Compact objects of spherical symmetry in beyond Horndeski theories

A. Bakopoulos${}^{\,(a)}$ 111Email<EMAIL_ADDRESS>C. Charmousis${}^{\,(b)}$
222Email<EMAIL_ADDRESS>P. Kanti${}^{\,(c)}$ 333Email:
<EMAIL_ADDRESS>N. Lecoeur${}^{\,(b)}$ 444Email<EMAIL_ADDRESS>

(a)Division of Applied Analysis, Department of Mathematics,
University of Patras, Rio Patras GR-26504, Greece

(b)Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

(c)Division of Theoretical Physics, Department of Physics,
University of Ioannina, Ioannina GR-45110, Greece

Abstract

We analyse in all generality beyond Horndeski theories of shift symmetry in a
static and spherically symmetric spacetime. By introducing four auxiliary
functions, we write the field equations in a particularly compact form. We
show that assuming additionally parity symmetry renders the system directly
integrable giving multiple families of black-hole solutions. These have
typically an asymptotically-flat Reissner-Nordstrom behaviour, and emerge in
the presence of a canonical kinetic term for the scalar field. In the absence
of parity symmetry, we present a general method which allows us to integrate
the field equations by choosing the form of only one coupling function and an
auxiliary quantity. This method leads to asymptotically flat and AdS black
hole solutions with differing properties. We finally discuss disformal
transformations within this context as a means of obtaining wormhole and black
hole solutions in different theories.

###### Contents

  1. 1 Introduction
  2. 2 Beyond Horndeski field equations and their analysis
  3. 3 The integrability of parity symmetric theories
    1. 3.1 Black holes with a canonical Kinetic term
    2. 3.2 Non-Homogeneous Black holes
  4. 4 Attacking the case of no parity symmetry
    1. 4.1 Parity breaking theories related to Kaluza-Klein reduction of Lovelock theory
    2. 4.2 Theories admitting solutions with non trivial asymptotics
  5. 5 Disformal transformations and solution generation techniques
  6. 6 Conclusions
  7. APPENDIX
  8. A Case of constant kinetic term
  9. B Additional Classes of Parity-Symmetric Solutions
  10. C Parity symmetry breaking with a beyond Horndeski term

##  1 Introduction

The impressive income of observational data for compact objects have initiated
a revolutionary epoch in the field of gravitational physics. Observations
range from gravitational wave signals of binary mergers of relatively few
solar masses (eg. [1], [2], [3]) to images of supermassive black holes
generated from radio-telescope networks [4] and trajectories of stars orbiting
the Galactic center [5], to name just a few. Early findings are compatible
with GR, raising however a number of questions like the nature of the
secondary object in a certain binary [2] and the existence or not of the
expected GR mass gap between neutron stars and black holes (for a recent
discussion, see [6] and references within). Observational data, evolving from
discovery towards precision, will permit us to probe additional gravitational
parameters, eventually checking the validity of no hair theorems, star
trajectory parameters and possibly discovering novel effects (for example
distinguishing a wormhole throat versus an event horizon [7], [8], [9]).

Scalar tensor theories, [10],[11], [12], [13], provide a robust and measurable
departure from GR and are therefore very interesting geometric modifications
of GR. Since GR is not a UV complete theory scalar tensor are expected to play
an important role as curvature effects become stronger. This is particularly
true for the smaller mass compact objects, in binaries for example, as their
radius of curvature can be quite large. There is additional motivation for
scalar tensor theories coming from dark energy, although applicability of data
is questionable on cosmological scales [14]. An adjacent motivation is that
most modified gravity theories admit a well defined scalar tensor limit. A
classic example is the Horndeski cubic galileon (see eg. [15], [16]), which
originates from the 5 dimensional DGP [17] braneworld model at the decoupling
limit [18] (first discovered in massive gravity [19]). Note that this model
[17], from the 4 dimensional perspective has an infinite number of degrees of
freedom, although as perceived from 5 dimensions it is a rather simple
gravitational setup. Scalar tensor theories are also directly obtained from
higher dimensional Lovelock theory [20] (see for example [21]) to scalar
tensor via Kaluza Klein compactification [22], [23] or again from effective
string theory actions [24], [25], [26], [27], [28], [29] and their black hole
solutions (see for example [30], [31], [32],[33], [34]).

By now there is a number of analytic and numerical black hole solutions in
scalar tensor theories most of which obtained upon breaking hypotheses to no
hair theorems (see for example [35], [36], [37], [38], [39], [40], [41], [42],
[43], [44]). Most analytic solutions found exist in shift and parity symmetric
scalar tensor theories typically including the K-essence (or $G_{2}$) and
$G_{4}$ galileons (1.1) (see for example [38], [45], [41] and references
within). It is quite interesting that shift symmetry allows a large class of
these to have linear time dependence first introduced in [46] and generalised
in [47]. This is associated to the presence of shift symmetry and a time like
Killing vector field for static metrics. It is found that black hole metrics
are generically close or identical to GR black hole solutions as the scalar
field paints the spacetime following a geodesic congruence [48]. This
construction extends even to the stealth Kerr solution for a particular DHOST
theory [48]. Disformal mappings starting with stealth GR metrics provide for
the first time an analytic solution of a rotating black hole which alters Kerr
spacetime while passing all actual observational constraints [49], [50], [51].

Once we give up on parity or shift symmetry it is far more involved to obtain
analytic solutions. Yet there is no physical reason to admit these symmetries;
on the contrary putting them aside allows us to explore quite different
spacetime geometries from GR often with quite different characteristics. The
first asymptotically flat analytic solution found [23] was one obtained from
Kaluza-Klein reduction originating from a Lovelock black hole [52], [53] with
a non trivial horizon geometry. Given its higher order nature this solution
did not have a mass term with Newtonian fall off. However, as noticed more
recently, an intriguing singular limit [54] of Lovelock theory and a careful
analysis of the resulting scalar tensor theory, done is several different
ways, [55], [56], [57], [58] gave three very interesting 4-dimensional black
hole solutions (see the review [59]) with interesting phenomenology [60], [6].
These solutions originate from theories with and without shift symmetry.
Stationary metrics are even harder to obtain in the absence of these
symmetries. Some important progress has been achieved for a conformally
coupled scalar field for type D metrics [61] and a rotating singular solution
[62].

What is missing in the literature is a unifying and general study of
spherically symmetric solutions for a given wide class of scalar tensor
theories. This is the task we will attempt in this paper putting together
different techniques and approaches in order to elaborate methods and
eventually discover new black hole and wormhole solutions which are static and
of spherical symmetry. Given the above discussion, we will also choose to
focus on non stealth solutions. As such we will consider a time independent
scalar field. In what follows we will consider shift symmetric Horndeski plus
beyond Horndeski theory. The former is parametrised by four functions
$\\{G_{i}:i=2,..,5\\}=\\{G_{2},G_{3},G_{4},G_{5}\\}$ functions of $\varphi$
and its kinetic density $X=-\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi$
:

$S_{\mathrm{H}}=\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\,\left(\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}+\mathcal{L}_{5}\right),$
(1.1)

with

$\displaystyle\mathcal{L}_{2}$ $\displaystyle=G_{2}(X),$ (1.2)
$\displaystyle\mathcal{L}_{3}$ $\displaystyle=-G_{3}(X)\,\Box\phi,$ (1.3)
$\displaystyle\mathcal{L}_{4}$
$\displaystyle=G_{4}(X)R+G_{4X}\left[(\Box\phi)^{2}-\nabla_{\mu}\partial_{\nu}\phi\,\nabla^{\mu}\partial^{\nu}\phi\right],$
(1.4)
$\displaystyle\begin{split}\mathcal{L}_{5}&=G_{5}(X)G_{\mu\nu}\nabla^{\mu}\partial^{\nu}\phi-\frac{1}{6}\,G_{5X}\big{[}(\Box\phi)^{3}-3\,\Box\phi\,\nabla_{\mu}\partial_{\nu}\phi\,\nabla^{\mu}\partial^{\nu}\phi\\\
&\quad+2\,\nabla_{\mu}\partial_{\nu}\phi\,\nabla^{\nu}\partial^{\rho}\phi\,\nabla_{\rho}\partial^{\mu}\phi\big{]}.\end{split}$
(1.5)

The latter is given by two additional higher order terms,

$\displaystyle\mathcal{L}^{\rm bH}_{4}$
$\displaystyle=F_{4}(X)\varepsilon^{\mu\nu\rho\sigma}\,\varepsilon^{\alpha\beta\gamma}_{\,\,\,\,\,\,\,\,\,\,\,\sigma}\,\partial_{\mu}\phi\,\partial_{\alpha}\phi\,\nabla_{\nu}\partial_{\beta}\phi\,\nabla_{\rho}\partial_{\gamma}\phi,$
(1.6) $\displaystyle\mathcal{L}^{\rm bH}_{5}$
$\displaystyle=F_{5}(X)\varepsilon^{\mu\nu\rho\sigma}\,\varepsilon^{\alpha\beta\gamma\delta}\,\partial_{\mu}\phi\,\partial_{\alpha}\phi\,\nabla_{\nu}\partial_{\beta}\phi\,\nabla_{\rho}\partial_{\gamma}\phi\,\nabla_{\sigma}\partial_{\delta}\phi.$
(1.7)

The beyond Horndeski terms parametrised by $F_{4}$ and $F_{5}$ are not
independent. They are related so as to evade the appearance of a ghost degree
of freedom [63]. This relation reads

$XG_{5X}F_{4}=3F_{5}(G_{4}-2XG_{4X})\,.$ (1.8)

In the next section we will set-up the problem and we will define special
variables which will collectively trace the effects of functionals in the
theory. We will see that depending on these one can have for example
homogeneous or non homogeneous black holes. We will write the field equations
in a compact and largely solvable form grouping together classes of Horndeski
and beyond Horndeski theories. Then in section 3 we will study the case of
parity symmetry. This case will be shown to be integrable in a sense that will
be made clear in the third section. We will give a method providing exact
solutions focusing on physically interesting cases involving eg., a canonical
kinetic term. We will argue that although wormhole solutions can be
constructed with relative ease their throats would be only theory dependant
and not mass dependant. In section 4 we will leave aside the parity
hypothesis. Focusing on the properties of the solutions originating from
higher dimensional Lovelock theories we will generalise these in two different
classes. Then in section 5 we will discuss disformal transformations of given
solutions obtaining black hole and wormhole geometries (for non parity
symmetric theories).

##  2 Beyond Horndeski field equations and their analysis

Throughout this article we will consider a spherically symmetric Anzatz,

$ds^{2}=-h(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}\,,$ (2.1)

with a static scalar $\phi=\phi(r)$. The independent field equations for this
symmetry can be obtained in all generality. They first appeared in the
Appendix of [41] and later presented in [64] with minor corrections.

It is particularly useful to write down the field equations for spherical
symmetry and for zero scalar charge in all generality by introducing auxiliary
functions $Z$ and $Y$ which are specific combinations of the theory functions.
Indeed we have,

$\displaystyle Z(X)=2XG_{4X}-G_{4}+4X^{2}F_{4}\,,$ (2.2) $\displaystyle
Y(X)=\frac{1}{2}(-2X)^{3/2}G_{5X}+3(-2X)^{5/2}F_{5}\,.$ (2.3)

These functions “include” the beyond Horndeski contribution to the relevant
Horndeski terms. $Z$ and $Y$ will appear quite naturally when we discuss
disformal transformations. With these definitions we now consider the
functionals

$\displaystyle{\cal{A}}=$ $\displaystyle
4rZ_{X}+\phi^{\prime}[r^{2}G_{3X}+G_{5X}(1-3f)-2XfG_{5XX}+12fX(5F_{5}+2XF_{5X})]$
$\displaystyle=$ $\displaystyle
4rZ_{X}+\phi^{\prime}(r^{2}G_{3X}+G_{5X})+2\sqrt{f}Y_{X}\,,$ (2.4)
$\displaystyle{\cal{B}}=$ $\displaystyle
rZ-f\phi^{\prime}XG_{5X}+12f\phi^{\prime}X^{2}F_{5}$ $\displaystyle=$
$\displaystyle rZ+\sqrt{f}Y\,,$ (2.5)

where we have used the no ghost condition (1.8) relating $F_{4}$ and $F_{5}$
for any beyond Horndeski theory. With these definitions the field equations
can be shown to take a surprisingly simplified form,

$\displaystyle
X^{\prime}{\cal{A}}=2\left(\frac{h^{\prime}}{h}-\frac{f^{\prime}}{f}\right){\cal{B}}\,,$
(2.6)
$\displaystyle\frac{h^{\prime}f}{2h}{\cal{A}}=G_{2X}r^{2}+2G_{4X}-2rf\phi^{\prime}G_{3X}-2fZ_{X}\,,$
(2.7) $\displaystyle
2f\frac{h^{\prime}}{h}{\cal{B}}=-G_{2}r^{2}-2G_{4}-2fZ\,.$ (2.8)

We can now make several general remarks on the form of the field equations
that will be crucial when we attack the general case. Indeed if our theory
enjoys parity symmetry ($G_{3}=G_{5}=F_{5}=0$) then the system is largely
manageable, as we will explain in the next section, as ${\cal{A}}$ and
${\cal{B}}$ are related and (2.6) is integrated directly. This is not the case
once $G_{3},G_{5}$ or $F_{5}$ are non zero and ${\cal{A}}$ and ${\cal{B}}$ are
now independent.

