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

$$\begin{aligned} S&=\int \textrm{d}^4 x\, \sqrt{-g}{{\mathcal {L}}}\nonumber \\&=\int \textrm{d}^4 x \, \sqrt{-g}\, \Big (\gamma \,(R-2\Lambda ) +\beta \,R^2 - \alpha \, C_{abcd}\, C^{abcd} \Big ), \end{aligned}$$

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 [12, 24] 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) \(\gamma \ne 0\))

$$\begin{aligned} R_{ab}-\Lambda \, g_{ab}=4k\, B_{ab}, \end{aligned}$$

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, 17, 19, 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 \(\Lambda \ne 0\) have been obtained in [17, 20, 25, 26].Footnote 1

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

$$\begin{aligned}{} & {} \textrm{d}s^2 = -h(\bar{r})\,\textrm{d}t^2+\frac{\textrm{d}\bar{r}^2}{f(\bar{r})} +\bar{r}^2 \textrm{d}\omega ^2, \nonumber \\{} & {} \quad \textrm{d}\omega ^2 =\frac{\textrm{d}x^2+\textrm{d}y^2}{\left( 1+\frac{\epsilon }{4} (x^2+y^2)\right) ^2} , \end{aligned}$$

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 \(\epsilon \le 0\) 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 \(R=\text {const},\) 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 Sect. 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 Sect. 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\(_2\times \)S\(^2\), AdS\(_2\times \)H\(^2\), 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.Footnote 2 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 Sect. 5).

Concluding comments are given in Sect. 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

$$\begin{aligned}&\gamma \left( R_{ab} - {\frac{1}{2}} R\, g_{ab}+\Lambda \,g_{ab}\right) -4 \alpha \,B_{ab}\nonumber \\&\quad +2\beta \left( R_{ab}-\tfrac{1}{4}R\, g_{ab} +g_{ab}\, \Box - \nabla _b \nabla _a\right) R = 0 , \end{aligned}$$

where \(B_{ab}\) is the Bach tensor

$$\begin{aligned} B_{ab} \equiv \left( \nabla ^c \nabla ^d + {\frac{1}{2}} R^{cd} \right) C_{acbd}, \end{aligned}$$

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

$$\begin{aligned}&g^{ab}B_{ab}=0 , \quad B_{ab}=B_{ba} ,\nonumber \\&\nabla ^b B_{ab}=0 , \quad B_{ab}=\Omega ^{-2}\tilde{B}_{ab}. \end{aligned}$$

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.Footnote 3 Then the trace of (4) reduces to \(\gamma (R-4\Lambda )=0\). Assuming \(\gamma \ne 0\) (to keep the Einstein–Hilbert term in the action (1)) we thus have

$$\begin{aligned} R=4\Lambda , \end{aligned}$$

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

$$\begin{aligned} k\equiv \frac{\alpha }{\gamma +8\beta \Lambda } \qquad \qquad \qquad \qquad \qquad (\gamma +8\beta \Lambda \ne 0) . \end{aligned}$$

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, \({{\mathcal {H}}}'' +2\epsilon =0\) (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 [16, 19, 35]. This enables one to describe such spacetimes in the form

$$\begin{aligned} \textrm{d}s^2 \equiv \Omega ^2(r) \,\textrm{d}s^2_\textrm{K}= \Omega ^2(r) \Big [\,\textrm{d}\omega ^2 -2\,\textrm{d}u\,\textrm{d}r+{\mathcal {H}}(r)\,\textrm{d}u^2 \,\Big ], \end{aligned}$$

for which the resulting field equations are considerably simpler.

