General Relativistic Static Fluid Solutions with Cosmological Constant
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
KKLT an Overview
KKLT an overview THESIS submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in PHYSICS Author : Maurits Houmes Student ID : s1672789 Supervisor : Prof. dr. K.E. Schalm 2nd corrector : Prof. dr. A. Achucarro´ Leiden, The Netherlands, 2020-07-03 KKLT an overview Maurits Houmes Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands 2020-07-03 Abstract In this thesis a detailed description of the KKLT scenario is given as well as as a comparison with later papers critiquing this model. An attempt is made to provide a some clarity in 17 years worth of debate. It concludes with a summary of the findings and possible directions for further research. Contents 1 Introduction 7 2 The Basics 11 2.1 Cosmological constant problem and the Expanding Universe 11 2.2 String theory 13 2.3 Why KKLT? 20 3 KKLT construction 23 3.1 The construction in a nutshell 23 3.2 Tadpole Condition 24 3.3 Superpotential and complex moduli stabilisation 26 3.4 Warping 28 3.5 Corrections 29 3.6 Stabilizing the volume modulus 30 3.7 Constructing dS vacua 32 3.8 Stability of dS vacuum, Original Considerations 34 4 Points of critique on KKLT 39 4.1 Flattening of the potential due to backreaction 39 4.2 Conifold instability 41 4.3 IIB backgrounds with de Sitter space and time-independent internal manifold are part of the swampland 43 4.4 Global compatibility 44 5 Conclusion 49 Bibliography 51 5 Version of 2020-07-03– Created 2020-07-03 - 08:17 Chapter 1 Introduction During the turn of the last century it has become apparent that our uni- verse is expanding and that this expansion is accelerating. -
Quantum Theory of Scalar Field in De Sitter Space-Time
ANNALES DE L’I. H. P., SECTION A N. A. CHERNIKOV E. A. TAGIROV Quantum theory of scalar field in de Sitter space-time Annales de l’I. H. P., section A, tome 9, no 2 (1968), p. 109-141 <http://www.numdam.org/item?id=AIHPA_1968__9_2_109_0> © Gauthier-Villars, 1968, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincaré, Section A : Vol. IX, n° 2, 1968, 109 Physique théorique. Quantum theory of scalar field in de Sitter space-time N. A. CHERNIKOV E. A. TAGIROV Joint Institute for Nuclear Research, Dubna, USSR. ABSTRACT. - The quantum theory of scalar field is constructed in the de Sitter spherical world. The field equation in a Riemannian space- time is chosen in the = 0 owing to its confor- . mal invariance for m = 0. In the de Sitter world the conserved quantities are obtained which correspond to isometric and conformal transformations. The Fock representations with the cyclic vector which is invariant under isometries are shown to form an one-parametric family. They are inequi- valent for different values of the parameter. However, its single value is picked out by the requirement for motion to be quasiclassis for large values of square of space momentum. -
(Anti-)De Sitter Space-Time
Geometry of (Anti-)De Sitter space-time Author: Ricard Monge Calvo. Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Advisor: Dr. Jaume Garriga Abstract: This work is an introduction to the De Sitter and Anti-de Sitter spacetimes, as the maximally symmetric constant curvature spacetimes with positive and negative Ricci curvature scalar R. We discuss their causal properties and the characterization of their geodesics, and look at p;q the spaces embedded in flat R spacetimes with an additional dimension. We conclude that the geodesics in these spaces can be regarded as intersections with planes going through the origin of the embedding space, and comment on the consequences. I. INTRODUCTION In the case of dS4, introducing the coordinates (T; χ, θ; φ) given by: Einstein's general relativity postulates that spacetime T T is a differential (Lorentzian) manifold of dimension 4, X0 = a sinh X~ = a cosh ~n (4) a a whose Ricci curvature tensor is determined by its mass- energy contents, according to the equations: where X~ = X1;X2;X3;X4 and ~n = ( cos χ, sin χ cos θ, sin χ sin θ cos φ, sin χ sin θ sin φ) with T 2 (−∞; 1), 0 ≤ 1 8πG χ ≤ π, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π, then the line element Rµλ − Rgµλ + Λgµλ = 4 Tµλ (1) 2 c is: where Rµλ is the Ricci curvature tensor, R te Ricci scalar T ds2 = −dT 2 + a2 cosh2 [dχ2 + sin2 χ dΩ2] (5) curvature, gµλ the metric tensor, Λ the cosmological con- a 2 stant, G the universal gravitational constant, c the speed of light in vacuum and Tµλ the energy-momentum ten- where the surfaces of constant time dT = 0 have metric 2 2 2 2 sor. -
Coordinate Systems in De Sitter Spacetime Bachelor Thesis A.C
