hepth/0407050
Sequences of Bubbles and Holes:
New Phases of KaluzaKlein Black Holes
Henriette Elvang, Troels Harmark, Niels A. Obers
Department of Physics, UCSB
Santa Barbara, CA 93106, USA
The Niels Bohr Institute
Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
, ,
We construct and analyze a large class of
exact five and sixdimensional regular and static solutions of
the vacuum Einstein equations. These solutions describe sequences
of KaluzaKlein bubbles and black holes, placed alternately so
that the black holes are held apart by the bubbles. Asymptotically
the solutions are Minkowskispace times a circle,
i.e. KaluzaKlein space, so they are part of the phase
diagram introduced in hepth/0309116. In particular, they
occupy a hitherto unexplored region of the phase diagram, since
their relative tension exceeds that of the uniform black string.
The solutions contain bubbles and black holes of various
topologies, including sixdimensional black holes with ring
topology and tuboid topology .
The bubbles support the ’s of the horizons against
gravitational collapse. We
find two maps between solutions, one that relates five and
sixdimensional solutions, and another that relates solutions in
the same dimension by interchanging bubbles and black holes. To
illustrate the richness of the phase structure and the
nonuniqueness in the phase diagram, we consider in
detail particular examples of the general class of solutions.
Contents
 1 Introduction
 2 Review of the phase diagram
 3 The static KaluzaKlein bubble
 4 Generalized Weyl solutions
 5 Fivedimensional bubbleblack hole sequences
 6 Sixdimensional bubbleblack hole sequences
 7 Solutions with equal temperatures
 8 Properties of specific solutions
 9 Conclusions
 A Details on the solution
1 Introduction
In fourdimensional vacuum gravity, a black hole in an asymptotically flat spacetime is uniquely specified by the ADM mass and angular momentum measured at infinity [1, 2, 3, 4]. Uniqueness theorems [5, 6] for dimensional () asymptotically flat spacetimes state that the only static black holes in pure gravity are given by the SchwarzschildTangherlini black hole solutions [7]. However, in pure gravity there are no uniqueness theorems for nonstatic black holes with , or for black holes in spacetimes with nonflat asymptotics. On the contrary, there are known cases of nonuniqueness. An explicit example of this occurs in five dimensions for stationary solutions in an asymptotically flat spacetime: for a certain range of mass and angular momentum there exist both a rotating black hole with horizon [8] and rotating black rings with horizons [9].
The topic of this paper is static black hole spacetimes that are asymptotically Minkowski space times a circle , in other words, we study static black holes^{1}^{1}1We use “black hole” to denote any black object, no matter its horizon topology. in KaluzaKlein theory. For brevity, we generally refer to these solutions as KaluzaKlein black holes. Changing the boundary conditions from asymptotically flat space to asymptotically opens up for a rich spectrum of black holes. As we shall see, the nonuniqueness of KaluzaKlein black holes goes even further than for black holes in asymptotically flat space .
Much numerical [10, 11, 12, 13, 14, 15, 16] and analytical [17, 18, 19, 20, 21, 22, 23] work has been done to investigate the “phase space” of black hole solutions in KaluzaKlein theory. Part of the motivation has been by the wish to reveal the endpoint for the classical evolution of the unstable uniform black string [24].
Recently, Refs. [19, 21] proposed a phase diagram as part of a program for classifying all black hole solutions of KaluzaKlein theory. The input for the phase diagram consist of two physical parameters that are measured asymptotically: the dimensionless mass , where is the ADM mass and is the proper length of the KaluzaKlein circle at infinity, and the relative tension [19, 21, 25]. This is the tension per unit mass of a string winding the KaluzaKlein circle. The phase diagram makes it possible to illustrate the different branches of solutions and exhibit their possible relationships. The main purpose of this paper is to construct and analyze a large class of exact five and sixdimensional KaluzaKlein black hole solutions occupying an hitherto unexplored region of the phase diagram.
In KaluzaKlein spacetimes, it is wellknown that there exist both uniform and nonuniform black strings with the same mass [26, 10, 11, 27, 13, 16] (see also [28, 29, 30]). There is also a family of topologically spherical black holes localized on the KaluzaKlein circle [17, 14, 15, 22, 23]. These solutions, however, can be told apart at infinity, because they exist for different values of the relative tension . We show in this paper that there is nonuniqueness of black holes in KaluzaKlein theory, and we argue that for a certain open set of values of and there is even infinite nonuniqueness of KaluzaKlein black holes. Infinite nonuniqueness has been seen before in [31] for black rings with dipole charges in asymptotically flat fivedimensional space. The solutions we present here are, on the contrary, solutions of pure gravity and the nonuniqueness involves regular spacetimes with multiple black holes. While some configurations have black holes whose horizons are topologically spheres, we also encounter black rings with horizon topologies (for ), and in six dimensions a black tuboid with horizon.
