Non-uniform Black Strings Research Articles

The effect of temperature gradient on the heavy quark–antiquark potential using a gravity dual model

Abstract Thermal systems have traditionally been modeled via Euclideanized space by analytical continuation of time to an imaginary time. We extend the concept to static thermal gradients by recasting the temperature variation as a variation in the Euclidean metric. We apply this prescription to determine the quark–antiquark potential in a system with a thermal gradient. A naturally occurring QCD medium with thermal gradients is a quark–gluon plasma (QGP). However, the QGP evolves in time. Hence, we use a quasi-stationary approximation, which is applicable only if the rate of time evolution is slow. The application of our proposal to a quark–antiquark potential in QGP can be seen as a step towards a more exact theory that would incorporate time-varying thermal gradients. The effect of a static temperature gradient on the quark–antiquark potential is analyzed using a gravity dual model. A non-uniform black string metric is developed by perturbing the Schwarzschild metric, which allows us to incorporate the temperature gradient in the dual anti-de Sitter space. Finally, an expression for the quark–antiquark potential in the presence of a static temperature gradient is derived.

Thermal phase transition in Yang-Mills matrix model

We study the bosonic matrix model obtained as the high-temperature limit of two-dimensional maximally supersymmetric SU(N) Yang-Mills theory. So far, no consensus about the order of the deconfinement transition in this theory has been reached and this hinders progress in understanding the nature of the black hole/black string topology change from the gauge/gravity duality perspective. On the one hand, previous works considered the deconfinement transition consistent with two transitions which are of second and third order. On the other hand, evidence for a first order transition was put forward more recently. We perform high-statistics lattice Monte Carlo simulations at large N and small lattice spacing to establish that the transition is really of first order. Our findings flag a warning that the required large-N and continuum limit might not have been reached in earlier publications, and that was the source of the discrepancy. Moreover, our detailed results confirm the existence of a new partially deconfined phase which describes non-uniform black strings via the gauge/gravity duality. This phase exhibits universal features already predicted in quantum field theory.

Topology-changing horizons at large D as Ricci flows

The topology-changing transition between black strings and black holes localized in a Kaluza-Klein circle is investigated in an expansion in the inverse of the number of dimensions D. Performing a new kind of large-D scaling reduces the problem to a Ricci flow of the near-horizon geometry as it varies along the circle direction. The flows of interest here simplify to a non-linear logarithmic diffusion equation, with solutions known in the literature which are interpreted as the smoothed conifold geometries involved in the transition, namely, split and fused cones, which connect to black holes and non-uniform black strings away from the conical region. Our study demonstrates the adaptability of the 1/D expansion to deal with all the regimes and aspects of the static black hole/black string system, and provides another instance of the manner in which the large D limit reduces the task of solving Einstein’s equations to a simpler but compelling mathematical problem.

Critical Kaluza-Klein black holes and black strings in D = 10

We construct static vacuum localized black holes and non-uniform black strings in ten spacetime dimensions, where one of the dimension is compactified on a circle. We study the phase diagram of black objects with these boundary conditions, especially near the critical point where localized black holes and non-uniform black strings merge. Remarkably, we find that the merger happens at a cusp in the phase diagram. We verify that the critical geometry is controlled by a Ricci-flat double-cone as previously predicted. However, unlike the lower dimensional cases, we find that physical quantities approach to their critical values according to a power law plus a logarithmic correction. We extract the critical exponents and find very good agreement with the predictions from the double-cone geometry. According to holography, localized black holes and black strings are dual to thermal states of (1 + 1)-dimensional SU(N) maximal Super-Yang Mills theory compactified on a circle; we recover and extend the details of the (recently found) 1st order phase transition in this system from the gravity side.

Phases and stability of non-uniform black strings

We construct solutions of non-uniform black strings in dimensions from D ≈ 9 all the way up to D = ∞, and investigate their thermodynamics and dynamical stability. Our approach employs the large-D perturbative expansion beyond the leading order, including corrections up to 1/D4. Combining both analytical techniques and relatively simple numerical solution of ODEs, we map out the ranges of parameters in which non-uniform black strings exist in each dimension and compute their thermodynamics and quasinormal modes with accuracy. We establish with very good precision the existence of Sorkin’s critical dimension and we prove that not only the thermodynamic stability, but also the dynamic stability of the solutions changes at it.

