# Scalarization of Chern-Simons Kerr Black Hole Solutions and Wormholes

Nikos Chatzifotis<EMAIL_ADDRESS>Physics Department, School of
Applied Mathematical and Physical Sciences, National Technical University of
Athens, 15780 Zografou Campus, Athens, Greece.  Panos Dorlis
<EMAIL_ADDRESS>Physics Department, School of Applied Mathematical and
Physical Sciences, National Technical University of Athens, 15780 Zografou
Campus, Athens, Greece.  Nick E. Mavromatos  Physics Department, School of
Applied Mathematical and Physical Sciences, National Technical University of
Athens, 15780 Zografou Campus, Athens, Greece.  Physics Department, King’s
College London, Strand, London WC2R 2LS, UK.  Eleftherios Papantonopoulos
<EMAIL_ADDRESS>Physics Department, School of Applied Mathematical and
Physical Sciences, National Technical University of Athens, 15780 Zografou
Campus, Athens, Greece.

###### Abstract

Chern-Simons gravity rotating Kerr-type black hole solutions are revisited
from the point of view of their scalarization, namely examining the back
reaction of pseudoscalar axion fields on the rotating geometry. To lowest
order in an angular momentum expansion, and a long range approximation for
both the axion field configuration and its back reaction onto the metric, such
solutions had been discussed for the first time in the context of string
theory. In this work, we extend the analysis to give analytic expressions for
slowly rotating black holes outside the horizon, which formally include an all
order expansion in inverse powers of the radial distance from the centre of
the black hole, which allows to approach arbitrarily close the horizon. We
also discuss in some detail the way Chern-Simons gravity violates the energy
conditions, which leads to secondary axionic hair of the rotating black hole
solutions. This discussion is extended to studies of wormhole solutions, which
we construct in this article via the thin-shell procedure and we demonstrate
that the presence of the axion field enforces the two black hole solutions to
be counter-rotating and for slow rotation, there is no backreaction of the
Cotton tensor on the thin-shell of the wormhole.

††preprint: KCL-PH-TH/2022-03

##  I Introduction

Recently there is a lot of activity studying modified gravity theories which
are resulted from high order derivative corrections inspired from string
theory, from the presence of high curvature terms and from scalar fields
coupled to gravity, known as scalar-tensor theories. An important question in
all such theories concerns the behaviour of black hole solutions. This
question acquires important physical significance, given the rôle of such
black holes in producing gravitational waves (GW) during non-spherically
symmetric coalescence or collapse [1]. The GW, in return, can be used as
sensitive probes of the modified gravity scenarios which the black holes in
question constitute solutions to, given that the features and profile of the
GW in general depend on the underlying gravitational theory.

One of the basic feature of these modified gravity theories is the rôle of
scalar fields in such black hole solutions. In most of these theories the
scalar field backreacks on the geometry, the so-called process of
scalarization, and dresses the black hole with hair. In the context of string
theory [2, 3, 4, 5, 6] slowly rotating Kerr-type black holes acquire scalar
hair due to the presence of the so-called string model independent axion field
[7], which in the context of the (3+1)-dimensional effective low energy string
inspired gravitational theory, obtained after string compactification, is dual
to the field strength of the spin-one antisymmetric tensor field of the
massless gravitational multiplet. On the other hand, dilaton scalar field
secondary hair in non rotating black holes in dilaton-Gauss-Bonnet (GB) string
inspired gravity has been studied in [8]. In both examples, Kerr and non
rotating string inspired black holes, the existence of secondary scalar field
hair is allowed as a result of the violation of the weak energy conditions.

Gravity theories that contain higher powers of curvature are the Chern-Simons
(CS) theories which are consistent Lanczos-Lovelock gravity theories in higher
than four dimensions. CS black holes are special solutions of these theories
resulting in second order field equations for the metric with well defined AdS
asymptotic solutions [9]. The CS black holes in d-dimensions were studied in
[10] and calculating analytically the quasinormal modes of the scalar
perturbations the stability of these solutions was investigated (for a review
on CS black holes see [11]).

The action for CS modified gravity is defined by the sum of the Einstein-
Hilbert action and a new parity violating, four-dimensional correction.
Interest in the model spiked when it was found that string theory unavoidingly
requires such a correction to remain mathematically consistent. In the
perturbative string sector, such a correction is necessary by the Green-
Schwarz anomaly canceling mechanism upon four-dimensional compactification. In
general, such a correction arises in the presence of Ramond-Ramond scalars due
to duality symmetries. This parity violating axion field can dress a slowing
rotating black hole with (secondary) scalar hair. One of the first works that
evaluated the axion field hair that dresses a slow rotating black hole, as a
result of the coupling of the axion field to a Lorentz CS term, was presented
in [4, 3]. This result suggested that non-minimal gravitational couplings may
produce interesting new effects in black hole backgrounds. This solution was
further extended in [6] where it was found that the charges introduced by the
axion hair are determined by the mass, angular momentum and gauge charges of
the background rotating black hole. Allowing the axion field to be dynamical a
solution describing a rotating black hole in the small coupling slow rotation
limit was found in [12]. Static and rotating black string solutions in
dynamical CS modified gravity were studied in [13, 14].

In scalar-tensor theories [15] hairy black hole solutions were generated when
the scalar field coupled to gravity backreacts to the background metric. The
asymptotic properties of the spacetime is vital for the existence of hairy
black holes. One of the first hairy black hole was discussed in [16] but it
was found that it was unstable [17]. Introducing a scale through the presence
of a cosmological constant, regular hairy black hole solutions were found in
asymptotically AdS/dS spacetimes [18, 19, 20, 21, 22, 23, 24, 25]. If matter
interacts with black holes non-trivial scalar field can be supported in their
exterior region. Various hairy black holes were found in the context scalar-
tensor Horndeski theories [26] in which matter is coupled to the Einstein
tensor [27, 28, 29, 30, 31, 32, 33].

If the scalar field is directly coupled to curvature invariants hairy black
holes can also be generated. In this case the scalar hair is maintained by the
interaction with the spacetime curvature. One high curvature correction is the
Gauss-Bonnet (GB) term and to avoid topological invariant behaviour the scalar
field has to be coupled to the GB term in four dimensions [34]. These gravity
theories are known as extended scalar-tensor-Gauss-Bonnet (ESTGB) theories and
were studied extensively in the literature [35]-[42] in a attempt to get hairy
black hole solutions. For certain classes of the scalar coupling function it
was shown that we have spontaneous scalarization of black holes [43]-[49]. The
scalarization procedure were applied in ESTGB gravity theories with various
asymptotic spacetimes. in asymptotical flat spacetime black hole solutions and
compact objects [50]-[56] and also in AdS/dS spacetimes [57, 58, 59, 60, 61]
and in a black hole background with different curvature topologies has been
studied in [62, 63]. The connections of asymptotically AdS black holes
scalarization with holographic phase transitions in the dual boundary theory
was studied in [64, 65]. Recently the spontaneous scalarization in f(R)
gravity theories was discussed in [66].

There is a strong connection of the violation of the energy conditions with
the formation of scalar hair outside the horizon of a black hole. In both CS
and GB extended scalar-tensor gravity theories [8] there are negative
contribution of the higher-curvature terms on the effective stress tensor of
the theory which leads to the violation of energy conditions as we will
discuss in the following. Such violation of the energy conditions also
characterises the CS-axion string-inspired running-vacuum cosmology [67, 68].
This is a very interesting effect of the scalarization of black holes. It
suggests an interplay between the curvature and matter, the spacetime geometry
is deformed around the black hole because of the presence of matter which
interacts with curvature.

In this work in the generic CS gravity we will look for rotating black hole
solutions in the presence of axion fields, which backreact on the geometry and
source the anomalous terms. Our aim is to extent previous results [4, 5, 6,
69, 70] in which the existing solutions truncated to long range outside the
horizon, in the sense that there was no attempt to express formally the
solution in terms of inverse powers of the radial distance from the centre of
the black hole. Our solutions of the axionic field backreacting to a slow
rotating background metric will allow us to go arbitrarily close to the
horizon of the hairy black. In this way we will be able to study the behaviour
of axionic matter outside the horizon of a slow rotating Kerr-type metric and
see how this matter distribution violates the energy conditions and
investigate the possibility of the formation of a wormhole.

In particular, we shall discuss in detail the violation of energy conditions
that allow for the existence of secondary axion hair, following the discussion
in [8] and motivated by the violation of the energy conditions of our slow
rotating Kerr-type black hole solution with axionic hair, we will investigate
if a wormhole solution can be constructed. The new feature of our solution is
the presence of the axionic matter or the associated Cotton tensor, and using
the gluing procedure that connects via a wormhole throat two oppositely slowly
rotating Kerr black holes, we will study if the axionic matter backreacts on
the whormhole geometry to the first order in the angular parameter.

The work is organized as follows. In Section II, we review the construction of
the CS term and set up the Lagrangian and equations of motion for the graviton
and axion fields in CS gravity, which shall lead to the local rotating black-
hole solutions. We also study the properties of the Cotton tensor which
appears in the variation of the CS term, and which plays a central rôle in our
approach. In Section III we discuss local solutions of the Einstein and axion
equations of motion, of rotating Kerr-type black-hole in the slow rotation
approximation, that is to lowest order in an expansion in powers of the
angular momentum or spin of the black hole. In Section IV we discuss in detail
the energy conditions of the slow rotating black hole solution and demonstrate
the violation of the NEC condition, which leads to the existence of axionic
hair. In Section V we construct wormhole solutions via the thin shell
junctioning procedure, and we discuss possible backreaction on the axionic
field on the thin shell. Finally conclusions and outlook are discussed in
Section VI. Some technical aspects of our approach, concerning the black hole
solution and the wormhole construction, are discussed in the Appendices.

##  II Chern-Simons Gravitational Theory

In this Section we will review the Chern-Simons gravitational theory
discussing mainly local solution of this theory. Consider an axion matter
field coupled to the gravitational CS topological term known as the Pontryagin
density term [71, 72]. This term, henceforth denoted as $R_{CS}$, is
constructed from the contraction of the Riemann tensor to the dual Riemann
tensor and reads

$R_{CS}=\frac{1}{2}R^{\mu}_{\,\,\,\nu\rho\sigma}\widetilde{R}^{\nu\,\,\,\,\rho\sigma}_{\,\,\,\mu},$
(1)

where the symbol $\widetilde{(\dots)}$ denotes the dual of the Riemann tensor,
defined as

$\widetilde{R}_{\alpha\beta\gamma\delta}=\frac{1}{2}R_{\alpha\beta}^{\,\,\,\,\,\,\,\,\rho\sigma}\varepsilon_{\rho\sigma\gamma\delta},$
(2)

with
$\varepsilon_{\rho\sigma\kappa\lambda}=\sqrt{-g(x)}\,\hat{\epsilon}_{\rho\sigma\kappa\lambda}$
the covariant Levi-Civita under the convention that the symbol
$\hat{\epsilon}_{0123}=1$, etc.

The action that is giving the coupling of the axion field to the $R_{CS}$ term
in presence of its kinetic energy is given by

$S=\int
d^{4}x\,\sqrt{-g}\,\left[\frac{R}{2\kappa^{2}}-\frac{1}{2}(\partial_{\mu}b)(\partial^{\mu}b)-A\,b\,R_{CS}\right]=\int
d^{4}x\,\sqrt{-g}\left[\frac{R}{2\kappa^{2}}-\frac{1}{2}(\partial_{\mu}b)(\partial^{\mu}b)\right]-\int
d^{4}xAb\hat{R}_{CS}\leavevmode\nobreak\ ,$ (3)

where $\hat{R}_{CS}$ is the Chern-Simons term with the flat Levi-Civita tensor
$\hat{\epsilon}$, $\kappa=M_{\rm Pl}^{-1}$ is the inverse of the reduced
Planck mass, $b$ is a pseudoscalar denoting the axion matter field and $A$ is
the coupling parameter of the axion field to the Chern-Simons term with
dimension of length. Notice that the action (3) is shift-symmetric under the
transformation $b\to b+c$, because we have do not introduced any axion $b$
potential or mass term. Such actions are inspired from microscopic string-
theory models. In the case the action (3) is derived from the corresponding
low-energy string effective to quadratic order in derivatives [2, 4], after
appropriate string compactification, the axion field $b$ is the dual of the
field strength of the spin one antisymmetric tensor field in the bosonic
massless gravitational multiplet of the string, the so-called string-model
independent axion [7], and the parameter $A$ is fixed in terms of the string
Regge slope $\alpha^{\prime}=M_{s}^{-2}$, where $M_{s}$ denotes the string
scale, as follows [6]

$\displaystyle\displaystyle
A=\sqrt{\frac{2}{3}}\frac{\alpha^{\prime}}{48\kappa}\leavevmode\nobreak\ ,$
(4)

and is of order $\mathcal{O}\left(\frac{M_{p}}{M_{s}^{2}}\right)$. The CS
terms are accompanied by the axion field $b$, which, as we shall discuss
below, is in the weak approximation for slowly-rotating black holes of large
mass ${\mathcal{M}}$ compared to the Planck scale,

$\displaystyle{\mathcal{M}\,\kappa\gg 1}\,,$ (5)

which we assume throughout. Hence, by selecting appropriately small angular
momentum we guarantee the validity of our slow-rotation approximations in
which we keep terms of up to linear order in the black hole angular momentum.

