# On the Generalization of the Kruskal-Szekeres Coordinates: A Global
Conformal Charting of the Reissner–Nordström Spacetime

Ali<EMAIL_ADDRESS>Dejan<EMAIL_ADDRESS>1HEPCOS,
Physics Department, SUNY at Buffalo, Buffalo, New York, USA

###### Abstract

The Kruskal-Szekeres coordinates construction for the Schwarzschild spacetime
could be viewed geometrically as a squeezing of the $t$-line associated with
the asymptotic observer into a single point, at the event horizon $r=2M$.
Starting from this point, we extend the Kruskal charting to spacetimes with
two horizons, in particular the Reissner-Nordström manifold,
$\mathcal{M}_{RN}$. We develop a new method for constructing Kruskal-like
coordinates and find two algebraically distinct classes charting
$\mathcal{M}_{RN}$. We pedagogically illustrate our method by constructing two
compact, conformal, and global coordinate systems labeled $\mathcal{GK_{I}}$
and $\mathcal{GK_{II}}$ for each class respectively. In both coordinates, the
metric differentiability can be promoted to $C^{\infty}$. The conformal metric
factor can be explicitly written in terms of the original $t$ and $r$
coordinates for both charts.

##  I Introudction

Reissner–Nordström (RN) spacetime is a unique, static, spherically symmetric
and asymptotically flat solution to the coupled set of Maxwell equations and
Einstein Field equations. It describes the spacetime with the mass $M$,
measured in the asymptotic region, and a static spherical electric field
sourced by the charge $Q$ in the background, with the corresponding non-zero
stress-energy tensor. Spherical-like coordinates, $(t,r,\theta,\phi)$, known
as the Reissner–Nordström coordinates are the natural coordinates to represent
the metric tensor $g_{\mu\nu}$ [1, 2, 3, 4, 5]. This chart could be assigned
to an asymptotic observer, named Bob, at $r\rightarrow\infty$ equipped with a
clock measuring the time $t$. The RN metric in units $(c=G=1)$ can be written
as

$\mathrm{d}S_{RN}^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)\mathrm{d}t^{2}+\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}\mathrm{~{}d}r^{2}+r^{2}\left(\mathrm{~{}d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right).$
(1)

This coordinate system is ill-defined at two hypersurfaces (horizons). Similar
to the Schwarzschild spacetime, the coordinate singularity $g_{tt}=0$ locates
the Killing horizons of the spacetime related to the Killing vector
$\partial_{t}$.

$\begin{gathered}g_{tt}\left(r_{\pm}\right)=0,\\\
r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}.\end{gathered}$ (2)

For the non-extremal case, $M>Q$, the Reissner–Nordström black hole has an
inner $r_{-}$ and outer $r_{+}$ horizon, which makes its interior somewhat
similar to the interior of the Kerr Spacetime [6, 7]. Further, in the region
$E_{-}=\\{r|0<r<r_{-}\\}$ the metric will have the same signature as in the
region $E_{+}=\\{r|r_{+}<r<\infty\\}$. Consequently, the physical point-like
singularity at $r=0$ is timelike in nature, in drastic disagreement with the
Schwardchild spacelike singularity. The metric is dynamical in the trapped and
anti-trapped regions $E=\\{r|r_{-}<r<r_{+}\\}$, since the $r$ coordinate is
timelike due to the flip of the metric signature in these coordinates [4].

One way to illustrate the incompleteness of this chart around the black hole
horizons is by examining Bob’s clock to time out his girlfriend Alice who is,
for unclear reasons, freely falling towards the RN black hole’s outer horizon.
While Alice measures a finite amount of time, $\Delta\tau$, using her own
clock in her rest frame, Bob measures a significantly dilated duration of
time, $\Delta t$, by timing Alice’s worldline. In other words, Bob will never
see Alice crossing the outer event horizon in his lifetime. Therefore, better
charts are needed there [8].

Finding a new charting system to penetrate null hypersurfaces in different
spacetimes is a long-standing business. Novikov coordinates [9], Lemaître
coordinates [10], Gullstrand–Painlevé coordinates [11, 12],
Eddington–Finkelstein coordinates [13], and Kruskal–Szekeres coordinates [14,
15, 16] are all examples of charts developed to overcome the incompetents of
the Schwarzchild coordinates near the horizon. Some of them have been
generalized to Reissner–Nordström [3] and Kerr spacetimes [17, 18]. Most of
them were constructed by studying time-like and null-like geodesic behavior
around those black holes. However, here we will be adopting a more algebraic
approach based on a geometrical interpretation of the problem analogous to the
one found in [8].

Large astrophysical black holes are expected to be electrically neutral, given
that our universe is electrically neutral at large scales [19]. One exception
may be small primordial black holes that did not live long enough to get
neutralized. Another exception might be black holes charged under some other
hidden $U(1)$ gauge group different from electromagnetism [20]. In addition,
even a small amount of charge on a large black hole could be important when we
encounter certain phenomena such as cosmic rays. This provides enough
motivation to study RN black holes not only for academic interests but also
from a phenomenological point of view [21]. On the other hand, studying the
causal structure of the RN black hole, which is entirely different from the
one associated with the Schwardchild spacetime, is important since it shares
some generic features with other types of black holes with two horizons, e.g.
the Kerr black hole which is much more relevant in astrophysical situations
[7, 22].

Since rotating and charged black holes share a similar interior structure,
this makes constructing Penrose diagrams for the RN metric a cumbersome task
on its own. For example, Klösch and Strobl managed to provide _non-conformal_
global coordinates for the extreme and non-extreme RN spacetimes [23].
However, the attempts to construct conformal global coordinates were so far
based on patching two sets of the Kruskal–Szekeres coordinates
$\mathcal{K_{\pm}}$, where each set is well-behaved on one horizon $r_{\pm}$
while it fails on the other one $r_{\mp}$. This makes the region of validity
for each chart $\mathcal{E_{+}}=E_{+}\cup\\{r|r=r_{+}\\}$ and
$\mathcal{E_{-}}=E_{-}\cup\\{r|r=r_{-}\\}$ respectively. Switching between the
two charts was the key to covering the whole RN manifold and constructing a
global Penrose diagram [24, 25, 22]. Such patched Penrose diagrams, found in
[4] for example, will still prove inconvenient if we want to study geodesics
across the horizons [26]. To overcome this obstacle, we need to construct a
global conformal coordinate system.

