DOCSLIB.ORG

Gravity in Two Spacetime Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gravity in Two Spacetime Dimensions Gravity in Two Spacetime Dimensions Der Fakult¨at f¨ur Mathematik, Informatik und Naturwissenschaften der Rheinisch–Westf¨alischen Technischen Hochschule Aachen vorgelegte Habilitationsschrift zur Erlangung der venia legendi arXiv:hep-th/0011240v1 27 Nov 2000 von Dipl.–Ing. Dr. Thomas Strobl aus Wien Contents 1 Introduction 7 2 2d geometry and gravitational actions 11 2.1 Fromfourtolowerdimensions . 11 2.2 Purelygeometrical2dactions . 14 2.3 Einstein–Cartanvariables . 17 2.4 Extension to theories with nontrivial torsion . 19 2.5 WhysomanydifferentLagrangians . 22 2.6 Why consider low–dimensional, topological models . 24 2.7 Role of the dilaton within the present chapter . 27 3 Generalized dilaton theories and matter actions 29 3.1 Generalized dilaton action replacing the Einstein–Hilbert term . 29 3.1.1 Spherically reduced gravity . 29 3.1.2 String inspired dilaton gravity . 32 3.1.3 Mostgeneraldilatontheory . 36 3.1.4 Two possible routes to a simplified, equivalent model . 36 3.1.4.1 Conformal rescalings of the metric . 37 3.1.4.2 Equivalence to theories with torsion . 40 3.2 Matteractions ............................ 42 3.3 Generalfieldequations ........................ 45 3.4 More on dimensionally reduced theories . 48 3.5 Extensionto2dsupergravity . 51 4 2d gravity–Yang–Mills systems in terms of Poisson σ–models 55 4.1 Poissonmanifolds .......................... 56 4.2 Defining the σ–model ........................ 60 4.3 Generalized dilaton gravity as Poisson σ–model .......... 64 4.4 Extensiontoinclude2dsupergravity . 66 3 4 4.5 Includingmatterfields ........................ 71 4.5.1 Additional Yang–Mills fields . 71 4.5.2 Fermionicmatter ....................... 72 4.5.3 Scalarfields .......................... 73 5 Classical solutions on a local level 75 5.1 Pure 2d Gravity–Yang–Mills models . 75 5.1.1 Gaugefixing method, torsion–freetheories . 77 5.1.1.1 Attainability of certain gauge conditions . 77 5.1.1.2 Metrical variables . 79 5.1.1.3 Transforming back to original dilaton variables . 80 5.1.1.4 In Einstein–Cartan variables . 82 5.1.2 Targetspacemethod,thegeneralcase . 83 5.1.2.1 Torsion–free theories . 83 5.1.2.2 Theories with general torsion dependence . 84 5.1.2.3 Solution in the neighborhood of saddle-points . 88 5.1.2.4 Inclusion of Yang–Mills fields . 90 5.1.2.5 The model (2.21) . 91 5.1.3 Supergravity ......................... 92 5.2 Dilaton theories with matter . 96 5.2.1 Solvable generalizations of the CGHS model . 96 5.2.1.1 No gauge fields (α0 =0):.............. 98 5.2.1.2 Inclusion of gauge fields (arbitrary α0) ...... 101 5.2.2 Generalized dilaton gravity with chiral matter . 103 6 Classical solutions on a global level 107 6.1 Maximal Extensions for all Gravity–Yang–Mills systems . 109 6.1.1 Generalities on maximal extensions . 109 6.1.2 Some remarks onCarter-Penrose diagrams . 110 6.1.2.1 Further preliminaries . 113 6.1.3 Extremals ........................... 114 6.1.4 Construction of the Building Blocks, Carter-Penrose Dia- grams ............................. 116 6.1.5 Maximal Extension and Saddle-Point Charts . 123 6.1.6 Recipeandexamples . 129 6.1.7 Remarksonglobalcoordinates . 138 6.2 All global solutions of (all 2d) Gravity–Yang–Mills models . 139 5 6.2.1 Introductoryremarks . 139 6.2.1.1 Motivation . 140 6.2.1.2 Organization . 141 6.2.1.3 Anticipation of some of the results . 142 6.2.2 Cut-and-paste technique for the KV model . 143 6.2.2.1 The moduli space for M Σ IR ........ 147 ∼ × 6.2.3 A global σ-modelapproach. 148 6.2.4 Systematic grouptheoreticalapproach . 152 6.2.4.1 Preliminaries . 152 6.2.4.2 Thesymmetrygroup . 155 6.2.4.3 Subgroupsandfactorspaces . 162 6.2.4.4 Morethanonegenerator . 171 6.2.4.5 Remarks on the Constant Curvature Case . 180 6.2.5 Remarksonkinks ...................... 