January 20, 2021

Lovelock black holes with maximally symmetric horizons

Hideki Maeda, Steven Willison and Sourya Ray

Centro de Estudios Científicos (CECS), Casilla 1469, Valdivia, Chile

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Abstract

We investigate some properties of -dimensional spacetimes having symmetries corresponding to the isometries of an -dimensional maximally symmetric space in Lovelock gravity under the null or dominant energy condition. The well-posedness of the generalized Misner-Sharp quasi-local mass proposed in the past study is shown. Using this quasi-local mass, we clarify the basic properties of the dynamical black holes defined by a future outer trapping horizon under certain assumptions on the Lovelock coupling constants. The vacuum solutions are classified into four types: (i) Schwarzschild-Tangherlini-type solution; (ii) Nariai-type solution; (iii) special degenerate vacuum solution; (iv) exceptional vacuum solution. The conditions for the realization of the last two solutions are clarified. The Schwarzschild-Tangherlini-type solution is studied in detail. We prove the first law of black-hole thermodynamics and present the expressions for the heat capacity and the free energy.

###### Contents

- 1 Introduction
- 2 Preliminaries
- 3 Generalized Misner-Sharp quasi-local mass
- 4 Dynamical black holes
- 5 Vacuum solutions
- 6 Schwarzschild-Tangherlini-type black holes
- 7 Summary
- A Tensor decomposition
- B Lovelock gravity with a unique maximally symmetric vacuum
- C Schwarzschild-Tangherlini-type solution with electric charge

## 1 Introduction

Einstein’s general theory of relativity is the most successful theory of gravity which describes the movement of the planets in the solar system as well as the dynamics of the expanding universe. On the other hand, higher-dimensional gravity has been a prevalent subject for the last decade. This is strongly motivated by Superstring/M-theory which predicts the existence of extra dimensions in our universe. On the more Relativity side, this prevalence was triggered by the remarkable discovery of the black-ring solution in five dimensions [1] which explicitly shows the richer structure of the higher-dimensional spacetime. Undoubtedly, the most exciting prediction of general relativity is the existence of spacetime regions from which nothing can escape, namely black holes. Black holes carry a lot of information about the gravity theory in question, and the study of such objects is expected to yield insights into the fundamental properties of gravity.

The Lovelock Lagrangian is the most natural generalization of the Einstein-Hilbert Lagrangian for general relativity in arbitrary dimensions without torsion such that the resulting field equations are of second-order [2]. It consists of a sum of dimensionally extended Euler densities and reduces to the Einstein-Hilbert Lagrangian with a cosmological constant in four dimensions. The second-order field equations in Lovelock gravity give rise to the ghost-free nature of the theory. In string theory, the Einstein-Hilbert Lagrangian, which is the Lovelock term linear in the curvature tensor, is realized as the lowest order term in the Regge slope expansion of strings. However, the forms of the higher-curvature terms appearing as the next-order stringy corrections depend on the type of string theories. (See [3], for example.) Among them, the second-order Lovelock term, called the Gauss-Bonnet term, appears in heterotic string theory [4]. Apart from anything else, Lovelock gravity is of potential importance to give insights how special four-dimensional spacetime is.

In dimensions greater than four, the Lovelock Lagrangian contains many coupling constants, one for each term of particular order in the curvature. This in turn implies that there are in general more than one maximally symmetric vacuum solution such as Minkowski, de Sitter (dS), or anti-de Sitter (AdS). Interestingly, when we tune the coupling constants to admit a unique vacuum, the Lagrangian of maximal order in the curvature has an additional symmetry in odd dimensions and becomes a gauge theory for the Poincaré, dS, or AdS group, where the last two are the smallest nontrivial choices of such required groups containing the translation symmetry group on a pseudosphere. This resulting theory is called Chern-Simons gravity and has been studied with keen interest due to aesthetic reasons. (See [5] for a review.)

