Abstract
We study static black holes in quadratic gravity with planar and hyperbolic symmetry and nonextremal horizons. We obtain a solution in terms of an infinite powerseries 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\(_2\times \)S\(^2\), AdS\(_2\times \)H\(^2\), and flat spacetime). In the special case of toroidal black holes with \(\Lambda =0\), we also discuss how the Bach parameter needs to be finetuned to ensure that the metric does not blow up near infinity and instead matches asymptotically a Ricciflat solution.
Similar content being viewed by others
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 2spheres, 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 antide 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
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 nonvanishing cosmological constant \(\Lambda \) have been studied in [17, 20]. Although these nonSchwarzschild (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 nohair 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) \(\gamma \ne 0\))
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 nonvanishing 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 nearhorizon 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 finetune the parameters \(\bar{r}_0\) and b using numerical methods, thus effectively ending up with a oneparameter 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 nonzero cosmological constant or nonspherical 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
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 conformaltoKundt 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 nohair theorem, the resulting simplification of the field equations will enable us to obtain recurrent formulas for series coefficients of the metric functions in powerseries 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 conformaltoKundt 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 Frobeniuslike approach to solve the equations in the vicinity of a generic hypersurface of a constant radius. We use infinite powerseries expansions and the indicial equations to determine the possible leading powers of solutions. In particular, some of these solutions admit extremal or nonextremal horizons.
Section 4 then focuses on the study of solutions with nonextremal 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 finetuning the Bach parameter, a quadraticgravity black hole with \(\Lambda =0\) will asymptote an (appropriate) Ricciflat metric near spatial infinity. We will give evidence to support this expectation by finetuning a planar blackhole solution (\(\epsilon =0\)) with vanishing \(\Lambda \), using the polynomial expansion.^{Footnote 2} In contrast, finetuning 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
where \(B_{ab}\) is the Bach tensor
which is traceless, symmetric, conserved, and wellbehaved under a conformal transformation \(g_{ab}=\Omega ^2 \tilde{g}_{ab}\):
For fourdimensional (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 \(\gamma (R4\Lambda )=0\). Assuming \(\gamma \ne 0\) (to keep the Einstein–Hilbert term in the action (1)) we thus have
and the field equations (4) simplify to (2), where we have defined
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 finetuned 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 ConformaltoKundt ansatz
We are interested in static blackhole solutions with spherical, hyperbolic, or planar symmetry. Instead of the standard Schwarzschild coordinates, throughout the paper, we will mostly employ the conformaltoKundt form of the metric introduced in [16, 19, 35]. This enables one to describe such spacetimes in the form
for which the resulting field equations are considerably simpler.
The metric (9) admits a gauge freedom
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]
giving rise to the lineelement (3) with
where a prime denotes differentiation with respect to r.
3 Field equations and classes of powerseries 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)
The nonvanishing components of the Weyl and Bach tensors are, respectively,
and
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
The Bach tensor of the full metric can be obtained by a rescaling (6)
while \({C^a}_{\,bcd}={C^{\textrm{K}a}}_{\ bcd}\), as well known.
The Ricci squared, Bach, and Weyl invariants read^{Footnote 4}
where the two independent components of the Bach tensor, \({\mathcal {B}}_1(r)\) and \({\mathcal {B}}_2(r)\), are
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
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
As in [19, 20], let us introduce a conserved (\(\nabla ^b J_{ab}\equiv 0\)) symmetric tensor \(J_{ab}\)
The nontrivial 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
3.4 Classes of powerseries solutions
Let us assume that the metric functions \(\Omega (r) \) and \({\mathcal {H}}(r)\) in (9) can be expanded as infinite power series in \({\Delta \equiv rr_0}\) around a hypersurface \(r=r_0\), i.e.,
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 \(\Lambda \)).
In the rest of the paper, we will study the [0, 1] case, which corresponds to spacetimes admitting a nonextremal 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 nonextremal 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 nonextremal Killing horizon, located at \(r=r_0\). Let us thus relabel
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
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 onshell, 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
In order to have an outer black hole horizon, we thus need to take
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
and then the lowest nontrivial order of (33) implies
At any arbitrary higher order, one finds that the [0, 1] solution to Eqs. (32), (33) is given by the recurrent formulas
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
where we have introduced a dimensionless Bach parameter b by
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
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 blackhole horizon, it will be constrained by
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., \( \epsilon \Lambda a_{0}^2+b\ne 0\))
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 nonextremal black hole horizon must now satisfy
In particular, for \(\Lambda >0\), only spherical black holes (\(\epsilon =+1\)) will be possible.
With \(b=0\), the coefficients (41), (42) reduce to
Note, in particular, that the \(c_i\) sequence is truncated as
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
Employing the gauge freedom (10), we can set
(which also means \(a_1=a_0^2\)), reducing the metric functions to
Using (3), (12), it is easy to see that this solution corresponds to the wellknown [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
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 timedependent 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 blackhole horizon located elsewhere when \(\epsilon \Lambda >0\), cf. [7, 8, 40] for details).
