DOCSLIB.ORG

Strength of the Null Singularity Inside Black Holes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Strength of the Null Singularity Inside Black Holes PHYSICAL REVIEW D, VOLUME 60, 104033 Strength of the null singularity inside black holes Lior M. Burko Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125 ͑Received 3 May 1999; published 26 October 1999͒ We study analytically the Cauchy horizon singularity inside spherically symmetric charged black holes, coupled to a spherical, self-gravitating, minimally coupled, massless scalar field. We show that all causal geodesics terminate at the Cauchy horizon at a null singularity, which is weak according to the Tipler classi- fication. The singularity is also deformationally weak in the sense of Ori. Our results are valid at arbitrary points along the null singularity, in particular at late retarded times, when nonlinear effects are crucial. ͓S0556-2821͑99͒04920-6͔ PACS number͑s͒: 04.70.Bw, 04.20.Dw I. INTRODUCTION spherical charged black hole, the main features of the CH singularity were first found analytically for simplified models The issue of spacetime singularities—which are known to based on null fluids ͓10–12͔, and later confirmed numeri- inevitably occur inside black holes under very plausible as- cally for a model with a self-gravitating scalar field ͓13,14͔. sumptions ͓1͔—is an intriguing puzzle of physics. The laws Expressions for the divergence rate of the blueshift factors of physics, as we presently understand them ͑e.g., classical for that model, which are valid everywhere along the CH, general relativity͒, are presumably invalid at singularities. were found analytically in Ref. ͓15͔. Those expressions are Instead, some other theories ͑e.g., quantum gravity͒, as yet exact on the CH as functions of retarded time. However, they unknown, are expected to take over from general relativity are only asymptotic expressions as functions of advanced and control the spacetime structure. The general relativistic time ͑see below͒. predictions are nevertheless of the greatest importance, as The strength of the null singularity is of crucial impor- they reveal the spacetime structure under extreme conditions tance for the question of the hypothetical possibility of hy- in the strong-field regime. Of particular interest is the possi- perspace travel through the CH of black holes. A necessary bility that there are two distinct ways in which general rela- condition for this possibility to be realized is that physical tivity can fail at different types of singularities: for one type objects would traverse the CH peacefully. Because the CH is of singularity the failure is through infinite destructive ef- known to be a curvature singularity, it is necessary that the fects on physical objects, whereas for the other type the fail- singularity would be weak according to the Tipler classifica- ure is through the breakdown of predictability. tion of singularity strengths. For the toy model of a spherical Until recently, the only known generic singularity in gen- charged black hole, which we shall study here, the properties eral relativity was the Belinsky-Khalatnikov-Lifshitz ͑BKL͒ of the CH singularity which have been found in Refs. ͓11– singularity ͓2͔. According to the BKL picture, spacetime de- 15͔ are all consistent with the picture of a Tipler weak sin- velops a succession of Kasner epochs in which the axes of gularity. However, the weakness of the singularity was dem- contraction and expansion change directions chaotically. onstrated only for the simplified Ori model ͓12͔ and at This succession ends at unbounded oscillations at a spacelike asymptotically early times for spinning black holes ͓5͔, singularity, which is unavoidably destructive for any physi- where there are still no strong nonlinear effects, such as fo- cal object—a strong singularity. In the last several years, cusing of the null generators of the CH, which are crucial at however, evidence has been accumulating that the BKL sin- later times. In the context of spherical charged black holes gularity is not the only type of singularity which may evolve and a self-gravitating scalar field, several important features in general relativity from generic initial data. of the spacetime structure have been found in fully nonlinear The new type of singularity forms at the Cauchy horizon numerical simulations. Specifically, it was shown that for ͑CH͒ of spinning or charged black holes. ͑For a recent re- any point along the CH singularity there existed coordinates view see Ref. ͓3͔.