Variation of the action with respect to the metric and the axion field yields
the following equations of motion

$\displaystyle
G_{\mu\nu}=\kappa^{2}T^{b}_{\mu\nu}+4\kappa^{2}AC_{\mu\nu}\leavevmode\nobreak\
,$ (6) $\displaystyle\square b=A\,R_{CS}\leavevmode\nobreak\ ,$ (7)

where $T^{b}_{\mu\nu}$ is the stress energy-momentum tensor associated with
the kinetic term of a matter field,

$T^{b}_{\mu\nu}=\nabla_{\mu}b\nabla_{\nu}b-\frac{1}{2}g_{\mu\nu}(\nabla
b)^{2}\leavevmode\nobreak\ .$ (8)

The quantity $C_{\mu\nu}$ is the Cotton tensor derived from the metric
variation of $bR_{CS}$ (see Appendix (A)) and reads [71]

$C_{\mu\nu}=-\frac{1}{2}\nabla^{\alpha}\left[(\nabla^{\beta}b)\widetilde{R}_{\alpha\mu\beta\nu}+(\nabla^{\beta}b)\widetilde{R}_{\alpha\nu\beta\mu}\right]\leavevmode\nobreak\
.$ (9)

We notice that, unlike the gauge anomaly term, the gravitational CS term is
not topological, in the sense of its variation with respect to the metric
contributing to the total stress energy-momentum tensor. The Cotton tensor
satisfies a covariant non-conservation law [71, 72],

$\nabla_{\mu}C^{\mu\nu}=-\frac{1}{4}(\nabla^{\nu}b)R_{CS}\leavevmode\nobreak\
.$ (10)

This result can be straightforwardly recovered using the equations of motion.
Indeed, the equations of motion for the axion (7) are simply the Bianchi
identity of the gravitational equations. Indeed, under
$\nabla^{\mu}G_{\mu\nu}=0$, where $G_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,R$ is the Einstein tensor, and noting that
$\nabla^{\mu}T_{\mu\nu}=\square b\nabla^{\nu}b$, we can immediately find that

$\displaystyle
0=\nabla^{\mu}G_{\mu\nu}=\kappa^{2}\nabla^{\mu}T^{b}_{\mu\nu}+4\kappa^{2}A\nabla^{\mu}C_{\mu\nu}\,,$
(11)

from which,

$\displaystyle 0=\kappa^{2}\square
b\nabla^{\nu}b+4\kappa^{2}A\,\nabla^{\mu}\,C_{\mu\nu}\,\overset{\eqref{Axion}}{\implies}A\,\,R_{CS}\nabla^{\nu}b=-4A\,\nabla^{\mu}\,C_{\mu\nu}\implies\nabla^{\mu}\,C_{\mu\nu}=-\frac{1}{4}(\nabla^{\nu}b)\,R_{CS}\leavevmode\nobreak\
.$ (12)

Note that the conservation of the matter stress tensor, $T^{b}_{\mu\nu}$ is
violated, since $\nabla^{\mu}T^{b}_{\mu\nu}$ is non zero, being proportional
to the CS anomalous term (cf. (12))

$\displaystyle\nabla^{\mu}\,T^{b}_{\mu\nu}=-4\,A\,\nabla^{\mu}\,C_{\mu\nu}=A\,\frac{1}{4}\,(\nabla^{\nu}b)\,R_{CS}\leavevmode\nobreak\
.$ (13)

This implies an exchange of energy between matter (the axion field $b$) with
the gravitational anomaly for spacetime backgrounds for which the latter is
non trivial, such as gravitational wave perturbations [73, 68] or rotating
black holes [4, 5, 6], of interest in the current study. There is no issue of
the validity of general covariance though, given that (13) has been derived
using self-consistently the covariant formalism. Moreover, in the specific
context of string theory, the axion $b$ is itself part of the gravitational
multiplet of the string, which the graviton belongs to, and hence an exchange
of energy with the CS anomaly refers to interactions between members of that
multiplet.111In this respect, we mention that in the cosmological model of
[68], the gravitational CS anomalies have been assumed to characterise only
the very early universe, where only fields from the string gravitational
multiplet are assumed to exist as external classical fields. Matter, including
radiation and chiral fermion fields, are assumed to be generated in that
cosmology at the end of the inflationary period, which itself is due to a
condensation of GW that lead in turn to the formation a CS anomaly condensate,
contributing a de Sitter term in the effective action, which dominates in the
early universe and leads to inflation without the need of inflaton fields,
simply being due to gravitational non-linearities. The primordial CS terms
(which could be due to both, primordial GW as well as rotating primordial
black holes), though, are cancelled by the gravitational anomalies generated
by the chiral fermionic matter at the end of the inflationary period, and in
this sense the post inflationary cosmological evolution does not exhibit the
aforementioned axionic energy exchange with gravity.

##  III The search for local solutions: slowly rotating Kerr-type black holes

Since the axion matter field is a pseudoscalar, it imposes an axial symmetry
on the underlying spacetime. Naturally, if we consider a static and
spherically symmetric spacetime, the $R_{CS}$ term vanishes identically and
the axion dynamics need to be absent. As such, in a spherically symmetric
background, our theory is reduced to GR. However, the presence of the axion
field $b$ imposes axial symmetry and therefore we may search for rotating
compact object solutions. To get an exact rotating black hole solution we will
constrain ourselves to slowly rotating spacetimes. We shall perform our
analysis to leading order in the angular momentum parameter $a$ of the slowly-
rotating Kerr-type spacetime, and suppress any contributions of order
$\mathcal{O}(a^{2})$.

The metric ansatz we are going to consider is the following

$ds^{2}=-G(r)dt^{2}+F(r)dr^{2}-2r^{2}a\sin^{2}\theta
W(r)dtd\phi+r^{2}d\Omega^{2}\leavevmode\nobreak\ .$ (14)

From the axion equation of motion (7) we find that the CS term up to first
order in $a$ yields

$A\,R_{CS}=-aA\cos\theta
W^{\prime}\frac{4F^{2}G^{2}+rGF^{\prime}(rG^{\prime}-2G)+F(-4G^{2}+r^{2}(G^{\prime})^{2}+2rG(G^{\prime}-rG^{\prime\prime}))}{F^{2}G^{2}r^{2}\sqrt{FG}}\leavevmode\nobreak\
,$ (15)

while the left hand side of the axion equation (7) reads

$\square
b=\frac{1}{F}\left[\partial_{r}^{2}+\left(\frac{2}{r}-\frac{F^{\prime}}{F}+\frac{G^{\prime}}{2G}\right)\right]b+\frac{1}{r^{2}}\frac{1}{\sin\theta}\partial_{\theta}\left[\sin\theta\partial_{\theta}b\right]\leavevmode\nobreak\
,$ (16)

where prime denotes differentiation with respect to $r$. Using a separation of
variables and noticing that the right hand side of the axion equation (16) is
just the $L^{2}$ operator of the angular momentum of $b$, we deduce, on
account of equations (7), (15) and (16), that the axion field $b$ may be
written as

$b=aAu(r)P_{1}(\cos\theta)\leavevmode\nobreak\ ,$ (17)

where $P_{1}$ denotes the Legendre polynomial of the first order, which
straightforwardly cancels the angular dependence on the axionic equation of
motion. We note, however, that the axion field is of minimum order
$\mathcal{O}(a)$. Focusing on the Einstein equations (6), we find that this
implies that the $tt$ and $rr$ component of the gravitational equations of
motion are satisfied in vacuum, since the corresponding stress energy tensor
components are of order $\mathcal{O}(a^{2})$, which we ignore in our
approximation. This naturally means that

$G(r)=\frac{1}{F(r)}=1-\frac{2M}{r}\leavevmode\nobreak\ \,,\quad M\equiv
G\,\mathcal{M}\,,$ (18)

with $G=8\pi\,\kappa^{-2}$ the Newton’s gravitational constant in
(3+1)-dimensional spacetimes.

Therefore, any backreaction on our spacetime will be encoded in the off-
diagonal component of the metric. To this end, we consider the redefinition of

$W(r)=\frac{2M}{r^{3}}+w(r)\leavevmode\nobreak\ ,$ (19)

in order to extract the corrections, $w(r)$, on the recovered slowly rotating
Hartle-Thorne spacetime (14), which are enforced by the axionic contribution.
Using equations (17), (18) and (19), we consider the $t\phi$ component of the
gravitational equations of motion (6). In particular, we find that [71]

$G_{t\phi}=\kappa^{2}T_{t\phi}+4A\kappa^{2}C_{t\phi}\leavevmode\nobreak\ ,$

where the $t\phi$ component of the Cotton tensor $C_{t\phi}$ has the form

$C_{t\phi}=-\frac{3AM(r-2M)(ru^{\prime}-u)}{r^{5}}a\sin^{2}\theta+\mathcal{O}(a^{2})\leavevmode\nobreak\
.$ (20)

These lead to

$\left(\frac{u}{r}\right)^{\prime}=-\frac{1}{24A^{2}\kappa^{2}M}(r^{4}w^{\prime})^{\prime}\leavevmode\nobreak\
.$

Integrating the above expression and minimizing the corresponding integration
constant, which would otherwise lead to a divergent axion, we find that

$u(r)=-\frac{r^{5}w^{\prime}}{24A^{2}\kappa^{2}M}\leavevmode\nobreak\ .$ (21)

Therefore, we have now fixed all the degrees of freedom of our theory except
the corrections on the off-diagonal component $w$, which can be found via the
axionic equation of motion, by virtue of equation (21). In particular,
plugging our results on the axion equation (7) we find the following
differential equation

$r^{11}(r-2M)w^{\prime\prime\prime}+2r^{10}(6r-11M)w^{\prime\prime}+(28r^{10}-50Mr^{9}-576A^{2}\kappa^{2}M^{2}r^{4})w^{\prime}+3456A^{2}\kappa^{2}M^{3}=0\leavevmode\nobreak\
.$ (22)

To solve the above equation we consider a series expansion on the correction
function $w$. In particular, in order the radial component of the axion field
asymptotically to vanish, $w^{\prime}(r)$ needs to be at least of order
$\mathcal{O}(r^{-5})$ and consequently $w(r)$ to be at least of order
$\mathcal{O}(r^{-4})$, which in view of equation (19) implies that
asymptotically the leading order of $g_{t\phi}$ is that of the Hartle-Thorne
spacetime [1], i.e. asymptotically the spacetime coincides with the slowly
rotated Kerr metric (14). We define $w$ in a non-closed form as

$w(r)=\sum_{n=4}^{\infty}\frac{d_{n}M^{n-2}}{r^{n}}\leavevmode\nobreak\ ,$
(23)

where $M^{n-2}$ is introduced in order to keep the coefficients $d_{n}$
dimensionless. Hence, the problem of finding the geometric correction $w(r)$
has been reduced to the determination of the coefficients $d_{n}$. Making use
of (23), we find after some algebra that equation (22) reads

$\displaystyle
3456A^{2}\kappa^{2}M^{3}-162M^{7}d_{9}+256M^{7}d_{8}+8M^{2}d_{4}r^{5}+\sum_{n=-4}^{-1}\frac{M^{n+7}\left[-(n+3)(n+6)(n+9)d_{n+9}+2(n+4)^{2}(n+8)d_{n+8}\right]}{r^{n}}$
(24)
$\displaystyle+\sum_{n=1}^{\infty}\frac{M^{n+7}\left[-(n+3)(n+6)(n+9)d_{n+9}+2(n+4)^{2}(n+8)d_{n+8}\right]+576A^{2}\kappa^{2}M^{n+3}(n+3)d_{n+3}}{r^{n}}=0\leavevmode\nobreak\
.$

In order for the left hand site of this equation to be zero for any $r$ all
the coefficients of the powers $r^{n}$ have to vanish. This leads us to the
following equations

$\displaystyle d_{4}=0\leavevmode\nobreak\ ,$ (25) $\displaystyle
256d_{8}-162d_{9}=-3456\frac{A^{2}\kappa^{2}}{M^{4}}$
$\displaystyle-(n+3)(n+6)(n+9)d_{n+9}+2(n+4)^{2}(n+8)d_{n+8}=0\;\;,\;\;\text{for}\;n=-1,-2,-3,-4$
$\displaystyle-(n+3)(n+6)(n+9)d_{n+9}+2(n+4)^{2}(n+8)d_{n+8}+576\frac{A^{2}\kappa^{2}}{M^{4}}(n+3)d_{n+3}=0\;\text{for}\;n\geq
1\leavevmode\nobreak\ .$

Performing the shift of $n\rightarrow n-9$, the last equation yields to

$d_{n}=\frac{2(n-5)^{2}(n-1)}{n(n-6)(n-3)}d_{n-1}+\frac{576A^{2}\kappa^{2}}{n(n-3)M^{4}}d_{n-6},\;\;\text{for}\;n\geq
10\leavevmode\nobreak\ .$ (26)

The above recursive equation requires $d_{4,5,6,7,8,9}$ to be known in order
to calculate all the other coefficients. From (25) we find the following set
of equations

$\displaystyle d_{4}=d_{5}=0\leavevmode\nobreak\ ,$ (27) $\displaystyle-28$
$\displaystyle d_{7}+48d_{6}=0\leavevmode\nobreak\ ,$ $\displaystyle-80$
$\displaystyle d_{8}+126d_{7}=0\leavevmode\nobreak\ ,$ $\displaystyle 256$
$\displaystyle
d_{8}-162d_{9}=-3456\frac{A^{2}\kappa^{2}}{M^{4}}\leavevmode\nobreak\ .$

Thus, we are left with four unknown coefficients, but with three equations.
This means that we cannot completely determine the series coefficients from
the differential equation (22).