Recently in [27], Farshid proposed a _smoothing_ technique that could be used
to provide a $C^{2}$-conformal global chart for the RN spacetime, and pointed
out the possibility of generalizing the method to spherically symmetric
spacetimes. The method used was reported to be a generalization of one used
in[22] aiming to promote the differentiability of the map. One can also find
Penrose diagrams constructed using this method in [28]. The central idea of
this work was to find coordinates that extrapolate to each of the
Kruskal–Szekeres coordinates $\mathcal{K_{\pm}}$ when approaching the horizon
located at $r=r_{\pm}$. In addition, the smoothing was achieved through the
use of the bump functions [29, 30]. A similar technique was used by Schindler
in [31, 32], designed to provide a global chart regular for a special class of
spherically symmetric spacetimes with multiple horizons. The reader can also
find a comprehensive summary of the Penrose diagram theory in chapter one of
Schindler’s doctoral thesis [33].

In this work, we will define a new procedure that can produce compact,
conformal, and global (CCG) charts that are valid at both the inner and outer
horizons of RN spacetime, and for which the metric tensor is $C^{\infty}$.
Using this procedure we will provide two CCG coordinate systems for the RN
spacetime, which we label as type-I and type-II coordinates, based on their
class. Moreover, coordinates provided in [27] could be thought of as
coordinates of type-II. Our method makes no underlying assumptions about the
nature of the spacetime, other than it possesses two horizons. Therefore, to
facilitate future applications of this procedure, we will present here a
detailed pedagogical approach.

The structure of this paper is as follows. In section (II.1), we begin by
reformulating the core idea of the Kruskal chart, and then revisit the Kruskal
charting of the Schwarzchild (II.2) and the RN (II.4) spacetimes. In section
(III), the main procedure for constructing generalized Kruksal charts is
presented. The type-I and type-II coordinates as well as their relaxed
versions for RN spacetime are given in (IV.1) and (IV.2). Finally, we discuss
the outcome of the analysis and possible future work in section (V)

##  II Preliminary

###  II.1 Kruskal–Szekeres coordinates

Kruskal–Szekeres coordinates represent a maximal CCG chart for the
Schwarzschild metric and has been studied extensively in the literature [15,
16, 14, 34, 35]. Their global nature is attributed to two features: (i) they
can cover the null sphere located at radius $r=2M$ which Bob will fail to
chart, and (ii) it is a maximal extension of the Schwarzchild chart
representing two copies of the Schwarzschild universe. The metric written in
the spherical-like coordinate known as the Schwarzschild Spacetime
$(t,r,\theta,\phi)$ 333Since examining the behavior and possible problems of
the spherical coordinates as $r\rightarrow\infty$ falls beyond the scope of
this work, the angular dependence $(\theta,\phi)$ will be neglected from now
on for simplicity. where $t\in\mathbb{R}$, $r\in\mathbb{R}_{+}\char
92\relax\\{0\\}$, $\theta\in(0,\pi)$, and $\phi\in[0,2\pi)$ takes the well-
known form

$dS_{Sch}^{2}=\left(\frac{r-2M}{r}\right)\left\\{-dt^{2}+dr_{*}^{2}\right\\}=\frac{1}{r(r_{*})}\left(r(r_{*})-2M\right)dS_{Con}^{2},$
(3)

where $dS_{Sch}^{2}$ and $dS_{Con}^{2}$ stand for the Schwarzschild and
conformal metric respectively. Here, $r_{*}$ is defined444Usually, the
constant of integration in defining the tortoise coordinate, $r_{*}$, is
chosen to be $-2Mln(2M)$ in order to maintain dimensionless quantity inside
the natural logarithm. Here, for simplicity we omit this step.  as follows

$exp\left(r_{*}\right)=exp\left(r\right)\left|r-2M\right|^{2M}.$ (4)

It is worth emphasizing that the map from $r$-coordinate to its tortoise
version $r_{*}$ is bijective and its inverse is differentiable on each of
$\mathcal{S_{+}}$ and $\mathcal{S_{-}}$ separately. This is obviously due to
the modulus included in the definition of these coordinates in equation (4).

A rigorous procedure would involve solving the Einstein Field Equations in
Kruskal coordinates (which is the _top-down_ approach as in [8, 36]555The
conformal factor in these references is written in terms of $r$, however, it
is more instructive to think of $r(U,V)$ as a function of $U$ and $V$, and not
as the areal coordinate $r$.) by means of null-casting and redefining the null
coordinates. Since the Schwarzschild coordinates cover only the regions
$\mathcal{S_{-}}=\\{r|0<r<2M\\}$ and $\mathcal{S_{+}}=\\{r|2M<r<\infty\\}$ of
one universe of the Kruskal metric, trying to map the local chart to the
global one (i.e. the _bottom-up_ approach) is not quite rigorous, because the
map between the two charts as well as the Jacobian, Hessian, and the higher-
versions of it will be singular at the event horizon $r=2M$[37].

Nevertheless, we seek a global chart in which the metric is at least $C^{2}$
everywhere on the manifold in order to satisfy the coupled Field Equations
which contain first and second derivatives of the metric. Thus, we can apply
this bottom-up approach (as in most of the General Relativity textbooks [2,
38]) by studying the limit at $r=2M$ and analytically continuing the metric
there. At the end, the metric $g_{\mu\nu}$ must be written explicitly in the
Kruskal coordinates $(T,R,\theta,\phi)$ only. In this paper, we will follow
the bottom-up approach to find the generalized Kruksal coordinates which chart
the whole RNspacetime. Taking the Kruskal charting of the Schwarzschild black
hole as our guide, we review the traditional derivation of the Kruskal
coordinates.

###  II.2 Construction of the Kruskal coordinates: Schwardchild Spacetime

We begin by mapping the Schwarzschild coordinates to intermediate null
coordinates first, in particular the retarded ($u$) and advanced ($v$) time
coordinates, defined as $u=t-r_{*}$ and $v=t+r_{*}$. To handle the coordinate
singularity of the former at the horizon, $r=2M$, the null freedom is used to
map the latter set to another set of the null coordinates using $u\rightarrow
U\equiv h(u)$ and $v\rightarrow V\equiv k(v)$. This gives

$dS_{con}^{2}=-dudv=-\dfrac{dUdV}{\dfrac{dh}{du}\dfrac{dk}{dv}}\equiv-\dfrac{Q(U,V)dUdV}{r(U,V)-2M},$
(5)

where $Q(U,V)$ is at least $C^{2}$-function
$\mathcal{S}=\mathcal{S_{+}}\cup\mathcal{S_{-}}\cup\\{r|r=2M\\}$. This is
achieved by employing the definition of $r_{*}$. A sufficient coordinate
transformation is given by

