DOCSLIB.ORG

Dimensional Charged Gravastars ∗ Farook Rahaman A, ,A.A.Usmanib,Saibalrayc, Safiqul Islam A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dimensional Charged Gravastars ∗ Farook Rahaman A, ,A.A.Usmanib,Saibalrayc, Safiqul Islam A View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Physics Letters B 717 (2012) 1–5 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb The (2 + 1)-dimensional charged gravastars ∗ Farook Rahaman a, ,A.A.Usmanib,SaibalRayc, Safiqul Islam a a Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India b Department of Physics, Aligarh Muslim University, Aligarh 202 002, Uttar Pradesh, India c Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700 010, West Bengal, India article info abstract Article history: This is a continuation and generalization of our earlier work on gravastar in (2 + 1) anti-de Sitter Received 30 May 2012 space–time to (2 + 1)-dimensional solution of charged gravastar. Morphologically this gravastar contains Received in revised form 28 July 2012 three regions, namely: (i) charged interior, (ii) charged shell and (iii) electrovacuum exterior. We have Accepted 6 September 2012 studied different characteristics in terms of Length and Energy, Entropy, and Junction conditions of the Available online 10 September 2012 spherical charged distribution. It is shown that the present model of charged gravastar is non-singular Editor: M. Trodden and represents itself an alternative of black hole. © Keywords: 2012 Elsevier B.V. Open access under CC BY license. Gravitation Equation of state Charged gravastar 1. Introduction space–time was (3 + 1)-dimensional. The outer region of the Ra- haman et al. [17] model of gravastar corresponds to the exte- Recently the study of gravastars, the gravitational vacuum star, rior (2 + 1) anti-de Sitter space–time of BTZ-type black holes as has become a subject of considerable interest as it was proposed presented by Bañados, Teitelboim and Zanelli [18]. Therefore, the as an alternative to black holes. Mazur and Mottola [1,2] first pro- above two works demand that one should investigate a (2 + 1)- posed a new type of solution for the endpoint of a gravitational dimensional solution for charged gravastar. This is the motivation collapse in the form of cold, dark and compact objects. Therefore, of our present investigation. physically this was an extension of the concept of Bose–Einstein In favour of inclusion of charge in stellar distribution it has condensate to gravitational systems. The Mazur–Mottola model [1, been argued in the work of Usmani et al. [16] that compact stars 2] contains an isotropic de Sitter vacuum in the interior, while tend to assemble a net charge on the surface [19–23]. This facil- the exterior is defined by a Schwarzschild geometry, separated by itates stability of a fluid sphere by avoiding gravitational collapse a thin shell of stiff matter implying that the configuration of a and hence singularity [19,20,24–26]. In this connection we would gravastar has three different regions with different equations of like to mention the interesting charged model of Horvat [13] which state (EOS) [3–15], designated by: (I) Interior: 0 r < r1, p =−ρ; represents a gravastar where the analysis has carried out within Is- (II) Shell: r1 < r < r2, p =+ρ; and (III) Exterior: r2 < r, p = ρ = 0. rael’s thin shell formalism and the continuous profile approach. It is argued that the presence of matter on the thin shell of thick- In the present investigation we have followed the mechanism ness r2 − r1 = δ is essential to achieve the required stability of of Mazur and Mottola [1,2] in the framework of Einstein–Maxwell systems under expansion by exerting an inward force to balance formalism. To solve the specified equations, related to the regions the repulsion from within. designated as Interior, Shell and Exterior, we have considered ap- Usmani et al. [16] proposed a new model of a gravastar ad- propriate equations of state (EOS), viz. ρ =−p (dark energy), mitting conformal motion by assuming a charged interior. Their ρ = p (stiff fluid) and ρ = p = 0 (dust) respectively. Under this re- exterior was defined by a Reissner–Nordström line element instead gional classification of the spherical configurations we have solved of Schwarzschild’s one. Later on Rahaman et al. [17] designed a the Einstein–Maxwell field equations in the specific cases. There- neutral spherically symmetric model of gravastar in (2 + 1) anti- after we have characterized the interfaces and shell in terms of de Sitter space–time