However, we can extract a constraint from the weak field limit, i.e. the case
where the metric does not backreact on the axionic field. This case is valid
in the asymptotic region of our spacetime. Then, the metric solution is found
to be the slowly rotated Kerr spacetime in a first order approximation scheme
on the angular momentum. Plugging the slowly rotated Kerr metric to the
axionic equation of motion (7), we find the differential equation

$-2u(r)+2(r-M)u^{\prime}(r)+(r^{2}-2Mr)u^{\prime\prime}(r)=\frac{144M^{2}}{r^{5}}\leavevmode\nobreak\
,$ (28)

which is solved in the Appendix B to yield

$u(r)=-\frac{5}{4Mr^{2}}-\frac{5}{2r^{3}}-\frac{9M}{2r^{4}}\leavevmode\nobreak\
.$ (29)

Consequently, the asymptotic behavior determines the first terms of the radial
part of the axion. Of course, this result has to be compatible with (21),
which under relation (23) reads

$u(r)=\frac{1}{24A^{2}\kappa^{2}M}\left[4d_{4}M^{2}+\frac{5d_{5}M^{3}}{r}+\frac{6d_{6}M^{4}}{r^{2}}+\frac{7d_{7}M^{5}}{r^{3}}+\frac{8d_{8}M^{6}}{r^{4}}\right]+\mathcal{O}(1/r^{5})\leavevmode\nobreak\
.$ (30)

Performing a simple matching of the coefficients between equations (30) and
(29), we may deduce that the first coefficients of (23) read

$d_{4}=d_{5}=0\;\;,\;\;d_{6}=-5\gamma^{2}\;\;,\;\;d_{7}=-\frac{60\gamma^{2}}{7}\;\;,\;\;d_{8}=-\frac{27\gamma^{2}}{2}\;\;,\;\;d_{9}=0,\;\;\;\text{where}\;\;\;\gamma^{2}=\frac{A^{2}\kappa^{2}}{M^{4}}\leavevmode\nobreak\
.$ (31)

We note here that an important consequence of the above result is that the
relation (23) produces coefficients that contain higher even orders of
$\gamma$ by virtue of (26)

$d_{10}=d_{11}=0\;\;,\;\;d_{12},\,d_{13},\,d_{14}\,\sim\,-\gamma^{4}\;\;,\;....\;d_{21}\sim-\gamma^{4}-\gamma^{6}\;....,$
(32)

where $\sim$ denotes positive proportionality numerical coefficients, which we
omit for brevity, as we only want to indicate the pertinent order in $\gamma$
for the various $d_{n}$ coefficients. In Appendix C we demonstrate the
convergence of the relevant series (23) for every real value of the constant
$\gamma\in\mathbb{R}$. From relations (31), (32) (and (26)) we observe that
all coefficient are non positive, an important result we shall make use in the
following.

Then our scalarized slowly rotating Kerr metric reads [4, 5, 69]

$ds^{2}=-\left(1-\frac{2M}{r}\right)\,dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}\,d\,\Omega^{2}-2\,r^{2}\,a\,\sin^{2}\theta\,W(r)\,dt\,d\phi\leavevmode\nobreak\
,$ (33)

where the off-diagonal correction term reads

$W(r)=\frac{2M}{r^{3}}-\frac{A^{2}\kappa^{2}(189M^{2}+120Mr+70r^{2})}{14r^{8}}+\mathcal{O}(A^{2n}),\,\,\,\,\,\,\text{with}\,\,\,n={\rm
positive\leavevmode\nobreak\ integer}\geq 2\,,$ (34)

where the higher order terms $\mathcal{O}(A^{2n})$ can be found by using (26).
We remark at this point that such terms can become small perturbations of the
explicitly written expressions if we assume, in a self-consistent manner with
our slow rotation approximation, black hole solutions with sufficiently large
mass $M$. Even in the case of [67, 68] which is characterised by large
$A\,\kappa^{2}\gg 1$, the terms of $\mathcal{O}(A^{2n})$ in (34) will be
suppressed by inverse powers of $M$ (cf. (26)), and hence can be tuned to be
small by appropriately arranging $M$.

We stress that, although in this work we concentrated to lowest-order in an
angular-momentum expansion $a$ for the black hole solution, our solution goes
beyond the truncation considered in [69] as far as the incorporation of terms
inversely proportional to powers of the radial distance from the centre is
concerned. We include arbitrary order of such terms in an analytic fashion, as
follows from the above considerations, and thus we can approach the horizon of
the black hole from the exterior region arbitrarily close.222In the same
spirit, the work of [70], which considers $\mathcal{O}(a^{2})$ corrections in
the black-hole angular momentum, also uses truncated expressions, and various
other approximations. We plan to extend our analysis to such cases as well, in
the near future, going beyond these approximations.

The corresponding axionic contribution can now be found by using (17) and (21)
in the same order of $A$ as

$b=aA\cos\theta\left(-\frac{5}{4Mr^{2}}-\frac{5}{2r^{3}}-\frac{9M}{2r^{4}}\right)+\mathcal{O}(A^{m}),\,\,\,\,\,\,\text{for}\,\,\,m=2n+1,\,\,n\in\mathbb{Z}^{+},$
(35)

with $\mathbb{Z}^{+}$ the set of positive integers. This also yields the non-
zero component of the Cotton tensor (20) as

$\displaystyle
4\kappa^{2}A\,{C}_{t\phi}=a\frac{r-2M}{2}\sum_{n=4}^{\infty}\frac{n(n-3)d_{n}M^{n-2}}{r^{n+1}}\sin^{2}(\theta)\leavevmode\nobreak\
,$ (36)

which is negative for all $r>2M$, that is outside the horizon, due to the fact
that $d_{n}\leq 0$ for all $n$ (cf. (31) and (32)). Testing the result from
the Komar integral [1, 76] associated with the polar isometry denoted by the
Killing vector $\xi=\partial_{\phi}$, we find that

$J=\int\star d\xi=aM\leavevmode\nobreak\ ,$ (37)

which means that the axionic black-hole hair, proportional to $A\,a$, is a
secondary charge of the scalarized black hole, since it is not independent
from the defining parameters of the black hole, i.e. the mass and the angular
momentum as it happens in scalarized black hole models.

##  IV Violation of energy conditions for rotating black holes in Chern-Simons
gravity with axion Hair

In this Section we will discuss the violation of the energy conditions and
their connection to the no-hair theorem for the axionic CS slowly rotating
black holes we discussed in the previous Section. According to Bekenstein [74]
minimally as well as non-minimally [75] coupled scalar fields that produce an
energy-momentum tensor that obeys the WEC cannot be candidates for dressing a
black hole with scalar hair. In the following, we present the violation of the
NEC in our CS scalar gravity black hole solution (which implies the violation
of all energy conditions, see the discussion in Appendix D), and then present
a discussion on what one can conclude from such a violation as far as the
secondary axionic hair is concerned, for the case of slowly rotating black
holes (33).

Consider the action (3), whose equations of motion (6), (7), have been solved
above in the slowly rotating limit, with the solution to first order in the
small angular momentum parameter $a$ given by the slowly rotating Kerr-type
black hole (33). From the gravitational equations (6), the conserved energy-
momentum tensor is

$T^{eff}_{\mu\nu}=\kappa^{2}T_{\mu\nu}+4\kappa^{2}A\,{C}_{\mu\nu}\leavevmode\nobreak\
.$ (38)

Up to first order in the angular momentum, $\mathcal{O}(a)$, the only
contribution comes by the $t\phi$ component, which on account of (36), reads

$T^{eff}_{t\phi}=4\kappa^{2}A\,{C}_{t\phi}=a\frac{r-2M}{2}\sum_{n=4}^{\infty}\frac{n(n-3)d_{n}M^{n-2}}{r^{n+1}}\sin^{2}(\theta)\leavevmode\nobreak\
.$ (39)

As mentioned previously, due to the negative sign of all $d_{n}$, the
contribution of the ${C}_{t\phi}$ component to the conserved energy momentum
tensor is clearly negative outside the horizon, $r>2M$. Moreover, for a
stationary observer333We refer here to a stationary observer, while not to a
static one as before, due to the fact the spacetime is stationary and not
static. However, the definition of this observer is the same as before, i.e. a
four velocity proportional to the timelike Killing vector induced to the
spacetime. the energy density is zero up to $\mathcal{O}(a)$. Thus, the
existence of hair is expected to occur due to the violation of the NEC (see
Eq. (113) of Appendix D).

To demonstrate the violation of the NEC, let us define the following future
directed null vectors

$l_{\pm}^{\mu}=\left(1,0,0,-\frac{g_{t\phi}}{g_{\phi\phi}}\pm\sqrt{\left(\frac{g_{t\phi}}{g_{\phi\phi}}\right)^{2}-\frac{g_{tt}}{g_{\phi\phi}}}\right)\leavevmode\nobreak\
.$ (40)

The contraction with $T^{eff}_{\mu\nu}$, gives us

$T^{eff}_{\mu\nu}l_{\pm}^{\mu}l_{\pm}^{\nu}=\pm\frac{a(r-2M)^{3/2}}{2}\sin(\theta)\sum_{n=4}^{\infty}\frac{n(n-3)d_{n}M^{n-2}}{r^{n+5/2}}+\mathcal{O}(a^{2})\leavevmode\nobreak\
.$ (41)

The above result implies: $T^{eff}_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}\leq 0$ and
$T^{eff}_{\mu\nu}l_{-}^{\mu}l_{-}^{\nu}\geq 0$, where the equality holds, in
both cases, at the horizon $r=2M$. Thus, the NEC is clearly violated in the
exterior of the the black hole horizon, except for the poles $\theta=0,\pi$
where the contraction vanishes.

This violation is attributed to the coupling of the axion with the Chern-
Simons topological term, as this coupling is responsible for the appearance of
the Cotton tensor in the gravitational equations of motion. Moreover, as the
coupling constant takes larger values, the NEC violation intensifies and
consequently the deformation of the spacetime becomes more important. In Fig.
1, we illustrate the precise way by means of which this violation occurs: near
the horizon, the violation takes its maximum value, while far away is
negligible. So, the deformation of the Hartle-Thorne geometry is important
near the horizon, while far away from it, is negligible. In addition, the NEC
violation becomes stronger as we increase the value of the dimensionless
parameter $\gamma$, which is the visualization of the statement that stronger
coupling implies more deformation.

Figure 1: Behavior of $\Lambda_{+}=T^{eff}_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}/a$
at the equitorial plane $\theta=\pi/2$ up to $\mathcal{O}(A^{2})$ with respect
to the radial coordinate.

From the above considerations there seems to be a strong relation between the
energy conditions and the hair of the black holes. From the gravitational
equations of motion, we know that the matter fields and the underlined
geometry of the spacetime are coupled, one affects the other. The energy
conditions are statements that give insights on this coupling. Specifically,
they shed light between properties of the matter fields and the deformation of
the spacetime geometry due to the matter fields. Thus, if the existence of
hair for the black holes is closely related to the violation of the energy
conditions, it is closely related on how the spacetime geometry is deformed
around the black holes.

Hairy black hole solutions are examples on how this spacetime deformation can
be achieved. As we already discussed in the case of the CS scalar tensor
gravity an axion hair has appeared, due to the coupling of the axion
pseudoscalar field with the CS topological term. Moreover, this coupling is
responsible for the violation of the NEC, while the violation becomes more
important as the coupling becomes stronger. In addition, the violation reaches
its maximum near the horizon and then it tends to zero. Thus, we can conclude
that the coupling with the curvature term deforms the geometry arround the
black hole in such a way that the axion hair is allowed. However, we should
remark at this point that, since the axion hair appears with a secondary
charge that backreacts to the metric, the independent parameters that describe
the black hole are still limited to the mass, angular momentum and gauge
charges, and hence the problem about the loss of information of the collapsed
matter hidden inside the black hole is not solved with solely such a secondary
hair.

##  V Wormhole solutions in axionic Chern-Simons theories

In this Section we will discuss the possibility of generating a wormhole by
the gluing procedure that connects via a wormhole throat two oppositely slowly
rotating Kerr-type black holes we found (33) with secondary axion hair. After
a review of the thin shell formalism, we will proceed to construct the
wormhole with the aim of examining whether the presence of the CS term,
responsible for the violation of the energy condition, suffices to stabilise
the wormhole.

###  V.1 The thin-shell formalism

The gluing procedure [76, 77, 78, 79] consists of considering that our
spacetime, $\mathcal{V}$, is split by an embedded hypersurface $\Sigma$ into
two regions, $\mathcal{V}^{+}$ and $\mathcal{V^{-}}$. We denote the metric of
$\mathcal{V}^{\pm}$ as $g_{\mu\nu}^{\pm}$ and equip it with coordinates
$x^{\alpha}_{\pm}$. In a similar manner, we equip both sides of $\Sigma$ with
coordinates $y^{i}$, where $i=(0,1,2)$. We wish to find the conditions that
must be put on the metrics of each spacetime, $\mathcal{V}^{\pm}$, to ensure
that the union of $g^{+}$ and $g^{-}$ forms a valid solution to the
gravitational field equations. We assume that the same coordinates $y^{i}$ can
be installed on both sides of the hypersurface and we choose $n^{\mu}$ to be
the unit normal to $\Sigma$ pointing from $\mathcal{V}^{-}$ to
$\mathcal{V}^{+}$. We will also suppose that a continuous coordinate system
$x^{\mu}$, distinct from $x^{\mu}_{\pm}$, can be introduced on both sides of
the hypersurface. These coordinates overlap with $x^{\mu}_{\pm}$ in an open
region of $\mathcal{V}^{\pm}$ that contains $\Sigma$.

