# Topological black holes in higher derivative gravity

Alena Pravdová pravdova.math.cas.cz Institute of Mathematics of the Czech
Academy of Sciences,
Žitná 25, 115 67 Prague 1, Czech Republic  Vojtěch Pravda pravda.math.cas.cz
Institute of Mathematics of the Czech Academy of Sciences,
Žitná 25, 115 67 Prague 1, Czech Republic  Marcello Ortaggio
ortaggio(at)math(dot)cas(dot)cz Institute of Mathematics of the Czech Academy
of Sciences,
Žitná 25, 115 67 Prague 1, Czech Republic

###### Abstract

We study static black holes in quadratic gravity with planar and hyperbolic
symmetry and non-extremal horizons. We obtain a solution in terms of an
infinite power-series expansion around the horizon, which is characterized by
two independent integration constants – the black hole radius and the strength
of the Bach tensor at the horizon. While in Einstein’s gravity, such black
holes require a negative cosmological constant $\Lambda$, in quadratic gravity
they can exist for any sign of $\Lambda$ and also for $\Lambda=0$. Different
branches of Schwarzschild-Bach-(A)dS or purely Bachian black holes are
identified which admit distinct Einstein limits. Depending on the curvature of
the transverse space and the value of $\Lambda$, these Einstein limits result
in (A)dS-Schwarzschild spacetimes with a transverse space of arbitrary
curvature (such as black holes and naked singularities) or in Kundt metrics of
the (anti-)Nariai type (i.e., dS${}_{2}\times$S2, AdS${}_{2}\times$H2, and
flat spacetime). In the special case of toroidal black holes with $\Lambda=0$,
we also discuss how the Bach parameter needs to be fine-tuned to ensure that
the metric does not blow up near infinity and instead matches asymptotically a
Ricci-flat solution.

##  1 Introduction

Black holes can be regarded as the most fundamental objects in gravity,
serving as theoretical laboratories to study various aspects of gravitational
theories. In general relativity, Hawking’s theorem states that spacelike cross
sections of an event horizon in a stationary asymptotically flat spacetime are
topologically 2-spheres, assuming also that the dominant energy condition
holds [1, 2]. By relaxing some of the assumptions in Hawking’s theorem, one
can obtain more general horizon geometries. For instance, in a locally
asymptotically anti-de Sitter (AdS) spacetime (where the asymptotic flatness
and dominant energy condition are both violated), it is possible to construct
topological black holes for which the spacelike cross section of the event
horizon can be a compact Riemann surface of any genus $g$ [3, 4, 5, 6, 7, 8].

Stemming from the early results of [9, 10], recently there has been a great
interest in the study of static, spherically symmetric black holes in
quadratic gravity [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], where
corrections quadratic in the curvature are added to the Einstein-Hilbert
action

$S{=\int{\rm d}^{4}x\,\sqrt{-g}{\cal L}}=\int{\rm
d}^{4}x\,\sqrt{-g}\,\Big{(}\gamma\,(R-2\Lambda)+\beta\,R^{2}-\alpha\,C_{abcd}\,C^{abcd}\Big{)}\,,$
(1)

where ${\gamma=1/(16\pi G)}$, $G$ is the Newtonian constant (we will use units
such that $G=1=c$), $\Lambda$ is the cosmological constant, and $\alpha$,
$\beta$ are coupling constants of the theory.

It is well known [22, 23] that all Einstein spaces $R_{ab}=\Lambda g_{ab}$
automatically solve the vacuum field equations of quadratic gravity. Vacuum
black holes appearing in Einstein’s gravity are thus in a sense trivial
solutions to quadratic gravity. However, recently it has been shown [11, 12]
that, besides the standard Schwarzschild black hole, quadratic gravity also
admits another static, spherically symmetric black hole solution over and
above Schwarzschild. Extensions of such black holes with a non-vanishing
cosmological constant $\Lambda$ have been studied in [17, 20]. Although these
non-Schwarzschild (or “Schwarzschild-Bach”) black holes of [11, 12] and [17,
20] have nontrivial Ricci tensor, the Ricci scalar $R$ is vanishing or
constant, respectively. In fact, the Ricci scalar is constrained by the trace
no-hair theorem of [24, 12] which states that for static, spherically
symmetric black holes in quadratic gravity, the Ricci scalar is either zero
(in the asymptotically flat case) or constant (assuming that $R$ is
sufficiently quickly approaching a constant at infinity) throughout the
spacetime. Furthermore, for $R=$const, the field equations of quadratic
gravity considerably simplify to (assuming in (1) $\gamma\neq 0$)

$R_{ab}-\Lambda\,g_{ab}=4k\,B_{ab}\,,$ (2)

where $B_{ab}$ is the Bach tensor (5) and the constant $k$ is defined by (8).

The Schwarzschild-Bach black holes are thus clearly distinguished by a non-
vanishing Bach tensor. In fact, it has been shown that for these black holes,
the vanishing of the Bach tensor on the horizon guarantees the vanishing of
the Bach tensor throughout the spacetime [16, 19, 17, 20]. These black holes
are therefore characterized by two parameters, namely, the radius of the black
hole $\bar{r}_{0}$ and a Bach parameter $b$ (denoted as $\delta$ in some of
the literature) measuring a deviation from the Schwarzschild solution and
related to the value of the Bach invariant $B_{ab}B^{ab}$ on the horizon. An
exact solution in a closed form describing this black hole is unknown – in the
$\Lambda=0$ case, evidence for the existence of this black hole has been
provided in [11, 12] by taking the first few terms in the near-horizon
expansion and numerically integrating the solution out from the horizon to
some point outside the horizon, before the numerical solution diverges (the
fourth order equations of motion are numerically unstable). As it turns out,
to ensure the asymptotic flatness of the spacetime (such as to kill the
growing Yukawa modes), one needs to fine-tune the parameters $\bar{r}_{0}$ and
$b$ using numerical methods, thus effectively ending up with a one-parameter
family of solutions [11, 12, 13, 15, 18, 21]. Very recently, it has been shown
that for a given $\bar{r}_{0}$ there exist at least two values of $b$ giving
an asymptotically flat black hole [21]. However, introducing a non-zero
cosmological constant or non-spherical horizon topologies into the picture may
have significant consequences on the physics of these black holes. Various
results for the case $\Lambda\neq 0$ have been obtained in [25, 26, 17,
20].111More results, including several exact solutions, are available in
special quadratic theories such as conformal or pure $R^{2}$ gravity for
spherical [27, 28, 29, 30] and topological [31, 32, 33, 25, 34] black holes.
These special theories will not be considered in the present paper.

In this work, we will broaden the search for black holes in quadratic gravity.
In addition to static black holes with spherical symmetry, we will also
include hyperbolic and planar symmetry. The corresponding metric ansatz thus
reads

${\rm d}s^{2}=-h(\bar{r})\,{\rm d}t^{2}+\frac{{\rm
d}\bar{r}^{2}}{f(\bar{r})}+\bar{r}^{2}{\rm d}\omega^{2},\ \ \ \ {\rm
d}\omega^{2}=\frac{{\rm d}x^{2}+{\rm
d}y^{2}}{\left(1+\frac{\epsilon}{4}(x^{2}+y^{2})\right)^{2}},$ (3)

where the transverse geometry is $S^{2}$, $E^{2}$, and $H^{2}$ for $\epsilon=$
$+1$, $0$, and $-1$, respectively. As in Einstein’s gravity, one can use the
metric (3) with $\epsilon=$ $0$ or $-1$ to construct topological black holes,
for which the horizon is a flat torus ($g=1$) or a Riemann surface of genus
$g>1$, respectively (such compactifications are discussed in [3, 4, 5, 6, 7,
8]).

It is well known that in Einstein’s gravity, black holes with $\epsilon\leq 0$
require $\Lambda<0$. Here we will show that these constraints do not apply to
quadratic gravity and that such black holes can exist for any sign of
$\Lambda$ as well as for $\Lambda=0$.

We will take the conformal-to-Kundt approach, recently employed in [16, 19]
and [17, 20], to study static, spherically symmetric black holes in quadratic
gravity with vanishing and nonvanishing $\Lambda$, respectively. Accordingly,
the standard metric (3) is rewritten in a form conformal to the Kundt metric.
This greatly simplifies the field equations of quadratic gravity, at the price
of working in somewhat physically less transparent coordinates. Together with
the assumption $R=$const, motivated by the trace no-hair theorem, the
resulting simplification of the field equations will enable us to obtain
recurrent formulas for series coefficients of the metric functions in power-
series expansions, and thus to include also analytical results in the study of
these black holes.