$\displaystyle U\equiv\nu exp\left(\dfrac{-u}{4M}\right),$ $\displaystyle
V\equiv\nu exp\left(\dfrac{v}{4M}\right),$ (6)

where

$\nu=\begin{dcases}+1&r>2M\\\ -1&r<2M\\\ \end{dcases},$ (7)

The signs $\pm$ are included to achieve the maximal analytical extension of
the metric. The product $UV$ is positive in the regions II and III, and
negative in the regions I and IV, following the convention given in [2]. The
$r$ coordinate is defined implicitly as

$UV=exp\left(\frac{r}{2M}\right)(r-2M).$ (8)

This equation can be explicitly solved for $r$ by employing the multi-valued
Lambert function $W$ [39, 40],

$r=2M\left[W\left(\frac{UV}{-2Me}\right)+1\right].$ (9)

Then, the Schwarzchild metric will have the following form in the new double
null coordinates

$dS_{Sch}^{2}=-\frac{16M^{2}e^{-\dfrac{r(U,V)}{2M}}}{r(U,V)}dUdV+r^{2}(U,V)d\Omega^{2}.$
(10)

Finally, the Kruskal coordinates $T_{KS}$ and $R_{KS}$ are related to the new
null coordinates through the following transformations

$\begin{array}[]{l}U\equiv\frac{1}{2}\left(T_{KS}-R_{KS}\right),\\\
V\equiv\frac{1}{2}\left(T_{KS}+R_{KS}\right).\end{array}$ (11)

It is worth writing the final version of the metric in the Kruskal coordinates
as

$\begin{gathered}dS_{Sch}^{2}=\frac{8Mexp\left(-W\left(T_{KS},R_{KS}\right)-1\right)}{W\left(T_{KS},R_{KS}\right)+1}(-dT_{KS}^{2}+dR_{KS}^{2})\\\
+4M^{2}\left(W\left(T_{KS},R_{KS}\right)+1\right)^{2}d\Omega^{2}.\end{gathered}$
(12)

As a cross-check, one could verify that the Einstein tensor $G_{\mu\nu}$
corresponding to the Kruskal metric is zero everywhere on the Schwarzschild
manifold, thus confirming that the stress-energy tensor $T_{\mu\nu}$ is
identically zero (as it must be for the Schwarzschild solution). This is true
despite the fact that taking the derivatives of the metric with respect to the
coordinates $(T,R)$ (using implicit differentiation with respect to $(t,r)$)
will be ill-defined at the event horizon. One could also verify that the maps
between the Kruskal and the Schwarzschild chart are diffeomorphic in the
regions $S_{+}$ and $S_{-}$ [37].

###  II.3 A geometric picture of the Kruskal charting

The procedure of constructing Kruskal coordinates for Schwarzschild spacetime
outlined in the previous section becomes limited when applied to spacetimes
with more than one horizon. To be able to resolve this obstacle, we re-
interpret the main premise of the construction. If Bob lived in a four-
dimensional Minkowski spacetime, his clock would be able to properly time the
events taking place there globally. However, once the spacetime is only
asymptotically Minkowskian, the chart will fail near the null hypersurfaces.
_But what if we start with a chart in the flat spacetime which is ill-defined
at the locations defining these null hypersurfaces?_ For example, we can
define a ”bad” chart $\mathcal{Z}$ in the conformal spacetime with the metric
$g^{Con}_{\mu\nu}$, in which any given time duration $\Delta\tau$ of Alice’s
trip to the $r=2M$666In our analysis the conformal spacetime is Minkowski, so
there are no horizons at the radius $r=2M$. is mapped to
$\Delta\tilde{t}\rightarrow 0$.

Apparently, there is a family of these “ _bad_ ” charts $\mathcal{Z}$ that
would be well defined on the physical spacetime, with the metric
$g_{\mu\nu}=\omega^{2}(x)g^{Con}_{\mu\nu}$, where $\omega(x)$ is the conformal
factor. They are only conditioned to contract the time interval $\Delta\tau$
at the same rate as the dilation of time in Bob’s frame. One can find an
equivalent argument in [8] that we quote here ” _A better coordinate system,
one begins to believe, will take these two points at infinity and spread them
out into a line in a new ( $r_{new},t_{new}$)-plane; and will squeeze the line
($r=2M,t$ from $-\infty$ to $\infty$) into a single point in the
($r_{new},t_{new}$)-plane_”.

As we will show here later, applying this simple argument to spacetimes with
more than one horizon would be a tedious algebraic task. Mathematically, the
fundamental premise of the construction is to find conformal coordinates
$\mathcal{Z}$ that generate poles of the same rank as zeros of the conformal
factor. Then as the zeros and poles cancel out, the physical metric in
$\mathcal{Z}$ will be CCG, in light of the bottom-up approach. In the next
subsections, we will review the Kruskal charting of the RN spacetime following
the notation in [3, 22].

###  II.4 Outer and inner Kruskal coordinatse: Reissner Nordstrom spacetime

One example where the standard Schwarzchild-like Kruskal charting will fail in
constructing CCG is the RN spacetime.

$\begin{gathered}dS_{RN}^{2}=\frac{\left(r-r_{+}\right)\left(r-r_{-}\right)}{r^{2}}\left\\{-dt^{2}+dr_{*}^{2}\right\\}=\frac{\left(r-r_{+}\right)\left(r-r_{-}\right)}{r^{2}}dS_{Con}^{2},\\\
dS_{Con}^{2}=-dudv,\end{gathered}$ (13)

where $dS_{RN}^{2}$ stands for RN metric, while $(u,v)$ represents the double
null coordinates constructed in the same manner as in the Schwardchild case.
The RN radial tortoise coordinate $r_{*}$ is defined as

$\begin{gathered}exp\left(\large
r_{*}\right)=exp\left(r\right)\left|r-r_{+}\right|^{\dfrac{\alpha_{+}}{2}}\left|r-r_{-}\right|^{-\dfrac{\alpha_{-}}{2}},\\\
\alpha_{+}\equiv\frac{2r_{+}^{2}}{r_{+}-r_{-}},\\\
\alpha_{-}\equiv\frac{2r_{-}^{2}}{r_{+}-r_{-}},\\\ \end{gathered}$ (14)

where $\alpha_{-}$ and $\alpha_{+}$ are the surfaces gravity at $r_{-}$ and
$r_{+}$ respectively.

Similar to the Schwardchild tortoise coordinate, $r_{*}(r)$ is bijective and
its inverse is differentiable on $E_{+}$, $E$, and $E_{-}$ separately.
However, there is a potential to solve explicitly for $r$ by employing
generalized Lambert functions $\mathcal{W}$ [41, 42, 43, 44]. Since this is a
tedious task on its own, we confine our analysis to the main objective, while
this step could be addressed in future work.

