- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in High Energy Physics

Volume 2017 (2017), Article ID 7943649, 6 pages

https://doi.org/10.1155/2017/7943649

## Cylindrically Symmetric, Asymptotically Flat, Petrov Type D Spacetime with a Naked Curvature Singularity and Matter Collapse

Department of Physics, Cotton College, Guwahati 781 001, India

Correspondence should be addressed to Faizuddin Ahmed

Received 1 December 2016; Revised 17 January 2017; Accepted 1 February 2017; Published 6 March 2017

Academic Editor: Elias C. Vagenas

Copyright © 2017 Faizuddin Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We present a cylindrically symmetric, Petrov type D, nonexpanding, shear-free, and vorticity-free solution of Einstein’s field equations. The spacetime is asymptotically flat radially and regular everywhere except on the symmetry axis where it possesses a naked curvature singularity. The energy-momentum tensor of the spacetime is that for an anisotropic fluid which satisfies the different energy conditions. This spacetime is used to generate a rotating spacetime which admits closed timelike curves and may represent a Cosmic Time Machine.

#### 1. Introduction

A spacetime is said to have singularities if it possesses inextensible curves with finite affine length. The singularities of spherically symmetric matter filled spacetimes can be recognized from the divergence of the energy density and curvature scalars,* for example*, the Kretschmann scalar and . These singularities are of two kinds, spacelike singularity (e.g., Schwarzschild singularity) and timelike singularity. In timelike singularities, two possibilities arise. (i) There are an even number of horizons around a timelike singularity (e.g., RN black hole). In that case, an observer cannot see this singularity from outside (the singularity is covered by an event horizon). (ii) There is no horizon around a timelike singularity and the latter one may be observable from the rest of the spacetime.

To counter the occurrence of timelike singularities which are not covered by an event horizon, Penrose [1–3] proposed a Conjecture known as Cosmic Censorship Conjecture. The Cosmic Censorship Conjecture (CCC) essentially states that no physically realistic collapse, evolving from a well-posed initial data set and satisfying the dominant energy condition, results in a singularity in the causal past of null infinity. According to Weak Cosmic Censorship, singularities have no effect on distant observers;* that is*, they cannot communicate with far-away observers. Till date, there are no mathematical theorems or proofs that support (or counter) this Conjecture. On the contrary, there is no mathematical reason that a naked singularity cannot exist in a solution of the field equations in GR. In literature, there are a number of examples of gravitational collapse which form a naked curvature singularity. The earliest model is the Lemaitre-Tolman-Bondi (LTB) [4–6] solution, a spherically symmetric inhomogeneous collapse of dust fluid that admits both naked and covered singularities. A small sample of gravitational collapse with the formation of a naked singularity is discussed in [7–16]. In this paper, we construct a nonvacuum metric that forms a naked curvature singularity. The presence of a naked singularity in a solution of Einstein’s field equations may break down the causality condition and may form closed timelike curves (CTCs). The possibility that a generic naked curvature singularity gives rise to a Cosmic Time Machine has been discussed by Clarke and de Felice [17] (see also [18–20]). A Cosmic Time Machine is a spacetime which is asymptotically flat and admits closed non-spacelike curves which extend to future infinity. In literature, there are a number of vacuum and nonvacuum spacetimes admitting CTCs which have been constructed without naked singularity. Small samples of these are in [21–33]. Recently, a vacuum spacetime with a naked curvature singularity admitting CTCs appeared in [34] which may represent a Cosmic Time Machine. One way of classifying causality violating spacetimes would be to categorize the metrics either as eternal time machines in which CTCs always exist (in this class would be [21, 22]) or as time machine spacetimes in which CTCs appear after a certain instant of time (in this class would be [25, 26]). However, many of the spacetimes admitting CTC suffer from one or more severe physical problems. The time machine spacetime discussed in [35–39] violated the weak energy condition (WEC) and the strong energy condition (SEC) is violated in [40–43]. The WEC states that for any physical (timelike) observer the energy density is nonnegative, which is the case for all known types of (classical) matter fields. Hawking proposed a Chronology Protection Conjecture [44] which states that the laws of physics will always prevent a spacetime from forming closed timelike curves (CTCs). But the general proof of the Chronology Protection Conjecture has not yet existed.