͒ The features of this singularity are mark- for which the metric coefficients were finite and the metric edly different from those of the BKL singularity: ͑i͒ It is null determinant was nondegenerate in an open neighborhood to ͑rather than spacelike͒, ͑ii͒ it is weak ͑according to Tipler’s the past ͓13,14͔. However, despite previous claims ͓5,14,16͔, classification ͓4͔͒, specifically, the tidal deformations which this still does not guarantee that the singularity is weak in the an extended physical object suffers upon approaching the Tipler sense ͓17͔. singularity are bounded. In the case of a spinning black hole, It is the purpose of this paper to present an analytical the evidence for the null and weak singularity has emerged demonstration of the weakness of the singularity for the from analytical perturbative ͓5,6͔ and nonperturbative ͓7͔ model of a spherical charged black hole with a self- analyses. In addition, the local existence and genericity of a gravitating, minimally coupled, massless, real scalar field. null and weak singularity in solutions of the vacuum Einstein Our results are valid at arbitrary points along the CH singu- equations was demonstrated in Ref. ͓8͔. This was more re- larity, in particular at late times, where strong nonlinear ef- cently demonstrated also in the framework of plane- fects ͑focusing of the null generators of the CH and growth symmetric spacetimes in Ref. ͓9͔. For the toy model of a of the blueshift factors͒ are crucial. In fact, our results are 0556-2821/99/60͑10͒/104033͑5͒/$15.0060 104033-1 ©1999 The American Physical Society LIORM.BURKO PHYSICAL REVIEW D 60 104033 ⌽ valid everywhere along the CH singularity, down to the ,V are exactly inversely proportional to r(u), in the follow- event where the generators of the CH are completely fo- ing sense. Consider two outgoing null rays, and let one ray ϭ ϭ cused, and the singularity becomes spacelike and Tipler be at u u1, say, and the other at u u2. The ratios ͓ ͔ ⌽ ⌽ strong 18 . We emphasize that although our discussion here r,V(2)/r,V(1) and ,V(2)/ ,V(1) approach r(u1)/r(u2)as ! is analytical, we do make assumptions which are based on V 0. Taking now u1 to be in the asymptotically early parts Ϸ ⌽ results obtained by numerical simulations. of the CH, where r(u1) rϪ , we find that both r,V and ,V are inversely proportional to r(u). As r(u) is monotonically II. STRENGTH OF THE SINGULARITY decreasing as a function of retarded time along the CH, we ⌽ find that r,V and ,V grow monotonically along the CH. This We write the general spherically symmetric line element growth is a nonlinear effect which indicates the strengthen- in the form ing of the singularity along the CH ͑although the singularity is still weak according to the Tipler classification; see be- ds2ϭϪf ͑u,v͒dudvϩr2͑u,v͒d⍀2, ͑1͒ low͒. where d⍀2ϭd␽2ϩsin2 ␽d␸2 is the line element on the unit All the nonzero components of the Riemann-Christoffel curvature tensor R␮␯␳␴ are given completely in terms of the two sphere. The coordinates u,v are any outgoing and ingo- ⌽ ing null coordinates, correspondingly. ͑Below, we shall spe- divergent blueshift factors r,V , ,V , and the finite quantities ⌽ cialize to a specific choice of gauge, and define a particular r,r,u , ,u , and F. Interestingly, R␮␯␳␴ does not depend on choice of an ingoing null coordinate.͒ We consider the class gradients of F. This can be understood from the following of scalar field perturbations which is inherent to any gravita- consideration. The tensor R␮␯␳␴ can be written as the sum of tional collapse process. These are the perturbations which the Weyl tensor, and another tensor which is built from the ͑ result from the evolution of nonvanishing multipole mo- Ricci and the metric tensors but not involving their deriva- ͒ ments during the collapse. When these perturbations propa- tives . In spherical symmetry the Weyl tensor is given com- gate outwards, they are partially reflected off the spacetime pletely in terms of the mass function, which is defined by ,␮ϭ Ϫ ϩ 2 2 curvature and captured by the black hole. This process re- r,␮r 1 2M(u,v)/r q /r , q being the charge of the sults in a scalar field, which at late advanced times decays black hole. In Kruskal-like coordinates the mass function ϭ ϩ ϩ 2 along the event horizon according to an inverse power of M(u,V) (r/2)(1 4r,ur,V /F) q /(2r), which depends ͑ advanced time. Specifically, we assume that the scalar field only on r,r,V ,r,u , and F. The divergence of the mass func- behaves along the event horizon at late times according to tion at the singularity, and consequently also the divergence Ϫ ⌽EHϰ(␬v ) n ͓19–21͔, where n is a positive integer which of curvature, is evident from the divergence of r,V and the e ͒ is related to the multipole moment under consideration. ͑We finiteness of r, r,u , and F. The Ricci tensor R␮␯ ϭ ⌽ ⌽ do not consider, however, other possible sources of pertur- 2 ,␮ ,␯ , and consequently R␮␯␳␴ is independent of gra- ͓ ͔ ͒ dients of F. bations 22 . By ve we denote the usual advanced time in the Eddington gauge, and rϮ ,␬ are the outer and inner ho- We find that the components of R␮␯␳␴ which have the ͑ rizons and the surface gravity of the latter, respectively, for a strongest divergence near the CH are RV␸V␸ and RV␽V␽ .