contrary to the former work where, as usual, Length, Energy and Entropy. The Junction conditions of the spher- ical charged distribution are imposed on the different regions by using Lanczos equations in (2 + 1)-dimensional space–time. The model thus obtained represents an alternative to black hole in * Corresponding author. E-mail addresses: [email protected] (F. Rahaman), [email protected] the form of charged gravastar as it is free from any singular- (A.A. Usmani), [email protected] (S. Ray), sofi[email protected] (S. Islam). ity. 0370-2693 ©2012ElsevierB.V. Open access under CC BY license. http://dx.doi.org/10.1016/j.physletb.2012.09.010 2 F. Rahaman et al. / Physics Letters B 717 (2012) 1–5 2. Einstein–Maxwell equations where q(r) is total charge of the sphere under consideration. For a charged fluid distribution, the generalized Tolman–Oppenheimer– We first consider the line element for the interior space–time Volkov (TOV) equation may be written as of a static spherically symmetric charged distribution of matter in 1 1 (2 + 1) dimensions in the form [27,28] (ρ + p)γ + p = r2 E2 , (14) 2 8πr2 2 γ (r) 2 λ(r) 2 2 2 ds =−e dt + e dr + r dθ . (1) which is the conservation equation in (2 + 1) dimensions. We note that the term inside the integral sign in Eq. (13) is The Hilbert action coupled to electromagnetism is given by λ(r) σ (r)e 2 , which is equivalent to the volume charge density. We √ − 3 R 2Λ 1 c will consider the volume charge density in polynomial function I = dx −g − F Fbc + Lm , (2) 16π 4 a of r. Hence we use the condition where L is the Lagrangian for matter. The variation with respect λ(r) n m σ (r)e 2 = σ0r , (15) to the metric gives the following self-consistent Einstein–Maxwell equations with cosmological constant Λ for a charged perfect fluid where n is arbitrary constant as polynomial index and the con- distribution stant σ0 is referred to the central charge density. By using the latter result of Eq. (15),oneobtainsfromEq.(13) 1 PF EM as Rab − Rgab + Λgab =−8π T + T . (3) 2 ab ab 4πσ0 n+1 The explicit forms of the energy momentum tensor (EMT) compo- E(r) = r , (16) n + 2 nents for the matter source (we assumed that the matter distribu- 4πσ0 + tion at the interior of the star is perfect fluid type) and electro- q(r) = rn 2. (17) magnetic fields are given by n + 2 Now, we write some consequences of the filed equations and TOV T PF = (ρ + p)u u + pg , (4) ab i k ik equation. Eq. (9) implies 1 1 EM c cd −λ(r) 2 T =− F Fbc − gab Fcd F , (5) e = M(r) − Λr , (18) ab 4π a 4 where where ρ, p, ui and Fab are, respectively, matter-energy density, fluid pressure and velocity three vector of a fluid element and r r electromagnetic field. Here, the electromagnetic field is related to M(r) = C − 16π rρ(r) dr − 2 rE2(r) dr (19) current three vector 0 0 J c = σ (r)uc, (6) is the active gravitational mass of the spherical distribution. Since, eλ(r) > 0 within the charged sphere of radius R as well as regular as at the origin, we demand that C > 0. Using Eqs. (10), (13) and the ab =− a above result (18), we finally obtain the TOV equation as F;b 4π J , (7) 2 (p + ρ)(8π p − E − Λ)r 1 where, σ (r) is the proper charge density of the distribution. In p =− + q2 . (20) = t M(r) − Λr2 8 r2 our consideration, the three velocity is assumed as ua δa and π consequently, the electromagnetic field tensor can be given as Following Mazur and Mottola [1,2] we consider a new (2 + 1)- dimensional charged perfect fluid configuration which has three = t r − r t Fab E(r) δaδb δaδb , (8) different regions with different equations of state: where E(r) is the electric field. =− The Einstein–Maxwell equations with a cosmological constant IInterior:0 r < R, ρ p; + = (Λ<0), for the space–time described by the metric (1) together II Shell: R r < R , ρ p; = = with the energy–momentum tensor given in Eqs. (4) and (5),yield III Exterior: r2 < r, ρ p 0. (rendering G = c = 1) Accordingly our configuration is supported by an interior region −λ λ e with equation of state p =−ρ. The shell of our configuration be- = 8πρ + E2 + Λ, (9) 2r longs to the interfaces, r = R and r = R + , where is the thick- − γ e λ ness of the shell. This thin shell, where the metric coefficients are = 8π p − E2 − Λ, (10) continuous, contains ultra-relativistic fluid of soft quanta obeying 2r equation of state ρ = p. The outer region of this gravastar corre- −λ e 1 1 γ 2 + γ − γ λ = 8π p + E2 − Λ, (11) sponds to the electrovacuum exterior solution popularly known as 2 2 2 the charged BTZ black hole space–time. − λ e 2 σ (r) = (rE) , (12) 3.