182 7 Towards quantum gravity 187 7.1 Hamiltonian formulation . 187 7.1.1 For a cylindrical spacetime . 188 7.1.2 Modifications for open spacetimes . 191 7.1.3 Abelianization and the reduced phase space . 195 7.1.3.1 Remarks on the variable p ............. 195 7.1.3.2 Global abelianization for Schwarzschild . 197 7.2 Dirac quantization of gravity–Yang–Mills models . 198 7.2.1 Physicalstates ........................ 198 7.2.1.1 The case of simply connected symplectic leaves . 201 7.2.1.2 Multiply connected leaves . 202 7.2.1.3 Lorentzian signature of the theory . 203 7.2.1.4 Comparison of classical and quantum results . 205 7.2.1.5 The missing kink number . 206 7.2.2 Furtherremarks ....................... 207 7.2.2.1 Comparing the Lorentzian and Euclidean theories 207 7.2.2.2 Modifications for open spacetimes . 208 7.2.2.3 The inner product and the issue of time . 209 7.3 Brief outlook on Hawking radiation . 210 Bibliography 213 6 Chapter 1 Introduction There is no doubt that, at the fundamental level, theoretical physics is in the need of reconciling General Relativity and quantum theory. This becomes obvious already from Einstein’s equations 8πG G = T . (1.1) µν c2 µν Here Gµν is the Einstein tensor, determined solely in terms of the metric g, which governs the ‘gravitational force’, and Tµν is the energy momentum tensor of matter fields. We know that matter fields are governed by a quantum field theory, the standard model of strong and electroweak interactions. Equations (1.1), however, contain purely classical, commuting quantities. On the other hand, Einstein’s equations show that, in the presence of matter, T = 0, the µν 6 metric g can not be strictly Minkowskian (since then G = 0); it can be so at µν 6 best to some approximation. A Minkowski background metric is used, however, in standard quantum field theory of elementary particle physics. Certainly, no current experiments in our laboratories come only close to ex- plicitly display the above contradiction experimentally. Still, from the point of view of a unified and mathematically consistent theory of all interactions, a new theory is needed. There is a variety of (at least apparently) different contemporary approaches to improve on this situation: It ranges from noncommutative geometry [1], over twistors [2], over the attempt to quantize Einstein gravity nonperturbatively in a canonical fashion [3], to the advent of string theory [4], to name just the most prominent directions. The at the moment probably most promising of these attempts is the string approach. But also string theory is surely not formulated in a fully satisfactory and self-consistent way yet. 7 8 Black holes appear to provide an important link between gravity and quan- tum theory. First, near a black hole (and in particular near the singularity) the gravitational field cannot be ignored or approximated sufficiently well by a flat, Minkowskian metric. Second, the discovery of Hawking radiation [5] demon- strated that quantum effects cannot be ignored for black holes; rather they lead to a (slow, but still nonvanishing) evaporation of the classically indissoluble ob- jects. Moreover, theoretical considerations allow to establish a striking analogy between the laws of thermodynamics and the laws governing black holes in classi- cal general relativity [6]. Further Gedankenexperiments [7] show that the second law of thermodynamics should be altered to include an additive contribution to the entropy, assigned to the black hole. Already on dimensional grounds, the 3 Planck length lP = ~G/c enters the generalized second law of thermodynam- ics postulated in thisp way. It is felt by many contemporary theoreticians that black holes will be the gate to some fundamentally new picture of our world, bringing together seemingly separate subjects such as gravity, quantum theory, and thermodynamics. As a side-remark let me mention that by now certainly black holes