In the general case when (2.6) is not integrable, we start by noting that the
simplifying assumption $f=h$ already constrains heavily the theory at hand.
Indeed, Eq. (2.6) imposes either $X$ to be constant or ${\cal{A}}=0$.

When $X$ is constant, then (2.7) and (2.8) must give an identical solution for
$h$. As we explicitly show in Appendix A, parity breaking theories render this
impossible whereas even the parity-symmetric theories fail to lead to a viable
solution. In fact, $X$ being a constant is a hypothesis trimmed to time (and
radially) dependant scalar fields yielding quite often stealth solutions (see
the nice general analysis in [47]). We will therefore not consider any further
the case of constant $X$.

We are therefore left with the latter condition, ${\cal{A}}=0$, which gives
that the right hand side of (2.7) has to be zero; which is of course rather
restrictive on the theory. If however this condition can be fulfilled, then
the system is integrated by solving (2.8). We see therefore that the analysis
is far more complex in parity asymmetric theories.

There is however an example analytic solution of Horndeski theory [55] that
follows the path we underlined above: For [55] we have the Horndeski theory
functions

$G_{2}=8\alpha X^{2},\;G_{3}=-8\alpha X,\;G_{4}=1+4\alpha
X,\;G_{5}=-4\alpha\ln X$

and for this case it turns out that the RHS of (2.9) is simply

$G_{2X}r^{2}+2G_{4X}-2rf\phi^{\prime}G_{3X}-2fZ_{X}=\frac{{\cal{A}}}{f\phi^{\prime}}=0$
(2.9)

giving that the system of equations is actually compatible. We will come back
and use this observation when studying parity breaking theories in section 4.

We will see throughout the following sections how to use equations (2.6)-(2.8)
to solve different cases. It is useful to note that the degeneracy condition
(1.8) relates $Z$ and $Y$ according to,

$Y=-\frac{Z}{2}\frac{G_{5X}(-2X)^{3/2}}{G_{4}-2XG_{4X}}.$ (2.10)

##  3 The integrability of parity symmetric theories

In this section, we will consider shift symmetric beyond Horndeski theories
with parity symmetry in $\phi$. We therefore take $G_{3}=G_{5}=F_{5}=0$.
Several analytic solutions have been found for parity preserving (beyond)
Horndeski theories mostly in the presence of a linear time-dependence [46],
[47] , [65], [66], [67],[68] but also for a static scalar field [69], [70],
[71], [72], [73]. Here, we will consider the resolution of such a theory with
a static scalar field in full generality, and provide a method which will
allow us to find concrete solutions. We will then pick out and look at certain
interesting cases as illustrative examples.

In the absence of odd parity terms and with our auxiliary variables at hand,
things greatly simplify since now ${\cal{A}}=4rZ_{X}$ and ${\cal{B}}=rZ$. It
is then straightforward to integrate the first field equation (2.6), to get

$Z^{2}f=\gamma^{2}h\,,$ (3.1)

where $\gamma$ is a real constant that sets the relation between $f$ and $h$.
We remind the reader that $Z$ gives the relation between $F_{4}$ and $G_{4}$
(2.2). The remaining field equations combine to give us

$r^{2}(ZG_{2})_{X}+2(G_{4}Z)_{X}=0$ (3.2)

and

$2\gamma^{2}(rh)^{\prime}+Z(G_{2}r^{2}+2G_{4})=0\,.$ (3.3)

The former equation is essentially a condition on the theory which, once the
theory is fixed, gives an algebraic relation determining $X$ (or equivalently
the scalar field $\phi$). Note in passing that $X$ is only function of $r$ as
the metric does not appear in this equation. The last equation (3.3) is a
first-order ODE for the metric function $h=h(r)$. Again, one has to fix the
theory, i.e. choose the form of the coupling functions $G_{2}$, $G_{4}$ and
$F_{4}$ (or $Z$), before proceeding to solve for $h$. The remaining metric
function $f=f(r)$ may then be determined through (3.1). For later use, we note
that the functional form of $Z$, i.e. choosing $Z=\gamma$ or $Z=Z(X)$, allows
for the emergence or not, respectively, of homogeneous solutions with $f=h$.

A general way to proceed in order to find explicit solutions is to consider an
arbitrary function ${\mathcal{G}}={\mathcal{G}}(X)$, such that

${\mathcal{G}}_{X}=\frac{\alpha r^{2}+\beta}{\epsilon r^{2}+\delta}$ (3.4)

and the field equation (3.2) are compatible555Note that $\alpha$, $\beta$,
$\epsilon$ and $\delta$ can in general be functions of $X$ but for simplicity
we take them here as constants.. Compatibility immediately gives the
conditions

$\displaystyle G_{2}Z=\epsilon{\mathcal{G}}-\alpha X+C\;,\quad
2G_{4}Z=\delta{\mathcal{G}}-\beta X+D\,,$ (3.5)

where $\epsilon$, $\delta$, $\alpha$, $\beta$, $C$ and $D$ are constants. The
latter equation (3.3) reduces to the form

$2\gamma^{2}(rh)^{\prime}+r^{2}(\epsilon{\mathcal{G}}-\alpha
X+C)+\delta{\mathcal{G}}-\beta X+D=0\,.$ (3.6)

In this form, (3.4) and (3.6) are the field equations conveniently written
using ${\mathcal{G}}$ while disentangling $r$ and $X$. Once we choose the
function ${\mathcal{G}}={\mathcal{G}}(X)$, (3.4) determines $X$, and thus
$\phi^{\prime}$, in terms of $r$. The same equation provides also
${\mathcal{G}}$ in terms of $r$, and then (3.6) is an explicit $r$-dependent
ODE giving the solution for $h$. The chosen form of ${\mathcal{G}}$ determines
through (3.5) also the theory, namely the forms of the coupling functions
$G_{2}$ and $G_{4}$ modulo $Z$. The latter quantity is fixed relative to the
desired relation between $f$ and $h$, namely $f=h$ or $f\neq h$. Choosing $Z$
appropriately, we find $f$ through (3.1) and fix the remaining coupling
function $F_{4}$ thus obtaining a full solution to the given theory.

There are numerous examples one can consider with relative ease, essentially
depending on whether (3.6) is integrable or not. In what follows, we will
focus on two particular cases, one with $Z=\gamma$ leading to a homogeneous
solution with $f=h$ and one with $Z=Z(X)$ for which a non-homogeneous solution
emerges.

###  3.1 Black holes with a canonical Kinetic term

We will first consider the case of a constant $Z$ and, in particular, set
$Z=\gamma$. Then, from (3.1), we immediately get $f=h$. Let us also choose a
linear ${\mathcal{G}}_{X}=2\mu X+\zeta$. Integrating the latter with respect
to $X$ and substituting in (3.5), we obtain the functional forms of $G_{2}$
and $G_{4}$, namely

$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\frac{\epsilon\mu}{\gamma}X^{2}+\frac{\epsilon\zeta-\alpha}{\gamma}X-2\Lambda\,,$
$\displaystyle G_{4}$ $\displaystyle=$
$\displaystyle\frac{\delta\mu}{2\gamma}X^{2}+\frac{\delta\zeta-\beta}{2\gamma}X+1\,.$
(3.7)

In the above, we have without loss of generality fixed the constant term in
the expression of $G_{2}$ to a vacuum cosmological constant $\Lambda$. We have
likewise fixed the corresponding term in the expression of $G_{4}$ to unity in
order to restore the Einstein-Hilbert term. Note that $G_{2}$ contains a
linear term in $X$, therefore the theory includes a canonical kinetic term for
the scalar field.

Since $Z$ determines the relation between $F_{4}$ and $G_{4}$ via (2.2),
making a choice for $Z$ imposes certain constraints on the form of these two
coupling functions. For example, in Horndeski theory where $F_{4}=0$,
solutions with $Z=\gamma$, or equivalently with $f=h$, are only possible for
$G_{4}=-\gamma+\sqrt{-2X}$. In beyond Horndeski theory, fixing $Z=\gamma$ and
employing $G_{4}$ given above, completely determines the form of $F_{4}$ as

$F_{4}=\frac{\gamma+1}{4X^{2}}+\frac{\beta-\delta\zeta}{8\gamma
X}-\frac{3\delta\mu}{8\gamma}\,.$ (3.8)

Turning now to the derivation of the solution, we first notice that (3.4) can
be easily solved for $X$ giving the result

$X(r)=\frac{(\alpha-\epsilon\zeta)r^{2}+\beta-\delta\zeta}{2\mu(\epsilon
r^{2}+\delta)}\,.$ (3.9)

Substituting $X$ and ${\mathcal{G}}$ into (3.3), we obtain, after a tedious
but straightforward integration, the solution

$h(r)=C_{1}+C_{2}r^{2}+C_{3}\frac{\arctan(\sqrt{\frac{\epsilon}{\delta}}r)}{\sqrt{\frac{\epsilon}{\delta}}r}-\frac{2M}{r},$
(3.10)

where

$C_{1}=-\frac{1}{\gamma}+\frac{(\epsilon\zeta-\alpha)[\delta(\epsilon\zeta+\alpha)-2\beta\epsilon]}{8\gamma^{2}\epsilon^{2}\mu}\;,\qquad
C_{3}=\frac{(\beta\epsilon-\delta\alpha)^{2}}{8\gamma^{2}\epsilon^{2}\mu\delta}\,,$

and $M$ is an integration constant666It is interesting to note that a
Newtonian fall-off arises for the mass term in parity preserving theories in
agreement with [74] which arrives at this conclusion using a generalised Kerr-
Schild method.. Also,

$C_{2}=\frac{\Lambda}{3\gamma}+\frac{(\epsilon\zeta-\alpha)^{2}}{24\mu\epsilon\gamma^{2}}$

stands for the effective cosmological constant which is corrected by the
linear term in $G_{2}$. If we wish this quantity to vanish without having to
fine-tune the parameters of the theory, we have to set independently
$\alpha-\epsilon\zeta=\Lambda=0$. We then find $C_{1}=-\frac{1}{\gamma}$ and
we take $\gamma=-1$ to attain asymptotic flatness. Note that in the Horndeski
theories considered in [69] and [70], solutions similar to (3.10) were
obtained, but with an always non-zero effective cosmological constant. Going
to beyond Horndeski theories seems to be the key point in order to attain
asymptotic flatness with such a profile.