By examining the tortoise coordinate definition, it is obvious that a zero at
$r_{\pm}$ is always coupled with a pole at $r_{\mp}$, hence it is not
straightforward to factor out a product of simple poles at $r_{+}$ and $r_{-}$
in the conformal metric. Nevertheless, it remains possible to construct
regular charts at one horizon that is ill-defined at the other. These
coordinates are regular in the domain $\mathcal{E_{+}}$ and $\mathcal{E_{-}}$,
respectively. The outer $\mathcal{K_{+}}$ and inner $\mathcal{K_{-}}$ Kruskal
coordinates are simply related to the “$-$” null-chart
$(\mathcal{U}_{-},\mathcal{V}_{-})$ and “$+$” null-chart
$(\mathcal{U}_{+},\mathcal{V}_{+})$ following the same definition in (11). We
will work with the following sign convention

$\begin{aligned} \mathcal{U}_{+}&=\nu_{+}U_{+},\quad\\\
U_{+}&=exp\left(\frac{-u}{\alpha_{+}}\right),\quad\quad\\\
\end{aligned}\begin{aligned} \mathcal{V}_{+}&=\nu_{+}V_{+}\\\
V_{+}&=exp\left(\frac{v}{\alpha_{+}}\right),\\\ \end{aligned}$ (15)

where

$\nu_{+}=\begin{dcases}+1&r>r_{+}\\\ -1&r<r_{+}\\\ \end{dcases},$ (16)

to represent the maximal analytical extension of these coordinates. Then the
$t$ and $r$ coordinates are characterized by the following curves in the
$(\mathcal{U}_{+},\mathcal{V}_{+})$-plane:

$\begin{gathered}\mathcal{U}_{+}\mathcal{V}_{+}=exp\left(\frac{r}{2\alpha_{+}}\right)\left(r-r_{+}\right)\left|r-r_{-}\right|^{-\alpha},\\\
\frac{\mathcal{V}_{+}}{\mathcal{U}_{+}}=\pm
exp\left(+\dfrac{2t}{\alpha_{+}}\right).\end{gathered}$ (17)

Similarly,

$\begin{aligned} \mathcal{U}_{-}&=\nu_{-}U_{-},\quad\\\
U_{-}&=exp\left(\frac{u}{\alpha_{-}}\right),\quad\quad\\\
\end{aligned}\begin{aligned} \mathcal{V}_{-}&=\nu_{-}V_{-},\quad\\\
V_{-}&=exp\left(\frac{-v}{\alpha_{-}}\right)\\\ \end{aligned}$ (18)

where

$\nu_{-}=\begin{dcases}+1&r>r_{-}\\\ -1&r<r_{-}.\end{dcases}$ (19)

The $t$ and $r$ curves in the $(\mathcal{U}_{-},\mathcal{V}_{-})$-plane are
defined as

$\begin{gathered}\mathcal{U}_{-}\mathcal{V}_{-}=exp\left(-\frac{r}{2\alpha_{-}}\right)\left(r-r_{-}\right)\left|r-r_{+}\right|^{-\bar{\alpha}},\\\
\frac{\mathcal{V}_{-}}{\mathcal{U}_{-}}=\pm
exp\left(-\dfrac{2t}{\alpha_{-}}\right),\end{gathered}$ (20)

Consequently, the metric in these “+” or “-” null-charts becomes

$\displaystyle dS_{RN}^{2}$
$\displaystyle=-\alpha_{\pm}\frac{\left(r-r_{+}\right)\left(r-r_{-}\right)}{r^{2}}\frac{d\mathcal{U}_{\pm}d\mathcal{V}_{\pm}}{\mathcal{U}_{\pm}\mathcal{V}_{\pm}}$
(21)
$\displaystyle=-\alpha_{+}\frac{exp\left(-\dfrac{2r}{\alpha_{+}}\right)}{r^{2}}\left(r-r_{-}\right)^{1+\alpha}d\mathcal{U}_{+}d\mathcal{V}_{+}$
$\displaystyle=-\alpha_{-}\frac{exp\left(\dfrac{2r}{\alpha_{-}}\right)}{r^{2}}\left(r_{+}-r\right)^{1+\bar{\alpha}}d\mathcal{U}_{-}d\mathcal{V}_{-},$

where777The extreme cases ($Q=M$ and $Q>M$) of the RN metric are not
considered here.

$\displaystyle\alpha\equiv\frac{\alpha_{-}}{\alpha_{+}}=\left(\frac{r_{-}}{r_{+}}\right)^{2}\rightarrow$
$\displaystyle 0<\alpha<1,$ (22)
$\displaystyle\bar{\alpha}\equiv\frac{\alpha_{+}}{\alpha_{-}}=\left(\frac{r_{+}}{r_{-}}\right)^{2}\rightarrow$
$\displaystyle 1<\bar{\alpha}.$

It is easy to check that the metric in “$+$” (“$-$”) null-coordinates is
regular at the outer (inner) horizon $r_{+}$($r_{-})$. However, the
coordinates fail 888$dS^{2}_{RN}=0$ at $r=r_{-}$ ($r_{+}$) in the “$+$”
(“$-$”) coordinates according to equation (21). at the inner (outer) horizon
$r_{-}$ ($r_{+}$). Moreover, the metric in the “$+$”-null coordinates is not
asymptotically flat in agreement with the Schwarzschild induced metric defined
on the hypersurfaces with fixed $\theta$ and $\phi$ in equation (12), where
the conformal factor approaches zero as $r\rightarrow\infty$. Nevertheless,
global Kruskal coordinates could be built by combining these two definitions
in (15, 18) together (see e.g. the work of Carter[24], Hamilton[22]
Schindler[31], and Farshid [27]). Although they all managed to find regular
metric across the horizon, still the metric is only $C^{2}$ in the former
case.

##  III Global Conformal Chart Criteria

We start our analysis by studying the conditions needed for a valid conformal
global chart. We want to map the double null coordinates $(u,v)$ to the global
double null coordinates $(\tilde{u},\tilde{v})$, while still maintaining the
light-cone invariance in the new coordinates $(\tilde{u},\tilde{v})$. The most
direct way to achieve that will be to use only the null-freedom as

$\displaystyle\tilde{u}\equiv h(u)\quad\rightarrow$ $\displaystyle
du=\frac{1}{\dfrac{dh}{du}}d\tilde{u},$ (23) $\displaystyle\tilde{v}\equiv
k(v)\quad\rightarrow$ $\displaystyle dv=\frac{1}{\dfrac{dk}{dv}}d\tilde{v}.$

To construct a well-defined chart on the entire Reissner–Nordström manifold,
we identify three distinct possibilities with reference to the singularity
structure of the term $\dfrac{dh}{du}\dfrac{dk}{dv}$, focusing on its behavior
at $r=r_{-}$ and $r=r_{+}$. The three options are:

  1. 1.

