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General Relativity As Taught in 1979, 1983 By
General Relativity As taught in 1979, 1983 by Joel A. Shapiro November 19, 2015 2. Last Latexed: November 19, 2015 at 11:03 Joel A. Shapiro c Joel A. Shapiro, 1979, 2012 Contents 0.1 Introduction............................ 4 0.2 SpecialRelativity ......................... 7 0.3 Electromagnetism. 13 0.4 Stress-EnergyTensor . 18 0.5 Equivalence Principle . 23 0.6 Manifolds ............................. 28 0.7 IntegrationofForms . 41 0.8 Vierbeins,Connections . 44 0.9 ParallelTransport. 50 0.10 ElectromagnetisminFlatSpace . 56 0.11 GeodesicDeviation . 61 0.12 EquationsDeterminingGeometry . 67 0.13 Deriving the Gravitational Field Equations . .. 68 0.14 HarmonicCoordinates . 71 0.14.1 ThelinearizedTheory . 71 0.15 TheBendingofLight. 77 0.16 PerfectFluids ........................... 84 0.17 Particle Orbits in Schwarzschild Metric . 88 0.18 AnIsotropicUniverse. 93 0.19 MoreontheSchwarzschild Geometry . 102 0.20 BlackHoleswithChargeandSpin. 110 0.21 Equivalence Principle, Fermions, and Fancy Formalism . .115 0.22 QuantizedFieldTheory . .121 3 4. Last Latexed: November 19, 2015 at 11:03 Joel A. Shapiro Note: This is being typed piecemeal in 2012 from handwritten notes in a red looseleaf marked 617 (1983) but may have originated in 1979 0.1 Introduction I am, myself, an elementary particle physicist, and my interest in general relativity has come from the growth of a field of quantum gravity. Because the gravitational inderactions of reasonably small objects are so weak, quantum gravity is a field almost entirely divorced from contact with reality in the form of direct confrontation with experiment. There are three areas of contact 1. In relativistic quantum mechanics, one usually formulates the physical quantities in terms of fields. A field is a physical degree of freedom, or variable, definded at each point of space and time. -
Riemann Surfaces
RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. -
Uniformization of Riemann Surfaces Revisited
UNIFORMIZATION OF RIEMANN SURFACES REVISITED CIPRIANA ANGHEL AND RARES¸STAN ABSTRACT. We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces. 1. INTRODUCTION Paul Koebe and shortly thereafter Henri Poincare´ are credited with having proved in 1907 the famous uniformization theorem for Riemann surfaces, arguably the single most important result in the whole theory of analytic functions of one complex variable. This theorem generated con- nections between different areas and lead to the development of new fields of mathematics. After Koebe, many proofs of the uniformization theorem were proposed, all of them relying on a large body of topological and analytical prerequisites. Modern authors [6], [7] use sheaf cohomol- ogy, the Runge approximation theorem, elliptic regularity for the Laplacian, and rather strong results about the vanishing of the first cohomology group of noncompact surfaces. A more re- cent proof with analytic flavour appears in Donaldson [5], again relying on many strong results, including the Riemann-Roch theorem, the topological classification of compact surfaces, Dol- beault cohomology and the Hodge decomposition. In fact, one can hardly find in the literature a self-contained proof of the uniformization theorem of reasonable length and complexity. Our goal here is to give such a minimalistic proof. Recall that a Riemann surface is a connected complex manifold of dimension 1, i.e., a connected Hausdorff topological space locally homeomorphic to C, endowed with a holomorphic atlas. Uniformization theorem (Koebe [9], Poincare´ [15]). Any simply-connected Riemann surface is biholomorphic to either the complex plane C, the open unit disk D, or the Riemann sphere C^. -
Uniformization of Riemann Surfaces
CHAPTER 3 Uniformization of Riemann surfaces 3.1 The Dirichlet Problem on Riemann surfaces 128 3.2 Uniformization of simply connected Riemann surfaces 141 3.3 Uniformization of Riemann surfaces and Kleinian groups 148 3.4 Hyperbolic Geometry, Fuchsian Groups and Hurwitz’s Theorem 162 3.5 Moduli of Riemann surfaces 178 127 128 3. UNIFORMIZATION OF RIEMANN SURFACES One of the most important results in the area of Riemann surfaces is the Uni- formization theorem, which classifies all simply connected surfaces up to biholomor- phisms. In this chapter, after a technical section on the Dirichlet problem (solutions of equations involving the Laplacian operator), we prove that theorem. It turns out that there are very few simply connected surfaces: the Riemann sphere, the complex plane and the unit disc. We use this result in 3.2 to give a general formulation of the Uniformization theorem and obtain some consequences, like the classification of all surfaces with abelian fundamental group. We will see that most surfaces have the unit disc as their universal covering space, these surfaces are the object of our study in 3.3 and 3.5; we cover some basic properties of the Riemaniann geometry, §§ automorphisms, Kleinian groups and the problem of moduli. 