We therefore focus on this asymptotically flat black hole metric with $f=h$,
which reads

$h(r)=1+\frac{(\beta-\delta\zeta)^{2}}{8\delta\mu}\frac{\arctan(\sqrt{\frac{\epsilon}{\delta}}r)}{\sqrt{\frac{\epsilon}{\delta}}r}-\frac{2M}{r}\,,$
(3.11)

with $\epsilon\delta>0$, while the scalar field satisfies the equation

$\phi^{\prime 2}(r)=-\frac{(\beta-\delta\zeta)}{\mu(\epsilon
r^{2}+\delta)}\,\frac{1}{h(r)}\,.$ (3.12)

Thus, the scalar field is real for $r$ spacelike, provided the quantities
$\delta\mu$ and $\left(\beta-\delta\zeta\right)$ have opposite signs. The
scalar field diverges at the point where $h(r)$ vanishes, however, the
fundamental scalar quantity of the theory $X\equiv-h\phi^{\prime 2}/2$ remains
everywhere finite. Note also that $\phi$ becomes trivial at asymptotic
infinity as expected. Taking the same limit of (3.11), we obtain a Reissner-
Nordstrom type of solution. The ADM mass is given by $M$ along with the vacuum
contribution, namely

$M_{tot}=M-\frac{\pi(\beta-\delta\zeta)^{2}}{32\delta\mu}\sqrt{\frac{\delta}{\epsilon}}\,.$

We observe that, even in the case where $M=0$, we do not obtain a vacuum
spacetime due to the contribution from the $\arctan$ term (see also [69] and
[70]). In fact, we see that we still have a black-hole horizon if
$-(\beta-\delta\zeta)^{2}/8<\delta\mu<0$. Otherwise, we have a naked
singularity since, at $r=0$, we have $h(0)\neq 1$ with a vacuum mass term due
to the non trivial scalar field. The asymptotic solution at large distance is
completed by a tidal charge term with

$Q^{2}=-\frac{(\beta-\delta\zeta)^{2}}{8\mu\epsilon}\,.$ (3.13)

For $M\neq 0$, there are straightforward constraints on its sign and magnitude
so as to ensure a positive $M_{tot}$. Note in particular that $\delta\mu>0$
only allows for positive values of $M$, while negative values are also
permitted if $\delta\mu<0$. Then, for $M>0$, there is a unique horizon and the
spacetime is a black hole, while for $M<0$, the spacetime describes either a
black hole with an outer and an inner horizon, or a naked singularity,
depending on the parameters of the theory and on the magnitude of $M$ compared
to the latter. Note that for $\delta\mu<0$, our solution possesses a robust
Reissner-Nordstrom asymptotic limit with $Q^{2}>0$.

With the above conditions, the solution (3.11)-(3.12) is therefore an
asymptotically flat black hole with secondary scalar hair. As mentioned
earlier, the $G_{2}$ term features in its expression a canonical kinetic
term777 Note that even in the absence of a linear term in $G_{2}$,
substituting $X^{2}\longrightarrow-X$ restores the canonical kinetic term
without changing the form of the equations and the metric solution itself.
This is due to the form of the equations (3.2-3.3) and is not generally valid
beyond parity symmetric theories. An interesting question arises in the
context of the above theory regarding black holes and no-hair theorems: under
what conditions on the theory can one include a canonical scalar kinetic term
while having a non trivial black hole? According to [73], in the case of
spherical symmetry this is indeed possible if $G_{4}$ has a $\sqrt{-X}$ term
included in its expression. Such a term was shown to provide a source term in
the scalar equation in order for $\phi^{\prime}$ to be non trivial [73], and
to have a similar effect to the one of the $\phi$-Gauss-Bonnet term discussed
in detail in [36]. Our analysis here has demonstrated that hairy black holes
can emerge for alternative forms of the coupling function $G_{4}$ while
$G_{2}$ continues to feature a canonical kinetic term. This is due to an
overall freedom regarding the choice of the form of the function
${\mathcal{G}}(X)$, which in turn defines the forms of $G_{2}$ and $G_{4}$.
Both the solution presented here and the BCL one [73] belong to the same class
of hairy black holes which emerge in the context of beyond Horndeski theories
described by the relations (3.5) but with different choices 888We can limit
ourselves to the Horndeski case with $F_{4}=0$ and obtain the BCL solution
[73] upon choosing ${\mathcal{G}}=\mu X+\zeta\sqrt{-2X}+\eta$ and fixing
accordingly the constants of the theory. for ${\mathcal{G}}(X)$. In support of
this, in Appendix B, we present additional examples of hairy black holes with
a Reissner-Nordstrom asymptotic behaviour (but with tidal rather than electric
charge); these solutions emerge in beyond Horndeski theories with different
forms of $G_{4}$ and $G_{2}$ functions but with always a canonical kinetic
term for the scalar field.

Before closing this section, let us take a brief look at the solutions above
in the presence of a positive or negative cosmological constant. In this case,
$C_{1}$ need not be fixed to unity as the $C_{2}r^{2}$-term is always dominant
for large enough $r$. We can rather choose to fix $C_{1}+C_{3}=1$ so as, for
$M=0$, the solution is regular at $r=0$, $h(0)=1+{\cal O}(r^{2})$. By fixing
the constants in such a way the solution close to $r=0$ has a (anti)de-Sitter
core and we have no longer a solid deficit angle. The vacuum solution is
therefore a regular but not maximally symmetric solution (de Sitter or anti de
Sitter) because of the presence of the scalar field which continues to be non-
trivial. In this case, we have therefore a soliton solution, i.e. an
everywhere regular solution (such solutions have been found in higher order
Proca theories [75]). Adding a non trivial mass term gives us a black hole of
zero electric or magnetic charge but with similar spacetime properties as the
(A)dS-Reisser-Nordstrom black hole.

###  3.2 Non-Homogeneous Black holes

In this case, we will assume that $Z=Z(X)$ and as a result obtain, via (3.1),
the relation

$f(r)=\frac{\gamma^{2}h(r)}{Z^{2}(X)}\,.$ (3.14)

The line-element then reads

$ds^{2}=-h(r)dt^{2}+\frac{Z(X)^{2}dr^{2}}{\gamma^{2}h(r)}+r^{2}d\Omega^{2}\,.$
(3.15)

There are numerous possibilities for the choice of $Z=Z(X)$ which in fact give
different types of black hole, wormhole or singular solutions. For a start if
$Z=Z(X)$ is infinite at a particular finite radius $r=r_{T}$ (which is not a
zero of $h$ and $f$) then $r=r_{T}$ is a possible wormhole throat (see [64]).
But it is now easy to see that since $X$ is only function of $r$, according to
(3.1), $Z$ can only be a function of $r$ and crucially not of the metric
functions $f$ and $h$. As such any throat will solely depend on the theory and
not on an independent integration constant (essentially the mass parameter)
which renders such wormholes eternal and fine tuned to the theory in question.
We will not consider such a possibility furthermore. On the other hand, by
choosing a smooth $Z(X)$ functional without zeros or singularities, we can
obtain a non-homogeneous black-hole geometry (depending on the properties of
$f$ and $h$).

Here, we will focus on the latter case and, for simplicity, assume again that
${\mathcal{G}}_{X}=2\mu X+\zeta$. This means that our ODE for $h$, (3.6), is
identical to the one found in the previous subsection giving again the
functional form (3.10) with the same ($C_{1},C_{2},C_{3}$) theory-dependent
constants. Similarly, (3.4) gives us $X$ as the same function of $r$, (3.9),
as before. We now choose an analytic and non-trivial $Z$ such that

$Z(X)=\gamma(1+X).$ (3.16)

Note that we have chosen $Z$ specifically aiming to obtain asymptotically flat
black-hole solutions. Indeed, the solution for $h(r)$, (3.10), reduces to the
asymptotically-flat Reissner-Nordstrom background, (3.11), under the same
choices made in the previous subsection, namely $\alpha-\zeta\epsilon=0$ and
$\gamma=-1$. In that case, from (3.9) we obtain

$X(r)=\frac{\beta-\delta\zeta}{2\mu(\epsilon r^{2}+\delta)}.$

Employing the above, the solution for $h(r)$, (3.11), is completed by the
expressions for $\phi^{\prime}(r)$ and $f(r)$ given by

$\phi^{\prime 2}(r)=-\frac{(\beta-\delta\zeta)}{\mu(\epsilon
r^{2}+\delta)}\,\frac{1}{f(r)}\,,\qquad f(r)=\frac{h(r)}{(1+X)^{2}}\,.$ (3.17)

We observe that $X$ goes to zero for large $r$ so that also $f(r)$ reduces
asymptotically to unity, a result which validates our choice (3.16) from
$Z(X)$. Since the solution for $h(r)$ is the same as in the previous
subsection, the spacetime geometry preserves all the characteristics discussed
there. Apart from the asymptotic flatness, the spacetime features a tidal
charge and one or two horizons or a naked singularity, following the
description given in the previous subsection.

The choice (3.16) for the functional $Z(X)$ fixes also the forms of the
coupling functions $G_{2}$, $G_{4}$ and $F_{4}$, through (3.5), as follows

$\displaystyle G_{2}$ $\displaystyle=\frac{\epsilon\mu X^{2}}{\gamma(1+X)}\,,$
$\displaystyle G_{4}$ $\displaystyle=\frac{\delta\mu
X^{2}+(\delta\zeta-\beta)X-2\gamma^{2}}{2\gamma(1+X)}\,,$ $\displaystyle
F_{4}$
$\displaystyle=\frac{\beta-\delta\zeta+X^{2}\left(2\gamma^{2}-\delta\mu\right)+X\left(-\beta+6\gamma^{2}+\delta(\zeta-3\mu)\right)}{8\gamma
X(X+1)^{2}}\,.$

In the above expressions, we have set, for simplicity, $C+\epsilon\eta=0$ and
$D+\delta\eta=-2\gamma^{2}$. At large distances, the above choices justify the
vanishing of the cosmological constant and ensure the restoration of the
Einstein-Hilbert term upon setting also $\gamma=-1$.

Having completed the presentation of black-hole solutions in the context of
the parity-symmetric beyond Horndeski theory and before moving to the non
parity-symmetric sector in the next section, we would like to stress that
solutions with an asymptotic (A)dS-Reissner-Nordstrom behaviour arise also in
cases where a mixed selection of coupling functions is made. In appendix C, we
study a particular case where we keep $G_{3}=G_{5}=0$ but nevertheless allow
for a non-vanishing $F_{5}$ term together with the $G_{2}$ and $G_{4}$
functions.

##  4 Attacking the case of no parity symmetry

When we give up parity symmetry, keeping $G_{3}$, $G_{5}$ and eventually
$F_{5}$ terms in the theory along with $G_{2}$, $G_{4}$ (and eventually
$F_{4}$), the field equations get considerably more difficult to tackle in all
generality. In the sense encountered in the previous section, integrability is
lost and most known black hole solutions have been found numerically. To our
knowledge, the only analytical solution is that of [55] valid in a parity
breaking but shift symmetric theory. In order to extend the existing analytic
results, we will try to generalize the theory appearing in [55], where the
solution obeys the condition $f=h$. As we commented also in section 2, the
field equation (2.6) dictates that the choice $f=h$ is possible only if
${\cal{A}}=0$ or $X^{\prime}=0$. The latter case was dealt with in appendix A
and shown not to lead to a viable solution. When is then the former choice,
${\cal{A}}=0$, compatible with the remaining 2 field equations, or, to set the
question in a different way, which theories allow for solutions with $f=h$?
The answer is quite simple given the form of our field equations (2.6)-(2.8).

Indeed, if we assume that $f=h$ and thus ${\cal{A}}=0$, then the RHS of (2.7)
must also vanish. In this case, we may impose the condition

$G_{2X}r^{2}+2G_{4X}-2rf\phi^{\prime}G_{3X}-2fZ_{X}=-\sqrt{f}{\cal{A}}Q,$
(4.1)

where $Q=Q(X)$ is an arbitrary function of $X$, in order for the system to be
well defined. Indeed ${\cal{A}}=0$ will be giving us the scalar field or
function $X$ algebraically similarly to (3.2) and will be solving 2 out of the
3 field equations. The above equation can be seen as a polynomial in powers of
$r$ (and $f$) with $X$-dependent coefficients. Employing the definition (2)
for ${\cal{A}}$, and matching the corresponding coefficients of $r$ and $f$,
(4.1) leads to the following universal constraints that must be valid for the
theory functions

$\displaystyle G_{2X}=-\sqrt{-2X}QG_{3X}=-2Q^{2}Z_{X},$ (4.2) $\displaystyle
2G_{4X}=-\sqrt{-2X}QG_{5X},$ (4.3) $\displaystyle Z_{X}=QY_{X}\,.$ (4.4)

Combining the relation (4.3) with the no-ghost constraint (2.10) gives

$Z=QY\left(1-\frac{G_{4}}{2XG_{4X}}\right).$ (4.5)

Then, the compatibility of (4.4) with (4.5) leads to the additional constraint

$Q_{X}Y=\left(QY\frac{G_{4}}{2XG_{4X}}\right)_{,X}\,.$ (4.6)

Let us then summarise: for the class of theories where the condition (4.1)
holds, the coupling functions $G_{2}$, $G_{3}$ and $G_{5}$ are given in terms
of $G_{4}$, $Q$ and $Y$ via999Instead of $Y$, which involves the odd coupling
functions $G_{5}$ and $F_{5}$, one may equivalently choose $Z$ as the third
variable, which features the even functions $G_{4}$ and $F_{4}$. (4.2)-(4.3).
Due to the additional constraint (4.6), only two of the latter three
quantities are independent and thus our solutions will be parametrised by two
free theory functions, say $G_{4}$ and $Q$.