Coordinate systems in De Sitter spacetime Bachelor thesis A.C. Ripken July 19, 2013 Radboud University Nijmegen Radboud Honours Academy Abstract The De Sitter metric is a solution for the Einstein equation with positive cosmological constant, modelling an expanding universe. The De Sitter metric has a coordinate sin- gularity corresponding to an event horizon. The physical properties of this horizon are studied. The Klein-Gordon equation is generalized for curved spacetime, and solved in various coordinate systems in De Sitter space. Contact Name Chris Ripken Email [email protected] Student number 4049489 Study Physicsandmathematics Supervisor Name prof.dr.R.H.PKleiss Email [email protected] Department Theoretical High Energy Physics Adress Heyendaalseweg 135, Nijmegen Cover illustration Projection of two-dimensional De Sitter spacetime embedded in Minkowski space. Three coordinate systems are used: global coordinates, static coordinates, and static coordinates rotated in the (x1,x2)-plane. Contents Preface 3 1 Introduction 4 1.1 TheEinsteinfieldequations . 4 1.2 Thegeodesicequations. 6 1.3 DeSitterspace ................................. 7 1.4 TheKlein-Gordonequation . 10 2 Coordinate transformations 12 2.1 Transformations to non-singular coordinates . ......... 12 2.2 Transformationsofthestaticmetric . ..... 15 2.3 Atranslationoftheorigin . 22 3 Geodesics 25 4 The Klein-Gordon equation 28 4.1 Flatspace.................................... 28 4.2 Staticcoordinates ............................... 30 4.3 Flatslicingcoordinates. 32 5 Conclusion 39 Bibliography 40 A Maple code 41 A.1 Theprocedure‘riemann’. 41 A.2 Flatslicingcoordinates. 50 A.3 Transformationsofthestaticmetric . ..... 50 A.4 Geodesics .................................... 51 A.5 TheKleinGordonequation . 52 1 Preface For ages, people have studied the motion of objects. These objects could be close to home, like marbles on a table, or far away, like planets and stars. -
Gravitational Decoupling in Higher Order Theories
S S symmetry Article Gravitational Decoupling in Higher Order Theories Joseph Sultana Department of Mathematics, Faculty of Science, University of Malta, MSD 2080 Msida, Malta; [email protected] Abstract: Gravitational decoupling via the Minimal Geometric Deformation (MGD) approach has been used extensively in General Relativity (GR), mainly as a simple method for generating exact anisotropic solutions from perfect fluid seed solutions. Recently this method has also been used to generate exact spherically symmetric solutions of the Einstein-scalar system from the Schwarzschild vacuum metric. This was then used to investigate the effect of scalar fields on the Schwarzschild black hole solution. We show that this method can be extended to higher order theories. In particular, we consider fourth order Einstein–Weyl gravity, and in this case by using the Schwarzschild metric as a seed solution to the associated vacuum field equations, we apply the MGD method to generate a solution to the Einstein–Weyl scalar theory representing a hairy black hole solution. This solution is expressed in terms of a series using the Homotopy Analysis Method (HAM). Keywords: Einstein–Weyl gravity; gravitational decoupling; black holes 1. Introduction Obtaining exact physically relevant solutions of Einstein’s field equations is by no means a straightforward task. The spherically symmetric perfect fluid analytic solutions of Einstein’s equations have been derived and analyzed in detail [1,2]. However, as soon as Citation: Sultana, J. Gravitational one relaxes the assumption of a perfect fluid source and considers more realistic sources, Decoupling in Higher Order Theories. then the problem of finding analytic solutions becomes intractable. -
A Brief Analysis of De Sitter Universe in Relativistic Cosmology
Munich Personal RePEc Archive A Brief Analysis of de Sitter Universe in Relativistic Cosmology Mohajan, Haradhan Assistant Professor, Premier University, Chittagong, Bangladesh. 18 June 2017 Online at https://mpra.ub.uni-muenchen.de/83187/ MPRA Paper No. 83187, posted 10 Dec 2017 23:44 UTC Journal of Scientific Achievements, VOLUME 2, ISSUE 11, NOVEMBER 2017, PAGE: 1-17 A Brief Analysis of de Sitter Universe in Relativistic Cosmology Haradhan Kumar Mohajan Premier University, Chittagong, Bangladesh E-mail: [email protected] Abstract The de Sitter universe is the second model of the universe just after the publications of the Einstein’s static and closed model. In 1917, Wilhelm de Sitter has developed this model which is a maximally symmetric solution of the Einstein field equation with zero density. The geometry of the de Sitter universe is theoretically more complicated than that of the Einstein universe. The model does not contain matter or radiation. But, it predicts that there is a redshift. This article tries to describe the de Sitter model in some detail but easier mathematical calculations. In this study an attempt has been taken to provide a brief discussion of de Sitter model to the common readers. Keywords: de Sitter space-time, empty matter universe, redshifts. 1. Introduction The de Sitter space-time universe is one of the earliest solutions to Einstein’s equation of motion for gravity. It is a matter free and fully regular solution of Einstein’s equation with positive 3 cosmological constant , where S is the scale factor. After the publications of the S 2 Einstein’s static and closed universe model a second static model of the static and closed but, empty of matter universe was suggested by Dutch astronomer Wilhelm de Sitter (1872–1934). -
Research Article Conformal Mappings in Relativistic Astrophysics