Load more

Recommended publications
  • Marcel Grossmann Awards
    MG15 MARCEL GROSSMANN AWARDS ROME 2018 ICRANet and ICRA MG XV MARCEL GROSSMANN AWARDS ROME 2018 and TEST The 15th Marcel Grossmann Meeting – MG XV 2nd July 2018, Rome (Italy) Aula Magna – University “Sapienza” of Rome Institutional Awards Goes to: PLANCK SCIENTIFIC COLLABORATION (ESA) “for obtaining important constraints on the models of inflationary stage of the Universe and level of primordial non-Gaussianity; measuring with unprecedented sensitivity gravitational lensing of Cosmic Microwave Background fluctuations by large-scale structure of the Universe and corresponding B- polarization of CMB, the imprint on the CMB of hot gas in galaxy clusters; getting unique information about the time of reionization of our Universe and distribution and properties of the dust and magnetic fields in our Galaxy” - presented to Jean-Loup Puget, the Principal Investigator of the High Frequency Instrument (HFI) HANSEN EXPERIMENTAL PHYSICS LABORATORY AT STANFORD UNIVERSITY “to HEPL for having developed interdepartmental activities at Stanford University at the frontier of fundamental physics, astrophysics and technology” - presented to Research Professor Leo Hollberg, HEPL Assistant Director Individual Awards Goes to LYMAN PAGE “for his collaboration with David Wilkinson in realizing the NASA Explorer WMAP mission and as founding director of the Atacama Cosmology Telescope” Goes to RASHID ALIEVICH SUNYAEV “for the development of theoretical tools in the scrutinising, through the CMB, of the first observable electromagnetic appearance of our Universe” Goes to SHING-TUNG YAU “for the proof of the positivity of total mass in the theory of general relativity and perfecting as well the concept of quasi-local mass, for his proof of the Calabi conjecture, for his continuous inspiring role in the study of black holes physics” Each recipient is presented with a silver casting of the TEST sculpture by the artist A.
    [Show full text]
  • Observational Cosmology - 30H Course 218.163.109.230 Et Al
    Observational cosmology - 30h course 218.163.109.230 et al. (2004–2014) PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Thu, 31 Oct 2013 03:42:03 UTC Contents Articles Observational cosmology 1 Observations: expansion, nucleosynthesis, CMB 5 Redshift 5 Hubble's law 19 Metric expansion of space 29 Big Bang nucleosynthesis 41 Cosmic microwave background 47 Hot big bang model 58 Friedmann equations 58 Friedmann–Lemaître–Robertson–Walker metric 62 Distance measures (cosmology) 68 Observations: up to 10 Gpc/h 71 Observable universe 71 Structure formation 82 Galaxy formation and evolution 88 Quasar 93 Active galactic nucleus 99 Galaxy filament 106 Phenomenological model: LambdaCDM + MOND 111 Lambda-CDM model 111 Inflation (cosmology) 116 Modified Newtonian dynamics 129 Towards a physical model 137 Shape of the universe 137 Inhomogeneous cosmology 143 Back-reaction 144 References Article Sources and Contributors 145 Image Sources, Licenses and Contributors 148 Article Licenses License 150 Observational cosmology 1 Observational cosmology Observational cosmology is the study of the structure, the evolution and the origin of the universe through observation, using instruments such as telescopes and cosmic ray detectors. Early observations The science of physical cosmology as it is practiced today had its subject material defined in the years following the Shapley-Curtis debate when it was determined that the universe had a larger scale than the Milky Way galaxy. This was precipitated by observations that established the size and the dynamics of the cosmos that could be explained by Einstein's General Theory of Relativity.