_Type-O_ : $\dfrac{dh}{du}\dfrac{dk}{dv}$ has a zero either at $r=r_{-}$ or
$r=r_{+}$. The regularity of the metric in the new $(\tilde{u},\tilde{v})$
coordinates would be achieved at $r=r_{-}$ or $r=r_{+}$ but not
simultaneously. “$\pm$” null coordinates are examples for this case. However,
generating nontrivial coordinates out of $(U_{\pm},V_{\pm})$ is possible999The
transformations that lead to such coordinates are expected to be more
complicated as they are restricted by the requirement to leave the singularity
structure invariant or to generate a decoupled zero at the other horizon.
101010For example, the transformation of the form $f(U)\equiv U^{p}$ and
$g(V)\equiv V^{p}$ where $p\in\mathbb{R}$ are allowed since they clearly
change the singularity structure of the term $\dfrac{dh}{du}\dfrac{dk}{dv}$..
This condition could be formulated as follows

$\frac{dh}{du}\frac{dk}{dv}=\left(r-r_{\pm}\right)\zeta\left(r_{*},t\right)$
(24)

  2. 2.

_Type-I_ : $\dfrac{dh}{du}\dfrac{dk}{dv}$ has a product of zeros at $r=r_{-}$
and $r=r_{+}$.

If we manage to factor out this product of zeros while keeping the associated
poles decoupled, then we will have a conformal global coordinate for the RN
spacetime. We will illustrate this case with an example in IV.1. This
condition can be formulated as

$\frac{dh}{du}\frac{dk}{dv}=\left(r-r_{+}\right)\left(r-r_{-}\right)\gamma\left(r_{*},t\right)$
(25)

  3. 3.

_Type-II_ : a sum of decoupled simple zeros at $r=r_{+}$ and $r=r_{-}$, each
coupled to a pole, and possibly zeros of constrained rank at $r=r_{-}$ for
former and $r=r_{+}$ for later. In principle, this mixture of poles and zeros
might be easier to find compared to _Type-I_ , however, the metric is expected
to be more complicated form-wise. We will illustrate this case with an example
in IV.2. This condition can be formulated as

$\displaystyle\frac{dh}{du}\frac{dk}{dv}=(r-r_{+}){M_{+}(r_{*},t)}+(r-r_{-})M_{-}(r_{*},t)+\beta(r_{*},t),$
(26)

The three differential equations listed above are sufficient to construct the
desired singularity structure in each case, while the constraints are encoded
within $\zeta$, $\gamma$, $M_{\pm}$ and $\beta$.

##  IV Constructing CCGs for Reissner–Nordström spacetime

###  IV.1 Type-I CCG Global Chart

As we mentioned before, just by looking at the definition of $r_{*}$, there is
no simple way of factorizing the zeros $(r-r_{+})$ or $(r-r_{-})$ without
invoking poles at $r_{-}$ or $r_{+}$. Still, we can consider combining
equations (20) and (17)

$\mathcal{U}_{+}\mathcal{U}_{-}\mathcal{V}_{+}\mathcal{V}_{-}=\frac{\left(r-r_{+}\right)}{\left|r-r_{+}\right|^{\bar{\alpha}}}\frac{\left(r-r_{-}\right)}{\left|r-r_{-}\right|^{\alpha}}exp\left({\dfrac{r}{2\alpha_{+}}}\right)exp\left(-\dfrac{r}{2\alpha_{-}}\right)$
(27)

This may give us a hint for how to find $(\tilde{u},\tilde{v})$ with the
desired map to fulfill the singularity structure of type-I. For example, we
can start with the following definitions of $\mathcal{GK_{I}}$

$\begin{aligned}
\frac{dh}{du}=\frac{\mu}{U_{+}^{-1}+U_{-}^{-1}},\quad\quad\quad\end{aligned}\begin{aligned}
\frac{dk}{dv}=\frac{\mu}{V_{+}^{-1}+V_{-}^{-1}},\quad\quad\quad\end{aligned}$
(28)

where

$\mu=\begin{dcases}+1&r>r_{+}\mid\hskip 2.84526pt\mid r<r_{-}\\\
-1&r_{-}<r<r_{+}.\\\ \end{dcases}$ (29)

The definition we give in (28) reduces to evaluating the $I_{1}$-integration
given here

$I_{1}=\int\frac{1}{x^{q}+1}dx,$ (30)

where $q>1$. This integration has an upper and lower bound, hence the sign
convention we use here will locate the inner horizon $r_{-}$, outer horizon
$r_{+}$ and the asymptotically flat region $r\rightarrow\infty$ according to
the choice of the reference point. We choose that point to be the outer
horizon $u\rightarrow-\infty$ ($v\rightarrow\infty$). Accordingly, we have a
monotonic map from $u$($v$), defined in any of the regions $E_{\pm}$ and $E$;
to $\tilde{u}$ ($\tilde{v}$). Moreover, the map from $(t,r)$ to the later
coordinates is also _globally monotonic_. Moreover, those coordinates have a
built-in _compact_ domain and, hence, could be used directly to construct
Penrose diagrams for the RN spacetime. The choice of signs $\mu$ does not harm
the continuity or differentiability of the map, still, it will result in a
uniform signature of the Generalized Kruskal Coordinate of type-I.
Nevertheless, it is sufficient if we have only a semi-positive conformal
factor. Accordingly, the metric in the CCG type-I coordinates will take the
following form.