The metric (9) admits a gauge freedom

$$\begin{aligned} r \rightarrow \lambda \,r+\upsilon , \quad u \rightarrow \lambda ^{-1}\,u , \end{aligned}$$

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

When \(\Omega '\ne 0\ne {\mathcal {H}}\) one can also define the standard Schwarzschild coordinates by [35]

$$\begin{aligned} \bar{r} = \Omega (r), \quad t = u - \int \frac{\textrm{d}r}{{\mathcal {H}}(r)} , \end{aligned}$$

giving rise to the line-element (3) with

$$\begin{aligned} h=-\Omega ^2\, {\mathcal {H}}, \quad f = -\left( \frac{\Omega '}{\Omega }\right) ^2{\mathcal {H}}, \end{aligned}$$

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 \(\textrm{d}s^2_\textrm{K}\) of (9) read (cf. [19] for \(\epsilon =1\) case)

$$\begin{aligned} R_{ru}^\textrm{K}= & {} -\frac{1}{2}\,{{\mathcal {H}}}'', \qquad \quad R_{uu}^\textrm{K}= -{{\mathcal {H}}} \,R_{ru}^\textrm{K}, \end{aligned}$$
$$\begin{aligned} R_{xx}^\textrm{K}= & {} R_{yy}^\textrm{K}= \epsilon g^\textrm{K}_{xx}, \quad R^\textrm{K}={{\mathcal {H}}}'' + 2 \epsilon . \end{aligned}$$

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

$$\begin{aligned} C_{ruru}^\textrm{K}= & {} {-\frac{1}{6} R^\textrm{K}}, \qquad \quad \qquad \qquad \qquad C_{riuj}^\textrm{K}= { \frac{1}{12} R^\textrm{K}\,g^\textrm{K}_{ij}}, \end{aligned}$$
$$\begin{aligned} C_{kilj}^\textrm{K}= & {} {\frac{1}{6}R^\textrm{K}\,(g^\textrm{K}_{kl} g^\textrm{K}_{ij}-g^\textrm{K}_{kj}g^\textrm{K}_{il}) }, \quad C_{uiuj}^\textrm{K}= -{{\mathcal {H}}}\, C^\textrm{K}_{riuj} \, \end{aligned}$$


$$\begin{aligned}{} & {} B_{rr}^\textrm{K}= {-\frac{1}{6}\,{{\mathcal {H}}}''''}, \nonumber \\{} & {} B_{ru}^\textrm{K}= { \frac{1}{12}\, \left( 2\,{{\mathcal {H}}}{{\mathcal {H}}}'''' +{{\mathcal {H}}}'{{\mathcal {H}}}''' -{\frac{1}{2}}{{{\mathcal {H}}}''}^2 +2\epsilon ^2\right) }, \end{aligned}$$
$$\begin{aligned}{} & {} B_{uu}^\textrm{K}= -{{\mathcal {H}}}\,B_{ru}^\textrm{K},\nonumber \\{} & {} B_{xx}^\textrm{K}= B_{yy}^\textrm{K}= { \frac{1}{12}}\,g^\textrm{K}_{xx}\, \left( {{\mathcal {H}}}{{\mathcal {H}}}''''+{{\mathcal {H}}}' {{\mathcal {H}}}'''-{\frac{1}{2}}{{{\mathcal {H}}}''}^2 +2\epsilon ^2\right) .\nonumber \\ \end{aligned}$$

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

$$\begin{aligned}{} & {} R_{rr} = -2\Omega ^{-2} \big (\Omega \Omega ''-2{\Omega '}^2\big ),\nonumber \\{} & {} R_{ru} = -\frac{1}{2}\Omega ^{-2} \big (\Omega ^2 {{\mathcal {H}}}\big )'', \end{aligned}$$
$$\begin{aligned}{} & {} R_{uu} = -{{\mathcal {H}}}\, {R}_{ru},\nonumber \\{} & {} R_{xx} = {R}_{yy} =\Omega ^{-2}g_{xx}^\textrm{K}\, \big [ \big ({{\mathcal {H}}} \Omega \Omega '\big )'+\epsilon \Omega ^2 \big ], \end{aligned}$$
$$\begin{aligned}{} & {} R = 6\Omega ^{-3} \big [({{\mathcal {H}}}\Omega ')' +{\frac{1}{6}} ({{\mathcal {H}}}''+2 \epsilon )\Omega \big ]. \end{aligned}$$

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

$$\begin{aligned} B_{ab} = \Omega ^{-2}B_{ab}^\textrm{K}, \end{aligned}$$

while \({C^a}_{\,bcd}={C^{\textrm{K}a}}_{\ bcd}\), as well known.