Load more

Recommended publications
  • Quantum-Corrected Rotating Black Holes and Naked Singularities in (2 + 1) Dimensions
    PHYSICAL REVIEW D 99, 104023 (2019) Quantum-corrected rotating black holes and naked singularities in (2 + 1) dimensions † ‡ Marc Casals,1,2,* Alessandro Fabbri,3, Cristián Martínez,4, and Jorge Zanelli4,§ 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, CEP 22290-180, Brazil 2School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland 3Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, C. Dr. Moliner 50, 46100 Burjassot, Spain 4Centro de Estudios Científicos (CECs), Arturo Prat 514, Valdivia 5110466, Chile (Received 15 February 2019; published 13 May 2019) We analytically investigate the perturbative effects of a quantum conformally coupled scalar field on rotating (2 þ 1)-dimensional black holes and naked singularities. In both cases we obtain the quantum- backreacted metric analytically. In the black hole case, we explore the quantum corrections on different regions of relevance for a rotating black hole geometry. We find that the quantum effects lead to a growth of both the event horizon and the ergosphere, as well as to a reduction of the angular velocity compared to their corresponding unperturbed values. Quantum corrections also give rise to the formation of a curvature singularity at the Cauchy horizon and show no evidence of the appearance of a superradiant instability. In the naked singularity case, quantum effects lead to the formation of a horizon that hides the conical defect, thus turning it into a black hole. The fact that these effects occur not only for static but also for spinning geometries makes a strong case for the role of quantum mechanics as a cosmic censor in Nature.
    [Show full text]
  • Traversable Wormholes and Regenesis
    Traversable Wormholes and Regenesis The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Gao, Ping. 2019. Traversable Wormholes and Regenesis. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:42029626 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Traversable Wormholes and Regenesis A dissertation presented by Ping Gao to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts April 2019 c 2019 | Ping Gao All rights reserved. Dissertation Advisor: Daniel Louis Jafferis Ping Gao Traversable Wormholes and Regenesis Abstract In this dissertation we study a novel solution of traversable wormholes in the context of AdS/CFT. This type of traversable wormhole is the first such solution that has been shown to be embeddable in a UV complete theory of gravity. We discuss its property from points of view of both semiclassical gravity and general chaotic system. On gravity side, after turning on an interaction that couples the two boundaries of an eternal BTZ black hole, in chapter 2 we find a quantum matter stress tensor with negative average null energy, whose gravitational backreaction renders the Einstein-Rosen bridge traversable. Such a traversable wormhole has an interesting interpretation in the context of ER=EPR, which we suggest might be related to quantum teleportation.