3.1. The Dirichlet Problem on Riemann surfaces In this section we recall some result from Complex Analysis that some readers might not be familiar with. More precisely, we solve the Dirichlet problem; that is, to find a harmonic function on a domain with given boundary values. This will be used in the next section when we classify all simply connected Riemann surfaces. -
Analytic Solutions of Partial Differential Equations
Analytic Solutions of Partial Differential Equations MATH3414 School of Mathematics, University of Leeds 15 credits • Taught Semester 1, • Year running 2003/04 • Pre-requisites MATH2360 or MATH2420 or equivalent. • Co-requisites None. • Objectives: To provide an understanding of, and methods of solution for, the most important • types of partial differential equations that arise in Mathematical Physics. On completion of this module, students should be able to: a) use the method of characteristics to solve first-order hyperbolic equations; b) classify a second order PDE as elliptic, parabolic or hyperbolic; c) use Green's functions to solve elliptic equations; d) have a basic understanding of diffusion; e) obtain a priori bounds for reaction-diffusion equations. Syllabus: The majority of physical phenomena can be described by partial differential equa- • tions (e.g. the Navier-Stokes equation of fluid dynamics, Maxwell's equations of electro- magnetism). This module considers the properties of, and analytical methods of solution for some of the most common first and second order PDEs of Mathematical Physics. In particu- lar, we shall look in detail at elliptic equations (Laplace?s equation), describing steady-state phenomena and the diffusion / heat conduction equation describing the slow spread of con- centration or heat. The topics covered are: First order PDEs. Semilinear and quasilinear PDEs; method of characteristics. Characteristics crossing. Second order PDEs. Classifi- cation and standard forms. Elliptic equations: weak and strong minimum and maximum principles; Green's functions. Parabolic equations: exemplified by solutions of the diffusion equation. Bounds on solutions of reaction-diffusion equations. Form of teaching • Lectures: 26 hours. 7 examples classes. -
Surfaces and Complex Analysis (Lecture 34)
Surfaces and Complex Analysis (Lecture 34) May 3, 2009 Let Σ be a smooth surface. We have seen that Σ admits a conformal structure (which is unique up to a contractible space of choices). If Σ is oriented, then a conformal structure on Σ allows us to view Σ as a Riemann surface: that is, as a 1-dimensional complex manifold. In this lecture, we will exploit this fact together with the following important fact from complex analysis: Theorem 1 (Riemann uniformization). Let Σ be a simply connected Riemann surface. Then Σ is biholo- morphic to one of the following: (i) The Riemann sphere CP1. (ii) The complex plane C. (iii) The open unit disk D = fz 2 C : z < 1g. If Σ is an arbitrary surface, then we can choose a conformal structure on Σ. The universal cover Σb then inherits the structure of a simply connected Riemann surface, which falls into the classification of Theorem 1. We can then recover Σ as the quotient Σb=Γ, where Γ ' π1Σ is a group which acts freely on Σb by holmorphic maps (if Σ is orientable) or holomorphic and antiholomorphic maps (if Σ is nonorientable). For simplicity, we will consider only the orientable case. 1 If Σb ' CP , then the group Γ must be trivial: every orientation preserving automorphism of S2 has a fixed point (by the Lefschetz trace formula). Because Γ acts freely, we must have Γ ' 0, so that Σ ' S2. To see what happens in the other two cases, we need to understand the holomorphic automorphisms of C and D. -
TEICHM¨ULLER SPACE Contents 1. Introduction 1 2. Quasiconformal