    [Show full text]
  • The Generic Black Hole Singularity Degree of Doctor of Philosophy
    University of Alberta The Generic Black Hole Singularity by Sharon Marijke Morsink 9 A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Theoret ical P hysics Depart ment of P hysics Edmonton, Alberta Spring 1997 Acquisitions and Acquisitions et BiMiqraphic Senrices , services bibfiogrsphiques The author has granted a non- L'auteur a accordé une licence non exclusive licence aIiowhg the exclusive permettant à la National Li"brary of Canada to Bibîiotbtqpe nationale du Canada de reptoduce, loan, dimi.or sell reproduire, prêter, distniuer ou copies of his/her thesis by any means vmQe des copies de sa thèse de and in any form or format. making quelque manière et sous quelque this thesis available to interesteci fome que ce soit pour mettre des persons. exemplauw de cette thése à la dispositon des personnes intéressées- The author retains ownership of t&e L'auteur conserve la propriété du copyright in Merthesis. Neither &oit d'auteur qui protège sa thèse. Ni the thesis nor substaritiai exnacts la thèse ni des extraits substantiels de fhm it may be printed or othewïse celle-ci ne doivent être imprimés ou reproduced with the author's autrement reproduits sans son permission. autorisation. Preface During my yeazs as a graduate student at the University of Alberta 1 have benefitted fkom colhborations with my supervisor, Werner bel, and my fellow graduate stu- dents, Warren Anderson, AEo Bonanno, Patrick Brady and Serge Droz. In order to reflect the collaborative nature of the research 1 have written the thesis in the first person plural.
    [Show full text]
  • Lorentz Lectures: Gravitational Waves
    The Warped Side of the Universe Kip Thorne Colloquium Ehrenfestii , University of Leiden, 16 Sept 2009 Lorentz Lectures: Gravitational Waves Understandable without prior knowledge of general relativity But knowledge of general relativity will help Slides available (pdf) late Thursday nights at http://www.its.caltech.edu/~kip/LorentzLectures/ 2 Phenomena and objects Made from warped space and time 3 Black Holes 4 Black Holeʼs Spacetime Geometry Curvature of Space Rotational Motion of Space Warping of Time 2 2 2 2 2 2 2 ds = grrdr +gθθ dθ +gφφ (d φ - ω dt) - α dt space curvature space rotation time warp 3-metric shift function lapse function 5 Kerr Metric Map of spacetime geometry for fast spinning hole a/M=0.998 Rate of Time flow 6 Other Examples • The Big Bang Singularity • Our Universe as a Brane in a . higher-dimensional bulk • Cosmic String C/R = 2π(1-4Gµ/c2) • Singularity inside a black hole • Naked Singularity • Wormhole • Gravitational Waves 7 Other Examples WHICH ARE REAL? WHAT ELSE? 8 Probing the Warped Side: Tools What kinds of objects might exist? » General Relativity Theory – Progress has slowed… » Numerical Relativity – Exciting new era… LIGO – Part 1 of Colloquium What kinds of objects do exist? » Electromagnetic observations –Limited information » Gravitational-Wave observations LISA –Ideal tool for probing the . Warped Side –Part 2 of Colloquium 9 Part 1 Numerical Relativity 10 The “Holy Grail”: Collisions of Black Holes - The most violent events in the Universe ~ 10 % of holes’ mass is converted to radiation [contrast with nuclear fusion: < 0.5 %] GW Luminosity ~ 0.1 Mc2 /(100 GM/c3) 2 24 4 =0.001 c /G ~ 10 Lsun ~ 10 LEM universe No Electromagnetic Waves emitted whatsoever - except from, e.g.
