even in this very classical situation the above general
statement has not been observed thus far. The special case when one of the
node systems, say $\mathbf{x}$, is the extremal (equioscillating) node system
$\mathbf{w}$, is well known and seems to be folklore. However, comparison of
interval maxima vectors belonging to two arbitrary node systems looks more
complicated and nothing was written about it in the literature, we could page
through.

Already the issue of kernels satisfying the “proportionality condition” is
rich enough to study them in detail. However, we also have initial results in
the more general situation when kernels are not so much related. In these
cases, however—somewhat similarly to the situation here regarding the periodic
case—some more care is needed concerning assumptions on the field $J$. More
precisely, we need not assume extra conditions for singular kernels, but when
the $K_{j}$ may be non-singular, then “joint singularity behavior” of
translates of kernels and the field $J$ itself need to be tuned. For these
questions we hope to return later.

##  11\. Acknowledgment

We are deeply indebted to V.V. Arestov, V.I. Berdyshev and M.V. Deykalova for
providing us useful references and advice regarding classical literature on
Chebyshev type problems and partial maxima functions. Also we are glad to
mention the inspiring atmosphere of the regular Stechkin Summer Schools-
Workshops, which provided us an ideal forum to present and discuss our ever
developing results with a generous and professional community. We have
benefited much from the comments and questions received there. We are also
grateful to the anonymous referee for the suggestions for the improvement of
the paper.

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Bálint Farkas
School of Mathematics and Natural Sciences,
University of Wuppertal
Gaußstraße 20
42119 Wuppertal, Germany

Béla Nagy
Department of Analysis
Bolyai Institute, University of Szeged
Aradi vértanuk tere 1
6720 Szeged, Hungary

Szilárd Gy. Révész
Alfréd Rényi Institute of Mathematics
Reáltanoda utca 13-15
1053 Budapest, Hungary