The Phase Transition between Caged Black Holes and Black Strings – A Review
Abstract:
Black hole uniqueness is known to fail in higher dimensions, and the multiplicity of black hole phases leads to phase transitions physics in General Relativity. The blackhole blackstring transition is a prime realization of such a system and its phase diagram has been the subject of considerable study in the last few years. The most surprising results seem to be the appearance of critical dimensions where the qualitative behavior of the system changes, and a novel kind of topology change. Recently, a full phase diagram was determined numerically, confirming earlier predictions for a merger of the blackhole and black string phases and giving very strong evidence that the endstate of the GregoryLaflamme instability is a black hole (in the dimension range ). Here this progress is reviewed, illustrated with figures, put into a wider context, and the still open questions are listed.
To Dorit, Inbal and Neta,
my wife and daughters
1 Introduction
In this introduction we will present the general background for studying General Relativity in higher dimensions and the novel field of phase transitions in General Relativity. We will list the systems known to exhibit phase transitions, and take the opportunity to discuss the rotating black ring before we proceed to concentrate on our system of choice, the blackhole blackstring transition.
Why study GR in higher dimensions? There are several good reasons to study General Relativity (GR) in higher dimensions, namely , where is the total spacetime dimension. From a theoretical point of view there is nothing in GR that restricts us to . On the contrary, the theory is independent of , and should be considered as a parameter. It is common practice in theoretical physics to explore large regions of parameter space of a theory in order to enhance its understanding, rather than restrict to the experimental values and GR should be no exception. For example, in the study of gauge theories it is standard to consider various possibilities for the gauge group and matter content which differ from the standard model.
Additional reasons to study higher dimensional GR include string theory and the phenomenological scenario of “large extra dimensions”, as we proceed to discuss. String theory has a “builtin” preference for higher dimensional spacetimes with 10 (the ”critical dimension”) or 11 dimensions, where the extra dimensions must be compactified. This preference originates in the cancellation of the conformal (quantum) anomaly in 10d which is necessary for the consistency of weakly coupled string theories. The “large extra dimensions” scenario (which is presumably string theory inspired) stresses the following important realizations: that to date gravity is measured only down to –1mm range (which is an “astronomically” poor resolution relative to the one we have for other forces), that it is quite consistent to assume the existence of a compact dimension(s) smaller than the experimental bound and that the situation can be rectified only by improving gravitational and accelerator experiments.
The novel feature  nonuniqueness of black objects. Often when we generalize a problem to allow for an arbitrary dimension the qualitative features do not change and thus the generalization does not produce “new physics”, even if the quantitative expressions are different. However, in GR we do find qualitative changes. If we roughly divide the field of General Relativity into black holes, gravitational waves and cosmology, we find a qualitative change in the first of these categories: one of the basic properties of 4d black holes changes, namely black hole uniqueness.^{1}^{1}1Other qualitative differences include the disappearance of stable circular orbits for (in Newtonian gravity), the absence of propagating gravitational waves in , and the BelinskiiKhalatnikovLifshitz (BKL) analysis of the approach to a spacelike singularity, where there is a critical dimension , such that for the system becomes nonchaotic (see the review [1] and references therein).
Here we should digress to make the distinction between two closely related black hole notions: “no hair” and “uniqueness” (see for example [2]). “No hair” denotes the feature that the space of black hole solutions has a small dimension usually parameterized by asympototically measured quantities (mass, angular momentum and electric charge, for example) much like macroscopic thermodynamical variables. In this respect a black hole strongly contrasts with a nonblackhole star which typically has a much larger number of characteristics such as its internal matter ingredients each with its own equation of state and spatial distribution possibly resulting in an unbounded number of independent multipoles for mass, charge and angular momentum. Whether this “nohair” property continues to hold for higher dimensional black holes could depend on the way one chooses to generalize it. If one generalizes “no hair” to mean that the solutions are determined in term of a small number of (not necessarily conserved) asymptotic data then it continues to hold in higher dimensions as far as we know. However, if one would choose the more restrictive definition which requires conserved charges then this property fails in higher dimensions as was demonstrated in the generalized rotating black ring [3, 4, 5], whose parameters include some nonconserved dipole charges.
“Uniqueness” on the other hand is the more specialized statement that a choice of all of these asymptotic black hole parameters selects a unique black hole rather than a discrete set. In other words, that only a single branch of solutions exists. In 4d uniqueness was proven to hold, namely that given the mass, charge and angular momentum (satisfying some inequalities to ensure the existence of a solution with no naked singularities) there is a unique black hole. However, the proof relies heavily on properties which are special to 4d: Hawking’s proof that the horizon topology has to be and the simplifying gauge choices of WeylPapapetrou and Ernst (see [6] for references to original papers and reviews and for a speculative generalization of uniqueness to higher dimensions). See [7, 8] for a determination of the allowed horizon topologies in certain .
The breakdown of black hole uniqueness in higher dimensions implies the coexistence of several phases with the same asymptotic charges on a nontrivial phase diagram. Phase transitions between the various phases should occur as parameters are changed. As always one may define the order of the phase transition. It could be a first order transition in which case it is triggered nonperturbatively by a competition of entropies between two phases which are separated by a finite distance in configuration space, or it could be of second or higher order, in which case it is triggered by perturbative tachyons and the transition is smooth (see subsection 3.2).
Such first order transitions would be accompanied by an exceptional release of energy, sometimes called a thunderbolt,^{2}^{2}2This term was introduced by [9] for a certain gravitational shock wave in the presence of a naked singularity and seems appropriate for the system under study as well. simply since the total mass of the final state must be lower than or equal to that of the initial state and the excess energy must be lost through radiation. Moreover, exact mass equality is highly unlikely, but rather a loss of mass is natural as spacetime would undergo violent changes including sometimes the rolldown of a tachyonic mode.
The two systems. To date we know of two systems with higher dimensional nonuniqueness resulting in nontrivial phase transition physics

