1 Introduction

BRXTH596

The Lovelock Black Holes

Cecilia Garraffo and Gaston Giribet

Brandeis Theory Group, Martin Fisher School of Physics

Brandeis University, Waltham, MA 02454-9110.

Instituto de Astronomía y Física del Espacio, CONICET

Ciudad Universitaria, C.C. 67 Suc. 28, 1428, Buenos Aires, Argentina.

Department of Physics, Universidad de Buenos Aires and CONICET

Ciudad Universitaria, Pabellón I, 1428. Buenos Aires, Argentina.

Centro de Estudios Científicos, CECS, Valdivia, Chile

Arturo Prat 514, Valdivia, Chile.

Lovelock theory is a natural extension of Einstein theory of gravity to higher dimensions, and it is of great interest in theoretical physics as it describes a wide class of models. In particular, it describes string theory inspired ultraviolet corrections to Einstein-Hilbert action, while admits the Einstein general relativiy and the so called Chern-Simons theories of gravity as particular cases. Recently, five-dimensional Lovelock theory has been considered in the literature as a working example to illustrate the effects of including higher-curvature terms in the context of AdS/CFT correspondence.

Here, we give an introduction to the black hole solutions of Lovelock theory and analyze their most important properties. These solutions can be regarded as generalizations of the Boulware-Deser solution of Einstein-Gauss-Bonnet gravity, which we discuss in detail here. We briefly discuss some recent progress in understading these and other solutions, like topological black holes that represent black branes of the theory, and vacuum thin-shell wormhole-like geometries that connect two different asymptotically de-Sitter spaces. We also make some comments on solutions with time-like naked singularities.

1 Introduction

Why higher-curvature corrections?

It is a common belief that General Relativity, despite its fabulous success in describing our Universe at middle and large scale, has to be corrected at short distance. In particular, the apparent tension between Einstein’s theory and quantum field theory supports the idea that General Relativity is merely an effective model that would be replaced in the UV regime by a different theory, and such a new theory would ultimately permit us to make sense of what we call Quantum Gravity. The natural scale at which one expects such short distance corrections to manifestly appear is the Planck scale , determined by the Newton’s coupling constant .

At present, the most successful candidate to represent a quantum theory of gravity is String Theory (or its mother theory, M-theory). In fact, one of the predictions of string theory is the existence of a massless particle of spin whose dynamics at classical level is governed by Einstein equations

(1)

In addition, string theory also predicts next-to-leading corrections to (1), which would be relevant at distances comparable with the typical length scale of the theory . These short-distance corrections are typically described by supplementing Einstein-Hilbert action by adding higher-curvature terms [3], correcting General Relativity in the UV regime. As a result, the stringy spin interaction turns out to be finite, and this raises the hope to finally have access to a consistent theory of quantum gravity.

To investigate black hole physics in higher-curvature gravity theories, the first question we have to answer is whether such theories actually induce short-distance modifications to the black hole geometry or not. Despite expectations that the inclusion of higher-curvature terms in the gravitational action yields modifications to General Relativity, it is not necessarily the case that such modifications manifestly appear in the static spherically symmetric sector of the space of solutions. In fact, as we will see below, Schwarzschild geometry usually resists modifications. In turn, first it is important to identify the theories of gravity that yield modifications to the spherically symmetric solution.

Schwarzschild metric as a persistent solution

To warm up, let us start by considering a very simple example of higher-curvature term. Consider the action

(2)

which corresponds to Einstein-Hilbert action in four dimensions augmented with the square of the curvature scalar, where is a coupling constant with dimensions of length. This action is a particular case of the so-called -gravity theories, which are defined by adding to the Einstein-Hilbert Lagrangian a function of the Ricci scalar . It is well known that -gravity theories are equivalent (after field redefinition that involves a conformal transformation) to General Relativity coupled to a scalar field , provided a suitable self-interaction potential that depends on the function (see [4] and references therein). In this sense, these theories are not different from particular models of quintessence. Here, we are interested in less simple models; however, let us consider (2) as the starting point of our discussion.

A remarkable point is that the theory defined by action (2) admits (Anti-) de Sitter-Schwarzschild metric as its static spherically symmetric solution. In particular, when the theory still admits the Schwarzschild solution even for , and it is due to the property .

The theory defined by action (2) is not the only theory of gravity that admits Schwarzschild metric as a persistent solution. Actually, this is a rather common feature of theories with higher-curvature terms. In the case of quadratic terms in four dimensions this is an indirect consequence of the Gauss-Bonnet theorem111The simplest pure gravitational theory that excludes Schwarzschild solution in four dimensions is a cubic contraction of the Weyl tensor [5]. In dimension , and because the Kretschmann invariant is independent from the quadratic scalars and , quadratic deformation of Einstein gravity may exclude Schwarzschild-Tangherlini solution.. A second example we can consider is Einstein gravity coupled to conformally invariant gravity; namely

(3)

where is a coupling constant and is the Weyl tensor, whose quadratic contraction reads

(4)

The equations of motion associated to this action read

(5)

where is the Bach tensor,

(6)

It is easy to show that, when , Scwarzschild metric solves equations (5) as well. This follows from the fact that Bach tensor (6) vanishes if Ricci tensor vanishes, and thus all solutions to General Relativity are also solutions of (5).