    [Show full text]
  • Final Fate of a Massive Star
    FINAL FATE OF A MASSIVE STAR Modern science has introduced the world to plenty of strange ideas, but surely one of the strangest is the fate of a massive star that has reached the end of its life. Having exhausted the fuel that sustained it for millions of years, the star is no longer able to hold itself up under its own weight, and it starts collapsing catastrophically. Modest stars like the sun also collapse but they stabilize again at a smaller size, whereas if a star is massive enough, its gravity overwhelms all the forces that might halt the collapse. From a size of Fig. 1 Black Hole (Artist’s conception) millions of kilometers across, the star crumples to a pinprick smaller than the dot on an “i.” What is the final fate of such massive collapsing stars? This is one of the most exciting questions in astrophysics and modern cosmology today. An amazing inter-play of the key forces of nature takes place here, including the gravity and quantum mechanics. This phenomenon may hold the key secrets to man’s search for a unified understanding of all forces of nature. It also may have most exciting connections and implications for astronomy and high energy astrophysics. This is an outstanding unresolved issue that excites physicists and the lay person alike. The story really began some eight decades ago when Subrahmanyan Fig. 2 An artist’s conception of Chandrasekhar probed the question of final fate of stars such as the Sun. He naked singularity showed that such a star, on exhausting its internal nuclear fuel, would stabilize as a `white dwarf', which is about a thousand kilometers in size.
    [Show full text]
  • Spacetime Warps and the Quantum World: Speculations About the Future∗
    Spacetime Warps and the Quantum World: Speculations about the Future∗ Kip S. Thorne I’ve just been through an overwhelming birthday celebration. There are two dangers in such celebrations, my friend Jim Hartle tells me. The first is that your friends will embarrass you by exaggerating your achieve- ments. The second is that they won’t exaggerate. Fortunately, my friends exaggerated. To the extent that there are kernels of truth in their exaggerations, many of those kernels were planted by John Wheeler. John was my mentor in writing, mentoring, and research. He began as my Ph.D. thesis advisor at Princeton University nearly forty years ago and then became a close friend, a collaborator in writing two books, and a lifelong inspiration. My sixtieth birthday celebration reminds me so much of our celebration of Johnnie’s sixtieth, thirty years ago. As I look back on my four decades of life in physics, I’m struck by the enormous changes in our under- standing of the Universe. What further discoveries will the next four decades bring? Today I will speculate on some of the big discoveries in those fields of physics in which I’ve been working. My predictions may look silly in hindsight, 40 years hence. But I’ve never minded looking silly, and predictions can stimulate research. Imagine hordes of youths setting out to prove me wrong! I’ll begin by reminding you about the foundations for the fields in which I have been working. I work, in part, on the theory of general relativity. Relativity was the first twentieth-century revolution in our understanding of the laws that govern the universe, the laws of physics.
    [Show full text]
  • Advanced Lectures on General Relativity
    Lecture notes prepared for the Solvay Doctoral School on Quantum Field Theory, Strings and Gravity. Lectures given in Brussels, October 2017. Advanced Lectures on General Relativity Lecturing & Proofreading: Geoffrey Compère Typesetting, layout & figures: Adrien Fiorucci Fonds National de la Recherche Scientifique (Belgium) Physique Théorique et Mathématique Université Libre de Bruxelles and International Solvay Institutes Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium Please email any question or correction to: [email protected] Abstract — These lecture notes are intended for starting PhD students in theoretical physics who have a working knowledge of General Relativity. The 4 topics covered are (1) Surface charges as con- served quantities in theories of gravity; (2) Classical and holographic features of three-dimensional Einstein gravity; (3) Asymptotically flat spacetimes in 4 dimensions: BMS group and memory effects; (4) The Kerr black hole: properties at extremality and quasi-normal mode ringing. Each topic starts with historical foundations and points to a few modern research directions. Table of contents 1 Surface charges in Gravitation ................................... 7 1.1 Introduction : general covariance and conserved stress tensor..............7 1.2 Generalized Noether theorem................................. 10 1.2.1 Gauge transformations and trivial currents..................... 10 1.2.2 Lower degree conservation laws........................... 11 1.2.3 Surface charges in generally covariant theories................... 13 1.3 Covariant phase space formalism............................... 14 1.3.1 Field fibration and symplectic structure....................... 14 1.3.2 Noether’s second theorem : an important lemma................. 17 Einstein’s gravity.................................... 18 Einstein-Maxwell electrodynamics.......................... 18 1.3.3 Fundamental theorem of the covariant phase space formalism.......... 20 Cartan’s magic formula...............................