We may therefore start our analysis by choosing $Q=Q(X)$ and $G_{4}$,
determining $Y$ via (4.6). Then, (4.2)-(4.4) will completely fix the beyond
Horndeski theory by providing the forms of the coupling functions $G_{2}$,
$G_{3}$, $G_{5}$, $F_{4}$ (via $Z$) and $F_{5}$ (via $Y$). This theory
choosing filter is clearly not the most general parity breaking theory but it
is clearly a general starting point admitting [55] as one particular solution
as we will see.

The field equations (2.6)-(2.8) for this special class of theories simplify
to:

$\displaystyle
X^{\prime}{\cal{A}}=2\left(\frac{h^{\prime}}{h}-\frac{f^{\prime}}{f}\right){\cal{B}}\,,$
(4.7)
$\displaystyle{\cal{A}}\left(\frac{h^{\prime}\sqrt{f}}{h}+2Q\right)=0\,,$
(4.8) $\displaystyle
2f\frac{h^{\prime}}{h}{\cal{B}}+G_{2}r^{2}+2G_{4}+2fZ=0\,,$ (4.9)

where (2.7) was rewritten with the help of (4.1). If we look for solutions
with $f=h$, then (4.7) leads immediately to ${\cal{A}}=0$; ${\cal{A}}$ can be
conveniently rewritten, by combining its definition (2) with the constraints
(4.2)-(4.3), as

${\cal{A}}=\frac{2Z_{X}}{Q\sqrt{f}}\left[(rQ+\sqrt{f})^{2}-\frac{G_{4X}}{Z_{X}}\right].$
(4.10)

The vanishing of the above combination will then give us the solution for $X$
(or the scalar field $\phi^{\prime}$) as a function of $f$. Note here a
crucial difference with parity symmetric theories where we saw that $X$ only
depended on $r$ (3.2) and not on the metric functions $f$ or $h$. Equation
(4.8) will be trivially satisfied, for $f=h$, whereas (4.9) will provide a
first-order differential equation for the sole metric function $f(r)$. If one
solves this latter ode a full solution will be known to this theory.

Let us note that, although the condition (4.1) was motivated by the assumption
that $f=h$, the field equations (4.7)-(4.9) allow also for the emergence of
solutions with $f\neq h$ within the same class of theories; in that case,
${\cal{A}}\neq 0$ and the solution for the scalar field follows instead from
the equation $\frac{h^{\prime}\sqrt{f}}{h}+2Q=0$ – provided that $Q(X)$ is not
trivial – leaving us with two ODEs, (4.7) and (4.9), to determine the two
unknown metric functions $h$ and $f$.

In the context of the present analysis, we will focus on homogeneous solutions
with $f=h$. As outlined above, the first step towards finding such a solution
is to choose the form of $Q(X)$. To this end, we set $Q(X)=\gamma\,(-2X)^{m}$,
with $\gamma$ a constant of dimension $2m-1$. This choice, in conjunction with
(4.6) and (4.4), entails the general solution for all $m$,

$Y(X)=c(-2X)^{1-m}\,G_{4X}\,G_{4}^{2m-1},\qquad
Z(X)=c\gamma\,G_{4}^{2m-1}(G_{4}-2XG_{4X})\,,$ (4.11)

where $c$ is a constant of integration and the product $\gamma c$ is
dimensionless. The above relations determine the functionals $Y$ and $Z$ in
terms of $G_{4}$. Fixing the latter as well as the value of $m$ will allow us
to completely fix the theory via (4.2)-(4.3) and then integrate (4.7)-(4.9) in
order to find the solution. For this purpose, in the next two subsections we
choose to study two separate cases: the case with $m=1/2$ leading to
$Q=\gamma\sqrt{-2X}$ and the case with $m=0$ corresponding to $Q=\gamma$ an
arbitrary constant. The first case is particularly interesting as it
corresponds to (Horndeski) theories related to the Kaluza Klein reduction of
Lovelock theories [22], [23], and includes the shift symmetric theory and
black hole found in [55].

###  4.1 Parity breaking theories related to Kaluza-Klein reduction of
Lovelock theory

If we choose $m=1/2$, then we obtain $Q=\gamma\sqrt{-2X}$, and $\gamma$ is a
dimensionless quantity. Subsequently, one finds:

$Y=c\;G_{4X}\sqrt{-2X},\qquad Z=\gamma c\,(G_{4}-2XG_{4X})\,.$

Without further ado we may now completely fix the theory by choosing
$G_{4}=1+\alpha\left(-2X\right)^{n}$. In fact the case $n=1,\gamma c=-1$ gives
the theory [55] that originates from higher dimensional Lovelock theory and in
particular the Kaluza Klein reduction of the Gauss-Bonnet term101010See also
[58] for an alternative interesting derivation. One can loosely relate $n+1$
to the order, in powers of curvature, of the higher order Lovelock
term111111Although a complete justification of this is beyond the scope of
this paper it is nevertheless an important characterisation of $n$ involved in
the higher dimensional reduction. For example, the $n=2$ theory can be seen to
be related to the third order Lovelock term $L_{3}$ [76] via Kaluza klein
reduction to 4 dimensions. In particular, for $n=2$, we have $G_{4}=1+\alpha
X^{2}$ and $G_{5}=X$ which interestingly gives the self tuning Paul term [15]
found in Fab 4 [77, 78].

The remaining coupling functions of the theory are then:

$\displaystyle G_{2}={}$ $\displaystyle 2\gamma^{3}c\alpha
n(2n-1)\frac{\left(-2X\right)^{n+1}}{n+1},$ $\displaystyle G_{3}={}$
$\displaystyle-2\gamma^{2}c\alpha(2n-1)\left(-2X\right)^{n},\quad
G_{5X}=\frac{4\alpha n}{\gamma}\left(-2X\right)^{n-2}\,,$ $\displaystyle
F_{4}={}$ $\displaystyle\frac{\gamma
c+1}{4X^{2}}\left(1+\alpha(1-2n)\left(-2X\right)^{n}\right),\quad
F_{5}=\frac{\gamma c+1}{3\gamma}\left(-2\alpha
n\right)\left(-2X\right)^{n-3}\,.$

As before, we have chosen not to include a cosmological constant in $G_{2}$.
The Horndeski case is readily identified with the choice $\gamma c=-1$ for
which both $F_{4}$ and $F_{5}$ vanish. In addition, the choices $n=1$ and
$\gamma=1$ correspond precisely to the case of [55] (for our conventions on
the coupling constants). Here, we will keep $n$ and $\gamma c$ arbitrary and
attempt to generalise, in the context of beyond Horndeski theory, the black-
hole solution found in [55].

The constraint $\mathcal{A}=0$, valid for all solutions with $f=h$, in
conjunction with (4.10) lead to the following relation for the scalar field:

$\phi^{\prime}=\frac{1-\sqrt{\gamma c(1-2n)f}}{r\gamma\sqrt{\gamma
c(1-2n)f}}\,.$ (4.12)

The above clearly demands that $n\neq 1/2$ and that we define a positive
function $F^{2}(r)\equiv\gamma c(1-2n)f>0$. Then, the differential equation
(4.9) for the metric function can be integrated once with respect to the
radial coordinate $r$ to give the following equation:

$(n+1)\left(\gamma^{3}c(1-2n)\right)^{n}r^{2n}\left(1-2n+F^{2}\right)+\alpha\left(1-F\right)^{2n}\left(1+2nF+F^{2}\right)-\lambda
r^{2n-1}=0\,,$ (4.13)

where $\lambda$ is an integration constant. The above is an algebraic equation
with $2(n+1)$ degree in $F$. For $n\geq 1$, integer or half-integer, it
becomes a polynomial equation in $F$. In the case where $n=1$, we easily
obtain the explicit result:

$f(r)=-\frac{1}{\gamma
c}-\frac{r^{2}\gamma^{2}}{\alpha}\left(1\pm\sqrt{1+\frac{\alpha\lambda}{r^{3}\gamma^{6}c^{2}}}\right).$
(4.14)

The above, as anticipated from our earlier comments, reduces to the black-hole
solution [55] when $\gamma c=-1$ (Horndeski case). Note that the extension of
the solution in beyond Horndeski picks up a solid angle deficit since $\gamma
c\neq-1$ while $\phi^{\prime}\neq 0$ for $M=0$. In other words if $\gamma
c\neq-1$ and $M=0$ we have a naked singularity at $r=0$ while the scalar field
also explodes there. It is only the Horndeski case which has a well defined
vacuum for $M=0$.

Let us now move on to the general $n$ case. Equation (4.13) implies the
following asymptotic behaviour for the metric function:

$f(r)=-\frac{1}{\gamma c}+\frac{\lambda}{(n+1)\gamma
c(1-2n)\left(\gamma^{3}c(1-2n)\right)^{n}}\frac{1}{r}+\mathcal{O}\left(\frac{1}{r^{2n}}\right).$
(4.15)

Employing the above into (4.12), we find in turn that the scalar field behaves
at infinity as

$\phi^{\prime}=\frac{1-\sqrt{2n-1}}{r\gamma\sqrt{2n-1}}+\mathcal{O}\left(\frac{1}{r^{2}}\right),$
(4.16)

which implies that $\phi$ diverges like $\ln(r)$ at infinity, except for $n=1$
where $\phi=\mathcal{O}\left(\frac{1}{r}\right)$. However, $-2X=f\phi^{\prime
2}$ always vanishes at infinity. As regards the metric, it is asymptoting
Minkowski spacetime only for the Horndeski case ($\gamma c=-1$)121212If
$\gamma c\neq-1$ the asymptotic metric is only locally asymptotically flat. By
this we mean that the spacetime curvature tensor asymptotes zero but one may
have a global deficit angle as in the case of the gravitating monopole
solution in GR [79]. In this case, we may restore a Schwarzschild-like
behaviour131313In fact, one can notice on (4.15) that the metric is almost
Schwarzschild at infinity for large values of $n$. if the integration constant
$\lambda$ is related to the mass $M$ by

$\lambda=2M(n+1)(1-2n)\left(\gamma^{2}(2n-1)\right)^{n}.$ (4.17)

In what follows we will focus on the Horndeski case, and thus take $\gamma
c=-1$, but allow for general $n$. We will also set $\gamma=1$, since $\gamma$
can always be absorbed into $\alpha$ via the rescaling
$\alpha\to\alpha/\gamma^{2n}$. All the above combined allow to write (4.13) in
the form:

$(n+1)\left(2n-1\right)^{n}r^{2n-1}\left[(2n-1)(2M-r)+rF^{2}\right]+\alpha\left(1-F\right)^{2n}\left(1+2nF+F^{2}\right)=0\,.$
(4.18)

Although we cannot solve analytically (4.18) for $n>1$, we can find a
perturbative solution assuming $\alpha\rightarrow 0$. To this end, we use the
expansion

$f(r)=1-\frac{2M}{r}+\sum_{i=1}^{\infty}\alpha^{i}f_{i}(r).$ (4.19)

By replacing the above expression in (4.18) and solving order by order, we can
determine the functions $f_{i}(r)$. For instance, the first-order correction
is found to be

$\displaystyle
f_{1}(r)=-\frac{2\,[1-k(r)]^{2n}\left[nr(k(r)+1)-2Mn+M\right]}{r^{2n+1}\,(n+1)\,(2n-1)^{n+1}}\,,$
(4.20)

where

$k(r)=\sqrt{(2n-1)\left(1-\frac{2M}{r}\right)}.$ (4.21)

The above perturbative form (4.19) for the metric function $f(r)$ is valid
over the entire radial regime in the small-$\alpha$ limit.