The rotating ring

The blackhole blackstring transition
The latter was chosen for a thorough study of its phase structure which is the subject of this review, presumably since it is somewhat simpler to analyze on account of the smaller number of metric functions and its higher degree of symmetry. Before proceeding to analyze it in detail, we discuss the other example, the black ring.
The rotating black ring lives in the flat (and topologically trivial) 5d background. Spherical rotating black holes solutions in higher dimensions, which generalize the 4d Kerr solution were already found in 1986 by Myers and Perry [10]. These solutions have an horizon topology in 5d and display a maximal angular momentum (at fixed mass). In 2001 Emparan and Reall [11] discovered a beautiful solution, the ring, with horizon topology , and with an angular momentum which is bounded from below, but not from above (at fixed mass). Figure 1 shows the regions on the angular momentum axis which are occupied by the various phases, and one notes that there is a middle region where three phases coexist – one black hole and two rings. Unlike the black string, rotating ring solutions are known only in 5d, presumably due to the special property that in 5d the centrifugal potential and the (Newtonian) gravitation potential have the same () dependence.
For some time it was not clear whether the black ring is stable, and despite some recent findings the issue is not settled yet. In 2004 charged rings were shown to be BPS [12] and hence plausibly “superstable” (that is, nonperturbatively stable). Although the stability of the original nonBPS ring of [11] is still undetermined (however, see [13] for interesting partial results on stability), there is comfort in knowing that some of its closest relatives which share many of their outstanding properties are plausibly stable. Soon after, several groups made progress in obtaining larger families of ring solutions, both BPS [14, 4, 5, 15] and nonBPS [16], all of them restricted to 5d. Moreover, the inclusion of nonconserved dipole moments in [5] demonstrates the “nohair” principle in higher dimensions must be generalized at least to allow for nonconserved quantities.
As the recent discovery of families of black rings demonstrates, the black ring may hold further surprises. In particular, we do not know the full parameter space for rings, and we know close to nothing about the associated phase transitions. Thus rotating rings constitute a promising and active field of research.
Outline. At this point we set aside the topic of black rings until the discussion section and we turn in the next section to the other example for nonuniqueness, the blackhole blackstring transition which is the main topic of this review. In section 2 the physical setup is described and the questions of interest are formulated. In section 3 we describe the analytic considerations that culminate in subsection 3.5 to a certain suggestive qualitative form of the phase diagram which is compared there with numerical data. Section 4 describes the quantitative tools that were employed in order to obtain solutions, including both numerical and analytic methods. Finally, related work is described in section 5 and we conclude with a summary of the results and a discussion of open questions in section 6.
2 Setup and formulation of questions
2.1 Background metric and phases
Background metric. We consider a background with extra compact dimensions. In such a background one expects to find several phases of black objects depending on the relative size of the object and the relevant length scales in the compact dimensions. For simplicity we discuss here pure GR (the only field is the metric) with no cosmological constant. Thus the backgrounds considered are of the form where is any dimensional compact Ricciflat manifold, is the number of extended spacetime dimensions, and the total spacetime dimension is .
The simplest compactifying manifold is a single compact dimension , and accordingly that was the considered in most of the research so far. was chosen not only for its simplicity but also since while more involved will have several phases of black objects, the phase transition physics between any two specific phases is expected to be essentially similar (generically) to the case. Some research was devoted to , the dimensional torus [17, 18], and we shall discuss it later. Other possibilities for include K3 and CalabiYau threefolds, as well as Ricciflat spaces which are not supersymmetric.
Thus we consider a background with a single compact dimension of size , namely , and (the lower bound on is set in order to avoid spacetimes with 2 or less extended spatial dimensions where the presence of a massive source is inconsistent with asymptotic flatness, see [19, 20, 21] for a limited analogue in ).
Black objects. The nonrotating black objects in which we are interested are static and spherically symmetric. Thus, the essential geometry is 2d after suppressing time and the angular coordinates in the extended dimensions. Our coordinates are defined in figure 2.
Such solutions are characterized by 3 dimensionful parameters and where is the mass of the black object (measured in the asymptotic dimensional spacetime, and the detailed expression is given in 18), and is Newton’s constant. These define a single dimensionless parameter ^{3}^{3}3We are dealing with classical GR, and thus we do not set .
(1) 
Alternatively one may use a different parameterization of the solutions such as replacing by , the inverse temperature.^{4}^{4}4More precisely in order to avoid using we define here , namely the period of the Euclidean time direction, and is the surface gravity. Correspondingly one may define another dimensionless parameter
(2) 
where the proportionality factor may be chosen later by convenience. In thermodynamic terms, the parameter (1) or (2) is the “control parameter” of the system, and the choice between these two depends on whether one prefers the microcanonical or canonical ensembles, respectively.
In this background one expects at least two phases of black object
solutions: when , namely the size of the black object
is small (compared to the size of the extra dimension) one
expects the region near the object to closely resemble a
dimensional black hole, while as one increases the mass one
expects that at some point the black hole will no longer fit in
the compact dimension and a black string, whose horizon winds
around the compact dimension will be formed. The precise distinction between these two phases
is give by
Definition:
We distinguish between the black hole (BH) and the black
string according to their horizon topology which is either spherical —
or cylindrical
— , respectively.
These phases are illustrated in figures
3,4,5(b).
We shall sometimes refer to such a black hole localized in a compact dimension as a
“caged black hole”.