    [Show full text]
  • Static Charged Fluid in (2+ 1)-Dimensions Admitting Conformal
    Accepted for publication in IJMPD STATIC CHARGED FLUID IN (2 + 1)-DIMENSIONS ADMITTING CONFORMAL KILLING VECTORS FAROOK RAHAMAN Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India [email protected] SAIBAL RAY Department of Physics, Government College of Engineering & Ceramic Technology, Kolkata 700 010, West Bengal, India [email protected] INDRANI KARAR Department of Mathematics, Saroj Mohan Institute of Technology, Guptipara, West Bangal, India [email protected] HAFIZA ISMAT FATIMA Department of Mathematics, The University of Lahore, Lahore, Pakistan [email protected] SAIKAT BHOWMICK Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India [email protected] arXiv:1211.1228v3 [gr-qc] 24 Feb 2014 GOURAB KUMAR GHOSH Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India [email protected] Received Day Month Year Revised Day Month Year Communicated by Managing Editor New solutions for (2 + 1)-dimensional Einstein-Maxwell space-time are found for a static spherically symmetric charged fluid distribution with the additional condition of allowing conformal killing vectors (CKV). We discuss physical properties of the fluid parameters. Moreover, it is shown that the model actually represents two structures, namely (i) Gravastar as an alternative of black hole and (ii) Electromagnetic Mass model depending on the nature of the equation state of the fluid. Here the gravitational mass originates from electromagnetic field alone. The solutions are matched with the exterior region of the Ba˜nados-Teitelboim-Zanelli (BTZ) type isotropic static charged black hole as a 1 Accepted for publication in IJMPD 2 consequence of junction conditions. We have shown that the central charge density is dependent on the value of M0, the conserved mass of the BTZ black hole.
    [Show full text]
  • Table of Contents (Print)
    PHYSICAL REVIEW D PERIODICALS For editorial and subscription correspondence, Postmaster send address changes to: please see inside front cover (ISSN: 1550-7998) APS Subscription Services P.O. Box 41 Annapolis Junction, MD 20701 THIRD SERIES, VOLUME 101, NUMBER 10 CONTENTS D15 MAY 2020 The Table of Contents is a total listing of Parts A and B. Part A consists of articles 101501–104021, and Part B articles 104022–109903(E) PART A RAPID COMMUNICATIONS Faithful analytical effective-one-body waveform model for spin-aligned, moderately eccentric, coalescing black hole binaries (7 pages) ........................................................................................... 101501(R) Danilo Chiaramello and Alessandro Nagar ARTICLES Study of the ionization efficiency for nuclear recoils in pure crystals (10 pages) .............................................. 102001 Y. Sarkis, Alexis Aguilar-Arevalo, and Juan Carlos D’Olivo Observation of a potential future sensitivity limitation from ground motion at LIGO Hanford (12 pages) ................ 102002 J. Harms, E. L. Bonilla, M. W. Coughlin, J. Driggers, S. E. Dwyer, D. J. McManus, M. P. Ross, B. J. J. Slagmolen, and K. Venkateswara Efficient gravitational-wave glitch identification from environmental data through machine learning (12 pages) ......... 102003 Robert E. Colgan, K. Rainer Corley, Yenson Lau, Imre Bartos, John N. Wright, Zsuzsa Márka, and Szabolcs Márka Limits on the stochastic gravitational wave background and prospects for single-source detection with GRACE Follow-On (6 pages) ........................................................................................................ 102004 M. P. Ross, C. A. Hagedorn, E. A. Shaw, A. L. Lockwood, B. M. Iritani, J. G. Lee, K. Venkateswara, and J. H. Gundlach Analysis and visualization of the output mode-matching requirements for squeezing in Advanced LIGO and future gravitational wave detectors (14 pages) ............................................................................................