$\displaystyle dS_{RN}^{2}$
$\displaystyle=-\frac{1}{r^{2}}\left\\{\left|r-r_{-}\right|^{\alpha+1}exp\left(-\frac{2r}{\alpha_{+}}\right)+\left|r-r_{+}\right|^{\bar{\alpha}+1}exp\left(\frac{2r}{\alpha_{-}}\right)\right.$
(31)
$\displaystyle\left.+2\cosh\left[t\left(\frac{1}{\alpha_{+}}+\frac{1}{\alpha_{-}}\right)\right]exp\left(r\left[\frac{-1}{\alpha_{+}}+\frac{1}{\alpha_{-}}\right]\right)\left|r-r_{+}\right|^{\frac{\bar{\alpha}+1}{2}}\left|r-r_{-}\right|^{\frac{\alpha+1}{2}}\right\\}d\tilde{u}d\tilde{v}$

The metric possesses a conformal factor resembling the sum of the conformal
factors of the $\mathcal{K_{\pm}}$ in addition to a new time-dependent term
that vanishes on both horizons. The conformal factor shown in equation (31) is
a semi-positive function over the domain of the $r$-coordinate, hence the
metric is well-behaved on both of the horizons and takes the following
asymptotic behavior as $r\rightarrow r_{+}$,

$dS_{RN}^{2}(r\rightarrow
r_{+})\rightarrow-\frac{exp\left(-\dfrac{2r}{\alpha_{+}}\right)}{r^{2}}\left(r-r_{-}\right)^{1+\alpha}d\tilde{u}d\tilde{v},$
(32)

Similarly as $r\rightarrow r_{-}$,

$dS_{RN}^{2}(r\rightarrow
r_{-})\rightarrow-\frac{exp\left(\dfrac{2r}{\alpha_{-}}\right)}{r^{2}}\left(r_{+}-r\right)^{1+\bar{\alpha}}d\tilde{u}d\tilde{v},$
(33)

Executing the integration in $I_{1}$ might be doable through the use of
Hypergeometric functions ${}_{2}F_{1}(a,b;c;x)$ [45, 46, 47] or generically
through expanding the function under the integration sign and then
integrating.

However, this is not the end of the story. Similar to the Schwarzschild case,
the Jacobian and its higher relatives will be undefined at the horizons, thus
it is not reliable to take the derivatives of the conformal factor implicitly.
One can also argue that there should be two kinks present at $r=r_{+}$ and
$r=r_{-}$, due to the existence of the modules in the conformal factor as well
as the monotonic map between $r$ and $(\tilde{u},\tilde{v})$. However, we can
get rid of these kinks, for instance, through the use of relaxation functions.

In short, another set of coordinates $(\tilde{u},\tilde{v})$ can be introduced
which will inherit the properties mentioned above and possess a relaxed (well-
behaved and smooth) conformal factor at $r_{+}$ and $r_{-}$. As a consequence,
the metric will be guaranteed to be $C^{\infty}$ in these new coordinates. We
choose the function $\tanh{x}$ to do this job. The relaxed coordinate
transformation is

$\begin{gathered}\frac{dh}{du}=\frac{\mu}{\tanh\left[U_{-}^{2}\right]U_{+}^{-1}+\tanh\left[U_{+}^{2}\right]U_{-}^{-1}},\\\
\frac{dk}{dv}=\frac{\mu}{\tanh\left[V_{-}^{2}\right]V_{+}^{-1}+\tanh\left[V_{+}^{2}\right]V_{-}^{-1}}.\end{gathered}$
(34)

The metric now becomes

$\displaystyle dS_{RN}^{2}$
$\displaystyle=-\frac{1}{r^{2}}\left\\{Q_{-}(r,t)\left|r-r_{-}\right|^{\alpha+1}exp\left(-\frac{2r}{\alpha_{+}}\right)+Q_{+}(r,t)\left|r-r_{+}\right|^{\bar{\alpha}+1}exp\left(\frac{2r}{\alpha_{-}}\right)\right.$
(35)
$\displaystyle\left.+\tilde{Q}(r,t)exp\left(r\left[\frac{-1}{\alpha_{+}}+\frac{1}{\alpha_{-}}\right]\right)\left|r-r_{+}\right|^{\frac{\bar{\alpha}+1}{2}}\left|r-r_{-}\right|^{\frac{\alpha+1}{2}}\right\\}d\tilde{u}d\tilde{v},$

where $Q_{-}$, $Q_{+}$, and $\tilde{Q}$ are defined as

$\displaystyle
Q_{-}(r,t)=\tanh\left[exp\left(\frac{2(t-r)}{\alpha_{-}}\right)\frac{\left|r-r_{-}\right|}{\left|r-r_{+}\right|^{\bar{\alpha}}}\right]\tanh\left[exp\left(\frac{-2(t+r)}{\alpha_{-}}\right)\frac{\left|r-r_{-}\right|}{\left|r-r_{+}\right|^{\bar{\alpha}}}\right]$
(36) $\displaystyle
Q_{+}(r,t)=\tanh\left[exp\left(\frac{2(r-t)}{\alpha_{+}}\right)\frac{\left|r-r_{+}\right|}{\left|r-r_{-}\right|^{\alpha}}\right]\tanh\left[exp\left(\frac{-2(-t+r)}{\alpha_{+}}\right)\frac{\left|r-r_{+}\right|}{\left|r-r_{-}\right|^{\alpha}}\right]$
$\displaystyle\tilde{Q}(r,t)=Q_{1}(r,t)exp\left(t\left(\frac{1}{\alpha_{+}}+\frac{1}{\alpha_{-}}\right)\right)+Q_{2}(r,t)exp\left(-t\left(\frac{1}{\alpha_{+}}+\frac{1}{\alpha_{-}}\right)\right)$
$\displaystyle
Q_{1}(r,t)=\tanh\left[exp\left(\frac{2(t-r)}{\alpha_{-}}\right)\frac{\left|r-r_{-}\right|}{\left|r-r_{+}\right|^{\bar{\alpha}}}\right]\tanh\left[exp\left(\frac{-2(-t+r)}{\alpha_{+}}\right)\frac{\left|r-r_{+}\right|}{\left|r-r_{-}\right|^{\alpha}}\right]$
$\displaystyle
Q_{2}(r,t)=\tanh\left[exp\left(\frac{-2(t+r)}{\alpha_{-}}\right)\frac{\left|r-r_{-}\right|}{\left|r-r_{+}\right|^{\bar{\alpha}}}\right]\tanh\left[exp\left(\frac{2(r-t)}{\alpha_{+}}\right)\frac{\left|r-r_{+}\right|}{\left|r-r_{-}\right|^{\alpha}}\right]$

This relaxed version of the conformal factor is guaranteed to be smooth and
semi-positive everywhere in coordinates $(\tilde{u},\tilde{v})$. Before we
move to construct the type-II coordinates $\mathcal{GK_{II}}$, there are three
features of the metric worth commenting on. First, the metric is not
asymptotically flat and is different from the $\mathcal{K_{\pm}}$ coordinates
where the induced metric on the submanifold $M_{2}=\mathcal{M}\char 92\relax
SO(3)$ is asymptotically vanishing. In $\mathcal{GK_{I}}$ coordinates the
induced metric on $M_{2}$ blows up. This is completely natural as the
coordinates are compact, hence the proper distance is invariant. Second, the
$\mathcal{GK_{I}}$ coordinates are dynamically casting the metric since the
conformal factor includes explicit time dependence after and before the
relaxation. This prevents the $r$ and $t$ from being related to
$(\tilde{u},\tilde{v})$ by simple transformation similar to (17,20). Third,
the integral $I_{2}$ defining $(\tilde{u},\tilde{v})$ is given by