The presence of a horizon is signified by the vanishing of the metric function
$f(r)$. Thus, setting $F(r)=0$ in (4.18) defines the horizon radius as the
value $r_{H}$ which satisfies the equation:

$(n+1)\left(2n-1\right)^{n+1}r_{H}^{2n-1}\left(r_{H}-2M\right)=\alpha\,.$
(4.22)

As expected, the coupling parameter $\alpha$ induces a deviation from the
Schwarzschild radius $r_{0}=2M$ (for clarity and in order to compare with the
GR limit, we restrict here to the case $M>0$). As above, one can therefore
write a series expansion for the horizon radius, in the limit
$\alpha\rightarrow 0$, of the form

$r_{H}=2M+\sum_{k=1}^{\infty}b_{k}\,\alpha^{k}\,.$

The coefficients $b_{k}$ can be found again order by order, with the first
three given by the expressions

$b_{1}=\frac{1}{(n+1)(2M)^{2n-1}(2n-1)^{n+1}},\quad
b_{2}=\frac{1-2n}{2M}b_{1}^{2},\quad
b_{3}=\frac{(2n-1)(3n-1)}{(2M)^{2}}b_{1}^{3}\,.$ (4.23)

One can check that the same results follow by demanding the vanishing of the
perturbative solution (4.19).

The presence of the coupling parameter $\alpha$ in (4.22) not only does change
the horizon radius $r_{H}$ compared to GR but it also determines the number of
roots of that equation, and thus the topological structure of spacetime.
Equation (4.22) has the simple form of a polynomial of order $2n$, and much
can be said about its real, positive roots. To this end, we define the
quantities:

$r_{n}=\left(1-\frac{1}{2n}\right)2M<2M,\qquad\alpha_{M}=-(n+1)\left(2n-1\right)^{3n}\left(\frac{M}{n}\right)^{2n}<0\,.$
(4.24)

Then, if $\alpha>0$, there is a unique horizon with $r_{H}>2M$. If
$\alpha_{M}<\alpha<0$, there are exactly two horizons with
$r_{H-}<r_{n}<r_{H+}<2M$. One has that $r_{H+}\to\left(2M\right)^{-}$ when
$n\to\infty$, and $r_{H-}\to 0^{+}$ when $\alpha\to 0^{-}$. If
$\alpha<\alpha_{M}$, the spacetime has no horizons. One can note that, as
$n\to\infty$,
$\alpha_{M}\sim-M\left(2n\right)^{n+1}\left(2M\right)^{2n-1}\exp\left(-3/2\right)\to-\infty$,
hence a larger and larger parameter space emerges for $\alpha$ which allows
for black-hole solutions. All the previous results are consistently
illustrated by the Lu-Pang case [55], which has $n=1$, where:

$r_{H\pm}=M\pm\sqrt{M^{2}+\frac{\alpha}{2}}\,.$ (4.25)

It is worth noting that the case $\alpha<0$ always leads, for any $n$, to a
positive $f$ at the origin: either with a black-hole metric function $f(r)$
changing sign twice if $\alpha>\alpha_{M}$, or with a soliton if
$\alpha<\alpha_{M}$. The behaviour of the metric function as $r\to 0$ can be
derived from equation (4.18), and reads:

$f(r)=\frac{1}{2n-1}-\frac{2}{2n-1}\left(\frac{M}{-\alpha}(2n-1)^{n+1}\right)^{\frac{1}{2n}}r^{1-\frac{1}{2n}}+\mathcal{O}\left(r^{1-\frac{1}{2n}}\right).$
(4.26)

We observe that a solid angle deficit emerges, which is however covered by the
horizons when $\alpha_{M}<\alpha<0$.

###  4.2 Theories admitting solutions with non trivial asymptotics

We will now consider the case $m=0$ which leads to $Q=\gamma$, with $\gamma$ a
constant of dimension $-1$. This case is not particularly physically motivated
as in the previous section, but has some simplifying mathematical properties.
We now obtain:

$Y=c(-2X)\frac{G_{4X}}{G_{4}},\qquad Z=\gamma(c+Y)\,.$ (4.27)

In order to fix the theory, we will take again
$G_{4}=1+\alpha\left(-2X\right)^{n}$. The remaining coupling functions of the
theory are now:

$\displaystyle G_{2}={}$
$\displaystyle-2\Lambda-2\gamma^{3}c\frac{1+\alpha(1-2n)\left(-2X\right)^{n}}{1+\alpha\left(-2X\right)^{n}}\,,\quad
G_{3X}=8\gamma^{2}c\alpha
n^{2}\frac{\left(-2X\right)^{n-3/2}}{\left(1+\alpha\left(-2X\right)^{n}\right)^{2}}\,,$
$\displaystyle G_{5X}={}$ $\displaystyle\frac{4\alpha
n}{\gamma}\left(-2X\right)^{n-3/2}\,,\quad F_{4}=\frac{\left(1+\gamma
c+\alpha\left(-2X\right)^{n}\right)\left(1+\alpha(1-2n)\left(-2X\right)^{n}\right)}{4X^{2}\left(1+\alpha\left(-2X\right)^{n}\right)},$
$\displaystyle F_{5}={}$
$\displaystyle\frac{-2n\alpha}{3\gamma}\left(-2X\right)^{n-5/2}\frac{1+\gamma
c+\alpha\left(-2X\right)^{n}}{1+\alpha\left(-2X\right)^{n}}\,,$

where a cosmological constant $\Lambda$ is now included in $G_{2}$. In this
case, we observe that, for $G_{4}=G_{4}(X)$, the functions $F_{4}$ and $F_{5}$
are always non-trivial, hence the $Q=\gamma$ case always belongs to beyond
Horndeski theories141414The opposite holds true only in the less interesting
$n=0$ case, and thus for a trivial $G_{4}$. Then, for generic coefficients we
obtain general relativity with an $F_{4}$ correction. The latter can be
eliminated upon making an appropriate choice of coefficients, namely $1+\gamma
c+\alpha=0$.. Note also that any $n$ leads to algebraic functions of $X$,
except $n=1/2$, which gives a logarithmic $G_{5}$ (i.e. a Gauss-Bonnet term)
and a logarithmic $G_{3}$.

Since we assume again that $f=h$, (4.7) dictates that $\mathcal{A}=0$ and
(4.10) gives:

$\left(r\gamma+\sqrt{f}\right)^{2}=\frac{G_{4}^{2}}{-2n\gamma c}\,,$ (4.28)

which implies that $-2n\gamma c>0$. Therefore, we can express $G_{4}$ directly
with respect to $r$ as $G_{4}=\beta\left(r\gamma+\sqrt{f}\right)$, with
$\beta=\pm\sqrt{-2n\gamma c}$. With this information at hand (4.9) can then be
integrated once directly to give the following third order polynomial equation
in $\sqrt{f}$:

$4ncf^{3/2}+3\left(\frac{\beta}{\gamma}+\left(2n-1\right)r\gamma
c\right)f+r^{2}\left(r\Lambda_{n}-3\beta\gamma\right)-\lambda=0\,.$ (4.29)

In the above, $\lambda$ is an integration constant with dimension 1, and we
have introduced the quantity
$\Lambda_{n}=\Lambda+\gamma^{3}c\left(1-2n\right)$ for simplicity.
Interestingly, the coupling parameter $\alpha$ of $G_{4}$ does not play any
role in (4.29). We notice that, due to the $f^{3/2}$ term, this equation gives
by construction the form of $f$ only in the spacetime regions where $f\geq 0$.
Let’s look for solutions where $f$ is positive at infinity. The asymptotic
behaviour is then an Anti-de Sitter one:

$f(r)=\Lambda_{f}r^{2}+\epsilon_{1}r+\epsilon_{0}+\frac{\epsilon_{-1}}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)$
(4.30)

where the coefficient $\Lambda_{f}>0$ satisfies the equation

$4nc\Lambda_{f}^{3/2}+3\left(2n-1\right)\gamma c\Lambda_{f}+\Lambda_{n}=0\,,$
(4.31)

and the coefficients $\epsilon_{i}$ are given by

$\displaystyle\epsilon_{1}={}$
$\displaystyle\Xi\left(\beta\gamma-\frac{\beta\Lambda_{f}}{\gamma}\right),$
(4.32) $\displaystyle\epsilon_{0}={}$
$\displaystyle\Xi\left(-\frac{\beta\epsilon_{1}}{\gamma}-\frac{nc\epsilon_{1}^{2}}{2\sqrt{\Lambda_{f}}}\right),$
(4.33) $\displaystyle\epsilon_{-1}={}$
$\displaystyle\Xi\left(-\frac{\beta\epsilon_{0}}{\gamma}-\frac{nc\epsilon_{1}\epsilon_{0}}{\sqrt{\Lambda_{f}}}+\frac{nc\epsilon_{1}^{3}}{12\Lambda_{f}^{3/2}}+\frac{\lambda}{3}\right).$
(4.34)

In the above, we have defined the quantity
$\Xi=\left[2nc\sqrt{\Lambda_{f}}+\left(2n-1\right)\gamma c\right]^{-1}$, for
simplicity. The equation (4.31) giving $\Lambda_{f}$ admits generic solutions
depending on the exact values and relative signs of $nc$ and $\Lambda_{n}$.

The presence of horizons with $r_{H}>0$ is linked again with the vanishing of
$f$, in which case (4.29) gives the constraint

$r_{H}^{2}\left(r_{H}\Lambda_{n}-3\beta\gamma\right)-\lambda=0\,.$ (4.35)

This third order polynomial can be easily analysed. We will consider the case
$\beta\gamma<0$ as the sign of this quantity does not affect the type of
spacetime that emerges. Then, defining
$\lambda_{M}=-4\left(\beta\gamma\right)^{3}/\Lambda_{n}^{2}$, we find that:

  * •

If $0<\lambda<\lambda_{M}$ and $\Lambda_{n}>0$, there is exactly one horizon.
Therefore the function $f$, which is positive at infinity, will change sign
once and be negative at the origin.

  * •

If $0<\lambda<\lambda_{M}$ and $\Lambda_{n}<0$, there will be exactly two
horizons $r_{H-}<2\beta\gamma/\Lambda_{n}<r_{H+}$: $f$ will be positive both
below $r_{H-}$ and above $r_{H+}$.

  * •

If $\lambda<0$ or $\lambda>\lambda_{M}$, there will be a unique horizon if
$\lambda\Lambda_{n}>0$ (thus a similar behaviour as for the first case), and
no horizon otherwise.

It is worth investigating also the behaviour of the metric function as $r\to
0$, in the cases where $f$ is positive there. In that case, its form is given
by:

$f(r)=\eta_{0}+\frac{\gamma^{2}c\left(1-2n\right)}{\beta+2n\gamma
c\sqrt{\eta_{0}}}\,\eta_{0}r+\mathcal{O}\left(r^{2}\right),$ (4.36)

provided that $\eta_{0}$ satisfies

$4nc\eta_{0}^{3/2}+\frac{3\beta}{\gamma}\eta_{0}-\lambda=0\,.$ (4.37)

The above equation admits solutions in various cases, for example by setting
$nc\lambda>0$. One sees that a de Sitter core arises for the case $n=1/2$
(i.e. the Gauss-Bonnet choice): demanding that $f(0)=\eta_{0}=1$, which fixes
$\lambda=2c+3\beta/\gamma$ from (4.37), the metric function as $r\rightarrow
0$ reads:

$f(r)=1+\frac{\beta\gamma^{2}}{\beta+\gamma
c}r^{2}-\frac{\Lambda\gamma}{3\left(\beta+\gamma
c\right)}r^{3}+\mathcal{O}\left(r^{4}\right).$ (4.38)

Meanwhile, from (4.31), the cosmological constant at infinity takes the simple
form:

$\Lambda_{f}=\left(-\frac{\Lambda}{2c}\right)^{2/3},$ (4.39)

since in this case $\Lambda_{n}=\Lambda$. If a black hole were to respect both
these forms at $r\to 0$ and $r\to\infty$, it would belong to the second case
described above. This is possible if $\gamma$, $c$ and $\Lambda$ exist such
that $\beta\gamma<0$ (by symmetry), $\Lambda<0$ and $0<\lambda<\lambda_{M}$,
which now reads:

$0<2c+\frac{3\beta}{\gamma}<\frac{4\gamma^{4}c\beta}{\Lambda^{2}}.$ (4.40)

For example, the choice $\gamma<0$, $\gamma c\equiv-\delta$ and
$\beta=\sqrt{\delta}$, gives a middle term positive for any $\delta>9/4$,
while the RHS is positive, and choosing a sufficiently small
$\left\lvert\Lambda\right\rvert$ ensures the fulfilment of this constraint.
The corresponding spacetime is a regular black hole with two horizons and an
AdS behaviour at infinity.