The Ricci squared, Bach, and Weyl invariants readFootnote 4

$$\begin{aligned} R_{ab}\, R^{ab}&= 4\Lambda ^2+16k^2\, B_{ab} B^{ab} , \end{aligned}$$
$$\begin{aligned} B_{ab}\, B^{ab}&= \tfrac{1}{72}\,\Omega ^{-8}\,\big [({\mathcal {B}}_1)^2 + 2({\mathcal {B}}_1+{\mathcal {B}}_2)^2\big ] , \end{aligned}$$
$$\begin{aligned} C_{abcd}\, C^{abcd}&= \tfrac{1}{3} \,\Omega ^{-4}\,\big ({{\mathcal {H}}}'' +2\epsilon \big )^2 , \end{aligned}$$

where the two independent components of the Bach tensor, \({\mathcal {B}}_1(r)\) and \({\mathcal {B}}_2(r)\), are

$$\begin{aligned} {\mathcal {B}}_1 \equiv {{\mathcal {H}}}{{\mathcal {H}}}'''', \quad {\mathcal {B}}_2 \equiv {{\mathcal {H}}}'{{\mathcal {H}}}''' -\tfrac{1}{2}{{{\mathcal {H}}}''}^2 +2\epsilon ^2. \end{aligned}$$

It is useful to note that \(B_{ab}=0\Leftrightarrow {\mathcal {B}}_1=0={\mathcal {B}}_2\). It can be also verified easily that \(C_{abcd}=0\Leftrightarrow {{\mathcal {H}}}'' +2\epsilon =0\).

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

$$\begin{aligned}&\Omega \Omega ''-2{\Omega '}^2 = \tfrac{1}{3}k\,{{\mathcal {H}}}'''' , \end{aligned}$$
$$\begin{aligned}&\big (\Omega ^2 {{\mathcal {H}}}\big )'' -2\Lambda \Omega ^4 = -\tfrac{2}{3}k \left( 2\,{{\mathcal {H}}}{{\mathcal {H}}}''''+{{\mathcal {H}}}'{{\mathcal {H}}}''' \right. \nonumber \\&\left. -{\frac{1}{2}}{{{\mathcal {H}}}''}^2 +2\epsilon ^2\right) , \end{aligned}$$
$$\begin{aligned}&\big ({{\mathcal {H}}}\Omega \Omega '\big )'+\epsilon \Omega ^2 -\Lambda \Omega ^4=\tfrac{1}{3}k \,\left( {{\mathcal {H}}}{{\mathcal {H}}}''''+{{\mathcal {H}}}'{{\mathcal {H}}}''' \right. \nonumber \\&\left. -{\frac{1}{2}}{{{\mathcal {H}}}''}^2 +2\epsilon ^2 \right) . \end{aligned}$$

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

$$\begin{aligned} {\mathcal {T}}\equiv {{\mathcal {H}}}\Omega ''+{{\mathcal {H}}}'\Omega ' +{\frac{1}{6}} ({{\mathcal {H}}}''+2\epsilon )\Omega ={\frac{2}{3}\Lambda \,\Omega ^3 } . \end{aligned}$$

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

$$\begin{aligned} J_{ab}\equiv R_{ab}-\frac{1}{2}R\, g_{ab}+\Lambda \, g_{ab} - 4k\, B_{ab}. \end{aligned}$$

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 \(\Omega '\not =0\), 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).Footnote 5

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

$$\begin{aligned}&\Omega \Omega ''-2{\Omega '}^2 = \tfrac{1}{3}k\,{{\mathcal {H}}}'''' , \end{aligned}$$
$$\begin{aligned}&\Omega \Omega '{{\mathcal {H}}}'+3\Omega '^2{{\mathcal {H}}}+\epsilon \Omega ^2 -\Lambda \Omega ^4\nonumber \\&\quad = \tfrac{1}{3}k \left( {{\mathcal {H}}}'{{\mathcal {H}}}'''-{\frac{1}{2}}{{{\mathcal {H}}}''}^2 +2\epsilon ^2 \right) . \end{aligned}$$

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.,

$$\begin{aligned} \Omega (r)=\Delta ^n \sum _{i=0}^\infty a_i \,\Delta ^{i}, \quad {\mathcal {H}}(r)=\Delta ^p \,\sum _{i=0}^\infty c_i \,\Delta ^{i} . \end{aligned}$$

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 \(\Lambda \)).