A crucial feature of the black hole spacetimes studied in this paper is that they all involve KaluzaKlein “bubbles of nothing”. Expanding KaluzaKlein bubbles were first studied by Witten in [32] as the endstate of the semiclassical decay of the KaluzaKlein vacuum . The bubble is the minimal area surface that arises as the asymptotic smoothly shrinks to zero at a nonzero radius. The expanding Witten bubbles are nonstatic spacetimes with zero ADM mass. The KaluzaKlein bubbles appearing in the solutions of this paper are on the other hand static with positive ADM mass. KaluzaKlein bubbles will be reviewed early in the present paper.
The first solution combining a black hole and a KaluzaKlein bubble was found by Emparan and Reall [33] as an example of an axisymmetric static spacetime in the class of generalized Weyl solutions. Later Ref. [34] studied spacetimes with two black holes held apart by a KaluzaKlein bubble and argued that the bubble balances the gravitational attraction between the two black holes, thus keeping the configuration in static equilibrium.
One natural question to ask is what the role of these solutions is in the phase diagram of KaluzaKlein black holes. To address this issue we recall that one useful property of the phase diagram is that physical solutions lie in the region [19]
(1.1) 
These bounds were derived using various energy theorems [35, 36, 19, 25]. However, so far only solutions in the lower region,
(1.2) 
have been discussed in connection to the phase diagram. This region includes the following three known branches:
An obvious question is thus whether there are KaluzaKlein black hole solutions occupying the upper region
(1.3) 
We will find in this paper that it is in fact the solutions involving KaluzaKlein bubbles that occupy this region. A special point in the phase diagram is the static KaluzaKlein bubble which corresponds to the single point in the phase diagram, where is the dimensionless mass of the static KaluzaKlein bubble.
More generally, we construct exact metrics for bubbleblack hole configurations with bubbles and black holes in dimensions. These are regular and static solutions of the vacuum Einstein equations, describing sequences of KaluzaKlein bubbles and black holes placed alternately, e.g. for we have the sequence:
We will call this class of solutions bubbleblack hole sequences and refer to particular elements of this class as solutions. This large class of solutions, which was anticipated in Ref. [33], includes as particular cases the , and solutions obtained and analyzed in [33, 34]. All of these solutions have .
Besides their explicit construction, we present a comprehensive analysis of various aspects of these bubbleblack hole sequences. This includes the regularity and topology of the KaluzaKlein bubbles in the sequences, the topology of the event horizons, and general thermodynamical properties. An important feature is that the solutions are subject to constraints enforcing regularity, but this leaves independent dimensionless parameters allowing for instance the relative sizes of the black holes to vary. The existence of independent parameters in each solution is the reason for the large degree of nonuniqueness in the phase diagram, when considering bubbleblack hole sequences.
The KaluzaKlein bubbles play a key role in keeping these configurations in static equilibrium: not only do they balance the mutual attraction between the black holes, they also balance the gravitational selfattraction of black holes with nontrivial horizon topologies such as black rings. A different example of fivedimensional multiblack hole spacetimes based on the generalized Weyl ansatz was studied in Ref. [37]. Those solutions differ from ours in that they are asymptotically flat, and instead of bubbles, the black holes are held in static equilibrium by struts due to conical singularities.^{2}^{2}2 In four dimensional asymptotically flat space, the analogue of the configuration in [37] is the IsraelKahn multiblack hole solution, where the gravitational attraction of the black holes is balanced by struts between the black holes (or cosmic strings extending out to infinity). In KaluzaKlein theory the black holes are balanced by the bubbles and the metrics are regular and free of conical singularities [33, 34]. We stress that when discussing nonuniqueness we always restrict ourselves to solutions that are regular everywhere on and outside the horizon(s); thus we do not consider solutions with singular horizons or solutions with conical singularities. All bubbleblack hole solutions discussed in this paper are regular.
For the simplest cases, we will plot the corresponding solution branches in the phase diagram, where they are seen to lie in the upper region (1.3). Moreover, these examples illustrate the richness of the phase structure and the nonuniqueness in the phase diagram.
The structure and main results of the paper are as follows. We introduce in Section 2 the phase diagram and explain how and are easily computed from the asymptotic behavior of the metric. We also briefly review the three known solution branches that occupy the region (1.2), i.e. the uniform and nonuniform black strings and the localized spherical black holes.
Section 3 provides a review of the static KaluzaKlein bubble. In particular we review the argument that the static bubbles are classically unstable, and decay by either expanding or collapsing. We find a critical dimension below which the mass of the static bubble is smaller than the GregoryLaflamme mass for the uniform black string. Hence, for the endstate of the static bubble decay can be expected to be the endstate of the uniform black string, rather than the black string itself.