Another example of a modified theory that admits Schwarzschild metric as a solution is the Jackiw-Pi theory [6]. This theory has recently attracted much attention [7]. It is defined by the action

(7)

where the function is a Lagrange multiplier that couples to the Pontryagin density , constructed via the dual curvature tensor

where is the volume 4-form. The inclusion of the non-dynamical field comes from the fact that the Pontryagin density is a total derivative. Action (7) is often called Chern-Simons modified gravity; however, this has to be distinguished from the Chern-Simons gravitational theories we will discuss in the section 2.

It is not hard to see that the equations of motion derived from action (7) are solved by the Schwarzschild metric. Actually, this is because the Pontryagin density of Schwarzschild metric vanishes. In contrast, Kerr metric has non-vanishing Pontryagin form, and thus it is not a solution of Jackiw-Pi theory. In fact, the rotating solution of this theory has not yet been found, and this represents an interesting open problem as the Jackiw-Pi theory is considered as a phenomenologically viable correction to Einstein theory.

Summarizing, there are several models that, while representing short distance corrections to General Relativity, still admit the Schwarzschild metric as an exact solution. In particular, this implies that such models can not be the solution to problems like the issue of the black hole singularity. On the other hand, there are other models which, still being integrable, do yield deviations from General Relativity solutions even in the static spherically symmetric sector. In this paper we will be concerned with one of such models. We will study a very special case of higher-curvature corrections to Einstein gravity in higher dimensions, and we will see that substantial modifications to Schwarzschild solution are found at short distances.

Higher-curvature terms in higher dimensions

In addition to higher-curvature corrections to Einstein theory, string theory makes other strong predictions about nature. Probably, the most important ones are the existence of supersymmetry and the existence of extra dimensions. In fact, one of the requirements for superstring theory to be consistent is the space-time to have dimensions; and we learn from our daily experience that six of these extra dimensions have to be hidden somehow.

This digression convinces us that studying higher-curvature modification of General Relativity in higher dimensions seems to be important to address the problem of quantum gravity, at least within the context of string theory. This is precisely the subject we will study here. More precisely, in this paper we will investigate how the string inspired higher-curvature corrections to Einstein-Hilbert action modify the black hole physics in the UV regime. This turns out to be a very important question since the black holes are known to be a fruitful arena to explore gravitational phenomena beyond the classical level.

The prototypical example we will analyze is -dimensional quadratic Lovelock Lagrangian. But, first, before introducing this theory, let us begin by considering a much more general example. Consider the action

(8)

where the constants and are the coupling constants for each quadratic term. The field equations obtained by varying the action (8) with respect to the metric read

Action (8) is the most general quadratic action one can write down in -dimensions. For , the Gauss-Bonnet theorem permits to fix without loss of generality. In , however, three quadratic invariants are needed to describe the most general Lagrangian of this type.

For generic values of the coupling constants and , the equations of motion (1) are fourth-order differential equations for the metric (i.e. there are terms prportional to , and ). Nevertheless, a remarkable property of (8) is that there exists one particular choice of the coupling constants , and that results in the cancellation of all these higher order terms, yielding second order differential equations. It is easy to see that this choice is , which only gives a non-trivial modification to Einstein theory for . Actually, in and this choice corresponds to Lovelock theory (see (16) below); namely

(10)

It is worth emphasizing that this choice of coupling constants that yields second order equations of motion is unique (up to a free parameter ). This feature also holds in dimensions, and is a consequence of a more general result known as the Lovelock theorem [1].

In this paper we will be mainly concerned with the theory defined by action (8) with . Besides the uniqueness of the choice , which already makes this model interesting in its own right, let us say that this is exactly the effective Lagrangian that appears in the low energy action of heterotic string theories in the appropriate frame (and in M-theory compactifications too).

Higher-curvature terms from string theory effective action

Now, let us sketch how the five-dimensional Lovelock theory arises in the low energy limit of M-theory (and, consequently, of string theory) when the theory is compactified from (resp. ) to .

M-theory is supposed to be an extension of string theory; a fundamental theory that, in certain regime, would flow to string theory [8].

This Mother-theory, if it exists, is yet to be found; nevertheless, we do know what it has to look like in certain low energy limit: it has to look like eleven-dimensional supergravity augmented with higher-curvature terms. That is, the bosonic sector of the M-theory effective action is given by the graviton (i.e. the metric) and the 3-form gauge field (with field strength ). Including the pure gravitational fourth order corrections (), this effective action takes the form222The eleven-dimensional Newton constant is given by the Planck scale . [9]

where the ellipses stand for the fermionic content and higher-order contributions. These higher order contributions include terms like () and also couplings of the form ( ); we will not consider these terms here. However, let us mention that the existence of the terms in the action above are related to the terms ( ) through supersymmetry, although indirectly.

The tensor in the third line of (1) is defined in terms of the way it acts on antisymmetric tensors of second rank, namely

where tr() refers to the trace of .

The term in the fourth line in (1) is actually one of the terms that appear in the Lagrangian of Lovelock theory (see (16) below, where this term is expressed in an alternative way). In contrast, the term in the third line, which is of the same order, does not correspond to a term in the Lovelock theory333Actually, while second-order terms of heterotic string theory expressed in a particular frame agree with the second-order term of the Lovelock theory, the fourth-order terms of Type IIA and IIB string theories (and M-theory) do not agree with the fourth-order term of the Lovelock theory..