are no more purely theoretical objects. There is already overwhelming experimental evidence for the existence of black holes in the center of many galaxies, including ours (cf., e.g., [8]). Also black holes have been found as partners in binary systems [9]. It is one of the famous results of General Relativity, called the no–hair theorem [10], that within Einstein–Maxwell theory all asymptotically flat, stationary black holes are determined uniquely by specification of their mass m, their charge q, and their angular momentum J. This scenario extends to some other matter couplings such as to the case of additional scalar fields (with convex potential). Still, the ‘no–hair conjecture’ does not hold in full generality: counter examples have been found recently (cf references in [10]) such as, e.g., in the Einstein–Yang– Mills theory with gauge group SU(2) (note that these solutions are unstable, however). Thus, (in those cases where the no–hair theorem or conjecture holds) the space of (stationary) black hole solutions (modulo gauge transformations) is finite dimensional only. The formulation of gravity in two spacetime dimensions allows to isolate pre- cisely these most interesting objects of General Relativity: Upon definition of ap- propriate two–dimensional gravitational actions, cf. chapters 2 and 3 below, the space of their classical solutions does not only contain (two–dimensional) black holes, it consists only of such spacetimes, except possibly for naked singularity 9 solutions and a vacuum ground state. Many of the considerations performed for their four–dimensional analogs, moreover, like the prediction of Hawking radia- tion as well as the analogy to thermodynamics, may be transferred also into the two–dimensional setting. Finally, also here the original theory is invariant with respect to diffeomorphisms (the group of coordinate transformations), such that a transition to the quantum theory is plagued by similar conceptual difficulties as in four spacetime dimensions (like the ‘problem of time’ [11] to name just one of them). During about the last decade, two–dimensional gravity models thus turned out as an interesting and promising theoretical laboratory to test ideas on how to approach quantization of gravity.
Load more

Recommended publications
  • Arxiv:0911.0334V2 [Gr-Qc] 4 Jul 2020
    Classical Physics: Spacetime and Fields Nikodem Poplawski Department of Mathematics and Physics, University of New Haven, CT, USA Preface We present a self-contained introduction to the classical theory of spacetime and fields. This expo- sition is based on the most general principles: the principle of general covariance (relativity) and the principle of least action. The order of the exposition is: 1. Spacetime (principle of general covariance and tensors, affine connection, curvature, metric, tetrad and spin connection, Lorentz group, spinors); 2. Fields (principle of least action, action for gravitational field, matter, symmetries and conservation laws, gravitational field equations, spinor fields, electromagnetic field, action for particles). In this order, a particle is a special case of a field existing in spacetime, and classical mechanics can be derived from field theory. I dedicate this book to my Parents: Bo_zennaPop lawska and Janusz Pop lawski. I am also grateful to Chris Cox for inspiring this book. The Laws of Physics are simple, beautiful, and universal. arXiv:0911.0334v2 [gr-qc] 4 Jul 2020 1 Contents 1 Spacetime 5 1.1 Principle of general covariance and tensors . 5 1.1.1 Vectors . 5 1.1.2 Tensors . 6 1.1.3 Densities . 7 1.1.4 Contraction . 7 1.1.5 Kronecker and Levi-Civita symbols . 8 1.1.6 Dual densities . 8 1.1.7 Covariant integrals . 9 1.1.8 Antisymmetric derivatives . 9 1.2 Affine connection . 10 1.2.1 Covariant differentiation of tensors . 10 1.2.2 Parallel transport . 11 1.2.3 Torsion tensor . 11 1.2.4 Covariant differentiation of densities .