Applications. Before proceeding to discuss the phases in more detail, let us mention some applications that contribute to its importance, beyond its considerable intrinsic value. In String Theory it has attracted continued interest, particularly regarding the thermodynamic phase diagram for various gravity theories and/or field theories [22, 23] which are related by dualities to the higher dimensional origin of brane solitons (such as the Mtheory origin of string branes), where the physics is significantly affected by the question whether they are localized in the compact dimensions or wrap them. Another field of application is black holes on braneworlds [24, 25], a problem closely related to the one discussed here, only the background in which the black objects live includes not only an extra dimension but also a “phenomenological” brane localized in that dimension and carrying the fields of the standard model.
We now proceed to discuss the two phases with more detail.
The black string. We can readily write down
solutions which describe uniform black strings (see figure
3)
(3) 
where is the metric on , which for our central example, an parameterized by the coordinate , is just
(4) 
and is the dimensional Schwarzschild black hole (also known as SchwarzschildTangherlini [26]), which is given by
(5) 
where
(6) 
is the metric on the sphere
(7) 
is related to the black hole mass, , via [10]
(8) 
where is the dimensional Newton constant, and
(9) 
These metrics are Ricci flat as a result of being a direct product of Ricci flat metrics. They are called “uniform” for being a direct product with (moreover, for the full metric is independent). Later we will encounter also nonuniform strings (see figure 5(b)). Note that for general (namely dim) these metrics actually describe branes rather than strings.
The uniform black string solution is valid for any (and fixed ). However, we shall soon see that for “thin” enough strings, namely small enough , an instability develops.
Caged black holes. One expects localized black hole (BH) solutions to exist (see figure 4), intuitively obtained by constructing a black hole locally without ever being “aware” of the compactness of the some of the dimensions, at least as long as the black hole is much smaller than the compact dimensions (and the number of extended spacetime dimension is to avoid problems with asymptotics) .
As the black hole grows it will start feeling the presence of the compact dimensions and it will deform accordingly. At some critical one may expect that the black hole will be too large to fit into , and so the mass of this phase will be bounded from above.
Unlike the uniform black string there is no explicit metric that we can write down. This situation was confronted by two methods: an analytic perturbative expansion [27, 28, 29, 30] and numerical analysis [31, 32, 33, 34]. Both techniques will be described in section 4, and here we only note that the analytic method is useful for small black holes (actually is the small parameter for the perturbation series), while for large black holes, where the interesting phase transition physics occurs the numerical methods are essential. The existence of both techniques created a healthy feedback where both methods were used to test and improve each other.
2.2 GregoryLaflamme instability
Gregory and Laflamme (GL, 1993 [35]) discovered that the uniform black string solution (5) develops a dependent metricinstability below a certain critical mass [35] (see figure 5). By a ‘‘metricinstability’’ one means that when one analyzes the spectrum of frequenciessquared for small perturbations around this background, a negative eigenvalue is found.^{5}^{5}5Such a metric instability is also known as a “tachyon”, where the latter term is used in a more general sense than the “usual” 4d tachyonic field. While one usually considers 4d tachyonic fields , whose Lagrangian behaves as and stands for a 3d spatial gradient, one also generalizes it to arbitrary spatial dimension, including spatial dimension 0, which is the case here, when we take to be the amplitude of the GL mode and , where , the inverse decay time, is to be defined shortly in figure 6. , where
In hindsight, this instability makes a lot of sense. In general, gravity has a tendency to clump matter. For example, a uniform distribution of gravitating matter (“gas”) is know to be unstable against the formation of inhomogeneities (the socalled “Jeans instability”): when an inhomogeneity forms the denser regions exert a stronger gravitational pull on their neighborhood, thereby triggering an unstable positive feedback. Similarly here, a long enough string “wants” to develop inhomogeneities (if it is short enough then it gets stabilized by the energetic costs of spatial gradients). Another perspective is to recall that the Schwarzschild black hole has negative specific heat (black hole thermodynamics). While this is not enough to destabilize a single black hole, it should certainly destabilize a homogeneous collection of black holes, namely a black string, which could increase its entropy by redistributing its mass nonuniformly. This intuition is the basis for the GubserMitra Correlated Stability Conjecture [36, 37] which states that a homogeneous black brane is (classically) perturbatively unstable if and only if the dimensionally reduced black hole is thermodynamically unstable (semiclassically)^{6}^{6}6See [38] for proofs of certain aspects of this conjecture.. From now on we continue to discuss only perturbative instabilities.
The main results of GL are summarized in figure 6 (taken from [35, 39]) which depicts the inverse decay times as a function of for total spacetime dimensions .
From figure 6 we see that the tachyonic mode appears for wavenumbers (at fixed ) which are lower than a critical wavenumber (which depends on ). In order to find it is not necessary to look at perturbations in dimensions, but rather it suffices to find the negative mode of the Euclidean dimensional Schwarzschild black hole. This mode, discovered by Gross, Perry and Yaffe [40] (in the 4d case) and hence denoted here by satisfies
(10) 
where is the Lichnerowicz operator for perturbations in the Schwarzschild background and is the negative eigenvalue. Given the marginally tachyonic mode is given by [41, 38]^{7}^{7}7Footnote 4 of [38] explains “This relationship (eq. 11  BK) was noted in [41], although the connection with classical stability was not appreciated.”
(11) 
In [42] the critical GL lengths were obtained for Schwarzschild black holes in various dimensions (see table 1). From these the high asymptotic form was extracted and later proven analytically in [17] to be
(12) 
This means that for large the black string becomes unstable at a compactification length when it is quite “fat” (namely ) and indicates that such a string would not decay into a black hole which would not “fit” inside the extra dimension.
At the GL mode is marginally tachyonic, namely a zeromode. Morse theory arguments strongly suggest^{8}^{8}8The phrase “strongly suggest” is used conservatively due to possible subtleties in the argument which are indicated in subsection 3.3 and were not explored in full rigor. However, it is the author’s opinion that Morse theory arguments essentially guarantee the existence of the nonuniform branch. that this zeromode produces a branch of solutions emanating from the GL point describing nonuniform strings due to the dependence of the GL mode.
d  4  5  6  7  8  9  10  11 

