# Black hole and wormhole solutions in Einstein-Maxwell-Scalar theory

Júlio C. Fabris<EMAIL_ADDRESS>Tales A. O. Gomes
<EMAIL_ADDRESS>Denis C. Rodrigues<EMAIL_ADDRESS>Núcleo Cosmo-
Ufes & Departamento de Física, Universidade Federal do Espírito Santo (UFES),
Av. Fernando Ferrari, 540, Vitória 29075-910, Brazil.

(August 27, 2024)

###### Abstract

We classified and studied the charged black hole and wormhole solutions in the
Einstein–Maxwell system in the presence of a massless, real scalar field. The
possible existence of charged black holes in general scalar–tensor theories
was studied in Bronnikov et al, 1999; black holes and wormholes exist for a
negative kinetic term for the scalar field. Using a conformal transformation,
the static, spherically symmetric possible structures in the minimal coupled
system are described. Besides wormholes and naked singularities, only a
restricted class of black hole exists, exhibiting a horizon with an infinite
surface and a timelike central singularity. The black holes and wormholes
defined in the Einstein frame have some specificities with respect to the non-
minimal coupling original frame, which are discussed in the text.

Keywords: black holes; wormholes; scalar-tensor theory; Einstein conformal
frame; scalar field; casual structure.

##  I Introduction

Black holes (BHs) are objects predicted by the general relativity (GR) theory.
Their main characteristic feature is the existence of an event horizon, a
hypersurface separating two regions, the internal one, which generally
contains a singularity, and the external one with the asymptotic spatial
infinity where the observer may be located. Even if physically and
conceptually a BH may be considered an intriguing object, with many subtle
properties, its characterization is very simple. It is conjectured that only
three parameters are enough to describe a BH: its mass, charge, and angular
momentum. This is the main content of the no hair conjecture: just three
numbers define the BH. In this sense, the BH may be considered as a simpler
object existing in nature. However, the possibility of existing BHs with
additional parameters, for example a scalar charge [1, 2], has been evoked in
the literature, but the stability of the resulting structure is not always
assured. Besides BHs, GRs admit solutions with no horizon and no singularity
and containing two asymptotic spatial regions connected by a throat in
spacetime. They have been named wormholes (WHs) since they exhibit a short
path between two distant regions. In general, a WH requires repulsive effects,
at least around the throat. Repulsive gravitational effects may be a problem
for its stability and also for the possibility to be traversable, especially
by a human being. It must be stressed that the stability of a BH is also
frequently problematic, especially when it contains some kind of hair.
However, the classical BH configurations represented by the Schwarzschild
solution (which is static and uncharged), the Reissner–Nordström solution
(static and charged), and the Kerr solution (rotating and uncharged) are
stable, except perhaps for their corresponding extreme cases. If in some
solutions, the event horizon is absent, the singularity becomes visible to an
external observer. This configuration is called a naked singularity (NS). They
are in general unstable, meaning that they may not exist in nature. The
conjecture asserting that a naked singularity may not exist in nature is
dubbed cosmic censorship. For some extensive descriptions of these objects and
their properties, see [3, 4, 5].

BHs have been considered for a long period as hypothetical objects. This
situation has changed drastically in recent times. The detection by the LIGO
project [6] of gravitational waves emitted by the coalescence of two BHs and
the first image of a BH revealed by the Event Horizon Telescope (EHT) [7] have
left little doubt about the existence of these objects. The characteristics of
the emissions coming from galaxies with an active nucleus (AGN) show also
clear evidence of the existence of supermassive black holes in their center.
The possibility of the existence of a central supermassive BH has been
extended essentially to almost all galaxies, including our galaxy. The mass of
an isolated BH (called an astrophysical BH, resulting from the collapse of
massive stars) is of the order of some tens of solar masses ($M_{\odot}$), but
the mass of the BHs in the center of galaxies spans from millions to billions
of solar mass. While the BHs have mounted to the status of a legitimate
astrophysical object, WHs remain a mathematical result of the GR theory, with
no clear evidence until now for their existence.

In the literature, there is a plethora of BH solutions found in many different
contexts, most of them extending the content of known classical solutions.
Many different fields have been considered as a source for the BH structure.
The classical BH solutions are vacuum solutions for massive, charged or
uncharged, rotating or non-rotating configurations. In these classical
solutions, only the RN solution requires a non-trivial right-hand side of the
GR equations due to the presence of the electromagnetic field. Other possible
energy–momentum tensors, containing gauge fields or scalar fields, may be
introduced (see [8] and references therein), leading generally to new BH or
even WH structures. Furthermore, non-minimal coupling between the fields
and/or with the geometry may be considered, leading to quite rich
configurations. Theories with extra dimensions may lead also to quite special
structures. However, the stability of each of these configurations must be
tested carefully. See [9] for a description of some of these structures.

Scalar fields are the simpler source that can be added in a gravitational
theory. However, scalar fields may favor the appearance of NSs instead of BHs,
unless this appears as a phantom field, that is a field with negative kinetic
energy, or if it is coupled in a non-trivial way with gravity and/or other
gauge fields. An example is the Einstein–Maxwell dilaton (EMD) system: they
contain BHs that are asymptotically flat, but only in the case that the scalar
field is coupled with the Maxwell field or if the scalar field is phantom [10,
11, 12, 13]. Of course, the presence of self-interaction for the scalar field,
represented by a potential term, $V(\phi)$, may change all these
considerations. However, it is not always clear how to motivate the presence
of the potential term. In what follows, we ignore this possibility.

BHs and WHs in the Brans–Dicke theory in the presence of an EM field were
extensively studied in [14]. In this theory, the EM field is coupled minimally
to gravity, but the scalar field is coupled non-minimally. All possible BH,
WH, and NS cases were identified. The BH solutions belong to the so-called
cold BHs: the horizon has an infinite area, and its corresponding Hawking
temperature is zero. They are in general unstable [15], even if they present
some interesting causal structures. Cold black hole are also present when the
EM field is absent in the non-minimal and minimal coupling of the scalar field
with gravity [16, 17, 18]. However, in all cases, the scalar field must have a
phantom character when transposed to the Einstein frame.

The goal of the present work is to revisit the problem treated in [14]. The
main difference is that the BH and WH solutions are identified in the Einstein
frame. Again, BHs and WHs will be found only if the scalar field is phantom;
otherwise, only NSs appear. To our knowledge, the BH and WH solutions
identified here are new, even if some of their features are similar to other
known solutions, especially those found in [14]. Besides, to complement the
analysis presented in [14], our goal was to stress the properties of the
solutions with their physical content and the role played by the conformal
transformation, and its main consequences, in passing from the Jordan to the
Einstein frame. For example, in contrast to the original Brans–Dicke case,
with its different types of BH structures, there are only two possible classes
of BHs, one with a double horizon and another one with a single horizon,
characteristics similar to the Reissner–Nordström BH, non-extreme and extreme.
In both cases, the central singularity is timelike (hence, repulsive), which
may again be related to the presence of the Maxwell field. Both solutions are
cold black holes, which may be related to the presence of a (phantom) scalar
field. The WH class of solutions requires also a phantom scalar field, but it
is regular without horizons or singularities.

In the next section, we derive the general solution. All derivations are very
similar to the calculations exhibit in [14]. In Section III, the different
solutions are classified. In Section IV, the NS, BH, and WH solutions are
identified. In Section V, we present our conclusions.

##  II Field Equations and General Solutions

Our starting point is the action of the Einstein–Maxwell theory with a
minimally coupled massless scalar field,

$S=\int
d^{4}x\sqrt{-g}(R-\epsilon\nabla_{\alpha}\Phi\nabla^{\alpha}\Phi+F_{\alpha\beta}F^{\alpha\beta}),$
(1)

where $R$ is the Ricci scalar, $\Phi$ is a massless, real scalar field, and
$F^{\mu\nu}$ is Maxwell’s tensor. Our conventions are the following: the
metric signature is $(+---)$; the Ricci tensor is
$R_{\mu\nu}=\partial_{\rho}\Gamma^{\rho}_{\mu\nu}-\partial_{\nu}\Gamma^{\rho}_{\mu\rho}+\Gamma^{\rho}_{\mu\nu}\Gamma^{\sigma}_{\rho\sigma}-\Gamma^{\rho}_{\mu\sigma}\Gamma^{\sigma}_{\nu\rho}$.
Using the canonical expression for the energy–momentum tensor,

$\displaystyle T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\sqrt{-g}{\cal
L}}{\delta g^{\mu\nu}},$ (2)

we obtain the energy–momentum tensor for the scalar field and for the
electromagnetic field, respectively:

$\displaystyle T^{\Phi}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\epsilon\biggr{(}\Phi_{;\mu}\Phi_{;\nu}-\frac{1}{2}g_{\mu\nu}\Phi_{;\rho}\Phi_{;\rho}\biggl{)},$
(3) $\displaystyle T^{F}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-2\biggr{(}F_{\mu\rho}F_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\biggl{)}.$
(4)

The parameter $\epsilon$ can assume the values $+1$, which gives us a positive
kinetic energy for the scalar field, which is called canonical, and $-1$,
which represents the “phantom” scalar field with negative kinetic energy. This
action describes the interaction between gravity and a massless scalar field
in the presence of an electromagnetic field in four dimensions. The field
equations generated by this action are:

$R_{\mu\nu}=\epsilon\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\left(2F_{\mu\lambda}F^{\lambda}_{\nu}-\frac{1}{2}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right),$
(5) $\nabla_{\alpha}F^{\alpha\beta}=0,$ (6)
$\nabla_{\alpha}\left[\frac{1}{2}\epsilon^{\alpha\beta\lambda\sigma}F_{\lambda\sigma}\right]=0,$
(7) $\nabla_{\alpha}\nabla^{\alpha}\Phi=0,$ (8)

where $\epsilon^{\alpha\beta\lambda\sigma}$ is the Levi-Civita symbol.

Before going on to the specific computations, we provide some words on the
choice of the action (1). Our goal is to explore the black hole and wormhole
structures in minimally coupled scalar and electromagnetic fields. In doing
so, we neglected two interesting cases. The first is the addition of a self-
interaction in the scalar sector that can be represented by a potential term
$V(\Phi)$. The reason was two-fold. First, we intended to keep contact with
the study carried out in [14], where a similar analysis was carried out in the
Jordan frame, in the context of the Brans–Dicke theory, which in its
traditional form has no potential for the scalar field. The Brans–Dicke
structure is connected to our action (1) by a conformal transformation.
Moreover, the presence of a potential, even in its simpler form representing a
massive field, spoils the possibility to have analytical solutions, and a
quite sophisticated numerical treatment is necessary, an approach that lies
beyond the scope of the present work. However, we stress the important
possible connections that the case with a potential $V(\Phi)$ has, for
example, with the $f(R)$ theories, which can be reformulated in a Brans–Dicke-
type theory with a suitable potential [19]. Another possibility, in
introducing a self-interaction in the scalar sector, is the inverse problem:
the potential is determined from a given solution, with some specific
features. This problem is interesting also, but the motivation for the
potential determined in this way is not always clear.

The second important restriction of the action (1) is the absence of a
coupling between the scalar field and the electromagnetic field. This case was
extensively studied in [10, 11, 12, 13]. These studies have explored all
possible solutions (always in the absence of a potential term), asymptotically
flat and non-flat, in the Einstein frame. In this sense, perhaps it would be
interesting to revert the problem and write the solutions in the original non-
minimal coupling (sometimes also called the string frame). However, this would
constitute another different problem, deserving a separate analysis.

Coming back to the action (1), in deriving the corresponding static,
spherically symmetric solutions, we followed closely the computation carried
out in [14]. We give some details of this computation in order to be as
complete as possible.