    [Show full text]
  • Einstein and Friedmann Equations
    Outline ● Go over homework ● Einstein equations ● Friedmann equations Homework ● For next class: – A light ray is emitted from r = 0 at time t = 0. Find an expression for r(t) in a flat universe (k=0) and in a positively curved universe (k=1). Einstein Equations ● General relativity describes gravity in terms of curved spacetime. ● Mass/energy causes the curvature – Described by mass-energy tensor – Which, in general, is a function of position ● The metric describes the spacetime – Metric determines how particles will move ● To go from Newtonian gravity to General Relativity d2 GM 8 πG ⃗r = r^ → Gμ ν(gμ ν) = T μ ν dt2 r 2 c4 ● Where G is some function of the metric Curvature ● How to calculate curvature of spacetime from metric? – “readers unfamiliar with tensor calculus can skip down to Eq. 8.22” – There are no equations with tensors after 8.23 ● Curvature of spacetime is quantified using parallel transport ● Hold a vector and keep it parallel to your direction of motion as you move around on a curved surface. ● Locally, the vector stays parallel between points that are close together, but as you move finite distances the vector rotates – in a manner depending on the path. Curvature ● Mathematically, one uses “Christoffel symbols” to calculate the “affine connection” which is how to do parallel transport. ● Christoffel symbols involve derivatives of the metric. μ 1 μρ ∂ gσ ρ ∂ gνρ ∂ g Γ = g + + σ ν σ ν 2 ( ∂ x ν ∂ xσ ∂ xρ ) ● The Reimann tensor measures “the extent to which the metric tensor is not locally isometric to that of Euclidean space”.
    [Show full text]
  • SPINORS and SPACE–TIME ANISOTROPY
    Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE{TIME ANISOTROPY University of Athens ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii - i ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. ii - iii ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics.
    [Show full text]
  • General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
    General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ).
    [Show full text]
  • Tensor-Spinor Theory of Gravitation in General Even Space-Time Dimensions
    Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor.
    [Show full text]
  • Static Solutions of Einstein's Equations with Spherical Symmetry
    Static Solutions of Einstein’s Equations with Spherical Symmetry IftikharAhmad,∗ Maqsoom Fatima and Najam-ul-Basat. Institute of Physics and Mathematical Sciences, Department of Mathematics, University of Gujrat Pakistan. Abstract The Schwarzschild solution is a complete solution of Einstein’s field equations for a static spherically symmetric field. The Einstein’s field equations solutions appear in the literature, but in different ways corresponding to different definitions of the radial coordinate. We attempt to compare them to the solutions with nonvanishing energy density and pressure. We also calculate some special cases with changes in spherical symmetry. Keywords: Schwarzschild solution, Spherical symmetry, Einstein’s equations, Static uni- verse. 1 Introduction Even though the Einstein field equations are highly nonlinear partial differential equations, there are numerous exact solutions to them. In addition to the exact solution, there are also non-exact solutions to these equations which describe certain physical systems. Exact solutions and their physical interpretations are sometimes even harder. Probably the simplest of all exact solutions to the Einstein field equations is that of Schwarzschild. Furthermore, the static solutions of Einstein’s equations with spherical symmetry (the exterior and interior Schwarzschild solutions) are already much popular in general relativity, solutions with spherical symmetry in vacuum are much less recognizable. Together with conventional isometries, which have been more widely studied, they help to provide a deeper insight into the spacetime geometry. They assist in the arXiv:1308.1233v2 [gr-qc] 2 May 2014 generation of exact solutions, sometimes new solutions of the Einstein field equations which may be utilized to model relativistic, astrophysical and cosmological phenomena.