    [Show full text]
  • La Constant Cosmològica I L'energia Fosca: Del Big Bang Al Futur De L
    Institute of Space Sciences Historia de la expansión del Universo y el concepto de Big Bang Emilio Elizalde Seminario Universidad de Zaragoza www.ice.csic.es/personal/elizalde/eli/eli.htm 21 Marzo, 2019 Name or Title or Xtra Some facts (a few rather surprising...) • Adam Riess, NP 2011, at Starmus (Tenerife), about Hubble: • “Hubble obtained the distances and redshifts of distant nebulae…” • “ Hubble discovered that the Universe was expanding …” • No mention to Vesto Slipher, Henrietta Leavitt, … • Brian Schmidt, NP 2011, at Starmus (Tenerife) & Lisa Randall, Harvard U, in Barcelona, about Einstein: SHOES- • “Einstein was the first to think about the possibility of a ‘dark energy’…” Supernovae • No mention to Fritz Zwicky, another extraordinary astronomer • Fritz Zwicky discovered dark matter in the early 1930s while studying how galaxies move within the Coma Cluster • He was also the first to postulate and use nebulae as gravitational lenses (1937) • How easily* brilliant astronomers get dismissed • How easily* scientific myths arise *in few decades Institute of Space Sciences Name or Title or Xtra Institute of Space Sciences La expansión del Universo: Uno de los descubrimientos más importantes de toda la Historia de la Humanidad Name or Title or Xtra Institute of Space Sciences Univers d’Anaximandre 610–546 BC Univers de Ptolomeu, s.II Univers de Copèrnic, Thomas Digges 1576 Diversos models d’Univers Name or Title or Xtra Institute of Space Sciences 1917 Universe : eternal Universe = Milky Way Universe: static why? Einstein field
    [Show full text]
  • Asymptotically Ads Spacetimes with a Timelike Kasner Singularity
    Prepared for submission to JHEP Asymptotically AdS spacetimes with a timelike Kasner singularity Jie Ren Racah Institute of Physics, The Hebrew University of Jerusalem, 91904, Israel E-mail: [email protected] Abstract: Exact solutions to Einstein’s equations for holographic models are presented and studied. The IR geometry has a timelike cousin of the Kasner singularity, which is the less generic case of the BKL (Belinski-Khalatnikov-Lifshitz) singularity, and the UV is asymptotically AdS. This solution describes a holographic RG flow between them. The solution’s appearance is an interpolation between the planar AdS black hole and the AdS soliton. The causality constraint is always satisfied. The entanglement entropy and Wilson loops are discussed. The boundary condition for the current-current correlation function and the Laplacian in the IR is examined. There is no infalling wave in the IR, but instead, there is a normalizable solution in the IR. In a special case, a hyperscaling-violating geometry is obtained after a dimensional reduction. arXiv:1603.08004v3 [hep-th] 1 Aug 2016 Contents 1 Introduction1 2 Anisotropic solution and boundary stress tensor3 3 Geodesics and extremal surfaces5 3.1 Geodesics5 3.2 Entanglement entropy and Wilson loops7 4 Two-point correlation functions 10 5 Relation to hyperscaling-violating geometries 12 6 Discussion 14 A Relation to other coordinates 16 B Notes for an integral 17 C Kaluza-Klein reduction on S1 17 1 Introduction The planar AdS black hole plays a basic role in the AdS/CFT correspondence, where a strongly coupled quantum field theory in the AdS boundary is mapped to a classical gravity in the bulk.