Einstein-Gauss-Bonnet black strings at large D

We study the black string solutions in the Einstein-Gauss-Bonnet(EGB) theory at large D. By using the 1/D expansion in the near horizon region we derive the effective equations that describe the dynamics of the EGB black strings. The uniform and non-uniform black strings are obtained as the static solutions of the effective equations. From the perturbation analysis of the effective equations, we find that thin EGB black strings suffer from the Gregory-Laflamme instablity and the GB term weakens the instability when the GB coefficient is small, however, when the GB coefficient is large the GB term enhances the instability. Furthermore, we numerically solve the effective equations to study the non-linear instability. It turns out that the thin black strings are unstable to developing inhomogeneities along their length, and at late times they asymptote to the stable non-uniform black strings. The behavior is qualitatively similar to the case in the Einstein gravity. Compared with the black string instability in the Einstein gravity at large D, when the GB coefficient is small the time needed to reach to final state increases, but when the GB coefficient is large the time to reach to final state decreases. Starting from the point of view in which the effective equations can be interpreted as the equations for the dynamical fluid, we evaluate the transport coefficients and find that the ratio of the shear viscosity and the entropy density agrees with that obtained previously in the membrane paradigm after taking the large D limit.

Critical behavior of the black hole/black string transition

We numerically construct static localized black holes in five and six spacetime dimensions which are solutions to Einstein’s vacuum field equations with one compact periodic dimension. In particular, we investigate the critical regime in which the poles of the localized black hole are about to merge. A well adapted multi-domain pseudo-spectral scheme provides us with accurate results and enables us to investigate the phase diagram of those localized solutions within the critical regime, which goes far beyond previous results. We find that in this regime the phase diagram possesses a spiral structure adapting to the one recently found for non-uniform black strings. When approaching the common endpoint of both phases, the behavior of physical quantities is described by complex critical exponents giving rise to a discrete scaling symmetry. The numerically obtained values of the critical exponents agree remarkably well with those derived from the double-cone metric.

Pseudo-spectral construction of non-uniform black string solutions in five and six spacetime dimensions

In this paper, we describe in detail a scheme for the construction of highly accurate numerical solutions to Einstein's field equations in five and six spacetime dimensions, corresponding to non-uniform black strings. The scheme consists of a sophistically adapted multi-domain pseudo-spectral method which incorporates a detailed understanding of the solution's behavior at the domain boundaries and at critical points. In particular, the five-dimensional case is exceedingly demanding, as logarithmic terms appear which need to be treated with special care. Our scheme resolves these issues, and permits the investigation of unprecedentedly strong deformations of the black string horizon. As a consequence, we are able to study in detail the critical regime in phase diagrams displaying characteristic thermodynamic quantities such as mass and entropy. Our results show typical spiral curves in such diagrams which provide strong support for previous numerical works.

Black rings at large D

We study the effective theory of slowly rotating black holes at the infinite limit of the spacetime dimension D. This large D effective theory is obtained by integrating the Einstein equation with respect to the radial direction. The effective theory gives equations for non-linear dynamical deformations of the slowly rotating black hole as effective equations. The effective equations contain the slowly rotating Myers-Perry black hole, slowly boosted black string, non-uniform black string and black ring as stationary solutions. We obtain an analytic solution of the black ring by solving the effective equations. Furthermore, by perturbation analysis of the effective equations, we find a quasinormal mode condition of the black ring in analytic way. As a result we confirm that thin black ring is unstable against non-axisymmetric perturbations. We also include 1/D corrections to the effective equations and discuss the effects by 1/D corrections.

Non-uniform black strings and the critical dimension in the 1/D expansion

Non-uniform black strings (NUBS) are studied by the large $D$ effective theory approach. By solving the near-horizon geometry in the $1/D$ expansion, we obtain the effective equation for the deformed horizon up to the next-to-next-to-leading order (NNLO) in $1/D$. We also solve the far-zone geometry by the Newtonian approximation. Matching the near and far zones, the thermodynamic variables are computed in the $1/D$ expansion. As the result, the large $D$ analysis gives a critical dimension $D_*\simeq13.5$ at which the translation-symmetry-breaking phase transition changes between first and second order. This value of $D_*$ agrees perfectly, within the precision of the $1/D$ expansion, with the result previously obtained by E. Sorkin through the numerical resolution. We also compare our NNLO results for the thermodynamics of NUBS to earlier numerical calculations, and find good agreement within the expected precision.

