Thin shells associated to black string spacetimes
Abstract
In this article, we study thin shells of matter connecting charged black string geometries with different values of the corresponding parameters. We analyze the matter content and the mechanical stability of the shells undergoing perturbations that preserve the cylindrical symmetry. Two different global configurations are considered: an interior geometry connected to an exterior one at the surface where the shell is placed, and two exterior geometries connected by a wormhole throat located at the shell position.
Keywords
General relativity Thin shells Cylindrical spacetimes1 Introduction
Thin matter layers (or thinshells) [1, 2, 3, 4] and their associated geometries appear in both cosmological and astrophysical frameworks. At a cosmological scale, the formalism used to define such layers has been applied in braneworld models, in which a spacetime is defined as the surface where two higher dimensional manifolds are joined (see for instance [5] and references therein). At an astrophysical level, such matter layers appear, for example, as models for stellar atmospheres, gravastars, etc. [6, 7]. Most thinshell models considered in the literature are associated to spherically symmetric geometries. However, cylindrical shells and the corresponding spacetimes are also of physical interest.
Cosmic strings [8] are topological defects which would result from symmetry breaking processes in the early Universe. Theories with only the presence of scalar fields predict the socalled global strings, while the addition of one or more gauge fields leads to the prediction of local or gauge strings. The possible important role of cosmic strings in the explanation of structure formation at cosmological scale [9] and, besides, the fact that it would be possible to detect their gravitational lensing effects [10, 11] led to a considerable amount of work devoted to their study [12]. A large proportion of the recent developments addressing cylindrical shells deals with geometries associated to cosmic strings [13, 14, 15, 16, 17, 18, 19, 20]. Other studies regarding cylindrical shells involve wormhole spacetimes, see for example [21, 22, 23, 24].
Black strings have a very different origin, as they would be associated to gravitational collapse. While for \(3+1\) dimensions, in the absence of a cosmological constant, the collapse of a cylindrical matter distribution does not lead to horizons, within the framework of a theory with a negative cosmological constant, i.e. \(\Lambda <0\), the appearence of an event horizon does take place [25]. In this sense, the situation is the same as in \(2+1\) dimensional gravity, where an analogous of the Schwarzschild solution does not exist for zero cosmological constant, but for \(\Lambda <0\) we have the well known Bañados–Teitelboim–Zanelli (BTZ) solution representing a threedimensional black hole [26]. In fact, the strong relation between these three and fourdimensional geometries was stressed in [27], where it was shown that the BTZ black hole solution can be translated into the black string geometry. Different aspects of shells connecting BTZ geometries were considered in [28, 29, 30, 31, 32]. Wormholes associated with charged black strings supported by shells with a Chaplygin equation of state were introduced in [33], while the linearized stability of thinshell wormholes related to non charged black strings has been recently analyzed in [34].
In the present work we address the study of shells associated to charged nonrotating black string geometries of the form (2). We will consider both the case of inner solutions joined to outer ones, and the case of traversable thinshell wormholes connecting two exterior regions of manifolds of this kind. We will analyze the properties of the matter supporting such spacetime geometries and the linearized mechanical stability of the shells undergoing perturbations which preserve the cylindrical symmetry. As usual, we adopt units such that \(c=G=1\).
2 Cylindrical shells
3 Stability analysis
4 Application to charged shells
In this section, we present examples of application of the formalism to both type I and II geometries. In all of them, the metrics adopted for the construction of the spacetimes correspond to black string solutions in Einstein–Maxwell theory, given by Eq. (2), with the same negative cosmological constant \(\Lambda \) in both regions \(\mathcal {M}_1\) and \(\mathcal {M}_2\) of the whole manifold \(\mathcal {M}\). We start by considering two cases of type I geometries, i.e. we construct spacetimes in which an interior region \(\mathcal {M}_1\) is joined by means of a shell \(\Sigma \) to an exterior one \(\mathcal {M}_2\) that extends to infinity in the radial coordinate. Then, we analyze two examples of type II spacetimes, corresponding to thinshell wormholes, with the throat located at the shell \(\Sigma \) which joins the two regions \(\mathcal {M}_1\) and \(\mathcal {M}_2\) that compose \(\mathcal {M}\).