.876  1.27  1.58  1.85  2.09  2.30  2.50  2.69  
d  12  13  14  15  19  29  49  99 
2.87  3.03  3.19  3.34  3.89  5.06  6.72  9.75 
The endstate. Whenever one discovers a perturbative tachyon indicating a decay, the question of its endstate is naturally raised. As the endstate configuration often lies away from the initial configuration a perturbative analysis does not suffice and one needs global information regarding all stable static solutions which is more difficult to obtain.
Gregory and Laflamme believed the endstate to be the black hole, both since that was the only other phase they knew about and since comparing entropies at small one notices that the black hole phase has superior entropy in this regime. Indeed for small , where the black hole is well approximated by a dimensional spherical black hole, the entropies scale as
(13) 
in units where . It is seen that the exponent is a monotonically decreasing function of its argument and hence the exponent is smaller for (the black hole) resulting in a larger area.
More recently Horowitz and Maeda [43] showed that the black string horizon cannot pinch in finite “horizon time” (namely, finite affine parameter along the horizon generators). They interpreted that as an indication that surprisingly a black hole could not be the endstate of decay and predicted instead the existence of a stable nonuniform string phase that would serve as an endstate. The argument for pinching in infinite horizon time relied on assuming the increasing area theorem for an event horizon and applying it to an area element at the “waist” – the inward collapsing region of the event horizon. The extension to the claim on the endstate involved estimates on why infinite horizon time should imply infinite asymptotic time (time for an asymptotic observer). While these claims stimulated much of the research reported here, and while numerical evidence lends support for “pinching in infinite horizon time” (see subsection 4.3), strong evidence against the endstate being a nonuniform string will be described as this review proceeds, implying that the endpoint is actually the black hole phase as originally argued by Gregory and Laflamme (at least in dimension ). See the summary section for a more complete discussion.
2.3 Issues
Let us formulate some major issues or questions regarding this phase transition. These issues may be roughly divided into two groups: static and time evolution.
The static issues include

Endstate of decay.

Qualitative form of the phase diagram including the determination of all static phases.

Detailed quantitative data on the phase diagram: the domain of existence of each phase and the determination of critical points.
During the last couple of years there was significant progress on the static issues, resulting also in the surprising discoveries of critical dimensions and a topology change. The deepest issues belong however to the time evolution

The spacetime structure, namely determination of the Penrose diagram, or an appropriate generalization thereof.

A naked singularity and a violation of Cosmic Censorship.
It is plausible that as the black string pinches a naked singularity is formed, naively because the singularity which “originally” winds the compact dimension gets “broken”, perhaps at the event of pinching. Another argument comes from the clash between the arguments of [43] and results on the system’s phase diagram [44, 34]. Possibilities include a problem with the assumption that there are no singularities strictly outside the horizon and an infinite duration with respect to “horizontime” (horizon affine parameter) while the asymptotictime duration is finite. Note that the initial conditions in this case are generic, unlike known examples of naked singularities.