We considered a static, spherically symmetric spacetime described by the
metric,

$ds^{2}={\,\rm e}^{2\gamma}dt^{2}-{\,\rm e}^{2\alpha}du^{2}-{\,\rm
e}^{2\beta}d\Omega^{2},$ (9)

where $\gamma$, $\alpha$, and $\beta$ are functions of the radial coordinate
$u$ only and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the
differential two-sphere. The non-vanishing terms of Ricci tensor for this
metric are,

$\displaystyle R_{00}$ $\displaystyle={\,\rm
e}^{-2\alpha+2\gamma}[\gamma^{\prime\prime}+(\gamma^{\prime}-\alpha^{\prime}+2\beta^{\prime})\gamma^{\prime}],$
(10) $\displaystyle R_{11}$
$\displaystyle=-\gamma^{\prime\prime}-2\beta^{\prime\prime}+(2\beta^{\prime}+\gamma^{\prime})\alpha^{\prime}-2(\beta^{\prime})^{2}-(\gamma^{\prime})^{2},$
(11) $\displaystyle R_{22}$ $\displaystyle=1+{\,\rm
e}^{-2\alpha+2\beta}[(\alpha^{\prime}-\gamma^{\prime}-2\beta^{\prime})\beta^{\prime}-\beta^{\prime\prime}],$
(12)

where primes denote the derivative with respect to the radial coordinate $u$.

For a point charge in a static spherically symmetric spacetime, the only non-
null terms of Maxwell’s tensor are,

$F_{10}=-F_{01}=E(u).$ (13)

Therefore, Maxwell’s Equation (6) gives us,

$E(u)=Q{\,\rm e}^{\alpha+\gamma-2\beta},$ (14)

where $Q$ is a constant of integration and can be interpreted as the electric
charge. On the other hand, since we are working with a static, spherically
symmetric spacetime, we can assume that the scalar field is a function of the
radial coordinate only ($\Phi\equiv\Phi(u)$). Thus, we obtain,

$\Phi^{\prime}=C{\,\rm e}^{\alpha-\gamma-2\beta},$ (15)

where $C$ is a constant of integration and may be interpreted as a scalar
charge.

The resulting equations, using the non-vanishing Ricci tensor components and
the solution for the electric field, are:

$\displaystyle\gamma^{\prime\prime}+(\gamma^{\prime}-\alpha^{\prime}+2\beta^{\prime})\gamma^{\prime}$
$\displaystyle=Q^{2}{\,\rm e}^{2\alpha-4\beta},$ (16)
$\displaystyle\gamma^{\prime\prime}+2\beta^{\prime\prime}-(2\beta^{\prime}+\gamma^{\prime})\alpha^{\prime}+2\beta^{\prime
2}+\gamma^{\prime 2}$ $\displaystyle=-\epsilon C^{2}{\,\rm
e}^{2\alpha-2\gamma-4\beta}+Q^{2}{\,\rm e}^{2\gamma-4\beta},$ (17)
$\displaystyle\beta^{\prime\prime}-(\alpha^{\prime}-\gamma^{\prime}-2\beta^{\prime})\beta^{\prime}$
$\displaystyle={\,\rm e}^{2\alpha-2\beta}-Q^{2}{\,\rm e}^{2\gamma-4\beta}.$
(18)

Equation (16) suggests that a suitable choice of the coordinates is the
harmonic coordinates, where $\alpha(u)=\gamma(u)+2\beta(u)$. The harmonic
coordinates are generally a convenient choice when scalar fields are present.
With this choice, the equations become:

$\gamma^{\prime\prime}=Q^{2}{\,\rm e}^{2\gamma},$ (19)
$\beta^{\prime\prime}-\beta^{\prime
2}-2\beta^{\prime}\gamma^{\prime}=-\frac{1}{2}\epsilon C^{2},$ (20)
$\gamma^{\prime\prime}+\beta^{\prime\prime}={\,\rm e}^{2\gamma+2\beta}.$ (21)

This last equation can be integrated, leading to,

$(\gamma^{\prime}+\beta^{\prime})^{2}={\,\rm e}^{2\gamma+2\beta}+k^{2}{\rm
sign}\,k,$ (22)

where $k$ is a constant of integration. Making the substitution
$\gamma+\beta=-\ln[s(u)]$, we have:

$s^{\prime 2}=1+k^{2}{\rm sign}\,k\,s^{2}.$ (23)

The solution for $s(u)$ will depend on the sign of $k$; thus, we may write
$s(u)\equiv s(k,u)$, and the solution will be:

$s(k,u)=\begin{cases}k^{-1}\sinh(ku),&k>0,\\\ u,&k=0,\\\
k^{-1}\sin(ku),&k<0.\end{cases}$ (24)

The solution for $\gamma$ is obtained by integrating Equation (19), which
leads to,

$\gamma^{\prime 2}=Q^{2}{\,\rm e}^{2\gamma}+\lambda^{2}{\rm sign}\,\lambda,$
(25)

where $\lambda$ is another constant of integration. To solve this equation, we
use the substitution $\gamma(u)=-\ln[h(u)]$. Thus,

$h^{\prime 2}=Q^{2}+\lambda^{2}{\rm sign}\,\lambda\,h^{2}.$ (26)

Comparing this equation to Equation (24), we obtain the relation
$h(u)=Qs(\lambda,u+u_{0})$.

Finally, the general solution for the metric will be,

$ds^{2}=\frac{Q^{-2}dt^{2}}{s^{2}(\lambda,u+u_{0})}-Q^{2}\frac{s^{2}(\lambda,u+u_{0})}{s^{2}(k,u)}\left(\frac{du^{2}}{s^{2}(k,u)}+d\Omega^{2}\right).$
(27)

The definition of the function $s(\lambda,u+u_{0})$ is similar to the
definition of $s(k,u)$ given above.

From Equation (20), we have the following relation for the integration
constant:

$k^{2}{\rm sign}\,k-\lambda^{2}{\rm sign}\,\lambda=\frac{\epsilon C^{2}}{2}.$
(28)

The scalar and electromagnetic fields are written, respectively, as:

$\Phi(u)=Cu,\quad\mbox{and}\quad E(u)=\frac{1}{Qs^{2}(\lambda,u+u_{0})}.$ (29)

Without losing generality, we can normalize $g_{00}=1$ at $u=0$ by imposing
the condition,

$s^{2}(\lambda,u_{0})=\frac{1}{Q^{2}}.$ (30)

We thus have four integration constants: $\lambda$, $k$, $Q$, and $C$. Another
important constant is the geometric mass $m$ of the configuration, which we
can obtain by comparing the asymptotic Equation (27) with the Schwarzschild
metric,

$\gamma_{sc}(r)=\frac{1}{2}\ln(1-\frac{2m}{r}).$ (31)

At the asymptotic $u\rightarrow 0$, it behaves as $r^{-1}$. Hence,

$\gamma_{sc}(u)=\frac{1}{2}\ln(1-2mu)\quad\rightarrow\quad\gamma_{sc}^{\prime}(0)=-m,$

and comparing with Equation (27),

$\gamma^{\prime}(u)=-\frac{s^{\prime}(\lambda,u+u_{0})}{s(\lambda,u+u_{0})}\Rightarrow
m=\frac{s^{\prime}(\lambda,u_{0})}{s(\lambda,u_{0})}.$ (32)

Now, we can connect all constants using the asymptotic behavior of (26) as
$u\rightarrow 0$, the relations (28,32), and the condition (30), obtaining,

$m^{2}-Q^{2}=\lambda^{2}{\rm sign}\,\lambda=k^{2}{\rm sign}\,k-\frac{\epsilon
C^{2}}{2}.$ (33)

Finally, we can write the metric in terms of $s(k,u)$ only, as:

$ds^{2}=\frac{s^{2}(\lambda,u_{0})dt^{2}}{s^{2}(\lambda,u+u_{0})}-\frac{s^{2}(\lambda,u+u_{0})}{s^{2}(\lambda,u_{0})s^{2}(k,u)}\left(\frac{du^{2}}{s^{2}(k,u)}+d\Omega^{2}\right).$
(34)

From here, the Reissner–Nordström solution of GR is recovered by putting
$C=0$; thus, $\lambda=k$. In this sense, we have three possible cases,
$\lambda=k>0$, $\lambda=k=0$, and $\lambda=k<0$, corresponding to the RN non-
extreme ($m>Q$), extreme ($m=Q$), and naked singularity ($m<Q$), respectively.
In each case, the familiar form of the RN solution is obtained with the
transformation:

$r=\frac{s(\lambda,u+u_{0})}{s(\lambda,u_{0})s(k,u)}.$ (35)

Another interesting limiting case is for $Q=0$, which is the scalar–vacuum
solution, obtained in [20]. The limit $Q\rightarrow 0$ must be taken
preserving the boundary condition (30). This leads to:

$\lambda\geq 0\quad\mbox{and}\quad u_{0}\rightarrow\infty\quad\Rightarrow\quad
s(\lambda,u+u_{0})\rightarrow e^{2\lambda u}.$ (36)

This solution was studied in detail in [16, 17].

Noticing that for $u\rightarrow 0$, we have $s(k,0)\rightarrow 0$ for all $k$,
we can identify two surfaces of interest from the metric (34). One surface is
at $u=-u_{0}$, where the metric terms behave as:

${\,\rm e}^{2\gamma}\rightarrow\infty,\quad\mbox{and}\quad{\,\rm
e}^{2\alpha},{\,\rm e}^{2\beta}\rightarrow 0,$

characterizing a central singularity, and the other is At $u=0$, where:

${\,\rm e}^{2\gamma}\rightarrow 1,\quad\mbox{and}\quad{\,\rm
e}^{2\alpha},{\,\rm e}^{2\beta}\rightarrow\infty,$

that is an asymptotic flat surface (Minkowski) expressed in the harmonic
coordinates. Therefore, the general solution (34) will have at least one
singularity and one asymptotic flat surface. The sign of $\lambda$ and $k$
will determine the combination of functions in the metric, according to
Equation (24). The structure of the spacetime may also have one or more
horizons, or even no horizon at all. In all possible cases, the solutions that
we look for are BHs and WHs.

##  III Classifying the Static, Spherically Symmetric Solutions

The metric (34) has different forms depending on the sign of the constant of
integration $\lambda$ and $k$. Each combination of signs will provide a
relation between the constants. In this section, we investigate each case
representing an independent solution. All possible relations are listed in
Tables 1 and 2. The features of these different cases depend on, crucially,
whether the scalar field is canonical ($\epsilon=1$, Table 1) or if it is
phantom ($\epsilon=-1$, Table 2).

Table 1: Possible relations between the constants of integration in the canonical sector, $\epsilon=+1$. The $*$ indicates the cases that give complex values for the constants, which are not of interest.  $\epsilon=+$ |  $k=0$ |  $k>0$ |  $k<0$
---|---|---|---
$\lambda=0$ |  $C=0$ |  $k^{2}=\frac{C^{2}}{2}$ |  * $k^{2}=-\frac{C^{2}}{2}$
$\lambda>0$ |  * $\lambda^{2}=-\frac{C^{2}}{2}$ |  $k^{2}-\lambda^{2}=\frac{C^{2}}{2}$ |  * $k^{2}+\lambda^{2}=-\frac{C^{2}}{2}$
$\lambda<0$ |  $\lambda^{2}=\frac{C^{2}}{2}$ |  $k^{2}+\lambda^{2}=\frac{C^{2}}{2}$ |  $\lambda^{2}-k^{2}=\frac{C^{2}}{2}$
Table 2: Possible relations between the constants for the phantom sector, $\epsilon=-1$. $*$ indicates the cases that give complex values for the constants, which are not of interest. $\epsilon=-$ |  $k=0$ |  $k>0$ |  $k<0$
---|---|---|---
$\lambda=0$ |  $C=0$ |  * $k^{2}=-\frac{C^{2}}{2}$ |  $k^{2}=\frac{C^{2}}{2}$
$\lambda>0$ |  $\lambda^{2}=\frac{C^{2}}{2}$ |  $\lambda^{2}-k^{2}=\frac{C^{2}}{2}$ |  $k^{2}+\lambda^{2}=\frac{C^{2}}{2}$
$\lambda<0$ |  * $\lambda^{2}=-\frac{C^{2}}{2}$ |  * $k^{2}+\lambda^{2}=-\frac{C^{2}}{2}$ |  $k^{2}-\lambda^{2}=\frac{C^{2}}{2}$

Assuming all constants to be real, some of these relations will not be valid
since they provide complex values for some of the constants. Excluding all the
cases where the constants assume complex values, we are left with the
independent solutions, which are listed in Tables 3 ($\epsilon=1$) and 4
($\epsilon=-1$).