    [Show full text]
  • Einstein's 1916 Derivation of the Field Equations
    1 Einstein's 1916 derivation of the Field Equations Galina Weinstein 24/10/2013 Abstract: In his first November 4, 1915 paper Einstein wrote the Lagrangian form of his field equations. In the fourth November 25, 1915 paper, Einstein added a trace term of the energy- momentum tensor on the right-hand side of the generally covariant field equations. The main purpose of the present work is to show that in November 4, 1915, Einstein had already explored much of the main ingredients that were required for the formulation of the final form of the field equations of November 25, 1915. The present work suggests that the idea of adding the second-term on the right-hand side of the field equation might have originated in reconsideration of the November 4, 1915 field equations. In this regard, the final form of Einstein's field equations with the trace term may be linked with his work of November 4, 1915. The interesting history of the derivation of the final form of the field equations is inspired by the exchange of letters between Einstein and Paul Ehrenfest in winter 1916 and by Einstein's 1916 derivation of the November 25, 1915 field equations. In 1915, Einstein wrote the vacuum (matter-free) field equations in the form:1 for all systems of coordinates for which It is sufficient to note that the left-hand side represents the gravitational field, with g the metric tensor field. Einstein wrote the field equations in Lagrangian form. The action,2 and the Lagrangian, 2 Using the components of the gravitational field: Einstein wrote the variation: which gives:3 We now come back to (2), and we have, Inserting (6) into (7) gives the field equations (1).
    [Show full text]
  • The Einstein Tensor Characterizing Some Riemann Spaces
    ••fc IC/93/204 Internal Report (Limited Distribution) International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE EINSTEIN TENSOR CHARACTERIZING SOME RIEMANN SPACES M. S. Rahman1 International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT A formal definition of the Einstein tensor is given. Mention is made of how this tensor plays a role of expressing certain conditions in a precise form. The cases of reducing the Einstein tensor to a zero tensor are studied on its merit. A lucid account of results, formulated as theorems, on Einstein symmetric and Einstein recurrent spaces is then presented. MIRAMARE - TRIESTE July 1993 Permanent address: Department of Mathematics, Jahangirnagar University, Savar, Dhaka-134;2, Bangladesh. 1 Introduction and Definition The Einstein tensor [5] is defined by This tensor is in covariant form E R (1) We note that Exjj — 0 which means that the divergence of the Einstein tensor vanishes. This equation which is very remarkably an identity in Riemannian geometry is impor- tant in the theory of Relativity. Einstein's tensor has proved of tremendous importance for subsequent researches both in Riemannian geometry and in Relativity. Hitherto, the tensor had been of interest to mathematicians and physicists, and simply for its own sake. Contracting (1) with gl> gives E = R-^R where we may call E the Einstein scalar curvature. It readily follows from (2) that in a V4 the algebraic sum of the Einstein scalar curva- ture and scalar curvature is zero, that is, E + R = 0. This paper illuminates the virtue of the Einstein tensor in a consistent manner and deals with its characterizing some Riemann spaces.
    [Show full text]
  • Canonical and Gravitational Stress-Energy Tensors
    Canonical and gravitational stress-energy tensors M. Leclerc Section of Astrophysics and Astronomy, Department of Physics, University of Athens, Greece April 11, 2006 Abstract It is dealt with the question, under which circumstances the canonical Noether stress-energy tensor is equivalent to the gravitational (Hilbert) tensor for general matter fields under the influence of gravity. In the framework of general relativity, the full equivalence is established for matter fields that do not couple to the metric derivatives. Spinor fields are included into our analysis by reformulating general relativity in terms of tetrad fields, and the case of Poincar´egauge theory, with an additional, independent Lorentz connection, is also investigated. Special attention is given to the flat limit, focusing on the expressions for the matter field energy (Hamiltonian). The Dirac- Maxwell system is investigated in detail, with special care given to the separation of free (kinetic) and interaction (or potential) energy. Moreover, the stress-energy tensor of the gravitational field itself is briefly discussed. PACS: 04.50.+h, 04.20.Fy 1 Introduction In textbooks on general relativity, the Hilbert stress-energy tensor is often presented as an improvement over the canonical Noether tensor, because it is automatically symmetric, while the Noether tensor has to be submitted to a relocalization if one insists on a symmetric tensor. That we have, on one hand, a symmetric tensor, and on the other hand, a tensor that is physically equivalent to a symmetric arXiv:gr-qc/0510044v6 25 Aug 2006 tensor, is thus well known. This, however, does still not proof that both tensors are indeed equivalent.