Evolution and End Point of the Black String Instability: Large D Solution.

We derive a simple set of nonlinear, (1+1)-dimensional partial differential equations that describe the dynamical evolution of black strings and branes to leading order in the expansion in the inverse of the number of dimensions D. These equations are easily solved numerically. Their solution shows that thin enough black strings are unstable to developing inhomogeneities along their length, and at late times they asymptote to stable nonuniform black strings. This proves an earlier conjecture about the end point of the instability of black strings in a large enough number of dimensions. If the initial black string is very thin, the final configuration is highly nonuniform and resembles a periodic array of localized black holes joined by short necks. We also present the equations that describe the nonlinear dynamics of anti-de Sitter black branes at large D.

Effective theory of black holes in the 1/D expansion

The gravitational field of a black hole is strongly localized near its horizon when the number of dimensions D is very large. In this limit, we can effectively replace the black hole with a surface in a background geometry (eg Minkowski or Anti-deSitter space). The Einstein equations determine the effective equations that this 'black hole surface' (or membrane) must satisfy. We obtain them up to next-to-leading order in 1/D for static black holes of the Einstein-(A)dS theory. To leading order, and also to next order in Minkowski backgrounds, the equations of the effective theory are the same as soap-film equations, possibly up to a redshift factor. In particular, the Schwarzschild black hole is recovered as a spherical soap bubble. Less trivially, we find solutions for 'black droplets', ie black holes localized at the boundary of AdS, and for non-uniform black strings.

Stable non-uniform black strings below the critical dimension

The higher-dimensional vacuum Einstein equation admits translationally non-uniform black string solutions. It has been argued that infinitesimally non-uniform black strings should be unstable in 13 or fewer dimensions and otherwise stable. We construct numerically non-uniform black string solutions in 11, 12, 13, 14 and 15 dimensions. Their stability is investigated using local Penrose inequalities. Weakly non-uniform solutions behave as expected. However, in 12 and 13 dimensions, strongly non-uniform solutions appear to be stable and can have greater horizon area than a uniform string of the same mass. In 14 and 15 dimensions all non-uniform black strings appear to be stable.

Einstein-Gauss-Bonnet black strings

We construct uniform black string solutions in Einstein-Gauss-Bonnet gravity for all dimensions d between five and ten and discuss their basic properties. Closed form solutions are found by taking the Gauss-Bonnet term as a perturbation from pure Einstein gravity. Nonperturbative solutions are constructed by solving numerically the equations of the model. The Gregory-Laflamme instability of the black strings is explored via linearized perturbation theory. Our results indicate that new qualitative features occur for d = 6, in which case stable, infinitely long black strings exist for large enough values of the Gauss-Bonnet coupling constant. For other dimensions, the black strings are dynamically unstable and have also a negative specific heat. We argue that this provides an explicit realization of the Gubser-Mitra conjecture, which links local dynamical and thermodynamic stability. Non-uniform black strings in Einstein-Gauss-Bonnet theory are also constructed in six spacetime dimensions.

A new approach to static numerical relativity and its application to Kaluza–Klein black holes

We propose a framework for solving the Einstein equation for static and Euclidean metrics. First, we address the issue of gauge-fixing by borrowing from the Ricci-flow literature the so-called DeTurck trick, which renders the Einstein equation strictly elliptic and generalizes the usual harmonic-coordinate gauge. We then study two algorithms, Ricci-flow and Newton's method, for solving the resulting Einstein–DeTurck equation. We illustrate the use of these methods by studying localized black holes and non-uniform black strings in five-dimensional Kaluza–Klein theory, improving on previous calculations of their thermodynamic and geometric properties. We study spectra of various operators for these solutions, in particular finding the negative modes of the Lichnerowicz operator. We classify the localized solutions into two branches that meet at a minimum temperature. We find good evidence for a merger between the localized and non-uniform solutions. We also find a narrow window of localized solutions that possess negative modes yet have positive specific heat.