In section 2, we present necessary background material, such as the field
equations of quadratic gravity following from the action (1), and the
conformal-to-Kundt approach to simplify them.

In section 3, the field equations are derived for the ansatz (3) reexpressed
in the Kundt coordinates. We then use a Frobenius-like approach to solve the
equations in the vicinity of a generic hypersurface of a constant radius. We
use infinite power-series expansions and the indicial equations to determine
the possible leading powers of solutions. In particular, some of these
solutions admit extremal or non-extremal horizons.

Section 4 then focuses on the study of solutions with non-extremal horizons,
which is the main focus of the present paper. The recurrent formulas for the
series coefficients are determined. Interestingly, depending on $\Lambda$ and
$\epsilon$, the Einstein limit of the black hole solutions constructed here
contains not only (A)dS-Schwarzschild black holes with a transverse space of
arbitrary curvature, but also naked singularities, Kundt metrics of the
(anti-)Nariai type – dS${}_{2}\times$S2, AdS${}_{2}\times$H2, and flat
spacetime. As in the spherical case [11, 12, 13, 15, 16, 17, 18, 19, 20, 21],
one might expect that, by fine-tuning the Bach parameter, a quadratic-gravity
black hole with $\Lambda=0$ will asymptote an (appropriate) Ricci-flat metric
near spatial infinity. We will give evidence to support this expectation by
fine-tuning a planar black-hole solution ($\epsilon=0$) with vanishing
$\Lambda$, using the polynomial expansion.222However, more solid evidence
would be to match the expansion in the vicinity of the horizon with the
asymptotic expansion in the form of logarithmic-exponential transseries (cf.
[15]). In contrast, fine-tuning is not necessary to obtain an asymptotically
Einstein spacetime in the case of nonvanishing $\Lambda$ within a certain
continuous range of parameters of the solution (see also sec. 5).

Concluding comments are given in section 5, also on some related results
obtained for the $\Lambda<0$ case in [25, 26].

##  2 Background

###  2.1 Quadratic gravity

The field equations following from action (1) read

$\gamma\left(R_{ab}-{\textstyle{\frac{1}{2}}}R\,g_{ab}+\Lambda\,g_{ab}\right)-4\alpha\,B_{ab}+2\beta\left(R_{ab}-\tfrac{1}{4}R\,g_{ab}+g_{ab}\,\Box-\nabla_{b}\nabla_{a}\right)R=0\,,$
(4)

where $B_{ab}$ is the Bach tensor

$B_{ab}\equiv\big{(}\nabla^{c}\nabla^{d}+{\textstyle{\frac{1}{2}}}R^{cd}\big{)}C_{acbd}\
,$ (5)

which is traceless, symmetric, conserved, and well-behaved under a conformal
transformation $g_{ab}=\Omega^{2}\tilde{g}_{ab}$:

$g^{ab}B_{ab}=0\,,\qquad B_{ab}=B_{ba}\,,\qquad\nabla^{b}B_{ab}=0\,,\qquad
B_{ab}=\Omega^{-2}\tilde{B}_{ab}\,.$ (6)

For four-dimensional (conformally) Einstein spacetimes, the Bach tensor
vanishes identically [27].

As discussed above, in this work we will restrict ourselves to solutions with
$R=$const.333This is not a restriction for Einstein-Weyl gravity, i.e., when
$\beta=0$, since it then clearly follows from (4). Then the trace of (4)
reduces to $\gamma(R-4\Lambda)=0$. Assuming $\gamma\neq 0$ (to keep the
Einstein-Hilbert term in the action (1)) we thus have

$R=4\Lambda,$ (7)

and the field equations (4) simplify to (2), where we have defined

$k\equiv\frac{\alpha}{\gamma+8\beta\Lambda}\qquad(\gamma+8\beta\Lambda\neq
0).$ (8)

For later purposes, let us note that solutions with vanishing Bach tensor
reduce to Einstein spacetimes (cf. (2)). The latter are of constant curvature
if, in addition, ${\mathcal{H}}^{\prime\prime}+2\epsilon=0$ (cf. (25)). We
excluded the specially fine-tuned case $\gamma+8\beta\Lambda=0$, for which the
field equations (with (7)) reduce to $\alpha B_{ab}=0$ (see, e.g., [35]), as
in conformal gravity – all static (spherical, hyperbolic, or planar) black
holes are already known in this case [27, 28, 29, 31]. If, in addition, also
$\alpha=0$, one has a special Einstein-$R^{2}$ gravity for which any metric
with $R=4\Lambda$ is a solution (see, e.g., [30, 32, 36, 37] for some
examples).

###  2.2 Conformal-to-Kundt ansatz

We are interested in static black-hole solutions with spherical, hyperbolic,
or planar symmetry. Instead of the standard Schwarzschild coordinates,
throughout the paper, we will mostly employ the conformal-to-Kundt form of the
metric introduced in [35, 16, 19]. This enables one to describe such
spacetimes in the form

${\rm d}s^{2}\equiv\Omega^{2}(r)\,{\rm d}s^{2}_{\hbox{\tiny
K}}=\Omega^{2}(r)\Big{[}\,{\rm d}\omega^{2}-2\,{\rm d}u\,{\rm
d}r+\mathcal{H}(r)\,{\rm d}u^{2}\,\Big{]}\,,$ (9)

for which the resulting field equations are considerably simpler.

The metric (9) admits a gauge freedom

$r\to\lambda\,r+\upsilon\,,\qquad u\to\lambda^{-1}\,u\,,$ (10)

where $\lambda\,,\upsilon$ are constants, i.e., it is invariant up to
rescaling $\mathcal{H}\to\lambda^{2}\mathcal{H}$.

When $\Omega^{\prime}\neq 0\neq\mathcal{H}$ one can also define the standard
Schwarzschild coordinates by [35]

$\bar{r}=\Omega(r)\,,\qquad t=u-\int\\!\frac{{\rm d}r}{\mathcal{H}(r)}\,,$
(11)

giving rise to the line-element (3) with

$h=-\Omega^{2}\,\mathcal{H},\quad
f=-\left(\frac{\Omega^{\prime}}{\Omega}\right)^{2}\mathcal{H},$ (12)

where a prime denotes differentiation with respect to $r$.

##  3 Field equations and classes of power-series solutions

###  3.1 Ricci, Weyl, and Bach tensors for the Kundt seed metric

The nontrivial Ricci tensor components and the Ricci scalar of the Kundt
background metric ${\rm d}s^{2}_{\hbox{\tiny K}}$ of (9) read (cf. [19] for
$\epsilon=1$ case)

$\displaystyle R_{ru}^{\hbox{\tiny K}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-\textstyle{\frac{1}{2}}\,{\cal
H}^{\prime\prime}\,,\qquad\quad\ R_{uu}^{\hbox{\tiny K}}=-{\cal
H}\,R_{ru}^{\hbox{\tiny K}}\,,$ (13) $\displaystyle R_{xx}^{\hbox{\tiny
K}}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!R_{yy}^{\hbox{\tiny
K}}=\epsilon g^{\hbox{\tiny K}}_{xx}\,,\qquad R^{\hbox{\tiny K}}={\cal
H}^{\prime\prime}+2\epsilon\,.$ (14)