Table 3: Possible independent solutions for $\epsilon=+1$. $\epsilon=+1$
---
$\lambda>0$ and $k>0$
$\lambda<0$ and $k>0$
$\lambda<0$ and $k=0$
$\lambda<0$ and $k<0$
$\lambda=0$ and $k>0$
Table 4: Possible independent solutions for $\epsilon=-1$. $\epsilon=-1$
---
$\lambda>0$ and $k>0$
$\lambda>0$ and $k=0$
$\lambda>0$ and $k<0$
$\lambda<0$ and $k<0$
$\lambda=0$ and $k<0$

###  III.1 Independent Solutions in the Canonical Sector

The canonical sector is given by a positive value for the kinetic energy of
the scalar field, $\epsilon=+1$. In this section, we analyze each independent
solution of Equation (34) listed in Table 3, knowing that all solutions have
the singular surface at $u=-u_{0}$ and are asymptotically flat at $u=0$. Thus,
we have:

  1. 1.

$k>\lambda>0$: In this case, the metric takes the form,

$ds^{2}=\frac{\lambda^{2}\sinh^{2}(\lambda
u_{0})dt^{2}}{\sinh^{2}[\lambda(u+u_{0})]}-\frac{k^{2}\sinh^{2}[\lambda(u+u_{0})]}{\lambda^{2}\sinh^{2}(\lambda
u_{0})\sinh^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sinh^{2}(ku)}+d\Omega^{2}\right),$
(37)

and the constants are related by:

$m^{2}-Q^{2}=\lambda^{2}=k^{2}-\frac{C^{2}}{2}.$ (38)

Therefore, we can see that for $u\rightarrow\infty$, we have ${\,\rm
e}^{\gamma}\rightarrow 0$. Thus, there may be a horizon at this surface.
However, analyzing the angular term of the metric, for large values of $u$, it
can be approximated to:

${\,\rm e}^{2\beta}\approx{\,\rm e}^{2(\lambda-k)u},$

and since $k>\lambda$, we have,

$\lim_{u\rightarrow\infty}{\,\rm e}^{2\beta(u)}=0.$ (39)

Hence, we have another singular surface. This case can only describe a naked
singularity with an asymptotic flat spacetime;

  2. 2.

$k>0>\lambda$: Using the appropriate functions for this case, the metric is
written as,

$ds^{2}=\frac{\sin^{2}(\lambda
u_{0})dt^{2}}{\sin^{2}[\lambda(u+u_{0})]}-\frac{k^{2}\sin^{2}[\lambda(u+u_{0})]}{\sin^{2}(\lambda
u_{0})\sinh^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sinh^{2}(ku)}+d\Omega^{2}\right),$
(40)

where:

$m^{2}-Q^{2}=-\lambda^{2}=k^{2}-\frac{C^{2}}{2}.$ (41)

Because the sine function has a finite range and it is a periodic function,
there is no surface where ${\,\rm e}^{\gamma}\rightarrow 0$; thus, it has no
horizon, but it has many singular points. The singular points of the first
term of the metric are at,

$u_{\lambda,n}=\pm\frac{\pi n}{\lambda}-u_{0},\quad n=0,1,2,3...$ (42)

Other interesting surfaces are at $u\rightarrow\pm\infty$, where we have:

$\lim_{u\rightarrow\pm\infty}{\,\rm e}^{2\beta(u)}=0,$ (43)

which describes another singular surface. This case has only naked
singularities;

  3. 3.

$\lambda<k=0$: The metric in this case takes the form,

$ds^{2}=\frac{\sin^{2}(\lambda
u_{0})dt^{2}}{\sin^{2}[\lambda(u+u_{0})]}-\frac{\sin^{2}[\lambda(u+u_{0})]}{\sin^{2}(\lambda
u_{0})u^{2}}\left(\frac{du^{2}}{u^{2}}+d\Omega^{2}\right).$ (44)

The constants of integration are related by,

$Q^{2}-m^{2}=\lambda^{2}=\frac{C^{2}}{2}.$ (45)

This case is exactly the same as the previous one, where there is no horizon,
and we also have the same singular points and local minimums;

  4. 4.

$\lambda<k<0$: Writing the metric for this case, we find,

$ds^{2}=\frac{\sin^{2}(\lambda
u_{0})dt^{2}}{\sin^{2}[\lambda(u+u_{0})]}-\frac{k^{2}\sin^{2}[\lambda(u+u_{0})]}{\sin^{2}(\lambda
u_{0})\sin^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sin^{2}(ku)}+d\Omega^{2}\right),$
(46)

with the relation,

$Q^{2}-m^{2}=\lambda^{2}=k^{2}+\frac{C^{2}}{2}.$ (47)

Again, the first term of the metric ${\,\rm e}^{2\gamma}$ is the same as the
last cases, but the angular term now has another sine function. Therefore, we
still have the singular points $u_{s,n}$, but now, we have different local
minimums. Another interesting surface in this case is,

$u_{k,n}=\pm\frac{\pi n}{k},\quad n=0,1,2,3...$ (48)

where,

$\lim_{u\rightarrow u_{k,n}}{\,\rm e}^{2\beta}\rightarrow\infty,$ (49)

indicating an asymptotic surface. Since, $|k|<|\lambda|$, we have that the
interval between the singular points $u_{s,n}$ is smaller than the interval
between the asymptotic regions $u_{k,n}$. Hence,

$|u_{\lambda,n+1}-u_{\lambda,n}|<|u_{k,n+1}-u_{k,n}|,$ (50)

and there is always a singular surface between the asymptotic surfaces. Thus,
this can only be a naked singularity.

  5. 5.

$k>\lambda=0$: Here, the metric is written as,

$ds^{2}=\frac{u_{0}^{2}dt^{2}}{(u+u_{0})^{2}}-\frac{k^{2}(u+u_{0})^{2}}{u_{0}^{2}\sinh^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sinh^{2}(ku)}+d\Omega^{2}\right),$
(51)

with the relation,

$m^{2}-Q^{2}=k^{2}-\frac{C^{2}}{2}=0.$ (52)

Here, we can identify that ${\,\rm e}^{\gamma}\rightarrow 0$ in the surface
$u=\pm\infty$, but one can verify that at this surface, the third criterion
for BH selection is not satisfied. Moreover, ${\,\rm e}^{2\beta}\rightarrow 0$
for those points. Thus, this is not a horizon. This solution describes also a
naked singularity.

###  III.2 Independent Solutions in the Phantom Sector

In this section, we analyze the phantom sector, where the kinetic energy of
the scalar field has negative energy, using the possible solutions listed in
Table 4:

  1. 1.

$\lambda>k>0$: In this case, we have the metric as,

$ds^{2}=\frac{\sinh^{2}(\lambda
u_{0})dt^{2}}{\sinh^{2}[\lambda(u+u_{0})]}-\frac{k^{2}\sinh^{2}[\lambda(u+u_{0})]}{\sinh^{2}(\lambda
u_{0})\sinh^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sinh^{2}(ku)}+d\Omega^{2}\right),$
(53)

with the relation:

$m^{2}-Q^{2}=\lambda^{2}=k^{2}+\frac{C^{2}}{2}.$ (54)

We can see that for $u\rightarrow\pm\infty$, we have ${\,\rm
e}^{\gamma}\rightarrow 0$, which is one condition for having a horizon.
Analyzing the angular term for large values of $u$, we can approximate it as:

${\,\rm e}^{2\beta}\approx{\,\rm e}^{2(\lambda-k)u}.$

Since we have that $\lambda>k$, the surface $u\rightarrow-\infty$ implies
${\,\rm e}^{2\beta}\rightarrow 0$. Therefore, it is a singular surface rather
than a horizon. On the other hand, for $u\rightarrow\infty$, we have ${\,\rm
e}^{\beta}\rightarrow\infty$, which characterizes a horizon with an infinite
surface area. Then, working in the range $-u_{0}<u<\infty$, this solution
indeed represents a BH. This solution can be mapped to one of the charged
Brans–Dicke BH (Case [1-] of [14]) through a conformal transformation. This is
expected since the conformal transformation that maps the Einstein frame in
the Jordan frame is given by an exponential of the scalar field $\Phi$, being
regular. However, the overall structure and features of the solution are
affected by this conformal transformation.

In fact, the conformal transformation connects the metric in the Einstein
frame (used here) and the metric in the Jordan frame, used in [14], according
to the relation,

$\displaystyle ds^{2}_{J}=\varphi^{-1}ds^{2}_{E},$ (55)

where the subscripts $E$ and $J$ indicate the Einstein and Jordan frames,
respectively, and $\varphi$ is the original Brans–Dicke field, which is
related to the gravitational coupling. The scalar field used here and the
original Brans–Dicke field obey the following relation:

$\displaystyle\Phi=\sqrt{|\frac{3}{2}+\omega|}\ln\varphi.$ (56)

In this way, the metric in the original frame that corresponds to the metric
(53) in the Einstein frame takes the form,

$\displaystyle
ds_{J}^{2}=e^{-\frac{Cu}{\sqrt{|\frac{3}{2}+\omega|}}}\biggr{\\{}\frac{\sinh^{2}(\lambda
u_{0})dt^{2}}{\sinh^{2}[\lambda(u+u_{0})]}-\frac{k^{2}\sinh^{2}[\lambda(u+u_{0})]}{\sinh^{2}(\lambda
u_{0})\sinh^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sinh^{2}(ku)}+d\Omega^{2}\right)\biggl{\\}}.$
(57)

The overall conformal factor introduces, with respect to the metric in the
Einstein frame, the constant $C$, implying new conditions to have a black
hole, that is to obtain $g_{00}\rightarrow 0$ with $g_{22}\neq 0$. Moreover,
the conditions to have an analytical extension of the metric beyond the
horizon are affected by the presence of the parameter $C$, leading to a large
spectrum of possibilities, with different structures. This remark applies to
other black hole and wormhole solutions discussed above;

  2. 2.

$\lambda>k=0$: The metric in this case is written as,

$ds^{2}=\frac{\sinh^{2}(\lambda
u_{0})dt^{2}}{\sinh^{2}[\lambda(u+u_{0})]}-\frac{\sinh^{2}[\lambda(u+u_{0})]}{\sinh^{2}(\lambda
u_{0})u^{2}}\left(\frac{du^{2}}{u^{2}}+d\Omega^{2}\right),$ (58)

where we have the relation:

$m^{2}-Q^{2}=\lambda^{2}=\frac{C^{2}}{2}.$ (59)

In the limit $u\rightarrow\pm\infty$, this case has ${\,\rm
e}^{2\gamma}\rightarrow 0$ and ${\,\rm e}^{2\beta}\rightarrow\infty$.
Therefore, for the appropriate coordinate range, this solution represents a BH
and is connected, by a conformal transformation, to the other charged
Brans–Dicke BH in the Jordan frame (Case [2-] of [14]). Again, the conformal
transformation maps the solution in one frame into the solution in the other
frame;

  3. 3.

