# Thin shells joining local cosmic string geometries

- 521 Downloads
- 3 Citations

## Abstract

In this article we present a theoretical construction of spacetimes with a thin shell that joins two different local cosmic string geometries. We study two types of global manifolds, one representing spacetimes with a thin shell surrounding a cosmic string or an empty region with Minkowski metric, and the other corresponding to wormholes which are not symmetric across the throat located at the shell. We analyze the stability of the static configurations under perturbations preserving the cylindrical symmetry. For both types of geometries we find that the static configurations can be stable for suitable values of the parameters.

## Keywords

Thin Shell Cosmic String Static Configuration Null Energy Condition Exotic Matter## 1 Introduction

Cosmic strings are topological defects which could be the result of symmetry breaking processes in the early Universe [1, 2]. Their possible important role in the explanation of structure formation at cosmological scale [3, 4, 5, 6, 7] and, besides, the fact that it would be possible to detect their gravitational lensing effects [8, 9] led to a considerable amount of work devoted to the study of their associated geometries. From a different point of view, the recent interest in cylindrically symmetric spacetimes was also driven by the present day theoretical framework in which open or closed strings constitute the most serious basis for a unification of the fundamental interactions (for instance; see Ref. [10]). Within cosmic string models, particular attention have deserved the so-called gauge or local cosmic strings, which are solutions of the equations of a two-component scalar field coupled to a gauge field. The spacetime outside the core of these strings is conical, that is, locally flat but with a deficit angle which is determined by their mass per unit length [11, 12]. The local flatness of the metric implies zero force on a static particle, but relativistic particles would be deflected, and moving strings would give rise to wakes behind them [3, 4, 5, 6, 7]. Thus, though cosmic strings are not supposed today to be the main source of the primordial cosmological matter fluctuations, they can still be of relevance as a secondary source of fluctuations [13, 14, 15, 16]. In the context of axisymmetric spacetimes, the simplicity and the physical interest of the local string metric would make it comparable to the Schwarzschild metric within spherically symmetric geometries.

Thin matter layers (“thin shells”) [17, 18, 19, 20] 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 [21] and references therein). At an astrophysical level, such matter layers appear, for example, as models for stellar atmospheres, gravastars, etc. [22, 23]. While most thin-shell models studied are associated to spherically symmetric geometries, cylindrical shells have also been considered of interest. In particular, traversable wormhole geometries [24, 25, 26, 27, 28, 29] supported by cylindrical thin shells have been recently studied [30, 31, 32, 33, 34, 35, 36, 37]. The mechanical stability analysis of thin-shell cylindrical wormholes which are symmetric across the throat was performed in Refs. [38, 39]. A well-known feature of wormholes in general relativity is the presence of exotic matter not satisfying the null energy condition at the throat or in the region close to it [24, 25, 26, 27, 28, 29]. Moreover, it has been shown that the fluid supporting spherically symmetric wormholes is exotic in any scalar-tensor theory of gravity with non-ghost massless scalar field and graviton, and also in *f*(*R*) gravity with non-ghost graviton [40, 41]. The nature of matter threading cylindrical wormholes was discussed in Refs. [37, 42]. Shells related to wormholes, not supporting them but allowing for flat asymptotics, were considered in [43]. Static cylindrical shells joining inner and outer regions have been associated to different matter contents and different backgrounds in Refs. [44, 45, 46]. Recently, conical spacetimes joined by static shells were the object of field theory calculations in curved space, in the context of a program developed to probe the global aspects of a geometry by the self-force on a charge (see Refs. [47, 48]).

Most preceding work deals with different geometries joined by a static shell, or the stability of wormholes which are symmetric across the throat. In the present article we consider shells joining two different conical geometries, that is, the geometries associated to two local cosmic strings with different mass per unit length. We study the properties of matter on the shells and their mechanical stability under perturbations preserving the symmetry. We analyze both the case of inner solutions joined to outer ones and the case of traversable thin-shell wormholes which are not symmetric across the throat, as those constituting the backgrounds in [47, 48]. As usual, we adopt units such that \(c=G=1\).