Finally, let us discuss the profile of the scalar field, the first derivative
of which follows from (4.28) and is given by:

$\phi^{\prime}=\frac{1}{\sqrt{f}}\left(\frac{\beta\left(r\gamma+\sqrt{f}\right)-1}{\alpha}\right)^{\frac{1}{2n}}\underset{r\to\infty}{=}\left(\frac{\beta\left(\gamma+\sqrt{\Lambda_{f}}\right)}{\alpha}\right)^{\frac{1}{2n}}\frac{r^{\frac{1}{2n}-1}}{\sqrt{\Lambda_{f}}}+\mathcal{O}\left(r^{\frac{1}{2n}-2}\right)\,.$
(4.41)

Above, we have also derived the asymptotic form of $\phi^{\prime}$ at radial
infinity: it diverges for $0<n<1/2$, while it converges to a constant value if
$n=1/2$, and to zero otherwise. However, whenever
$\gamma=-\sqrt{\Lambda_{f}}$, the above behaviour is modified and becomes:

$\phi^{\prime}=-\frac{\left(-1/\alpha\right)^{1/2n}}{r\gamma}+\mathcal{O}\left(\frac{1}{r^{3}}\right)$
(4.42)

such that $-2X=f\phi^{\prime 2}$ converges for any $n$. Given (4.31), the case
$\gamma=-\sqrt{\Lambda_{f}}$ corresponds to a unique relation between the
theory coefficients, namely $\Lambda=2\gamma^{3}c$, with $\gamma<0$. This
choice leads to a simplification of the asymptotic development (4.30):
$\epsilon_{1}$ and $\epsilon_{0}$ both vanish, consistently with (4.32) and
(4.33), while $\Lambda_{f}$ recovers the simple expression (4.39) for any
value of $n$. Finally, for $n=1/2$, a regular black hole can still be
obtained, since the constraint (4.40) is verified for any $\delta\equiv-\gamma
c\in\left]9/4,\delta_{M}\right[$, where $2\delta_{M}^{3/2}-3\delta_{M}-1=0$.

##  5 Disformal transformations and solution generation techniques

It is well known that a disformal transformation $D$ depending on $X$ takes a
solution of Horndeski theory to a solution of beyond Horndeski theory (see for
example [63], [80], [81]). In a recent publication [64], such techniques were
used to construct a traversable regular wormhole solution (see also [82],
[83]). Let us re-visit this construction here with the benefit of our
simplified field equations (2.6)-(2.8). Following [64], by barred quantities
we will be noting the seed “known” Horndeski solution. As such we have
$\bar{\phi},\bar{h},\bar{f}$ and of course
$\bar{X}=-\frac{1}{2}\bar{f}\bar{\phi^{\prime 2}}$ for some specific set of
coupling functions $\\{\bar{G_{i}}\\}$ in Horndeski theory. Via some $D(X)$
function we go to the “unknown” image metric which is a solution in beyond
Horndeski theory {$G_{i},F_{4},F_{5}$}, and reads

$g_{\mu\nu}=\bar{g}_{\mu\nu}-D(\bar{X})\,\partial_{\mu}\phi\,\partial_{\nu}\phi\,.$

Given that $\phi$ is only a function of $r$, we have immediately that
$\bar{\phi}=\phi$, $\bar{h}=h$ whereas the only terms that do change for the
image solution are the following

$f=\frac{\bar{f}}{1+2D\bar{X}},\qquad X=\frac{\bar{X}}{1+2D\bar{X}}\,.$ (5.1)

As we vary the disformability function $D$, we span all values of $f$ for the
given same $h$ and $\phi$ while at the same time we change the theory
according to specific transformation rules. For the case of spherical
symmetry, these rules read

$\displaystyle G_{4}$ $\displaystyle=$
$\displaystyle\frac{\bar{G_{4}}}{(1+2\bar{X}D)^{1/2}}\,,\qquad
G_{5X}=\frac{\bar{G}_{5\bar{X}}(1+2\bar{X}D)^{5/2}}{1-2\bar{X}^{2}D_{\bar{X}}}\,,$
(5.2) $\displaystyle F_{4}$ $\displaystyle=$
$\displaystyle(\bar{G_{4}}-2\bar{X}\bar{G}_{4\bar{X}})\frac{D_{\bar{X}}(1+2\bar{X}D)^{5/2}}{2(1-2\bar{X}^{2}D_{\bar{X}})}\,,$
(5.3) $\displaystyle F_{5}$ $\displaystyle=$
$\displaystyle\bar{X}\bar{G}_{5\bar{X}}\frac{D_{\bar{X}}(1+2\bar{X}D)^{7/2}}{6(1-2\bar{X}^{2}D_{\bar{X}})}\,.$
(5.4) $\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\frac{\bar{G_{2}}}{(1+2\bar{X}D)^{1/2}}\,,\qquad
G_{3X}=\bar{G}_{3\bar{X}}\frac{(1+2\bar{X}D)^{5/2}}{1-2\bar{X}^{2}D_{\bar{X}}}\,.$
(5.5)

In the above, we have used the relation

$D_{X}=D_{\bar{X}}\frac{(1+2\bar{X}D)^{2}}{1-2\bar{X}^{2}D_{\bar{X}}}\,,$
(5.6)

while the constraint (1.8) is verified.

Given our simplified general equations of motion (2.6)-(2.8), one can easily
find how our variables $Z,Y$, ${\cal{A}}$ and ${\cal{B}}$ transform under the
disformal transformation. We find

$\displaystyle Z$ $\displaystyle=$
$\displaystyle(1+2\bar{X}D)^{1/2}\bar{Z}\,,\quad
Y=(1+2\bar{X}D)^{1/2}\bar{Y}\,,\quad{\cal{B}}=(1+2\bar{X}D)^{1/2}\bar{{\cal{B}}}\,,$
(5.7) $\displaystyle{\cal{A}}$ $\displaystyle=$
$\displaystyle\frac{(1+2\bar{X}D)^{5/2}}{1-2\bar{X}^{2}D_{\bar{X}}}\bar{{\cal{A}}}+4\frac{(1+2\bar{X}D)^{3/2}}{1-2\bar{X}^{2}D_{\bar{X}}}(D+\bar{X}D_{\bar{X}})\,\bar{{\cal{B}}}\,.$
(5.8)

Hence, given a solution of $\\{\bar{G}_{i},\;i=2,3,4,5\\}$-Horndeski theory
with {$\bar{f},\bar{h},\bar{X}$}, we obtain a solution of beyond Horndeski
theory with $\\{G_{i},F_{4},F_{5},\;i=2,3,4,5\\}$ given by (5.2)-(5.5) for any
$D=D(X)$ with {$f,X$} given in (5.1) and $h=\bar{h}$. Using the above
transformation rules (5.7)-(5.8), it is straightforward to show that the field
equations remain invariant under an arbitrary disformal transformation. The
functional $D(X)$ determines the nature of the image solution. Let us examine
some simple non trivial examples.

As it was shown in [64], one can obtain a beyond-Horndeski wormhole solution
starting from a Horndeski black-hole solution. The seed solution found by Lu
and Pang [55] (see also [56], [57]) is described by the following Horndeski
theory functions

$G_{2}=8\alpha\bar{X}^{2},\;G_{3}=-8\alpha\bar{X},\;G_{4}=1+4\alpha\bar{X},\;G_{5}=-4\alpha\ln\bar{|X|}.$

The solution reads [55]

$\bar{h}(r)=\bar{f}(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha
M}{r^{3}}}\right),\quad{\rm
and}\quad\bar{\phi}^{\prime}=\frac{\sqrt{\bar{h}}-1}{r\sqrt{\bar{h}}},$ (5.9)

and describes a black hole with ADM mass $M$ and a non trivial scalar field,
with $\alpha$ being the constant coupling parameter of the theory. The
spacetime geometry is characterised by the roots of $\bar{h}$, located at
$r_{\pm}=M\pm\sqrt{M^{2}-\alpha}$, with the largest one being the event
horizon, $r_{h}=r_{+}$, whereby $\alpha\leq M^{2}$. Note that, strictly
speaking, the black hole inner horizon is ill defined as the scalar becomes
imaginary in the interior of the outer horizon151515This caveat can be
remedied by introducing linear time dependence [6] but we will stick to the
static case here for simplicity.

Indeed, under the disformal transformation (5.1), the metric functions and
scalar field become:

$h=\bar{h}\,,\qquad
f=\frac{\bar{h}}{1+2D(\bar{X})\bar{X}}\,,\qquad\phi=\bar{\phi}\,.$ (5.10)

The above functions comprise a solution to a beyond Horndeski theory, given by
(5.2-5.5) and parametrized by $D$. The new line-element of spacetime therefore
reads

$ds^{2}=-h(r)\,dt^{2}+\frac{[1+2D(\bar{X})\bar{X}]\,dr^{2}}{h(r)}+r^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\,.$
(5.11)

Taking into account the expression (5.9) for $\bar{\phi}^{\prime}$ in
Horndeski theory, the $\bar{X}$ function may be written as

$\bar{X}=-\frac{1}{2}\,\bar{h}\bar{\phi}^{\prime
2}=-\frac{1}{2}\,\frac{(\sqrt{\bar{h}}-1)^{2}}{r^{2}}\,,$ (5.12)

and hence the new metric function $f$ is defined from $r=r_{h}$ to radial
infinity. If we choose $D(\bar{X})=\beta\bar{X}^{\lambda}$ where $\lambda>-1$
then asymptotics are not spoiled while if, for example $\beta>0$, $f$ is
strictly positive till $r=r_{h}$. In this case therefore we have a black-hole
solution with $f\neq h$ in beyond Horndeski theory. Alternatively choosing $D$
so as to have an additional zero for $f$ at $r=r_{0}$ with $r_{0}>r_{h}$ one
can construct a wormhole solution.

If we define for simplicity the quantity $W=1+2D\bar{X}$, then a disformal
transformation of the form

$W^{-1}=1+\frac{1}{\lambda^{2}}\sum_{i=1}^{m}c_{i}\left(r_{0}\sqrt{-2(2n-1)\bar{X}}\right)^{i},$
(5.13)

where $(\lambda,\,r_{0},\,n,\,c_{i})$ are constants, may transform a Horndeski
black hole to a beyond Horndeski wormhole. The detailed process was presented
in [64] where the above transformation with $n=1$, $m=2$, $c_{1}=-2$ and
$c_{2}=1$ was applied to the Lu-Pang solution [55]. Let us now apply the above
transformation to the Horndeski black hole solution found in Section 4.1, i.e.
the solution described by Eq.(4.18). We consider the most simple case161616The
choice $c_{1}\neq 0$ may transform the Lu-Pang black hole ($n=1$) to a
wormhole but for $n>1$ adds a small contribution to the $1/r$ term at infinity
that spoils the ADM mass. where $m=2,\,c_{1}=0$, and $c_{2}=-1$. For this
choice, we find

$\bar{X}=\frac{1-\sqrt{(2n-1)h}}{r\sqrt{(2n-1)h}},\quad\text{and}\quad
W^{-1}=1-\frac{r_{0}^{2}(1-\sqrt{(2n-1)h})^{2}}{\lambda^{2}r^{2}}.$ (5.14)

For the existence of a wormhole, we demand that the function $W^{-1}$ has a
root at a value of the radial coordinate $r_{0}$ larger than $r_{h}$. The
equation $W^{-1}=0|_{r=r_{0}}$ then leads to the following condition

$h(r_{0})=\frac{(1\pm\lambda)^{2}}{2n-1}.$ (5.15)