$\lambda>0>k$: For this case, the metric takes the form,

$ds^{2}=\frac{\sinh^{2}(\lambda
u_{0})dt^{2}}{\sinh^{2}[\lambda(u+u_{0})]}-\frac{k^{2}\sinh^{2}[\lambda(u+u_{0})]}{\sinh^{2}(\lambda
u_{0})\sin^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sin^{2}(ku)}+d\Omega^{2}\right),$
(60)

with:

$m^{2}-Q^{2}=\lambda^{2}=-k^{2}+\frac{C^{2}}{2}.$ (61)

The radial function, ${\,\rm e}^{2\alpha}$, connects two spatial infinities,
which is a characteristic of WHs. Thus, for an appropriate range, excluding
the singularity at $u=-u_{0}$, this can be a wormhole;

  4. 4.

$k<\lambda<0$:

For this case we have,

$ds^{2}=\frac{\sin^{2}(\lambda
u_{0})dt^{2}}{\sin^{2}[\lambda(u+u_{0})]}-\frac{k^{2}\sin^{2}[\lambda(u+u_{0})]}{\sin^{2}(\lambda
u_{0})\sin^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sin^{2}(ku)}+d\Omega^{2}\right),$
(62)

with,

$Q^{2}-m^{2}=\lambda^{2}=k^{2}-\frac{C^{2}}{2}.$ (63)

The radial term of the metric function ${\,\rm e}^{2\beta}$ diverges at
$u=\frac{n\pi}{k}$, with $n$ any integer, while the first term, ${\,\rm
e}^{2\gamma}$, is singular at $u=\frac{n\pi}{\lambda}-u_{0}$. Since
$|k|>|\lambda|$, ${\,\rm e}^{2\beta}$ oscillates faster than ${\,\rm
e}^{2\gamma}$, thus we can always choose a coordinate range where we have two
spatial infinities and avoid the singularity at
$u=\frac{n\pi}{\lambda}-u_{0}$, therefore a WH solution;

  5. 5.

$k<\lambda=0$: The last independent solution is,

$ds^{2}=\frac{u_{0}^{2}dt^{2}}{(u+u_{0})^{2}}-\frac{k^{2}(u+u_{0})^{2}}{u_{0}^{2}\sin^{2}(ku)}\left(\frac{k^{2}du^{2}}{\sin^{2}(ku)}+d\Omega^{2}\right),$
(64)

with,

$Q^{2}-m^{2}=k^{2}-\frac{C^{2}}{2}=0.$ (65)

As in the previous cases where we have a sine function in ${\,\rm
e}^{2\beta}$, there are Minkowski asymptotics at $u=\frac{n\pi}{k}$ and
horizons only at $u=\pm\infty$. Thus, in the appropriate coordinate range,
excluding the singularity at $u=-u_{0}$, this is a WH solution.

##  IV Black Hole and Wormhole Solutions

In this section, we discuss the solutions that indeed have a black hole and
wormhole structure. The BH solutions have event horizons with an infinite
area, i.e., $g_{22}\rightarrow\infty$ as $u\rightarrow u_{h}$, which
corresponds to a zero Hawking temperature, and a finite proper time for
infalling particles to attain the horizon (classified as type B1 BHs in [14],
where the type B2 BHs were also defined, for which the horizon is at an
infinite geodesic distance). In general, the form of these solutions is not
familiar because these solutions are found in the harmonic gauge, while the
form of the most well-known solutions are in the quasi-global gauge. To study
the geometry, it is helpful to introduce a new radial coordinate, which is in
the quasi-global coordinate. Before that, we revise the conditions required
for a given solution to represent a BH or a WH.

###  IV.1 Criteria for Black Hole and Wormhole Selection

BH solutions are spacetimes with a singular surface bounded by an event
horizon, $u_{h}$. For a static, spherically symmetric spacetime, as Equation
(9), we have the following criteria to select the event horizon; see the
discussion in [14]. At this surface, $u=u_{h}$, we have:

  1. 1.

The timelike Killing vector becomes null, which means ${\,\rm
e}^{\gamma}\rightarrow 0$;

  2. 2.

The surface area of the horizon is always positive non-null, so ${\,\rm
e}^{2\beta}>0$;

  3. 3.

For an observer at rest, the horizon is invisible; thus, the integral

$t^{*}=\int{\,\rm e}^{\alpha-\gamma}du\rightarrow\infty\quad\mbox{as}\quad
u\rightarrow u_{h};$

  4. 4.

The Hawking temperature $T_{H}$ is finite. One may use the following
expression for the Hawking temperature of a surface $u=u_{h}$, in natural
units,

$T_{H}=\lim_{u\rightarrow
u_{h}}\frac{\kappa(u)}{2\pi},\qquad\kappa(u)\overset{\rm def}{=}{\,\rm
e}^{\gamma-\alpha}|\gamma^{\prime}|;$ (66)

  5. 5.

The Kretschmann scalar $\mathcal{K}$ is finite. We discuss more about this
scalar in Appendix A;

  6. 6.

The metric must admit an analytical extension beyond the horizon.

Wormholes are a spacetime containing two Minkowskian asymptotic regions
connected by a throat, which is characterized by a minimum of the areal
function ${\,\rm e}^{2\beta}$. The spacetime must be geodesically complete and
regular. This means that the Kretschmann scalar does not diverge anywhere.
Moreover, this spacetime does not contain horizons, implying that it can be
traversed from one asymptotic to the other, and vice versa.

###  IV.2 First Black Hole Solution

In Section III.2, we verified that there are two possible BH solutions. The
first black hole solution is Equation (53). The appropriate transformation for
this solution is given by,

${\,\rm e}^{-2ku}=1-\frac{2k}{\rho}=P(\rho),$ (67)

which allows us to write the metric in a more familiar form,

$ds^{2}=f(\rho)dt^{2}-\frac{d\rho^{2}}{f(\rho)}-\frac{P(\rho)}{f(\rho)}\rho^{2}d\Omega^{2},$
(68)

where:

$f(\rho)=\frac{4\lambda^{2}{P(\rho)}^{a}}{[(m+\lambda)-(m-\lambda){P(\rho)}^{a}]^{2}},\quad
a=\frac{\lambda}{k}\quad\mbox{and}\quad
m^{2}-Q^{2}=\lambda^{2}=k^{2}+\frac{C^{2}}{2}.$ (69)

The geometric mass is defined as $m=\lambda\coth(\lambda u_{0})$, and since it
must be always positive, we have that $u_{0}>0$. The geometric mass defined
previously coincides, following the usual definitions [21], with the ADM mass,
as can be explicitly verified. The coordinate transformation used here changes
the exponential character of the metric functions to a power law one. This
transformation allows describing the interior region of the black hole.
However, this implies fractional powers of negative numbers, which violates
the analyticity of the solution as the horizon is crossed. In order to avoid
the lost of analyticity, the allowed values for $a$ are:

$a=1,2,3,4,5,...$ (70)

The case $a=1$ corresponds to the non-extreme RN solution, since it leads to
$C=0$. This “quantization condition” leads to a discrete parametrization of
the BH solutions and is also present, similarly, in the Jordan frame [14].
Using the solutions for the scalar and electric field Equation (29) in the
harmonic gauge and the coordinate transformation, Equation (67), we have:

$\Phi(\rho)=-\frac{C}{2k}\ln|P(\rho)|\quad\mbox{and}\quad E(\rho)=Qf(\rho).$
(71)

Now, with the metric in the quasi-global coordinate, one can directly identify
one horizon at $\rho=2k$ and an asymptotic region at $\rho\rightarrow\infty$.
We can also verify that $f(\rho)\rightarrow 0$ as $\rho\rightarrow 0$, giving
us another horizon.

The singularities in this coordinate system are given by the points where
$f(\rho_{s})\rightarrow\pm\infty$,

${P(\rho_{s})}^{a}=\left(1-\frac{2k}{\rho_{s}}\right)^{a}=\left(\frac{m+\lambda}{m-\lambda}\right).$
(72)

The right hand-side of the above equation is always positive, since by
definition, $m>\lambda$. However, $P(\rho_{s})$ can admit negative and
positive roots or only positive roots depending on the value of $a$. Hence,
depending on $a$, if it is even or odd, the singularity will have different
positions, splitting the possible solutions into two cases. For both cases,
the singularity $f(\rho_{s})\rightarrow+\infty$ is timelike, which is
frequently a characteristic of charged BHs.

####  IV.2.1 Case of $a$ Odd

If $a$ assumes an odd value in the first solution of BHs, then we can directly
take the $a$th root of Equation (72) to obtain:

$1-\frac{2k}{\rho_{s}}=\left(\frac{m+\lambda}{m-\lambda}\right)^{1/a}\Rightarrow\rho_{s}=\frac{2k}{1-\left(\frac{m+\lambda}{m-\lambda}\right)^{1/a}},$
(73)

and since $m>\lambda$, the singularity is located at $\rho=\rho_{s}<0$.
Therefore, in this case, the singularity is bounded by two event horizons, one
at $\rho=0$ and another one at $\rho=2k$, where both surfaces have an infinite
area, $e^{2\beta}\rightarrow\infty$, and it is Minkowski at
$\rho\rightarrow\infty$. Hence, we have three regions, called III, II, and I,
respectively: $\rho_{s}<\rho<0$, $0<\rho<2k$, and $2k<\rho<\infty$. The
signature is $(+---)$ for Regions I and III, while it is $(-+--)$ for Region
II. The causal structure is similar to a non-extreme RN solution. In Figure 1,
we plot the effective potential of the geodesics equation and the
Carter–Penrose diagram (CPD) for this case.

Figure 1: Plot of the Effective potential for massless particles for the first
solution, in the case where $a$ assumes odd values (left-top figure), and a
zoom of the potential after the horizon (left-bottom figure). The vertical
lines in the left figure indicate the two event horizons of the solution.
Moreover, the right figure is the Carter–Penrose diagram drawing for this
case. The red lines are the singularities; the double lines are the horizons;
the blue lines are the possible geodesic curves. The value of the parameters
used were: $\lambda=27$, $k=9$, $m=30$, and $L=100$.

####  IV.2.2 Case of $a$ Even

Now if $a$ is even in the first solution, then the $a\mbox{th}$ root of
Equation (72) gives us:

$1-\frac{2k}{\rho_{s}}=\pm\left(\frac{m+\lambda}{m-\lambda}\right)^{1/a}\rightarrow\rho_{s,\pm}=\frac{2k}{1\pm\left(\frac{m+\lambda}{m-\lambda}\right)^{1/a}},$
(74)

where the $\pm$ sign is due to the fact that $P(\rho_{s})$ can be positive or
negative. In this sense, there are two singular surfaces, where one is the
same as in the previous case with $a$ odd, $\rho_{s}=\rho_{s,-}$, while the
other singular surface is located between the two horizons, $0<\rho_{s,+}<2k$.
Analyzing the surface $\rho=\rho_{s,+}$, we find that the geodesics ends at
this surface; thus, this solution consists of two regions,
$\rho_{s,+}<\rho<2k$ and $2k<\rho<\infty$, instead of three. In this case,
there is no change of signature from one region to the other. It has a causal
structure similar to the extreme RN solution. A plot of the effective
potential and the CPD for this case is shown in Figure 2.