Table 1 Values of [np] 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

$$\begin{aligned} r_h\equiv r_0 . \end{aligned}$$

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

$$\begin{aligned} \bar{r}_h\equiv \Omega (r_h)=a_0>0 , \end{aligned}$$

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 \(\Omega \rightarrow -\Omega \).

Before discussing the metric on-shell, let us note that, when \(a_1\ne 0\), the leading order behaviour of the metric functions h and f in the Schwarzschild coordinates (3) is given by

$$\begin{aligned}&h=-\frac{c_0a_{0}^2}{a_1}(\bar{r}-\bar{r}_h)+{{\mathcal {O}}} \left( (\bar{r}-\bar{r}_h)^2\right) ,\nonumber \\&f=-\frac{c_0a_1}{a_0^2}(\bar{r}-\bar{r}_h) +{{\mathcal {O}}}\left( (\bar{r}-\bar{r}_h)^2\right) . \end{aligned}$$

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

$$\begin{aligned} c_0 a_1<0, \end{aligned}$$

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 \(a_1>0\),Footnote 6 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

$$\begin{aligned} a_1=\frac{a_0}{3c_0}\left[ 2\Lambda a_0^2-(\epsilon +c_1)\right] , \end{aligned}$$

and then the lowest nontrivial order of (33) implies

$$\begin{aligned} c_2 =\frac{1}{6kc_0}\left[ 2k(c_1^2-\epsilon ^2) +a_0^2(2\epsilon -c_1-\Lambda a_0^2)\right] . \end{aligned}$$

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

$$\begin{aligned} c_{l+2}&=\frac{3}{k\,(l+3)(l+2)(l+1)l}\nonumber \\&\quad \times \sum ^{l}_{i=0}a_i \, a_{l+1-i}(l+1-i)(l-3i) \quad \forall \ l\ge 1, \end{aligned}$$
$$\begin{aligned} a_{l}&=\frac{1}{l^2c_0}\Bigg [\tfrac{2}{3} \Lambda \sum ^{l-1}_{j=0}{a_{l-1-j}}\sum ^{j}_{i=0}a_i \,a_{j-i}-\tfrac{1}{3}\,\epsilon a_{l-1}\nonumber \\&\quad -\sum ^{l}_{i=1}c_i\,a_{l-i} \left[ l(l-i)+\tfrac{1}{6}i(i+1)\right] \Bigg ] \quad \forall \ l\ge 2. \end{aligned}$$

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

$$\begin{aligned} B_{ab}\,B^{ab}(r_h)= & {} \left( \frac{\epsilon ^2-c_1^2 +3c_0c_2}{3 a_0^4} \right) ^2 =\left( \frac{c_1-2\epsilon +\Lambda a_0^2}{6k a_0^2}\right) ^2\nonumber \\= & {} \frac{b^2}{4 k^2 a_0^4 }, \end{aligned}$$
$$\begin{aligned} C_{abcd}\, C^{abcd}(r_h)= & {} \frac{4}{3 a_0^4}(\epsilon + c_1)^2, \end{aligned}$$

where we have introduced a dimensionless Bach parameter b by

$$\begin{aligned} b \equiv \frac{1}{3}\left( c_1-2\epsilon +\Lambda a_0^2\right) , \end{aligned}$$