4.1 Shells around vacuum (bubbles)
4.2 Shells around black strings

When \(\beta _h \le \beta _e\), or equivalently \(0< m_1 \le m_2 /4\), the qualitative behavior of the configurations is very similar to the one found for bubbles in the previous subsection, with the only important difference being that the shell radius should always be larger than the radius of the horizon of the black string, as it can be seen in Fig. 2 in which \(m_1 = 0.2 m_2\).

If \(\beta _e < \beta _h\), or equivalently \(m_2 /4< m_1 < m_2\), the modulus of the charge per length \(\lambda _2\) when the stability behavior has a transition moves to \(\lambda _c = (4m_1)^{1/6}(m_2  m_1)^{1/2}\), which is smaller than the extremal one \(\lambda _e\). This critical value \(\lambda _c\) corresponds to the charge per length for which the horizon \(\beta _{+} = \alpha r_{+}\) of the outer geometry used in the construction coincides with the event horizon \(\beta _h\) of the black string. The value of \(\lambda _c\) becomes smaller in case that \(m_1\) approaches to \(m_2\). These features are shown in Fig. 3, where \(m_1 = 0.5 m_2\).
4.3 Wormholes with the throat joining vacuum and nonvacuum regions
4.4 Wormholes symmetric across the throat
5 Summary
We have presented two classes of cylindrically symmetric thin shells and we have performed the stability analysis of the static configurations under perturbations preserving the symmetry. We have applied the formalism to the study of spacetimes associated to black strings in Einstein–Maxwell theory. In particular, we have mathematically constructed bubbles, thin shells surrounding black strings, and thinshell wormholes. In the first and second cases, we have found that stable configurations with normal matter are possible if the parameters of the model are suitably chosen. For wormholes, we have obtained that stable static solutions are possible for selected values of the parameters, but they always require exotic matter that violates the weak energy condition. As we have mentioned in the Introduction, the geometry adopted in our construction is closely related to the charged BTZ solution. Then, it is natural to compare our results with those previously obtained in \(2+1\) dimensions [28, 29]. For both shells around black holes as for shells supporting wormhole geometries in \(2+1\) dimensions, when the modulus of the charge grows, the stability regions enlarge their size and afterwards recover a form similar to that for null or low charges. However, this does not happen in \(3+1\) dimensions. On the other hand, the examples studied above of shells associated to black string spacetimes show a sort of improvement in the conditions for stability, when compared with the lower dimensional scenario. For null charge, while the spacetimes in \(2+1\) dimensions admitted stable configurations only for values of the parameter \(\eta \) larger than unity, now we have found stable configurations compatible with \(0\le \eta <1\). This is a positive feature as, at least for non exotic matter, \(\eta \) is usually understood as the squared velocity of sound on the shell, and \(\eta >1\) would imply a superluminal wave propagation. Though, in most of the examples studied in this work, the stability with a positive and small \(\eta \) requires a large modulus of the charge per length, close to the extremal one. However, we have found the particularly interesting case of charged shells around black strings with \(m_2/4<m_1 <m_2\), in which this transition in the behavior of the stability regions takes place for a modulus of the charge per length parametrically smaller than the extremal one. The effect of perturbations that break the cylindrical symmetry is yet to be investigated, because in some cases this type of perturbations can actually give rise to instabilities (e.g. Gregory–Laflamme instability [35]).
Footnotes
 1.
There is a third case corresponding to joining two interior submanifolds, i.e. \(\mathcal {M}_1=\{x^\alpha _1/0\le r_1\le a_1 \} ,\ \mathcal {M}_2=\{x^\alpha _2/0\le r_2\le a_2 \}\), that will be not considered in this work.
 2.
Note that in the presence of a non vanishing cosmological constant, the corresponding geometry is not that of the Minkowski spacetime.
Notes
Acknowledgements
This work has been supported by Universidad de Buenos Aires and CONICET.
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