A string theory contribution similar to that of the third and fourth lines of (1) also appears in ten dimensions [3]. This can be written as follows444Compactifying to four dimensions gives raise to the higher-curvature correction See [10] for a recent discussion on these quartic terms in four dimensions.

Now, let us analyze what happens when the M-theory effective action we discussed above (including the higher-curvature terms ) is compactified to five dimensions. Let us assume we reduce from to by compactifying six of the eleven dimensions in compact Calabi-Yau (CY) threefold. In that case, the effective action of the five-dimensional theory takes the form [11, 12]

(12)

where we used units such that , and where the coupling is a quantity that depends on the details of the internal CY manifold555 More precisely, are the components of the second Chern-class of the Calabi-Yau space, while are the so-called scalar components of the vector multiplet, which are proportional to the Kähler moduli of the Calabi-Yau; see also [13]. The quantity is given by the integral of the 6-dimensional extension of the 4-dimensional Euler characteristic over CY, namely . In addition, the dimensional reduction of terms gives raise to other corrections, like the shifting of the coefficient of the Einstein-Hilbert term..

In turn, we see that quadratic terms in (12) come from the terms666Let us be reminded of the fact that M-theory effective action also has other terms of the form TrTr . of (1). We observe that action (12) resembles a particular case of (8), namely the case with , identifying . This is precisely the theory we will study in this paper: the most general quadratic theory of gravity with equations of motion of second order, which, as we have just seen, arises as Calabi-Yau compactifications of M-theory. We already mentioned that a quadratic action similar to (12) also appears in the 1-loop corrected effective action of heterotic string theory. Written in the Einstein frame, the coupling of higher-curvature terms in the heterotic effective action is given by , where the dilaton field clearly contributes. Black holes solutions in dilatonic Einstein-Gauss-Bonnet theory were studied in Refs. [14, 15, 16].

Higher-curvature terms in AdS/CFT correspondence

Because an action like (12) also appears in the effective action of the heterotic string, it is also usually referred to as ”string inspired higher-curvature corrections”. In turn, it represents a nice model to explore the effects of next-to-lading contributions of string theory to gravitational physics. In particular, this five-dimensional (Lovelock) model of gravity was recently considered in the context of AdS/CFT holographic correspondence [17]. Actually, one of the applications of the Lovelock theory to AdS/CFT that has attracted attention recently was that of showing that the so-called Kovtun-Son-Starinets bound [18, 19] may be violated in a theory that contains higher-curvature corrections. The Kovtun-Son-Starinets bound (KSS) is the conjecture that states: the ratio between the shear viscosity to the entropy of all the materials obey the universal relation

(13)

In Refs. [20, 21, 22] it was observed that when action (8) with and is considered in asymptotically locally AdS space, then the conformal field theory (CFT) that would be dual to such a theory of gravity would satisfy

(14)

what then would violate (13) for . Therefore, the KSS bound would be violated for all the CFTs with a Einstein-Gauss-Bonnet gravity duals with positive , and this is precisely the sign of that comes from string theory.

The consideration of five-dimensional Lovelock theory as a working example to study the effects of including higher-curvature terms in AdS/CFT has been an active line of research in the last years. Just recently, very interesting papers discussing the interplay between causality and higher-curvature terms in the context of AdS/CFT appeared [23, 24]; Ref. [24] considers the case of Lovelock theory. Causality in the dual CFT constrains the value of the coupling of the quadratic Gauss-Bonnet term777It is usual to define the dimensionless parameter . In terms of this parameter, the permitted range reads .. The causality bound comes from demanding that the recidivist gravitons that hit back the boundary after a bulk excursion do not spoil locality in the CFT. The permitted range for the coupling turns out to be

(15)

The value (i.e. ), which is not in this range, corresponds to the Chern-Simons theory of gravity, which we will discuss in section 2. At this value, the ration would vanish (if it were the case that the theory at has a dual description too888We thank D. Hofman and J. Edelstein for conversations about the case .).

Other works discussing higher-curvature actions in the context of AdS/CFT appeared recently. See for instance [25], where holographic superconductors in five-dimensional Lovelock gravity are considered, showing that higher-curvature corrections affect the condensation phenomenon. Besides, the corrections in the non-relativistic version of AdS/CFT [28, 27] were also studied, and the Lovelock theory is also used in Ref. [26] as the working example for illustrating the renormalization of the dynamical exponent . Let us now move to discuss Lovelock theory in detail.

The Lovelock Theory of Gravity

Lovelock theory is the most general metric theory of gravity yielding conserved second order equations of motion in arbitrary number of dimensions . In turn, it is the natural generalization of Einstein’s general relativity (GR) to higher dimensions [1, 2]. In three and four dimensions Lovelock theory coincides with Einstein theory [29], but in higher dimensions both theories are actually different. In fact, for Einstein gravity can be thought of as a particular case of Lovelock gravity since the Einstein-Hilbert term is one of several terms that constitute the Lovelock action. Besides, Lovelock theory also admits other quoted models as particular cases; for instance, this is the case of the so called Chern-Simons gravity theories, which in a sense are actual gauge theories of gravity.