A thunderbolt and quantum gravity.
The decay is accompanied by a release of energy (after all, a tachyon is involved) in the form of radiation (see [45]). It is plausible that this radiation pulse is classically singular (a “thunderbolt”), perhaps due to its origin from the naked singularity. In such a case it is quite plausible that some knowledge of quantum gravity will be necessary in order to understand this outgoing radiation.
While there was much progress on the static issues, there was practically none on the time evolution issues, and these remain unsolved.
3 Qualitative features
In order to understand the phase transition physics and to resolve the issue of the endstate it suffices to map out all static and stable solutions of the system, since the endstate is certainly static and stable. But actually, Morse theory arguments will lead us to consider all static solutions whether stable or not, in order to take advantage of a “phase conservation rule” which is a powerful qualitative constraint on the form of the phase diagram. So we seek the phase diagram of all static solutions as a function of , and throughout this review we will restrict ourselves to static aspects of the system.^{9}^{9}9Except for subsection 4.3 where a simulated time evolution is described, and subsections 2.3,6.2 where the open questions are discussed.
In this section we seek to determine the phase diagram qualitatively, and the quantitative aspects will be described in the next section.
3.1 Order parameter
We wish to define an order parameter such that the uniform string will have and the emerging nonuniform branch (from the GL point) will have , namely should be a measure of nonuniformity^{10}^{10}10We use the notation for the general discussion of an order parameter, to distinguish it from the closely related perturbation parameter around uniform strings, which we denote by , that will be introduced later and which also satisfies that if and only if the string is uniform.. Actually, it is desirable to have both the black strings and the black holes at finite values of , motivated by the expectation for a merger of the two due to Morse theory arguments as will be explained in subsection 3.3.
It turns out that an asymptotic analysis of the metric and the associated charges furnishes physically meaningful candidates [46, 31]. However, it should be noted that the central discussion on the qualitative form of the phase diagram that will culminate in subsection 3.5 will be independent of this choice of the order parameter, and the discussion here is intended mainly to avoid any unnecessary vagueness that tends to lead to concerns, such as the very existence of an appropriate order parameter which is finite on both blackhole and blackstring.
For concreteness, we take the compactification manifold to be throughout this subsection. Far away from the black object the leading behavior of the radial coordinate is welldefined (by comparison with the flat geometry) and thus, as usual, it is possible to read the (ADM) mass of the object by the asymptotic behavior of the metric functions. One such asymptotic constant can be measured from the falloff of and for a spherical hole in a flat (and topologically trivial) background this would be the only independent asymptotic constant, and it would be proportional to the mass. Here there is one more asymptotic constant: the metric becomes independent (dependent modes are massive from the lower dimensional point of view and hence they decay like ) and thus it is sensible to perform a dimensional reduction asymptotically. After dimensional reduction , the size of the extra dimension, turns into a scalar field. Thus we expect two asymptotic charges – the mass and the scalar charge. (The latter is nonconserved, but is conventionally called “a charge”, presumably since it is the coefficient for the leading asymptotic falloff of a field just like the electric charge can be read from the falloff of the electrostatic potential. It is precisely this property of being the leading term in the asymptotic region which we are interested in.)
One can define the asymptotic charges from either the higher dimensional or from the dimensionally reduced points of view. In the higher dimension the metric defines two asymptotic constants ^{11}^{11}11Exactly two independent ones as discussed above.
(14) 
From the lower dimension point of view the definition for conforms with the standard definition of a scalar charge: One defines the scalar field from the component of the metric through , and the scalar charge through the asymptotic behavior of as (we are not careful here to fix constants in any particular way). Thus from (14) we see that .
We identify the total mass from the dimensionally reduced metric which is gotten by a Weyl rescaling of the metric and therefore , where is defined by . Identifying with and using (8) we get
(15) 
In addition to the mass, the asymptotic constants can be used to express another physical charge, the tension . In a satisfying analogy with the wellknown first law of gas thermodynamics
(16) 
where is defined here through are the energy, temperature, entropy, pressure and volume, respectively,
(17) 
Namely, the tension is defined to be the thermodynamic conjugate to , the size of the extra dimension (see [47, 48] for earlier and equivalent definitions of tension).
The thermodynamic charges are related to the asymptotic constants through
(18) 
This relation may derived either through the thermodynamic definitions or from the ‘‘method of equivalent sources’’^{12}^{12}12This is our own notation for this known method – see for instance [10], but we shall not attempt complete referencing for it., as we proceed to explain. The thermodynamic definition is fully specified by the gravitational (GibbonsHawking) action where is the inverse temperature and is the free energy.^{13}^{13}13The gravitational action is defined in 26 and discussed around it. Alternatively, in the “method of equivalent sources” one imagines that the asymptotic fields were generated by a weak stressenergy source and uses the linearized equations to infer the integrated stressenergy charges from the metric asymptotics. Note that the current expression for the mass (18) coincides with the mass read off the dimensionally reduced metric (15).
Let us gain some insight into the behavior of the tension and the scalar charge. For the uniform string metric (3,4) from its definition (14), and hence from (18). For the small black hole, on the other hand, one finds from the Newtonian approximation (76,62) that and . More precisely, to leading order the tension is proportional to : [27, 30]. See table 2 for a summary.
Uniform string  Small black hole  

Scalar charge  0  
Tension  0 
Inverting (18) we obtain
(19) 
Looking at the expression for the scalar charge we see that the mass tends to increase it, namely mass “wants to generate more space” for itself, while tension “wants” to contract the extra dimension. Thus we may say that for the uniform string () the tension has exactly the correct value to cancel the tendency of mass to expand the extra dimension.
In fact we empirically find that for all black hole and black string solutions their parameters lie between the uniform string and the small BH. This is only partly understood. Positive tension , was proven in analogy with the positive mass theorem [49, 50]. However, it is not clear so far why holds. Actually, when one considers also bubbles (for instance [51]) then is no longer positive. While would correspond to the bound it was argued in [46] (see also [52]) that the completely general bound is higher, namely . This bound is set by the bubble and is consistent with the Strong Energy Condition . ^{14}^{14}14The metric signature convention is “mostly plus”.
We may now define the order parameter. Since is zero exactly for the uniform string we can use some multiple of it. The natural choice is a dimensionless scalar charge, being either for the microcanonical ensemble or for the canonical ensemble. The dimensionless scalar charge has the additional advantage of placing all phases at finite values: not only is the uniform string at but also the small black hole is at finite value, namely , as can be seen from table 2. Alternatively, one may choose a dimensionless tension, such as , as an order parameter that vanishes not for uniform strings, but rather for black holes, and is closely related to and (18).
Here we note that the differential form of the first law of black hole thermodynamics (17) may be integrated using the scale invariance of GR. Namely, when one scales the lengths the mass and area (entropy) scale according to . Taking the differentials for these transformation with respect to at and substituting into (17) yields
(20) 
which is a useful formula known as “the integrated first law” or “Smarr’s formula” (shown in the current context in [46, 31]).
An interesting property of a phase diagram with this order parameter is that intersections in the phase diagram are constrained due to the first law [46].
3.2 Order of phase transition
In general, one of the basic properties of any phase transition is its order. In this subsection we first review black hole thermodynamics in the GibbonsHawking formalism and the general LandauGinzburg theory and then we summarize the results for the system under study.
The gravitational free energy. We will use the standard semiclassical^{15}^{15}15 From a practical point of view all the computations with this action are classical. enters only in the dictionary between the variables of such as (the periodicity of Euclidean time and the area), and the thermodynamic variables such as (the inverse temperature and the entropy). For example . GibbonsHawking [53] gravitational free energy given by the gravitational action (to be described below) evaluated on a ‘‘Euclidean section’’ of the metric. Since our solutions are static,^{16}^{16}16“Static” means by convention “nonrotating and timeindependent” or more formally, invariance not only under time translations but also under time reversal, so that . Equivalently, there exist hypersurfaces such that the Killing vector field is orthogonal to them, namely the hypersurfaces. it is straightforward to obtain a “Euclidean section” simply by taking the transformation . In a standard way, requiring the absence of conical singularities at the horizon fixes the period of Euclidean time to be , where is the surface gravity which is constant over the horizon by the zeroth law.^{17}^{17}17The zeroth law is derived by imposing the constraint , where is the Einstein tensor, is the coordinate normal to the horizon and are the coordinates tangent to the horizon, excluding , the time.
The gravitational action is given by the standard EinsteinHilbert action with an additional boundary term
(21) 
such that is stationary on solutions with respect to variations of the metric which preserve the boundary metric [53, 54].
(22) 
where is the Ricci scalar and can be defined by either of the following equivalent definitions^{18}^{18}18[53] uses definition (23), and the equivalence with definition (24) is used in a computation. I believe that the third and last definition is implied by the relation between the boundary conditions and the action.

(23) where is the trace of the second fundamental form for the embedding of the boundary in the manifold.