The non-vanishing components of the Weyl and Bach tensors are, respectively,

$\displaystyle C_{ruru}^{\hbox{\tiny K}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{\textstyle-\frac{1}{6}R^{\hbox{\tiny
K}}}\,,\qquad\qquad\qquad\ \ \ C_{riuj}^{\hbox{\tiny
K}}={\textstyle\frac{1}{12}R^{\hbox{\tiny K}}\,g^{\hbox{\tiny K}}_{ij}}\,,$
(15) $\displaystyle C_{kilj}^{\hbox{\tiny K}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{\textstyle\frac{1}{6}R^{\hbox{\tiny
K}}\,(g^{\hbox{\tiny K}}_{kl}g^{\hbox{\tiny K}}_{ij}-g^{\hbox{\tiny
K}}_{kj}g^{\hbox{\tiny K}}_{il})}\,,\quad C_{uiuj}^{\hbox{\tiny K}}=-{\cal
H}\,C^{\hbox{\tiny K}}_{riuj}\,$ (16)

and

$\displaystyle B_{rr}^{\hbox{\tiny K}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{\textstyle-\frac{1}{6}\,{\cal
H}^{\prime\prime\prime\prime}}\,,\qquad B_{ru}^{\hbox{\tiny
K}}={\textstyle\frac{1}{12}\,\big{(}2\,{\cal H}{\cal
H}^{\prime\prime\prime\prime}+{\cal H}^{\prime}{\cal
H}^{\prime\prime\prime}-{\textstyle\frac{1}{2}}{{\cal
H}^{\prime\prime}}^{2}+2\epsilon^{2}\big{)}}\,,$ (17) $\displaystyle
B_{uu}^{\hbox{\tiny K}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-{\cal H}\,B_{ru}^{\hbox{\tiny K}}\,,\qquad
B_{xx}^{\hbox{\tiny K}}=B_{yy}^{\hbox{\tiny
K}}={\textstyle\frac{1}{12}}\,g^{\hbox{\tiny K}}_{xx}\,\big{(}{\cal H}{\cal
H}^{\prime\prime\prime\prime}+{\cal H}^{\prime}{\cal
H}^{\prime\prime\prime}-{\textstyle\frac{1}{2}}{{\cal
H}^{\prime\prime}}^{2}+2\epsilon^{2}\big{)}\,.$ (18)

###  3.2 Ricci and Bach tensors for the full metric

The nontrivial Ricci tensor components and the Ricci scalar for the full
metric (9) are

$\displaystyle R_{rr}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-2\Omega^{-2}\big{(}\Omega\Omega^{\prime\prime}-2{\Omega^{\prime}}^{2}\big{)}\,,\qquad
R_{ru}=-\textstyle{\frac{1}{2}}\Omega^{-2}\big{(}\Omega^{2}{\cal
H}\big{)}^{\prime\prime}\,,$ (19) $\displaystyle R_{uu}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!-{\cal
H}\,{R}_{ru}\,,\qquad\qquad\qquad\quad\ \
R_{xx}={R}_{yy}=\Omega^{-2}g_{xx}^{\hbox{\tiny K}}\,\big{[}\big{(}{\cal
H}\Omega\Omega^{\prime}\big{)}^{\prime}+\epsilon\Omega^{2}\big{]}\,,$ (20)
$\displaystyle R\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!6\Omega^{-3}\big{[}({\cal
H}\Omega^{\prime})^{\prime}+{\textstyle\frac{1}{6}}({\cal
H}^{\prime\prime}+2\epsilon)\Omega\big{]}\,.$ (21)

The Bach tensor of the full metric can be obtained by a rescaling (6)

$B_{ab}=\Omega^{-2}B_{ab}^{\hbox{\tiny K}}\,,$ (22)

while $C^{a}_{\phantom{a}bcd}=C^{{\hbox{\tiny K}}a}_{\phantom{Ka}bcd}$, as
well known.

The Ricci squared, Bach, and Weyl invariants read444Note that the field
equations (2) have been used to obtain (23).

$\displaystyle R_{ab}\,R^{ab}$
$\displaystyle=4\Lambda^{2}+16k^{2}\,B_{ab}B^{ab}\,,$ (23) $\displaystyle
B_{ab}\,B^{ab}$
$\displaystyle=\tfrac{1}{72}\,\Omega^{-8}\,\big{[}(\mathcal{B}_{1})^{2}+2(\mathcal{B}_{1}+\mathcal{B}_{2})^{2}\big{]}\,,$
(24) $\displaystyle C_{abcd}\,C^{abcd}$
$\displaystyle=\tfrac{1}{3}\,\Omega^{-4}\,\big{(}{\mathcal{H}}^{\prime\prime}+2\epsilon\big{)}^{2}\,,$
(25)

where the two independent components of the Bach tensor, $\mathcal{B}_{1}(r)$
and $\mathcal{B}_{2}(r)$, are

$\mathcal{B}_{1}\equiv{\mathcal{H}}{\mathcal{H}}^{\prime\prime\prime\prime}\,,\qquad\mathcal{B}_{2}\equiv{\mathcal{H}}^{\prime}{\mathcal{H}}^{\prime\prime\prime}-\tfrac{1}{2}{{\mathcal{H}}^{\prime\prime}}^{2}+2\epsilon^{2}\,.$
(26)

It is useful to note that
$B_{ab}=0\Leftrightarrow\mathcal{B}_{1}=0=\mathcal{B}_{2}$. It can be also
verified easily that
$C_{abcd}=0\Leftrightarrow{\mathcal{H}}^{\prime\prime}+2\epsilon=0$.

###  3.3 Derivation and simplification of the field equations

Following [19, 20], in this section, we show that the field equations (2) for
the full metric (9) reduce to two coupled autonomous nonlinear differential
equations.

The nontrivial components $rr$, $ru$, and $xx$ of the field equations (2) read

$\displaystyle\Omega\Omega^{\prime\prime}-2{\Omega^{\prime}}^{2}$
$\displaystyle=\tfrac{1}{3}k\,{\mathcal{H}}^{\prime\prime\prime\prime}\,,$
(27)
$\displaystyle\big{(}\Omega^{2}{\mathcal{H}}\big{)}^{\prime\prime}-2\Lambda\Omega^{4}$
$\displaystyle=-\tfrac{2}{3}k\big{(}2\,{\mathcal{H}}{\mathcal{H}}^{\prime\prime\prime\prime}+{\mathcal{H}}^{\prime}{\mathcal{H}}^{\prime\prime\prime}-{\textstyle\frac{1}{2}}{{\mathcal{H}}^{\prime\prime}}^{2}+2\epsilon^{2}\big{)}\,,$
(28)
$\displaystyle\big{(}{\mathcal{H}}\Omega\Omega^{\prime}\big{)}^{\prime}+\epsilon\Omega^{2}-\Lambda\Omega^{4}$
$\displaystyle=\tfrac{1}{3}k\,\big{(}{\mathcal{H}}{\mathcal{H}}^{\prime\prime\prime\prime}+{\mathcal{H}}^{\prime}{\mathcal{H}}^{\prime\prime\prime}-{\textstyle\frac{1}{2}}{{\mathcal{H}}^{\prime\prime}}^{2}+2\epsilon^{2}\big{)}\,.$
(29)

The $yy$ component of (2) is identical to the $xx$ component and the $uu$
component is a multiple of the $ru$ component.

The trace (7) of the field equations takes the form

$\mathcal{T}\equiv{\mathcal{H}}\Omega^{\prime\prime}+{\mathcal{H}}^{\prime}\Omega^{\prime}+{\textstyle\frac{1}{6}}({\mathcal{H}}^{\prime\prime}+2\epsilon)\Omega={\textstyle\frac{2}{3}\Lambda\,\Omega^{3}}\,.$
(30)

As in [19, 20], let us introduce a conserved ($\nabla^{b}J_{ab}\equiv 0$)
symmetric tensor $J_{ab}$

$J_{ab}\equiv
R_{ab}-\textstyle{\frac{1}{2}}R\,g_{ab}+\Lambda\,g_{ab}-4k\,B_{ab}\,.$ (31)

The non-trivial components are $J_{rr}$, $J_{uu}=-{\mathcal{H}}\,J_{ru}$, and
$J_{xx}=J_{yy}$. The vacuum field equations (4), assuming $R=$const, then take
the form $J_{ab}=0$. When $\Omega^{\prime}\not=0$, one can show that once the
field equations $J_{rr}=0$ and $J_{ru}=0$ hold, then also $J_{xx}$ vanishes
(see Appendix C in [20] for a more detailed discussion in the $\epsilon=1$
case).555In this paper, we will not study non-Einstein spacetimes with
$\Omega=$const, which correspond to Kundt metrics.