## 2 Construction and stability

^{1}The deficit angle is proportional to the string mass per unit length, and it cannot be removed by a mere redefinition of the angular coordinate; this deficit angle would generate physical effects as double images resulting from the deflection of light, and matter density fluctuations in the form of wakes in the plane described by the string motion [3, 4, 5, 6, 7, 8, 9, 11, 12]. From these geometries, we take the radii \(a_{1,2}>0\) and we construct a geodesically complete manifold \(\mathcal {M}=\mathcal {M}_1 \cup \mathcal {M}_2\), which can take any of the following two forms:

^{2}In both cases, the submanifolds are joined at the hypersurface \(\Sigma \equiv \partial \mathcal {M}_1 \equiv \partial \mathcal {M}_2 \), which defines the shell

*a*. By taking the proper time derivative in Eq. (12) and using Eqs. (13) and (14), we obtain the conservation equation in the form

*a*. By introducing the equation of state relating the pressure \(p_\varphi \) and the energy density, the first order differential equation (19) can be integrated to obtain \(\sigma (a)\) from

## 3 Examples

In this section, we apply the formalism introduced above to examples: we construct a thin shell surrounding a local cosmic string and a cylindrical wormhole not symmetric across the throat corresponding to the shell.

### 3.1 Thin shells

### 3.2 Wormholes

*areal*condition, the area functions \(\mathcal{A}_{1,2}(r_{1,2})=2\pi \sqrt{g_{\varphi \varphi }^{1,2}(r_{1,2})g_{zz}^{1,2}(r_{1,2})}= 2\pi \omega _{1,2}r_{1,2}\) should increase at both sides of the throat. The second one [42], known as the

*radial*condition, only requires that the circular radius functions \(\mathcal{R}_{1,2}(r_{1,2}) = \sqrt{g_{\varphi \varphi }^{1,2}(r_{1,2})} = \omega _{1,2}r_{1,2}\) have a minimum at the throat. It is clear that the two possible flare-out conditions, which in our construction only differ by a constant factor, are satisfied because \(r_{1,2} \ge a_{1,2}\). This mathematical construction creates a complete manifold with two different regions at the sides of the throat corresponding to the conic submanifolds \(\mathcal {M}_1\) and \(\mathcal {M}_2\), determined by Eq. (2), which are joined to form a thin-shell wormhole. From Eqs. (15), (16), and (17), the static values of the energy density and pressures at the throat are

## 4 Summary

We have presented the matter characterization and the stability analysis of cylindrical shells joining two local cosmic string metrics. We have studied two types of global geometries, the first one associated to spacetimes with a thin shell separating an inner from an outer region, and the second one describing wormholes with a shell at the throat connecting two different geometries. In the case of wormholes, the fluid at the throat is always exotic, i.e. it does not satisfy the null and weak energy conditions. In the case of shells where an inner region joins an outer one, the matter can be ordinary or exotic, depending on the relation between the corresponding angle deficits. For the stability analysis of static configurations, we have adopted linearized equations of state that relate the pressures \(p_\varphi \) and \(p_z\) with the surface energy density \(\sigma \) at the shell, with coefficients \(\eta _\varphi \) and \(\eta _z\), respectively. We have found that the static configurations are stable if suitable values of the parameters are adopted, for both shells and wormholes. The pressure along the axis turns out to be not relevant for stability, and only the coefficient associated to the angular pressure enters the stability condition; this is reasonable for radial perturbations, i.e. perturbations which involve a change of the circumference perimeter are expected to be affected by forces in the angular direction. For wormholes to be stable the coefficient \(\eta _\varphi \) must always be negative; instead, in the case of shells separating an interior region from an exterior one, we find the desirable feature that they can be stable when \(\eta _\varphi >0\).

## Footnotes

- 1.
If \(\omega _{1,2}>1\) the geometry has a surplus angle, case which we do not consider here.