On the other hand, Lovelock theory resembles also string inspired models of gravity as its action contains, among others, the quadratic Gauss-Bonnet term, which is the dimensionally extended version of the four-dimensional Euler density. This quadratic term is present in the low energy effective action of heterotic string theory [30, 31, 32], and it also appears in six-dimensional Calabi-Yau compactifications of M-theory; see [13] and references therein. In [33] Zwiebach earlier discussed the quadratic Gauss-Bonnet term within the context of string theory, with particular attention on its property of being free of ghost about the Minkowski space. Besides, the theory is known to be free of ghosts about other exact backgrounds [34]. For a nice and concise review on stringy corrections to gravity actions [35, 36, 37] see the introduction of [38] and references therein. For interesting recent discussions on higher order curvature terms see [13, 10, 39, 40, 41] and related works.

The Lovelock theory represents a very interesting scenario to study how the physics of gravity results corrected at short distance due to the presence of higher order curvature terms in the action. In this paper we will be concerned with the black hole solutions of this theory, and we will discuss how short distance corrections to black hole physics substantially change the qualitative features we know from our experience with black holes in GR. So, let us introduce the Lovelock theory.

The Lagrangian of the theory is given as a sum of dimensionally extended Euler densities, and it can be written as follows999Here we are ignoring the boundary terms. We will consider these terms in section 2. [1, 2]

(16)

where the generalized Kronecker -function is defined as the antisymmetric product

(17)

Each term in (16) corresponds to the dimensional extension of the Euler density in dimensions101010The -dimensional Euler density is given by M, where, again, we are not considering the boundary terms., so that these only contribute to the equations of motion for . Consequently, without lack of generality, in (16) can be taken to be for even dimensions and for odd dimensions111111See [42] for a related discussion on gravitational dynamics and Lovelock theory..

The coupling constants in (16) have dimensions of [length], although it is convenient to normalize the Lagrangian density in units of the Planck scale . Expanding the product in (16) the Lagrangian takes the familiar form

(18)

where we see that coupling corresponds to the cosmological constant , while with are coupling constants of additional terms that represent ultraviolet corrections to Einstein theory, involving higher order contractions of the Riemann tensor . In particular, the second order term is precisely the Gauss-Bonnet term discussed above. The cubic term121212cf. [44], where it was shown that no unambiguous cubic terms arise in string theory effective action; in particular, the Lovelock cubic term is studied. Cubic terms are strongly constrained by supersymmetry. still has a moderate form [43], namely

(19)

The fourth order term coincides with the pure gravitational term in the last line of (1).

Even though the way of writing Lovelock action in its tensorial form (18)-(19) may result clear to introduce the theory, it is not the most efficient way for most of the calculations one usually deal with. A more convenient way of working out these expressions is to resort to the so-called first-order formalism, which turns out to be useful both for formal purposes and for practical ones. Nevertheless, it is important to point out that the first-order formalism is not necessarily equivalent to the second-order formalism, so it should not be regarded merely as a different nomenclature. In the first-order formalism, both the vielbein and the spin connection are considered as independent degrees of freedom, and the torsion acquires in general propagating degrees of freedom [45]. It is only in the torsion-free sector where both formulations are equivalent; notice that the vanishing torsion condition is always allowed by the equations of motion; see [46], see also [47]. We will make use of the first-order formalism at the end of section 2, as it is almost unavoidable in the discussion of Chern-Simons theory. However, with the intention to make the exposition as friendly as possible, we will avoid abstruse notation in the rest of the paper. In any case, since we could not afford to give all the definitions necessary to introduce the subject, we will assume the reader is familiarized with basic notions of the theory of gravity and with the standard nomenclature.

Overview

The paper is organized as follows. In section 2, we analyze the spherically symmetric black hole solutions in Lovelock theory [49, 50]. In five-dimensions this is given by the Boulware-Deser solution [34], whose most important properties we review. The special properties of electrically charged black holes [51, 52] are also briefly discussed. In one of the subsections of section 2, we extend the analysis to those black objects whose horizon geometries correspond to more general spaces of constant (but not necessarily positive) curvature [48, 53]. These are the so-called topological black holes, which can be thought of as black brane solutions of the theory. Also, we briefly review the most relevant features of the Lovelock black hole thermodynamics [54], focusing our attention on the qualitative features that have no analogue in GR. Throughout the discussion, the five-dimensional black hole of the Einstein-Gauss-Bonnet theory will serve as prototypical example. In section 3, we discuss the role of boundary terms [55] and the junction conditions these yield [56, 58, 57]. We show how solutions with non-trivial topology can be constructed by a method of a geometric surgery. Particular attention is focussed on vacuum wormhole solutions recently found [59, 60]. Finally, we study the spherically symmetric solutions that develop naked curvature singularities. We study these naked singularities with quantum probes and show that, in spite of the divergence in the curvature, these spaces are well-behaved within a quantum mechanical context.