(24) the derivative of the boundary volume with respect to a (proper length) shift in the normal direction.

is the boundary term obtained through integration by parts of such that contains only first derivatives of the metric and no second derivatives, namely where denotes here a generic metric element.
In asymptotically flat spaces as defined above (23) diverges and must be regularized. The standard regularization [53] is done by measuring relative to flat space. One chooses a large cutoff , where in our case the boundary is , where are the periods of the directions, respectively^{19}^{19}19 For large the black object metric is virtually independent, as we discuss above 14.. Next one subtract the action of a flat space with the same boundary, which in our case is , namely
(25) 
Finally one takes .
Combining this with (21,22,23), the free energy is finally given by
(26) 
which includes a bulk integral over , the Ricci scalar and a boundary integral over , where is the trace of the second fundamental form on the boundary, and is the same quantity for the reference flat space geometry.
LandauGinzburg theory. In a phase transition some derivative of the free energy is discontinuous and goes through a jump. The order of a phase transition is defined to be the order of this derivative. A first order transition is between two phases which are separated in configuration space and hence have different entropies (and other thermodynamic variables) and are therefore exothermic (involving latent heat) while for second order and higher the phases are continuously connected and there is no finite release of energy.
The LandauGinzburg theory of phase transitions [55] tells us how to infer the order of the transition from the local behavior of the free energy near the critical point , where the variables parameterize (a selected subset of) the configuration space, and may act as order parameters, and the variables are the control parameters, for instance the temperature. In our case, the dimensionless control parameter may be chosen as . Geometrically it is the ratio of the two asymptotic periods, while physically it is a dimensionless inverse temperature. Since the control parameter is essentially the temperature, we interpret the thermodynamics as taking place in the canonical ensemble, and accordingly, the name “free energy” is fitting. The relevant configuration variable is , the amplitude of the marginally tachyonic GL mode (11), so symbolically the perturbation is
(27) 
A generic phase transition is of first order as depicted in figure 7. ^{20}^{20}20See a discussion of the HawkingPage transition in subsection 3.3. However, in our case possesses a certain parity symmetry (which is nongeneric) which opens the possibility for higher order transitions, as we proceed to explain. We note from (27) that is complex and its phase is related to translations in the direction. Since the action is invariant under translations we have
(28) 
and the nonuniform phase spontaneously breaks this symmetry. From now on, without loss of generality, we consider to be real and omit the absolute value notation, so that is an even function, as claimed. This corresponds to fixing the translations in (27) through , and the sign reversal now amounts to shifting by half a period. with a real
Note that can be related to the order parameter defined in the previous section, the scalar charge . The latter being invariant under translations must be a function of as well. Since they both vanish for the uniform string we conclude that
(29) 
(there is a genericity assumption made here which is confirmed by calculations).
Having a marginally tachyonic mode appear at some critical value means that the quadratic term in has a zero at , namely
(30) 
where is some positive constant (assuming the phase is stable for as is our case). Now the LandauGinzburg theory suggests to expand the free energy at to higher orders in and to test whether the free energy has a minimum or not^{21}^{21}21In the general case, where may also have negative modes, the precise criterion for whether the transition is first order is whether the of the new phase increases. Note that in the expansion one may need to incorporate the backreaction. These issues are further discussed below. If has a minimum at , as in figure 8, then for two stable minima are created close by at small , the system will continuously evolve into these new phases and the transition must be second order (or higher). If however has a ‘‘direction of descent’’^{22}^{22}22A nonstandard term which we use to mean “some direction in configuration space where decreases”. at , as in figure 9, that means that since must be bounded from below there must be some other minimum at some finite value of , whose free energy is lower than the phase. Therefore the system underwent already a first order transition at some higher value of where the free energies of both phases were equal.
Therefore LandauGinzburg theory instructs us (in our case, where is even) to expand
(31) 
and to determine the sign of , the quartic coefficient at . Positive (negative) implies a second (first) order transition. Of course if happens to vanish one needs to compute higher orders, but this did not happen in this system.
Actually, so far we neglected all other modes except for the GL mode. When these are brought into consideration one finds it is required to compute first the (quadratic) backreaction of the GL mode and incorporate it into the computation of the quartic term in the action. The reason for taking the backreaction is that what we really want to know is whether the emergent phase from the critical point goes up or down in in the phase diagram, indicating a first or second order transition. To see that we denote the extra modes by and expand the free energy as follows ^{23}^{23}23The index runs over all the extra modes, namely the additional configuration variables. In our case it should really be the continuous variable and sums should be replaced by integrals, but we keep this notation for conceptual clarity. We expended up to quartic order in , incorporating the fact that will be second order in .
(32) 
The equation of motion for the is . We denote the solution by , where stands for backreaction. The equation of motion for is
(33) 
where is the backreaction that solves the equations of motion. Note that is computed without deviating from criticality .
Results. The determination of the order was first carried out by Gubser [56] for the background (see also Wiseman’s improvement [57]). Part of the original motivation there was to find a second order phase transition and hence together with it a branch of stable nonuniform strings emanating from the GL point, such as those predicted by [43]. However, the transition was found to be first order. Sorkin [42] generalized the method to the backgrounds and found a surprising critical dimension
first order second order .
The critical dimension ^{24}^{24}24Of course dimensions are integral, and the notation means only that the change in the order happens between and . is demonstrated in figure 10. Kudoh and Miyamoto [58] observed that the critical dimension depends on the ensemble: Sorkin’s critical dimension holds for the microcanonical ensemble, while in the canonical ensemble ^{25}^{25}25A black hole embedded in a heat bath. the critical dimension is lower by one. In subsection 6.1, “results”, we comment on the relation of these results with the predictions of [43].
Due to the importance of the critical dimension it is a good idea to develop some intuition about it. First, as discussed around equation (12) for high the critical GL string is quite “fat” and we expect that the string will not decay (directly) into the black hole, which would be “too big to fit” inside the extra dimension (see also [59] for a similar argument involving , which is defined below). Another indicator comes from comparing with the value of the approximate equal area point: where is the entropy of the uniform string and is BH area approximated by the small black hole expression. For a first order transition we expect but that holds only for .
Method. Gubser’s method [56] was to perturbatively follow the branch of nonuniform strings emanating from the GL critical point. The perturbation parameter is , the amplitude for the GL mode (27), and in order to determine the order of the transition is was necessary to compute some metric functions up to the third order in the perturbation parameter. As we mentioned above, to determine the order there is a somewhat more efficient method, namely it is actually enough to compute only the second order backreaction and substitute into the quartic part of the action (see appendix A in [44] and [18]). However, the longer computation naturally yields additional results not included in the shorter one.
In practice, when computing the backreaction we consider a continuum of modes, and therefore the discrete index in (32) and the discussion around it is replaced by the continuous variable , and the “field” is replaced by the all the fields in the problem. Moreover, inverting the linear operator to solve for means obtaining the fields by solving a second order ODE (obtained from linearizing the equations of motion) with a source term quadratic in the perturbation (the first order mode).
3.3 Morse theory
When one turns to consider the question of the endpoint for decay, or more generally of finding the phase diagram of all static phases, one is at first discouraged by the lack of any knowledge regarding the nonuniform strings and the big black holes. The most interesting question is to determine the qualitative features of the phase diagram. For that purpose one needs qualitative tools, and one such tool was given in [44] under the name “Morse theory”, fulfilling the intuition that generically a phase persists as the system is “deformed” by changing a parameter, and that the disappearance of a phase should require some special circumstances that are worth elucidating.
Indeed solutions of the Einstein equations are extrema of the gravitational action in the space of metrics, and as such are generically stable under perturbations. The topological theory of extrema of functions is wellknown and is called ‘‘Morse theory’’ and it includes the specification of the allowed transitions. ^{26}^{26}26Some readers may be familiar with the way Morse theory measures global properties of manifolds (Homology), but here we need a different aspect of the theory – local invariants of extrema under deformation of the function.
For other qualitative tools in the study of thermodynamics in the astrophysical context of selfgravitating systems see the review [60], and especially the closely related Poincaré method to determine the perturbative stability of phases just by looking at a certain kind of a phase diagram.
Actually, there is a subtlety in the identification of extrema of the action with solutions of the equations of motion due to gauge (diffeomorphism) redundancy, which we would like to mention. It is certainly true that solutions are extrema of the action. However we wish to consider the action as a function of metrics up to gauge invariance, namely as a function of the gaugefixed metric. Therefore the extremum equations should be supplemented by the gaugefixing constraints. Nevertheless, in this case it was found [57] that the constraints are actually implied by the extremum requirement through a combination of properly chosen boundary conditions and the constraints’ Bianchi identities. It could be that this is true more generally.
A lightning review of Morse theory (see section (3.2) of [44] for a somewhat longer introduction). For functions of one variable the ways in which an extrema can disappear are clear