Harrison transformation and charged black objects in Kaluza-Klein theory

We generate charged black brane solutions in D−dimensions in a theory of gravity coupled to a dilaton and an antisymmetric form, by using a Harrison-type transformation. The seed vacuum solutions that we use correspond to uplifted Kaluza-Klein black strings and black holes in (D−p)-dimensions. A generalization of the Marolf-Mann quasilocal formalism to the Kaluza-Klein theory is also presented, the global charges of the black objects being computed in this way. We argue that the thermodynamics of the charged solutions can be derived from that of the vacuum configurations. Our results show that all charged Kaluza-Klein solutions constructed by means of Harrison transformations are thermodynamically unstable in a grand canonical ensemble. The general formalism is applied to the case of nonuniform black strings and caged black hole solutions in D = 5,6 Einstein-Maxwell-dilaton gravity, whose geometrical properties and thermodynamics are discussed. We argue that the topology changing transition scenario, which was previously proposed in the vacuum case, also holds in this case. Spinning generalizations of the charged black strings are constructed in six dimensions in the slowly rotating limit. We find that the gyromagnetic ratio of these solutions possesses a nontrivial dependence on the nonuniformity parameter.

Black hole-black string phase transitions from hydrodynamics

We discuss the phase transitions between three states of a plasma fluid (plasma ball, uniform plasma tube, and non-uniform plasma tube), which are dual to the corresponding finite energy black objects (black hole, uniform black string, and non-uniform black string) localized in an asymptotically locally AdS space. Adopting the equation of state for the fluid obtained by the Scherk-Schwarz compactification of a conformal field theory, we obtain axisymmetric static equilibrium states of the plasma fluid and draw the phase diagrams with their thermodynamical quantities. By use of the fluid/gravity correspondence, we predict the phase diagrams of the AdS black holes and strings on the gravity side. The thermodynamic phase diagrams of the AdS black holes and strings show many similarities to those of the black hole-black string system in a Kaluza-Klein vacuum. For instance, the critical dimension for the smooth transition from the uniform to non-uniform strings is the same as that in the Kaluza-Klein vacuum in the canonical ensemble. The analysis in this paper may provide a holographic understanding of the relation between the Rayleigh-Plateau and Gregory-Laflamme instabilities via the fluid/gravity correspondence.

PHASES OF HIGHER DIMENSIONAL BLACK HOLES

We review some of the most striking properties of the phase diagrams of higher dimensional black holes in pure gravity. We focus on static black hole solutions with Kaluza–Klein asymptotics and stationary black hole solutions in flat Minkowski space. Both cases exhibit a rich pattern of interconnected phases and merger points with topology changing transitions. In the first case, the phase diagram includes uniform and non-uniform black strings, localized black holes and sequences of Kaluza–Klein bubbles. In the latter case, it includes Myers–Perry black holes, black rings, black saturns and pinched black holes.

Curvature driven diffusion, Rayleigh-Plateau instability, and Gregory-Laflamme instability

It can be expected that the respective endpoints of the Gregory-Laflamme black brane instability and the Rayleigh-Plateau membrane instability are related because the bifurcation diagrams of the black hole--black string system and the liquid drop--liquid bridge system display many similarities. In this paper, I investigate the nonlinear dynamics of the Rayleigh-Plateau instability in a range of dimensions, including the critical dimension at which the phase structure changes. I show that near the critical dimension and above, depending on a parameter in initial conditions an unstable cylinder will either pinch off or converge to an equilibrium state. The equilibrium state is apparently nonuniform but has a constant mean curvature everywhere. The results suggest that in the gravity side, near the critical dimension and above, the final state of an unstable black string (which is not too long) is a nonuniform black string. The equation of motion adopted to describe the dynamics is the surface diffusion equation, which was originally proposed to describe a grooving process of heated metal surfaces. An interesting correspondence between the diffusion dynamics and black hole (thermo)dynamics is discussed.

Interior of nonuniform black strings

We consider nonuniform black strings inside their event horizon. We present numerical evidence, that the singularity touches the horizon as the horizon topology changing transition is reached.