The bubbleblack hole sequences are constructed using the general Weyl ansatz of [33]. We review this method in section 4, where we also write down metrics for the simplest KaluzaKlein spacetimes and explain how to read of the asymptotic quantities using Weyl coordinates.
In Section 5 we construct the solution for the general bubbleblack hole sequence in five dimensions. We analyze the constraints of regularity, the structure of the KaluzaKlein bubbles, and the event horizons and their topology. We also compute the physical quantities relevant for the phase diagram and the thermodynamics. Section 6 provides a parallel construction and analysis for the sixdimensional bubbleblack hole sequences.
It is shown that the five and sixdimensional solutions are quite similar in structure and are in fact related by an explicit map. In particular, we find a map that relates the physical quantities, so that we can use it to obtain the phase diagram for the sixdimensional solutions from the fivedimensional one. This map is derived in Subsection 6.5.
For static spacetimes with more than one black hole horizon we can associate a temperature to each black hole by analytically continuing the solution to Euclidean space and performing the proper identifications needed to make the Euclidean solution regular where the horizon was located in the Lorentzian solution. The temperatures of the black holes need not be equal, and we derive a generalized Smarr formula that involves the temperature of each black hole. The Euclidean solution is regular everywhere only when all the temperatures are equal. It is always possible to choose the free parameters of the solution to give a oneparameter family of regular equal temperature solutions, which we shall denote by .
We show in Section 7 that the equal temperature solutions are of special interest for two reasons: First, the two solutions, and , are directly related by a double Wick rotation which effectively interchanges the time coordinate and the coordinate parameterizing the KaluzaKlein circle. This provides a duality map under which bubbles and black holes are interchanged. The duality also implies an explicit map between the physical quantities of the solutions, in particular between the curves in the phase diagram.
Secondly, we show that for a given family of solutions, the equal temperature solution extremizes the entropy for fixed mass and fixed size of the KaluzaKlein circle at infinity. For all explicit cases considered we find that the entropy is minimized for equal temperatures. This is a feature that is particular to black holes, independently of the presence of bubbles. As an analog, consider two Schwarzschild black holes very far apart. It is straightforward to see that for fixed total mass, the entropy of such a configuration is minimized when the black holes have the same radius (hence same temperature), while the maximal entropy configuration is the one where all the mass is located in a single black hole.
In Section 8 we consider in detail particular examples of the general five and sixdimensional bubbleblack hole sequences obtained in Sections 56. For these examples, we plot the various solution branches in the phase diagram and discuss the total entropy of the sequence as a function of the mass.
We find that the entropy of the solution is always lower than the entropy of the uniform black string of the same mass . We expect that all other bubbleblack hole sequences have entropy lower than the solution; we confirm this for all explicitly studied examples in Section 8. The physical reason to expect that all bubbleblack hole sequences have lower entropy than a uniform string of same mass, is that some of the mass has gone into the bubble rather than the black holes, giving a smaller horizon area for the same mass.
As an appetizer, and to give a representative taste of the type of results we find for the phase diagram, we show in Figure 1 the phase diagram for six dimensions (the phase diagram for five dimensions is similar). Here we briefly summarize what is shown in the plot. The horizontal line is the uniform black string branch. This branch separates the diagram into two regions: and .
The region has been the focus of many recent studies. The branch coming out of the uniform black string branch at (the GregoryLaflamme mass) is the nonuniform black string branch, which is reproduced here using numerical data courtesy Wiseman [11]. The curve starting at is the branch of spherical black holes localized on the KaluzaKlein circle. Here we plot the slope of the first part of the branch using the analytical results for small black holes [22].
It is the region that contains the bubbleblack hole sequences. The top point of the curves in this region is the static bubble solution located at . All solutions start out at this point and approach the uniform black string branch at as . The lowest lying of these curves is the solution with a single black ring (with horizon topology ) supported by a single KaluzaKlein bubble. There are also solutions lying “inside” the wedge bounded by the solution and the solution with one bubble balancing two equal size black rings with equal temperatures. The solutions in this wedge are all solutions where the two black rings are at different temperatures. We note that for any given , the solutions in the wedge provide a continuous set of bubbleblack hole sequences with the same mass. These solutions can be told apart since they have different values of . But even specifying both and does not give a unique solution. Though not visible in the figure, the top curve crosses into the wedge of solutions at the value . This curve is the one parameter family describing two equalsize KaluzaKlein bubbles supporting a black tuboid, a black hole with horizon topology .
We conclude in Section 9 with a discussion of our results and outlook for future developments. An appendix treats details of the analysis for the solution.
Notation
Throughout the paper we use to denote the spacetime dimension of the Minkowski part of the metric. The spacetimes are asymptotically , and we use for the dimension of the full spacetime.
2 Review of the phase diagram
In [19, 21] a program was set forth to categorize — in higherdimensional General Relativity — all static solutions of the vacuum Einstein equations that asymptote to for , with being the dimensional Minkowski spacetime. When an event horizon is present we call these solutions static neutral KaluzaKlein black holes, since is a KaluzaKlein type spacetime. In this section we review the ideas and results of [19, 21] that are important for this paper.
