Spacetime structure of 5D hypercylindrical vacuum solutions with tension
aa r X i v : . [ g r- q c ] S e p Spacetime structure of 5D hypercylindrical vacuum solutions with tension
Inyong Cho ∗ BK21 Physics Research Division and Department of Physics,Sungkyunkwan University, Suwon 440-746, Korea
Gungwon Kang † Korea Institute of Science and Technology Information,52-11 Eoeun-dong, Yuseong-gu, Daejeon 305-806, Korea
Sang Pyo Kim ‡ Department of Physics, Kunsan National University, Kunsan 573-701, Korea
Chul H. Lee § Department of Physics and BK21 Division of Advanced Research andEducation in Physics, Hanyang University, Seoul 133-791, Korea
We investigate geometrical properties of 5D cylindrical vacuum solutions with a transverse spher-ical symmetry. The metric is uniform along the fifth direction and characterized by tension andmass densities. The solutions are classified by the tension-to-mass ratio. One particular example isthe well-known Schwarzschild black string which has a curvature singularity enclosed by a horizon.We focus mainly on geometry of other solutions which possess a naked singularity. The light signalemitted by an object approaching the singularity reaches a distant observer with finite time, but isinfinitely red-shifted.
PACS numbers: 04.50.+h,04.70.-sKeywords: Black string, Singularity, Causal structure
I. INTRODUCTION
Black string solutions are higher dimensional blackhole spacetimes possessing “hypercylindrical” horizonswith/without compactification, instead of “spherical”ones. Recently, it has been of much interest studyingproperties of those spacetime backgrounds. Being differ-ent from that the stationary black hole with a sphericalhorizon topology is stable, the Schwarzschild black stringwas found to be unstable under small perturbations; itis the so-called Gregory-Laflamme (GL) instability [1].This instability has been studied in many ways after-wards. In particular, whether a perturbed black stringis fragmented into an array of small black holes, or endsup with a stable non-uniform black string has been a hotissue [2, 3]. The robustness of the GL instability has alsobeen studied in supergravity theories [4] as well as in gen-eral relativity with a negative cosmological constant [5].However, it is still not understood well what really causesthe GL instability.The Schwarzschild black string is a particular case ofthe 5D hypercylindrical vacuum solution. It is charac-terized by a single parameter (usually called M ) whilethe general solution has two parameters. The two- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] parameter solution was first found by Kramer [6] andwas manipulated in various ways by others in the liter-ature [7, 8, 9, 10, 11]. Although geometrical propetiesof this spacetime were studied in many works, most ofthe studies were in the context of the Kaluza-Klein tho-ery. Consequently, understanding of the geometry wasbased on the four-dimensional gravity with a scalar field.This caused many misleading interpretations for the fullfive-dimensional geometry of the solutions.Very recently, the physical meaning of the two pa-rameters was correctly interpreted for the first time inRef. [11] by one of us. The author considered the weak-field solutions of the Einstein field equations outside somematter distribution. He identified the two parameterswith “mass” and “tension” densities by matching withthe vacuum solutions at asymptotic region. When thetension-to-mass ratio is exactly one half, in particular, itcorresponds to the Schwarzschild black string.In this paper we investigate geometrical properties ofsuch “hypercylindrical” spacetime having arbitrary ten-sion in detail. The solutions are classified mainly by thetension-to-mass ratio a . The physical range of the ratiois 0 ≤ a ≤
2. Specific values a = 1 / a = 2 corre-spond to the well-known Schwarzschild black string andKaluza-Klein bubble [12]. We are particularly interestedin the other values of a in the range. The geometry pos-sesses a naked singularity. The light signal emitted by anobserver approaching the singularity escapes within finitetime, but is infinitely redshifted. There is no wormholestructure in the full five dimensional geometry.In Section II, we study the geometrical properties ofthe hypercylindrical solution. In Section III, we discussthe causal structure of the spacetime, and we concludein Section IV. II. GEOMETRICAL PROPERTIES