TEICHMULLER¨ SPACE MATTHEW WOOLF Abstract. It is a well-known fact that every Riemann surface with negative Euler characteristic admits a hyperbolic metric. But this metric is by no means unique { indeed, there are uncountably many such metrics. In this paper, we study the space of all such hyperbolic structures on a Riemann surface, called the Teichm¨ullerspace of the surface. We will show that it is a complete metric space, and that it is homeomorphic to Euclidean space. Contents 1. Introduction 1 2. Quasiconformal maps 2 3. Beltrami Forms 3 4. Teichm¨ullerSpace 4 5. Fenchel-Nielsen Coordinates 6 References 9 1. Introduction Much of the theory of Riemann surfaces boils down to the following theorem, the two-dimensional equivalent of Thurston's geometrization conjecture: Proposition 1.1 (Uniformization theorem). Every simply connected Riemann sur- face is isomorphic either to the complex plane C, the Riemann sphere P1, or the unit disk, D. By considering universal covers of Riemann surfaces, we can see that every sur- face admits a spherical, Euclidean, or (this is the case for all but a few surfaces) hyperbolic metric. Since almost all surfaces are hyperbolic, we will restrict our attention in the following material to them. The natural question to ask next is whether this metric is unique. We see almost immediately that the answer is no (almost any change of the fundamental region of a surface will give rise to a new metric), but this answer gives rise to a new question. Does the set of such hyper- bolic structures of a given surface have any structure itself? It turns out that the answer is yes { in fact, in some ways, the structure of this set (called the Teichm¨uller space of the surface) is more interesting than that of the Riemann surface itself. -
The Uniformization Theorem Donald E. Marshall the Koebe Uniformization Theorem Is a Generalization of the Riemann Mapping The- O
The Uniformization Theorem Donald E. Marshall The Koebe uniformization theorem is a generalization of the Riemann mapping The- orem. It says that a simply connected Riemann surface is conformally equivalent to either the unit disk D, the plane C, or the sphere C∗. We will give a proof that illustrates the power of the Perron method. As is standard, the hyperbolic case (D) is proved by constructing Green’s function. The novel part here is that the non-hyperbolic cases are treated in a very similar manner by constructing the Dipole Green’s function. A Riemann surface is a connected Hausdorff space W , together with a collection of open subsets Uα ⊂ W and functions zα : Uα → C such that (i) W = ∪Uα (ii) zα is a homeomorphism of Uα onto the unit disk D, and −1 (iii) if Uα ∩ Uβ = ∅ then zβ ◦ zα is analytic on zα(Uα ∩ Uβ). The functions zα are called coordinate functions, and the sets Uα are called coordinate disks. We can extend our collection of coordinate functions and disks to form a base for −1 D 1 the topology on W by setting Uα,r = zα (r ) and zα,r = r zα|Uα,r , for r < 1. We can also compose each zα with a M¨obius transformation of the disk onto itself so that we can assume that for each coordinate disk Uα and each p0 ∈ Uα there is a coordinate function zα with zα(p0) = 0. For the purposes of the discussion below, we will assume that our collection {zα,Uα} includes all such maps, except that we will further assume that each Uα has compact closure in W , by restricting our collection. -
Parametrix Construction of the Transition Probability Density of the Solution to an SDE Driven by $\Alpha $-Stable Noise
Parametrix construction of the transition probability density of the solution to an SDE driven by α-stable noise Victoria Knopova∗ and Alexei Kulik∗∗ Abstract Let L := −a(x)(−∆)α/2 + (b(x), ∇), where α ∈ (0, 2), and a : Rd → (0, ∞), b : Rd → Rd. Under certain regularity assumptions on the coefficients a and b, we associate Rd 2 Rd with the C∞( )-closure of (L,C∞( )) a Feller Markov process X, which possesses a transition probability density pt(x,y). To construct this transition probability density and to obtain the two-sided estimates on it, we develop a new version of the parametrix method, which even allows us to handle the case 0 < α ≤ 1 and b 6= 0, i.e. when the gradient part of the generator is not dominated by the jump part. Keywords: Pseudo-differential operator, generator of a Markov process, transition prob- ability density, martingale problem, SDE, Levi’s parametrix method. MSC 2010: Primary: 60J35. Secondary: 60J75, 35S05, 35S10, 47G30. 1 Introduction Let Z(α), α ∈ (0, 2), be a symmetric α-stable process in Rd; that is, a Lévy process with the characteristic function (α) α Eei(ξ,Zt ) = e−t|ξ| , ξ ∈ Rd. (α) (α) It is well known that the generator L of the semigroup (Pt )t≥0, where (α) x (α) arXiv:1412.8732v3 [math.PR] 27 Nov 2017 E Pt f(x)= f(Zt ), 2 Rd admits on C∞( ) the representation (α) cα L f(x)= P.V. f(x + u) − f(x) d+α du. (1.1) Rd |u| Z ∗V.M. -
The Automorphism Groups on the Complex Plane Aron Persson