However, for generic coupling constants, the higher curvature terms in the field equations in Lovelock gravity make the analyses cumbersome. In order to reduce the complexity, until now the research on Lovelock gravity has been focused on spacetimes with high degrees of symmetry. Especially, -dimensional spacetimes having symmetries corresponding to the isometries of an -dimensional maximally symmetric space have been intensively investigated. There is a Schwarzschild-Tangherlini-type vacuum black-hole solution and the corresponding Jebsen-Birkhoff theorem was established for generic coupling constants [6]. Based on this solution, the thermodynamical aspects of static black holes have been investigated [7, 8, 9]. (See [10] for a review of Lovelock black holes.) Thermodynamical properties of more general concepts of horizons in Lovelock gravity were also studied [11, 12, 13, 14, 15, 16]. Very recently, it was shown that, unlike the four-dimensional Schwarzschild black hole, sufficiently small higher-dimensional vacuum Lovelock black holes are dynamically unstable in asymptotically flat spacetime against tensor-type and scalar-type gravitational perturbations in even and odd dimensions, respectively [17]. (See [18] for the charged case.) This remarkable result could suggest that the four-dimensional spacetime is special.

At present, there are many important open problems in less symmetric spacetimes in Lovelock gravity. For example, the counterpart of the Myers-Perry rotating vacuum solution is yet to be found [19]. (A five-dimensional rotating vacuum solution was found in the case of Chern-Simons gravity [20].) Strong results in symmetric spacetimes will provide a firm basis for the study of less symmetric spacetimes. The present paper is written for this purpose. In fact, the dynamical aspects of Lovelock black holes have not been fully investigated so far. In the quadratic case, namely in Einstein-Gauss-Bonnet gravity, it was shown in the symmetric spacetime that the results in general relativity can be generalized in a unified manner by introducing a well-defined quasi-local mass [21, 22, 23, 24, 25]. This allows to classify the solutions depending on whether they have a general relativistic limit or not, and it was shown that the solution which has a general relativistic limit has similar properties as in general relativity, while the other has pathological properties. In the present paper, those results are extended to general Lovelock gravity. Using the quasi-local mass in Lovelock gravity proposed in [22] , we prove a number of propositions in a similar fashion to the general relativistic case. We also present a number of new results.

The rest of the present paper is constituted as follows. In the following section, we give an introduction to Lovelock gravity and our spacetime ansatz. In Section III, we study the properties of the generalized Misner-Sharp quasi-local mass proposed in [22]. Section IV focuses on the study of dynamical black holes defined by a future outer trapping horizon. We also present several exact solutions with various kinds of matter fields as concrete examples of dynamical spacetimes with trapping horizons. In section V, we classify all the vacuum solutions. In section VI, we study the static vacuum black hole in more detail. Concluding remarks and discussions including future prospects are summarized in Section VII. In appendix A, we present expressions for the decomposition of geometric tensors in our symmetric spacetime. In appendix B, we give a brief comment on the case with a unique maximally symmetric vacuum. In appendix C, we discuss about the Schwarzschild-Tangherlini-type solution with electric charge.

Our basic notation follows [26]. The conventions for curvature tensors are and . The Minkowski metric is taken to be the mostly plus sign, and Roman indices run over all spacetime indices. We adopt the units in which only the -dimensional gravitational constant is retained.

Note added:

At the final stage of the present work we were informed that X.O. Camanho and J.D. Edelstein were working on a similar subject [27].

## 2 Preliminaries

### 2.1 Lovelock gravity

We begin with an introduction to Lovelock gravity. The Lovelock action in -dimensional spacetime is given by

(2.1) | ||||

(2.2) |

where . We assume without any loss of generality and make no assumptions about the signs of unless otherwise mentioned. The symbol denotes a totally anti-symmetric product of Kronecker deltas, normalized to take values and [2, 28], defined by

(2.3) |

is a coupling constant with dimension . is related to the cosmological constant by . The general Lovelock Lagrangian density is given by an arbitrary linear combination of dimensionally continued Euler densities. In even dimension , the variation of the -dimensional Euler density is a total derivative and does not contribute to the field equations and in any dimension , the -dimensional Euler density vanishes identically. Therefore the sum truncates at a finite order. The first four Lovelock Lagrangians are explicitly shown as

(2.4) | ||||

(2.5) | ||||

(2.6) | ||||

(2.7) |

In even dimensions, the contribution to the action of the -th order Lagrangian becomes a topological invariant and does not contribute to the field equations.