The first field equation $J_{rr}=0$ reduces to (27). Substituting for
${{\mathcal{H}}^{\prime\prime\prime\prime}}$ from (27), the equation
$J_{ur}=0$ can be simplified and we arrive at the following final form of the
field equations

$\displaystyle\Omega\Omega^{\prime\prime}-2{\Omega^{\prime}}^{2}=$
$\displaystyle\ \tfrac{1}{3}k\,{\mathcal{H}}^{\prime\prime\prime\prime}\,,$
(32) $\displaystyle\Omega\Omega^{\prime}{\mathcal{H}}^{\prime}+3\Omega^{\prime
2}{\mathcal{H}}+\epsilon\Omega^{2}-\Lambda\Omega^{4}=$ $\displaystyle\
\tfrac{1}{3}k\big{(}{\mathcal{H}}^{\prime}{\mathcal{H}}^{\prime\prime\prime}-{\textstyle\frac{1}{2}}{{\mathcal{H}}^{\prime\prime}}^{2}+2\epsilon^{2}\big{)}\,.$
(33)

###  3.4 Classes of power-series solutions

Let us assume that the metric functions $\Omega(r)$ and $\mathcal{H}(r)$ in
(9) can be expanded as infinite power series in ${\Delta\equiv r-r_{0}}$
around a hypersurface $r=r_{0}$, i.e.,

$\Omega(r)=\Delta^{n}\sum_{i=0}^{\infty}a_{i}\,\Delta^{i}\,,\qquad\mathcal{H}(r)=\Delta^{p}\,\sum_{i=0}^{\infty}c_{i}\,\Delta^{i}.$
(34)

Substituting these expansions into the field equations (32) and (33) and
comparing the leading terms leads to constraints on possible values of $n$ and
$p$. In Table 1, we summarize the classes allowing for a vanishing $\Lambda$
or an arbitrary $\Lambda$. Note that there exist further classes allowing only
for certain discrete nonzero values of $\Lambda$ – these are not included in
the table and will be studied elsewhere. The case $\epsilon=+1$ has been
already analyzed in [19, 20] (including the discrete values of $\Lambda$).

Case | Class $[n,p]$ | $\Lambda$ | $\epsilon$
---|---|---|---
I | $[-1,2]$ | 0 | $\not=0$
| $[-1,3]$ | 0 | 0
| $[0,p>2]$ | 0 | 0
II | $[0,0]$ | any | any
| $[0,1]$ | any | any
| $[1,0]$ | any | any
III | $[-1,0]$ | any | any
| $[0,2]$ | any | $\not=0$
Table 1: Values of $[n,p]$ compatible with indicial equations following from
(32)–(34). Certain further cases allowing only for discrete nonzero values of
$\Lambda$ are not included in the table. See [16, 17, 19, 20] for details in
the case $\epsilon=+1$.

In the rest of the paper, we will study the $[0,1]$ case, which corresponds to
spacetimes admitting a non-extremal Killing horizon. For certain ranges of
parameters, this can be interpreted as a black hole horizon. The remaining
cases (as well as additional classes obtained using asymptotic expansions in
negative powers of $r$) will be studied elsewhere.

##  4 Case $[0,1]$: black holes with a non-extremal horizon

###  4.1 Preliminaries

From now on, we focus on solutions for which $n=0$ and $p=1$ in (34). This
means that we are expanding the metric near a non-extremal Killing horizon,
located at $r=r_{0}$. Let us thus relabel

$r_{h}\equiv r_{0}.$ (35)

Because of the freedom (10), the particular value $r_{0}$ has no physical
meaning. However, in the physical coordinates (11), (3), the horizon radius is
given by

$\bar{r}_{h}\equiv\Omega(r_{h})=a_{0}>0,$ (36)

which is a dimensionful scale set by $a_{0}$ (which is effectively an
integration constant, see the following for more comments). Without loss of
generality, we have fixed the sign of $a_{0}$ using the invariance of (9)
under $\Omega\to-\Omega$.

Before discussing the metric on-shell, let us note that, when $a_{1}\neq 0$,
the leading order behaviour of the metric functions $h$ and $f$ in the
Schwarzschild coordinates (3) is given by

$h=-\frac{c_{0}a_{0}^{2}}{a_{1}}(\bar{r}-\bar{r}_{h})+{\cal
O}\left((\bar{r}-\bar{r}_{h})^{2}\right),\qquad
f=-\frac{c_{0}a_{1}}{a_{0}^{2}}(\bar{r}-\bar{r}_{h})+{\cal
O}\left((\bar{r}-\bar{r}_{h})^{2}\right).$ (37)

In order to have an outer black hole horizon, we thus need to take

$c_{0}a_{1}<0\,,$ (38)

which ensures that both $h$ and $f$ are positive in the exterior region
$\bar{r}>\bar{r}_{h}$ in the vicinity of $\bar{r}=\bar{r}_{h}$ (negative $h$
and $f$ would correspond, e.g., to an inner or a cosmological horizon). Note
that, when $a_{1}>0$,666This can be always achieved using the gauge
transformation (10). The special case $a_{1}=0$ leads to fractional steps in
the expansion of the metric functions $f(\bar{r})$ and $h(\bar{r})$ in the
physical coordinates (3), and will not be considered in the following (see
[19, 20] for the case $\epsilon=+1$). near and across the horizon $\bar{r}$ is
monotonically increasing with $r$, while for $a_{1}<0$, $\bar{r}$ is
monotonically decreasing with $r$. Therefore $\partial_{r}$ is outward/inward
according to the sign of $a_{1}$.

###  4.2 General solution

The lowest nontrivial order of the trace equation (30) gives

$\displaystyle a_{1}=\frac{a_{0}}{3c_{0}}\left[2\Lambda
a_{0}^{2}-(\epsilon+c_{1})\right]\,,$ (39)

and then the lowest nontrivial order of (33) implies

$c_{2}=\frac{1}{6kc_{0}}\left[2k(c_{1}^{2}-\epsilon^{2})+a_{0}^{2}(2\epsilon-
c_{1}-\Lambda a_{0}^{2})\right]\,.$ (40)

At any arbitrary higher order, one finds that the $[0,1]$ solution to eqs.
(32), (33) is given by the recurrent formulas

$\displaystyle c_{l+2}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{3}{k\,(l+3)(l+2)(l+1)l}\,\sum^{l}_{i=0}a_{i}\,a_{l+1-i}(l+1-i)(l-3i)\qquad\forall\
l\geq 1\,,$ (41) $\displaystyle a_{l}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{l^{2}c_{0}}\Bigg{[}\tfrac{2}{3}\Lambda\sum^{l-1}_{j=0}{a_{l-1-j}}\sum^{j}_{i=0}a_{i}\,a_{j-i}-\tfrac{1}{3}\,\epsilon
a_{l-1}-\sum^{l}_{i=1}c_{i}\,a_{l-i}\left[l(l-i)+\tfrac{1}{6}i(i+1)\right]\Bigg{]}\
\ \ \forall\ l\geq 2\,.$ (42)

The three parameters $a_{0}$, $c_{0}$, and $c_{1}$ remain arbitrary and can be
thought of as integration constants.

It is also useful to observe that the Bach and Weyl invariants (24), (25) at
${r=r_{h}}$ read

$\displaystyle B_{ab}\,B^{ab}(r_{h})$ $\displaystyle=$
$\displaystyle\left(\frac{\epsilon^{2}-c_{1}^{2}+3c_{0}c_{2}}{3a_{0}^{4}}\right)^{2}=\left(\frac{c_{1}-2\epsilon+\Lambda
a_{0}^{2}}{6ka_{0}^{2}}\right)^{2}=\frac{b^{2}}{4k^{2}a_{0}^{4}}\,,\ $ (43)
$\displaystyle C_{abcd}\,C^{abcd}(r_{h})$ $\displaystyle=$
$\displaystyle\frac{4}{3a_{0}^{4}}(\epsilon+c_{1})^{2}\,,$ (44)

where we have introduced a dimensionless Bach parameter $b$ by

$b\equiv\frac{1}{3}\left(c_{1}-2\epsilon+\Lambda a_{0}^{2}\right)\,,$ (45)

which measures the strength of the Bach tensor at the horizon (it is
proportional to $\mathcal{B}_{2}(r_{h})$, see (26)).