The $x^{\mu}$ coordinate system will be used only qualitatively on the
derivation of the junctioning conditions. We note that the $\Sigma$
hypersurface is to be pierced by a congruence of geodesics that intersect it
orthogonally. We take $\lambda$ to denote proper time (or proper distance) and
we adjust the parametrization such that the geodesics cross the hypersurface
at $\lambda=0$ and that $\lambda$ is negative in $\mathcal{V}^{-}$ and
positive in $\mathcal{V}^{+}$. Therefore, a point $p\in\mathcal{V}^{\pm}$
identified by the coordinates $x^{\mu}$ is linked to $\Sigma$ by a member of
the congruence, and $\lambda(x^{\mu})$ is the proper time (or proper distance)
from $\Sigma$ to $p$ along this geodesic. As such, a displacement away from
the hypersurface along one of the geodesics may be described by

$n^{\mu}=\frac{dx^{\mu}}{d\lambda}\implies
n_{\mu}=\epsilon\partial_{\mu}\lambda\implies\partial_{\mu}=\epsilon
n_{\mu}\partial_{\lambda}\leavevmode\nobreak\ ,$ (42)

where $n_{\mu}$ is normalized to $n_{\mu}n^{\mu}=\epsilon=\pm 1$. In order to
perform the match of $\mathcal{V}^{+}$ and $\mathcal{V}^{-}$ through the
hypersurface $\Sigma$, we need to consider the distribution function
formalism. In particular, we denote $\Theta(\lambda)$ as the Heaviside
distribution function for which
$\partial_{\lambda}\Theta(\lambda)=\delta(\lambda)$, where $\delta(\lambda)$
is the Dirac distribution function. In general, we will express a quantity $A$
as

$A(x)=A^{+}(x^{+})\Theta(\lambda)+A^{-}(x^{-})\Theta(-\lambda)\leavevmode\nobreak\
,$ (43)

where the indices $\pm$ indicate that the value of $A$ in the region
$\mathcal{V}^{\pm}$. The jump of $A$ across $\Sigma$ is denoted by

$[A]=A(\mathcal{V}^{+})|_{\Sigma}-A(\mathcal{V}^{-})|_{\Sigma}\leavevmode\nobreak\
.$ (44)

Finally, we also note the relations

$[n^{\mu}]=[e^{\mu}_{i}]=0\leavevmode\nobreak\ ,$ (45)

where $\displaystyle e^{\mu}_{i}=\frac{\partial x^{\mu}}{\partial y^{i}}$ is
the pullback operator performing the mapping
$\mathcal{V}^{\pm}\rightarrow\Sigma$. $[n^{\mu}]=0$ stems from (42) and the
fact that $x^{\mu}$ was assumed to be continuous across $\Sigma$, while
$[e^{\mu}_{i}]=0$ stems from the fact that $\Sigma$ is equipped with the same
coordinates $y^{i}$ on both sides. Making use of the above we may extract the
first junctioning condition. In particular, we note that the total spacetime
metric of $\mathcal{V}$ is expressed as

$g_{\mu\nu}=g^{+}_{\mu\nu}\Theta(\lambda)+g^{-}_{\mu\nu}\Theta(-\lambda)\leavevmode\nobreak\
,$ (46)

while its derivative reads

$\partial_{\rho}g_{\mu\nu}=\partial^{+}_{\rho}g^{+}_{\mu\nu}\Theta(\lambda)+\partial^{-}_{\rho}g^{-}_{\mu\nu}\Theta(-\lambda)+\epsilon
n_{\rho}\delta(\lambda)[g_{\mu\nu}]\leavevmode\nobreak\ .$ (47)

In the construction of the gravitational tensors we will encounter terms of
the form $(\partial g)^{2}$, which will yield terms of the form
$\delta^{2}(\lambda)$, which are ill-defined. This means that, in order for
the gluing procedure to work, we need the first junctioning condition to read

$[g_{\mu\nu}]=0\leavevmode\nobreak\ .$ (48)

We note that the total spacetime metric admits the decomposition
$g_{\mu\nu}=h_{\mu\nu}+\epsilon n_{\mu}n_{\nu}$, where $h_{\mu\nu}$ is the
first fundamental form of the hypersurface defined in the $x^{\mu}$ coordinate
system. As such, the first junctioning condition may be reexpressed via (45)
as $[g_{\mu\nu}]=[h_{\mu\nu}]$, which by virtue of the pullback operators
$h_{ij}=h_{\mu\nu}e^{\mu}_{i}e^{\nu}_{j}$ reads

$[h_{ij}]=0\leavevmode\nobreak\ .$ (49)

As a general guideline, we note that any quantity $A$ that is expressed in the
form of (43), will yield a non-trivial $[A]$ term when differentiated, which
needs to be tested for consistency in the construction of the equations of
motion. By virtue of (47) and (48), the affine connection is easily found to
be

$\Gamma^{\alpha}_{\,\,\,\beta\gamma}=\Theta(\lambda)\Gamma^{+\alpha}_{\,\,\,\beta\gamma}+\Theta(-\lambda)\Gamma^{-\alpha}_{\,\,\,\beta\gamma}\leavevmode\nobreak\
,$ (50)

and its derivative similarly reads

$\partial_{\delta}\Gamma^{\alpha}_{\,\,\,\beta\gamma}=\Theta(\lambda)\partial_{\delta}\Gamma^{\alpha}_{+\beta\gamma}+\Theta(-\lambda)\partial_{\delta}\Gamma^{\alpha}_{-\beta\gamma}+\epsilon\delta(\lambda)[\Gamma^{\alpha}_{\,\,\,\beta\gamma}]n_{\delta}\leavevmode\nobreak\
.$ (51)

Therefore, the Riemann tensor reads

$R^{\alpha}_{\,\,\,\beta\gamma\delta}=\Theta(\lambda)R^{\alpha}_{+\beta\gamma\delta}+\Theta(-\lambda)R^{\alpha}_{-\beta\gamma\delta}+\delta(\lambda)\mathfrak{R}^{\alpha}_{\,\,\,\beta\gamma\delta}\leavevmode\nobreak\
,$ (52)

where

$\mathfrak{R}^{\alpha}_{\,\,\,\beta\gamma\delta}=\epsilon\left([\Gamma^{\alpha}_{\,\,\,\beta\delta}]n_{\gamma}-[\Gamma^{\alpha}_{\,\,\,\beta\gamma}]n_{\delta}\right)\leavevmode\nobreak\
.$ (53)

We see that the Riemann tensor is properly defined as a distribution, but the
$\delta$ function represents a curvature singularity at $\Sigma$. The second
junction condition usually seeks to eliminate this term in GR, but we also
have the contribution of the Cotton tensor to consider. We note that a
physical interpretation can nevertheless be given to the singularity. It
corresponds to the curvature induced by a thin-shell of matter, which is used
to construct the wormhole by the gluing of the two spacetimes. A second note
is that, since $\mathfrak{R}$ is the difference of connections, it forms a
tensor. As such, it needs to be expressed in a tensorial quantity, which is
calculated in Appendix E, and reads

$\mathfrak{R}^{\alpha}_{\,\,\,\beta\gamma\delta}=(n_{\beta}n_{\gamma}[K^{\alpha}_{\delta}]-n_{\beta}n_{\delta}[K^{\alpha}_{\gamma}]-[K_{\beta\delta}]n^{\alpha}n_{\gamma}+[K_{\beta\gamma}]n^{\alpha}n\delta)\leavevmode\nobreak\
.$ (54)

In a similar fashion, following the analysis in Appendix E, we find that

$G_{\mu\nu}=G_{\mu\nu}^{+}\Theta(\lambda)+G_{\mu\nu}^{-}\Theta(-\lambda)+\delta(\lambda)\mathfrak{G}_{\mu\nu}\leavevmode\nobreak\
,$ (55)

where

$\mathfrak{G}_{\alpha\beta}=-\epsilon([K_{\alpha\beta}]-[K]g_{\alpha\beta})-[K]n_{\alpha}n_{\beta}\leavevmode\nobreak\
.$ (56)

We may now move on to the stress-energy tensor components of our theory.

###  V.2 The junctioning conditions of Chern-Simons gravity

We will firstly consider the trivial case of $T_{\mu\nu}$. We note that the
axionic field is to be expressed by virtue of (43) as

$b=b^{+}\Theta(\lambda)+b^{-}\Theta(-\lambda)\leavevmode\nobreak\ .$ (57)

Naturally, in order for $\displaystyle
T_{\mu\nu}=\nabla_{\mu}b\nabla_{\nu}b-\frac{1}{2}g_{\mu\nu}(\nabla b)^{2}$ to
not contain any terms of the form $\delta(\lambda)^{2}$, we need to impose
that

$\partial_{\mu}b=\partial^{+}_{\mu}b^{+}\Theta(\lambda)+\partial^{-}_{\mu}b^{-}\Theta(\lambda)+\epsilon\delta(\lambda)[b]\implies[b]=0\leavevmode\nobreak\
,$ (58)

which means that the jump of the axionic field vanishes identically.
Therefore, it is quite easy to deduce that

$T_{\mu\nu}=T^{+}_{\mu\nu}\Theta(\lambda)+T^{-}_{\mu\nu}\Theta(-\lambda)\leavevmode\nobreak\
,$ (59)

which means that the stress-energy tensor of the axion does not contribute to
equations of motion of the thin-shell, i.e. the terms with $\delta(\lambda)$.
It is in this sense that in order for the wormhole construction to work in GR,
a thin-shell of matter must be inserted by hand in order for the second
junctioning condition, $\mathfrak{G}_{\alpha\beta}=0$, to be violated.

We will now analyse the effects of the Cotton tensor. We firstly note that, by
virtue of
$\displaystyle\left|g\right|=-\frac{1}{4}\hat{\varepsilon}^{\mu\nu\rho\sigma}\hat{\varepsilon}^{\alpha\beta\gamma\delta}\,g_{\mu\alpha}\,g_{\nu\beta}\,g_{\rho\gamma}\,g_{\sigma\delta}$
and the metric (46), we find that

$\left|g\right|=\left|g^{+}\right|\Theta(\lambda)+\left|g^{-}\right|\Theta(-\lambda)\leavevmode\nobreak\
,$ (60)

which yields

$\varepsilon_{\mu\nu\rho\sigma}=\varepsilon^{+}_{\mu\nu\rho\sigma}\Theta(\lambda)+\varepsilon^{-}_{\mu\nu\rho\sigma}\Theta(-\lambda)\leavevmode\nobreak\
.$ (61)

To simplify the calculations, we rewrite the Cotton tensor as

$C_{\mu\nu}=\,\nabla_{\alpha}D^{\alpha}_{\,\,\,\mu\nu}\leavevmode\nobreak\ ,$
(62)

where

$D^{\alpha}_{\,\,\,\mu\nu}=\left(-\frac{\partial^{\xi}b}{2}(\widetilde{R}^{\alpha}_{\,\,\,\mu\xi\nu}+\widetilde{R}^{\alpha}_{\,\,\,\nu\xi\mu})\right)\leavevmode\nobreak\
,$ (63)

where by virtue of relations (52), (53) and (61), the dual Riemann reads

$\widetilde{R}^{\alpha}_{\,\,\,\mu\xi\nu}=\Theta(\lambda)\widetilde{R}^{\alpha}_{+\mu\xi\nu}+\Theta(-\lambda)\widetilde{R}^{\alpha}_{-\mu\xi\nu}+\delta(\lambda)\mathfrak{R}^{\alpha}_{\,\,\,\mu\gamma\delta}\varepsilon^{\gamma\delta}_{\,\,\,\,\,\xi\nu}\leavevmode\nobreak\
.\\\ $ (64)

Therefore, the tensorial quantity of (63) is decomposed as

$D^{\alpha}_{\,\,\,\mu\nu}=D^{\alpha}_{+\mu\nu}\Theta(\lambda)+D^{\alpha}_{-\mu\nu}\Theta(-\lambda)+\delta(\lambda)\left(-\frac{\partial^{\xi}b}{2}\mathfrak{R}^{\alpha}_{\,\,\,\mu\gamma\delta}\varepsilon^{\gamma\delta}_{\,\,\,\,\,\xi\nu}+(\mu\leftrightarrow\nu)\right)\leavevmode\nobreak\
.$ (65)

Since the Cotton tensor is found by the divergence of
$D^{\alpha}_{\,\,\,\mu\nu}$, the calculation will yield components of the form
$\partial_{\lambda}\delta(\lambda)$, which are ill-defined. In order to avoid
this and make the procedure work, we impose that

$\left(-\frac{\partial^{\xi}b}{2}\mathfrak{R}^{\alpha}_{\,\,\,\mu\gamma\delta}\varepsilon^{\gamma\delta}_{\,\,\,\,\,\xi\nu}+(\mu\leftrightarrow\nu)\right)=0\leavevmode\nobreak\
.$ (66)

Computing now the divergence of $D^{\alpha}_{\,\,\,\mu\nu}$ to calculate the
Cotton tensor, we immediately find that

$C_{\mu\nu}=C^{+}_{\mu\nu}\Theta(\lambda)+C^{-}_{\mu\nu}\Theta(-\lambda)+\epsilon\delta(\lambda)n_{\alpha}[D^{\alpha}_{\,\,\,\mu\nu}]=C^{+}_{\mu\nu}\Theta(\lambda)+C^{-}_{\mu\nu}\Theta(-\lambda)+\delta(\lambda)\mathfrak{C}_{\mu\nu}\leavevmode\nobreak\
,$ (67)