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 196385, 12 pages http://dx.doi.org/10.1155/2013/196385 Research Article Conformal Mappings in Relativistic Astrophysics S. Hansraj, K. S. Govinder, and N. Mewalal Astrophysics and Cosmology Research Unit, School of Mathematics, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa Correspondence should be addressed to K. S. Govinder; [email protected] Received 1 March 2013; Accepted 10 June 2013 Academic Editor: Md Sazzad Chowdhury Copyright © 2013 S. Hansraj et al. 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. We describe the use of conformal mappings as a mathematical mechanism to obtain exact solutions of the Einstein field equations in general relativity. The behaviour of the spacetime geometry quantities is given under a conformal transformation, and the Einstein field equations are exhibited for a perfect fluid distribution matter configuration. The field equations are simplified andthenexact static and nonstatic solutions are found. We investigate the solutions as candidates to represent realistic distributions of matter. In particular, we consider the positive definiteness of the energy density and pressure and the causality criterion, as well as the existence of a vanishing pressure hypersurface to mark the boundary of the astrophysical fluid. 1. Introduction By considering the case of a uniform density sphere, Schwarzschild [3]foundauniqueinteriorsolution.However, The gravitational evolution of celestial bodies may be mod- the problem of finding all possible solutions for a nonconstant eled by the Einstein field equations. -
Cosmological Spacetimes with Λ > 0
Cosmological Spacetimes with ¤ > 0 Gregory J. Galloway¤ Department of Mathematics University of Miami 1 Introduction This paper is based on a talk given by the author at the BEEMFEST, held at the University of Missouri, Columbia, in May, 2003. It was indeed a great pleasure and priviledge to participate in this meeting honoring John Beem, whose many oustanding contributions to mathematics, and leadership in the ¯eld of Lorentzian geometry has enriched so many of us. In this paper we present some results, based on joint work with Lars Andersson [2], concerning certain global properties of asymptotically de Sitter spacetimes satisfying the Einstein equations with positive cosmological constant. There has been increased interest in such spacetimes in recent years due, ¯rstly, to observations concerning the rate of expansion of the universe, suggesting the presence of a positive cosmological constant in our universe, and, secondly, due to recent e®orts to understand quan- tum gravity on de Sitter space via, for example, some de Sitter space version of the AdS/CFT correspondence; cf., [4] and references cited therein. In fact, the results discussed here were originally motivated by results of Witten and Yau [21, 5], and related spacetime results on topological censorship [11], pertaining to the AdS/CFT correspondence. These papers give illustrations of how the geometry and/or topol- ogy of the conformal in¯nity of an asymptotically hyperbolic Riemannian manifold, or an asymptotically anti-de Sitter spacetime, can inuence the global structure of the manifold. In this vein, the results described here establish connections between the geometry/topology of conformal in¯nity and the occurence of singularities in asymptotically de Sitter spacetimes. -
Generalized De Sitter Space
Generalized De Sitter Space Pedro F. Gonz´alez-D´ıaz Centro de F´ısica “Miguel Catal´an”, Instituto de Matem´aticas y F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficas, Serrano 121, 28006 Madrid (SPAIN) (September 14, 1999) This paper deals with some two-parameter solutions to the spherically symmetric, vacuum Einstein equations which, we argue, are more general than the de Sitter solution. The global structure of one such spacetimes and its extension to the multiply connected case have also been investigated. By using a six-dimensional Minkowskian embedding as its maximal extension, we check that the thermal properties of the considered solution in such an embedding space are the same as those derived by the usual Euclidean method. The stability of the generalized de Sitter space containing a black hole has been investigated as well by introducing perturbations of the Ginsparg-Perry type in first order approximation. It has been obtained that such a space perdures against the effects of these perturbations. PACS numbers: 04.20.Jb, 04.62.+v, 04.70.Dy I. INTRODUCTION equations can be written as µν µν In some respects de Sitter space is to cosmology what R =Λg , (1.1) hydrogen atom is to atomic physics. In most practical sit- with Λ the cosmological constant. Assuming then stat- uations they both become the first example where any re- icness for a general spherically symmetric metric spective new formalism, description or theory is brought to be checked out. Actually, since the very moment it 2 A 2 B 2 2 2 ds = e dt + e dr + r dΩ2, (1.2) was discovered one finds it difficult to encounter a paper − 2 2 2 2 on cosmology in which no mention is made of de Sitter with dΩ2 = dθ + sin θdφ the metric on the unit three- universe. -
Axisymmetric Spacetimes in Relativity