2 The Lovelock black holes

Spherically symmetric black hole solutions

Let us first consider the theory in five dimensions. Since in the term does not contribute to the equations of motion, the five-dimensional Lovelock theory basically corresponds to Einstein gravity coupled to the dimensional extension of the four dimensional Euler density, i.e. the theory that is usually referred as Einstein-Gauss-Bonnet theory (EGB). The spherically symmetric static solution of EGB theory was obtained by Boulware and Deser in Ref. [34]. The metric takes the simple form [61]

(20)

where is the metric of a unitary -sphere, and where the metric function is given by

(21)

with . Here we used the standard convention , , and, besides, we have set the Newton constant to a specific value for short. From (21) we notice that there exist two different branches of solutions to the spherically symmetric ansatz (20), namely and , and this reflects the fact that the equations of motion give a differential equation quadratic in the metric function . As usual, the parameter arises here as an integration constant, and it corresponds to the mass of the solution131313For the discussion on the computation of charges in this theory see the list of references [62, 64, 63, 65, 66, 67, 68, 69, 70, 72]; see also [73, 74, 75, 76, 154]., up to the factor we absorbed141414More precisely, in the definition of we absorbed a factor where is the surface of the -sphere, and where is the Newton constant, given by , which has been fixed to a specific values such that . in .

It is worth mentioning that (20)-(21) is the most general spherically symmetric solution to EGB theory, provided the fact that the metric is smooth everywhere and that the parameters and are generic enough. In turn, a Birkhoff theorem holds for this model [77, 78, 79, 5]. It is important to emphasize that for very particular choices of the set of parameters , degeneracy in the space of solutions can appear, and in those special cases the Birkhoff’s theorem can be circumvented; see [78] for a very interesting discussion. To our knowledge, the most complete analysis of the EGB analogue of Birkhoff’s theorem was performed in [70], where the Nariai-type solutions [80] where also discussed.

If , the solution corresponding to in (21) may represent a black hole solution whose horizon, in the case , is located at . On the other hand, as long as and , the branch has no horizon but presents a naked singularity at .

Solutions and have substantially different behaviors, and only one of them tends to the GR solution in the small limit. In fact, in the limit the branch looks like

(22)

where we see it approaches the five-dimensional (Anti)-de Sitter-Schwarzschild-Tangherlini solution [81]. On the other hand, in the limit the solution corresponding to the branch behaves like

(23)

and we see it acquires a large effective cosmological constant term . In particular, this implies that microscopic (A)dS space-time is a solution of the theory even for . This feature was expressed by Boulware and Deser [34] by saying that EGB theory has its own cosmological constant problem, with . In a sense, the branch is commonly believed to be a false vacuum of the theory, and it is known to present ghost instabilities [34]; see also [82].

The branch , on the other hand, is well-behaved, and it represents short distance corrections to GR black holes (22). While at short distances the black hole solutions of both theories are substantially different due to the effects of the Gauss-Bonnet term, in the large distance regime the Lovelock black hole (20) with behaves like a GR black hole whose parameters and get corrected by finite- subleading contributions ; for instance, the parameter of the mass term gets corrected yielding the effective mass . In the large limit, the next-to-leading -dependent contribution to (22) goes like .

The damping of this additional term, which in dimensions goes like , is actually strong, and, for distance large enough, it is negligible even in comparison with semiclassical corrections to the metric due to field theory backreaction, which typically go like (for instance, see [83]).

All these features are essentially due to the nature of the Gauss-Bonnet term, and also hold in higher dimensions. In fact, it is straightforward to generalize solution (20) to the case of EGB gravity in dimensions, and the metric is seen to adopt a very similar form [34]. Actually, it is given by simply replacing the element of the -sphere in (20) by the element of the unitary -sphere , and by replacing the piece in (21) by .

In spite of the non-polynomial form of (21), the horizon structure of Boulware-Deser solution is quite simple, and in dimensions the horizon location is given by the roots of the polynomial

(24)

where has been appropriately rescaled by a  -dimensional constant factor.

From (24) we observe that the five-dimensional case is actually a remarkable example since, among other special features, it allows to have massive solutions with naked singularities. We mentioned above that if and the black hole horizon is located at , and this implies a lower bound for the spherical solution not to develop a naked singularity, namely . That is, for we do find naked singularities even for the well-behaved branch with positive . For the model with a second order term this only occurs in . In seven dimensions, for instance, the Boulware-Deser solution with develops horizons at and then the horizon always exists provided , . Naked singularities in dimensions usually arise when a term of order is present in the action. So, for the EGB theory this only occurs for .

Another special feature of the (uncharged) five-dimensional case is that the metric (20) turns out to be finite at the origin, namely . Nevertheless, the curvature still diverges at the origin, although not in a dramatic way. We will return to this point in the last section where we will discuss naked singularities.

It could be important to mention that the analysis of the dynamical stability of EGB black holes is also special for . The stability analysis under tensor mode perturbations has been explored recently, and it has been shown that the EGB theory exhibits some differences with respect to Einstein theory; at least, it seems to be the case for sufficiently small values of mass in five and six dimensions [84] where instabilities arise; see also Refs. [85, 87, 86, 88]. In this sense, the cases and are special ones. See Ref. [89] for an interesting recent discussion. On the other hand, let us be reminded of the fact that in dimensions the Lovelock action (16) presents also additional terms of higher order , so that in the Boulware-Deser black hole geometry (20)-(21) only corresponds to a very special example of Lovelock black hole.