The Automorphism Groups on the Complex Plane Aron Persson VT 2017 Essay, 15hp Bachelor in Mathematics, 180hp Department of Mathematics and Mathematical Statistics Abstract The automorphism groups in the complex plane are defined, and we prove that they satisfy the group axioms. The automorphism group is derived for some domains. By applying the Riemann mapping theorem, it is proved that every automorphism group on simply connected domains that are proper subsets of the complex plane, is isomorphic to the automorphism group on the unit disc. Sammanfattning Automorfigrupperna i det komplexa talplanet definieras och vi bevisar att de uppfyller gruppaxiomen. Automorfigruppen på några domän härleds. Genom att applicera Riemanns avbildningssats bevisas att varje automorfigrupp på enkelt sammanhängande, öppna och äkta delmängder av det komplexa talplanet är isomorf med automorfigruppen på enhetsdisken. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Group and Set Theory 3 2.2. Topology and Complex Analysis 5 3. The Definition of the Automorphism Group 9 4. Automorphism Groups on Sets in the Complex Plane 17 4.1. The Automorphism Group on the Complex Plane 17 4.2. The Automorphism Group on the Extended Complex Plane 21 4.3. The Automorphism Group on the Unit Disc 23 4.4. Automorphism Groups and Biholomorphic Mappings 25 4.5. Automorphism Groups on Punctured Domains 28 5. Automorphism Groups and the Riemann Mapping Theorem 35 6. Acknowledgements 43 7. References 45 1. Introduction To study the geometry of complex domains one central tool is the automorphism group. Let D ⊆ C be an open and connected set. The automorphism group of D is then defined as the collection of all biholomorphic mappings D → D together with the composition operator. -
HARMONIC MORPHISMS BETWEEN RIEMANNIAN MANIFOLDS by Bent FUGLEDE
Ann. Inst. Fourier, Grenoble 28, 2 (1978), 107-144, HARMONIC MORPHISMS BETWEEN RIEMANNIAN MANIFOLDS by Bent FUGLEDE Introduction. The harmonic morphisms of a Riemannian manifold, M , into another, N , are the morphisms for the harmonic struc- tures on M and N (the harmonic functions on a Riemannian manifold being those which satisfy the Laplace-Beltrami equation). These morphisms were introduced and studied by Constantinescu and Cornea [4] in the more general frame of harmonic spaces, as a natural generalization of the conformal mappings between Riemann surfaces (1). See also Sibony [14]. The present paper deals with harmonic morphisms between two Riemannian manifolds M and N of arbitrary (not necessarily equal) dimensions. It turns out, however, that if dim M < dim N , the only harmonic morphisms M -> N are the constant mappings. Thus we are left with the case dim M ^ dim N . If dim N = 1 , say N = R , the harmonic morphisms M —> R are nothing but the harmonic functions on M . If dim M = dim N = 2 , so that M and N are Riemann surfaces, then it is known that the harmonic morphisms of (1) We use the term « harmonic morphisms » rather than « harmonic maps » (as they were called in the case of harmonic spaces in [4]) since it is necessary (in the present frame of manifolds) to distinguish between these maps and the much wider class of harmonic maps in the sense of Eells and Sampson [7], which also plays an important role in our discussion. 108 B. FUGLEDE M into N are the same as the conformal mappings f: M -> N (allowing for points where df= 0). -
Riemannian Geometry
Appendix A Riemannian Geometry In this appendix besides introducing notation and basic definitions we recollect some of the more technical results in Riemannian geometry we exploited in these lecture notes. We freely refer to the excellent exposition of the theory provided by [4, 8], and to [27] for further details. A.1 Notation In what follows, Vn will always denote a C1 compact n–dimensional manifold, (n 3). Unless otherwise stated, the manifold Vn will be supposed oriented, connected, and without boundary. Let .E; Vn;/be a smooth vector bundle over Vn with projection map W E ! Vn. When no confusion arises the space of smooth 1. n; / n 1. n; / sections fs 2 C V E j ı s D idV g will simply be denoted by C V E instead of the standard .E/ (to avoid confusion (Fig. A.1) with the already too numerous s appearing in Riemannian geometry: Christoffel symbols, harmonic n i n coordinates, ...).If U V is an open set, we let fx giD1 be local coordinates for @ @=@ i n 1. ; n/ the points p 2 U. The vector fields f i WD x giD1 2 C U TV provide the n local (positively oriented) coordinate basis for TpV , p 2 U. The corresponding n 1 dual basis for the cotangent spaces Tp V , is provided by the set of –forms i n 1. ; n/ 1. n; 2 n/ fdx giD1 2 C U T V . In particular, the metric tensor g 2 C V ˝S T V , 1. n; 2 n/ where C V ˝S T V denotes the set of smooth symmetric bilinear forms over n i k V , has the local coordinates representation g D gikdx ˝dx , where gik WD g.@i;@k/, and the Einsteinp summation convention is in effect.