Annihilation. A maximum and a minimum can coalesce under continuous deformation and disappear into a monotonous function. We call this the basic 1d vertex ^{27}^{27}27Here we introduce the term “vertex” to mean the event when two or more extrema of a function coincide as a deformation parameter is varied. Such an event looks like a collision of phases in a phase diagram and the name comes from the analogy with the Feynman diagram vertex at the collision point between two or more particle worldlines. of “annihilation” – see figure 7.

Runaway. An extrema of a function may run away to infinity either in or in during a finite range of deformation parameter. In this paper we shall find “annihilation” explanations for changing phases, and thus we will not need to resort to “runaway” explanations, even though they are certainly a logical possibility.
When one considers a function of several variables, , one may get a simple generalization of the basic 1d vertex by adding spectating negative directions as well as positive directions, namely is a 1d function such as the one depicted in figure 7, and are positive constants. Now, the minimum in figure 7 turns into an extremum with negative modes, while the maximum has negative directions. Therefore we obtain, what we call “the basic vertex” , where
Basic vertex: two extrema with and negative directions may annihilate.
While for generic extrema the Hessian is nondegenerate and is welldefined, one may wish to know the rules for more general vertices. Indeed, Morse theory can be phrased as saying that the most general vertex is a coincidence of several of these basic vertices, thereby justifying our use of the adjective “basic”.
The conclusion and some reservations. We see that a stable phase () is allowed to disappear at the expense of “annihilating” with an unstable phase with one negative mode (). The latter in turn can disappear by annihilating against either or phases and so one. This is exactly the “phase conservation rule” [44] which we were seeking. It sets a strong qualitative constraint on the existence of phases. However, the price to be paid is that all phases must now be mapped out, not only the stable ones.
As shown in figures 8,9 this rule determines the stability of the nonuniform string emanating from the GL point. One may ask where this phase might end. From the phase conservation rule we conclude that the simplest way to satisfy it, without requiring any additional phases, is that the nonuniform string phase would annihilate against the black hole phase [44] at a point on the phase diagram that we call “the merger”. Taking into account the existence of a critical dimension we find that depending on the dimension we get two possible behaviors: for the nonuniform black string, which is unstable for small nonuniformity, annihilates with the black hole (which is stable when it is small) while for a stable nonuniform string transforms into a stable BH, and () in the canonical (microcanonical ensemble). This is our main conclusion from Morse theory and we stress again that it seems to be the simplest scenario, but others cannot be excluded.
The rigor of the prediction of a phase merger, even if intuitive and clear, is questionable due to the following observation. In the next section we will see that the Euclidean versions of the black hole and the string have different topologies and hence their metrics would be expected to live in different, disconnected spaces of metrics, and it wouldn’t make sense for phases to move from one space to the other. Nevertheless, we shall take the prediction above seriously and look whether these two spaces of metrics are in some sense glued together. Indeed we shall find a continuous transition (and in finite distance) between the two spaces, which we view as an important confirmation for the consistency of the picture. However, the way in which the spaces are glued is still poorly understood, and the gluing may very well be nonsmooth as well as involve the infinite dimensionality of the space of metrics in an essential way, in which case the validity of the Morse theory argument is not selfevident. At this point, I consider the conclusion above to be essentially correct and justified if not rigorously a priori then a posteriori by the agreement of the predicted and the numerically computed phase diagrams.
An example: the HawkingPage transition. The reader familiar with the phase transition between thermal AntideSitter (AdS) and large black holes in AdS, known as the “HawkingPage transition” [61], may benefit from applying the Morse theory ideas above to that context.
In the HawkingPage transition in its canonical ensemble setting, one considers a spacetime with a negative cosmological constant whose boundary is that same as that of thermal AdS, namely a large spatial sphere of Radius times a compact circle of Euclidean time of size . The dimensionless control parameter is , and as usual is the inverse temperature. In 4d Hawking and Page found that several phases exist in this system as follows. For the only phase which exists is thermal AdS, AdS filled with a thermal gas of radiation. For two additional phases show up, the small and large AdS black holes (inside a thermal bath), the large (small) black hole being thermodynamically stable (unstable) due to its positive (negative) specific heat. However, the black holes’ free energy is inferior to that of thermal AdS. Finally at the large black hole dominates.^{28}^{28}28Another transition is expected at which is of quantum nature and will not be discussed here.
In the current context, this phase transition, which is a first order phase transition is described in figure 11 (compare with the general first order transition of figure 7). For there is a single minimum for the free energy which is thermal AdS. At two phases are “pair created” (or “annihilated” if one approached from below) through our “basic vertex”: a stable large black hole and an unstable small black hole, but thermal AdS still has lower free energy. Here we are assuming the GubserMitra Correlated Stability Conjecture [36, 37] to infer the perturbative stability or instability from the sign of the specific heat and the associated thermodynamic stability of the black holes. Then at a first order phase transition occurs when the free energies of thermal AdS and the large BH become equal. Finally, for smaller the large BH dominates.
3.4 Merger point
In the last subsection we saw how Morse theory makes it plausible that the nonuniform string phase merges with the BH phase. We shall first encounter a problem for this picture, namely topological differences between the two phases. However, we are familiar with some topology changes such as the flop and the conifold, and one of the surprising results of the research on this system is the emergence of a novel type of topology change, called the “merger” transition in [44].
A topology change. Intuitively the transition from black string to black hole involves a region where the horizon becomes thinner and thinner as a parameter is changed until it pinches and the horizon topology changes. We call this region “the waist” and this process is described in the upper row of figure 12 using the coordinates defined in figure 2. It is important to remember that all metrics under consideration are static and that they change as we change an external parameter, not time. Since the metrics are static we may as well consider their Euclidean versions (this point was discussed in the second paragraph of section 3.2).
We shall now demonstrate that this merger transition involves a local topology change of the Euclidean manifold. Let us zoom in around a very thin waist, whereby all the scales of the problem such as and are very large and irrelevant, identifying what may be called the local geometry at the waist. Consider a codimension 1 surface within the local geometry but far away from the waist, such as the one denoted by a dashed line in all three geometries in the top row of figure 12. ^{29}^{29}29In other words, if we denote by the radius of the (minimal) angular sphere () at the waist, then we wish to consider the fixed surface where . Actually, this surface is the asymptotic boundary of the local geometry.
The topology of this asymptotic boundary is given by as we proceed to explain. The angular piece, , is obvious while the requires explanation. One should remember that in our figures, such as figure 12, we suppress not only the angular coordinates but also the time variable. In the Euclidean continuation the Euclidean time must be periodic in order to avoid a conical singularity at the horizon, and moreover, the proper size of this circle vanishes at the horizon ( at the horizon). Thus the circle fibration of Euclidean time ^{30}^{30}30which is topologically trivial since for all . over an interval (the dashed line in figure 12), such that the fibred circle shrinks on the edges ^{31}^{31}31More formally, contracts to a point. exactly produces a topological . This is just like the fibration of the surface of the earth by latitude lines, which shrink at the poles (see figure 13).
Naturally, the topology of this boundary surface is constant as local changes occur near the waist. Next we notice that in the blackhole phase the is contractible (onto the exposed axis), while in the string phase the is contractible. Therefore the local topology of (Euclidean) spacetime is changing, not only the horizon topology. Thus, the topology change can be modelled by a “pyramid” (just like the conifold transition) – see the lower row of figure 12: the rectangular basis of the pyramid denotes the asymptotic boundary where each edge represents one of the sphere factors, and the pyramid’s truncated apex encodes which one of the spheres is contractible and which one remains of finite size and is noncontractible. By the nature of topology, in order to change it there must be at least one singular solution along the way (with at least one singular point). The simplest possibility, (which is also realized in the conifold) is to assume that the the singular geometry is the cone over .
It is suggestive that the local spacetime topology change is also accompanied by a change in the global topology. I believe this is the case for the following reason. The black string solution is contractible to (after contracting the cigar), and thus its elementary noncontractible cycles are and . For the black hole on the other hand the noncontractible cycles seem to be and : the is the horizon, the is the compact direction and the is the axis connecting the poles together with the time fibration. It is seen that there are several topological differences between the geometries, for instance, the black hole has a 2d topologically nontrivial cycle while the string does not.
Cones. Figure 12 encodes the topological nature of the merger. But is it really true that this can be realized with Ricciflat metrics?
It is easy to write down a Ricci flat metric for the singular solution, the cone (see the middle of figure 12). Actually this can be done for a somewhat more general cone, the cone over with no additional “cost”. The metric is
(34) 
where the coordinate measures the distance from the tip of the cone, is as usual the total spacetime dimension and the constant prefactors are essential for Ricciflatness. Note that is the singular tip of the cone, unless (or ) when it becomes the smooth origin of in spherical coordinates.
In order to exhibit “smooth cone” metrics which approach the singular cone from both sides of the transition (see right and left portions of figure 12) one may use the following ansatz
(35) 
with boundary conditions at
(36) 
such that becomes noncontractible while joins with to make a smooth neighborhood of the origin of .
Using (85) for the Ricci scalar of a fibration and after integration by parts one gets the action for
(37) 
Einstein’s equations are
(38) 
These equations are very similar to the equations encountered in the BelinskiiKhalatnikovLifshitz (BKL) analysis of the approach to a spacelike singularity (see the recent excellent “Cosmological Billiard” review [1]). Although the general (and often singular) qualitative behavior of these equations as for arbitrary initial conditions was not obtained in [44],^{32}^{32}32One can prove useful theorems for the evolution of the volume factor using the geometric analogue of the “ctheorem” (I thank J. Maldacena for pointing this). From (3.4) we have Hence if (or equivalently ) is somewhere decreasing it must continue to decrease monotonically. At the same time is monotonic. which together with the previous result guarantees that it was checked that for the boundary condition (36) can be expanded in a Taylor series around and the recurrence equations for the Taylor coefficients could be solved without encountering an obstruction.
Once a single smoothed cone solution is available, constructing a full family that approaches the singular cone is a matter of simply rescaling it. As illustrated in figure 14, since away from the smoothed tip the smoothed cone asymptotes to a cone, a geometry which is scale invariant, then after rescaling there is a natural way to identify the asymptotic cones, thereby specifying the way to take the limit over the family of rescaled metrics.
Altogether we succeeded in realizing the local topology change encoded in figure 12 by a family of Ricciflat metrics. I consider this nontrivial property to be strong evidence for the merger picture as presented in subsection 3.3.
I would like to stress some of the assumptions involved in locally modelling by cones