Spherically symmetric solutions in higher dimensions containing an arbitrary higher order terms in (16) can be implicitly found by solving a polynomial equation of degree whose solutions give the metric function ; this was originally noticed by Wheeler in [49, 50]. Moreover, several explicit examples containing arbitrary amount of terms are also known. These correspond to particular choices of the couplings in (16). One of these explicitly solvable cases corresponds to the Chern-Simons theory, which exists in odd dimensions. We will briefly discuss this special case below. A remarkable fact is that in the case a term of the Lovelock expansion (16) is considered in the action, then the spherically symmetric solution may still take a very simple expression, and, depending on the coupling constants , it may merely correspond to replacing the square root in (21) by a power ; see [90, 91, 92, 93] for explicit examples.

Adding electric charge

On the other hand, it is quite remarkable that electrically charged black hole solutions in Lovelock theory also present a very simple form. The solutions charged under both Maxwell and Born-Infeld electrodynamics have been known for long time [51, 52], and these solutions were reconsidered recently [94]. In general, the metric function of a charged solution takes the form (21) but replacing the mass parameter by a mass function that depends on the radial coordinate . Function depends on the particular electromagnetic Lagrangian one considers. In the case of Maxwell theory, and in five dimensions, this function is given by the energy contribution , where represents the electric charge of the black hole, and where the UV cut-off in the integral is absorbed in the definition of the additive constant . More precisely, for charged black holes in Einstein-Gauss-Bonnet-Maxwell theory we have , as it was originally noticed by Wiltshire [51]. On the other hand, in the case of black holes charged under Born-Infeld theory, the function is given by

(25)

where the is the Born-Infeld parameter, according to the standard form of the Lagrangian .  In the large limit , and then the metric approaches the charged solution for the Maxwell-Einstein-Gauss-Bonnet theory,

(26)

As expected, the five-dimensional Reissner-Nordström black hole is recovered in the large regime for the case .

Charged solutions of Lovelock theory coupled to Born-Infled electrodynamics present curious features that are not present in the case of Einstein-Maxwell theory. Perhaps the most relevant one is the existence of single-horizon charged solutions [94]. Besides, Lovelock black holes charged under Maxwell electrodynamics, and for certain values of the coupling constants , can develop curvature singularities at fixed values of the radial coordinate [93], making necessary to exclude a region of the space. This kind of divergence is usually called branch singularity, and it can also be present in uncharged solutions, as it happens for solutions of EGB gravity with and , [95, 96].

As in the case of Hoffmann’s solution in Born-Infeld-Einstein [97] theory, the Lovelock black holes charged under Born-Infled theory induce a contribution to the mass coming from the finite concentration of electromagnetic energy around the singularity. Of course, this happens because both theories coincides at large distances. For finite values of , has a large distance behavior that induces a mass contribution . In particular, this implies that, for certain range of and , naked singularities in five dimensions may arise even for values of the effective mass grater than . Notice that the cosmological constant term also acquires a -dependent contribution .

In the next subsection we will consider a generalization of the black hole solutions reviewed here. We will discuss extended black objects in EGB theory.

Topological black holes

One of the interesting aspects of Lovelock theory is that it admits another class of black objects, whose horizons are not necessarily positive curvature hypersurfaces [53]. These solutions are usually called topological black holes, and their metric are obtained by replacing the -sphere in (20) by a base manifold of constant (but not necessarily positive) curvature, provided a suitable shifting in the metric function . Namely, these solutions read151515These are analogues of the topological black holes previously known in four-dimensions, which, at constant hypersurfaces, correspond to fibrations of (closed) base manifolds with non-trivial topology.

(27)

where the metric function is now given by with , being its sign that of the curvature of the horizon hypersurface, whose line element is . For the Boulware-Deser solution (20)-(21) is recovered. In general, the base manifold here may be given by a more general constant curvature space: For instance, it can be given by the product of hyperbolic spaces for the case of negative curvature , or merely by a flat space piece . In turn, solutions (27) correspond to black brane type geometries. Such black objects represent fibrations over constant curvature -dimensional hypersurfaces, implying that the event horizon, in the cases it exists, is not necessarily a compact simply connected manifold.

Consider for example the five-dimensional EGB theory with negative cosmological constant , and its black brane solution of the form

(28)

with

(29)

where . These objects (brane-like configurations and topological black holes) have attracted some attention recently due to their curious properties, and, more recently, these were considered in applications to AdS/CFT; see for instance [21, 22].

In [98], an exhaustive classification of static topological black hole solutions of five-dimensional Lovelock theory was presented. The authors considered an ansatz such that spacelike sections are given by warped product of the radial coordinate and an arbitrary base manifold , and they showed that, for values of the coupling constant generic enough, the base manifold must be necessarily of constant curvature, and then the solutions of the theory reduce to the topological extension of the Boulware-Deser metric of the form (27). In addition, they showed that for the special case where the coupling is appropriately tuned in terms of the cosmological constant , then the base manifold could admit a wider class of geometries, and such enhancement of the freedom in choosing allows to construct very curious solutions with non-trivial topology. We will return to this point in section 2.

The existence of black holes with generic horizon structure was also analyzed in [99], where selection criteria for the base manifold were discussed161616The authors of [99] derived a necessary constraint to be obeyed by the Euclidean manifold that is candidate to represent a horizon geometry of a black hole solution in -dimensional Einstein-Gauss-Bonnet theory. They proved that such a -manifold has to obey the equation , where is the Weyl tensor in dimensions., and the authors concluded that sensible physical models strongly restrict most of the examples of exotic black holes with non-constant curvature horizons. Moreover, the different horizon structures were also studied in [48, 96] together with its relation to the asymptotic behavior of the corresponding solutions; see also [100, 101, 102, 103, 104]. Recently, the electrically charged topological black hole solutions were also analyzed, both for the case of the second order Lovelock theory in [95, 100] and for the case of the third order171717Recently, references [106, 107, 108, 109, 110, 111] discussed other classes of solutions. We will not comment on these solutions here. Lovelock theory in [103].