    [Show full text]
  • Numerical Relativity Beyond Astrophysics 2
    Numerical Relativity Beyond Astrophysics David Garfinkle Dept. of Physics, Oakland University, Rochester, MI 48309, USA Michigan Center for Theoretical Physics, Randall Laboratory of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA E-mail: [email protected] Abstract. Though the main applications of computer simulations in relativity are to astrophysical systems such as black holes and neutron stars, nonetheless there are important applications of numerical methods to the investigation of general relativity as a fundamental theory of the nature of space and time. This paper gives an overview of some of these applications. In particular we cover (i) investigations of the properties of spacetime singularities such as those that occur in the interior of black holes and in big bang cosmology. (ii) investigations of critical behavior at the threshold of black hole formation in gravitational collapse. (iii) investigations inspired by string theory, in particular analogs of black holes in more than 4 spacetime dimensions and gravitational collapse in spacetimes with a negative cosmological constant. arXiv:1606.02999v1 [gr-qc] 9 Jun 2016 Numerical Relativity Beyond Astrophysics 2 1. Introduction General relativity is Einstein’s theory of gravity. For weak gravitational fields the predictions of general relativity are well approximated by those of Newtonian gravity, so the main applications of general relativity are to those astrophysical situations where gravity is strong: neutron stars, black holes, and the big bang. However, general
    [Show full text]
  • Strength of the Null Singularity Inside Black Holes
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Strength of the null singularity inside black holes Lior M. Burko Theoretical Astrophysics 130{33, California Institute of Technology, Pasadena, California 91125 (July 20, 1999) We study analytically the Cauchy horizon singularity inside spherically-symmetric charged black holes, coupled to a spherical self-gravitating, minimally-coupled, massless scalar field. We show that all causal geodesics terminate at the Cauchy horizon at a null singularity, which is weak according to the Tipler classification. The singularity is also deformationally-weak in the sense of Ori. Our results are valid at arbitrary points along the null singularity, in particular at late retarded times, when non-linear effects are crucial. PACS number(s): 04.70.Bw, 04.20Dw I. INTRODUCTION the local existence and genericity of a null and weak sin- gularity in solutions of the vacuum Einstein equations The issue of spacetime singularities – which are known was demonstrated in [8]. This was more recently demon- to inevitably occur inside black holes under very plausible strated also in the framework of plane-symmetric space- assumptions [1] – is an intriguing puzzle of physics. The times in [9]. For the toy model of a spherical charged laws of physics, as we presently understand them (e.g., black hole, the main features of the CH singularity were classical General Relativity), are presumably invalid at first found analytically for simplified models based on singularities. Instead, some other theories (e.g., quan- null fluids [10–12], and later confirmed numerically for a tum gravity), as yet unknown, are expected to take over model with a self-gravitating scalar field [13,14].
    [Show full text]
  • Dynamics and Exact Bianchi I Spacetimes in Einstein–Æther Scalar field Theory
    Eur. Phys. J. C (2020) 80:589 https://doi.org/10.1140/epjc/s10052-020-8148-7 Regular Article - Theoretical Physics Dynamics and exact Bianchi I spacetimes in Einstein–æther scalar field theory Andronikos Paliathanasis1,2,a, Genly Leon3,b 1 Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, 5090000 Valdivia, Chile 3 Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280, Antofagasta, Chile Received: 20 April 2020 / Accepted: 15 June 2020 / Published online: 29 June 2020 © The Author(s) 2020 Abstract We determine exact and analytic solutions of the dominated period. These anisotropies are observed in the gravitational field equations in Einstein–aether scalar model cosmic microwave background, which support the idea that field with a Bianchi I background space. In particular, we the spacetimes become isotropic ones by evolving in time consider nonlinear interactions of the scalar field with the [3–5]. In addition a recent detailed study by using the X-ray aether field. For the model under consideration we can write clusters challenged the isotropic scenario [6] and supported the field equations by using the minisuperspace description. the anisotropic cosmological scenario. The point-like Lagrangian of the field equations depends on The spatial homogeneous but anisotropic spacetimes are three unknown functions. We derive conservation laws for the known as either Kantowski-Sachs or Bianchi cosmologies. field equations for specific forms of the unknown functions The isometry group of Kantowsky-Sachs spacetime is R × such that the field equations are Liouville integrable.
    [Show full text]
  • Semiclassical and Quantum Behavior of the Mixmaster Model in the Polymer Approach for the Isotropic Misner Variable
    Eur. Phys. J. C manuscript No. (will be inserted by the editor) Semiclassical and quantum behavior of the Mixmaster model in the polymer approach for the isotropic Misner variable C. Crinòa,1, G. Montanib,1,2, G. Pintaudi c,1 1Dipartimento di Fisica (VEF), P.le A. Moro 5 (00185) Roma, Italy 2ENEA, Fusion and Nuclear Safety Department, C.R. Frascati, via E. Fermi 45, 00044 Frascati (Roma), Italy November 6, 2018 Abstract We analyze the semi-classical and quantum be- singularity constitutes, along with the same behavior recov- havior of the Bianchi IX Universe in the Polymer Quantum ered in Bianchi type VIII, the prototype for a more general Mechanics framework, applied to the isotropic Misner vari- feature, characterizing a generic inhomogeneous cosmolog- able, linked to the space volume of the model. The study ical model [10] (see also [6, 11–15]). is performed both in the Hamiltonian and field equations The original interest toward the Bianchi IX chaotic cos- approaches, leading to the remarkable result of a still sin- mology was due to the perspective, de facto failed, to solve gular and chaotic cosmology, whose Poincaré return map the horizon paradox via such a statistical evolution of the asymptotically overlaps the standard Belinskii-Khalatnikov- Universe scale factors, from which derives the name, due to Lifshitz one. In the quantum sector, we reproduce the orig- C. W. Misner, of the so-called Mixmaster model. However, inal analysis due to Misner, within the revised Polymer ap- it was clear since from the very beginning, that the chaotic proach and we arrive to demonstrate that the quantum num- properties of the Mixmaster model had to be replaced, near bers of the point-Universe still remain constants of motion.