    [Show full text]
  • Arxiv:Gr-Qc/0112028V3 28 May 2002
    Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE–TIME ANISOTROPY arXiv:gr-qc/0112028v3 28 May 2002 University of Athens, 2002 ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii ... iii ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. iv - v ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics.
    [Show full text]
  • Curvature and the Stress-Energy Tensor
    Orthonormal Tetrads, continued Here’s another example, that combines local frame calculations with more global analysis. Suppose you have a particle at rest at infinity, and you drop it radially into a Schwarzschild black hole. What is the radial velocity as seen by a local static observer at radius r? The particle being at rest at infinity means that its total energy is ut = −1. Radial motion has θ and φ components zero, so u2 = −1 means r t u ur + u ut = −1 g (ur)2 + gttu2 = −1 rr t (1) (ur)2/(1 − 2M/r) − 1/(1 − 2M/r) = −1 (ur)2 = 2M/r . r Therefore, u = dr/dτ = q2M/r. The radial velocity seen by a local static observer is vrˆ = urˆ/utˆ rˆ = −u /utˆ − rˆ r t = e ru /(e tˆut) (2) −1/2 r −1/2 = −(1 − 2M/r) u /[(1 − 2M/r) ut] = q2M/r . Therefore, the locally measured radial velocity is just the same as the Newtonian expression, when Schwarzschild coordinates are used. By comparison, the radial velocity as measured at infinity is dr vr = = ur/ut = ur/(gttu )=(1 − 2M/r)ur = (1 − 2M/r) 2M/r . (3) dt t q This drops to zero at the horizon. Note that there is one factor of (1 − 2M/r)1/2 from the redshift and one from the change in the radial coordinate. Geodesic Deviation and Spacetime Curvature Previously we talked about geodesics, the paths of freely falling particles. We also indicated early on that the only “force” that gravity can exert on a particle is a tidal force.
    [Show full text]
  • Curvature and the Einstein Equation
    curvature.nb 1 Curvature and the Einstein Equation This is the Mathematica notebook Curvature and the Einstein Equation available from the book website. From a given ¢ £ ¡ metric g , it computes the components of the following: the inverse metric, g , the Christoffel symbols or affine connection, ¢ ¥ ¦ § ¢ £ © ¥ ¦ © ¦ ¥ © ¥ ¦ ¤ £ £ 1 £ ¨ ¨ ¨ ¨ 2 g g g g , © © © ( stands for the partial derivative x ), the Riemann tensor, ¢ ¥ ¦ § © ¦ ¢ ¥ © ¢ ¥ ¦ ¥ ¢ ¦ ¥ ¦ ¢ ¤ ¤ ¤ ¤ ¤ ¤ £ £ £ £ £ R , the Ricci tensor ¥ ¦ § ¢ ¥ ¦ R R ¢ , the scalar curvature, ¥ ¦ § ¥ ¦ R g R , and the Einstein tensor, ¥ ¦ ¥ ¦ ¥ ¦ § 1 ¨ ¨ ¨ ¨ G R 2 g R. ¥ ¦ You must input the covariant components of the metric tensor g by editing the relevant input line in this Mathematica notebook. You may also wish to change the names of the coordinates. Only the nonzero components of the above quantities are displayed as the output. All the components computed are in the coordinate basis in which the metric was specified. Clearing the values of symbols: First clear any values that may already have been assigned to the names of the various objects to be calculated. The names of the coordinates that you will use are also cleared. In[1]:= Clear coord, metric, inversemetric, affine, riemann, ricci, scalar, einstein, r, , , t Setting the dimension: The dimension n of the spacetime (or space) must be set: In[2]:= n 4 Out[2]= 4 Defining a list of coordinates: The example given here is the Schwarzschild metric. The coordinate choice of Schwarzschild is appropriate for this spheri- cally symmetric spacetime. curvature.nb 2 In[3]:= coord r, , , t Out[3]= r, , , t You can change the names of the coordinates by simply editing the definition of coord, for example, to coord = {x, y, z, t}, when another set of coordinate names is more appropriate.
    [Show full text]