#### 2. Analysis of the Spacetime

Consider the following cylindrically symmetric metric in coordinates:where the units are chosen such that and . The coordinates are labelled as , , , and . The ranges of the coordinates are , , and and is periodic coordinate: . The metric is Lorentzian with signature and the determinant of the corresponding metric tensor isThe nonzero components of the Einstein tensor areThe scalar curvature invariants are given byThese curvature scalars and first-order differential invariants blow up (or diverge) at , indicating the existence of a naked curvature singularity. Since the singularity occurs without an event horizon, the Cosmic Censorship Conjecture has no physical interest. These curvature scalars vanish rapidly at spatial infinity,* that is*, , indicating that the spacetime is asymptotically flat radially.

#### 3. Cylindrical Symmetry and Petrov Type of the Spacetime

The spacetime represented by (1) is cylindrically symmetric as it is clear from the following. Consider the Killing vector which has the normal formIts covector form isThe vector (5) satisfies the Killing equation . The spacetime (1) that has an axially symmetric axis is assured by the condition [45–49]as , where we have chosen the radial coordinate such that the symmetry axis is located at . In addition to this, the spacetime admits another spacelike Killing vector which, with the axial Killing vector , generates a two-dimensional isometry group and they commute. Thus the isometry group is Abelian and the presented spacetime is cylindrically symmetric [46, 47]. However, the spacetime is not regular near to the axis as it fails to satisfy the regularity condition (or elementary flatness condition) [46–49], namely,in the limit at the rotation axis,* that is*, . The appearance of spacetime singularities on the axis can be considered as representing the existence of some kind of sources [49–53].

For the classification of metric (1), we construct the following set of a null tetrad and they arewhere . The set of null tetrads above are such that the metric tensor for the line element (1) can be expressed asThe vectors (9) are null vectors and are orthogonal except for and . Using this null tetrad above, we have calculated the five Weyl scalars, of which onlyis nonvanishing, while . The metric is clearly of type D in the Petrov classification scheme. The nonvanishing Weyl scalar vanishes rapidly at spatial infinity along the radial direction,* that is*, , again indicating the asymptotic flatness.

#### 4. Matter Field Distribution of the Spacetime and Kinematic Parameters

For spacetime (1), we consider the energy-momentum tensor for an anisotropic fluid given bywhere is the unit timelike four-velocity vector and is the unit spacelike vector along the radial direction . Here the physical parameters , , and are the energy density, the radial pressure, and the tangential pressure of the anisotropic fluid, respectively.

For metric (1), the timelike unit four-velocity vector is defined byAnd the spacelike unit vector along the radial direction isThe nonzero components of the energy-momentum tensor (12) using (13) and (14) areand the trace of the energy-momentum tensor (12) isEinstein’s field equations (taking cosmological constant ) are given bywhere is the Einstein tensor and is the energy-momentum tensor. Thus we have from the field equations (17) using (3), (15), and (16)which are diverging at , a fact that indicates the collapse of anisotropic fluid on the symmetry axis. The matter distribution satisfies the different energy conditions [54]:The kinematic parameters, the* expansion *, the* acceleration* vector , the* shear tensor *, and the* vorticity tensor * associated with the fluid four-velocity vector are defined bywhere is the projection tensor. For spacetime (1), these parameters have the following expressions:The magnitude of the acceleration vector is .

#### 5. Generating a Rotating Spacetime Admitting CTCs: A Cosmic Time Machine

One can easily generate a rotating spacetime mixing of the term from metric (1) by the following transformations, keeping and the same:From metric (1), we getwhere is periodic coordinate and .

Consider an azimuthal curve defined by , where , , and are constants. From (23), we haveThese curves are null (or null geodesics) at , spacelike throughout , but become timelike for , which indicates the presence of CTCs. Hence, CTCs form at a definite instant of time satisfying . The above analysis is valid provided that the CTCs evolve from an initially spacelike hypersurface [26]. A hypersurface is spacelike provided for , timelike provided for , and null hypersurface provided for . From metric (23), we found thatThus, the hypersurface is spacelike () and can be chosen as initial hypersurface over which the initial data may be specified. There is a Cauchy horizon at for any such initial hypersurface . Hence, the spacetime evolves from an initial spacelike hypersurface in a causally well-behaved manner, up to a moment,* that is*, a null hypersurface , and the formation of CTC takes place from causally well-behaved initial conditions. Assuming that the evolution beyond the Cauchy horizon proceeds in an analytic manner, we recover the region as well, and CTCs appear.