    [Show full text]
  • Singularity Theorems a Critical Appraisal
    Singularity Theorems A critical appraisal José M M Senovilla UPV/EHU, Bilbao, Spain Institute of Theoretical Physics, Warsaw, 11th December 2020 Senovilla / 1/50 Outline 1 Introduction 2 The Penrose singularity theorem Curvature condition Incompleteness Trapped spheres 3 Singularity theorems Ideas behind the theorems Contemporary singularity theorems 4 The devil is in the detail Uncovering the mysteries of black holes Curvature divergence? Singularity-free spacetimes Remarks on extensions BKL or null singularity? 5 Conclusions and final comments Senovilla / 2/50 Introduction Penrose’s singularity theorem won the 2020 Nobel prize in Physics! It is a well deserved award. The theorem 1 is sheer beauty 2 contained novel ingredients along with fruitful ideas 3 prompted multiple developments in theoretical relativity 4 involved stunning physical consequences (incompleteness, failure of the theory?) In particular, the notion of trapped spheres is fundamental, a key idea in black hole physics, numerical relativity, mathematical relativity, cosmology, gravity analogs, etc. Its influence is endless, and its prolific range of applications keeps growing Senovilla Introduction/ 3/50 Modern GR As I proposed some time ago (and has been echoed by the Nobel Committee) singularity theorems constitute the first post-Einsteinian content of relativity. What had happened before the theorem (deflection of light, universal expansion, Cauchy’s problem, ADM formulation, Kerr’s solution, Wheeler-de Witt equation, ...) as well as most of what came next (background radiation, gravitational lenses, radiation in binary systems, direct observation of waves, etc) All of this had been explicitly predicted, known, anticipated, or foreseen in one way or another by Einstein. In contrast, the singularity theorems and their consequences were (surely) not even suspected by the founder of GR.
    [Show full text]
  • Multi-Scale Gravity and Cosmology
    July 20, 2013 JCAP 12 (2013) 041 [arXiv:1307.6382] Multi-scale gravity and cosmology Gianluca Calcagni Instituto de Estructura de la Materia, CSIC, Serrano 121, Madrid, 28006 Spain E-mail: [email protected] Abstract. The gravitational dynamics and cosmological implications of three classes of recently introduced multi-scale spacetimes (with, respectively, ordinary, weighted and q- derivatives) are discussed. These spacetimes are non-Riemannian: the metric structure is accompanied by an independent measure-differential structure with the characteristics of a multi-fractal, namely, different dimensionality at different scales and, at ultra-short distances, a discrete symmetry known as discrete scale invariance. Under this minimal paradigm, five general features arise: (a) the big-bang singularity can be replaced by a finite bounce, (b) the cosmological constant problem is reinterpreted, since accelerating phases can be mimicked by the change of geometry with the time scale, without invoking a slowly rolling scalar field, (c) the discreteness of geometry at Planckian scales can leave an observable imprint of loga- rithmic oscillations in cosmological spectra and (d) give rise to an alternative mechanism to inflation or (e) to a fully analytic model of cyclic mild inflation, where near scale invariance of the perturbation spectrum can be produced without strong acceleration. Various properties of the models and exact dynamical solutions are discussed. In particular, the multi-scale geometry with weighted derivatives is shown to be a Weyl
    [Show full text]