The general idea is to define a “phase diagram” and plot in it the branches of different types of static KaluzaKlein black holes. The physical parameters used in defining such a phase diagram should be measurable at asymptotic infinity. In [19, 20, 21] it was suggested that besides the proper length of the at infinity, the relevant physical parameters are the mass and the tension , which can be defined for any solution asymptoting to . The tension was defined in [38, 39, 19, 20, 25].
Let the be parameterized by the coordinate , which we take to have period . Define to be Cartesian coordinates for , so that the radial coordinate is . We consider black holes localized in , so the asymptotic region is defined by , and we write the asymptotic behavior of the metric components and as
(2.1) 
for . Note that we have chosen such that the period of is the proper length of the at infinity. It was shown in [19, 20, 25] that the ADM mass and the tension along the direction can be computed from the asymptotic metric as
(2.2) 
where sphere. is the surface volume of the
Since for given we wish to compare solutions with the same mass, it is natural to normalize the mass with respect to . We then work with the rescaled mass and the relative tension , which are dimensionless quantities defined as
(2.3) 
Not all values of and correspond to physically reasonable solutions. We have from the Weak Energy Condition, and must satisfy the bounds [19]
(2.4) 
The lower bound comes from positivity of the tension [35, 36]. The upper bound is due to the Strong Energy Condition. There is a more physical way to understand the upper bound: for a solution with the gravitational force on a test particle at infinity is attractive, while it would be repulsive if .
The aim of the work initiated in [19, 21] is to plot all static vacuum solutions that asymptote to in the phase diagram. In other words, we categorize all these solutions according to their physical parameters measured at asymptotic infinity. In this way one can get an overview of the possible solutions and one can for example see for a given mass what possible branches of solutions are available.
Previously only solutions with have been considered for the phase diagram. We focus on that part of the phase diagram in the remainder of this section; the rest of the paper will discuss solutions in the region of the phase diagram.
According to our present knowledge, the solutions with all have a local symmetry and two possible topologies for the event horizons: 1) , which we call black holes on cylinders, and 2) , which we call black strings.
There are three known branches of solutions:

Uniform black string branch. The metric for the uniform black string is constructed as the dimensional Schwarzschild metric times a circle:
(2.5) We note that , so by (2.3) a uniform black string has . Gregory and Laflamme [24, 40] discovered that the uniform string is classically unstable for , and the critical mass can be obtained numerically for each dimension . In Table 1 we list the explicit values of for . The uniform black strings are believed to be classically stable for .

The nonuniform black string branch. This branch was discovered in [27, 10]. For the beginning of the branch was studied in [10], and for a large piece of the branch was found numerically by Wiseman [11]. Recently, Sorkin [16] studied the nonuniform strings for general dimensions . The nonuniform string branch starts at with in the uniform string branch and then it has decreasing and increasing for . Sorkin [16] found that for it has instead decreasing and , which means that we have a critical dimension at where the physics of the nonuniform string branch changes. For the nonuniform black string has lower entropy than the uniform black string with the same mass , while for the nonuniform string has the higher entropy [16].
All three branches mentioned above can be described with the same ansatz for the metric. This ansatz was proposed in [17, 18], and proven in [12, 21].
In Figure 2 we display for the known solutions with in the phase diagram. The nonuniform black string branch was drawn in [19] using the data of [11]. We included in the right part of Figure 2 the copies of the black hole on cylinder branch and the nonuniform string branch [28, 21].^{3}^{3}3 We find the ’th copy of that solution by “repeating” the solution times on the circle [28, 21]. Furthermore, if the original solution is in the point of the phase diagram, then the ’th copy will be in the point [21].
For the temperature and entropy we use the rescaled temperature and entropy defined by
(2.6) 
The dimensionless quantities , , , and are connected through the Smarr formula [19, 20]
(2.7) 
We also have the first law of thermodynamics [19, 20]
(2.8) 
Using the two relations (2.7) and (2.8) one can pick a curve in the phase diagram and integrate the thermodynamics just from the points in the diagram alone [19]. In this way, the phase diagram contains the essential information about the thermodynamics of each branch. Note that for this argument we assumed that there was only one black hole present, so that there was only one temperature in the system. Later in this paper we present solutions with several disconnected event horizons, which can have different temperatures.
3 The static KaluzaKlein bubble
Static KaluzaKlein bubbles belong to the class of the solutions we wish to categorize in KaluzaKlein theory, and they turn out to play a crucial role for the solutions in the region of the phase diagram. In this section, we introduce KaluzaKlein bubbles and describe in detail the properties of static KaluzaKlein bubbles.