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

$$\begin{aligned}&a_{1} = -\frac{a_{0}}{c_{0}}\left[ (\epsilon -\Lambda a_{0}^2)+b\right] , \end{aligned}$$
$$\begin{aligned}&a_{2} \!=\! +\frac{a_{0}}{c_{0}^2}\left[ (\epsilon -\Lambda a_{0}^2)^2 \!+\!b \left( 2 \epsilon + a_0^2 \left( \frac{1}{8 k} -\frac{7 \Lambda }{3} \right) \right) + b^2\right] , \end{aligned}$$
$$\begin{aligned}&c_1 = 2\epsilon - \Lambda a_0^2 + 3b , \end{aligned}$$
$$\begin{aligned}&c_2 = - \frac{ 1}{6 k c_0 } \left[ 2 k \left( \epsilon ^2 -\left( 3 b - a_0^2 \Lambda +2 \epsilon \right) ^2\right) +3 b a_0^2 \right] , \end{aligned}$$
$$\begin{aligned}&c_3 = \frac{a_0^4 b (3-8k\Lambda )}{96 k^2 c_0^2}. \end{aligned}$$

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 Sect. 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 Sect. 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 Sect. 4.4.1 below). Because of (36), (38), and (46), at an outer black-hole horizon, it will be constrained by

$$\begin{aligned} \epsilon -\Lambda a_{0}^2+b>0 . \end{aligned}$$

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 \(a_1\ne 0\), i.e., \( \epsilon -\Lambda a_{0}^2+b\ne 0\))

$$\begin{aligned} h&=\frac{a_0 c_0^2}{\epsilon -\Lambda a_{0}^2+b} \bar{\Delta } -\frac{c_0^2}{(\epsilon -\Lambda a_{0}^2+b)^3}\nonumber \\&\quad \times \left[ (\epsilon -\Lambda a_{0}^2)\epsilon +b\left( 3\epsilon -\frac{7}{3}\Lambda a_{0}^2+\frac{1}{8k}a_{0}^2 \right) +2b^2\right] \bar{\Delta }^2+\cdots , \end{aligned}$$
$$\begin{aligned} f&=\frac{\epsilon -\Lambda a_{0}^2+b}{a_0}\bar{\Delta } -\frac{1}{a_{0}^2(\epsilon -\Lambda a_{0}^2+b)}\nonumber \\&\quad \times \left[ (\epsilon -\Lambda a_{0}^2)\epsilon +b\left( 3\epsilon -\Lambda a_{0}^2-\frac{3}{8k} a_0^2 \right) +2b^2\right] \bar{\Delta }^2 +\cdots , \end{aligned}$$

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 spacetimes, for which the Bach tensor is necessarily zero. Because of (51), solutions with a non-extremal black hole horizon must now satisfy

$$\begin{aligned} \epsilon -\Lambda a_{0}^2>0 . \end{aligned}$$

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

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

$$\begin{aligned}{} & {} a_i=a_0\,\bigg (-\frac{\epsilon -\Lambda a_0^2}{c_0}\bigg )^i \quad \forall \ i \ge 0, \end{aligned}$$
$$\begin{aligned}{} & {} c_1=2\epsilon -\Lambda a_0^2, \nonumber \\{} & {} c_2 = \frac{1}{3c_0} (\epsilon -\Lambda a_0^2) (3\epsilon -\Lambda a_0^2), \qquad c_i=0 \quad \forall \ i \ge 3. \nonumber \\ \end{aligned}$$

Note, in particular, that the \(c_i\) sequence is truncated as

$$\begin{aligned} {\mathcal {H}}(r)= c_0(r-r_h)+c_1(r-r_h)^2+c_2(r-r_h)^3 , \end{aligned}$$

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 \(\epsilon -\Lambda a_0^2\ne 0\), the \(a_i\) sequence is a geometric series, giving rise to

$$\begin{aligned} \Omega (r)&= a_0 \,\sum _{i=0}^\infty \,\left( -(\epsilon -\Lambda a_0^2) \frac{\Delta }{c_0}\right) ^i =\frac{a_0\,c_0}{c_0+(\epsilon -\Lambda a_0^2)\Delta }\nonumber \\&=\frac{a_0\,c_0}{(\epsilon -\Lambda a_0^2)(r-r_h)+c_0}. \end{aligned}$$