The gravitational equation following from this action is given by

(2.8) |

where is the energy-momentum tensor for matter fields obtained from and

(2.9) | ||||

(2.10) |

The tensor is given from . is satisfied for . There is an identity between the Lovelock Lagrangian and the Lovelock tensor:

(2.11) |

The field equations (2.8) contain up to the second derivatives of the metric. The first four Lovelock tensors are explicitly shown as

(2.12) | ||||

(2.13) | ||||

(2.14) | ||||

(2.15) |

### 2.2 Lovelock tensors in the symmetric spacetime

Consider an -dimensional spacetime to be a warped product of an -dimensional maximally symmetric space with sectional curvature and a two-dimensional orbit spacetime under the isometries of . is the unit metric on . The curvature of the -dimensional maximally symmetric space is given by

(2.16) |

where the superscript means the geometrical quantity on . We also assume that is compact in order to have a finite value for certain physical quantities.

For such a spacetime, without loss of generality the line element may be given by

(2.17) |

where and is a scalar on . Using the expressions of the decomposed geometric tensors given in Appendix A, the components of the -th order Lovelock tensor tangent to and the -th order Lovelock Lagrangian are respectively given by

(2.18) | ||||

(2.19) |

where

(2.20) |

Here is a metric compatible linear connection on , , and . The contraction was taken over the two-dimensional orbit space and is the Ricci scalar on .

The component of the -th order Lovelock tensor is given by

(2.21) |

where

(2.22) |

With Eqs. (2.18), (2.19), and (2.21), we can confirm the identity (2.11). The component of the left-hand side of the field equation (2.8) is

(2.23) |

where

(2.24) | ||||

(2.25) |

We note that for even by definition.

Hence, the most general energy-momentum tensor compatible with this spacetime symmetry governed by Lovelock equations is given by

(2.26) |

where and are a symmetric two-tensor and a scalar on , respectively.

We note that the results in the present paper are valid for and when we set .

## 3 Generalized Misner-Sharp quasi-local mass

In this section, we show that the generalized Misner-Sharp quasi-local mass proposed in [22] is certainly a natural counterpart of the Misner-Sharp mass in general relativity [29]. The generalized Misner-Sharp mass in Lovelock gravity was defined as

(3.1) |

where denotes the area of [22]. In the four-dimensional spherically symmetric case without a cosmological constant, reduces to the Misner-Sharp quasi-local mass [29]. We assume the reality of throughout the paper.

### 3.1 Unified first law

First, it is shown that the components of the Lovelock equation on can be written in the form of the so-called unified first law using . This unified first law was first shown in [30] adopting a coordinate system on . Here we present a tensorial proof.

###### Proposition 1

(Unified first law in Lovelock gravity.) The following unified first law holds:

(3.2) |

where

(3.3) | ||||

(3.4) |

Proof. The component of the field equation (2.8) on is

(3.5) |

The following expressions are useful:

(3.6) | ||||

(3.7) |

The derivative of gives

(3.8) | ||||

(3.9) |

where we used Eq. (3.6). Using the fied equation (2.8), we obtain the unified first law (3.2) from Eq. (3.9).

### 3.2 Quasi-local mass and the Kodama vector

In this subsection we show that the generalized Misner-Sharp mass is interpreted as a locally conserved quantity. We define the locally conserved current vector as

(3.10) | ||||

(3.11) |

Here , and is a volume element of . is the Kodama vector [32] and satisfies

(3.12) |

Hence, it is timelike (spacelike) in the untrapped (trapped) region. is divergence-free and generates a preferred time evolution vector field in the untrapped region. (See also [33].) In a static spacetime, reduces to the hypersurface-orthogonal timelike Killing vector. Also, is divergence-free because of the identity . Indeed, is a quasi-local conserved quantity associated with a locally conserved current , which is seen in the following expression of :

(3.13) |

(Similar expressions to (3.10) and (3.13) were obtained also in the stationary case given in section 7 in [34].) The above equation is obtained as follows. Using the identity , it follows from equation (3.9) that

(3.14) |

Furthermore, using the identity we obtain

(3.15) |

Hence, we obtain

(3.16) |

which shows Eq. (3.13). From Eq. (3.13), we immediately obtain

(3.17) |

which implies that is conserved along . As a consequence, the integral of over some spatial volume with boundary gives as an associated charge as , where is a directed surface element on . (See section III in [22].)