- 2.
There is a third case corresponding to joining two interior submanifolds, i.e. \(\mathcal {M}_1=\{x^\alpha _1/0\le r_1\le a_1 \} ,\ \mathcal {M}_2=\{x^\alpha _2/0\le r_2\le a_2 \}\); this will not be considered in this work.

## Notes

### Acknowledgments

This work has been supported by CONICET and Universidad de Buenos Aires.

## References

- 1.T.W.B. Kibble, Phys. Rept.
**67**, 183 (1980)ADSMathSciNetCrossRefGoogle Scholar - 2.A. Vilenkin, E.P.S. Shellard,
*Cosmic Strings and Other Topological Defects*(Cambridge University Press, Cambridge, 1994)zbMATHGoogle Scholar - 3.
- 4.N. Turok, R.H. Brandenberger, Phys. Rev. D
**33**, 2175 (1986)ADSCrossRefGoogle Scholar - 5.H. Sato, Prog. Theor. Phys.
**75**, 1342 (1986)ADSCrossRefGoogle Scholar - 6.T. Vachaspati, Phys. Rev. Lett.
**57**, 1655 (1986)ADSCrossRefGoogle Scholar - 7.T. Vachaspati, A. Vilenkin, Phys. Rev. Lett.
**67**, 1057 (1991)ADSCrossRefGoogle Scholar - 8.A. Vilenkin, Astrophys. J.
**282**, L51 (1984)ADSMathSciNetCrossRefGoogle Scholar - 9.J.R. Gott III, Astrophys. J.
**288**, 422 (1985)ADSMathSciNetCrossRefGoogle Scholar - 10.A. Vilenkin, in
*Inflating Horizons of Particle Astrophysics and Cosmology*, ed. by H. Suzuki, J. Yokoyama, Y. Suto, K. Sato (Universal Academy Press, Tokyo, 2006). arXiv:hep-th/0508135 - 11.A. Vilenkin, Phys. Rev. D
**23**, 852 (1981)ADSCrossRefGoogle Scholar - 12.W.A. Hiscock, Phys. Rev. D
**31**, 3288 (1985)ADSMathSciNetCrossRefGoogle Scholar - 13.J. Magueijo, A. Albrecht, D. Coulson, P. Ferreira, Phys. Rev. Lett.
**76**, 2617 (1996)ADSCrossRefGoogle Scholar - 14.U.L. Pen, U. Seljak, N. Turok, Phys. Rev. Lett.
**79**, 1611 (1997)ADSCrossRefGoogle Scholar - 15.A. Nayeri, R.H. Brandenberger, C. Vafa, Phys. Rev. Lett.
**97**, 021302 (2006)ADSCrossRefGoogle Scholar - 16.R.J. Danos, R.H. Brandenberger, G. Holder, Phys. Rev. D
**82**, 023513 (2010)ADSCrossRefGoogle Scholar - 17.N. Sen, Ann. Phys. (Leipzig)
**378**, 365 (1924)ADSCrossRefGoogle Scholar - 18.K. Lanczos, Ann. Phys. (Leipzig)
**379**, 518 (1924)ADSCrossRefGoogle Scholar - 19.G. Darmois, Mémorial des Sciences Mathématiques, Fascicule XXV (Gauthier-Villars, Paris, 1927), Chapter 5Google Scholar
- 20.
- 21.S.C. Davis, Phys. Rev. D