For definiteness, using eqs. (39), (40) and the recurrent relations (41),
(42), the first few coefficients expressed in terms of free parameters
$a_{0}$, $c_{0}$, and $b$ read

$\displaystyle a_{1}=-\frac{a_{0}}{c_{0}}\left[(\epsilon-\Lambda
a_{0}^{2})+b\right]\,,$ (46) $\displaystyle
a_{2}=+\frac{a_{0}}{c_{0}^{2}}\left[(\epsilon-\Lambda
a_{0}^{2})^{2}+b\left(2\epsilon+a_{0}^{2}\left(\frac{1}{8k}-\frac{7\Lambda}{3}\right)\right)+b^{2}\right]\,,$
(47) $\displaystyle c_{1}=2\epsilon-\Lambda a_{0}^{2}+3b\,,$ (48)
$\displaystyle
c_{2}=-\frac{1}{6kc_{0}}\left[2k\left(\epsilon^{2}-\left(3b-a_{0}^{2}\Lambda+2\epsilon\right)^{2}\right)+3ba_{0}^{2}\right]\,,$
(49) $\displaystyle c_{3}=\frac{a_{0}^{4}b(3-8k\Lambda)}{96k^{2}c_{0}^{2}}\,.$
(50)

To summarize, the above solution contains three integration constants $a_{0}$,
$c_{0}$, and $b$, along with the expansion radius $r_{0}=r_{h}$, the sign of
the curvature of the transverse space $\epsilon$, and the constants of the
theory $k$ and $\Lambda$. However, thanks to the gauge freedom (10), the
number of physical parameters boils down to two – essentially, the mass and
the Bach parameter.

We will show below in section 4.3 that if $b=0$, then $B_{ab}=0$ everywhere
(i.e., not just at the horizon), in which case the solution becomes Einstein
(cf. also section 2.1). The parameter $b$ thus measures how the solution
departs from a (topological) Schwarzschild–(A)dS black hole and plays the role
of a gravitational (Bachian) “hair” (see also section 4.4.1 below). Because of
(36), (38) and (46), at an outer black-hole horizon, it will be constrained by

$\epsilon-\Lambda a_{0}^{2}+b>0\,.$ (51)

From (12), using (46)–(48), the first two leading terms in the expansion of
the metric functions $h(\bar{r})$, $f(\bar{r})$ around any horizon
$\bar{r}_{h}=a_{0}$ read (assuming $a_{1}\neq 0$, i.e., $\epsilon-\Lambda
a_{0}^{2}+b\neq 0$)

$\displaystyle h=\frac{a_{0}c_{0}^{2}}{\epsilon-\Lambda
a_{0}^{2}+b}\bar{\Delta}-\frac{c_{0}^{2}}{(\epsilon-\Lambda
a_{0}^{2}+b)^{3}}\left[(\epsilon-\Lambda
a_{0}^{2})\epsilon+b\left(3\epsilon-\frac{7}{3}\Lambda
a_{0}^{2}+\frac{1}{8k}a_{0}^{2}\right)+2b^{2}\right]\bar{\Delta}^{2}+\ldots\,,$
(52) $\displaystyle f=\frac{\epsilon-\Lambda
a_{0}^{2}+b}{a_{0}}\bar{\Delta}-\frac{1}{a_{0}^{2}(\epsilon-\Lambda
a_{0}^{2}+b)}\left[(\epsilon-\Lambda
a_{0}^{2})\epsilon+b\left(3\epsilon-\Lambda
a_{0}^{2}-\frac{3}{8k}a_{0}^{2}\right)+2b^{2}\right]\bar{\Delta}^{2}+\ldots\,,$
(53)

where $\bar{\Delta}=\bar{r}-\bar{r}_{h}$. Note that in the gauge
$a_{1}=a_{0}^{2}$ (then $c_{0}=-(\epsilon-\Lambda a_{0}^{2}+b)/a_{0}$ from
(46)), $f$ and $h$ coincide at the leading order, and $f=h$ in the limit
$b\rightarrow 0$.

###  4.3 Identifying the background Einstein spacetimes ($b=0$)

Let us now discuss the subclass of solutions for which the Bach parameter $b$
vanishes. As we will show below, this consists of two families of Einstein’s
spacetimes, for which the Bach tensor is necessarily zero. Because of (51),
solutions with a non-extremal black hole horizon must now satisfy

$\epsilon-\Lambda a_{0}^{2}>0\,.$ (54)

In particular, for $\Lambda>0$, only spherical black holes ($\epsilon=+1$)
will be possible.

With $b=0$, the coefficients (41), (42) reduce to

$\displaystyle a_{i}=a_{0}\,\bigg{(}\\!-\frac{\epsilon-\Lambda
a_{0}^{2}}{c_{0}}\bigg{)}^{i}\qquad\forall\ i\geq 0\,,\quad$ (55)
$\displaystyle c_{1}=2\epsilon-\Lambda a_{0}^{2},\qquad
c_{2}=\frac{1}{3c_{0}}(\epsilon-\Lambda a_{0}^{2})(3\epsilon-\Lambda
a_{0}^{2})\,,\qquad c_{i}=0\quad\forall\ i\geq 3\,.$ (56)

Note, in particular, that the $c_{i}$ sequence is truncated as

$\mathcal{H}(r)=c_{0}(r-r_{h})+c_{1}(r-r_{h})^{2}+c_{2}(r-r_{h})^{3}\,,$ (57)

with (56). There appear two possibilities depending on whether
$\epsilon-\Lambda a_{0}^{2}=0$ or not.

####  4.3.1 Generic case $\epsilon-\Lambda a_{0}^{2}\not=0$: (A)dS-
Schwarzschild metric

Assuming $\epsilon-\Lambda a_{0}^{2}\neq 0$, the $a_{i}$ sequence is a
geometric series, giving rise to

$\Omega(r)=a_{0}\,\sum_{i=0}^{\infty}\,\Big{(}-(\epsilon-\Lambda
a_{0}^{2})\frac{\Delta}{c_{0}}\Big{)}^{i}=\frac{a_{0}\,c_{0}}{c_{0}+(\epsilon-\Lambda
a_{0}^{2})\Delta}=\frac{a_{0}\,c_{0}}{(\epsilon-\Lambda
a_{0}^{2})(r-r_{h})+c_{0}}\,.$ (58)

Employing the gauge freedom (10), we can set

$a_{0}=-\frac{1}{r_{h}}\,,\qquad c_{0}=r_{h}\epsilon-\frac{\Lambda}{r_{h}}$
(59)

(which also means $a_{1}=a_{0}^{2}$), reducing the metric functions to

${\bar{r}}=\Omega(r)=-\frac{1}{r}\,,\qquad\mathcal{H}(r)=\frac{\Lambda}{3}-\epsilon
r^{2}-\left(\frac{\Lambda}{3}-\epsilon
r_{h}^{2}\right)\frac{r^{3}}{r_{h}^{3}}\,.$ (60)

Using (3), (12), it is easy to see that this solution corresponds to the well-
known [38, 39] (A)dS-Schwarzschild solution with a transverse space of
arbitrary curvature, for which the only integration constant is usually
rewritten as $2m=\left(\frac{\Lambda}{3}-\epsilon
r_{h}^{2}\right)\frac{1}{r_{h}^{3}}$, and the metric functions in the physical
coordinates (3) take the form

$h({\bar{r}})=f({\bar{r}})^{-1}=\epsilon-\frac{2m}{{\bar{r}}}-\frac{\Lambda}{3}{\bar{r}}^{2}.$
(61)

In addition to the standard spherical Kottler metric (for $\epsilon=+1$, cf.,
e.g., [40, 41]), it describes topological black holes in Einstein’s gravity
[3, 4, 5, 6, 7, 8]. Here, the condition (54) implies, for example, the known
fact that black holes with $\epsilon\leq 0$ require $\Lambda<0$. When equality
holds (i.e., $\epsilon-\Lambda a_{0}^{2}=0$) one obtains the standard
extremality condition for Einstein black holes [40, 7, 8].777 Note indeed
that, although we have obtained the closed form (60) of the solution under the
assumption $\epsilon-\Lambda a_{0}^{2}\neq 0$, the metric defined by (60)
admits also the possibility $\epsilon-\Lambda a_{0}^{2}=0$, giving rise to a
spacetime with a degenerate horizon at $r=r_{h}$ when $\Lambda\neq 0$, and to
a flat spacetime when $\Lambda=0$. In the usual form of the solution expressed
in terms of $m$ (cf. above), one can further set $\Lambda=0=\epsilon$ while
keeping $m\neq 0$, which corresponds to a naked singularity. In the case
$\epsilon-\Lambda a_{0}^{2}<0$, the metric is time-dependent in the vicinity
of the horizon in the exterior region (cf. (52), (53) with $b=0$) so that the
hypersurface $\bar{r}=\bar{r}_{h}$ cannot be an outer black hole horizon
(nevertheless, the spacetime can contain an outer black-hole horizon located
elsewhere when $\epsilon\Lambda>0$, cf. [40, 7, 8] for details).