Figure 2: Plot of the effective potential for massless particles (left figure)
and the Carter–Penrose diagram (right figure) for the first solution with $a$
even. The vertical line indicates the event horizons. In the CPD, the red
lines are the singularities, the double lines are the horizons, and the blue
lines are the possible geodesic curves. Here, we used the following values for
the parameters: $\lambda=6$, $k=3$, $m=10$, $L=100$.

###  IV.3 Second Black Hole Solution

The second black hole solution in the phantom sector is Equation (58), and the
appropriate coordinate transformation for this case is:

$x=\frac{1}{u}.$ (75)

As a consequence, the metric becomes,

$ds^{2}=h(x)dt^{2}-\frac{dx^{2}}{h(x)}-\frac{x^{2}}{h(x)}d\Omega^{2},$ (76)

with:

$h(x)=\frac{4\lambda^{2}{\,\rm
e}^{-2\frac{\lambda}{x}}}{[(m+\lambda)-(m-\lambda){\,\rm
e}^{-2\frac{\lambda}{x}}]^{2}}.$ (77)

The geometric mass $m$ has the same definition as in the first BH solution,
and the constants are now related by:

$m^{2}-Q^{2}=\lambda^{2}=\frac{C^{2}}{2}.$ (78)

The solution in this case has one parameter less than the first solution,
since we fixed $k=0$; thus, the parameter $a$, which is a relation between
$\lambda$ and $k$, is not defined here. We can also see that the
transformation used to obtain Equation (76), giving an extension beyond the
horizon, keeps the exponential nature of the functions in the metric; hence,
there is no need for a “quantization condition” to extend the solution to the
interior region. The scalar and electric fields here are given by:

$\Phi(x)=\frac{C}{x}\quad\mbox{and}\quad E(x)=Qh(x).$ (79)

In this coordinate system, we can notice that this solution is Minkowski as
$x\rightarrow\infty$, while a horizon is present at $x=0$. The singularity is
located at $x=x_{s}$, where $x_{s}$ satisfies the equation:

${\,\rm e}^{-2\frac{\lambda}{x_{s}}}=\frac{m+\lambda}{m-\lambda}\Rightarrow
x_{s}=\frac{-2\lambda}{\ln\left(\frac{m+\lambda}{m-\lambda}\right)}.$ (80)

We can use the definition of the geometric mass, $m=\lambda\coth(\lambda
u_{0})$, to write:

$x_{s}=-\frac{1}{u_{0}},$ (81)

and since $u_{0}>0$, we have that $x_{s}<0$. As for the previous case, the
singularity here is timelike. Therefore, this solution, similar as in the
first solution with $a$ even, has two regions, $x_{s}<x<0$ and $0<x<\infty$,
connected by one event horizon at $x=0$. Figure 3 shows a plot of the
effective potential and the CPD of this solution.

Figure 3: Plot of the effective potential for massless particles for the
second solution (left figure). The vertical line indicates the event horizon
of the solution. Furthermore, the right figure is the CPD for this solution,
where the red lines are the singularities, the double lines are the horizons,
and the blue lines are the possible geodesic curves. For this plot, the values
for the parameters were: $\lambda=1.5$, $u_{0}=0.3$, and $L=10$.

###  IV.4 Wormhole Solutions

In the phantom sector (Section III.2), we verified that there are three
possible wormhole solutions. The metric for these WHs can be written formally
in a unified way by using the transformation,

$u=\frac{1}{k}\arccot{|ky|},$ (82)

leading to,

$ds^{2}=A_{n}(y)dt^{2}-\frac{dy^{2}}{A_{n}(y)}-{r_{n}(y)}^{2}d\Omega^{2}\quad\mbox{with}\quad{r_{n}(y)}^{2}=\frac{y^{2}+1}{A_{n}(y)},$
(83)

where the index $n$ corresponds to each wormhole solution. The scalar and
electric fields here are,

$\Phi(y)=\frac{C}{k}\arccot{ky},\quad\mbox{and}\quad E(y)=QA_{n}(y).$ (84)

The metric (83) admits an analytical extension, so the range for the radial
coordinate is $-\infty<y<\infty$, where $y\rightarrow\pm\infty$ corresponds to
the two flat spatial infinities and the throat is at $y=0$, with a radius of
$r_{n}(0)$. For each of the three possible cases, the function $A_{n}$ assumes
a different form.

  * •

First wormhole (60): In this case, we have the function:

$A_{1}(y)=\frac{4\lambda^{2}{\,\rm
e}^{-2\frac{\lambda}{k}\arccot{ky}}}{[(\lambda+m)+(\lambda-m){\,\rm
e}^{-2\frac{\lambda}{k}\arccot{ky}}]^{2}},$ (85)

where:

$m^{2}-Q^{2}=\lambda^{2}=\frac{C^{2}}{2}-k^{2}.$ (86)

Since $\lambda$ is a real constant, the above relation gives us $\lambda>m$,
for a non-null electric charge. The radius of the throat in this case is:

$r_{1}(0)=\frac{(\lambda+m)+(\lambda-m){\,\rm
e}^{-2\frac{\lambda\pi}{2k}}}{2\lambda{\,\rm e}^{-\frac{\lambda\pi}{2k}}}.$
(87)

The effective potential of the geodesic motion for massless particles, has a
local minimum at the throat and a maximum around it, where it is possible to
have a photon sphere. In Figure 4, there is a plot of $A_{1}$, the radius
$r_{1}$, and the effective potential. In the limit $Q\rightarrow 0$, which
corresponds to $\lambda\rightarrow m$, the metric (85) takes the form of:

$ds^{2}={\,\rm e}^{-2mu}dt^{2}-{\,\rm e}^{2mu}dy^{2}-{\,\rm
e}^{2mu}(y^{2}+1)d\Omega^{2}.$ (88)

The above metric is known as the wormhole of the anti-Fisher solution [20],
and for $m=0$, it becomes the Ellis wormhole [22]. A more detailed discussion
about these solutions can be found in [23, 15];

  * •

Second wormhole (62): The function for this case has the form:

$A_{2}(y)=\frac{\lambda^{2}{\sin(a\arccot{ky})}^{-2}}{[m+\lambda\cot(a\arccot{ky})]^{2}},$
(89)

where:

$Q^{2}-m^{2}=\lambda^{2}=k^{2}-\frac{C^{2}}{2}\quad\mbox{and}\quad
a=\frac{\lambda}{k}.$ (90)

Here, we have the condition that $|k|>|\lambda|$, which gives us that $a<1$.
The radius of the throat is:

$r_{2}(0)=\frac{m\sin(\frac{a\pi}{2})+\lambda\cos(\frac{a\pi}{2})}{\lambda}.$
(91)

The plot of the effective potential for geodesic motion for massless particles
(Figure 5) shows that there is the possibility for photons to orbit the
throat;

  * •

Third wormhole (64): Here, we have the function in the form:

$A_{3}(y)=\frac{k^{2}}{[k+m\arccot{ky}]^{2}},$ (92)

where:

$Q^{2}-m^{2}=k^{2}-\frac{C^{2}}{2}=0.$ (93)

At the throat, $y=0$, the radius is:

$r_{3}(0)=1+\frac{m\pi}{2k}.$ (94)

The effective potential in this case also allows photon spheres, as we can see
in Figure 6. The metric in this case also becomes the Ellis wormhole [22] for
$m=0$.

Figure 4: Plot of the function $A_{1}(y)$, the radius $r_{1}(y)$, and the
effective potential, respectively, for the first wormhole solution, Equation
(85). The values used for the parameters in this plot were: $\lambda=1$,
$k=3$, $m=2$, and $L=1$.

Figure 5: Plot of the function $A_{2}(y)$, the radius $r_{2}(y)$, and the
effective potential, respectively, for the second WH solution, Equation (89).
The values used for the parameters in this plot were: $\lambda=1$, $k=3$,
$m=2$, and $L=1$.

Figure 6: Plot of the function $A_{3}(y)$, the radius $r_{3}(y)$, and the
effective potential, respectively, for the third WH solution, Equation (92).
The values used for the parameters in this plot were: $k=3$, $m=1$, and $L=1$.

##  V Final Comments and Conclusions

We analyzed, in this work, the black hole and wormhole configurations of the
Einstein–Maxwell system with a massless scalar field. Such a structure can be
obtained from the Brans–Dicke theory in the presence of the Maxwell field by a
conformal transformation. Wormholes’ and black holes’ charged solutions in the
Brans–Dicke theory were studied in detail in [14]. In the Brans–Dicke theory
(Jordan frame) as in the Einstein–Maxwell-scalar system (Einstein frame)
studied here, black holes and wormholes can be obtained only when the scalar
field is phantom, that is, it has negative kinetic energy.

The conformal transformation maps the seven main classes of solutions obtained
in [14] (each of them containing sub-cases according to the conditions on the
parameters) to all solutions obtained here. In the present case, in the
Einstein frame, we have ten possible solutions, listed in Tables 3 and 4. In
all cases, there is a map directly to their counterparts in the Jordan frame.
In [14],for a canonical scalar field, in both frames, only naked singularities
are present. In the phantom sector, the possibilities are much richer, with BH
and WH configurations. The possibilities are richer in the Jordan frame due to
the presence of the conformal factor. For example, for $\lambda>k>0$, in the
Jordan frame, there are four possible cases, while in the Einstein frame, just
two cases. Moreover, in the Brans–Dicke original frame, the central
singularity can be timelike or spacelike, while in the Einstein frame, both BH
cases exhibit a timelike central singularity. The WH solutions in one case are
mapped into WH solutions in the other frame.

Although we can map all solutions obtained here to the ones obtained in the
Jordan frame, these frames are not completely equivalent. The presence of a
conformal factor in the metric changes the geodesics followed by particles
depending on the choice of the frames: a geodesic in the Jordan frame is not
necessarily a geodesic in the Einstein frame and vice versa. Even though, the
type of geodesic curves obtained here for the two BH solution, as shown in
Figures 1–3, are also present in the Jordan frame, which, however, contains in
general more different types of curves than those found in the Einstein frame.

The main properties of the black hole and wormhole solutions described here do
not depend on the choice of the coordinate system, for example the fact that
the black hole horizons have an infinite area. However, we must keep in mind
that some coordinate systems may cover only a portion of the complete
manifold, as happens if we fix the areal function as in the traditional
Schwarzschild coordinates for which $e^{2\beta}=r^{2}$.

The different parameters of the model obey a “quantization condition”, mainly
due to the analytical extension beyond the horizon. These quantization
conditions are different in the Jordan and Einstein frame due to the conformal
factor. In the Einstein frame, the “quantization condition” is given by
Equation (70), which leads to a different causal structure: one that is
similar to the non-extreme Reissner–Nordström solution, the case of $a$ odd,
while the other is similar to the extreme Reissner–Nordström solution, the
case of $a$ even. In both cases, the central singularity is timelike, which
seems to be a consequence of the presence of the electric field, as in the
Reissner–Nordström case. However, the horizons, in one case and in the other,
have an infinite area, being an example of the so-called cold black holes.
This may be considered as a consequence of the presence of a phantom scalar
field.

The studies of the stability problem for the solutions found here are a
necessary new step. For an appropriate analysis of the problem, the use of the
gauge-invariant quantities is required, as well as determining the stability
of the model. From the previous results obtained in the literature for similar
configurations [23, 15, 24], we may expect that the solutions are unstable. We
remark that the BHs found here have some similarities with the RN black holes
for which previous studies have revealed the presence of instabilities [25,
26] mainly in the Cauchy horizon. However, the stability of objects such as
black holes and wormholes must be studied case by case. We remark that the
stability analysis implies writing a Schrödinger-type equation with an
effective potential. This effective potential is generally singular at the
minimum of the areal function $e^{2\beta}$. This implies that a numerical
analysis is necessary in order to complete the stability study. We hope to
present such a numerical analysis in future work.