KaluzaKlein bubbles were discovered by Witten in [32], where it was explained that (in the absence of fundamental fermions) the KaluzaKlein vacuum is semiclassically unstable to creation of expanding KaluzaKlein bubbles. The expanding KaluzaKlein bubble solutions can be obtained by a double Wick rotation of the 5D Schwarzschild solution and the resulting spacetime is asymptotically . The KaluzaKlein bubble is located at the place where the direction smoothly closes off: this defines a minimal twosphere in the spacetime, i.e. a “bubble of nothing”. In this expanding KaluzaKlein bubble solution the minimal twosphere expands until all of the spacetime is gone. Since it is not a static solution, the expanding KaluzaKlein bubble is not in the class of solutions we consider for the phase diagram. We comment further on the expanding KaluzaKlein bubble below.
However, there exist also static KaluzaKlein bubbles. A dimensional static KaluzaKlein bubble solution can be constructed by adding a trivial timedirection to the Euclidean section of a dimensional Schwarzschild black hole. This gives the metric
(3.1) 
Clearly, this solution is static. It has a minimal sphere of radius located at . To avoid a conical singularity at , we need to be periodic with period
(3.2) 
With this choice of the KaluzaKlein circle shrinks smoothly to zero as , and the location becomes a point with respect to the and directions. Around the location the spacetime is locally of the form , where is the time, the twoplane is parameterized by and , and the becomes the minimal sphere at . Clearly, there are no boundaries at and no spacetime for , which is the reason for calling the KaluzaKlein bubble a “bubble of nothing”. Note that the bubble spacetime is a regular manifold with topology , where the is noncontractible.
Due to the periodicity of we see that the static bubble (3.1) asymptotically (i.e. for ) goes to . Thus, it belongs to the class of solutions we are interested in, i.e. static pure gravity solutions that asymptote to , and we can plot the solution in the phase diagram. Using (2.1) we read off and . From (2.3) and (3.2) we then find the dimensionless mass and the relative binding energy to be
(3.3) 
where we named this special value of as . Note that the static bubble only exists at one point in the diagram. This is because the relation between and in (3.2) means that once is chosen then all parameters are fixed. We note that since the static KaluzaKlein bubble has , it precisely saturates the upper bound in (2.4). Indeed, the Newtonian gravitational force on a test particle at infinity is zero.
The static KaluzaKlein bubble is known to be classically unstable. This can be seen from the fact that the static bubble is the Euclidean section of the Schwarzschild black hole times a trivial time direction. The Euclidean flat space (hot flat space) is semiclassically unstable to nucleation of Schwarzschild black holes. This was shown by Gross, Perry, and Yaffe [41], who found that the Euclidean Lichnerowicz operator for the Euclidean section of the four dimensional Schwarzschild solution with mass has a negative eigenvalue: with . The Lichnerowicz equation for the perturbations of the Lorentzian static bubble spacetime is (in the transverse traceless gauge), so taking the ansatz , the Lichnerowicz equation requires , ie. . So this is an instability mode of the static bubble, and the perturbation causes the bubble to either expand or collapse exponentially fast.
That the static KaluzaKlein bubble is classically unstable poses the question: what does it decay to? We discuss this in the following.
Note first that the static bubble is massive, while the expanding Witten bubble obtained as the double Wick rotation of the Schwarzschild black hole is massless. Therefore the instability of the static bubble does not connect it directly to the Witten bubble. Furthermore, if is the size of the circle at infinity, the minimal radius of the Witten bubble is , and the radius of the static bubble is . Therefore we have
(3.4) 
and this means that for given size of the KaluzaKlein circle at infinity, the radius of the static KaluzaKlein bubble is smaller than the minimal radius of the expanding KaluzaKlein bubble.
In the following, we discuss initial data describing massive bubbles that are initially expanding or collapsing, and we shall see that for given size of the KaluzaKlein circle the initially expanding bubbles have larger radii than the static bubble, and collapsing bubbles have smaller radii. Presumably the collapse of a massive bubble results in a black hole or black string, so we shall also compare the mass of the KaluzaKlein bubble to the GregoryLaflamme mass .
Initial data for massive expanding and collapsing bubbles
We now consider a subclass of a more general family of initial data for fivedimensional vacuum bubbles found by Brill and Horowitz [42]. The metric on the initial surface is
(3.5) 
with for arbitrary parameters and satisfying . The bubble is located at the positive zero of , , and by fixing the period of to be we avoid a conical singularity at . The ADM mass is , so
(3.6) 
In the following we assume that . In this case we find . Since this initial data exists for masses less than or equal to static bubble mass , it is not inconceivable that the evolution of this initial data can guide us about the possible endstates of the decay of the static KaluzaKlein bubble.