The most general form of the static metric for thetransverse spherically symmetric static spacetime witha translational symmetry along the fifth spatial directionin five dimensions may be written as ds = − F dt + G (cid:2) dρ + ρ (cid:0) dθ + sin θdφ (cid:1)(cid:3) + Hdz . (1)Here F , G and H are functions of the “isotropic” radialcoordinate ρ only. Note that the fifth direction is notassumed to be flat in general, i.e., H = 1. If we includea constant momentum flow along the z direction, the g tz component is not zero in general. Such a stationary so-lution was considered in Refs. [7, 13]. Time-dependentsolutions in a separable form were found in Ref. [14].A class of solutions allowing the z -dependence was alsoconsidered in Ref. [15]. By taking double Wick rotations,i.e., t → iz and z → it in the metric (1), one can easilysee that, given a solution with F and H , the metric with F and H being exchanged is a solution as well.The hypercylindrical type of system has been studiedby a number of people, and its vacuum solution has beenobtained in various forms. The solutions are basicallytwo-parameter solutions. The interpretation of these twointegration constants has not been given properly for along time. In Ref. [7] these constants were related tothe gravitational mass and scalar charge. Davidson andOwen [9] defined two kinds of mass parameters in thecontext of Kaluza-Klein dimensional reduction. Namely,they defined the gravitational mass parameter by con-sidering the asymptotic behavior of the four-dimensionaleffective metric, and speculated that the other mass pa-rameter is somehow related to the Kaluza-Klein electriccharge.It is Ref. [11] in which the physical meaning of these in-tegration constants was correctly given. Using the metricansatz (1), the solutions are given by F ( ρ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Kρ Kρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − a ) √ a − a +1) , (2) G ( ρ ) = (cid:18) Kρ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Kρ − Kρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a +1) √ a − a +1) − , (3) H ( ρ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Kρ Kρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a − √ a − a +1) . (4)In Ref. [11], the author considered weak-field nonvacuumsolutions of five dimensional Einstein equations for a sys-tem of matter distribution characterized by the mass andtension densities having the same spherical and transla-tional symmetries mentioned above. By comparing them with the asymptotic behaviors of the metric componentsabove at spatial infinity ( ρ ≫ K ), he found the rela-tionship between these two integration constants and thephysical parameters of the linear-mass density λ and thelinear-tension density τ , a = τλ , (5) K = r − a + a G λ. (6)Actually, this identification is a sort of analogy in thesense that the internal-vacuum region is replaced by acompact matter having the same symmetries. Therefore,the mass and tension in this analogy are contributionsfrom the matter stress-energy inside, but are not puregravitational contributions. The rigorous definitions ofmass and tension densities for gravitational fields them-selves can be found in Refs. [16, 17, 18] where the ADM-tension density is associated with the asymptotic spatial-translation symmetry along the z direction in much thesame way as that the ADM-mass density is associatedwith the asymptotic time-translation symmetry. Suchdefinitions give the same relationship above.The geometrical properties of the spacetime under con-sideration are very different depending on the value of thetension-to-mass ratio a . ( K can be absorbed to the ra-dial coordinate ρ .) The particular case of a = 1 / ds = − (cid:18) − K/ρ