Finally, it is important to give some words about the thermodynamics of the
black hole solutions found here. They belong to the class of the so-called
cold black holes because, in principle, their Hawking temperature is zero. It
is more appropriate to say that maybe it is not possible to attribute
thermodynamics properties to such black holes, in a sense similar to what
happens with the extreme RN black holes; see [27] and the references therein.
Strictly speaking, we can only state that cold black holes have zero surface
gravity $\kappa$, as it can be verified by computing,

$\displaystyle\kappa=\frac{1}{2}\frac{g_{00}^{\prime}}{\sqrt{g_{00}g_{11}}}\biggl{|}_{u_{h}}.$
(95)

It is quite direct to verify that $\kappa$ is, for the cases considered here,
equal to zero. However, a proper semiclassical evaluation of quantum fields in
the spacetime of cold black holes reveals that all the computation is ill
defined. Hence, it is not clear that it is possible to define the
thermodynamic quantities as for the usual black holes. Especially, the entropy
of a black hole with zero temperature and/or an infinite surface horizon is
not unambiguously defined; see [28, 29]. To our knowledge, the thermodynamics
of such objects remains an open problem.

Acknowledgements: We thank CNPq (Brasil) and FAPES (Brasil) for partial
financial support.

##  Appendix A Kretschmann Scalar

When studying any spacetime, it is above all important to know whether it is
regular, which means that all curvature invariants are finite at all its
points, or it contains curvature singularities at which at least one such
invariant is infinite. In many cases, it is most helpful to check the
finiteness of the Kretschmann scalar, defined as:

$\mathcal{K}=R^{\mu\nu\lambda\sigma}R_{\mu\nu\lambda\sigma}.$ (96)

For a spherically symmetric metric, the Kretschmann scalar is the sum of
squares of all nonzero components of the Riemann tensor [9],

$\mathcal{K}=4(K_{1})^{2}+8(K_{2})^{2}+8(K_{3})^{2}+4(K_{4})^{2},$ (97)

where:

$\displaystyle K_{1}$ $\displaystyle={R^{01}}_{01}=-{\,\rm
e}^{-\alpha-\gamma}\left(\gamma^{\prime}{\,\rm
e}^{\gamma-\alpha}\right)^{\prime},$ (98) $\displaystyle K_{2}$
$\displaystyle={R^{02}}_{02}={R^{03}}_{03}=-{\,\rm
e}^{-2\alpha}\beta^{\prime}\gamma^{\prime},$ (99) $\displaystyle K_{3}$
$\displaystyle={R^{12}}_{12}={R^{13}}_{13}=-{\,\rm
e}^{-\alpha-\beta}\left(\beta^{\prime}{\,\rm
e}^{\beta-\alpha}\right)^{\prime},$ (100) $\displaystyle K_{4}$
$\displaystyle={R^{23}}_{23}={\,\rm e}^{-2\beta}-{\,\rm
e}^{-2\alpha}\beta^{\prime 2};$ (101)

It is significant that all $K_{i}$ are invariant under the reparametrizations
of the coordinate; in other words, they behave as scalars at such
transformations. Since the scalar $\mathcal{K}$ is a sum of squares, for its
finiteness, it is necessary and sufficient that all its components $K_{i}$ are
finite.

It must be mentioned here that curvature singularities are not the only type
of singularities that can appear in physically relevant spacetimes. In the
most general form, a singularity is defined as a point or a set of points
where geodesics terminate at a finite value of their affine parameter,
characterizing geodesic incompleteness.

###  A.1 Kretschmann for the First Black Hole Solution

Using the first solution (Equation (53)), one can find the scalars (98–101)
as:

$K_{1}=-\frac{8\lambda^{3}\left[4\lambda(\lambda^{2}-m^{2}){P(\rho)}^{2a}-(m+\lambda)^{2}(k-\rho+\lambda){P(\rho)}^{a}+(m-\lambda)^{2}(k-\rho-\lambda){P(\rho)}^{3a}\right]}{\rho^{4}{P(\rho)}^{2}\left[(m+\lambda)-(m-\lambda){P(\rho)}^{a}\right]^{4}},$
(102)
$K_{2}=\frac{4\lambda^{3}\left[2\lambda(\lambda^{2}-m^{2}){P(\rho)}^{2a}-(m+\lambda)^{2}(k-\rho+\lambda){P(\rho)}^{a}+(m-\lambda)^{2}(k-\rho-\lambda){P(\rho)}^{3a}\right]}{\rho^{4}{P(\rho)}^{2}\left[(m+\lambda)-(m-\lambda){P(\rho)}^{a}\right]^{4}},$
(103)
$K_{3}=\frac{4\lambda^{2}\left[2(\lambda^{2}-m^{2})(k^{2}-2\lambda^{2}){P(\rho)}^{2a}+(m+\lambda)^{2}((k-\rho)\lambda+k^{2}){P(\rho)}^{a}+(m-\lambda)^{2}((\rho-k)\lambda+k^{2}){P(\rho)}^{3a}\right]}{\rho^{4}{P(\rho)}^{2}\left[(m+\lambda)-(m-\lambda){P(\rho)}^{a}\right]^{4}},$
(104) $\small
K_{4}=\frac{4\lambda^{2}\left[(2\lambda^{2}(k^{2}+m^{2}-\lambda^{2})-2k^{2}m^{2}){P(\rho)}^{2a}+(m+\lambda)^{2}((\lambda+k)^{2}-2\lambda\rho){P(\rho)}^{a}+(m-\lambda)^{2}((\lambda-k)^{2}+2\lambda\rho){P(\rho)}^{3a}\right]}{\rho^{4}{P(\rho)}^{2}\left[(m+\lambda)-(m-\lambda){P(\rho)}^{a}\right]^{4}}.$
(105)

Figure 7: Plot of the Kretschmann scalar for the first solution (Equation
(68)) with $a$ odd (left figure) and with $a$ even (right figure). The
vertical lines in the left figure indicate the horizons, while in the right
figure, the left and right vertical lines are the horizons and the one in the
middle is the second singularity of the case of $a$ even, showing that the
solution ends there, excluding the inner horizon. The values of the parameter
used here are the same as in the plots of Figures 1 and 2.

###  A.2 Kretschmann for the Second Black Hole Solution

For the second solution (Equation (58)), the scalars $K_{i}$ take the form:

$K_{1}=\frac{-8\lambda^{3}{\,\rm
e}^{\frac{2\lambda}{x}}\left\\{(m+\lambda)^{2}(x-\lambda){\,\rm
e}^{\frac{4\lambda}{x}}-\left[4\lambda(m+\lambda){\,\rm
e}^{\frac{2\lambda}{x}}+(x+\lambda)(m-\lambda)\right](m-\lambda)\right\\}}{x^{4}\left[(m+\lambda){\,\rm
e}^{\frac{2\lambda}{x}}-(m-\lambda)\right]^{4}},$ (106)
$K_{2}=\frac{4\lambda^{3}{\,\rm e}^{\frac{2\lambda}{x}}\left[(m+\lambda){\,\rm
e}^{\frac{2\lambda}{x}}+(m-\lambda)\right]\left[(x-\lambda)(m+\lambda){\,\rm
e}^{\frac{2\lambda}{x}}-(x+\lambda)(m-\lambda)\right]}{x^{4}\left[(m+\lambda){\,\rm
e}^{\frac{2\lambda}{x}}-(m-\lambda)\right]^{4}},$ (107)
$K_{3}=\frac{-4\lambda^{3}{\,\rm
e}^{\frac{6\lambda}{x}}\left\\{4\lambda(m^{2}-\lambda^{2}){\,\rm
e}^{-\frac{2\lambda}{x}}+\left[(m-\lambda)^{2}{\,\rm
e}^{-\frac{4\lambda}{x}}-(m+\lambda)^{2}\right]x\right\\}}{x^{4}\left[(m+\lambda){\,\rm
e}^{\frac{2\lambda}{x}}-(m-\lambda)\right]^{4}},$ (108)
$K_{4}=\frac{8\lambda^{3}{\,\rm
e}^{\frac{6\lambda}{x}}\left\\{(m-\lambda)^{2}\left(x+\frac{\lambda}{2}\right){\,\rm
e}^{-\frac{4\lambda}{x}}+(m+\lambda)\left[(m-\lambda)\lambda{\,\rm
e}^{-\frac{2\lambda}{x}}-(m+\lambda)\left(x-\frac{\lambda}{2}\right)\right]\right\\}}{x^{4}\left[(m+\lambda){\,\rm
e}^{\frac{2\lambda}{x}}-(m-\lambda)\right]^{4}},$ (109)

Figure 8: Plot of the Kretschmann scalar for the second solution Equation
(76), where the vertical lines indicate the horizons. The value of the
parameters are the same as in Figure 3.

###  A.3 Kretschmann for Wormhole Solutions

The Kretschmann scalar for the WH solutions has several terms, which are not
suitable to write here. However, we plot this scalar for each WH solution. The
plot shows us that the Kretschmann scalar, for all cases, is finite
everywhere.

Figure 9: Plot of the Kretschmann scalar for the first, second, and third WH
solution (Equations (85), (89) and (92), respectively). The value of the
parameters for each case is the same as in the plots of Figures 4–6,
respectively.

##  Appendix B General Form of Geodesic Equations

In this Appendix, we discuss the geodesic equations, using the Lagrangian
formalism. The Lagrangian of any spacetime can be written as:

$\mathcal{L}=\frac{1}{2}g^{\mu\nu}\dot{x}_{\mu}\dot{x}_{\nu}=\frac{1}{2}\mathcal{E},$
(110)

where:

$\mathcal{E}=\begin{cases}+&1\quad\mbox{for timelike geodesics},\\\
&0\quad\mbox{for null geodesics},\\\ -&1\quad\mbox{for spacelike
geodesics}.\end{cases}.$ (111)

For a static, spherically symmetric spacetime (Equation (9)), the Lagrangian
takes the form:

${\,\rm e}^{2\gamma(u)}{\dot{t}}^{2}-{\,\rm
e}^{2\alpha(u)}{\dot{u}}^{2}-{\,\rm
e}^{2\beta(u)}\left[{\dot{\theta}}^{2}+\sin^{2}\theta{\dot{\phi}}^{2}\right]=\mathcal{E}.$
(112)

The symmetries of the system allow us to define the following conserved
conjugate momentum,

$p_{t}=\frac{\partial\mathcal{L}}{\partial\dot{t}}={\,\rm
e}^{2\gamma}\dot{t}=E,$ (113)
$p_{\phi}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=-{\,\rm
e}^{2\beta}\sin^{2}\theta\dot{\phi}=L,$ (114)

where $E$ and $L$ are the constants of integration. Substituting these
expressions back into Equation (112),

$E^{2}e^{-2\gamma}-e^{2\alpha}{\dot{u}}^{2}-e^{2\beta}{\dot{\theta}}^{2}-\frac{L^{2}e^{-2\beta}}{\sin^{2}\theta}=\mathcal{E}.$
(115)

For simplicity, and in connection with the symmetry of the problem, we assumed
that the geodesic is located in the equatorial plane ($\theta=\frac{\pi}{2}$),
leading to,

$e^{2\alpha+2\gamma}\dot{u}^{2}=E^{2}-\mathcal{E}e^{2\gamma}-L^{2}e^{2\gamma-2\beta}.$
(116)

### Geodesic Equation for the General Solution

Now, for the general solution (Equation (27)), the geodesic equation becomes:

$\frac{\dot{u}^{2}}{{s(k,u)}^{4}}=E^{2}-\mathcal{E}\frac{{s(\lambda,u_{0})}^{2}}{{s(\lambda,u+u_{0})}^{2}}-L^{2}\frac{{s(\lambda,u_{0})}^{4}{s(k,u)}^{2}}{{s(\lambda,u+u_{0})}^{4}},$
(117)

where we can use the following transformation,

$\rho=\int\frac{du}{{s(k,u)}^{2}}\Rightarrow\dot{\rho}=\frac{\dot{u}}{{s(k,u)}^{2}},$
(118)

to obtain the geodesic equation in the form of an energy conservation law for
a particle moving in a potential field,

$\dot{\rho}^{2}=E^{2}-V_{eff}(u),$ (119)

with:

$V_{eff}(u)=\frac{{s(\lambda,u_{0})}^{2}}{{s(\lambda,u+u_{0})}^{2}}\left(\mathcal{E}+L^{2}\frac{{s(\lambda,u_{0})}^{2}{s(k,u)}^{2}}{{s(\lambda,u+u_{0})}^{2}}\right),$
(120)

playing the same role for geodesic motion as the potential in classical
mechanics for a one-dimensional motion of a point particle: the motion is only
possible in a region where $E^{2}\geq V_{eff}(u)$, while the values of the
coordinate at which $E^{2}=V_{eff}(u)$ correspond to turning points.