    [Show full text]
  • Holography of the BTZ Black Hole, Inside and out Arxiv:1307.7738V2
    UMD-PP-013-008 Holography of the BTZ Black Hole, Inside and Out Anton de la Fuente and Raman Sundrum Maryland Center for Fundamental Physics Department of Physics University of Maryland College Park, 20782 MD Abstract We propose a 1 + 1 dimensional CFT dual structure for quantum gravity and matter on the extended 2 + 1 dimensional BTZ black hole, realized as a quotient of the Poincaré patch of AdS3. The quotient spacetime includes regions beyond the singularity, "whiskers", containing timelike and lightlike closed curves, which at first sight seem unphysical. The spacetime includes the usual AdS-asymptotic boundaries outside the horizons as well as boundary components inside the whiskers. We show that local boundary correlators with some endpoints in the whisker regions: (i) are a protected class of amplitudes, dominated by effective field theory even when the associated Witten diagrams appear to traverse the singularity, (ii) describe well-defined diffeomorphism-invariant quantum gravity amplitudes in BTZ, (iii) sharply probe some of the physics inside the horizon but outside the singularity, and (iv) are equivalent to correlators of specific non-local CFT operators in the standard thermofield entangled state of two CFTs. In this sense, the whisker regions can be considered as purely auxiliary spacetimes in which these useful non-local CFT correlators can be rendered as local boundary correlators, and their diagnostic value more readily understood. Our results follow by first performing a novel reanalysis of the Rindler view of standard AdS/CFT duality on the Poincaré patch of AdS, followed by exploiting the simple quotient structure of BTZ which turns the Rindler horizon into the BTZ black hole horizon.
    [Show full text]
  • A Treatise on Astronomy and Space Science Dr
    SSRG International Journal of Applied Physics Volume 8 Issue 2, 47-58, May-Aug, 2021 ISSN: 2350 – 0301 /doi:10.14445/23500301/ IJAP-V8I2P107 © 2021 Seventh Sense Research Group® A Treatise on Astronomy and Space Science Dr. Goutam Sarker, SM IEEE, Computer Science & Engineering Department, National Institute of Technology, Durgapur Received Date: 05 June 2021 Revised Date: 11 July 2021 Accepted Date: 22 July 2021 Abstract — Astronomy and Space Science has achieved a VII details naked singularity of a rotating black hole. tremendous enthuse to ardent humans from the early Section VIII discusses black hole entropy. Section IX primitive era of civilization till now a days. In those early details space time, space time curvature, space time pre-history days, primitive curious men looked at the sky interval and light cone. Section X presents the general and and got surprised and overwhelmed by the obscure special theory of relativity. Section XI details the different uncanny solitude of the universe along with all its definitions of time from different viewpoint as well as terrestrial bodies. This only virulent bustle of mankind different types of time. Last but not least Section XII is the regarding the outer space allowed him to open up conclusion which sums up all the previous discussions. surprising and splendid chapter after chapter and quench his thrust about the mysterious universe. In the present II. PRELIMINARIES OF ASTRONOMY paper we are going to discuss several important aspects According to Big Bang Theory [1,3,4,8], our universe and major issues of astronomy and space science. Starting originated from initial state of very high density and from the formation of stars it ended up to black holes – temperature.
    [Show full text]
  • Conformal Field Theory and Black Hole Physics
    CONFORMAL FIELD THEORY AND BLACK HOLE PHYSICS Steve Sidhu Bachelor of Science, University of Northern British Columbia, 2009 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfilment of the Requirements for the Degree MASTER OF SCIENCE Department of Physics and Astronomy University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Steve Sidhu, 2012 Dedication To my parents, my sister, and Paige R. Ryan. iii Abstract This thesis reviews the use of 2-dimensional conformal field theory applied to gravity, specifically calculating Bekenstein-Hawking entropy of black holes in (2+1) dimen- sions. A brief review of general relativity, Conformal Field Theory, energy extraction from black holes, and black hole thermodynamics will be given. The Cardy formula, which calculates the entropy of a black hole from the AdS/CFT duality, will be shown to calculate the correct Bekenstein-Hawking entropy of the static and rotating BTZ black holes. The first law of black hole thermodynamics of the static, rotating, and charged-rotating BTZ black holes will be verified. iv Acknowledgements I would like to thank my supervisors Mark Walton and Saurya Das. I would also like to thank Ali Nassar and Ahmed Farag Ali for the many discussions, the Theo- retical Physics Group, and the entire Department of Physics and Astronomy at the University of Lethbridge. v Table of Contents Approval/Signature Page ii Dedication iii Abstract iv Acknowledgements v Table of Contents vi 1 Introduction 1 2 Einstein’s field equations and black hole solutions 7 2.1 Conventions and notations . 7 2.2 Einstein’s field equations .