It can be shown for static spacetimes that is vanishing and the unified first law (3.2) takes a simpler form. Using Eq. (3.7), we obtain

(3.18) |

Using the identity , we obtain

(3.19) |

Comparing Eqs. (3.18) and (3.19), we conclude that in static spacetimes, in which is a Killing vector and hence . This results surely implies that represents an energy flux.

### 3.3 Global mass, monotonicity, positivity, and rigidity

In this subsection, we show several important properties of the generalized Misner-Sharp mass. First it is shown that the quasi-local mass asymptotes to the Arnowitt-Deser-Misner (ADM) mass, calculated using the same geometrical formula as used in general relativity, in asymptotically flat spacetime. (The proof is similar to Proposition 2 in [22] since the higher-order terms converge to zero more rapidly at spatial infinity. )

###### Proposition 2

(Asymptotic behavior in asymptotically flat spacetime.) In an -dimensional asymptotically flat spacetime, coincides with the higher-dimensional ADM mass at spatial infinity.

Hereafter, we adopt the double-null coordinates on as

(3.20) |

Null vectors and are taken to be future-pointing. The area expansions along two independent future-directed null vectors and are defined as

(3.21) | ||||

(3.22) |

where is the area of the symmetric subspace defined by Eq. (3.4). They are used in the following definition.

###### Definition 1

A trapped (untrapped) surface is a compact -surface with . A trapped (untrapped) region is the union of all trapped (untrapped) surfaces. A marginal surface is a compact -surface with .

In this section, we fix the orientation of the untrapped surface by and , i.e., and are ingoing and outgoing null vectors, respectively. With this orientation, the Kodama vector

(3.23) |

is future-pointing in the untrapped region, where we used and .

In the double null coordinates, the quasi-local mass is expressed as

(3.24) |

while the unified first law (3.2) is written as

(3.25) | ||||

(3.26) |

The null energy condition for the matter field implies

(3.27) |

while the dominant energy condition implies

(3.28) |

The quasi-local mass has the following monotonic property independent of the signs of the Lovelock coupling constants .

###### Proposition 3

(Monotonicity.) Under the dominant energy condition, is non-decreasing (non-increasing) in any outgoing (ingoing) spacelike or null direction on an untrapped surface.

Since the equations (3.25) and (3.26) do not contain the Lovelock coupling constants explicitly, the proof of the above proposition is the same as the Proposition 4 in [22].

Next, we show the positivity of in the spherically symmetric spacetime ().

###### Proposition 4

(Positivity.) Suppose . If the dominant energy condition holds on an untrapped spacelike hypersurface with a regular center in spherically symmetric spacetime, then .

Proof. The point where is called center if it defines the boundary of . A central point is called regular if

(3.29) |

holds around the center and singular otherwise, where a constant is assumed to be non-zero. ( and are satisfied for Minkowski, which has a regular center.) Hence, the regular center is surrounded by untrapped surfaces. From Eq. (3.24), we obtain

(3.30) |

around the regular center. From Eq. (3.30), we obtain

(3.31) | ||||

(3.32) |

around the regular center. By Eqs. (3.31) and (3.32) and Proposition 3, the inequality is satisfied under the dominant energy condition since the regular center is surrounded by untrapped surfaces. As a result, by Eq. (3.30), is non-negative around the regular center. Then, the proposition follows from Proposition 3.

Proposition 4 shows the positivity of in the untrapped region with a regular center. On the other hand, we may obtain a positive lower bound for if there is a marginal surface. The quasi-local mass on the marginal surface is given by

(3.33) |

where . Using the Proposition 3, the following mass inequality is shown [35];

###### Proposition 5

(Mass inequality.) If the dominant energy condition holds, then holds on an untrapped spacelike hypersurface of which the inner boundary is a marginal surface with radius . For and , this lower bound is positive for and non-negative for .

Using Propositions 2, 4, and 5, we can show the following positive mass theorem in asymptotically flat spherically symmetric spacetime in Lovelock gravity.

###### Proposition 6

(Positivity of the ADM mass.) Suppose and and the dominant energy condition is satisfied in a spherically symmetric, asymptotically flat, and regular spacetime. Then, the ADM mass is non-negative.