####  4.3.2 Special case $\epsilon-\Lambda a_{0}^{2}=0$: direct product
Einstein spacetimes

When $\epsilon-\Lambda a_{0}^{2}=0$ one simply obtains $a_{i}=0$ for all
$i>0$, $c_{1}=\epsilon$ and $c_{i}=0$ for all $i>1$, i.e.,

$\Omega=a_{0},\qquad\mathcal{H}=c_{0}(r-r_{h})+\epsilon(r-r_{h})^{2},$ (62)

which describes an Einstein spacetime of the Kundt class in the form of a
direct product metric of the (anti-)Nariai type (namely dS${}_{2}\times$S2,
flat space, or AdS${}_{2}\times$H2, depending on the value of $\epsilon$). The
horizon at $r=r_{h}$ is a Killing horizon, but not a black hole one (cf.
(54)). With a coordinate transformation (cf. [38]), one can always set
$c_{0}=0=r_{h}$. This class of metrics is related to the near-horizon geometry
of the extremal limits of the black holes of section 4.3.1 (cf., e.g., the
review [42] and references therein).

One might wonder why, although the field equations (32), (33) can be solved
exactly in the Einstein limit, we have not recovered here all Einstein
spacetimes of the form (9) (or (3)) satisfying $\epsilon-\Lambda a_{0}^{2}=0$.
For example, in the $\epsilon=0=\Lambda$ case, such a solution is the AIII
metric of [43] (see [38, 39] for further references and a physical
interpretation) for which $\Omega\propto 1/r$ and $\mathcal{H}\propto r^{3}$,
which represents a naked singularity located at $r\rightarrow\infty$ (cf. also
footnote 7). Such a type of solutions does not appear here because, in this
section, we have considered the Einstein limit only of the [0,1] class –
which, by construction, contains a Killing horizon – and thus horizonless
metrics belonging to other cases in Table 1 may not appear.

###  4.4 More general solutions: black holes with nonvanishing Bach tensor

####  4.4.1 Generic case $\epsilon-\Lambda a_{0}^{2}\neq 0$

As mentioned above, the general quadratic-gravity solution of section 4.2 is
non-Einstein when $b\neq 0$. In this section, we will study a subset of black
holes obeying (51) which admit the (A)dS-Schwarzschild metric (60) as a
$b\rightarrow 0$ limit (cf. section 4.3.1). See the end of this section for a
discussion of the Einstein limit for the distinct cases $\epsilon>\Lambda
a_{0}^{2}$ and $\epsilon<\Lambda a_{0}^{2}$.

In order to express the metric functions $\Omega(r)$ and $\mathcal{H}(r)$ as
the (A)dS-Schwarzschild background plus a quadratic-gravity correction (cf.
(67), (68)), we reparametrize the series coefficients $a_{i}$ and $c_{i}$ by
introducing coefficients $\alpha_{i},\gamma_{i}$ as in [35, 17, 16, 19]. Using
again the gauge (59), one obtains

$\displaystyle a_{i}\\!\\!\\!$ $\displaystyle\equiv$
$\displaystyle\\!\\!\\!\tilde{a}_{i}-\frac{b}{r_{h}}\,\frac{\alpha_{i}}{(-r_{h}\rho)^{i}}\,,\qquad\hbox{where}\quad\tilde{a}_{i}\equiv
a_{i}(b=0)=\frac{1}{(-r_{h})^{1+i}}\ \ \ i\geq 0\,,$ (63) $\displaystyle
c_{1}\\!\\!\\!$ $\displaystyle\equiv$
$\displaystyle\\!\\!\\!2\epsilon-\frac{\Lambda}{r_{h}^{2}}+3b\,\gamma_{1},\quad
c_{2}\equiv\frac{3\epsilon
r_{h}^{2}-\Lambda}{3r_{h}^{3}}+3b\,\frac{r_{h}\gamma_{2}}{\epsilon
r_{h}^{2}-\Lambda},\ \quad c_{i}\equiv
3b\,\frac{\gamma_{i}\,r_{h}^{i-1}}{(\epsilon r_{h}^{2}-\Lambda)^{i-1}}\ \
\quad i\geq 3,$ (64)

where

$\displaystyle{\rho}$
$\displaystyle\equiv\epsilon-\frac{\Lambda}{r_{h}^{2}}\,,$ (65)

and

$\displaystyle\alpha_{0}\equiv
0\,,\quad\alpha_{1}=1\,,\quad\gamma_{1}=1\,,\quad\gamma_{2}=\frac{1}{3}\Big{[}4\epsilon-\frac{1}{r_{h}^{2}}\Big{(}2\Lambda+\frac{1}{2k}\Big{)}+3b\Big{]}\,.$
(66)

The remaining coefficients $\alpha_{i}$ and $\gamma_{i}$ will be specified
shortly.

The functions ${\Omega}$ and ${\mathcal{H}}$ then read

$\displaystyle\Omega(r)$
$\displaystyle=-\frac{1}{r}-\frac{b}{r_{h}}\sum_{i=1}^{\infty}\alpha_{i}\Big{(}\,\frac{r_{h}-r}{{\rho}\,r_{h}}\Big{)}^{i}\,,$
(67) $\displaystyle\mathcal{H}(r)$
$\displaystyle=(r-r_{h})\bigg{[}\,\epsilon\frac{r^{2}}{r_{h}}-\frac{\Lambda}{3r_{h}^{3}}\left(r^{2}+rr_{h}+r_{h}^{2}\right)+3b\,{\rho}\,r_{h}\sum_{i=1}^{\infty}\gamma_{i}\Big{(}\,\frac{r-r_{h}}{{\rho}\,r_{h}}\Big{)}^{i}\,\bigg{]}\,,$
(68)

indeed explicitly expressing the Bachian part of the metric as a correction to
the (A)dS-Schwarzschild background, as desired.

Using (41) and (42), the coefficients $\alpha_{l},\gamma_{l+1}$ for ${l\geq
2}$ are given by the recurrent relations

$\displaystyle\alpha_{l}=\,\frac{1}{l^{2}}\Bigg{[}-\frac{2\Lambda}{3r_{h}^{2}}\,\sum_{j=0}^{l-1}\sum_{i=0}^{j}\left[\alpha_{l-1-j}{\rho}^{j}+\big{(}{\rho}^{l-1-j}+b\,\alpha_{l-1-j}\big{)}\big{(}\alpha_{i}{\rho}^{j-i}+\alpha_{j-i}({\rho}^{i}+b\,\alpha_{i})\big{)}\right]$
$\displaystyle\hskip
39.83385pt-\frac{1}{3}\alpha_{l-2}(2\epsilon+{\rho}){\rho}(l-1)^{2}+\alpha_{l-1}\left[\frac{\epsilon}{3}+({\epsilon}+{\rho})\big{(}l(l-1)+\frac{1}{3}\big{)}\right]$
$\displaystyle\hskip
39.83385pt-3\sum_{i=1}^{l}(-1)^{i}\,\gamma_{i}\,({\rho}^{l-i}+b\,\alpha_{l-i})\left[l(l-i)+\frac{1}{6}i(i+1)\right]\Bigg{]}\,,$
$\displaystyle\gamma_{l+1}=\,\frac{(-1)^{l}}{kr_{h}^{2}\,(l+2)(l+1)l(l-1)}\sum_{i=0}^{l-1}\big{[}\alpha_{i}{\rho}^{l-i}+\alpha_{l-i}\big{(}{\rho}^{i}+b\,\alpha_{i}\big{)}\big{]}(l-i)(l-1-3i)\,\quad\forall\quad
l\geq 2\,.$ (69)