4.3.2 Special case \(\epsilon \Lambda a_0^2=0\): directproduct 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.,
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 nearhorizon 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 quadraticgravity solution of Sect. 4.2 is nonEinstein 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 quadraticgravity 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
where
and
The remaining coefficients \(\alpha _i\) and \(\gamma _i\) will be specified shortly.
The functions \({\Omega }\) and \({{\mathcal {H}}}\) then read
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
Explicitly, the first few terms then read
which leads to
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))
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 quadraticgravity blackhole horizon reduces to the (A)dS–Schwarzschild blackhole 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 (blackhole) horizon depending on the values of parameters \(\epsilon \), \(\Lambda \), and \(r_h\). More precisely, the blackhole 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 quadraticgravity 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.
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).
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 (\(\epsilon =0\)) blackhole 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.
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 finetune 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 finetuning 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 Ricciflat 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 quadraticgravity 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 finetuning would be to match the expansion in the vicinity of the horizon with an asymptotic expansion in the form of logarithmicexponential transseries (cf. [15]) in the physical coordinates (3).
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 blackhole solutions of the most general fourdimensional quadraticgravity theory (1) with a nonzero Einstein term (\(\gamma \ne 0\)), under the assumption \(R=\)const (motivated by [12]). We have presented a solution representing black holes possessing a nonextremal compact horizon of arbitrary topology. The solution is given in terms of an infinite powerseries expansion (based on a Frobeniuslike 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 higherderivative “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 finetuned, 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 finetuning – 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.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There is no data associated to this purely theoretical work.]
Notes
However, more solid evidence would be to match the expansion in the vicinity of the horizon with the asymptotic expansion in the form of logarithmicexponential transseries (cf. [15]).
This is not a restriction for Einstein–Weyl gravity, i.e., when \(\beta =0\), since it then clearly follows from (4).
In this paper, we will not study nonEinstein spacetimes with \(\Omega =\)const, which correspond to Kundt metrics.
This 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 \(\epsilon =+1\)).
Note indeed that, although we have obtained the closed form (60) of the solution under the assumption \(\epsilon \Lambda a_0^2\ne 0\), 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 \(\Lambda \ne 0\), 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 \(m\ne 0\), which corresponds to a naked singularity.
References
S.W. Hawking, Black holes in general relativity. Commun. Math. Phys. 25, 152–166 (1972)
S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of SpaceTime (Cambridge University Press, Cambridge, 1973)
J.P.S. Lemos, Twodimensional black holes and planar general relativity. Class. Quantum Gravity 12, 1081–1086 (1995)
C.G. Huang, C.B. Liang, A torus like black hole. Phys. Lett. A 201, 27–32 (1995)
S. Åminneborg, I. Bengtsson, S. Holst, P. Peldan, Making antide Sitter black holes. Class. Quantum Gravity 13, 2707–2714 (1996)
R.B. Mann, Pair production of topological antide Sitter black holes. Class. Quantum Gravity 14, L109–L114 (1997)
L. Vanzo, Black holes with unusual topology. Phys. Rev. D 56, 6475–6483 (1997)
D.R. Brill, J. Louko, P. Peldán, Thermodynamics of (3 + 1)dimensional black holes with toroidal or higher genus horizons. Phys. Rev. D 56, 3600–3610 (1997)
K.S. Stelle, Renormalization of higherderivative quantum gravity. Phys. Rev. D 16, 953–969 (1977)
K.S. Stelle, Classical gravity with higher derivatives. Gen. Relativ. Gravit. 9, 353–371 (1978)
H. Lü, A. Perkins, C.N. Pope, K.S. Stelle, Black holes in higher derivative gravity. Phys. Rev. Lett. 114, 171601 (2015)