Remembering that $s(k,0)=0$, independent of the chosen sign of $k$, this
describes two surfaces, $u=0$ and $u=-u_{0}$, the asymptotic flat one and the
singularity. In the asymptotic region, the effective potential is a constant,
while at the singularity, it diverges, $V_{eff}(-u_{0})\rightarrow\infty$,
which agrees with the expected behavior at these surfaces.

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# Rethinking the Domain Gap in Near-infrared Face Recognition

Michail Tarasiou1 and Jiankang Deng1 and Stefanos Zafeiriou1
1 Imperial College London

###### Abstract

Heterogeneous face recognition (HFR) involves the intricate task of matching
face images across the visual domains of visible (VIS) and near-infrared
(NIR). While much of the existing literature on HFR identifies the domain gap
as a primary challenge and directs efforts towards bridging it at either the
input or feature level, our work deviates from this trend. We observe that
large neural networks, unlike their smaller counterparts, when pre-trained on
large scale homogeneous VIS data, demonstrate exceptional zero-shot
performance in HFR, suggesting that the domain gap might be less pronounced
than previously believed. By approaching the HFR problem as one of low-data
fine-tuning, we introduce a straightforward framework: comprehensive pre-
training, succeeded by a regularized fine-tuning strategy, that matches or
surpasses the current state-of-the-art on four publicly available benchmarks.
Corresponding codes can be found at
`https://github.com/michaeltrs/RethinkNIRVIS`.

##  I INTRODUCTION

Face recognition (FR) is one of the most important and well-studied fields in
computer vision [39, 1]. It was for many years one of the main driving forces
for the development of new lines of research in machine learning and was one
of the first wins of Deep Neural Networks (DNNs) versus human perception [28].
Nowadays, FR technologies are widely adopted from cell-phones Face ID sensors
to border control and immigration to name just a few. The most adopted and
used systems currently operate with NIR images due to their high robustness to
illumination changes.

Heterogeneous face recognition (HFR) [31, 36, 16, 11, 12] is becoming
essential in modern FR systems. While Near-Infrared (NIR) sensors are
frequently used to capture face images during deployment, these images
(probes) often need to be compared to a pre-existing face database (gallery)
captured in the Visible (VIS) spectrum. Therefore, there’s a pressing need for
systems to effectively match faces across NIR and VIS modalities, highlighting
the importance and growing interest in HFR. Most published HFR works suggest
the presence of a domain gap as one of the main challenges in HFR [11, 12] and
propose techniques to bridge that gap.

We follow a fundamentally different approach. Motivated by the perceptual
similarities between VIS and NIR imagery (Fig.1) and the richness of VIS FR
datasets (Fig.2) we employ transfer learning for solving the HFR problem. Our
main observations and contributions are the following:

Figure 1: Face photo captured under visible and infrared light [17]. The
infrared spectrum can be divided into four sub-bands: NIR (0.75–1.4$\mu m$),
SWIR (1.4–3$\mu m$), MWIR (3–8 $\mu m$), and LWIR (8–15$\mu m$) [30]. The
spectral sensitivity of NIR imagery is much closer to that of the VIS spectrum
opposed to images captured at the far end of the IR spectrum.

Figure 2: Size of FR datasets ($\\#$images, $\\#$identities). The average size
of NIR-VIS datasets is three orders of magnitude smaller than RGB datasets.

  1. 1.

Domain gap: we have determined that large CNNs, when pre-trained on extensive
VIS data, show remarkable zero-shot performance in NIR-VIS HFR, even
outperforming current benchmarks. This observation contrasts the prevailing
HFR narrative of a large domain gap and has been missed by the HFR literature
which has focused exclusively on training smaller models that do not exhibit
this behaviour.

  2. 2.

VIS pre-training: based on the above finding, we shift our focus towards
harnessing large-scale VIS data for HFR and introduce pre-training strategies
which lead to demonstrably improved zero-shot performance.

  3. 3.

NIR-VIS fine-tuning: standard fine-tuning is found to disrupt the embedding
space developed during pre-training. A simple method is presented that does
not only rectify previous issues but also sets new performance benchmarks on
four public NIR-VIS HFR datasets. Furthermore, through harnessing large-scale
VIS data during fine-tuning, we find further improvements in sensor
generalization performance.

##  II RELATED WORK

Primer on face recognition. Recent years have witnessed a number of
advancements in deep face recognition [32, 29, 22, 34, 8], the majority of
which are based on the evolution of training loss functions. Most of the early
works rely on metric-learning based loss [5, 29], however, these methods are
usually inefficient on large-scale training datasets, suffering from the
combinatorial explosion in the number of sample combinations. Therefore,
research attention has moved to margin-based classification loss functions
that aim to enhance intra-class compactness and inter-class separability [35,
22, 33, 34, 8]. NIR-VIS heterogeneous face recognition.  There are two
dominant approaches in the modern deep HFR literature: 1) Image synthesis
methods propose to solve the HFR problem by bridging the domain gap at the
level of model inputs, by learning to translate faces across domains [26, 38,
16]. 2) Domain-invariant feature learning methods [27, 25, 13] aim at
extracting facial identity features which are invariant to the source image
domain, thus, bridging the domain gap at the level of extracted features.
Among these, [11, 12] choose an unconditional generative model trained to
generate paired NIR-VIS images from random noise and generate a large amount
of training samples which are used to train a network to learn a domain
invariant feature space. To the best of our knowledge, the current state-of-
the-art in HFR is achieved by [23], who reconstruct 3D face shape and
reflectance from a large 2D facial dataset and transform the VIS reflectance
to NIR reflectance in order to generate large-scale photorealistic data in the
NIR and VIS spectra for further fine-tuning. Transfer learning aims at
improving a learner’s performance on a target task and data domain pair by
“transferring” the knowledge already learned through training in different but
somehow related source task and domain pair [24]. Transfer learning through
reusing classifier weights has been extensively used as a means for knowledge
distillation [2] including works on FR [9]. However, transfer learning for FR
typically involves transferring to a different set of identities which
discards the possibility of reusing classifier weights. To avoid this issue,
[40] pre-compute the classifier as the mean per-class embedding of the pre-
trained backbone and freeze these values to fine-tune the backbone for
homogeneous FR. Additionally, they do not allow model parameters to deviate
significantly from pre-trained values through an L2 regularization term.

Figure 3: Proposed pre-training and fine-tuning with a subspace classifier for
HFR. (top) we utilize both source and target data, use augmentation from
Eq.(1) and train with a joint set of identities, (bottom) we initialize all
modules from pre-trained counterparts, feed both source and target data to our
backbone, freeze both linear classifier weights, and train with the combined
loss presented in Eq.(2).

##  III METHOD

In contrast to VIS images, the use of NIR cameras is not ubiquitous,
discarding the possibility of gathering large-scale NIR imagery data from the
public domain. This showcases the important role of large-scale VIS data as a
source of pre-training data. A schematic overview of the proposed framework is
presented in Fig. 3.

###  III-A Pre-training with Large Scale VIS Data

To achieve strong HFR performance a model needs to be able to achieve feature
invariance for both VIS and NIR modalities. Current FR models trained on
large-scale VIS datasets have arguably achieved very strong performances [8,
34]. Thus, we assume that pre-training on large VIS data is enough to learn a
robust embedding space for the VIS modality and focus our attention on
improving downstream transfer ability with regard to NIR images. Each face
image can be decomposed into three color channels $x=\\{x^{R},x^{G},x^{B}\\}$
each of which is an intensity map of captured light at each respective
spectral range. However, not all (R, G, B) channels share the same
similarities with the NIR channel, the spectral sensitivity of the R channel
has significantly higher overlap with the NIR spectral range than the B, G
channels as shown in Fig.1. Motivated by this observation we are using the red
channel as a means of shifting VIS images closer in appearance to the NIR
spectrum through the following augmentation:

$x=\\{(x^{R},x^{G},x^{B}),(x^{R},x^{R},x^{R})\\},p=0.5$ (1)

Furthermore, we can optionally combine the source (VIS) and target (NIR-VIS)
data for pre-training. In doing so we not only inject some target data
knowledge during pre-training but also obtain a classifier checkpoint
containing information about target identities which can be utilized directly
during fine-tuning.

###  III-B Fine-Tuning on Target NIR-VIS Data

Fine-tuning DNNs directly for downstream tasks has been shown to potentially
reduce performance in low data regimes [20], an observation which is also
verified in section IV-B. While a pre-trained backbone transfers significant
prior knowledge, FR classifier weights are typically initialized randomly and
trained together with the backbone despite potentially having a larger
capacity. We propose two techniques for transferring knowledge for FR
classifiers. First, given the strong zero-shot performance of VIS pre-trained
models, it is reasonable to assume that the encoded representations of NIR-VIS
data will also form compact clusters, the centers of which are expected to be
strong identity predictors. We thus employ the mean identity embeddings [40]
as classifier values for HFR. Second, assuming both source and target data are
available, we pre-train with both datasets and keep only the subspace of the
classifier that corresponds to target identities. In doing so our target class
centers fit well with respective identities and by explicitly comparing them
with source centers during pre-training we end up with a more robust target
embedding space. In both cases, we change the regularization scheme employed
in [40]. Since there is a domain gap between source and target data we opt for
a regularization scheme that does not penalize deviation from pre-trained
parameter values. Instead, we reuse source data during fine-tuning and learn a
simultaneously good solution for both HFR and homogeneous FR while placing no
explicit constraint on model parameters.

$L_{finetune}=L_{cls}^{NIR-VIS}+\lambda L_{cls}^{preVIS}$ (2)

##  IV Experiments

TABLE I: FR and HFR datasets used in experiments.

Database | Domain | $N_{images}$ |  $N_{subjects}$ (eval) | $Year$
---|---|---|---|---
Oulu-CASIA [3] | NIR-VIS | 7,680 | 80 (40) | 2009
BUAA [6] | NIR-VIS | 2.7k | 150 (40) | 2012
CASIA 2.0 [21] | NIR-VIS | 17.5k | 725 (358) | 2013
LAMP-HQ [37] | NIR-VIS | 73.6k | 573 (273) | 2019
MS1Mv3 [14, 10] | VIS | 5.1M | 93k | 2020

TABLE II: Backbone Architectures used in experiments.