9, tr' Axisymmetric Spacetimes in Relativity S. P. Drake Department of Physics and Mathematical Physics îåäi,î31 tÏfåi" Australia 22 July 1998 ACKNO\MTEDGEMENTS It is an unfortunate fact that only my name Soes on this thesis' This is not the work of one person, it is an accumulation of the labor of many' AII those who have helped me in ways that none of us could image should be acknowledged, but how. None of this would have been possible were it not for the wisdom and guidance of my supervisor Peter Szekeres. I have often pondered over the difficulty of supervising students. It must be heart-bleaking to watch as students make necessary mistakes. Patience I'm sure must be the key' So for his wisdom, patience and kindness I am deeply indebted to Peter Szekeres. Without the love and support of my family this thesis would never have begun, much less been completed, to them I owe my life, and all that comes with it. It would take too long to write the names of all those I wish to thank. Those who are special to me, have helped me through so much over the years will receive the thanks in Person. I would like to the department of physics here at Adelaide where most of my thesis work was done. I would like to thank the department of physics at Melbourne university, where I did my undergraduate degree and began my PhD. I would like to thank the university of Padova for there hospitality dur- ing my stay there. -
Yet Another Family of Diagonal Metrics for De Sitter and Anti-De Sitter Spacetimes
Yet another family of diagonal metrics for de Sitter and anti-de Sitter spacetimes Jiˇr´ıPodolsk´y,OndˇrejHruˇska Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, Prague, V Holeˇsoviˇck´ach 2, 180 00 Praha 8, Czech Republic [email protected] and [email protected] July 11, 2017 Abstract In this work we present and analyze a new class of coordinate representations of de Sitter and anti-de Sitter spacetimes for which the metrics are diagonal and (typically) static and axially symmetric. Contrary to the well-known forms of these fundamental geometries, that usually correspond to a 1 + 3 foliation with the 3-space of a constant spatial curvature, the new metrics are adapted to a 2 + 2 foliation, and are warped products of two 2-spaces of constant curvature. This new class of (anti-)de Sitter metrics depends on the value of cosmological constant Λ and two discrete parameters +1; 0; −1 related to the curvature of the 2-spaces. The class admits 3 distinct subcases for Λ > 0 and 8 subcases for Λ < 0. We systematically study all these possibilities. In particular, we explicitly present the corresponding parametrizations of the (anti- )de Sitter hyperboloid, visualize the coordinate lines and surfaces within the global conformal cylinder, investigate their mutual relations, present some closely related forms of the metrics, and give transformations to standard de Sitter and anti-de Sitter metrics. Using these results, we also provide a physical interpretation of B-metrics as exact gravitational fields of a tachyon. 1 Introduction Almost exactly 100 years ago in 1917 Willem de Sitter published two seminal papers [1, 2] in which he presented his, now famous, exact solution of Einstein's gravitational field equations without matter but with a positive cosmological constant Λ. -
Generalized Relativistic Anisotropic Compact Star Models by Gravitational Decoupling
Eur. Phys. J. C (2019) 79:85 https://doi.org/10.1140/epjc/s10052-019-6602-1 Regular Article - Theoretical Physics Generalized relativistic anisotropic compact star models by gravitational decoupling S. K. Maurya1,a, Francisco Tello-Ortiz2,b 1 Department of Mathematics and Physical Science, College of Arts and Science, University of Nizwa, Nizwa, Sultanate of Oman 2 Departamento de Física, Facultad de ciencias básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile Received: 29 November 2018 / Accepted: 15 January 2019 / Published online: 29 January 2019 © The Author(s) 2019 Abstract In this work we have extended the Maurya-Gupta main ingredient of this scheme is to add an extra source to isotropic fluid solution to Einstein field equations to an aniso- the energy momentum tensor of a spherically symmetric and tropic domain. To do so, we have employed the gravitational static spacetime through a dimensionless constant β decoupling via the minimal geometric deformation approach. m = ˜ m + βθm, The present model is representing the strange star candidate Tn Tn n (1) LMC X-4. A mathematical, physical and graphical analy- ˜ m sis, shown that the obtained model fulfills all the criteria where Tn represents a perfect fluid matter distribution and θ m to be an admissible solution of the Einstein field equations. n is the extra source which encodes the anisotropy and Specifically, we have analyzed the regularity of the metric can be formed by scalar, vector or tensor fields. It is worth ˜ m potentials and the effective density, radial and tangential pres- mentioning that Tn does not necessarily should corresponds sures within the object, causality condition, energy condi- to a perfect fluid, in fact it can describes, for example, an tions, equilibrium via Tolman–Oppenheimer–Volkoff equa- anisotropic fluid, a charged fluid, etc.