The formation of CTCs is thus identical to the Misner space. The Misner space is a two-dimensional locally flat metric with peculiar identifications. The metric of Misner space in 2D [55] iswhere but the coordinate is periodic. The curves defined by , where is a constant, are timelike and closed (since is periodic); thus CTCs formed. Note that the metric of Misner space (26) can be obtained from the Minkowski metric,by applying a similar transformation (22).

#### 6. Conclusion

In this paper, we present a cylindrically symmetric metric solution of the field equations which is regular everywhere except on the symmetry axis where it possesses a naked curvature singularity. The spacetime is of type D in the Petrov classification and is asymptotically flat radially and its matter-energy content is an anisotropic fluid that satisfies the different energy conditions. The nonzero kinematic parameters associated with the fluid four-velocity vector; namely, the acceleration diverges on the symmetry axis and vanishes at large radial distance. Similarly, the physical parameters, the energy density, the radial pressure, and the tangential pressure of the anisotropic fluid, behave in the same way. Finally, we generate a rotating spacetime admitting CTCs and these curves evolve from an initial spacelike hypersurface in a causally well-behaved manner. The possibility that a naked curvature singularity gives rise to a Cosmic Time Machine has been discussed by Clarke and his collaborators [17]. The presented cylindrically symmetric spacetime may represent such a Cosmic Time Machine.

#### Disclosure

The author’s permanent address is Hindustani Kendriya Vidyalaya, Dinesh Ojha Road, Guwahati 781 005, India.

#### Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

#### References

- R. Penrose, “Gravitational collapse: the role of general relativity,”
*Revista del Nuovo Cimento*, vol. 1, pp. 252–276, 1969. View at Google Scholar - R. Penrose, “Singularities and time-asymetry,” in
*General Relativity: An Einstein Centenary Survey*, S. W. Hawking and W. Israel, Eds., pp. 581–638, Cambridge University Press, Cambridge, UK, 1979. View at Google Scholar - R. Penrose, “The question of cosmic censorship,” in
*Black Holes and Relativistic Stars*, R. M. Wald, Ed., chapter 5, Chicago University Press, Chicago, Ill, USA, 1994. View at Google Scholar - G. Lemaitre, “L'univers en expansion,”
*Annales de la Societe Scientifique de Bruxelles A*, vol. 53, p. 51, 1933, Translated by M. A. H. MacCallum, “The expanding Universe”,*General Relativity & Gravitation*, vol. 29, no. 5, p. 641, 1997. View at Google Scholar - R. C. Tolman, “Effect of inhomogeneity on cosmological models,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 20, no. 3, pp. 169–176, 1934. View at Publisher · View at Google Scholar - H. Bondi, “Spherically symmetric models in general relativity,”
*Monthly Notices of the Royal Astronomical Society*, vol. 107, no. 5-6, p. 410, 1947. View at Google Scholar - A. Papapetrou, “Formation of a singularity and causality,” in
*A Random Walk in Relativity and Cosmology*, N. Dadhich, J. K. Rao, J. V. Narlikar, and C. V. Vishveshwara, Eds., pp. 184–191, John Wiley & Sons, New York, NY, USA, 1985. View at Google Scholar - P. C. Vaidya, “‘Newtonian’ time in general relativity,”
*Nature*, vol. 171, pp. 260–261, 1953. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Christodoulou, “Violation of cosmic censorship in the gravitational collapse of a dust cloud,”
*Communications in Mathematical Physics*, vol. 93, no. 2, pp. 171–195, 1984. View at Publisher · View at Google Scholar · View at MathSciNet - P. S. Joshi,
*Global Aspects in Gravitation and Cosmology*, vol. 87 of*International Series of Monographs on Physics*, The Clarendon Press, Oxford, UK, 1993. View at MathSciNet - P. S. Joshi,
*Singularities, Black Holes and Cosmic Censorship*, IUCAA Publication, Pune, India, 1997. - S. S. Deshingkar, P. S. Joshi, and I. H. Dwivedi, “Physical nature of the central singularity in spherical collapse,”
*Physical Review D*, vol. 59, no. 4, Article ID 044018, 1999. View at Publisher · View at Google Scholar - K. S. Govinder and M. Govender, “Gravitational collapse of null radiation and a string fluid,”
*Physical Review D*, vol. 68, no. 2, Article ID 024034, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - S. Barve, T. P. Singh, C. Vaz, and L. Witten, “A simple derivation of the naked singularity in spherical dust collapse,”
*Classical and Quantum Gravity*, vol. 16, no. 6, p. 1727, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. J. Clarke, “A title of cosmic censorship,”
*Classical and Quantum Gravity*, vol. 11, no. 6, pp. 1375–1386, 1994. View at Publisher · View at Google Scholar - P. S. Joshi and I. H. Dwivedi, “Initial data and the end state of spherically symmetric gravitational collapse,”
*Classical and Quantum Gravity*, vol. 16, no. 1, pp. 41–59, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. J. Clarke and F. de Felice, “Globally noncausal space-times. II. Naked singularities and curvature conditions,”
*General Relativity and Gravitation*, vol. 16, no. 2, pp. 139–148, 1984. View at Publisher · View at Google Scholar · View at MathSciNet - F. de Felice, “Cosmic time machines,” in
*Birth of the Universe and Fundamental Physics*, vol. 455 of*Lecture Notes in Physics*, pp. 99–102, Springer, Berlin, Germany, 1995. View at Publisher · View at Google Scholar - F. de Felice, “Naked singularities, cosmic time machines and impulsive events,”
*Nuovo Cimento B*, vol. 122, p. 481, 2007. View at Google Scholar - F. de Felice, “Cosmic time machines: the causality issue,”
*EPJ Web of Conferences*, vol. 58, Article ID 01001, 4 pages, 2013. View at Publisher · View at Google Scholar - K. Gödel, “An example of a new type of cosmological solutions of Einstein's field equations of gravitation,”
*Reviews of Modern Physics*, vol. 21, pp. 447–450, 1949. View at Publisher · View at Google Scholar · View at MathSciNet - W. J. van Stockum, “IX.—The gravitational field of a distribution of particles rotating about an axis of symmetry,”
*Proceedings of the Royal Society of Edinburgh*, vol. 57, pp. 135–154, 1938. View at Publisher · View at Google Scholar - F. J. Tipler, “Rotating cylinders and the possibility of global causality violation,”
*Physical Review. D. Particles and Fields. Third Series*, vol. 9, pp. 2203–2206, 1974. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - I. Gott, “Closed timelike curves produced by pairs of moving cosmic strings: exact solutions,”
*Physical Review Letters*, vol. 66, no. 9, pp. 1126–1129, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Ori, “Formation of closed timelike curves in a composite vacuum/dust asymptotically flat spacetime,”
*Physical Review D. Particles, Fields, Gravitation, and Cosmology*, vol. 76, no. 4, Article ID 044002, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - A. Ori, “A class of time-machine solutions with a compact vacuum core,”
*Physical Review Letters*, vol. 95, no. 2, Article ID 021101, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - W. B. Bonnor, “An exact, asymptotically flat, vacuum solution of Einstein's equations with closed timelike curves,”