Employing the gauge freedom (10), we can set

$$\begin{aligned} a_0 =-\frac{1}{r_h} , \quad c_0 =r_h \epsilon -\frac{\Lambda }{r_h} \end{aligned}$$

(which also means \(a_1=a_0^2\)), reducing the metric functions to

$$\begin{aligned} {\bar{r}}=\Omega (r) = -\frac{1}{r}, \quad {\mathcal {H}}(r) = \frac{\Lambda }{3} - \epsilon r^2 - \left( \frac{\Lambda }{3} - \epsilon r_h^2 \right) \frac{r^3}{r_h^3} . \end{aligned}$$

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 \(2\,m=\left( \frac{\Lambda }{3} - \epsilon r_h^2 \right) \frac{1}{r_h^3}\), and the metric functions in the physical coordinates (3) take the form

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

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 \(\epsilon \le 0\) require \(\Lambda <0\). When equality holds (i.e., \(\epsilon -\Lambda a_{0}^2=0\)) one obtains the standard extremality condition for Einstein black holes [7, 8, 40].Footnote 7 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. [7, 8, 40] 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.,

$$\begin{aligned} \Omega = a_0 , \quad {\mathcal {H}}= c_0(r-r_h)+\epsilon (r-r_h)^2 , \end{aligned}$$

which describes an Einstein spacetime of the Kundt class in the form of a direct product metric of the (anti-)Nariai type (namely dS\(_2\times \)S\(^2\), flat space, or AdS\(_2\times \)H\(^2\), 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 Sect. 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 \(\epsilon -\Lambda a_0^2\ne 0\)

As mentioned above, the general quadratic-gravity solution of Sect. 4.2 is non-Einstein when \(b\ne 0\). 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. Sect. 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 [16, 17, 19, 35]. Using again the gauge (59), one obtains

$$\begin{aligned} a_i\equiv & {} \tilde{a}_i-\frac{b}{r_h} \,\frac{\alpha _i}{(-r_h\rho )^i},\qquad \hbox {where}\nonumber \\ \tilde{a}_i\equiv & {} a_i(b=0)=\frac{1}{(-r_h)^{1+i}} \quad i\ge 0, \end{aligned}$$
$$\begin{aligned} c_1\equiv & {} 2\epsilon -\frac{\Lambda }{r_h^2} + 3b\,\gamma _1, \quad c_2 \equiv \frac{3 \epsilon r_h^2 - \Lambda }{3 r_h^3} +3b \, \frac{r_h\gamma _2}{\epsilon r_h^2-\Lambda },\nonumber \\ c_i\equiv & {} 3b\, \frac{\gamma _i \,r_h^{i-1}}{(\epsilon r_h^2 -\Lambda )^{i-1}}\quad i \ge 3, \end{aligned}$$


$$\begin{aligned} {\rho }\equiv \epsilon -\frac{\Lambda }{r_h^2}, \end{aligned}$$


$$\begin{aligned}&\alpha _0\equiv 0, \quad \alpha _1 = 1 , \quad \gamma _1=1,\nonumber \\&\quad \gamma _2 = \frac{1}{3}\Big [4\epsilon - \frac{1}{r_h^2} \Big (2\Lambda +\frac{1}{2k}\Big ) +3b\Big ] . \end{aligned}$$

The remaining coefficients \(\alpha _i\) and \(\gamma _i\) will be specified shortly.