If we demand that $\lambda$ is positive, we must choose the $(-)$ sign in the
above equation since, for $r>r_{h}$, $0<h<1$ and therefore
$1-\sqrt{2n-1}<\lambda<1$. For $\lambda=1$, we find $h(r_{0})=0$ which means
that the throat coincides with the horizon of the black hole ($r_{0}=r_{h}$)
while for $\lambda=1-\sqrt{2n-1}$ we have $h(r_{0})=1$ or equivalently
$r_{0}\rightarrow\infty$, and the throat radius is pushed to infinity. The
throat radius of the wormhole satisfies the following equation

$\displaystyle(n+1)\,(2n-1)^{n}\,r_{0}^{2n-1}\left[\,M(4n-2)+r_{0}((\lambda-2)\lambda-2n+2)\right]$
$\displaystyle+\alpha\lambda^{2n}\,\left[\,\lambda\,(\lambda-2)-2n\,(\lambda-1)+2\right]=0.$
(5.16)

While the above equation cannot be solved analytically for $n>1$, as was also
the case with (4.18), we can find an approximate solution when
$\alpha\rightarrow 0$. From Eq. (4.18), it is easy to verify that in that
limit the metric function $h$ acquires a Schwarzschild form

$h(r)=1-\frac{2M}{r}+\mathcal{O}(\alpha)\,,$ (5.17)

while, from $f=h/W$ and (5.14), we find

$f(r)=\left(1-\frac{2M}{r}\right)\left[1-\frac{r_{0}^{2}}{\lambda^{2}r^{2}}\left(1-\sqrt{(2n-1)\left(1-\frac{2M}{r}\right)}\right)^{2}\,\right]+\mathcal{O}(\alpha).$
(5.18)

From the condition $f(r_{0})=0$, and excluding the root that corresponds to
the horizon ($r_{0}=2M$) and the negative root, we find that the throat radius
has the following simple form:

$r_{0}=\frac{(4n-2)M}{2(n-1)+\lambda(2-\lambda)}+\mathcal{O}(\alpha).$ (5.19)

It thus depends on the mass $M$ and scale parameter $\lambda$, and receives
contributions from the coupling constant $\alpha$ of the theory.

The constructed wormhole solution is a solution to beyond Horndeski theory.
The exact form of the coupling functions readily follow by employing the
Horndeski functions $G_{i}$ presented in section 4.1 and the transformation
rules (5.2)-(5.5) in conjunction with the expression of the quantity
$1+2D\bar{X}$ given in (5.13).

##  6 Conclusions

Horndeski and beyond Horndeski theories are generalised theories of gravity
which incorporate standard GR but, at the same time, contain an additional
scalar degree of freedom. As a result, they provide an excellent framework for
the study of new solutions describing a plethora of compact objects, i.e.
black holes, wormholes or regular solutions, with a non-trivial scalar field.
The action functional of beyond Horndeski theory is characterised by six
arbitrary coupling functions between the scalar field and gravitational
quantities, and leads to a set of field equations of increased complexity.
Solving analytically these equations is not an easy task with a relatively
small number of such solutions having been found so far, especially in the
case of flat asymptotics. In the present work, we have considered a general
class of scalar-tensor (beyond Horndeski) theories, and formulated different
techniques for deriving analytic solutions describing static, compact objects
of spherical symmetry.

To this end, we have introduced four auxiliary functions ($Z$, $Y$,
${\cal{A}}$, ${\cal{B}}$) in terms of which the field equations for the metric
functions and scalar field assume a particularly simple form. In the case of
shift symmetric, parity preserving theories, parametrised by three coupling
functions, ($G_{2}$, $G_{4}$, $F_{4}$), we have demonstrated that the set of
field equations is integrable and may, upon appropriate choices, lead to a
variety of physically interesting, explicit black-hole solutions. To this end,
we have presented several classes of black holes with an asymptotically-flat,
Reissner-Nordstrom behaviour, either homogeneous or non-homogeneous.
Interestingly, all of these solutions arise in the context of beyond Horndeski
theories containing a canonical kinetic term for the scalar field.

We subsequently turned to the case of non-parity preserving theories and
assumed the presence of all six coupling functions, ($G_{2}$, $G_{3}$,
$G_{4}$, $G_{5}$, $F_{4}$, $F_{5}$). Although integrability seems to be lost
in this case, we developed a filtering technique for a sub-class of theories
which allows us to solve the set of field equations upon choosing the form of
only one coupling function, $G_{4}$, and an auxiliary quantity $Q$. The
technique was based on the quest for homogeneous solutions but allows also for
the emergence of non-homogeneous black holes. To simplify the analysis, we
focused on the former case with $f=h$, and presented two indicative sets of
analytic black-hole solutions. The first emerges in the context of a theory
which is related to the Kaluza-Klein reduction of Lovelock theory, and is a
generalisation of the asymptotically-flat solution [55] to other Horndeski and
beyond Horndeski theories. In this case, the spacetime may in fact admit one,
two or no horizons depending on the parameters of the solution, with a regular
behaviour of the metric function arising at the origin in the last two cases.
The second set of black-hole solutions presented here leads instead to an AdS
asymptotic behaviour but exhibits a similar horizon structure as the previous
solution; in the cases with two or no horizons, a de Sitter regular core was
interestingly found at the origin.

A different technique was employed in the last part of our work in order to
construct non-homogeneous black holes and also a different type of
gravitational solutions, namely a wormhole. For the construction of wormhole
solutions, the method of disformal transformation was applied to a seed black-
hole solution with the disformability function chosen so that the metric
function $f$ has a root at a value of the radial coordinate larger than the
black-hole horizon. In a previous work [64], such a disformal transformation
was applied to a Horndeski solution, i.e. the Lu-Pang black-hole solution
[55], and a wormhole was constructed with attractive characteristics, such as
a regular scalar field and no need for exotic matter. Here, to illustrate the
generality of the construction, we have proposed an alternative form for the
disformability function and applied it to the asymptotically-flat families of
black-hole solutions found in Section 4. This action leads again to a wormhole
solution with a non-trivial, regular scalar field and a throat radius
depending on the mass of the seed solution and coupling parameters of the
theory.

Our analysis demonstrates that the beyond Horndeski theory leads to a variety
of explicit solutions describing compact objects, black holes and wormholes.
These solutions may be derived either by direct integration methods as the
general ones we have exposed in sections 3 and 4, or by construction
techniques such as the application of a disformal transformation to a
previously known, seed solution. Interestingly, construction techniques of the
Kerr-Schild type have been shown to lead to non trivial regular black holes
but only for the case of DHOST theories [74], [84]. Hence it is not clear if
such vacuum solutions exist in beyond Horndeski theories. Our analysis has
been general but not exhaustive. For example, although our analysis in section
4 makes some progress in the direction of non-homogeneous black holes without
parity symmetry, no explicit solutions were found and it would be interesting
to look further into this question as parity breaking theories seem to be
particularly interesting departures from GR. As such it would be interesting
in particular to find solutions within theories where homogeneous solutions
are already known (as depicted in section 4) as uniqueness theorems such as
Birkhoff’s theorem are not valid in scalar tensor theories and solutions of
spherical symmetry, within the same theory, may be in competition in a sense
similar to scalarisation for example.

Another interesting point our analysis has outlined is the existence or not of
admissible wormhole solutions in parity symmetric theories. Our general
analysis in section 3 shows that the shape function responsible for the
presence of a throat, is independent of a (mass) integration constant. This is
quite opposite to what happens for parity breaking theories [64] where it was
shown that the throat disappears whenever mass is set to zero. This seems like
a good feature of a wormhole, similar to what happens for most black hole
solutions. It is intriguing that for parity symmetric theories, the throat
would be independent of an integration constant appearing in the solution and
in a certain sense such a throat would be an eternal throat. It would be
present for a given theory at a certain (theory dependent) size whatever the
mass of the solution. The admissibility of such a throat deserves maybe
further study.

Last but not least, we have considered theories with shift symmetry. This is
clearly a mathematically, rather than physically, motivated assumption.
Generically there is no reason that such a symmetry would exist in scalar
tensor theories and recent considerations, of Kaluza-Klein reduction from
Lovelock theory [21] have shown particularly interesting theories which do not
have shift symmetry and contain interesting black hole solutions [58], [59].
It would be interesting to study spherically symmetric solutions of such or
neighbouring theories with lesser symmetry or none at all.

## Acknowledgements

We are very happy to thank Eugeny Babichev and Karim Noui for encouraging and
useful discussions throughout the course of this work. The work of N.L. is
supported by the doctoral program CDSN ENS Lyon. A.B., P.K. and C.C. happily
acknowledge networking support by the GWverse COST Action CA16104, “Black
holes, gravitational waves and fundamental physics.” C.C. in particular thanks
the Department of Physics in the University of Ioannina for hospitality during
the course of this work.

## APPENDIX

##  Appendix A Case of constant kinetic term

If we assume that $h=f$, the right-hand-side of (2.6) vanishes, and we should
have either $X^{\prime}=0$ or ${\cal{A}}=0$. Here, we investigate the first
case, i.e. $X^{\prime}=0$ or equivalently $X=const.$. This easily leads to
$\phi^{\prime}=d/\sqrt{f}$, where $d$ is a constant, which upon integration,
defines the solution for the scalar field. The remaining two equations (2.7)
and (2.8) should provide the same solution for the metric function $f(r)$,
thus they should be of the same form for reasons of consistency. Using the
solution for $\phi^{\prime}$, these take the form

$\displaystyle
f^{\prime}\,\left[2rZ_{X}+\frac{d}{2\sqrt{f}}\,(r^{2}G_{3X}+G_{5X})+\sqrt{f}Y_{X}\right]$
$\displaystyle=G_{2X}r^{2}+2G_{4X}-2rd\sqrt{f}G_{3X}-2fZ_{X},$ (A.1)
$\displaystyle f^{\prime}\,(2rZ+2\sqrt{f}Y)$
$\displaystyle=-G_{2}r^{2}-2G_{4}-2fZ.$ (A.2)

Since for $X=const$, all $G_{i}$ functions, as well as $Z$ and $Y$, are also
constant, the comparison of the above two equations leads to the constraints

$Z_{X}=\lambda Z\,,\quad Y_{X}=2\lambda Y\,,\quad r^{2}G_{3X}+G_{5X}=0\,,$
(A.3) $G_{2X}=-\lambda G_{2}\,,\quad G_{4X}=-\lambda G_{4}\,,\quad
G_{3X}=0\,.$ (A.4)

The above indeed leads to the trivialisation of the odd functions $G_{3}$ and
$G_{5}$. From the ghost constraint (2.32), we then obtain that

$F_{5}=0\,,\quad{\rm or}\quad G_{4}=2XG_{4X}\,.$ (A.5)

The second choice is incompatible with the constraint on $G_{4}$ derived
earlier and thus is discarded. The first choice removes all odd functions from
the theory and leads to $Y=0$. Then, integrating (A.2), we obtain a
Schwarzschild-(A)dS solution of the form

$f(r)=\frac{G_{4}}{Z}-\frac{2M}{r}-\frac{G_{2}}{6Z}\,r^{2}\,.$ (A.6)

In order to restore asymptotic flatness, in the absence of the cosmological
constant term, we need to assume that $G4=Z$ which however is also
incompatible with the constraints derived earlier unless $X$ itself is
trivial.