One of the most interesting aspects of these objects with non-trivial horizon geometries is that they enable us to construct a very simple class of Kaluza-Klein black holes with interesting properties from the four-dimensional viewpoint. For instance, such a solution was recently studied by Maeda and Dadhich in Ref. [112]. These Kaluza-Klein black holes are given by a product MH between a four-dimensional manifold M and a -dimensional hyperbolic space H. It turns out that the four-dimensional piece of the geometry asymptotically approaches the charged black hole in locally AdS space. In turn, the Gauss-Bonnet term acts by emulating the Reissner-Nordström term for large , while it changes the geometry at short distances [113, 114, 115]. In addition to these solutions, other exotic Kaluza-Klein Lovelock black hole solutions with arbitrary order terms of the form and for a specific values of the coefficients  were studied in [116]. These black holes are different from those studied in [112], and are obtained by considering black -brane geometries of the form MT in the Lovelock theory with and . These solutions exist for even, and, in addition, the horizon structure also depends on . Analogous toric compactifications of the form MT were studied in [117], and warped brane-like configurations were also discussed in both [116] and [117].

It was shown in [116] that, in spite of the difference between Lovelock theory and Einstein theory, the qualitative features of thermodynamic stability of brane-like configurations in both theories are considerable similar, although the higher order terms can be seen to contribute. For example, the thermodynamical analogue of Gregory-Laflamme transition between black hole and black string configurations was discussed in [116]. Extended string-like objects in Lovelock theory and their thermodynamics were also discussed in [118, 119, 154]. We discuss black hole thermodynamics in the next subsection.

Thermodynamics

The purpose of this section is to describe the general aspects of black hole thermodynamics in Lovelock theory. In fact, one of the most interesting features of the Lovelock theory regards the thermodynamics of its black hole solutions. This is because it is in the analysis of the black hole thermodynamics where the substantial differences between Lovelock theory and Einstein theory manifest themselves.

Pioneer works where the Lovelock black hole thermodynamics was discussed in detail are references [120, 121]; see also [122, 123, 124]. In [54], Jacobson and Myers derived a close expression for the entropy of these solutions in dimensions, and they showed that the entropy of these black holes does not satisfy the area law, but contains additional terms that are given by a sum of intrinsic curvature invariants integrated over the horizon.

The thermodynamics of charged solutions was originally studied by Wiltshire in Refs. [51, 52], while the thermodynamics of topological black holes was studied more recently, in Refs. [125, 48, 53]. The study of charged topological black holes in presence of cosmological constant was addressed in [126], where the most general solution of this type in EGB theory was obtained. References [127, 128, 129] also analyze topological black holes and their thermodynamics; see also [102, 130].

The aim of this section is to discuss the more relevant thermodynamical features of Lovelock solutions. To do this, we will consider again the five-dimensional case (20)-(21). Actually, besides it represents a simple instructive example, the five-dimensional case is also special in what concerns thermodynamical properties. It is the best example to see that substantial differences between Lovelock gravity and Einstein gravity exist.

It is easy to verify that the Hawking temperature associated to the solution in with is given by

(30)

Then, we see that, as expected, (30) behaves like the Hawking temperature of a GR solution if the black hole is large enough, , going like . On the other hand, temperature tends to zero for small values of , going like . This implies that the specific heat changes its sign at length scales of order , and a direct consequence of this phenomenon is that five-dimensional Lovelock black holes turn out to be thermodynamically stable, as they yield eternal remnants. This can be easily verified by considering the rate of thermal radiation which goes like , behaving like at short distances.

Nevertheless, it is worth pointing out that for dimension the functional form of the temperature is substantially different from the case , as it includes an additional term which is actually proportional to . The general formula reads

(31)

which implies that, in , the short distance limit is given by , and the specific heat is then negative. This is the reason why the thermodynamic behavior of higher dimensional Einstein-Gauss-Bonnet black holes turns out to be more similar to that in Einstein theory if . In general, eternal black holes arise in dimensions if an -order term is present in the action.

So, let us return to our instructive example of five dimensions. The entropy associated to (30) is given by

(32)

from what we observe that black holes of Lovelock theory do not in general obey the Bekenstein-Hawking area law. Actually, some particular solutions, corresponding to topological black holes with flat horizon geometry , do obey the area law [131, 130], but it is not the case for spherically symmetric static solutions. A very interesting discussion on the area law181818In [105] other corrections to area law were studied. The authors thank S. Shankaranarayanan for pointing out this references to them. is that of Ref. [104], where a version of the area law for symmetric dynamical black holes defined by a future outer trapping horizon was derived. There, the authors discussed the differences between the branches of solutions with GR limit and those without it, and argue how for the latter one still can define a concept of increasing dynamical entropy.