The functions \({\Omega }\) and \({{\mathcal {H}}}\) then read

$$\begin{aligned} \Omega (r)&= -\frac{1}{r}-\frac{b}{r_h}\sum _{i=1}^\infty \alpha _i\Big (\,\frac{r_h-r}{{\rho }\,r_h}\Big )^i , \end{aligned}$$
$$\begin{aligned} {\mathcal {H}}(r)&= (r-r_h)\nonumber \\&\quad \times \bigg [\,\epsilon \frac{r^2}{r_h} -\frac{\Lambda }{3r_h^3} \left( r^2+rr_h+r_h^2\right) \nonumber \\&\quad +3b\,{\rho }\,r_h\sum _{i=1}^\infty \gamma _i \Big (\,\frac{r-r_h}{{\rho }\,r_h}\Big )^i\,\bigg ] , \end{aligned}$$

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 \({l \ge 2}\) are given by the recurrent relations

$$\begin{aligned} \alpha _{l}&= \frac{1}{l^2}\Bigg [-\frac{2\Lambda }{3r_h^2} \,\sum _{j=0}^{l-1}\sum _{i=0}^{j}\nonumber \\&\quad \left[ \alpha _{l-1-j}{\rho }^j + \big ({\rho }^{l-1-j} +b\,\alpha _{l-1-j}\big )\right. \nonumber \\&\quad \times \left. \big (\alpha _i {\rho }^{j-i} +\alpha _{j-i}({\rho }^i+b\,\alpha _{i})\big )\right] \nonumber \\&\quad -\frac{1}{3}\alpha _{l-2}(2\epsilon +{\rho }){\rho }(l-1)^2 +\alpha _{l-1}\nonumber \\&\quad \times \left[ \frac{\epsilon }{3}+({\epsilon }+{\rho }) \left( l(l-1)+\frac{1}{3}\right) \right] \nonumber \\&\quad -3\sum _{i=1}^{l}(-1)^i\,\gamma _i\,({\rho }^{l-i} +b\,\alpha _{l-i})\left[ l(l-i)+\frac{1}{6}i(i+1)\right] \Bigg ],\nonumber \\ \gamma _{l+1}&= \, \frac{(-1)^{l}}{kr_h^2\,(l+2)(l+1)l(l-1)} \nonumber \\&\quad \times \sum _{i=0}^{l-1}\big [\alpha _i {\rho }^{l-i}+\alpha _{l-i} \big ({\rho }^i+b\,\alpha _i\big ) \big ](l-i)(l-1-3i) \, \quad \forall \quad l\ge 2. \end{aligned}$$

Explicitly, the first few terms then read

$$\begin{aligned} \alpha _2&= 2\epsilon -\left( \frac{7}{3}\Lambda -\frac{1}{8k}\right) \frac{1}{r_h^2}+b , \nonumber \\ \alpha _3&= \frac{1}{9}\left[ 25\epsilon ^2+\left( \frac{29}{8k} -\frac{179}{3}\Lambda \right) \frac{\epsilon }{r_h^2}\right. \nonumber \\&\quad \left. +\left( \frac{1}{16k^2}-\frac{77}{24k}\Lambda +\frac{298}{9}\Lambda ^2\right) \frac{1}{r_h^4}\right] \nonumber \\&\quad +\frac{1}{9}\left[ 23\epsilon +\left( \frac{35}{8k} -\frac{104}{3}\Lambda \right) \frac{1}{r_h^2}\right] \, b+\frac{7}{9}\,b^2 , \quad \ldots , \end{aligned}$$
$$\begin{aligned} \gamma _3&= \frac{1}{96k^2r_h^4} \left( 1-\frac{8k}{3}\Lambda \right) , \nonumber \\ \gamma _4&= \frac{1}{18kr_h^2}\left[ \frac{\epsilon ^2}{5} +\left( -\frac{1}{4k}+\frac{4}{15}\Lambda \right) \frac{\epsilon }{r_h^2}-\frac{1}{160k^2r_h^4}\right. \nonumber \\&\quad \left. -\frac{1}{45r_h^4}\left( 14\Lambda ^2-\frac{75}{8k}\Lambda \right) \right] \nonumber \\&\quad +\frac{1}{720kr_h^2}\left[ 16\epsilon +\left( -\frac{13}{k} +\frac{56}{3}\Lambda \right) \frac{1}{r_h^2}\right] \,b \nonumber \\&\quad +\frac{1}{90kr_h^2}\,b^2 , \quad \ldots , \end{aligned}$$