$I_{2}=\int\frac{dx}{\tanh\left(x^{2}\right)x^{q+1}+\tanh\left(x^{-2q}\right)},$
(37)

The $q>1$ cases could be evaluated numerically, however, analytical methods
could still be helpful in studying the relation between $\mathcal{K_{\pm}}$
and $\mathcal{GK_{I}}$ at any point. This could be achieved for example by
employing series expansion, as mentioned earlier. Moreover, if we manage to
invert equations (34) to solve explicitly for the null coordinates in terms of
$\mathcal{GK_{I}}$, then we could employ the generalized Lambert function to
solve for $(t,r)$ explicitly as well. Such an expansion is expected to recover
equations (20) and (17) near to the horizons $r=r_{-}$ and $r=r_{+}$,
respectively.

###  IV.2 Type-II Global Chart

While constructing $\mathcal{GK_{I}}$, a simple zero at each horizon $r_{\pm}$
was a coupled one at the other horizon $r_{\mp}$. This product of zeros had a
semi-positive regular amplitude everywhere as shown in equation (31) or (35).
However, for $\mathcal{GK_{II}}$ we will have a different singularity
structure that serves the same purpose: sum of two zeros at each horizon
$r_{\pm}$, each coupled to a semi-positive singular amplitude at the other
horizon $r_{\mp}$ that is singular at the other horizon. In principle, this
class of charts should contain families of coordinates at which the
coordinates themselves are extrapolation between two Outer and Inner Kruksal
coordinates. In light of this statement, the chart given in [27] plausibly
belongs to that class.

The conformal metric will have a simple pole at $r=r_{\pm}$ coupled to
$M_{\pm}(r_{*},t)$, while $\beta\left(r_{*},t\right)$ is effectively a
residual term for completeness. $M_{\pm}(r_{*},t)$ and $\beta(r_{*},t)$ are
satisfying the following constraints. As $r\rightarrow r_{\pm}$

$\displaystyle M_{\pm}\left(r_{*},t\right)$ $\displaystyle\rightarrow
constant,$ (38) $\displaystyle M_{\mp}\left(r_{*},t\right)$
$\displaystyle\rightarrow 0,$ $\displaystyle\beta\left(r_{*},t\right)$
$\displaystyle\rightarrow 0.$

Alternatively, we can restate the first constraint as: $M_{\pm}$ must have no
overall pole at $r_{+}$ ($r_{-}$). Later, through this analysis, we will learn
that $\beta$ will be the key to finding the global conformal charts in this
procedure for the type-II coordinates. Given equation (14), we can rewrite
this in terms of the $\mathcal{K_{\pm}}$ or the double null coordinates as
follows

$\displaystyle\frac{dh}{du}\frac{dk}{dv}=\dfrac{exp\left(\dfrac{2(r_{*}-r_{+})}{\alpha_{+}}\right)}{\left|r-r_{-}\right|^{\alpha}}M_{+}(r_{*},t)+\dfrac{exp\left(-\dfrac{2(r_{*}-r_{-})}{\alpha_{-}}\right)}{\left|r-r_{+}\right|^{\bar{\alpha}}}M_{-}(r_{*},t)+\beta(r_{*},t),$
(39)

Revisiting the condition in equation (38), $M_{+}$ ($M_{-}$) must have zeros
at $r=r_{-}$ ($r=r_{+}$) of rank higher than $\alpha$ ($\bar{\alpha}$)
respectively. Searching for solutions for equation (39) could be more fruitful
if we were able to find functions $M_{\pm}$ and $\beta$ with $(r_{*}\pm t)$
dependence. Accordingly, the residual term $\beta$ could be used to easily
factorize the right-hand side of the equation (39) into a product of $u$\- and
$v$-dependent functions. The task of generating a solution to equation (39) is
not trivial, but if we find $M_{\pm}(u,v)$ and $\beta(u,v)$, this will boost
our progress towards achieving this task. The easiest hint we can get from the
form of that equation is to try to construct $M_{+}(M_{-})$ from the
$\mathcal{K_{+}}$ ($\mathcal{K_{-}}$). Following this logic, using the trial
and error method, we learn that if we define $M_{\pm}$ as

$\begin{aligned}
M_{+}\equiv\frac{\mu\mu}{\left(1+U_{+}^{1+2\bar{\alpha}}\right)\left(1+V_{+}^{1+2\bar{\alpha}}\right)},\quad\quad\quad\end{aligned}\begin{aligned}
M_{-}\equiv\frac{\mu\mu}{\left(1+U_{-}^{2}\right)\left(1+V_{-}^{2}\right)},\end{aligned}$
(40)

we can find $\beta$ that can do the factorization for us

$\beta\equiv\frac{\mu\mu
U_{+}V_{-}}{\left(1+U_{+}^{1+2\bar{\alpha}}\right)\left(1+V_{-}^{2}\right)}+\frac{\mu\mu
U_{-}V_{+}}{\left(1+V_{+}^{1+2\bar{\alpha}}\right)\left(1+U_{-}^{2}\right)}.$
(41)

This will leave us eventually with the following choices for $\dfrac{dh}{du}$
and $\dfrac{dk}{dv}$

$\displaystyle\frac{dh}{du}$
$\displaystyle=\mu\left[\frac{U_{+}}{1+U_{+}^{1+2\bar{\alpha}}}+\frac{U_{-}}{1+U_{-}^{2}}\right]$
(42) $\displaystyle\frac{dk}{dv}$
$\displaystyle=\mu\left[\frac{V_{+}}{1+V_{+}^{1+2\bar{\alpha}}}+\frac{V_{-}}{1+V_{-}^{2}}\right]$

One more time, the choice we made for $\mathcal{GK_{II}}$ coordinates is
naturally compact which means we can use those coordinates directly to build
the Penrose diagrams. Nonetheless, the integration in terms of $(u,v)$ is
significantly simpler than the one used in the example we give for the
$\mathcal{GK_{I}}$. Again, our choice of integration reference point will be
the outer horizon $r_{+}$. We can now write the metric