Let us now compute $\mathfrak{C}_{\mu\nu}$. We note that

$\displaystyle[\mathcal{D}^{\alpha\mu\nu}]$
$\displaystyle=\mathcal{D}^{\alpha\mu\nu}_{+}|_{\Sigma}-\mathcal{D}^{\alpha\mu\nu}_{-}|_{\Sigma}$
$\displaystyle=\left(\frac{-\partial^{+}_{\beta}b^{+}}{2}\widetilde{R}_{+}^{\alpha\mu\beta\nu}+\frac{-\partial^{+}_{\beta}b^{+}}{2}\widetilde{R}_{+}^{\alpha\nu\beta\mu}-\frac{-\partial^{-}_{\beta}b^{-}}{2}\widetilde{R}_{-}^{\alpha\mu\beta\nu}-\frac{-\partial^{-}_{\beta}b^{-}}{2}\widetilde{R}_{-}^{\alpha\nu\beta\mu}\right)|_{\Sigma}$
$\displaystyle=-\frac{1}{2}\left[\partial_{\beta}b\,\widetilde{R}^{\alpha\mu\beta\nu}\right]-\frac{1}{2}\left[\partial_{\beta}b\,\widetilde{R}^{\alpha\nu\beta\mu}\right]\leavevmode\nobreak\
,$

which by $[n_{\alpha}]=0$, yields that

$\mathfrak{C}_{\mu\nu}=-\frac{1}{2}\left[\partial^{\beta}b\,\widetilde{R}^{\alpha}_{\,\,\,\mu\beta\nu}n_{\alpha}\right]-\frac{1}{2}\left[\partial^{\beta}b\,\widetilde{R}^{\alpha}_{\,\,\,\nu\beta\mu}n_{\alpha}\right]\leavevmode\nobreak\
.$ (68)

We note that in contrast to $\mathfrak{G}_{\mu\nu}$, $\mathfrak{C}_{\mu\nu}$
is not tangent to the hypersurface. In particular,

$\mathfrak{C}_{\mu\nu}n^{\mu}=-\frac{1}{2}\left[\partial^{\beta}b\,\widetilde{R}^{\alpha}_{\,\,\,\nu\beta\mu}n^{\mu}n_{\alpha}\right],\qquad\mathfrak{C}_{\mu\nu}n^{\nu}=-\frac{1}{2}\left[\partial^{\beta}b\,\widetilde{R}^{\alpha}_{\,\,\,\mu\beta\nu}n^{\nu}n_{\alpha}\right]\leavevmode\nobreak\
.$

By making use of the symmetries of the dual Riemann, the only way that
$\mathfrak{C}_{\mu\nu}$ is tangent to the hypersurface is if
$\partial_{\beta}b=f(x)n_{\beta}$, i.e. the derivative of axion field is
normal to the hypersurface. However, by virtue of the fact that the axionic
field of our solution has angular dependence as well, this also can’t be the
case. Therefore, we need to insert a thin-shell with a non-trivial matter
stress-tensor on our action in order for the gluing to work, since there is no
way to cancel out the normal components of $\mathfrak{C}_{\mu\nu}$.444This
means that one needs to consider the full Gauss-Codazzi equations, [76], in
order to completely determine the dynamics of the thin-shell.

Therefore, we deduce that the modified Lanczos equations [79] in CS gravity
read

$\mathfrak{G}_{\mu\nu}-4A\kappa^{2}\mathfrak{C_{\mu\nu}}=\kappa^{2}S_{\mu\nu}\leavevmode\nobreak\
,$ (69)

where $S_{\mu\nu}$ is the surface stress-energy tensor of the thin-shell
expressed in the total spacetime coordinates $x^{\mu}$.

###  V.3 The gluing of the axionic black holes

We move on to the construction of thin-shell wormholes by slowly rotating
axionic black holes. The metrics on each manifold $\mathcal{V}^{\pm}$ read
according to our axionic solution (33),

$ds^{2}_{\pm}=-f_{\pm}dt_{\pm}^{2}+\frac{dr^{2}_{\pm}}{f_{\pm}}-2r^{2}_{\pm}\sin^{2}\theta_{\pm}a_{\pm}W_{\pm}dt_{\pm}d\phi_{\pm}+r^{2}_{\pm}d\Omega_{\pm}^{2}=-f_{\pm}dt_{\pm}^{2}+\frac{dr^{2}_{\pm}}{f_{\pm}}+r_{\pm}^{2}d\theta^{2}_{\pm}+r^{2}_{\pm}\sin^{2}\theta_{\pm}(d\phi_{\pm}-a_{\pm}W_{\pm}dt_{\pm})^{2}$

where the second equality is derived from the fact that $\mathcal{O}(a^{2})$
is a trivial contribution by virtue of the black hole being in the slow
rotation limit. We can simplify calculations by transforming the metric in a
co-rotating frame by

$d\psi=d\phi-aW(R)dt\leavevmode\nobreak\ ,$ (70)

which yields that

$ds^{2}_{\pm}=-f_{\pm}dt_{\pm}^{2}+\frac{dr^{2}_{\pm}}{f_{\pm}}+r_{\pm}^{2}d\theta^{2}_{\pm}+r^{2}_{\pm}\sin^{2}\theta_{\pm}(d\psi_{\pm}-a_{\pm}\Delta
W_{\pm}dt_{\pm})^{2}\leavevmode\nobreak\ ,$ (71)

where $\Delta W=W(r)-W(R)$. Therefore, the total space coordinates,
$x^{\mu}_{\pm}$ are $(t,r,\theta,\psi)_{\pm}$. We consider now the embedding
map $t_{\pm}=T_{\pm}(\tau),r_{\pm}=R_{\pm}(\tau)$, where $\tau$ is the proper
time of an observer comoving with the hypersurface. Then, each corresponding
induced metric reads

$\displaystyle
ds^{2}_{-}|_{\Sigma}=-\left[F(R_{-})\dot{T}_{-}^{2}-\frac{\dot{R}_{-}^{2}}{F(R_{-})}\right]d\tau^{2}+R_{-}^{2}(\tau)d\Omega^{2}\leavevmode\nobreak\
,$ (72) $\displaystyle
ds^{2}_{+}|_{\Sigma}=-\left[F(R_{+})\dot{T}_{+}^{2}-\frac{\dot{R}_{+}^{2}}{F(R_{+})}\right]d\tau^{2}+R_{+}^{2}(\tau)d\Omega^{2}\leavevmode\nobreak\
,$ (73)

where $F_{\pm}=1-2M/R_{\pm}(\tau)$ and the dot denotes differentiation with
respect to $\tau$. In order to perform the matching between the two induced
metrics, we set that the induced metric $h_{ij}$, reads

$ds^{2}_{\Sigma}=-d\tau^{2}+R^{2}(\tau)d\theta^{2}+R^{2}(\tau)\sin^{2}\theta
d\psi^{2}\leavevmode\nobreak\ ,$ (74)

i.e., the hypersurface coordinates $y^{i}$, are $(\tau,\theta,\psi)$ where,
naturally $\theta=\theta_{+}=\pi-\theta_{-}$. The matching of the bulk metrics
on the hypersurface implies that $R_{+}=R_{-}=R$ for the throat radius, while
$\dot{T}_{\pm}$ is found to satisfy

$F_{\pm}\dot{T_{\pm}}=\beta_{\pm}=\sqrt{\dot{R}_{\pm}^{2}+F_{\pm}}\leavevmode\nobreak\
.$ (75)

We consider that the two glued black holes are of the same mass, which yields
that $F_{+}=F_{-}\implies T_{+}=T_{-}\implies\beta_{+}=\beta_{-}$. We note
that by virtue of the co-rotating coordinate system, we can not yet have a
relation for the rotation of the two black holes.

We have to derive the pullback operators, $e^{\mu}_{i}$ and the unit normal
one-form, $n_{\mu}$. In particular, the four-velocity of an observer comoving
with the throat of the wormhole reads

$e^{\mu}_{\tau}=\left(\frac{\beta}{F},\dot{R},0,0\right)\leavevmode\nobreak\
.$ (76)

This means that, as expected, the observer comoving with the hypersurface has
vanishing $\psi$-velocity. On the other hand, the unit normal vector can be
readily found to yield

$n_{\mu}^{\pm}=\pm\left(-\dot{R},\frac{\beta}{F},0,0\right)\leavevmode\nobreak\
.$ (77)

We note that $n_{\mu}e^{\mu}_{\tau}=0$ and $n_{\mu}n^{\mu}=1$ are indeed
verified. Under (76) and (77), we may find that the extrinsic curvature of the
hypersurface $\displaystyle
K^{\pm}_{ij}=-n_{\alpha}^{\pm}\left(\frac{\partial^{2}x^{\alpha}}{\partial
y^{i}\partial y^{j}}+\Gamma^{\alpha}_{\,\,\,\beta\gamma}\frac{\partial
x^{\beta}}{\partial y^{i}}\frac{\partial x^{\gamma}}{\partial y^{j}}\right)$
reads

$K^{\pm}_{ij}=\pm\begin{pmatrix}-\frac{\dot{\beta}}{\dot{R}}&0&-\frac{a_{\pm}R^{2}\sin^{2}\theta
W^{\prime}}{2}\\\ 0&R\beta&0\\\ -\frac{a_{\pm}R^{2}\sin^{2}\theta
W^{\prime}}{2}&0&R\sin^{2}\theta\beta\end{pmatrix}$ (78)

where prime denotes differentiation with respect to $R$.

In order to compute the contribution of the Cotton tensor on the hypersurface,
we need to compute the terms of (68),
$\displaystyle-\frac{1}{2}\partial^{\beta}b\widetilde{R}^{\alpha}_{\,\,\,\mu\beta\nu}n_{\alpha}+(\mu\leftrightarrow\nu)$,
on each spacetime and then calculate their difference, if any, on the
hypersurface. In order to do that, we recall that the axionic solution is of
the form

$b_{\pm}=a_{\pm}Au_{\pm}\cos\theta_{\pm}\leavevmode\nobreak\ ,$ (79)

where $u$ is the radial component. In order for $[b]=0$ to be satisfied, under
the constraint that $\theta_{+}=\pi-\theta_{-}$, we immediately find that

$a_{+}=-a_{-}\leavevmode\nobreak\ ,$ (80)

which means that the two black holes need to be counter-rotating. The counter-
rotating property has been previously considered in [80, 81]. Our reasoning
here, however, is rather different. The result is enforced a priori, in order
for the axion to not contain any discontinuities on the throat. We note that
in a faster rotating paradigm, the axion would contain higher orders in the
angular momentum and Legendre polynomials of higher rank. This could in
principle differentiate the result of $[b]\implies a_{+}=-a_{-}$ to a much
more complicated expression with richer physical properties. Under the
constraint that the two black holes are actually counter-rotating, and then we
find that the Cotton tensor contribution on the hypersurface vanishes
identically, i.e.

$\mathfrak{C}_{\mu\nu}=0\leavevmode\nobreak\ .$ (81)

Therefore, by considering the addition of a thin-shell of matter, which
enables the construction of the wormhole via the Lanczos equations, we find
that, in the slowly rotating paradigm, the surface stress-energy tensor yields

$S_{ij}=-\frac{\epsilon}{\kappa^{2}}([K_{ij}]-[K]h_{ij})\leavevmode\nobreak\
,$ (82)

which is the familiar expression of the GR case. Calculating the surface
stress-energy tensor yields that

$S_{ij}=\frac{-\epsilon}{\kappa^{2}}\begin{pmatrix}4\frac{\beta}{R}&0&0\\\
0&-R^{2}\left(\frac{2\beta}{R}+\frac{2\dot{\beta}}{\dot{R}}\right)&0\\\
0&0&-R^{2}\sin^{2}\theta\left(\frac{2\beta}{R}+\frac{2\dot{\beta}}{\dot{R}}\right)\end{pmatrix}$
(83)

by virtue of the fact that two black holes are counter-rotating. Any
subsequent analysis on this case will not yield any fruitful results, since
(83) is the same surface stress-energy tensor as in the Schwarzschild case [1,
82, 83, 84]. We may conclude that the backreaction of the Cotton tensor on the
thin-shell is not giving any new result in the case of the slowly rotating
black hole to first order in the angular parameter $\mathcal{O}(a)$, and thus
one needs to consider faster rotating paradigms in order to test for the
stability of the resulting thin-shell wormhole.555We mention that the
contributions of the variation of the CS term on the hypersurface (i.e. the
appropriate surface terms associated with the Cotton tensor (20)), computed in
Appendix F, are of order $\mathcal{O}(a^{2})$ in the (small) angular momentum
parameter $a$. As such, there is no back reaction of the axion field (and
hence Cotton tensor) on the wormhole thin shell.

##  VI Conclusions and Outlook

In this work we studied the scalarization of a slowing rotating black hole in
the presence of an axion field coupled to Chern-Simons term. We found an exact
slow rotating Kerr-type black hole solution dressed with axionic hair. Our
solution is expressed in terms of the inverse powers of the radial distance
from the centre of the black hole. This behaviour of the black hole solution
allows us to go arbitrarily close to the horizon of the hairy black hole. In
this way we were able to study the properties of the axionic matter
distribution outside the slow rotation Kerr-type black hole.