Explicitly, the first few terms then read

$\displaystyle\alpha_{2}=2\epsilon-\left(\frac{7}{3}\Lambda-\frac{1}{8k}\right)\frac{1}{r_{h}^{2}}+b\,,$
$\displaystyle\alpha_{3}=\frac{1}{9}\left[25\epsilon^{2}+\left(\frac{29}{8k}-\frac{179}{3}\Lambda\right)\frac{\epsilon}{r_{h}^{2}}+\left(\frac{1}{16k^{2}}-\frac{77}{24k}\Lambda+\frac{298}{9}\Lambda^{2}\right)\frac{1}{r_{h}^{4}}\right]$
$\displaystyle\qquad+\frac{1}{9}\left[23\epsilon+\left(\frac{35}{8k}-\frac{104}{3}\Lambda\right)\frac{1}{r_{h}^{2}}\right]\,b+\frac{7}{9}\,b^{2}\,,\quad\ldots\,,$
(70)
$\displaystyle\gamma_{3}=\frac{1}{96k^{2}r_{h}^{4}}\left(1-\frac{8k}{3}\Lambda\right)\,,$
$\displaystyle\gamma_{4}=\frac{1}{18kr_{h}^{2}}\left[\frac{\epsilon^{2}}{5}+\left(-\frac{1}{4k}+\frac{4}{15}\Lambda\right)\frac{\epsilon}{r_{h}^{2}}-\frac{1}{160k^{2}r_{h}^{4}}-\frac{1}{45r_{h}^{4}}\left(14\Lambda^{2}-\frac{75}{8k}\Lambda\right)\right]$
$\displaystyle\qquad+\frac{1}{720kr_{h}^{2}}\left[16\epsilon+\left(-\frac{13}{k}+\frac{56}{3}\Lambda\right)\frac{1}{r_{h}^{2}}\right]\,b+\frac{1}{90kr_{h}^{2}}\,b^{2}\,,\quad\ldots\,,$
(71)

which leads to

$\displaystyle\Omega(r)$
$\displaystyle=-\frac{1}{r}+b\frac{(r-r_{h})}{{\rho}\,r_{h}^{2}}-\frac{b}{r_{h}}\left[2{\rho}+\frac{1}{24kr_{h}^{2}}(3-8k\Lambda)+b\right]\Big{(}\,\frac{r-r_{h}}{{\rho}\,r_{h}}\Big{)}^{2}+\cdots\,,$
(72) $\displaystyle\mathcal{H}(r)$
$\displaystyle=(r-r_{h})\bigg{\\{}\,\epsilon\frac{r^{2}}{r_{h}}-\frac{\Lambda}{3\,r_{h}^{3}}\left(r^{2}+rr_{h}+r_{h}^{2}\right)$
$\displaystyle\hskip
51.21495pt+3b\,{\rho}\,r_{h}\left[\Big{(}\,\frac{r-r_{h}}{{\rho}\,r_{h}}\Big{)}+\frac{1}{3}\Big{[}4\epsilon-\frac{1}{r_{h}^{2}}\Big{(}2\Lambda+\frac{1}{2k}\Big{)}+3b\Big{]}\Big{(}\,\frac{r-r_{h}}{{\rho}\,r_{h}}\Big{)}^{2}+\cdots\right]\,\bigg{\\}}\,.$
(73)

In the physical coordinates (3), using (12) and the gauge (59), the first two
leading terms in the expansion of the metric functions $h(\bar{r})$,
$f(\bar{r})$ around the horizon $\bar{r}_{h}=a_{0}$ are (cf. (52), (53))

$\displaystyle f\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\frac{\epsilon+b-\Lambda\bar{r}^{2}_{h}}{\bar{r}_{h}}\bar{\Delta}-\frac{1}{\epsilon+b-\Lambda\bar{r}^{2}_{h}}\left[-\frac{3b}{8k}+(b+\epsilon)\frac{\epsilon+2b-\Lambda\bar{r}^{2}_{h}}{\bar{r}^{2}_{h}}\right]\bar{\Delta}^{2}+\dots\,,$
(74) $\displaystyle h\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\frac{(\epsilon-\Lambda\bar{r}^{2}_{h})^{2}}{{\bar{r}_{h}\left(\epsilon+b-\Lambda\bar{r}^{2}_{h}\right)}}\bar{\Delta}-\frac{(\epsilon-\Lambda\bar{r}^{2}_{h})^{2}}{\bar{r}^{2}_{h}\left(\epsilon+b-\Lambda\bar{r}^{2}_{h}\right)^{3}}\left[b\left(\frac{1}{8k}-\frac{7\Lambda}{3}\right)-\epsilon\Lambda+\frac{1}{\bar{r}^{2}_{h}}(\epsilon+b)(\epsilon+2b)\right]\bar{\Delta}^{2}+\dots\,.$
(75)

Notice that $f\neq h$, except in the Einstein limit $b=0$. The first two
orders of such an expansion were also displayed in [25] using a different
gauge.

Let us now discuss the Einstein limit in both cases $\epsilon-\Lambda
a_{0}^{2}>0$ and $\epsilon-\Lambda a_{0}^{2}<0$ separately.

  * •

Case $\epsilon-\Lambda a_{0}^{2}>0$: (topological) Schwarzschild-Bach-(A)dS
black holes
In this case, in order to satisfy (51), $b$ is bounded from below by
$\tilde{b}\equiv\Lambda a_{0}^{2}-\epsilon<0$. Thus we can approach
$b\rightarrow 0$ from both sides while keeping the inequality (51) satisfied.
Therefore, in this limit, the quadratic-gravity black-hole horizon reduces to
the (A)dS-Schwarzschild black-hole horizon and we will refer to them as
(topological) Schwarzschild-Bach-(A)dS black holes.

  * •

Case $\epsilon-\Lambda a_{0}^{2}<0$: (topological) Schwarzschild-Bach-(A)dS
black holes and purely Bachian (topological) black holes
In this case, in order to satisfy (51), necessarily $b>\Lambda
a_{0}^{2}-\epsilon>0$. Thus $b$ cannot approach $0$ without violating (51) –
at some point, one reaches a critical value $b_{0}=\Lambda
a_{0}^{2}-\epsilon>0$ corresponding to $a_{1}=0$ and therefore the metric
functions $h(\bar{r})$ and $f(\bar{r})$ cannot be expressed as power series in
$\bar{\Delta}$ with integer powers, see [19, 20] for the case $\epsilon=+1$.
Nevertheless, in the Kundt coordinates, the expressions (67), (68) still hold
even for $b=b_{0}$ and the limiting procedure can be performed. For all $0\leq
b<b_{0}$, the horizon at $\bar{r}=\bar{r}_{h}$ is now a cosmological or an
inner horizon. The $b\rightarrow 0$ limit (cf. section 4.3.1) gives (A)dS-
Schwarzschild metric with a cosmological/inner horizon at
$\bar{r}=\bar{r}_{h}$ which may or may not admit another (black-hole) horizon
depending on the values of parameters $\epsilon$, $\Lambda$, and $r_{h}$. More
precisely, the black-hole horizon can appear either for $\Lambda>0$ and
$\epsilon>0$ or $\Lambda<0$ and $\epsilon<0$ with additional conditions on the
parameters given in [40] and [7, 8], respectively. Einstein limits of these
quadratic-gravity black holes are thus either (A)dS-Schwarzschild black holes
or naked singularities. If the limit is the (A)dS-Schwarzschild black hole, we
will refer to this black hole as a (topological) Schwarzschild-Bach-(A)dS
black hole. If the limit is a naked singularity, we will refer to this black
hole a purely Bachian (topological) black hole.

Figure 1:  Series (67) and (68) for the metric functions ${\cal H}(r)$ (left)
and $\Omega(r)$ (right) for the first 20 (red), 50 (orange), 100 (green), and
200 (blue) terms and a numerical solution (black) of (32) and (33) for values
$\epsilon=0$, $\Lambda=0.2$, $r_{h}=-1$, $k=0.5$, $b=0.3$ (recall the gauge
(59) is used here). From further analysis (not represented in the above
figures, cf. [17, 20] in the case $\epsilon=+1$), the coefficients
$\alpha_{i}$ and $\gamma_{i}$ seem to be approaching geometric series for
large values of $i$. This allows us to estimate the radius of the convergence
and the interval of convergence indicated, in both graphs, by the two vertical
dashed lines. Note that, in the interval of convergence, $\bar{r}=\Omega(r)$
decreases as $r$ grows. The horizon at $r=-1$ separates a static, outer region
($r<-1$, ${\cal H}<0$) from a time-dependent, inner one ($r>-1$, ${\cal
H}>0$). While it is straightforward to estimate the lower bound of the
interval of convergence in the physical coordinate ${\bar{r}}=\Omega(r)$
(using the right figure), it is difficult to estimate the upper bound, since
it is determined by a (possible) intersection of $\Omega(r)$ with the left
vertical dashed line. The numerical solution blows up precisely in the
vicinity of the left vertical dashed line, thus making an accurate estimate
impossible from such a graph.