H. Lü, A. Perkins, C.N. Pope, K.S. Stelle, Spherically symmetric solutions in higher derivative gravity. Phys. Rev. D 92, 124019 (2015)
H. Lü, A. Perkins, C.N. Pope, K.S. Stelle, Lichnerowicz modes and black hole families in Ricci quadratic gravity. Phys. Rev. D 96, 046006 (2017)
K. Kokkotas, R.A. Konoplya, A. Zhidenko, NonSchwarzschild blackhole metric in four dimensional higher derivative gravity: analytical approximation. Phys. Rev. D 96, 064007 (2017)
K. Goldstein, J.J. Mashiyane, Ineffective higher derivative black hole hair. Phys. Rev. D 97, 024015 (2018)
J. Podolský, R. Švarc, V. Pravda, A. Pravdová, Explicit black hole solutions in higherderivative gravity. Phys. Rev. D 98, 021502 (2018)
R. Švarc, J. Podolský, V. Pravda, A. Pravdová, Exact black holes in quadratic gravity with any cosmological constant. Phys. Rev. Lett. 121, 231104 (2018)
A. Bonanno, S. Silveravalle, Characterizing black hole metrics in quadratic gravity. Phys. Rev. D 99, 101501 (2019)
J. Podolský, R. Švarc, V. Pravda, A. Pravdová, Black holes and other exact spherical solutions in quadratic gravity. Phys. Rev. D 101, 024027 (2020)
V. Pravda, A. Pravdová, J. Podolský, R. Švarc, Black holes and other spherical solutions in quadratic gravity with a cosmological constant. Phys. Rev. D 103, 064049 (2021)
Y. Huang, D.J. Liu, H. Zhang, Novel black holes in higher derivative gravity. JHEP 02, 057 (2023)
H.A. Buchdahl, On Eddington’s higher order equations of the gravitational field. Proc. Edinb. Math. Soc. 8, 89–94 (1948)
H.A. Buchdahl, A special class of solutions of the equations of the gravitational field arising from certain gaugeinvariant action principles. Proc. Natl. Acad. Sci. USA 34, 66–68 (1948)
W. Nelson, Static solutions for fourth order gravity. Phys. Rev. D 82, 104026 (2010)
H. Lü, Y. Pang, C.N. Pope, J.F. VázquezPoritz, AdS and Lifshitz black holes in conformal and Einstein–Weyl gravities. Phys. Rev. D 86, 044011 (2012)
Z.Y. Fan, H. Lü, Thermodynamical first laws of black holes in quadraticallyextended gravities. Phys. Rev. D 91, 064009 (2015)
H.A. Buchdahl, On a set of conforminvariant equations of the gravitational field. Proc. Edinb. Math. Soc. 10, 16–20 (1953)
R.J. Riegert, Birkhoff’s theorem in conformal gravity. Phys. Rev. Lett. 53, 315–318 (1984)
P.D. Mannheim, D. Kazanas, Exact vacuum solution to conformal Weyl gravity and galactic rotation curves. Astrophys. J. 342, 635–638 (1989)
S. Deser, B. Tekin, Shortcuts to high symmetry solutions in gravitational theories. Class. Quantum Gravity 20, 4877–4883 (2003)
D. Klemm, Topological black holes in Weyl conformal gravity. Class. Quantum Gravity 15, 3195–3201 (1998)
R.G. Cai, Y. Liu, Y.W. Sun, A Lifshitz black hole in four dimensional \(R^2\) gravity. JHEP 10, 080 (2009)
G. Cognola, O. Gorbunova, L. Sebastiani, S. Zerbini, Energy issue for a class of modified higher order gravity black hole solutions. Phys. Rev. D 84, 023515 (2011)
G. Cognola, M. Rinaldi, L. Vanzo, S. Zerbini, Thermodynamics of topological black holes in \(R^2\) gravity. Phys. Rev. D 91, 104004 (2015)
V. Pravda, A. Pravdová, J. Podolský, R. Švarc, Exact solutions to quadratic gravity. Phys. Rev. D 95, 084025 (2017)
E. AyónBeato, A. Garbarz, G. Giribet, M. Hassaïne, Analytic Lifshitz black holes in higher dimensions. JHEP 04, 030 (2010)
S.H. Hendi, B. Eslam Panah, S.M. Mousavi, Some exact solutions of F(R) gravity with charged (a)dS black hole interpretation. Gen. Relativ. Gravit. 44, 835–853 (2012)
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein’s Field Equations, 2nd edn. (Cambridge University Press, Cambridge, 2003)
J.B. Griffiths, J. Podolský, Exact SpaceTimes in Einstein’s General Relativity (Cambridge University Press, Cambridge, 2009)
G.W. Gibbons, S.W. Hawking, Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738–2751 (1977)
S.W. Hawking, D.N. Page, Thermodynamics of black holes in antide Sitter space. Commun. Math. Phys. 87, 577–588 (1983)
H.K. Kunduri, J. Lucietti, Classification of nearhorizon geometries of extremal black holes. Living Rev. Relativ. 16, 8 (2013)
J. Ehlers, W. Kundt, Exact solutions of the gravitational field equations, in Gravitation: An Introduction to Current Research. ed. by L. Witten (Wiley, New York, 1962), pp.49–101
Acknowledgements
This work has been supported by the Czech Academy of Sciences (RVO 67985840) and research grant GAČR 1909659S.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP^{3}. SCOAP^{3} supports the goals of the International Year of Basic Sciences for Sustainable Development.
About this article
Cite this article
Pravdová, A., Pravda, V. & Ortaggio, M. Topological black holes in higher derivative gravity. Eur. Phys. J. C 83, 180 (2023). https://doi.org/10.1140/epjc/s10052023113389
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052023113389