$\displaystyle dS_{RN}^{2}$ $\displaystyle=-\frac{1}{r^{2}}\left\\{\
A_{+}^{-1}(r,t)+A_{-}^{-1}(r,t)+A^{-1}(r,t)\right\\}^{-1}d\tilde{u}d\tilde{v}$
(43)

where $A_{\pm}(r,t)$ and $A(r,t)$ are defined as follows

$\displaystyle A_{+}(r,t)$ $\displaystyle\equiv
exp\left(-\frac{2r}{\alpha_{+}}\right)\left|r-r_{-}\right|^{\alpha+1}+exp\left(\frac{2r}{\alpha_{-}}\right)\left|r-r_{+}\right|^{1+2\bar{\alpha}}\left|r-r_{-}\right|^{-1}$
(44) $\displaystyle
2\cosh\left[t\left(\frac{1}{\alpha_{+}}+\frac{2}{\alpha_{-}}\right)\right]exp\left(r\left[-\frac{1}{\alpha_{+}}+\frac{2}{\alpha_{-}}\right]\right)\left|r-r_{+}\right|^{\bar{\alpha}+\frac{1}{2}}\left|r-r_{-}\right|^{\frac{\alpha}{2}}$
$\displaystyle A_{-}(r,t)$ $\displaystyle\equiv
exp\left(\frac{2r}{\alpha_{-}}\right)\left|r-r_{+}\right|^{\bar{\alpha}+1}+exp\left(-\frac{2r}{\alpha_{-}}\right)\left|r-r_{-}\right|^{2}\left|r-r_{+}\right|^{-\bar{\alpha}+1}$
$\displaystyle+2\cosh\left[\frac{2t}{\alpha_{-}}\right]\left|r-r_{-}\right|\left|r-r_{+}\right|$
$\displaystyle A(r,t)$ $\displaystyle\equiv 2\cosh\left[\kappa
t\right]\exp(-\bar{\kappa}r)\left|r-r_{+}\right|^{\frac{1+\bar{\alpha}}{2}}|r-r_{-}|^{\frac{1+\alpha}{2}}$
$\displaystyle+2\cosh\left[\bar{\kappa}t\right]exp\left(-\kappa
r\right)\left|r-r_{+}\right|^{\frac{1-\bar{\alpha}}{2}}|r-r_{-}|^{\frac{3+\alpha}{2}}$
$\displaystyle+2\cosh\left[\frac{-t}{\alpha_{-}}\right]exp\left(\frac{3r}{\alpha_{-}}\right)\left|r-r_{+}\right|^{\frac{2+3\bar{\alpha}}{2}}|r-r_{-}|^{\frac{-1}{2}}$
$\displaystyle+2\cosh\left[\frac{-3t}{\alpha_{-}}\right]exp\left(\frac{r}{\alpha_{-}}\right)\left|r-r_{+}\right|^{\frac{2+\bar{\alpha}}{2}}|r-r_{-}|^{\frac{1}{2}},$

with the following limits

$\displaystyle A_{+}\left(r\rightarrow r_{+},t\right)\rightarrow
exp\left(-\frac{2r}{\alpha_{+}}\right)\left|r-r_{-}\right|^{\alpha+1},$ (45)
$\displaystyle A_{+}\left(r\rightarrow r_{\pm},t\right)\rightarrow
exp\left(\frac{2r}{\alpha_{-}}\right)\left|r-r_{+}\right|^{\bar{\alpha}+1},$
$\displaystyle A_{\pm}\left(r\rightarrow r_{\mp},t\right)\rightarrow\infty,$
$\displaystyle A\left(r_{\rightarrow}r_{\pm},t\right)\rightarrow\infty.$

Consequently, the metric will take the following asymptotic limit

$\displaystyle dS_{RN}^{2}(r\rightarrow r_{+})$
$\displaystyle=-\frac{e^{\frac{-2r}{\alpha_{+}}}|r-r_{-}|^{1+\alpha}}{r^{2}}d\tilde{u}d\tilde{v},$
(46) $\displaystyle dS_{RN}^{2}(r\rightarrow r_{-})$
$\displaystyle=-\frac{e^{\frac{2r}{\alpha_{-}}}|r-r_{+}|^{1+\bar{\alpha}}}{r^{2}}d\tilde{u}d\tilde{v}.$

##  V Discussion and Conclusion

After reinterpreting the premises of the Kruskal charting of the Schwarzschild
spacetime, we were able to provide a new approach to chart the
Reissner–Nordström spacetime featuring two horizons. The technique itself
showed to be employable in two distinctive ways, resulting in two families of
charting systems: conformal global _type-I_ and _type-II_ charts. In both
cases, the asymptotic form of the metric approaches the form of metric written
in terms of _Type-O_ charts. We illustrated the success of the provided
technique by constructing compact conformal global coordinates of type-I
$\mathcal{GK_{I}}$ and of type-II $\mathcal{GK_{II}}$ for the RN spacetime.
The price we pay for covering the whole spacetime with two horizons with only
one chart is time dependence.

After the construction, one could conclude that the metric is only $C^{1}$
since the map between $r$ and each of type-I $\mathcal{GK_{I}}$ and of type-II
$\mathcal{GK_{II}}$ is monotonic and smooth, and also due to the behavior of
the conformal map written in terms of $t$ and $r$ functions. Consequently, for
both type-I and type-II charts, the metric becomes $C^{\infty}$ only if we add
an extra step in the procedure. This additional step aims to promote
differentiability if and only if the metric has kinks and was applied by
employing relaxing functions to modify the charts at the kinks’ locations. As
expected, it is complicated to write the generalized Kruskal coordinates
$(\tilde{u},\tilde{v})$ explicitly in terms of the RN coordinates $(t,r)$,
related through equation ($\ref{radialplus},\ref{radialminus}$). However, we
hinted that this could be achieved by utilizing the generalized Lambert
function $\mathcal{W}$ in similar manner to the use of the Lambert function
$W$ in Schwarzchild case.

We proved that the domain of $r$ can be globally and monotonically mapped to
$\mathcal{GK_{I}}$ and $\mathcal{GK_{II}}$ for any curve of constant $t$. We
analyzed some aspects of the integral equation relating the null coordinates
to $\mathcal{GK_{I}}$ and $\mathcal{GK_{II}}$. Finally, we demonstrated that
the smoothing technique developed in [27] could be thought of as a special
case of the type-II family of coordinates. Since the Kerr and RN spacetimes
share some similarities (i.e. two horizons), our technique might be applicable
to the former case given that there is no underlying assumption about the type
of spacetime in hand.

###### Acknowledgements.

We wish to thank Mahmoud<EMAIL_ADDRESS>and Wei-
<EMAIL_ADDRESS>Lin for the useful discussions on various
aspects of the analysis provided in this article. We are also grateful to Sam
Powers for many valuable comments on the draft. D.S. is partially supported by
the US National Science Foundation, under Grants No. PHY-2014021 and
PHY-2310363.

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