For the initial data (3.5), Corley and Jacobson [43] showed that depending on the mass of the bubble and its size (or alternatively, the size of the at infinity), the bubbles are initially going to expand or collapse according to the initial acceleration of the bubble area which is given by
(3.7) 
(the derivatives are with respect to proper time). The relation (3.7) shows that for the bubble is initially expanding, but for it is initially collapsing. We note that for , we have and , so in this case the initial data describes an initially nonaccelerating bubble. It is clear that (3.5) with is initial data for the static KaluzaKlein bubble.
It may appear surprising that there are expanding as well as collapsing bubbles with the same mass . The difference relies not on the mass, but in the radius of the bubble compared to the size of the circle at infinity. Consider for the initial data (3.5) the size of the bubble relative to the size of the KaluzaKlein circle:
(3.8) 
For the static bubble (), we have . Initially expanding bubbles () have and initially collapsing bubbles () have . So by (3.8) we see that bubbles that are bigger than the static bubble are going to expand and bubbles that are smaller are going to collapse.
In an interesting paper [44], Lehner and Sarbach recently studied numerically the evolution of the initial data (3.5) (as well as more general bubbles). They found that initially expanding bubbles continue to expand, and the smaller the mass of the bubble, the more rapidly the area of the bubble grows as a function of proper time. By studying such massive expanding bubbles, one can hope to learn about the endstate of the instability of perturbations of the static bubble.
The instability can also cause the static KaluzaKlein bubble to collapse and thereby decay to another static solution, presumably one with an event horizon.^{4}^{4}4For vacuum solutions, the initial data for an initially collapsing bubble requires a positive mass. With strong gauge fields, negative mass bubbles can also initially collapse, but after a while the collapse is halted and the bubbles bounce and begin to expand again [44]. The numerical results [44] show that for an initially collapsing massive bubble, the collapse continues until at some point an apparent horizon forms. A comparison of curvature invariants suggests that the resulting black hole is a uniform black string [44].
Is the endstate of the instability of the static KaluzaKlein bubble then a black string? One would expect the endpoint of a classical evolution to be a classically stable configuration, so since the uniform black string is classically unstable for we must compare the mass of bubble to the GregoryLaflamme mass .
Comparison of KaluzaKlein bubble mass and GregoryLaflamme mass
The mass of the static KaluzaKlein bubble is given in (3.3), and for we can use the GregoryLaflamme masses computed in [24, 40]. In Table 1 we list the approximate values of and . We see that for the static bubble mass is always lower than the GregoryLaflamme mass , so in the microcanonical ensemble the static KaluzaKlein bubble cannot decay to the uniform black string branch but must decay to another branch of solutions, possibly the black hole on cylinder branch.
For higher , we can use the results of [16]. Here it is found that the GregoryLaflamme mass is for all to a good approximation. We have used this to plot in Figure 3 the static KaluzaKlein bubble mass versus the GregoryLaflamme mass , and also listed the approximate values for the dimensionless masses in Table 1. We see from Table 1 and Figure 3 that is greater than for . Therefore, for it is possible that the endpoint of the classical decay of the static KaluzaKlein bubble is a uniform black string.
The fact that we have a critical dimension at () is interesting in view of the recent results of [16] showing that the nonuniform black string branch starts having decreasing as decreases for , ie. with a critical dimension (). Moreover, in Ref. [29] the critical dimension () appeared in studying the stability of the cone metric, as a model for the black holeblack string transition. It would therefore be interesting to examine whether there is any relation between these critical dimensions.
As we have seen, the classical instability of the static KaluzaKlein bubble causes the bubble to either expand or collapse. For fivedimensional KaluzaKlein spacetimes, there exists initial data [42] for massive bubbles that are initially expanding or collapsing [43], and numerical studies [44] shows us that there exist massive expanding bubbles and furthermore the numerical analysis indicates that contracting massive bubbles collapse to a black hole with an event horizon.
If the unstable mode of the static bubble preserves the translational invariance around the direction, the endstate of a collapsing bubble would be expected to be a uniform black string. Consider then more general perturbations causing the decay of the static bubble. For we found , so here it is possible that the static bubble decays to a stable uniform black string. Moreover, for the nonuniform strings have higher entropy than the uniform strings for a given [16], so here it is possible that a nonuniform string can be the endpoint of the bubble decay.
However, for , we have seen that the mass of the bubble lies in the range for which the uniform black string is classically unstable, and therefore we do not expect the uniform black string to be the endstate of the decay of the static bubble. It seems therefore plausable that for the instability of the bubble develops inhomogeneities in the direction so that the likely endstate of the bubble decay is a black hole localized on the KaluzaKlein circle (ie. a solution on the black hole on cylinder branch).
Summary
The static KaluzaKlein bubble is massive and exists at a single point in the phase diagram. It is classically unstable and will either expand or collapse. In the latter case, the endstate is presumably an object with an event horizon. For we have argued that it cannot be the uniform black string, but should be whatever is the endstate of the uniform black string.
