Analytic approximation to 5 dimensional black holes with one compact dimension

Abstract

We study black hole solutions in R 4 × S 1 space, using an expansion to fourth order in the ratio of the radius of the horizon, µ, and the circumference of the compact dimension, L. A study of geometric and thermodynamic properties indicates that the black hole fills the space in the compact dimension at ǫ(µ/L) 2 ≃ 0.1. At the same value of ǫ the entropies of the uniform black string and of the black hole are approximately equal. PACS numbers: 04.50.+h, 04.70.Bw, 04.70.Dy I. INTRODUCTIONIf the topology of space-time is R 4 × S 1 then the only known exact solution representing a black object is the uniform black string with horizon topology S 2 × S 1 [1]. Though this solution exists for all values of the mass it is unstable below a critical value, M GL , as shown by Gregory and Laflamme [2].Horowitz and Maeda [3] argued that a uniform black string cannot change its topology into a black hole in finite affine time, making the possibility of such a transition questionable. They suggested the possibility of a transition to a nonuniform black string. Gubser [4] showed the existence of nonuniform black string solutions. The non-uniform black string solution in 6 dimensions was investigated numerically [5] for a range of the mass values above M GL .Nonuniform black string configurations do not exist for masses below the Gregory-Laflamme point, in the region where the uniform black string solution is unstable [5]. A natural candidate for a black object in this mass range is a black hole. Unfortunately, no exact black hole solutions are known in a 5 (or more) dimensional space with a compactified dimension. Still, on the basis of physical intuition, a black hole solution should exist for very small values of the mass. When the radius of the horizon is much smaller than the size of the compactified dimension, i.e. when M → 0, the black hole should be unaware of the compactification. Then a Myers-Perry [6] solution should become asymtotically exact. Indeed, a numerical solution, extending the Myers-Perry solution to larger values of the mass was found by Harmark and Obers [7,8] and Kudoh and Wiseman [9].Using general arguments Kol [10] suggested that the black hole branch and the nonuniform black string branch meet at a point when the black hole fills the compact dimension. Further arguments for such a transition were presented in [11,12].Recently, we have studied a related problem, namely the existence of black holes in 14] theories. We used [15,16] an approximation scheme based on the expansion of solutions in the ratio of the radius of the horizon of the black hole to the ADS curvature. In this paper we employ a similar strategy by expanding the metric and other relevant quantities in the ratio of the two natural lengths associated with a black hole configuration, (i) the five dimensional Schwarzschild radius, µ, associated with the mass µ = 8G 5 M/(3π) and (ii) the compactification length, L, defined as the proper circumference of the compact dimension in the region far away from the mass. The dimensionless ratio of the...

Black Holes in Higher-Dimensional Gravity

“…For small µ, the impressively accurate data of [48] are consistent with the analytical results of [31,32,33]. The results of [48] for d = 5 are displayed in a (µ, n) phase diagram in Figure 1.…”

“…Then one may attempt to find the solution in a perturbative expansion around this (limiting) known solution. This method, called matched asymptotic expansion [31,32,33,34,35,30], has been very successful. It applies to problems that contain two (or more) widely separated scales.…”

“…The starting point in this construction is the fact that as µ → 0, one has n → 0 so that the localized black hole solution becomes more and more like a (d+1)-dimensional Schwarzschild black hole in this limit. For d = 4, the second order correction to the metric and thermodynamics have been studied in [33]. More generally, the second order correction to the thermodynamics was obtained in Ref.…”

“…But there are many more phases of KK black holes, which in recent years have been uncovered by a combination of perturbative techniques (matched asymptotic expansion), numerical methods and exact solutions. These phases include non-uniform black strings (see [37,38,39,40,41,42] for numerical results), localized black holes (see [43,31,32,34,33,44,35,45] for analytical results and [46,47,48] for numerical solutions) and bubble-black hole sequences [49]. Here recent progress [35] includes the construction of small mass multi-black hole configurations localized on the circle which in some sense parallel the multi-black hole configurations obtained for rotating black holes.…”

“…This includes the explicit construction of thin black rings in six and higher dimensions [30] based on a perturbative technique known as matched asymptotic expansion [31,32,33,34,35]. Furthermore, in Ref.…”