Model | input size | params (M) | FLOPS (G)
---|---|---|---
MFN [4] | $112\times 112$ | 10.48 | 0.23
LC29 [36] | $128\times 128$ | 10.48 | 3.70
IR18 [15] | $112\times 112$ | 24.03 | 2.62
IR50 [15] | $112\times 112$ | 43.59 | 6.32
IR100 [15] | $112\times 112$ | 65.15 | 12.12

Datasets.  Information on the datasets used is presented in Table I. We use
the MS1Mv3 dataset [10] for RGB pre-training and the respective folds from
four publicly available HFR datasets for fine-tuning and evaluation. Data pre-
processing.  Following common practice for FR, we obtain normalized face crops
by aligning all faces to a pre-defined template [22, 8, 34], using five facial
landmarks extracted by RetinaFace [7]. Models. All employed models are
presented in Table II. Out of these, LC29 [36] has been explicitly proposed
for HFR. Training. We employ ArcFace [8] as the margin based FR loss. We pre-
train for 24 epochs, batch size 512, $\lambda=0.1$. We fine-tune for 20 epochs
of target data using m=0.6, starting with learn rate $10^{-4}$ which we decay
by 0.1 at epochs 10, 15, 20, batch size 64 keeping batch size for source data
at 512. All training takes place on $\times 8$ Nvidia V100 GPUs.

###  IV-A Zero-shot performance from VIS pre-training

We begin by assessing the zero-shot performance of RGB pre-trained FR models
in HFR without further fine-tuning, presented in Table III. It is observed
that larger architectures (IR50, IR100) behave qualitatively differently from
smaller ones, having very strong performance despite the domain shift in stark
opposition to very clear performance degradation for smaller models. This
finding suggests that there exists adequate information in large-scale FR
datasets to bridge the domain gap to NIR, however, this is not typically
observed due to the small model capacity typically used in previous studies.
Our proposed method for enhanced pre-training through augmentation offers
clear performance gains for smaller architectures and less so for larger
models. Finally, we observe that including target data in the pre-train set is
enough to bridge a significant portion of the performance gap between the
zero-shot and fine-tuned models.

TABLE III: Zero-shot NIR-VIS performance after pre-training (TAR@FAR=$10^{-4}$). $\dagger$ fold-1, * with target train data. Model |  Lamp-HQ $\dagger$ |  CASIA 2.0 $\dagger$ | Oulu-CASIA | BUAA
---|---|---|---|---
| base | \+ red aug. | +target* | base | \+ red aug. | +target* | base | \+ red aug. | +target* | base | \+ red aug. | +target*
MFN | 87.91 | 88.68 | 96.90 | 95.05 | 95.75 | 98.26 | 84.60 | 88.36 | 92.75 | 96.70 | 96.73 | 98.44
LC29 | 84.93 | 86.37 | 98.17 | 95.84 | 95.97 | 99.49 | 89.09 | 89.41 | 93.51 | 96.19 | 96.24 | 99.03
IR18 | 93.04 | 93.23 | 98.92 | 97.88 | 98.76 | 99.51 | 92.72 | 94.80 | 95.74 | 98.05 | 98.45 | 99.37
IR50 | 99.03 | 99.16 | 99.84 | 99.89 | 99.90 | 99.97 | 99.52 | 98.76 | 99.61 | 99.84 | 99.61 | 100.0
IR100 | 99.60 | 99.65 | 99.89 | 99.93 | 99.97 | 99.98 | 99.82 | 99.87 | 99.75 | 99.92 | 99.81 | 100.0
TABLE IV: Performance on NIR-VIS public datasets (TAR@FAR=$10^{-4}$) after (top) naive (pre-train/fine-tune), (bottom) regularized fine-tuning with either mean or subspace classifier and regularization scheme ([40], $\lambda$={0,1}). $\dagger$ fold 1. | Model |  Lamp-HQ$\dagger$ |  CASIA 2.0$\dagger$ | Oulu-CASIA | BUAA
---|---|---|---|---|---
|  | ✘/✓ | ✓/✘ | ✓/✓ | ✘/✓ | ✓/✘ | ✓/✓ | ✘/✓ | ✓/✘ | ✓/✓ | ✘/✓ | ✓/✘ | ✓/✓
Naive | MFN | 0.14 | 87.91 | 92.75 | 1.61 | 95.75 | 81.54 | 3.28 | 84.60 | 60.46 | 0.15 | 96.70 | 97.58
IR18 | 3.98 | 93.04 | 94.77 | 0.73 | 97.88 | 86.10 | 8.20 | 92.72 | 54.94 | 0.19 | 98.05 | 93.80
IR50 | 68.47 | 99.03 | 97.72 | 50.50 | 99.89 | 94.30 | 12.59 | 99.52 | 95.1 | 90.81 | 99.84 | 99.61
IR100 | 71.52 | 99.60 | 96.85 | 52.35 | 99.93 | 93.86 | 4.08 | 99.82 | 92.76 | 89.80 | 99.92 | 99.77
|  |  RCT [40] |  $\lambda$=0 |  $\lambda$=1 |  RCT [40] |  $\lambda$=0 |  $\lambda$=1 |  RCT [40] |  $\lambda$=0 |  $\lambda$=1 |  RCT [40] |  $\lambda$=0 |  $\lambda$=1
Mean | MFN | 99.12 | 99.50 | 99.34 | 99.52 | 99.61 | 99.58 | 94.05 | 93.65 | 96.61 | 98.64 | 99.23 | 99.52
IR18 | 99.67 | 99.77 | 99.70 | 99.83 | 99.89 | 99.90 | 96.59 | 95.82 | 96.96 | 99.61 | 100.0 | 99.84
IR50 | 99.91 | 99.93 | 99.91 | 99.98 | 99.98 | 99.98 | 99.88 | 99.85 | 99.88 | 100.0 | 100.0 | 100.0
IR100 | 99.93 | 99.93 | 99.93 | 99.98 | 99.98 | 99.98 | 99.97 | 99.97 | 99.97 | 100.0 | 100.0 | 100.0
Subspace | MFN | 99.34 | 99.69 | 99.54 | 99.60 | 99.68 | 99.68 | 95.12 | 95.73 | 97.16 | 99.52 | 99.34 | 99.41
IR18 | 99.73 | 99.78 | 99.76 | 99.86 | 99.90 | 99.92 | 96.58 | 96.78 | 99.45 | 99.70 | 99.92 | 99.95
IR50 | 99.91 | 99.95 | 99.93 | 99.98 | 99.98 | 99.98 | 99.88 | 99.90 | 99.96 | 100.0 | 100.0 | 100.0
IR100 | 99.93 | 99.93 | 99.94 | 99.98 | 99.98 | 99.98 | 99.97 | 99.97 | 99.97 | 100.0 | 100.0 | 100.0
TABLE V: Comparison with state-of-the-art. LC29 architecture with mean
embedding classifier and $\lambda=0$. Folds 1-10.

Method CASIA 2.0 $\dagger$ Lamp-HQ $\dagger$ Oulu-CASIA BUAA FAR=$10^{-4}$
$10^{-3}$ Rank-1 FAR=$10^{-4}$ FAR=$10^{-3}$ Rank-1 FAR=$10^{-3}$ Rank-1
FAR=$10^{-3}$ Rank-1 LAMP-HQ [37] - 98.2 ± 0.2 99.2 ± 0.0 - 78.2 ± 3.0 97.3 ±
0.2 89.0 100.0 93.4 98.8 DFAL [19] - 98.7 ± 0.2 99.1 ± 0.2 - - - 93.8 100.0
99.2 100.0 OMDRA [18] - 99.4 ± 0.2 99.6 ± 0.1 - - - 92.2 100.0 99.7 100.0 DVG-
Face [12] 99.2 ± 0.1 99.9 ± 0.0 99.9 ± 0.1 - - - 97.3 100.0 99.1 99.9 LC-29
[23] 99.90 ± 0.06 100.0 ± 0.0 99.9 ± 0.1 98.6 ± 0.4 99.4 ± 0.3 99.1 ± 0.3 99.1
100.0 99.8 100.0 LC-29 (ours) 99.9 $\pm$ 0.1 99.95 $\pm$ 0.02 100.0 99.35
$\pm$ 0.2 99.87 $\pm$ 0.05 100.0 99.62 100.0 99.90 100

TABLE VI: Cross dataset evaluation (TAR@FAR=$1^{-4}$). Pre-trained MFN is fine-tuned with mean classifier ($\lambda$=0 /$\lambda$=1). $\dagger$ Fold 1. | Evaluation
---|---
Training |  |  Lamp-HQ $\dagger$ |  CASIA 2.0 $\dagger$ | Oulu-Casia | BUUA
Lamp-HQ $\dagger$ |  99.50 / 99.34 |  99.17 / 99.35 |  85.57 / 92.81 |  92.91 / 97.66
CASIA 2.0 $\dagger$ |  88.30 / 91.63 |  99.61 / 99.58 |  82.35 / 92.88 |  91.27 / 98.28
Oulu-Casia |  74.79 / 87.37 |  84.49 / 96.88 |  93.65 / 96.61 |  87.79 / 96.50
BUUA |  86.18 / 88.31 |  96.86 / 98.16 |  84.13 / 91.31 |  99.23 / 99.52
| no fine-tune | 88.68 | 95.75 | 88.36 | 96.73

###  IV-B HFR Fine-Tuning Performance

In Table IV (top) we present experimental results on naively fine-tuning to
target HFR data through randomly initializing classifier weights and end-to-
end training. More specifically, we evaluate model performance with or without
pre-training or fine-tuning. We observe that without pre-training all models
perform substantially worse than pre-trained counterparts, in particular, the
smaller architectures fail to learn any discriminative features. Thus, pre-
training appears to be crucial for learning useful representations from small
HFR datasets. Additionally, naive target set fine-tuning appears to destroy
the embedding space learned during pre-training and lead to performance
degradation. This is always the case for IR50, IR100, and almost always for
IR18 and MFN. In Table IV (bottom) we present experimental results for
regularized fine-tuning methods. We observe clear performance gains as most
models reach performances close to 100$\%$ for most datasets and are never
found to degrade performance compared to no fine-tuning. We additionally find
that regularization w.r.t. parameter values of pre-trained network (RCT) does
not help and is almost always suboptimal compared to no regularization
($\lambda=0$). This can be explained by the NIR-VIS domain gap as RCT was
proposed for homogeneous data. Our proposed regularization ($\lambda=1$) is
found to be somewhat less performant for the more diverse datasets (Lamp-HQ
and CASIA) but offers important gains for the less diverse ones (Oulu-Casia
and BUUA). In most cases tested our subspace classifier outperforms mean
embedding, albeit at the added cost of target-specific pre-training. Further
benefits of our fine-tuning method can be observed in Table VI where we
perform cross-dataset evaluation among the four HFR datasets. Similarly, we
note that apart from Lamp-HQ and CASIA, $\lambda=1$ outperforms $\lambda=0$ in
every case, with very large performance differences in nondiagonal elements
that have been trained and evaluated in different datasets. Lastly, in Table V
we present a comparison with state-of-the-art methods for HFR. A LC29 model is
pre-trained with red channel augmentation, no target data, and fine-tuned with
a mean embedding classifier and $\lambda=0$. We observe similar performance
for CASIA 2.0 and significant gains for all other datasets. Importantly, our
framework is conceptually much simpler than competing methods which rely on
expensive processes for generating synthetic data or employ complex
architectures.

##  V Conclusion and Future Work

In this paper, we presented a simple method consisting of strong pre-training,
followed by regularized fine-tuning, that demonstrated robust performance in
HFR. Our experiments further revealed that large-scale models, in particular,
showcase significant zero-shot performances compared to their smaller
counterparts. This suggests that VIS data alone carry ample information to
effectively address the HFR problem. While knowledge distillation (KD) might
seem like a natural progression given these findings, our initial experiments
with this technique did not yield the anticipated results, which could be
attributed to various factors, including the intricacies of the HFR problem.
Future work might focus on refining KD techniques applicable to HFR.

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