We discussed in detail the violation of energy conditions that allowed the
existence of secondary axion hair of our slow rotating Kerr-type black hole
solution. There is strong relation between the energy conditions and the
formation of the axionic hair distribution outside the horizon of our black
hole solution. We found that because of the presence of the axion coupling to
the Chern-Simons term the null energy condition is violated, while the
violation becomes more important as the coupling becomes stronger and the
violation reaches its maximum near the horizon and then it tends to zero.
Thus, we can conclude that the coupling with the curvature term deforms the
geometry around the black hole in such a way that the axion hair is allowed.

Motivated by the violation of the null energy condition, we studied the
generation of a wormhole, by the gluing procedure, that connects via a
wormhole throat our two slowly rotating Kerr-type black holes with secondary
axion hair. We found that the condition for the axion field to not contain any
discontinuities on the throat, enforces the two black hole solutions to be
counter-rotating. For the slowly rotating to first order in the angular
parameter $\mathcal{O}(a)$ our black hole solution, we did not find any
backreaction of the Cotton tensor on the thin-shell of the wormhole.

It would be interesting to extent our study to higher order $\mathcal{O}(a)$
in the angular momentum. We expect to find a much richer structure of the
axionic hairy Kerr-type black hole. Then calculating the energy conditions we
will get a better understanding of the interplay of the axionic matter to
curvature outside the black hole horizon. Also we expect to find that there
will be a backreaction of the Chern-Simons term on the throat of the wormhole
and then it will be possible to test the stability of the resulting thin-shell
wormhole.

## Acknowledgements

We wish to thank Ioannis Kavvadias for fruitful discussions related to
material presented in Appendix C. The work of N.E.M is supported in part by
the UK Science and Technology Facilities research Council (STFC) under the
research grant ST/T000759/1. N.E.M. also acknowledges participation in the
COST Association Action CA18108 Quantum Gravity Phenomenology in the
Multimessenger Approach (QG-MM).

## APPENDICES

##  Appendix A Mathematical properties of the Chern-Simons action and the
Cotton tensor

In our analysis here we follow the Lorentzian signature, where the metric has
one negative eigenvalue, $(-,+,+,+)$ and the Riemann tensor is expressed as

$\displaystyle
R^{\rho}_{\,\,\,\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\,\,\,\sigma\nu}+\Gamma^{\rho}_{\,\,\,\xi\mu}\Gamma^{\xi}_{\,\,\,\sigma\nu}-\dots\,.$
(84)

The Cotton tensor, in the way defined in [71], follows from the infinitesimal
variation with respect to the graviton field of the gravitational anomaly CS
term in the action (3) with respect to $g_{\mu\nu}$. Using the definitions and
conventions of the Riemann tensor (84) and its dual,

$\widetilde{R}^{\alpha}_{\,\,\,\beta\gamma\delta}=\frac{1}{2}R^{\alpha}_{\,\,\,\beta\rho\sigma}\varepsilon^{\rho\sigma}_{\,\,\,\,\,\,\gamma\delta}\,,$

where $\varepsilon_{\rho\sigma\kappa\lambda}$ is the covariant Levi-Civita
tensor under the convention $\varepsilon_{0123}=\sqrt{-g}$, which implies
$\displaystyle\varepsilon^{0123}=\frac{-1}{\sqrt{-g}}$, we obtain

$\displaystyle\delta\,\Big{(}A\,\int
d^{4}x\,\sqrt{-g}\,b(x)\,R^{\mu\nu\rho\sigma}(x)\,\widetilde{R}_{\mu\nu\rho\sigma}(x)\Big{)}\equiv
4\,A\,\int d^{4}x\,\sqrt{-g}\,C^{\mu\nu}(x)\,\delta g_{\mu\nu}(x)=-4\,A\,\int
d^{4}x\,\sqrt{-g}\,C_{\mu\nu}(x)\,\delta g^{\mu\nu}(x)\leavevmode\nobreak\ ,$
(85)

with $\delta g_{\mu\nu}$ denoting the variation of the covariant metric
tensor, and we defined the variation $\delta$ of the CS terms with respect to
it. The quantity $C_{\mu\nu}$ is the Cotton tensor given by

$C_{\mu\nu}=-\frac{1}{2}\nabla^{\alpha}\left[(\nabla^{\beta}b)\widetilde{R}_{\alpha\mu\beta\nu}+(\nabla^{\beta}b)\widetilde{R}_{\alpha\nu\beta\mu}\right]\leavevmode\nobreak\
.$ (86)

In arriving at (85) we took into account the infinitesimal gravitational
variation of the Riemann tensor (84), the Bianchi identity
$\varepsilon^{\beta\alpha\rho\sigma}\,\nabla_{\alpha}\,R_{\rho\sigma}^{\,\,\,\,\,\,\,\mu\nu}=0$,
and ignored surface terms at infinity, arising from appropriate integrations
by parts in (85).

From the form of the Cotton tensor (86), it is straightforward to derive that
it is traceless, $g^{\mu\nu}\,C_{\mu\nu}=0\leavevmode\nobreak\ $

$\displaystyle
C_{\mu\nu}=-\frac{1}{4}\nabla_{\alpha}\left[(\nabla^{\beta}b)(R^{a}_{\,\,\,\mu\gamma\delta}\varepsilon^{\gamma\delta}_{\,\,\,\,\,\,\beta\nu}+R^{a}_{\,\,\,\nu\gamma\delta}\varepsilon^{\gamma\delta}_{\,\,\,\,\,\,\beta\mu})\right]$
$\displaystyle\implies
g^{\mu\nu}C_{\mu\nu}=\frac{1}{2}\nabla_{\alpha}\left[(\nabla_{\beta}b)(R^{a}_{\,\,\,\mu\gamma\delta})\varepsilon^{\mu\gamma\delta\beta}\right]=0\leavevmode\nobreak\
.$ (87)

The last equality stems from the Riemann tensor symmetries. In particular,
recall that $R^{\alpha}_{\,\,\,[\mu\nu\rho]}=0$, where $[\dots]$ denotes
complete antisymmetrization of the indices, which immediately implies

$R^{\alpha}_{\,\,\,\kappa\lambda\xi}\,\varepsilon^{\kappa\lambda\xi\sigma}=0\leavevmode\nobreak\
.$ (88)

A second important property is that its covariant derivative satisfies (10),
which we prove below, for completeness

$\displaystyle\nabla_{\mu}C^{\mu\nu}$
$\displaystyle=-\frac{1}{2}\nabla_{\mu}\nabla_{\rho}\left[(\nabla_{\sigma}b)(\widetilde{R}^{\rho\mu\sigma\nu}+\widetilde{R}^{\rho\nu\sigma\mu})\right]$
$\displaystyle=-\frac{1}{2}\nabla_{\mu}\nabla_{\rho}\left[(\nabla_{\sigma}b)\widetilde{R}^{\rho\mu\sigma\nu}\right]-\frac{1}{2}\nabla_{\mu}\nabla_{\rho}\left[(\nabla_{\sigma}b)\widetilde{R}^{\rho\nu\sigma\mu}\right]$
$\displaystyle=-\frac{1}{2}\left(\nabla_{[\mu}\nabla_{\rho]}\right)\left[(\nabla_{\sigma}b)\widetilde{R}^{\rho\mu\sigma\nu}\right]-\frac{1}{2}(\nabla_{\mu}\nabla_{\rho}-\nabla_{\rho}\nabla_{\mu})\left[(\nabla_{\sigma}b)\widetilde{R}^{\rho\nu\sigma\mu}\right]-\frac{1}{2}\nabla_{\rho}\nabla_{\mu}\left[(\nabla_{\sigma}b)\widetilde{R}^{\rho\nu\sigma\mu}\right]$
$\displaystyle=-\frac{1}{2}(\nabla_{\mu}\nabla_{\rho}-\nabla_{\rho}\nabla_{\mu})\left[(\nabla_{\sigma}b)(\widetilde{R}^{\rho\nu\sigma\mu}+\frac{1}{2}\widetilde{R}^{\rho\mu\sigma\nu})\right]$
$\displaystyle=-\frac{\nabla_{\sigma}b}{2}\left[-R_{\xi\mu}\left(\widetilde{R}^{\xi\nu\sigma\mu}+\frac{\widetilde{R}^{\xi\mu\sigma\nu}}{2}\right)+R^{\nu}_{\,\,\,\xi\mu\rho}\left(\widetilde{R}^{\rho\xi\sigma\mu}+\frac{\widetilde{R}^{\rho\mu\sigma\xi}}{2}\right)+R_{\xi\rho}\left(\widetilde{R}^{\rho\nu\sigma\xi}+\frac{\widetilde{R}^{\rho\xi\sigma\nu}}{2}\right)\right]$
$\displaystyle=-\frac{\partial_{\sigma}b}{2}\left[R^{\nu}_{\,\,\,\xi\mu\rho}\left(\widetilde{R}^{\rho\xi\sigma\mu}+\frac{\widetilde{R}^{\rho\mu\sigma\xi}}{2}\right)\right]=-\frac{\partial_{\sigma}b}{4}\left[\widetilde{R}^{\rho\xi\sigma\mu}(R^{\nu}_{\,\,\,\xi\mu\rho}-R^{\nu}_{\,\,\,\rho\mu\xi})+R^{\nu}_{\,\,\,\xi\mu\rho}\widetilde{R}^{\rho\mu\sigma\xi}\right]$
$\displaystyle=-\frac{\partial_{\sigma}b}{4}\left[\widetilde{R}^{\rho\xi\sigma\mu}R^{\nu}_{\,\,\,\mu\xi\rho}+R^{\nu}_{\,\,\,\xi\mu\rho}\widetilde{R}^{\rho\mu\sigma\xi}\right]=-\frac{\partial_{\sigma}b}{2}\widetilde{R}^{\rho\xi\sigma\mu}R^{\nu}_{\,\,\,\mu\xi\rho}=-\frac{\partial_{\sigma}b}{2}\widetilde{R}^{\rho\,\,\,\sigma\mu}_{\,\,\,\xi}R^{\xi\,\,\,\nu}_{\,\,\,\rho\,\,\,\mu}=-\frac{1}{4}(\partial^{\nu}b)R_{CS}\leavevmode\nobreak\
.$

where $R_{CS}$ is defined in (1), and in the last line we used the identity

$\widetilde{R}^{\rho\,\,\,\sigma\mu}_{\,\,\,\xi}R^{\xi\,\,\,\nu}_{\,\,\,\rho\,\,\,\mu}=\frac{1}{4}g^{\sigma\nu}\widetilde{R}^{\rho\,\,\,\lambda\mu}_{\,\,\,\xi}R^{\xi}_{\,\,\,\rho\lambda\mu}=\frac{1}{2}g^{\sigma\nu}R_{CS}\leavevmode\nobreak\
.$ (89)

##  Appendix B Solution of the differential equation (28)

In this Appendix, we consider the solution of the differential equation (28).
Naturally, the first step is to solve the homogeneous term of the ODE,

$-2u(r)+2(r-M)u^{\prime}(r)+(r^{2}-2Mr)u^{\prime\prime}(r)=0\leavevmode\nobreak\
.$ (90)

We note that a particular solution is $\displaystyle
u_{1}=c_{1}\left(\frac{r-M}{M}\right)$. We may use this solution to simplify
the ODE via $u=u_{1}(r)z(r)$. Then, our simplified expression reads

$\frac{1}{M}\left[2(M^{2}-4Mr+2r^{2})z^{\prime}+r(2M^{2}-3Mr+r^{2})z^{\prime\prime}\right]=0\leavevmode\nobreak\
,$ (91)

which implies that

$\displaystyle(\ln z^{\prime})$
$\displaystyle=\int\frac{8Mr-2M^{2}-4r^{2}}{(r-2M)(r-M)r}dr=-\ln[r(r-2M)(r-M)^{2}]+c_{2}$
$\displaystyle\implies z^{\prime}$
$\displaystyle=\frac{c_{2}}{r(r-2M)(r-M)^{2}}$ $\displaystyle z$
$\displaystyle=\frac{c_{2}}{(r-M)M^{2}}+\frac{c_{2}}{2M^{3}}\ln\left[1-\frac{2M}{r}\right]+c_{3}\leavevmode\nobreak\
,$

which means that the homogeneous solution to the ODE is

$u_{h}=c_{1}u_{1}(r)+c_{2}u_{2}(r)\leavevmode\nobreak\ ,$ (92)

where

$u_{1}(r)=\left(\frac{r-M}{M}\right)\leavevmode\nobreak\ ,\qquad
u_{2}(r)=1+\frac{1}{2M}(r-M)\ln\left[1-\frac{2M}{r}\right]\leavevmode\nobreak\
.$ (93)

In order to find the complete solution, we will make use of the method of
variation of parameters. We consider that the general solution of the
differential equation is expressed as

$u(r)=C_{1}(r)u_{1}(r)+C_{2}(r)u_{2}(r)\leavevmode\nobreak\ .$ (94)

Then, $C_{1}(r),C_{2}(r)$ can be solved by the system

$\displaystyle
C^{\prime}_{1}(r)u_{1}(r)+C^{\prime}_{2}(r)u_{2}(r)=0\leavevmode\nobreak\ ,$
(95) $\displaystyle
C^{\prime}_{1}(r)u^{\prime}_{1}(r)+C^{\prime}_{2}(r)u^{\prime}_{2}(r)=\frac{144M^{2}}{r^{5}(r^{2}-2Mr)}\leavevmode\nobreak\
.$ (96)