**67**, 024030 (2003)ADSMathSciNetCrossRefGoogle Scholar - 22.P.R. Brady, J. Louko, E. Poisson, Phys. Rev. D
**44**, 1891 (1991)ADSMathSciNetCrossRefGoogle Scholar - 23.M. Visser, D.L. Wiltshire, Class. Quant. Grav.
**21**, 1135 (2004)ADSMathSciNetCrossRefGoogle Scholar - 24.M.S. Morris, K.S. Thorne, Am. J. Phys.
**56**, 395 (1988)ADSMathSciNetCrossRefGoogle Scholar - 25.M.S. Morris, K.S. Thorne, U. Yurtsever, Phys. Rev. Lett
**61**, 1446 (1988)ADSCrossRefGoogle Scholar - 26.V.P. Frolov, I.D. Novikov, Phys. Rev. D
**42**, 1057 (1990)ADSMathSciNetCrossRefGoogle Scholar - 27.M. Visser,
*Lorentzian Wormholes*(AIP Press, New York, 1996)Google Scholar - 28.D. Hochberg, M. Visser, Phys. Rev. D
**56**, 4745 (1997)ADSMathSciNetCrossRefGoogle Scholar - 29.M. Visser, S. Kar, N. Dadhich, Phys. Rev. Lett.
**90**, 201102 (2003)ADSMathSciNetCrossRefGoogle Scholar - 30.E.F. Eiroa, C. Simeone, Phys. Rev. D
**70**, 044008 (2004)ADSMathSciNetCrossRefGoogle Scholar - 31.C. Bejarano, E.F. Eiroa, C. Simeone, Phys. Rev. D
**75**, 027501 (2007)ADSCrossRefGoogle Scholar - 32.M.G. Richarte, C. Simeone, Phys. Rev. D
**79**, 127502 (2009)ADSCrossRefGoogle Scholar - 33.E.F. Eiroa, C. Simeone, Phys. Rev. D
**81**, 084022 (2010) [Erratum-ibid.**90**, 089906 (2014)]Google Scholar - 34.M. Sharif, M. Azam, JCAP
**1304**, 023 (2013)ADSMathSciNetCrossRefGoogle Scholar - 35.M.G. Richarte, Phys. Rev. D
**88**, 027507 (2013)ADSCrossRefGoogle Scholar - 36.S. Habib Mazharimousavi, M. Halilsoy, Z. Amirabi, Eur. Phys. J. C
**74**, 2889 (2014)ADSCrossRefGoogle Scholar - 37.C. Simeone, Int. J. Mod. Phys. D
**21**, 1250015 (2012)ADSMathSciNetCrossRefGoogle Scholar - 38.S. Habib Mazharimousavi, M. Halilsoy, Z. Amirabi, Phys. Rev. D
**89**, 084003 (2014)ADSCrossRefGoogle Scholar - 39.E.F. Eiroa, C. Simeone, Phys. Rev. D
**91**, 064005 (2015)ADSMathSciNetCrossRefGoogle Scholar - 40.K.A. Bronnikov, A.A. Starobinsky, Mod. Phys. Lett. A
**24**, 1559 (2009)Google Scholar - 41.K.A. Bronnikov, M.V. Skvortsova, A.A. Starobinsky, Grav. Cosmol.
**16**, 216 (2010)ADSMathSciNetCrossRefGoogle Scholar - 42.K.A. Bronnikov, J.P.S. Lemos, Phys. Rev. D
**79**, 104019 (2009)ADSCrossRefGoogle Scholar - 43.K.A. Bronnikov, V.G. Krechet, J.P.S. Lemos, Phys. Rev. D
**87**, 084060 (2013)ADSCrossRefGoogle Scholar - 44.J. Bicak, M. Zofka, Class. Quant. Grav.
**19**, 3653 (2002)ADSMathSciNetCrossRefGoogle Scholar - 45.M. Arik, O. Delice, Gen. Relat. Gravit.
**37**, 1395 (2005)ADSMathSciNetCrossRefGoogle Scholar - 46.M. Zofka, J. Bicak, Class. Quant. Gravit.
**25**, 015011 (2008)MathSciNetCrossRefGoogle Scholar - 47.E. Rubín de Celis, O. Santillán, C. Simeone, Phys. Rev. D
**86**, 124009 (2012)ADSCrossRefGoogle Scholar - 48.E. Rubín de Celis, Eur. Phys. J. C
**76**, 92 (2016)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}