Notice that the second term in the right hand side of (32) implies that if then the entropy turns out to be negative for sufficiently small black holes191919Refs. [133, 134] discuss related features. The authors thank S. Odintsov for pointing out these references to them.. This was discussed in [132], where it was argued there that an additive ambiguity in the definition of the entropy could be a solution for the negative entropy contributions; see also the related discussion in [48]. In any case, the theory for negative values of the coupling constant is somehow pathological in several respects. It not only gives negative contributions to the entropy, but also ghost instabilities and strange causal structure arise if . We will not consider the negative values of here.

Because of the current interest in black hole thermodynamics of higher order theories, we consider convenient to mention that the entropy function formalism, recently proposed by A. Sen [135] within the context of the attractor mechanism, works nicely for the case of Lovelock black holes. In particular, this was recently studied in [136] for the case of EGB black holes, and it was explicitly shown that (32) is recovered by analyzing the near horizon geometry. A rather general analysis was presented in Ref. [137]. Very interesting discussions are those of Refs. [138, 139].

The thermodynamic properties of topological black holes are also very interesting; see for instance [140, 130]. As we already mentioned, it can be shown that those black objects whose horizons are of zero curvature do obey the area law for the entropy density. For instance, consider the black brane geometry (28), which is solution of the theory with negative cosmological constant, . It is straightforward to check that the Hawking temperature of this solution is given by

(33)

and that the area formula for the entropy density does hold in this special case. Remarkably, identical expression for the temperature is obtained in the particular case of the Chern-Simons theories of gravity, which we discuss in the next subsection.

Chern-Simons black holes

Now, let us move on, and analyze a very particular case of Lovelock theory which exist in odd dimensions. This is the so-called Chern-Simons gravity (CS), and can be thought of as a higher-dimensional generalization of the Chern-Simons description of three-dimensional Einstein gravity [141]. Basically, these theories are those particular cases of Lovelock Lagrangian (16) that admit a formulation in terms of a Chern-Simons action. As we will discuss, these models are given by a very precise choice of the set of coefficients .

To discuss CS gravity theories202020It is worth pointing out that the CS theories we are referring to herein are different to those discussed in Refs. [142, 143]. it is convenient to resort to the first-order formalism which, in spite of its advantage, it is paradoxically avoided in physics discussions. So, let us first review some basic notions: Consider the vielbein , which defines the metric as , where we are using the standard notation such that the greek indices correspond to the space-time while the latin indices are reserved for the tangent space. Now, consider the 1-form associated to the vielbein, defined by , and the corresponding 1-form associated to the spin connection , defined by . These quantities enable us to define the so-called curvature 2-form, which is given by

and is related to the Riemann tensor by . The torsion-free condition is then given by

In this language, local Lorentz invariance of the theory is expressed in terms of the covariant derivative

(34)

where represent the parameters of the transformation.

The remarkable fact is that, for particular cases of the action (16), if the coupling constants are chosen appropriately, the theory exhibits an additional local symmetry. For instance, if we consider the case , such additional symmetry turns out to be given by the invariance of the Lagrangian density under the gauge transformation

(35)

That is, the CS theory possesses a local symmetry under gauge transformation , with being a parameter. This is actually an off-shell local gauge symmetry of the theory (16) that arises for special choices of the coupling constants , as far as the boundary conditions are also chosen in the appropriate way. Besides, it can be easily verified that transformation (35), once considered together with (34), satisfies the Poincaré algebra  , and this is why these theories are usually referred as Poincaré-Chern-Simons gravitational theories [144]; see also [46] for an excellent introduction to Chern-Simons gravity.

So, let us specify which are the theories that possess the gauge symmetry like212121Notice that, as mentioned, (35) is the transofrmation that corresponds to the case . The analogous tranformation for the case takes a slightly different form, see [46]. (34)-(35), namely the CS theories. To do this, first it is convenient to rewrite the Lovelock Lagrangian. In the first-order formalism, the Lovelock action corresponding to (16) in dimensions can be written as

(36)

where correspond to independent coefficients that are a rearrangement of the coefficients . In (36), the convention is such that the coupling has been set to (or, alternatively speaking, it has been absorbed in the definition of the curvature ), so that in this notation we have , and .

It is worth noticing that, in order to represent the most general form of (16), the coefficients in (36) should be allowed to take complex values. In fact, Lovelock action (16) with real coefficients can correspond to (36) with imaginary . An example is given by the five-dimensional theory whose action reads , which leads to the particular form of (16) where no Einstein-Hilbert contribution is present, but only the cosmological constant and the Gauss-Bonnet term appear, with for a real .

The CS gravity theories, however, are given by real values of . More precisely, CS theory correspond to the special case where the coupling in (36) combine to give only one value for the effective cosmological constant . In terms of the Lagrangian density (16) this corresponds to taking the coupling constants to be for, while is given by the cosmological constant . It is important to mention that (36) corresponds to the case of negative cosmological constant, which yields the CS theory with the AdS group (i.e. the group ) as the one that generates the gauge symmetry. The case of positive is simply obtained by changing , while the Poincaré invariant theory is obtained through the Inonu-Winger contraction of (A)dS group; see [46] for details. An example of Poincaré invariant CS is given by the Lagrangian containing only the quadratic Gauss-Bonnet term in five dimensions, without the Einstein-Hilbert term and with .

As it is well known, an example of the CS gravity theory is given by three-dimensional Einstein theory, whose action222222For simplicity here we have fixed the Newton constant according to .,