Therefore,

$\begin{bmatrix}C_{1}^{\prime}\\\
C_{2}^{\prime}\end{bmatrix}=\frac{1}{\mathcal{W}}\begin{bmatrix}u_{2}^{\prime}&-u_{2}\\\
-u_{1}^{\prime}&u_{1}\end{bmatrix}\begin{bmatrix}0\\\
\frac{144M^{2}}{r^{5}(r^{2}-2Mr)}\end{bmatrix}\leavevmode\nobreak\ ,$ (97)

where $\mathcal{W}$ denotes the Wronskian of our solutions. The system can be
easily solved to yield that

$\displaystyle C_{1}(r)$
$\displaystyle=\frac{81M}{2r^{4}}-\frac{5}{r^{3}}-\frac{15}{4Mr^{2}}-\frac{15}{4M^{2}r}+\ln\left(1-\frac{2M}{r}\right)\frac{6(4r-3M)}{r^{4}}-\frac{15}{8M^{3}}\ln\left(1-\frac{2M}{r}\right)+c_{1}\leavevmode\nobreak\
,$ (98) $\displaystyle C_{2}(r)$
$\displaystyle=\frac{36M}{r^{4}}-\frac{48}{r^{3}}+c_{2}\leavevmode\nobreak\ .$
(99)

Making use of (94), we find that the complete solution reads

$u(r)=-c_{1}+c_{2}-\frac{15}{4M^{3}}-\frac{9M}{2r^{4}}-\frac{5}{2r^{3}}-\frac{5}{4Mr^{2}}+c_{1}\frac{r}{M}+\ln\left(1-\frac{2M}{r}\right)\left[\frac{15}{8M^{3}}-\frac{c_{2}}{2}\right]+\ln\left(1-\frac{2M}{r}\right)\left[\frac{c_{2}r}{2M}-\frac{15r}{8M^{4}}\right]\leavevmode\nobreak\
.$ (100)

In order to cancel the divergent terms, we fix the integration constants to
$c_{1}=0$ and $\displaystyle c_{2}=\frac{15}{4M^{3}}$ and we find that the
asymptotic solution of the axion reads

$u(r)=-\frac{9M}{2r^{4}}-\frac{5}{2r^{3}}-\frac{5}{4Mr^{2}}\leavevmode\nobreak\
.$ (101)

##  Appendix C Proof of Convergence of the series (23)

The relevant series appears in the $g_{t\phi}$ element of the metric of our
black hole solution (14), see (19), (23):

$g_{t\phi}=r^{2}\,\left(-\frac{2M}{r^{3}}-w(r)\right)a\;\sin^{2}(\theta)\equiv\left(-\frac{2M}{r}-\widetilde{w}(r)\right)a\;\sin^{2}(\theta),\quad\widetilde{w}(r)=\sum_{n=4}^{\infty}\frac{d_{n}M^{n-2}}{r^{n-2}}\,,$
(102)

where $d_{n}$ is determined by the following recurrence relation (26):

$d_{n}=\frac{2(n-5)^{2}(n-1)}{n(n-6)(n-3)}d_{n-1}+\frac{576\gamma^{2}}{n(n-3)}d_{n-6},\;\;\text{for}\;n\geq
10\leavevmode\nobreak\ ,$ (103)

and initial conditions (31):

$d_{4}=d_{5}=0\;\;,\;\;d_{6}=-5\gamma^{2}\;\;,\;\;d_{7}=-\frac{60\gamma^{2}}{7}\;\;,\;\;d_{8}=-\frac{27\gamma^{2}}{2}\;\;,\;\;d_{9}=0$
(104)

with $\gamma\in\mathbb{R}$. We restrict ourselves in the exterior to horizon
region $r\geq 2M$.

For notational convenience and brevity we redefine $r\rightarrow r/M$, hence

$\widetilde{w}(r)=\sum_{n=4}^{\infty}\frac{d_{n}}{r^{n-2}}.$ (105)

Below we shall prove the convergence of this sum for all $r\geq 2$ for all
$\gamma\in\mathbb{R}$.

To this end, we first note that $d_{n}\leq 0\;,\,\forall n$. Thus, we define
the sequence $c_{n}=-d_{n}$, for which $c_{n}\geq 0,\;\forall n$. Then, in
terms of $c_{n}$, we have:

$\widetilde{w}(r)=-\sum_{n=4}^{\infty}\frac{c_{n}}{r^{n-2}}$ (106)

where

$c_{n}=a_{n}c_{n-1}+b_{n}c_{n-6},\;\;\text{for}\;n\geq 10\leavevmode\nobreak\
,$ (107)

with

$a_{n}=\frac{2(n-5)^{2}(n-1)}{n(n-6)(n-3)}\,\quad
b_{n}=\frac{576\gamma^{2}}{n(n-3)},$ (108)

and initial conditions:

$c_{4}=c_{5}=0\;\;,\;\;c_{6}=5\gamma^{2}\;\;,\;\;c_{7}=\frac{60\gamma^{2}}{7}\;\;,\;\;c_{8}=\frac{27\gamma^{2}}{2}\;\;,\;\;c_{9}=0$
(109)

Let us define the sequence:

$\tilde{\Sigma}_{N}=\sum_{n=4}^{N}\frac{c_{n}}{r^{n-2}}$

As $c_{n}/r^{n-2}$ are non-negative, $\tilde{\Sigma}_{N}$ is an increasing
sequence, meaning $\tilde{\Sigma}_{N+1}\geq\tilde{\Sigma}_{N},\;\forall
N\in\mathbb{N}$.

For increasing sequences, the following theorem exists [85]:

Theorem: An increasing sequence tends either to a finite limit or to
$+\infty$.

Hence, a necessary and sufficient condition for the convergence of
$\tilde{\Sigma}$ is the demonstration that it is bounded, i.e. that there
exists a finite, positive number $\mathcal{N}$, such that:

$\tilde{\Sigma}=\sum_{n=4}^{\infty}\frac{c_{n}}{r^{n-2}}\leq\mathcal{N}$

Statement 1:  If $\sum_{n=4}^{\infty}\frac{c_{n}}{2^{n-2}}$ converges, then
$\sum_{n=4}^{\infty}\frac{c_{n}}{r^{n-2}}\;\text{converges}\;\;\forall r\geq
2$.

###### Proof.

Suppose that $\sum_{n=4}^{\infty}\frac{c_{n}}{2^{n-2}}\leq\mathcal{K}$, where
$\mathcal{K}$ finite. For $r>2\rightarrow 1/r^{n-2}<1/2^{n-2}\rightarrow
c_{n}/r^{n-2}\leq c_{n}/2^{n-2}\;\forall n\geq 4$, where the equality holds in
the case of $c_{n}=0$. Thus,

$\displaystyle\sum_{n=4}^{\infty}\frac{c_{n}}{r^{n-2}}\leq\sum_{n=4}^{\infty}\frac{c_{n}}{2^{n-2}}\quad\Rightarrow\quad\sum_{n=4}^{\infty}\frac{c_{n}}{r^{n-2}}\leq\mathcal{K}$

which means that $\sum_{n=4}^{\infty}\frac{c_{n}}{r^{n-2}}$ converges $\forall
r>2$. ∎

Hence, according to the above statement 1, we should establish the convergence
of the infinite series

$\Sigma=\sum_{n=4}^{\infty}\frac{c_{n}}{2^{n-2}}$

For $a_{n}$ and $b_{n}$, we have:

$\displaystyle 1)\;\;a_{n}\rightarrow 2,\;\text{as}\;n\rightarrow+\infty$
$\displaystyle 2)\;\;b_{n}\rightarrow 0,\;\text{as}\;n\rightarrow+\infty$

i.e. $a_{n}$ and $b_{n}$ both converge.

Below, we use the mathematical result that a sequence, say $s_{n}$, is
convergent, i.e. tends to a finite limit $s$ as $n\rightarrow\infty$, implies
that $s_{n}$ is bounded [85].

This implies that, since $a_{n},b_{n}$ are convergent, in view of the above
result, they are bounded, meaning that there exist $k_{1},k_{2}$, such that

$\left|a_{n}\right|\leq k_{1}\;\;and\;\;\left|b_{n}\right|\leq
k_{2}\;\;\forall n\in\mathbb{N}$ (110)

Statement 2: The sequence $c_{n}$ is bounded by induction.

###### Proof.

Suppose that there exists a subsequence $c_{N-6},...c_{N-1}$, for some $N$,
that is bounded, i.e. there exists $N^{\prime}$, such that

$\left|c_{N-6}\right|,\left|c_{N-5}\right|,...,\left|c_{N-1}\right|\leq
N^{\prime}$

.
Then, using the triangle inequality

$\displaystyle\left|c_{N}\right|\leq\left|a_{N}\right|\left|c_{N-1}\right|+\left|b_{N}\right|\left|c_{N-6}\right|\quad\Rightarrow$
$\displaystyle\left|c_{N}\right|\leq(k_{1}+k_{2})N^{\prime}\quad\Rightarrow$
$\displaystyle\left|c_{N}\right|\leq\widetilde{\mathcal{M}}\,,$

where $\widetilde{\mathcal{M}}=(k_{1}+k_{2})\,N^{\prime}$ finite.

Thus, as the corresponding subsequence exists for $N=10$, as $c_{4},...,c_{9}$
are finite, by induction, there exists finite $\mathcal{D}$, such that

$\left|c_{n}\right|\leq\mathcal{D},\;\forall n\geq 10$

concluding that $c_{n}$ is bounded $\forall n\geq 4$. ∎

Thus, as $c_{n}$ is bounded and non-negative, there exists
$\mathcal{D}\,>\,0$, such that:

$0\leq c_{n}\leq\mathcal{D}\;,\forall\;n\geq 4$

Thence,

$\Sigma=\sum_{n=4}^{\infty}\frac{c_{n}}{2^{n-2}}\leq\mathcal{D}\sum_{n=4}^{\infty}\left(\frac{1}{2}\right)^{n-2}$

or

$\Sigma\leq\mathcal{D}\sum_{n=2}^{\infty}\left(\frac{1}{2}\right)^{n}$

For the geometric series, we know that

$\sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x},\qquad\text{for}\;-1<x<1$

Thus, for $x=1/2$, it is easy to show that:

$\sum_{n=2}^{\infty}\left(\frac{1}{2}\right)^{n}=\frac{1}{2}$

which implies

$\Sigma=\sum_{n=4}^{\infty}\frac{c_{n}}{2^{n-2}}\leq\frac{\mathcal{D}}{2},$

i.e., the infinite series $\Sigma$ is bounded.

This proves the required result, that the sum $\widetilde{w}(r)$, and thus
$w(r)$ ((23)), converges $\forall r\geq 2$ and $\forall\gamma\in\mathbb{R}$.

##  Appendix D The energy conditions for Einstein spaces

Here we review the various energy conditions that characterise gravitational
theories in Einstein spacetimes. Suppose that $T_{\mu\nu}$ refers to the
conserved energy-momentum, obeying the Einstein gravitational equations of
motion

$R_{\mu\nu}=\kappa^{2}\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\leavevmode\nobreak\
.$ (111)

Notice that if one considers higher-curvature modified gravities, such as
scalar-Gauss-Bonnet or axion-Chern-Simons gravities, the contributions of the
modifications are absorbed in the definition of $T_{\mu\nu}$, viewed as an
effective stress tensor. With this understanding, we review the following
energy conditions.

  * •

Weak Energy Condition (WEC) : The energy density as measured by any observer
with a timelike four-velocity $t^{\mu}$, is non-negative, which formally can
be expressed as

$T_{\mu\nu}t^{\mu}t^{\nu}\geq
0\;,\;\;\forall\;t:\;\;t^{\mu}t_{\mu}<0\leavevmode\nobreak\ .$ (112)

  * •

Null Energy Condition (NEC): expresses the requirement that the geometry has a
focusing (attractive) effect on null geodesics,

$T_{\mu\nu}l^{\mu}l^{\nu}\geq
0\;,\;\;\forall\;l:\;\;l^{\mu}l_{\mu}=0\leavevmode\nobreak\ ,$ (113)

where $l^{\mu}$ is any null four-vector.

We can understand the condition by noting that the Raychadhuri equation
describing the focusing of null geodesics is

$\frac{d}{d\tau}\theta_{l}=-R_{\mu\nu}l^{\mu}l^{\nu}+...$

where the parameter $\theta$ in Raychadhuri equation has the interpretation of
the rate of change of the congruences’ cross sectional volume. Thus, for a
focusing effect on the null geodesics: $R_{\mu\nu}l^{\mu}l^{\nu}\geq 0$.
Hence, the gravitational equations of motion yield the same condition for the
conserved energy-momentum tensor.

  * •

Strong Energy Condition (SEC): expresses the requirement that the geometry has
a focusing (attractive) effect on timelike geodesics.

$T_{\mu\nu}t^{\mu}t^{\nu}\geq\frac{1}{2}T\,g_{\mu\nu}\,t^{\mu}t^{\nu}\;,\;\;\forall\;t:\;\;t^{\mu}t_{\mu}<0\leavevmode\nobreak\
,$ (114)

where $T$ is the gravitational trace of the stress tensor $T\equiv
g^{\mu\nu}\,T_{\mu\nu}$. The condition is understood again by noting that the
Raychadhuri equation describing the focusing of timelike geodesics is

$\frac{d}{d\tau}\theta_{t}=-R_{\mu\nu}t^{\mu}t^{\nu}+...$

Thus, for a focusing effect on the timelike geodesics:
$R_{\mu\nu}t^{\mu}t^{\nu}\geq 0$. Hence, the gravitational equations of motion
(111) yield (114) for the conserved energy-momentum tensor.