which leads to

$$\begin{aligned} \Omega (r)&=-\frac{1}{r}+b\frac{(r-r_h)}{{\rho }\,r_h^2}\nonumber \\&\quad -\frac{b}{r_h}\left[ 2{\rho }+\frac{1}{24 k r_h^2}(3-8k\Lambda ) +b\right] \Big (\,\frac{r-r_h}{{\rho }\,r_h}\Big )^2+\cdots , \end{aligned}$$
$$\begin{aligned} {\mathcal {H}}(r)&= (r-r_h)\bigg \{\,\epsilon \frac{r^2}{r_h} -\frac{\Lambda }{3\,r_h^3}\left( r^2+rr_h+r_h^2\right) \nonumber \\&\quad +3b\,{\rho }\,r_h\left[ \Big (\,\frac{r-r_h}{{\rho }\,r_h}\Big ) +\frac{1}{3}\Big [4\epsilon -\frac{1}{r_h^2} \Big (2\Lambda +\frac{1}{2k}\Big )+3b\Big ]\right. \nonumber \\&\quad \left. \times \Big (\,\frac{r-r_h}{{\rho }\,r_h}\Big )^2 +\cdots \right] \,\bigg \}. \end{aligned}$$

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))

$$\begin{aligned} f&=\frac{\epsilon +b-\Lambda \bar{r}^2_h}{\bar{r}_h}\bar{\Delta } -\frac{1}{\epsilon +b-\Lambda \bar{r}^2_h}\nonumber \\&\quad \times \left[ -\frac{3b}{8k}+(b+\epsilon )\frac{\epsilon +2b -\Lambda \bar{r}^2_h}{\bar{r}^2_h}\right] \bar{\Delta }^2+\dots , \end{aligned}$$
$$\begin{aligned} h&=\frac{(\epsilon -\Lambda \bar{r}^2_h )^2}{{\bar{r}_h \left( \epsilon +b-\Lambda \bar{r}^2_h\right) }}\bar{\Delta } -\frac{(\epsilon -\Lambda \bar{r}^2_h)^2}{\bar{r}^2_h \left( \epsilon +b-\Lambda \bar{r}^2_h\right) ^3}\nonumber \\&\quad \times \left[ b\left( \frac{1}{8k}-\frac{7\Lambda }{3}\right) -\epsilon \Lambda +\frac{1}{\bar{r}^2_h}(\epsilon +b)(\epsilon +2b)\right] \bar{\Delta }^2+\cdots . \end{aligned}$$

Notice that \(f\ne h\), 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 \(0\le b<b_0\), the horizon at \(\bar{r}=\bar{r}_h\) is now a cosmological or an inner horizon. The \(b\rightarrow 0\) limit (cf. Sect. 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.

Fig. 1
figure 1

Series (67) and (68) for the metric functions \({{\mathcal {H}}}(r)\) (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 (\(r<-1\), \({{\mathcal {H}}}<0\)) from a time-dependent, inner one (\(r>-1\), \({{\mathcal {H}}}>0\)). 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 Fig. 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 Sect. 5 for further comments).

Fig. 2
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 Sect. 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 Fig. 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 Sect. 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 Fig. 3.

Fig. 3
figure 3

Series (41) and (42) for the metric functions \({{\mathcal {H}}}(r)\) (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 (\(r>-1\), \({{\mathcal {H}}}<0\)) from a time-dependent, inner one (\(r<-1\), \({{\mathcal {H}}}>0\))

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 \(\bar{r}\rightarrow +\infty \). 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 Fig. 4. The asymptotic Ricci-flat spacetime (i.e., an AIII metric [38, 39, 43]) is characterized by \(h(\bar{r})\propto f(\bar{r})= 2m/{\bar{r}}\) (the parameter m is determined by the parameters 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).

Fig. 4
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\(_2\times \)S\(^2\), flat space, or AdS\(_2\times \)H\(^2\), depending on the value of \(\epsilon \), see Sect. 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 (\(\gamma \ne 0\)), 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 Fig. 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 Sect. 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.