Proof. First let us consider the case where there is no marginal surface in the spacetime. If there is a regular center, the ADM mass is non-negative by Proposition 4. If there is no regular center, there is at least one wormhole throat on a spacelike hypersurface, where

(3.34) |

is satisfied with a radial spacelike vector . This equation gives

(3.35) |

Since , or is satisfied. Because the asymptotically flat region consists of untrapped surfaces, there is at least one marginal surface on a spacelike hypersurface by the mean value theorem in the latter case. Hence, both of them reduce to the case with a marginal surface on a spacelike hypersurface. If there is a marginal surface on a spacelike hypersurface, the ADM mass is positive because of Proposition 5 and the fact that the asymptotically flat region consists of untrapped surfaces.

In addition to the monotonicity and positivity, has the following rigidity property. This proposition is shown for non-negative Lovelock coupling constants and claims that if the generalized Misner-Sharp mass is vanishing somewhere, the spacetime is locally vacuum. (The proof is similar to Proposition 1 in [24].)

###### Proposition 7

(Rigidity.) Under the dominant energy condition with , , and , if is satisfied, then holds there.

We have shown that is a well-defined quasi-local mass and a natural generalization of the Misner-Sharp mass in general relativity. In the next section, we use to evaluate the mass of a dynamical black hole.

### 3.4 Branches of solutions

Actually, the Lovelock equations admit multiple branches of solutions. In order to see this, we define the following function:

(3.36) |

with is equivalent to Eq. (3.1) and consistent with the Lovelock equations. Here we assume . The function is regarded as a function of with coefficients depending on . Hence, if has real solutions of for a given value of , there are real branches of solutions there.

If we have , there is at least one real solution for independent of the Lovelock coupling constants by the mean value theorem. Hence, the following proposition is shown.

###### Proposition 8

(Existence of real solutions.) Let be the largest integer with non-zero in and suppose . Then, the field equations admit real metric for any given value of if is odd.

Indeed, in the second-order Lovelock gravity (), namely in Einstein-Gauss-Bonnet gravity, the metric of the Boulware-Deser-Wheeler solution with negative mass can be complex for some value of [36, 37]. The above proposition claims that at least one of three branches is real in the cubic Lovelock vacuum solution () [38]. We should note that although there is a real branch for each value of , it is not necessarily true that the same branch would be real for all values of . Therefore, we cannot guarantee that there exists a single spacetime which extends over all values of . In general Lovelock gravity, the term with the highest power in is in odd dimensions, while it is in even dimensions (since ). Thus, the following corollary is given from Proposition 8.

###### Corollary 1

In general Lovelock gravity in dimensions with , the field equations admit a real metric for any given value of if or , where is an integer.

In contrast, in the case where or , may not have a real solution for some and there the metric becomes complex and unphysical. We will see this in the Schwarzschild-Tangherlini-type solution in Proposition 18.

## 4 Dynamical black holes

In this section, we study the properties of dynamical black holes defined by a future outer trapping horizon. Trapping horizons were defined by Hayward [41, 42]. Our system contains those in [11, 12, 13, 14] as special cases with a particular type of matter field. In this section, we don’t use the orientation in the previous section. Instead, we set on the marginal surface without loss of generality. A marginal surface is classified as follows.

###### Definition 2

A marginal surface is future if , past if , bifurcating if , outer if , inner if and degenerate if .

A future (past) marginal surface means that the outgoing (ingoing) null rays are marginally trapped, namely instantaneously parallel on the horizon, on the marginal surface. A Kodama vector (3.11) is future-pointing (past-pointing) on the future (past) marginal surface.

###### Definition 3

A trapping horizon is the closure of a hypersurface foliated by future or past, outer or inner marginal surfaces.

Among all the classes, a future outer trapping horizon defines a dynamical black hole because it means that the ingoing null rays are converging (), while the outgoing null rays are instantaneously parallel on the horizon () and diverging just outside the horizon and converging just inside () [41, 42]. On the other hand, a past outer trapping horizon defines a dynamical white hole. Accordingly, we call the closure of a hypersurface foliated by bifurcating or degenerate marginal surfaces, a bifurcating trapping horizon or a degenerate trapping horizon, respectively.

### 4.1 Properties of trapping horizon

In this subsection, we show some basic properties of trapping horizons. The following lemma is essential in the proofs of the later propositions.

###### Lemma 1

Under the null energy condition with for and ,