It is important to emphasize that the static KaluzaKlein bubble does not have any event horizon. This also means that it does not have entropy or temperature (i.e. the temperature is zero). However, in the following sections we discuss solutions in five and six dimensions with both KaluzaKlein bubbles and event horizons present. In sections 56, we shall see that in the region of the phase diagram, all known solutions describe combinations of KaluzaKlein bubbles and black hole event horizons, and for each value of in the range there exist continuous families of such solutions.
4 Generalized Weyl solutions
In this section we review the generalized Weyl solutions. We use this method in sections 5 and 6 to find exact solutions describing sequences of KaluzaKlein bubbles and black holes. We examine the asymptotics of the generalized Weyl solutions and show how to read off the physical quantities from the asymptotic metric. Furthermore, as a warmup to the following sections, we discuss the uniform black string and the static KaluzaKlein bubble metrics in Weyl coordinates.
4.1 Review of generalized Weyl solutions
Emparan and Reall showed in [33] that for any dimensional static spacetime with additional commuting orthogonal Killing vectors, i.e. with a total of commuting orthogonal Killing vectors, the metric can be written in the form
(4.1) 
where and . For the metric (4.1) to be a solution of pure gravity, i.e. of the Einstein equations without matter, the potentials , , must obey
(4.2) 
and are therefore axisymmetric solutions of Laplace’s equation in a threedimensional flat Euclidean space with metric
(4.3) 
The potentials are furthermore required to obey the constraint
(4.4) 
Given the potentials , , the function is determined, up to a constant, by the integrable system of differential equations
(4.5) 
Therefore, we can find solutions to the Einstein equations by first solving the Laplace equations (4.2) for the potentials , , subject to the constraint (4.4), and subsequently solve (4.5) to find .
In four dimensions this method of finding static axisymmetric solutions was pioneered by Weyl in [45]. Emparan and Reall then generalized Weyl’s results to higher dimensions in [33]. We refer therefore to solutions of the kind described above as generalized Weyl solutions.
In general, sources for the potentials at lead to naked singularities, so we consider only sources at . The location corresponds to a straight line in the unphysical threedimensional space with metric (4.3) mentioned above. The constraint (4.4) then means that the total sum of the potentials is equivalent to the potential of an infinitely long rod of zero thickness lying along the axis at . In the unphysical threedimensional space (4.3), this infinite rod has mass per unit length, with Newton’s constant in this space set to one. We demand furthermore that for a given value of there is only one rod, except in isolated points. Thus, we build solutions by combining rods of mass per unit length for the different potentials under the restrictions that the rods do not overlap and that they add up to the infinite rod.
We use here and in the following the notation that denotes a rod from to .
We now review which rod sources to use for the potentials to write familiar static axisymmetric solutions in the generalized Weyl form:

Minkowski space. No source for the potential of the direction, but an infinite rod for the potential of the direction.^{5}^{5}5Minkowski space can also be constructed from other rod configurations, see [33].

A Schwarzschild black hole. A finite rod for the potential of the direction, and two semiinfinite rods and for the potential of the direction. This is illustrated in the left part of Figure 4. Here with being the mass of the black hole and the fourdimensional Newton’s constant.

Minkowski space. Two semiinfinite rods, one rod for the potential in the direction, and the other rod for the potential for the direction. No sources for the potential .

A Schwarzschild black hole. The rod configuration is illustrated in the right part of Figure 4. It consists of a finite rod for the potential in the direction, a semiinfinite rod for of direction, and another semiinfinite rod for of the direction. Here with being the mass of the black hole and the fivedimensional Newton’s constant.
Furthermore, we review in detail particular solutions that are crucial to this paper in Section 4.2.
As will become clearer in the following we have, for our purposes at least, the rule of thumb that a semiinfinite or infinite rod gives rise to a rotational axis (so that the corresponding coordinate becomes an angle in the metric), a finite rod in the timedirection gives rise to an event horizon, while a finite rod in the spatial directions results in a static KaluzaKlein bubble.
4.2 KaluzaKlein spacetimes as generalized Weyl solutions
In this section we show how we can write the five and sixdimensional KaluzaKlein spacetimes and as generalized Weyl solutions, and we explain how to read off the physical quantities for generalized Weyl solutions asymptoting to these spacetimes. We furthermore describe the uniform black string branch and the static KaluzaKlein bubbles in five and six dimensions, since this clarifies our use of the generalized Weyl solution technique, and also since these solutions will be the buildingblocks of the solutions presented below. We begin in five dimensions and then move on to six dimensions.
Fivedimensional KaluzaKlein spacetime and asymptoting solutions
For we have the generalized Weyl ansatz Eq. (4.1) which we write as
(4.6) 
Notice that we have renamed and .