*Classical and Quantum Gravity*, vol. 19, no. 23, pp. 5951–5957, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. B. Bonnor, “Closed timelike curves in classical relativity,”
*International Journal of Modern Physics D*, vol. 12, no. 09, pp. 1705–1708, 2003. View at Publisher · View at Google Scholar - W. B. Bonnor and B. R. Steadman, “Exact solutions of the Einstein-Maxwell equations with closed timelike curves,”
*General Relativity and Gravitation*, vol. 37, no. 11, pp. 1833–1844, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - C. W. Misner and A. H. Taub, “A singularity-free empty Universe,”
*Soviet Physics—JETP*, vol. 28, no. 1, p. 122, 1969. View at Google Scholar - R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,”
*Physical Review Letters*, vol. 11, pp. 237–238, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Sarma, M. Patgiri, and F. U. Ahmed, “Pure radiation metric with stable closed timelike curves,”
*General Relativity and Gravitation*, vol. 46, article no. 1633, 2014. View at Publisher · View at Google Scholar · View at Scopus - F. Ahmed, B. Bikash Hazarika, and D. Sarma, “The anti-de Sitter spacetime as a time machine,”
*European Physical Journal Plus*, vol. 131, no. 7, article no. 230, 2016. View at Publisher · View at Google Scholar · View at Scopus - D. Sarma, F. Ahmed, and M. Patgiri, “Axially symmetric, asymptotically flat vacuum metric with a naked singularity and closed timelike curves,”
*Advances in High Energy Physics*, vol. 2016, Article ID 2546186, 4 pages, 2016. View at Publisher · View at Google Scholar - M. S. Morris, K. S. Thorne, and U. Yurtsever, “Wormholes, time machines, and the weak energy condition,”
*Physical Review Letters*, vol. 61, no. 13, pp. 1446–1449, 1988. View at Publisher · View at Google Scholar · View at Scopus - M. S. Morris and K. S. Thorne, “Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity,”
*American Journal of Physics*, vol. 56, no. 5, pp. 395–412, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Alcubierre, “The warp drive: hyper-fast travel within general relativity,”
*Classical and Quantum Gravity*, vol. 11, no. 5, pp. L73–L77, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. E. Everett, “Warp drive and causality,”
*Physical Review. D. Third Series*, vol. 53, no. 12, pp. 7365–7368, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - A. E. Everett and T. A. Roman, “Superluminal subway: the Krasnikov tube,”
*Physical Review. D*, vol. 56, no. 4, pp. 2100–2108, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Soen and A. Ori, “Improved time-machine model,”
*Physical Review D*, vol. 54, no. 8, pp. 4858–4861, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - K. D. Olum, “Ori-Soen time machine,”
*Physical Review D*, vol. 61, no. 12, Article ID 124022, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - A. Ori and Y. Soen, “Causality violation and the weak energy condition,”
*Physical Review D*, vol. 49, no. 8, pp. 3990–3997, 1994. View at Publisher · View at Google Scholar · View at Scopus - A. Ori, “Must time-machine construction violate the weak energy condition?”
*Physical Review Letters*, vol. 71, p. 2517, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - S. W. Hawking, “Chronology protection conjecture,”
*Physical Review D*, vol. 46, no. 2, pp. 603–611, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - B. Carter, “The commutation property of a stationary, axisymmetric system,”
*Communications in Mathematical Physics*, vol. 17, pp. 233–238, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mars and J. M. M. Senovilla, “Axial symmetry and conformal Killing vectors,”
*Classical and Quantum Gravity*, vol. 10, no. 8, pp. 1633–1647, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Carot, J. M. Senovilla, and R. Vera, “On the definition of cylindrical symmetry,”
*Classical and Quantum Gravity*, vol. 16, no. 9, pp. 3025–3034, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Wang, “Critical collapse of a cylindrically symmetric scalar field in four-dimensional Einstein's theory of gravity,”
*Physical Review D*, vol. 68, no. 6, Article ID 064006, 12 pages, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt,
*Exact Solutions of Einstein's Field Equations*, Cambridge University Press, Cambridge, UK, 2003. View at Publisher · View at Google Scholar - W. B. Bonnor, “Physical interpretation of vacuum solutions of Einstein's equations. Part I. Time-independent solutions,”
*General Relativity and Gravitation*, vol. 24, no. 5, pp. 551–574, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - W. B. Bonnor, J. B. Griffiths, and M. A. MacCallum, “Physical interpretation of vacuum solutions of Einstein's equations. II. Time-dependent solutions,”
*General Relativity and Gravitation*, vol. 26, no. 7, pp. 687–729, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. F. da Silva, L. Herrera, F. M. Paiva, and N. O. Santos, “The Levi-Civita space-time,”
*Journal of Mathematical Physics*, vol. 36, no. 7, pp. 3625–3631, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Haggag and F. Desokey, “Perfect-fluid sources for the Levi-Civita metric,”
*Classical and Quantum Gravity*, vol. 13, no. 12, p. 3221, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - S. W. Hawking and G. F. Ellis,
*The Large Scale Structure of Space-Time*, Cambridge University Press, Cambridge, UK, 1973. View at MathSciNet - C. W. Misner, “Taub-NUT space as a Counter-example to almost anything,” in
*Relativity Theory and Astrophysics I: Relativity and Cosmology*, J. Ehlers, Ed., vol. 8 of*Lectures in Applied Mathematics*, p. 160, American Mathematical Society, 1967. View at Google Scholar