Let us conclude this section by presenting an example of a purely Bachian
(toroidal) black hole with $\epsilon=0$, $\Lambda=0.2$, $r_{h}=-1$, $k=0.5$,
$b=0.3$ in Figure 1, where the series approximation is depicted together with
a numerical solution. Note that black holes of the form (3) with $\epsilon=0$
and $\Lambda>0$ are not allowed in Einstein’s gravity.

Figure 2 shows that, within a certain continuous range of parameters, black
holes of this section with $\Lambda<0$ are asymptotically AdS (here depicted
for the spherical case $\epsilon=1$). This is in agreement with previous
numerical results of [25] obtained in the case of Einstein-Weyl gravity (see
section 5 for further comments).

Figure 2:  Functions $f(\bar{r})/\bar{r}^{2}$ and $h(\bar{r})/\bar{r}^{2}$ for
spherical Schwarzschild-Bach-AdS black holes obtained from the first 200 terms
of series (41) and (42) for parameters $r_{h}=-1$, $k=1/3$, $b=1/5$,
$\Lambda=-3$, $\epsilon=1$ (recall the gauge (59) is used here where, in
general, $h\not=f$ asymptotically; however, this could be remedied by a gauge
transformation $t\rightarrow\sigma t$). Further calculations suggest that,
also within a certain range of parameters of the solution, fine-tuning of the
Bach parameter $b$ is not necessary to obtain an asymptotically AdS spacetime
(see section 5 for further comments). This observation for the quadratic-
gravity black holes is in agreement with previous numerical results of [25]
obtained in the case of Einstein-Weyl gravity. In contrast, in the $\Lambda=0$
case, fine-tuning is necessary (see Figure 4 and the numerical results of [11,
12, 13, 15, 18, 21]).

####  4.4.2 Special case $\epsilon-\Lambda a_{0}^{2}=0$

In the special case $\epsilon-\Lambda a_{0}^{2}=0$, solutions $[0,1]$ with
series expansions (34) and coefficients $a_{i}$ and $c_{i}$ given in section
4.2 describe purely Bachian black holes provided condition (51) is satisfied,
i.e., $b>0$. This will be assumed in what follows. In the physical coordinates
(3), the first few orders of the metric functions $h$ and $f$ are given by
(52) and (53), respectively.

In particular, within this special class, one can have black holes with
$\epsilon=0=\Lambda$. By contrast, recall that Einstein’s gravity with
$\Lambda=0$ does not allow for flat ($\epsilon=0$) black-hole horizons in
vacuum (3) (cf. (54)). This is thus a new feature of quadratic gravity and
planar horizons (and compactifications thereof) with $\Lambda=0$ are now
allowed. For the special case of Weyl conformal gravity (i.e.,
$\gamma=0=\beta$), this was noted already in [31]. Coefficients $a_{i}$ and
$c_{i}$ for these black holes can be obtained by substituting
$\epsilon=0=\Lambda$ in (41) and (42). An example of such a solution is given
in Figure 3.

Figure 3:  Series (41) and (42) for the metric functions ${\cal H}(r)$ (left)
and $\Omega(r)$ (right) for the first 20 (red), 50 (orange), 100 (green), and
200 (blue) terms for values $\epsilon=0$, $\Lambda=0$, $r_{h}=-1$, $k=1/2$,
$c_{0}=-1$, $a_{0}=1$, $b=1/5$. Similarly as in Fig. 1, the coefficients
$a_{i}$ and $c_{i}$ seem to be approaching geometric series for large values
of $i$, which can be used to estimate the interval of convergence, denoted by
the vertical dashed lines. The horizon at $r=-1$ separates a static, outer
region ($r>-1$, ${\cal H}<0$) from a time-dependent, inner one ($r<-1$, ${\cal
H}>0$).

However, similarly as in the case of spherical horizons with $\Lambda=0$ [11,
12, 13, 15, 16, 18, 19, 21], it turns out that, for generic values of the
parameters $a_{0}$ and $b$, the metric functions $h$ and $f$ diverge as
$\bar{r}\to+\infty$. In order to remedy this one needs to fine-tune the
parameter $b$ (for any given $a_{0}$), such that in the weak gravity regime,
one recovers a solution of general relativity. We have obtained the evidence
that this is indeed possible by performing fine-tuning using the first 100
terms in the series expansion of the solution (instead of using a numerical
solution, as was done in [11, 12, 13, 15, 18, 21]) – this is shown in Figure
4. The asymptotic Ricci-flat spacetime (i.e., an AIII metric [43, 38, 39]) is
characterized by $h(\bar{r})\propto f(\bar{r})=2m/{\bar{r}}$ (the parameter
$m$ is determined by the parametres of the quadratic-gravity solution, $a_{0}$
and $b$; the equality $h=f$ could be achieved by an appropriate gauge
transformation $t\rightarrow\sigma t$). A more precise approach to fine-tuning
would be to match the expansion in the vicinity of the horizon with an
asymptotic expansion in the form of logarithmic-exponential transseries (cf.
[15]) in the physical coordinates (3).

Figure 4: Function $h(\bar{r})$ for $b=1.575812$ (a blue dotted curve),
$1.575811$ (blue), $1.5757$ (red), and $1.5756$ (red dotted) and the general
relativistic solution $h(\bar{r})=2\tilde{m}/{\bar{r}}$ for $2\tilde{m}$=0.32
(black dashed) for values $\epsilon=0$, $\Lambda=0$, $r_{h}=-1$, $k=1/2$,
$c_{0}=-1$, $a_{0}=1$ (note that, in this gauge, $h\not=f$ asymptotically).
Fine-tuning is done using the first 100 terms in the power-series expansions
of the metric functions.

The Einstein limit of these spacetimes belongs to the Kundt class in the form
of a direct product metric of the (anti-)Nariai type (namely
dS${}_{2}\times$S2, flat space, or AdS${}_{2}\times$H2, depending on the value
of $\epsilon$, see section 4.3.2.

##  5 Conclusions

We have studied static black-hole solutions of the most general four-
dimensional quadratic-gravity theory (1) with a non-zero Einstein term
($\gamma\neq 0$), under the assumption $R=$const (motivated by [12]). We have
presented a solution representing black holes possessing a non-extremal
compact horizon of arbitrary topology. The solution is given in terms of an
infinite power-series expansion (based on a Frobenius-like approach) around
the horizon. Several different branches of the solution have been identified,
which admit different Einstein limits – accordingly, they can thus be
interpreted as either higher-derivative “corrections” to Einstein black holes,
or as purely Bachian black holes (for which the horizon does not survive the
Einstein limit).

Although the general solution contains two independent integration constants
(i.e., the black hole radius and the Bach parameter), for the special case of
toroidal black holes with $\Lambda=0$, we have given evidence that solutions
of physical interest (i.e., matching asymptotically an Einstein solution) need
to be fine-tuned, such that there is in fact only one free parameter. This
resembles corresponding results obtained using numerical methods in the case
of spherical, asymptotically flat black holes [11, 12, 13, 15, 18, 21].
Further investigation in this direction will be of interest and will play an
important role in the study of thermodynamics of these topological black
holes. On the other hand, for theories with $\Lambda<0$ and $k>-3/(4\Lambda)$,
numerical results of [25] indicate that (for a given radius) there exists an
open interval of values of $b$ such that these black holes are asymptotically
AdS, with no need of any fine-tuning – while the behavior becomes
asymptotically Lifshitz at the extremes of that interval. This behaviour is
preserved also in full quadratic gravity, provided the Ricci scalar is
constant (cf. eqs. (2), (8), and Figure 2). The first law of the
thermodynamics for these black holes was studied in [26] and contrasted with
the asymptotically flat case [11, 12, 13, 15, 18, 21].

Finally, it is worth emphasizing that the simplified form of the field
equations and the summary of classes of solutions provided in section 3 is of
interest also in a broader context. Based on this, other types of solutions
(such as extremal black holes, naked singularities, and wormholes) exist and
will be studied elsewhere.

## Acknowledgments

This work has been supported by the Czech Academy of Sciences (RVO 67985840)
and research grant GAČR 19-09659S.

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