    [Show full text]
  • Black Hole As a Wormhole Factory
    Physics Letters B 751 (2015) 220–226 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Black hole as a wormhole factory ∗ Sung-Won Kim a, Mu-In Park b, a Department of Science Education, Ewha Womans University, Seoul, 120-750, Republic of Korea b Research Institute for Basic Science, Sogang University, Seoul, 121-742, Republic of Korea a r t i c l e i n f o a b s t r a c t Article history: There have been lots of debates about the final fate of an evaporating black hole and the singularity Received 22 September 2015 hidden by an event horizon in quantum gravity. However, on general grounds, one may argue that a black Received in revised form 15 October 2015 1/2 −5 hole stops radiation at the Planck mass (hc¯ /G) ∼ 10 g, where the radiated energy is comparable Accepted 16 October 2015 to the black hole’s mass. And also, it has been argued that there would be a wormhole-like structure, Available online 20 October 2015 − known as “spacetime foam”, due to large fluctuations below the Planck length (hG¯ /c3)1/2 ∼ 10 33 cm. In Editor: M. Cveticˇ this paper, as an explicit example, we consider an exact classical solution which represents nicely those two properties in a recently proposed quantum gravity model based on different scaling dimensions between space and time coordinates. The solution, called “Black Wormhole”, consists of two different states, depending on its mass parameter M and an IR parameter ω: For the black hole state (with ωM2 > 1/2), a non-traversable wormhole occupies the interior region of the black hole around the singularity at the origin, whereas for the wormhole state (with ωM2 < 1/2), the interior wormhole is exposed to an outside observer as the black hole horizon is disappearing from evaporation.
    [Show full text]
  • Rotating Charged Black String and Three Dimensional Black Holes
    CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN Document Server Rotating Charged Black String and Three Dimensional Black Holes Jos´eP.S.Lemos1 Departamento de Astrof´ısica Observat´orio Nacional – CNPq Rua General Jos´e Cristino 77, 20921, Rio de Janeiro, RJ, Brazil, & Departamento de F´ısica Instituto Superior T´ecnico Av. Rovisco Pais 1, 1096, Lisboa, Portugal Vilson T. Zanchin 2 Departamento de F´ısica-CCNE Universidade Federal de Santa Maria 97119-900 Santa Maria, RS, Brazil 1e-mail:[email protected] 2Present address: Department of Physics, Brown University, Providence, R.I. 02912. e-mail:[email protected] Abstract Einstein-Maxwell equations with a cosmological constant are analyzed in a stationary spacetime admitting an isometry group R×U(1). We find charged rotating black string solutions in a spacetime asymptotically anti-de Sitter in the radial direction. The mass (M), angular momentum (J) and charge (Q) line densities are defined using the Hamiltonian formalism of Brown and York. It is shown through dimensional reduction that M, J and Q are respec- tively the mass, angular momentum and charge of a related three dimensional black hole. The black string solutions are classified according to the charge and angular momentum parameters. The causal structure of each case is studied in detail for different ranges of the parameters and the Penrose dia- grams are shown. There are many similarities between the charged rotating black string and the Kerr-Newman spacetime. The solution has Cauchy and event horizons, ergosphere, timelike singularities, closed timelike curves, and extremal cases.
    [Show full text]