##  Appendix B Additional Classes of Parity-Symmetric Solutions

The solution (3.10) derived in section 3.1 in the context of the parity-
symmetric sector of beyond Horndeski theory is only one characteristic example
of a broad class of black-hole solutions which may be found in this theory.
Insisting on having a canonical kinetic term for the scalar field in the
theory, one may consider the following forms of the parity-symmetric coupling
functions $G_{2}$ and $G_{4}$

$\displaystyle G_{2}$ $\displaystyle=-2\Lambda-\alpha X+\delta X^{m},$ (B.1)
$\displaystyle G_{4}$ $\displaystyle=\zeta+\beta X^{n},$ (B.2)

where the power coefficients $m$ and $n$ can be in general different from 2.
We will assume again that $Z=\gamma$, and therefore look for black-hole
solutions with $f=h$. Knowing $Z$ and $G_{4}$ allows us to determine the
remaining parity-symmetric function of the theory, namely

$F_{4}=\frac{\beta(1-2n)X^{n}+\zeta+\gamma}{4X^{2}}.$ (B.3)

A large class of physically interesting black-hole solutions may be
analytically determined in the case where $n=m$. Then, (3.2) yields the form

$\phi^{\prime 2}=\frac{1}{h(r)}\left|\frac{\alpha
2^{n-1}r^{2}}{n\,(2\beta+\delta r^{2})}\right|^{\frac{1}{n-1}},$ (B.4)

for the first derivative of the scalar field. For arbitrary integer, positive,
values of $n>1$, (3.3) leads in turn to the solution for the metric function

$\displaystyle h(r)$ $\displaystyle=-\frac{\zeta}{\gamma}+\frac{\Lambda
r^{2}}{3\gamma}+X(r)\left(\frac{2\beta}{\delta}+r^{2}\right)\frac{\alpha(n-1)}{6n\gamma}+\frac{\lambda}{2\gamma
r}$ $\displaystyle-\frac{(n-1)\alpha\beta}{3n\delta\gamma}\left(\frac{\alpha
r^{2}}{2n\beta}\right)^{1/n-1}{\rm{}_{2}F_{1}}\left[\frac{1}{n-1},\frac{n+1}{2(n-1)};\frac{3n-1}{2(n-1)};-\frac{\delta
r^{2}}{2\beta}\right],$ (B.5)

where $\lambda$ is an integration constant and with ${\rm{}_{2}F_{1}}$ we
denote the Hypergeometric function ${\rm{}_{2}F_{1}}(a,b;c;r)$. For the
special case of $n=m=1$, the solution for $h(r)$ may be written as a
polynomial and reduces to the Schwarzschild-(A)dS solution for $\zeta=-\gamma$
with a trivial scalar field. For $n=m=2$, (B.5) is expressed in terms of the
$\arctan$ function, as also did the solution of section 3.1 where quadratic
expressions were similarly assumed for $G_{2}$ and $G_{4}$. For $n=m=3$, the
metric function is written in terms of radicals whereas, for $n=m>3$, the
hypergeometric function cannot be written in terms of elementary functions.

Nevertheless, for all values of $n=m>1$, the solutions exhibit the same
behaviour at asymptotic infinity. There, the solution (B.5) reduces to the
form

$h(r)\simeq
1-\frac{\Lambda_{eff}\,r^{2}}{3}-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\cdots,$
(B.6)

thus describing an (A)dS-Reissner-Nordstrom solution under the identifications

$\beta=-\frac{n\delta(\zeta+\gamma)}{\alpha}\left(\frac{n\delta}{\alpha}\right)^{1/n-1},$
(B.7)
$M=-\frac{\lambda}{4\gamma}-\frac{\alpha\beta(n-1)}{3n\delta\gamma}\left(\frac{\alpha}{n\delta}\right)^{1/n-1}\sqrt{\frac{2\pi\beta}{\delta}}\,\frac{\Gamma\left[\frac{3n-1}{2(n-1)}\right]}{\Gamma\left[\frac{1}{n-1}\right]}\,.$
(B.8)

The first relation ensures the asymptotic flatness of spacetime, in the
absence of a cosmological constant, while the second determines the mass of
the black hole in terms of the integration constant $C$ as well as the
remaining parameters of the theory. The effective cosmological constant turns
out to be given by the combination

$\Lambda_{eff}\equiv-\frac{\Lambda}{\gamma}-\frac{\alpha(n-1)}{2n\gamma}\left(\frac{\alpha}{n\delta}\right)^{1/n-1},$
(B.9)

and is sourced by both the bare term $\Lambda$ and the canonical kinetic term
of the scalar field appearing in the expression of $G_{2}$. Therefore, a
(beyond) Horndeski theory with a canonical kinetic term for the scalar field
generically leads to a non-asymptotic flat spacetime. Asymptotically-flat
black-hole solutions may of course arise if the parameters of the theory are
chosen so that the right-hand-side of the above equation vanishes. Finally,
the solution is “dressed” with a tidal charge given by

$Q^{2}\equiv-\frac{\alpha\beta^{2}}{\gamma\delta^{2}(n-1)}\left(\frac{\alpha}{n\delta}\right)^{1/n-1},$
(B.10)

which is also an inherent feature of our solutions for generic values of the
parameters of the theory.

Black-hole solutions with either an exact or an asymptotic (A)dS-Reissner-
Nordstrom behaviour arise also in the case where $m\neq n$ in (B.1)-(B.2). For
instance, if we keep $m=2$ as above but instead set $n=1$, so that $G_{4}$ is
linear in $X$, (3.2) gives the result

$\phi^{\prime 2}=\frac{2\beta-r^{2}\alpha}{\delta r^{2}h(r)},$ (B.11)

while (3.3) leads to the following solution for the metric function

$h=1-\frac{2M}{r}+\frac{2\delta(\gamma+\zeta)^{2}}{\gamma\alpha
r^{2}}+\Bigl{(}\Lambda+\frac{\alpha^{2}}{8\delta}\Bigr{)}\,\frac{r^{2}}{3\gamma}.$
(B.12)

The above is an exact (A)dS-Reissner-Nordstrom solution where we have made the
choice $\beta=-2\delta(\gamma+\zeta)/\alpha$. It is a solution of the beyond
Horndeski theory since $F_{4}$ is still given by (B.3) with $n=1$. For $m=2$
and $n=3$, we also obtain a black-hole solution with a non-trivial $\phi$ and
a metric function expressed in terms of polynomials and radicals; however, for
brevity, we refrain from showing here its explicit from as its asymptotic
(A)dS-Reissner-Nordstrom behaviour is similar to the one presented in (B.6).

A particularly simple configuration arises if we set $\delta=0$ in (B.1).
Then, the coupling function $G_{2}$ is given only in terms of the bare
cosmological constant $\Lambda$ and a canonical kinetic term. The profile of
the scalar field is still given by (B.4) with $\delta=0$. It is easy to see
that $\phi^{\prime}$ remains regular everywhere only for $n<2$. As noted
earlier, the case $n=1$ leads to the pure Schwarzschild-(A)dS solution, and
the same holds for the case with $n=0$. The case with $n=1/2$ does lead to a
black-hole solution with a non-trivial scalar field described by the equation

$\phi^{\prime 2}=\frac{2\beta^{2}}{\alpha^{2}r^{4}h}\,,$ (B.13)

and a metric function given by the expression

$h=-\frac{\zeta}{\gamma}-\frac{2M}{r}+\frac{\Lambda
r^{2}}{3\gamma}+\frac{\beta^{2}}{2\alpha\gamma}\,\frac{1}{r^{2}}.$ (B.14)

Upon choosing $\zeta=-\gamma$, asymptotic flatness is restored and the
solution describes again an (A)dS-Reissner-Nordstrom spacetime. However, this
is a solution to Horndeski theory since the choices $n=1/2$ and
$\zeta=-\gamma$ trivialise the function $F_{4}$ according to (B.3). This
solution has been studied in detail in [73]. In order to keep $F_{4}$ in the
theory, though, we choose instead $n=1/3$. Then, (3.2) gives

$\phi^{\prime
2}=\left(\frac{2}{3}\,\frac{\beta}{\alpha}\right)^{3/2}\,\frac{1}{r^{3}h}\,,$
(B.15)

and leads to a vanishing $\phi^{\prime}$ at infinity. The last equation,
(3.3), yields for the metric function the exact expression

$h=1-\frac{2M}{r}+\frac{\Lambda
r^{2}}{3\gamma}-\frac{2\beta}{3\gamma}\,\sqrt{\frac{\beta}{\alpha}}\,\frac{\ln
r}{r},$ (B.16)

where we have again set $\zeta=-\gamma$. The above expression describes a
spacetime which is asymptotically (A)dS with the cosmological constant
coinciding with the bare one $\Lambda$. The Schwarzschild term proportional to
the mass $M$ of the black hole is now supplemented by an unusual logarithmic
term which although vanishes at infinity it does so slower than the mass term.
This solution arises in the context of beyond Horndeski theory since our
choices leave a non-vanishing $F_{4}$ function, i.e.
$F_{4}=\beta/(12X^{5/3})$.

##  Appendix C Parity symmetry breaking with a beyond Horndeski term

We will now study a case which stands between parity-symmetric and parity-
breaking beyond Horndeski theories. To this end, we will keep the assumption
of a parity-symmetric Horndeski sector and thus set $G_{3}=G_{5}=0$. Then, the
no-ghost condition (1.8) leads to $0=F_{5}\left(G_{4}-2XG_{4X}\right)$. If one
sets $G_{4}(X)=\sqrt{-\beta X}$, where $\beta>0$, the no-ghost condition is
satisfied without taking $F_{5}=0$, that is without killing the non-parity
symmetric part of the beyond Horndeski sector. However, for simplicity, we may
consider $F_{4}=0$. Then, from the definitions (2.2)-(2.3), we readily find

$Z(X)=0\,,\qquad Y(X)=3\left(-2X\right)^{5/2}F_{5}\,,$ (C.1)

whereas the definitions (2)-(2.5) lead to

$\mathcal{A}=2\sqrt{f}Y_{X}\,,\qquad\mathcal{B}=\sqrt{f}Y\,.$ (C.2)

Then, the field equation (2.6) integrates to

$f=\frac{\gamma h}{Y}\,,$ (C.3)

where $\gamma$ is an integration constant. Here, we will look for homogeneous
black-hole solutions with $f=h$. Then, the aforementioned equation leads to
$Y=\gamma$, which completely determines $F_{5}$, namely
$F_{5}=\frac{\gamma}{3\left(-2X\right)^{5/2}}$. For $Y=\gamma$, we also obtain
$\mathcal{A}=0$ and $\mathcal{B}=\sqrt{f}\gamma$ from (C.2). Then, the
remaining equations of motion (2.7)-(2.8) become:

$\displaystyle 0=G_{2X}r^{2}-\frac{\beta}{\sqrt{-\beta X}}\,,$ (C.4)
$\displaystyle 2\gamma\sqrt{f}f^{\prime}=-G_{2}r^{2}-2\sqrt{-\beta X}\,.$
(C.5)

Setting the form of the last coupling function $G_{2}$ will fix the theory,
and provide the solution to the scalar field and metric function via
(C.4)-(C.5), respectively.

We will study in detail the case where $G_{2}(X)=\eta^{2}X+p\sqrt{-X}$, where
$(\eta,p)$ are constant coefficients. In this form of $G_{2}$, the canonical
kinetic term of the scalar field is supplemented by an additional $\sqrt{-X}$
term. Equation (C.4) now determines $X$

$X=-\frac{\left(2\sqrt{\beta}+pr^{2}\right)^{2}}{4\eta^{4}r^{4}}\,,$ (C.6)

and the scalar field itself through the definition $X=-f\phi^{\prime 2}/2$.
Using the forms of $G_{2}$ and $X$, the latter equation (C.5) takes the form

$2\gamma\sqrt{f}f^{\prime}+\frac{4\beta+p^{2}r^{4}+4\sqrt{\beta}pr^{2}}{4\eta^{2}r^{2}}=0\,.$
(C.7)

The above integrates to

$f(r)=\frac{1}{4}\left(\frac{12\beta-p^{2}r^{4}-12\sqrt{\beta}pr^{2}+24\gamma\eta^{2}\lambda
r}{2\gamma\eta^{2}r}\right)^{2/3}\,,$ (C.8)

where $\lambda$ is an integration constant. The expansion at infinity is

$\displaystyle
f(r)=\left(-\frac{p^{2}}{16\gamma\eta^{2}}\right)^{2/3}r^{2}+\frac{\sqrt{\beta}\left(-\frac{\sqrt{2}\,p^{2}}{\gamma\eta^{2}}\right)^{2/3}}{p}+\frac{2\lambda}{\left(-\frac{p^{2}}{2\gamma\eta^{2}}\right)^{1/3}r}-\frac{3\beta\left(-\frac{\sqrt{2}\,p^{2}}{\gamma\eta^{2}}\right)^{2/3}}{p^{2}r^{2}}+\mathcal{O}\left(\frac{1}{r^{3}}\right)$
(C.9)

which describes an (A)dS-Reissner-Nordstrom background for $p>0$ and
$\gamma<0$. Note that the ADM mass

$M=2\lambda\left(\frac{2|\gamma|\eta^{2}}{p^{2}}\right)^{1/3}$ (C.10)

depends on the integration constant $\lambda$, and is therefore a free
parameter of the solution, while the effective cosmological constant and tidal
charge are fully determined by the coupling parameters
($\beta,\eta,p,\gamma$).

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