  * •

Dominant Energy Condition (DEC): This energy condition refers to the current
density: $P^{\alpha}=-T^{\alpha}\\!_{\beta}t^{\beta}$, i.e. the energy-
momentum current as seen by an observer with 4-velocity $t^{\mu}$. It is
essentially the statement that the speed of the flow of energy should not
exceed that of light, i.e. $P^{\alpha}$ should be causal and future directed
for all timelike and future directed $t^{\alpha}$. Since $t^{\alpha}$ is
timelike and future directed, the above conditions are mathematically
expressed as follows

$P_{\alpha}t^{\alpha}\leq 0\;\;\text{and}\;\;P_{\alpha}P^{\alpha}\leq
0\leavevmode\nobreak\ ,$

which yields the DEC as a set of the two requirements

$T_{\mu\nu}t^{\mu}t^{\nu}\geq
0\;\;\text{and}\;\;T_{\mu\nu}T^{\mu}\\!_{\alpha}t^{\nu}t^{\alpha}\leq
0\;\;\forall\;t:\;\;t_{\mu}t^{\mu}<0\leavevmode\nobreak\ .$ (115)

We note that the first of the above is just
$T_{\alpha\beta}t^{\alpha}t^{\beta}\geq 0$, that is, the WEC.

One can show the relations between the various energy conditions, as a
“hierarchy of the validity” between them

$\displaystyle(DEC)\implies(WEC)\implies(NEC)$ (116)
$\displaystyle(SEC)\implies(NEC)$ $\displaystyle\text{(NEC) violation}\;\,{\rm
implies}\,\,\text{(DEC), (SEC), (WEC) violation}$

where the arrow notation $\implies$ implies the validity of a condition.

Thus, the NEC (113) is the weakest energy condition, because if this condition
is satisfied, then it is implied that the rest are saitisfied. However, if we
look the relations according to their violation, the NEC is the strongest one,
because if this condition is violated all of the other conditions are
violated, too. This is the case for the Kerr black-hole solution (33) of our
SC-axion gravitational theory (3) [71, 72], implying the existence of axionic
hair [4, 5, 6, 69, 70].

##  Appendix E Computation of the extrinsic-curvature junctioning quantities
$\mathfrak{R}^{\alpha}_{\,\,\,\beta\gamma\delta}$ in the wormhole

The fact that the metric is continuous across $\Sigma$ in the coordinates
$x^{\mu}$ implies that its tangential derivatives also must be continuous.
This means that if $\partial_{\gamma}g_{\alpha\beta}$ is to be discontinuous,
the discontinuity must be directed along the normal vector $n^{\alpha}$. As
such, there exists a tensor field $q_{\alpha\beta}$ such that

$[\partial_{\gamma}g_{\alpha\beta}]=n_{\gamma}q_{\alpha\beta}\implies
q_{\alpha\beta}=\epsilon[\partial_{\gamma}g_{\alpha\beta}]n^{\gamma}\leavevmode\nobreak\
.$ (117)

Making use of (117) under the first junctioning condition which yields
$[g^{\alpha\beta}]=0$, we may find that

$[\Gamma^{\alpha}_{\,\,\,\beta\gamma}]=\frac{1}{2}(q^{\alpha}_{\,\,\,\beta}n_{\gamma}+q^{\alpha}_{\,\,\,\gamma}n_{\beta}-q_{\beta\gamma}n^{\alpha})\leavevmode\nobreak\
.$ (118)

This means that

$\mathfrak{R}^{\alpha}_{\,\,\,\beta\gamma\delta}=\frac{\epsilon}{2}(q^{\alpha}_{\,\,\,\delta}n_{\beta}n_{\gamma}-q^{\alpha}_{\,\,\,\gamma}n_{\beta}n_{\delta}-q_{\beta\delta}n^{\alpha}n_{\gamma}+q_{\beta\gamma}n^{\alpha}n_{\delta})\leavevmode\nobreak\
.$ (119)

Now that we have a tensorial quantity for $\mathfrak{R}$, we may immediately
note that the Einstein tensor reads

$G_{\alpha\beta}=G^{+}_{\alpha\beta}\Theta(\lambda)+G^{-}_{\alpha\beta}\Theta(-\lambda)+\delta(\lambda)\mathfrak{G}_{\alpha\beta}\leavevmode\nobreak\
,$ (120)

where

$\mathfrak{G}_{\alpha\beta}=\frac{\epsilon}{2}(q_{\mu\alpha}n^{\mu}n_{\beta}+q_{\mu\beta}n^{\mu}n_{\alpha}-qn_{\alpha}n_{\beta}-\epsilon
q_{\alpha\beta}-(q_{\mu\nu}n^{\mu}n^{\nu}-\epsilon
q)g_{\alpha\beta})\leavevmode\nobreak\ .$ (121)

From the form of $\mathfrak{G}_{\alpha\beta}$, we note that it is symmetric,
as expected, and that it is also tangent to the hypersurface, since
$\mathfrak{G}_{\alpha\beta}n^{\alpha}=0$. Therefore, it admits the
decomposition of

$\mathfrak{G}^{\alpha\beta}=\mathfrak{G}^{ij}e^{\alpha}_{i}e^{\beta}_{j}\leavevmode\nobreak\
.$ (122)

Evaluating the decomposition we find

$\displaystyle\mathfrak{G}_{ij}$
$\displaystyle=\mathfrak{G}_{\alpha\beta}e^{\alpha}_{i}e^{\beta}_{j}$
$\displaystyle=\frac{\epsilon}{2}(-\epsilon
q_{\alpha\beta}e^{a}_{i}e^{\beta}_{j}+\epsilon q_{\mu\nu}(g^{\mu\nu}-\epsilon
n^{\mu}n^{\nu})h_{ij})$
$\displaystyle=\frac{1}{2}(-q_{\alpha\beta}e^{a}_{i}e^{\beta}_{j}+q_{\mu\nu}h^{\mu\nu}h_{ij})\leavevmode\nobreak\
.$

We now note that $q_{\alpha\beta}e^{\alpha}_{i}e^{\beta}_{j}$ is just the jump
of the extrinsic curvature across the hypersurface. Indeed, calculating

$\displaystyle[K_{ij}]$
$\displaystyle=[(\nabla_{\alpha}n_{\beta})e^{\alpha}_{i}e^{\beta}_{j}]$
$\displaystyle\overset{(\ref{w.4})}{=}[(\nabla_{\alpha}n_{\beta})]e^{\alpha}_{i}e^{\beta}_{j}$
$\displaystyle=-\left([\Gamma^{\gamma}_{\,\,\,\alpha\beta}]n_{\gamma}\right)e^{\alpha}_{i}e^{\beta}_{j}$
$\displaystyle\overset{(\ref{B.2})}{=}-\left(\frac{1}{2}(q^{\gamma}_{\,\,\,\beta}n_{\alpha}+q^{\gamma}_{\,\,\,\alpha}n_{\beta}-q_{\alpha\beta}n^{\gamma})\right)n_{\gamma}e^{\alpha}_{i}e^{\beta}_{j}$
$\displaystyle=\frac{\epsilon}{2}q_{\alpha\beta}e^{\alpha}_{i}e^{\beta}_{j}\leavevmode\nobreak\
,$

we find that

$[K_{ij}]=\frac{\epsilon}{2}q_{\alpha\beta}e^{\alpha}_{i}e^{\beta}_{j},\qquad[K]=\frac{\epsilon}{2}h^{mn}(q_{\alpha\beta}e^{\alpha}_{m}e^{\beta}_{n})\leavevmode\nobreak\
.$ (123)

Therefore, since $K_{\mu\nu}n^{\mu}=0$, i.e. the extrinsic curvature of the
hypersurface is tangent to the hypersurface, as is well known, we find by the
decomposition $K_{ij}=K_{\mu\nu}e^{\mu}_{i}e^{\mu}_{j}$ that

$q_{\alpha\beta}=\frac{2}{\epsilon}[K_{\alpha\beta}]\leavevmode\nobreak\ ,$
(124)

which yields that

$\mathfrak{R}^{\alpha}_{\,\,\,\beta\gamma\delta}=(n_{\beta}n_{\gamma}[K^{\alpha}_{\delta}]-n_{\beta}n_{\delta}[K^{\alpha}_{\gamma}]-[K_{\beta\delta}]n^{\alpha}n_{\gamma}+[K_{\beta\gamma}]n^{\alpha}n\delta)\leavevmode\nobreak\
,$ (125)

which was used in the text.

##  Appendix F Chern-Simons (surface) contributions on the Wormhole thin shell

We consider the CS term of the action (3)

$I_{CS}=\int d^{4}x\,\sqrt{-g}\,bR\widetilde{R}\leavevmode\nobreak\ .$ (126)

Upon taking the variation of the $bR\widetilde{R}$ term we find that

$\delta I_{CS}\sim 2\int
d^{4}x\,\sqrt{-g}\,bR^{\alpha}_{\,\,\,\beta\kappa\lambda}\varepsilon^{\kappa\lambda\rho\sigma}(\nabla_{\rho}\delta\Gamma^{\beta}_{\,\,\,\alpha\sigma})\leavevmode\nobreak\
,$ (127)

whose surface term is

$\int
d\Sigma_{\rho}bR^{\alpha}_{\,\,\,\beta\kappa\lambda}\varepsilon^{\kappa\lambda\rho\sigma}(\delta\Gamma^{\beta}_{\,\,\,\alpha\sigma})\leavevmode\nobreak\
.$ (128)

Recalling that $\delta\Gamma$ reads under $g^{\mu\nu}\rightarrow
g^{\mu\nu}+\delta g^{\mu\nu}$,

$\delta\Gamma^{\beta}_{\,\,\,\alpha\sigma}=-\frac{1}{2}\left[g_{\mu\sigma}\nabla_{\alpha}(\delta
g^{\mu\beta})+g_{\mu\alpha}\nabla_{\sigma}(\delta
g^{\mu\beta})-g_{\alpha\nu}g_{\sigma\mu}\nabla^{\beta}(\delta
g^{\mu\nu})\right]\leavevmode\nobreak\ ,$ (129)

we have that

$\displaystyle\int
d\Sigma_{\rho}bR^{\alpha}_{\,\,\,\beta\kappa\lambda}\varepsilon^{\kappa\lambda\rho\sigma}(\delta\Gamma^{\beta}_{\,\,\,\alpha\sigma})$
$\displaystyle=-\int
d\Sigma_{\rho}b\widetilde{R}^{\alpha\,\,\,\rho\sigma}_{\,\,\,\beta}g_{\mu\sigma}\nabla_{\alpha}(\delta
g^{\mu\beta})-\int
d\Sigma_{\rho}b\widetilde{R}^{\alpha\,\,\,\rho\sigma}_{\,\,\,\beta}g_{\mu\alpha}\nabla_{\sigma}(\delta
g^{\mu\beta})+\int
d\Sigma_{\rho}b\widetilde{R}^{\alpha\,\,\,\rho\sigma}_{\,\,\,\beta}g_{\alpha\mu}g_{\sigma\nu}\nabla^{\beta}(\delta
g^{\mu\nu})$ $\displaystyle=-\int
d\Sigma_{\rho}b\widetilde{R}^{\alpha\,\,\,\rho}_{\,\,\,\beta\,\,\,\mu}\nabla_{\alpha}(\delta
g^{\mu\beta})-\int
d\Sigma_{\rho}b\widetilde{R}^{\,\,\,\,\,\,\rho\alpha}_{\mu\beta}\nabla_{\alpha}(\delta
g^{\mu\beta})+\int
d\Sigma_{\rho}b\widetilde{R}^{\,\,\,\alpha\rho}_{\mu\,\,\,\,\,\,\beta}\nabla_{\alpha}(\delta
g^{\mu\beta})$ $\displaystyle=\int
d\Sigma_{\rho}b(\widetilde{R}^{\,\,\,\alpha\rho}_{\mu\,\,\,\,\,\,\nu}-\widetilde{R}^{\alpha\,\,\,\rho}_{\,\,\,\nu\,\,\,\mu})\nabla_{\alpha}(\delta
g^{\mu\nu})$ $\displaystyle=\int
d^{3}x\sqrt{h}n^{\rho}b(\widetilde{R}_{\mu\beta\rho\nu}+\widetilde{R}_{\nu\beta\rho\mu})g^{\alpha\beta}\nabla_{\alpha}(\delta
g^{\mu\nu})\leavevmode\nobreak\ .$

Since $\delta g^{\mu\nu}$ is zero on $\Sigma$, $\nabla_{\alpha}\delta
g^{\mu\nu}=\partial_{\alpha}\delta g^{\mu\nu}$. Also, by the decomposition of
$g^{\alpha\beta}=h^{\alpha\beta}+\epsilon n^{\alpha}n^{\beta}$, the tangential
derivative of the variation of the metric also vanishes on $\Sigma$.
Therefore,

$\displaystyle\int
d\Sigma_{\rho}bR^{\alpha}_{\,\,\,\beta\kappa\lambda}\epsilon^{\kappa\lambda\rho\sigma}(\delta\Gamma^{\beta}_{\,\,\,\alpha\sigma})$
$\displaystyle=\int
d^{3}x\sqrt{h}n^{\rho}b(\widetilde{R}_{\mu\beta\rho\nu}+\widetilde{R}_{\nu\beta\rho\mu})\epsilon
n^{\beta}n^{\alpha}\partial_{\alpha}(\delta g^{\mu\nu})\leavevmode\nobreak\ .$

Computing the interior of the integral for our case, it is found that it is of
order $\mathcal{O}(a)^{2}$ or higher, without even using the constraint to
calculate it on the hypersurface.

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