The singular solution has a single singular point.

The singular solution is continuously self similar (CSS).
Both assumptions are reasonable and minimal: there could be more than a single singularity, but there is at least one, and the local singular solution must forget the “long distance” scales and hence it would be scalefree, and the simplest way to obtain that is if the solution is selfsimilar. Continuous selfsimilarity would be the simplest possibility and in [44] it was found to lead to a local isometry enhancement where the coordinate (see figure 2) conspires with to make a round . Indeed, a Numeric study [62], limited by numerical resolution, found consistent evidence for this cone structure. More recently it was claimed in [63] that in a certain range of dimensions () continuous selfsimilarity (CSS) is in fact (spontaneously) broken into discrete selfsimilarity (DSS) which requires to replace the cones by certain “wiggly cones” as local models of the merger point.
Tachyons on cones. It turns out that the cones may have tachyons and that their existence surprisingly depends on a critical dimension as we proceed to show – see also [44] and references therein including [64].
The dangerous mode is a function which inflates slightly one of the spheres while shrinking the other. The ansatz for the perturbation is
(39) 
A priori one could start with two separate scale functions, one for each sphere, but the constraint relates them as above (after ignoring the trivial perturbation which represents translation) – for more details see [44] eq. (6.6) and below.
The quadratic part of the action, disregarding an overall constant is
(40) 
and through the change of variables it can be recast to have a canonical kinetic term
(41) 
The equation of motion for , namely the zero mode equation, is
(42)  
(43) 
The solutions are
(44) 
The expression (44) for , the characteristic exponents, reveals a critical dimension [44]
(45) 
such that for are complex while for they are real. Complex characteristic exponents (for ) imply that (the real part of) the zero mode has infinitely many nodes (zeroes) equally spaced in with “logperiod” . The presence of infinitely many nodes for the zero mode implies the presence of infinitely many tachyons, just like the number of nodes of the zeroenergy solution to a Schrödinger equation counts the number of negative energy states. In [63] these logperiodicity and tachyons (for ) were interpreted as an indication for a spontaneous breaking of continuous selfsimilarity (CSS) into discrete selfsimilarity (DSS). For on the other hand, and there are no nodes nor tachyons.
Note that is the total spacetime dimension, and it is independent of and separately. When we wish to distinguish this critical dimension from others we shall denote it by .
One can view the zeromode equation (42) in a wider context by considering the eigenvalue problem
(46) 
where is the same as in eq. (43), is the eigenvalue,^{33}^{33}33This is unrelated to the introduces in subsection 3.2 to represent the amplitude of the perturbation, and will not be used elsewhere in this review. and we note that (46) reduces to (42) upon setting .
The eigenvalue problem (46) is in Schrödinger form, and we may apply known results. For potentials of the general form it is wellknown that while the classical energy is unbounded from below, the quantum problem may have a ground state as long as the potential is not “too negative”. For instance, for we get the Hydrogen atom. More generally the spectrum is bounded from below for , while for the critical value , which is our concern here, the prefactor becomes dimensionless and the potential is conformally invariant. Due to scale invariance the spectrum is constrained to be invariant under positive rescaling of the eigenvalues. Now itself exhibits a critical value, namely , such that the spectrum is bounded from below and actually nonnegative only for (since the solution has no nodes – see for example [65]). Equating we arrive once more at as in (45).
Some moduli space properties. The merger transition lies in finite distance in moduli space (actually, it would not deserve to be called a “topology change” otherwise, since it would require infinite “resources” to be reached.) The argument was unpublished so far and here it is supplemented as appendix B. Moreover, the appearance of a kink^{34}^{34}34More formally, a discontinuity of the tangent to the phase line. in the phase diagram at merger was predicted in section (5.3) of [44], and the numerics indeed seem to exhibit some sort of a kink – see the next section.
3.5 Phase diagrams – predictions and data
We may now assemble all the theoretical input and draw the predicted qualitative form of the phase diagram for various spacetime dimensions – see figure 15. These figures apply with small changes to both the microcanonical ensemble, which is the usual physical setting where energy is conserved, and to the canonical ensemble, namely black holes immersed in a heat bath.
The vertical axis is the dimensionless control parameter defined by either the dimensionless mass (1) in the microcanonical ensemble or by the dimensionless inverse temperature (2) in the canonical. The horizontal axis is the order parameter chosen in subsection 3.1 to be the dimensionless scalar charge (see especially eq. (14) and the next to last paragraph in that subsection).
The local structure around the GL point is determined by the order of the transition as discussed in subsection 3.2: for the GL vertex is first order [56] just like figure 9, while for it is a second order vertex just like figure 8. , the critical dimension depends on the ensemble – in the microcanonical case it is [42] while in the canonical case it is [58].
Finally, we connect the black hole and black string phases, as suggested by the central conclusion of subsection 3.3, based on Morse theory arguments and further justified by presenting a novel topology change in subsection 3.4.
The diagrams in figure 15 were constructed by attempting to draw the simplest diagram consistent with the data and assumptions in the last two paragraphs. The nontrivial nature of these diagrams is best illustrated by the various other possibilities that were considered in the literature, see for example the six scenarios in [66], section 6.
The critical dimension (45) affecting the stability of the cone in the critical merger solution is not visible in figure 15. According to the picture that emerges from [63], there is a single branch of nonuniform strings, irrespective of (and being unstable as long as the transition is first order), but for this solution approaches a discretely selfsimilar (DSS) solution near the singularity, while for it approaches a continuously selfsimilar (CSS) solution there. While the qualitative predicted form is similar, the kink at merger is likely to be different.
Another point to note is the critical point GL’ in figure 15 which is there to remind us that each (nonuniform) solution has “harmonies” or “copies” gotten by trivially fitting several cycles inside , namely replacing for any in the solutions. In particular the GL point has these copies. However, these copies of the phase diagram are decoupled (except for their connection with the uniform string) and therefore there is no need to draw them. See also [66].
Numerical data. Clearly, the predicted phase diagrams in figure 15, were not proven here, but rather argued to be the simplest possibility which is consistent with certain carefully analyzed arguments, some of them not fully understood yet. As such it suggests to be tested by actually obtaining these solutions. Indeed, one of the joys of this problem is the feedback between theory^{35}^{35}35The word “theory” is used here to loosely mean all the considerations that one can apply before any exact solutions are available and numerics, which is in many ways like the classical feedback between experiment and theory which we sorely miss.
Recently the numerical determination of the phase diagram in 6d was all but completed [34] – see figure 16. A visualization of the merger transition through embedding diagrams is shown in figure 17.
We see perfect agreement with the predicted diagram in figure 15 with (which basically appeared already in [44]), especially regarding the prediction for a “merger” of the black hole and the string phases, and the absence of a stable nonuniform phase (predicted by [43]). We view this as a vindication of the picture presented here. Additional interesting features of the numeric figure are a kink at merger,^{36}^{36}36However, the geometry of the horizon and axis seems to merge rather smoothly. perhaps related to the predicted kink (see the last paragraph in the previous subsection), and the location of the merger at roughly a local maximum on the diagram.
4 Obtaining solutions
In the previous section we examined the qualitative features of the phase diagram. Now we turn to the more quantitative aspects, those required in order to obtain solutions for static black objects.
4.1 2d gravitostatics
Counting degrees of freedom. The most general metric which is static^{16}^{16}footnotemark: 16 and spherically symmetric is
(47) 
where all functions are defined on the plane, is an arbitrary metric on the plane and since the metrics are static we might as well work with Euclidean signature.^{37}^{37}37See also the discussion above 21.
Altogether the problem is defined in the 2d plane and the field content is the metric and two scalars . That means that we can write down a 2d action for these fields without loosing any of the equations of motion. The action is
(48)  
where ^{38}^{38}38See the definition below (8). (see appendix A for useful formulae to determine this and related actions). The total number of fields is : 3 for the 2d metric and 2 for the two scalars. Two fields may be eliminated by a choice of coordinates in the plane which leaves us with three fields. As we proceed we shall review some of the gauges that were used. are the 2d Ricci scalar, the volume element, and the area of the unit sphere
If we formally compute the number of “dynamic” or “physical” degrees of freedom we get a total of : is for the metric degrees^{39}^{39}39Attributing degrees of freedom to gravitons in dimensions. and is for the two scalars. So far nobody succeeded in reducing the problem to a single field, and it is not clear whether that is possible or not, but there is a clever ansatz due to Harmark and Obers which reduces the problem to two fields [67], (see (63,65) and thereabout) . Morally speaking, one may hope that the equations for the three fields, if not reducible to a single field, could at least be separated such that first one solves two equations for “diffeo gauge fields” and only then a single equation is solved for the “physical field”.
Note that if we relax the static requirement the number of degrees of freedom increases – see subsection 4.3.
Constraints and boundary conditions. Today numerical relativists perform full 3+1 dimensional simulations with some success. One would think that simulating a static problem, namely “gravitostatics”, would be wellunderstood by now, but this turned out not to have been the case. The main conceptual hurdle which was necessary to cross was the treatment of constraints and boundary conditions. This problem was solved by Wiseman in the 2d case [57], as we now describe.
Relaxation and electrostatics. Since Newtonian gravitostatics is equivalent to electrostatics it is useful to recall the method there. In electrostatics one wishes to find the electrostatic potential defined over some domain, satisfying

The Poisson equation
(49) where is the given charge density distribution

Boundary conditions: Dirichlet, Neumann or some mix.
A successful numerical algorithm to solve this problem is the relaxation method. This method is very physical in the sense that it has some similarities with the way in which an excited field settles down or “relaxes” as a function of time to a static solution. In the relaxation method one chooses a grid, consisting of points , and then one starts with an initial field configuration which satisfies the boundary conditions. At each step is modified according to a local rule to create a sequence which converges to the solution as . More specifically, once chooses a discretization of the Laplacian and solve for . For example, if one uses a square grid with spacing and chooses the following discretization
(50) 
then after discretizing (49) using (50) and solving for we find
(51) 
Actually, one can go further and introduce a “relaxation speed parameter”, , defined by
(52) 