We begin by describing the KaluzaKlein spacetime as a generalized Weyl solution. This corresponds to the potentials
(4.7) 
We see that this is an infinitely long rod for the potential, i.e. in the direction. Note that as can be checked from (4.5). Making the coordinate transformation
(4.8) 
we get fourdimensional Minkowskispace in spherical coordinates times a circle
(4.9) 
We see that is the circle direction which we take to be periodic with period .
We now consider generalized Weyl solutions that asymptote to the fivedimensional KaluzaKlein spacetime . The asymptotic region of a generalized Weyl solution is the region .
In Section 2 we explained how to read off the rescaled mass and the relative tension for static solutions asymptoting to the fivedimensional KaluzaKlein spacetimes . From (2.3) we see that we need to read off , , and in order to find and . While the circumference is clearly the period of we can read off and from the potentials and as
(4.10) 
for . Using this in (2.3) with then gives and .
Uniform black string and static KaluzaKlein bubble in
In Section 2 we reviewed the uniform black string in , in particular the metric was given in (2.5). The fivedimensional uniform black string metric () can be written in Weyl coordinates by choosing the potentials
(4.11) 
with
(4.12) 
This means that we have a finite rod for the potential of the direction, no rod sources for the potential of the direction, and two semiinfinite rods and for the potential of the direction. We have depicted this rod configuration in the left part of Figure 5. Using (4.10) we see that and . We have
(4.13) 
as can be checked using (4.5). If we make the coordinate transformation
(4.14) 
and set we get back the metric (2.5) for the uniform black string with , which explicitly exhibits the spherical symmetry.
To get instead a static KaluzaKlein bubble as a generalized Weyl solution we can make a double Wick rotation of the and directions. This gives the potentials
(4.15) 
with and given by (4.12). This means that we have two semiinfinite rods and in the direction and a finite rod in the direction. We have depicted this rodconfiguration in the right half of Figure 5. The function is again given by (4.13). To avoid a conical singularity at for , where the orbit of shrinks to zero, we need to fix the period of to be . Using (4.10) we see that and , so and by Eq. (2.3). If we make the coordinate transformation (4.14) we get the explicitly spherically symmetric metric (3.1) for the static KaluzaKlein bubble metric with .
Sixdimensional KaluzaKlein spacetime and asymptoting solutions
For we have the generalized Weyl ansatz Eq. (4.1) which we write as
(4.16) 
Notice that we have renamed , and .
Written as a generalized Weyl solution the KaluzaKlein spacetime corresponds to the potentials
(4.17) 
We see that this is a semiinfinite rod sourcing the potential for the direction and a semiinfinite rod for the potential for the direction. We have
(4.18) 
as can be verified using (4.5). Making the coordinate transformation
(4.19) 
where , we see that we get fivedimensional Minkowskispace in spheroidal coordinates times a circle
(4.20) 
We consider now sixdimensional generalized Weyl solutions that asymptote to . We can read off and from the asymptotics of the potentials since we have
(4.21) 
for . Using this with (2.3) for we get and .
Uniform black string and static KaluzaKlein bubble in
The uniform black string as a generalized Weyl solution has the potentials
(4.22) 
with
(4.23) 
The source configuration for the potentials are therefore a finite rod for , no source the potential , a semiinfinite rod for , and a semiinfinite rod for . We have depicted this rod configuration in the left half of Figure 6. Using (4.21) we see that and . The function is given by
(4.24) 
as can be checked using (4.5). Notice Eq. (4.24) reduces to Eq. (4.18) for . If we set and make the coordinate transformation
(4.25) 
with , we get the metric (2.5) for the uniform black string metric, which explicitly exhibits the spherical symmetry.
If we make a double Wick rotation of the and directions we get the static KaluzaKlein bubble. This corresponds to the potentials
(4.26) 
This means we have no rod sources for , a finite rod for , a semiinfinite rod for , and a semiinfinite rod for . We have depicted this rodconfiguration in the right part of Figure 6. The function is again given by (4.24). We need to avoid a conical singularity at for . Using (4.21) we see that and , so and by Eq. (2.3). If we make the coordinate transformation (4.25) we get the explicitly spherically symmetric metric (3.1) for the static KaluzaKlein bubble metric with .
5 Fivedimensional bubbleblack hole sequences
In this section we derive the metrics for bubbleblack hole sequences in five dimensions, using the generalized Weyl construction reviewed above. We discuss some general aspects, such as regularity, topology of the KaluzaKlein bubbles and event horizons, and the asymptotics of the solution. Specific cases as well as further general physical properties of these fivedimensional solutions are presented in Section 8.
5.1 Five dimensional solutions
In this section we construct fivedimensional solutions with static KaluzaKlein bubbles and black holes. We use method of the generalized Weyl solutions reviewed in Section 4 to construct the solutions. This means we use the ansatz
(5.1) 
for the metric. The black holes and bubbles are placed alternately along the axis in the Weyl coordinates, like pearls on a string, for instance,