K/ρ (cid:19) dt + (cid:18) Kρ (cid:19) ( dρ + ρ d Ω ) + dz , (7)= − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r d Ω + dz , (8)where M = 2 K = G λ and r = ρ (1 + K/ρ ) is the usual“isentropic” circumference radius. The case of a = 2stands for the so-called Kaluza-Klein bubble [12].A priori there is no restriction on the parameter val-ues. However, if we assume the positivity of the massdensity, we have λ ≥
0. In Ref. [19] it was proved thatthe pure gravitational contribution to the tension is posi-tive definite. Therefore, the physical range for the valuesof the tension would be τ ≥ a ≥ τ ≤ λ (i.e., a ≤
2) infive dimensions. We apply the same upper bound for thegravitational contribution to the tension although thereis no definite proof for it in the literature as long as weknow. Therefore, one may assume that the physical range r/K ρ/K
FIG. 1: Plot of r vs. ρ . From the top down, the curves showthe typical shape corresponding to the case of 1 / < a < a = 1 / ,
2, and 0 < a < / of the tension parameter is0 ≤ τ ≤ λ (i.e., 0 ≤ a ≤ . (9)Based on some desirable cosmological behavior, David-son and Owen [9] speculated that the physical choice is a < /
2. On the other hand, Ponce de Leon [20] claimedthat − < a < / R ABCD R ABCD diverges at ρ = K except for a = 1 / a = 2. Therefore, a curvaturesingularity locates there.Now let us consider the isentropic radius r defined by r = ρ p G ( ρ ) = ( ρ + K ) ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ + Kρ − K (cid:12)(cid:12)(cid:12)(cid:12) ( a +1) √ a − a +1) − . (10)The shape of r ( ρ ) is classified into three types dependingon the scale of a as shown in Fig. 1. As we can see fromthe figure, there exist two copies of the spacetime sepa-rated by ρ = K . Both copies share the same geometrysince the metric is invariant under the transformation ρ → K/ρ . Therefore, we shall focus on the region of ρ ≥ K from now.For the Schwarzschild black string ( a = 1 / ρ = K surface corresponds to the event horizon at r = 2 M (=4 K ) from Eq. (8). The ρ -coordinate cannot describe theinterior region of the horizon while the r -coordinate cando. The Kaluza-Klein bubble ( a = 2) solution is relatedto the Schwarzschild black string simply by double Wickrotations.For 0 < a < /
2, the space for r = [0 , ∞ ) is well de-scribed by ρ = [ K, ∞ ), and those two radial coordinateshave a one-to-one correspondence.For 1 / < a <
2, the relation between r and ρ is verypeculiar. For a given value of r , ρ is double valued. There exists an extremum at ρ + = a + 1 + p (2 a − − a ) p a − a + 1) K. (11)The region below r ( ρ + ) is not covered by the ρ -coordinate. The S surface area at z = constant hy-persurface is A ( ρ ) = 4 πr . For 1 / < a <
2, this area isinfinite at ρ = K , reaches minimum at ρ = ρ + , and thenincreases again. Therefore, it looks like a wormhole ge-ometry at ρ + and attracted people’s attention in earlierworks.The proper length along the fifth direction is given by L ( ρ ) = Z z +1 z √ Hdz = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Kρ Kρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a − √ a − a +1) . (12)As ρ decreases, it monotonically shrinks down to zero at ρ = K for a > / a < / z = 1 is then given by A total ( ρ ) = A ( ρ ) × L ( ρ ) = 4 πr √ H = 4 π ( ρ + K ) ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Kρ Kρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − √ a − a +1) . (13)Since the exponent above is always positive except for a = 1 /
2, the total area turns out to be zero at ρ = K for a = 1 /
2, and increases monotonically as ρ increases.For a = 1 /
2, the exponent becomes zero and the surfacearea is finite. Therefore, although the submanifold at z = constant looks like a wormhole geometry as explainedabove, the full geometry including the z direction is notthat of a wormhole spacetime. III. CAUSAL STRUCTURE
In this section we discuss the causal structure of thegiven geometry. The interesting things would occur at ρ = K where the singularity is located, and at ρ + wherethe S surface area becomes minimum.Let us consider radial motions at a z = constant sub-manifold. The metric becomes ds = − F dt + Gdρ = − F ( dt + dρ ∗ ) ( dt − dρ ∗ ) , (14)where the tortoise coordinate ρ ∗ is defined as dρ ∗ = r GF dρ. (15)The ingoing- and outgoing-null coordinates are definedrespectively as v = t + ρ ∗ and u = t − ρ ∗ . (16) tρ = K ρ