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Semi-Riemann Geometry and General Relativity

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Semi-Riemann Geometry and General Relativity Semi-Riemann Geometry and General Relativity Shlomo Sternberg September 24, 2003 2 0.1 Introduction This book represents course notes for a one semester course at the undergraduate level giving an introduction to Riemannian geometry and its principal physical application, Einstein’s theory of general relativity. The background assumed is a good grounding in linear algebra and in advanced calculus, preferably in the language of differential forms. Chapter I introduces the various curvatures associated to a hypersurface embedded in Euclidean space, motivated by the formula for the volume for the region obtained by thickening the hypersurface on one side. If we thicken the hypersurface by an amount h in the normal direction, this formula is a polynomial in h whose coefficients are integrals over the hypersurface of local expressions. These local expressions are elementary symmetric polynomials in what are known as the principal curvatures. The precise definitions are given in the text.The chapter culminates with Gauss’ Theorema egregium which asserts that if we thicken a two dimensional surface evenly on both sides, then the these integrands depend only on the intrinsic geometry of the surface, and not on how the surface is embedded. We give two proofs of this important theorem. (We give several more later in the book.) The first proof makes use of “normal coor- dinates” which become so important in Riemannian geometry and, as “inertial frames,” in general relativity. It was this theorem of Gauss, and particularly the very notion of “intrinsic geometry”, which inspired Riemann to develop his geometry. Chapter II is a rapid review of the differential and integral calculus on man- ifolds, including differential forms,the d operator, and Stokes’ theorem. Also vector fields and Lie derivatives. At the end of the chapter are a series of sec- tions in exercise form which lead to the notion of parallel transport of a vector along a curve on a embedded surface as being associated with the “rolling of the surface on a plane along the curve”. Chapter III discusses the fundamental notions of linear connections and their curvatures, and also Cartan’s method of calculating curvature using frame fields and differential forms. We show that the geodesics on a Lie group equipped with a bi-invariant metric are the translates of the one parameter subgroups. A short exercise set at the end of the chapter uses the Cartan calculus to compute the curvature of the Schwartzschild metric. A second exercise set computes some geodesics in the Schwartzschild metric leading to two of the famous predictions of general relativity: the advance of the perihelion of Mercury and the bending of light by matter. Of course the theoretical basis of these computations, i.e. the theory of general relativity, will come later, in Chapter VII. Chapter IV begins by discussing the bundle of frames which is the modern setting for Cartan’s calculus of “moving frames” and also the jumping off point for the general theory of connections on principal bundles which lie at the base of such modern physical theories as Yang-Mills fields. This chapter seems to present the most difficulty conceptually for the student. Chapter V discusses the general theory of connections on fiber bundles and then specialize to principal and associated bundles. 0.1. INTRODUCTION 3 Chapter VI returns to Riemannian geometry and discusses Gauss’s lemma which asserts that the radial geodesics emanating from a point are orthogo- nal (in the Riemann metric) to the images under the exponential map of the spheres in the tangent space centered at the origin. From this one concludes that geodesics (defined as self parallel curves) locally minimize arc length in a Riemann manifold. Chapter VII is a rapid review of special relativity. It is assumed that the students will have seen much of this material in a physics course. Chapter VIII is the high point of the course from the theoretical point of view. We discuss Einstein’s general theory of relativity from the point of view of the Einstein-Hilbert functional. In fact we borrow the title of Hilbert’s paper for the Chapter heading. We also introduce the principle of general covariance, first introduce by Einstein, Infeld, and Hoffmann to derive the “geodesic principle” and give a whole series of other applications of this principle. Chapter IX discusses computational methods deriving from the notion of a Riemannian submersion, introduced and developed by Robert Hermann and perfected by Barrett O’Neill. It is the natural setting for the generalized Gauss- Codazzi type equations. Although technically somewhat demanding at the be- ginning, the range of applications justifies the effort in setting up the theory. Applications range from curvature computations for homogeneous spaces to cos- mogeny and eschatology in Friedman type models. Chapter X discusses the Petrov classification, using complex geometry, of the various types of solutions to the Einstein equations in four dimensions. This classification led Kerr to his discovery of the rotating black hole solution which is a topic for a course in its own. The exposition in this chapter follows joint work with Kostant. Chapter XI is in the form of a enlarged exercise set on the star operator. It is essentially independent of the entire course, but I thought it useful to include, as it would be of interest in any more advanced treatment of topics in the course. 4 Contents 0.1 Introduction . 2 1 The principal curvatures. 11 1.1 Volume of a thickened hypersurface . 11 1.2 The Gauss map and the Weingarten map. 13 1.3 Proof of the volume formula. 16 1.4 Gauss’s theorema egregium. 19 1.4.1 First proof, using inertial coordinates. 22 1.4.2 Second proof. The Brioschi formula. 25 1.5 Problem set - Surfaces of revolution. 27 2 Rules of calculus. 31 2.1 Superalgebras. 31 2.2 Differential forms. 31 2.3 The d operator. 32 2.4 Derivations. 33 2.5 Pullback. 34 2.6 Chain rule. 35 2.7 Lie derivative. 35 2.8 Weil’s formula. 36 2.9 Integration. 38 2.10 Stokes theorem. 38 2.11 Lie derivatives of vector fields. 39 2.12 Jacobi’s identity. 40 2.13 Left invariant forms. 41 2.14 The Maurer Cartan equations. 43 2.15 Restriction to a subgroup . 43 2.16 Frames. 44 2.17 Euclidean frames. 45 2.18 Frames adapted to a submanifold. 47 2.19 Curves and surfaces - their structure equations. 48 2.20 The sphere as an example. 48 2.21 Ribbons . 50 2.22 Developing a ribbon. 51 2.23 Parallel transport along a ribbon. 52 5 6 CONTENTS 2.24 Surfaces in R3............................. 53 3 Levi-Civita Connections. 57 3.1 Definition of a linear connection on the tangent bundle. 57 3.2 Christoffel symbols. 58 3.3 Parallel transport. 58 3.4 Geodesics. 60 3.5 Covariant differential. 61 3.6 Torsion. 63 3.7 Curvature. 63 3.8 Isometric connections. 65 3.9 Levi-Civita’s theorem. 65 3.10 Geodesics in orthogonal coordinates. 67 3.11 Curvature identities. 68 3.12 Sectional curvature. 69 3.13 Ricci curvature. 69 3.14 Bi-invariant metrics on a Lie group. 70 3.14.1 The Lie algebra of a Lie group. 70 3.14.2 The general Maurer-Cartan form. 72 3.14.3 Left invariant and bi-invariant metrics. 73 3.14.4 Geodesics are cosets of one parameter subgroups. 74 3.14.5 The Riemann curvature of a bi-invariant metric. 75 3.14.6 Sectional curvatures. 75 3.14.7 The Ricci curvature and the Killing form. 75 3.14.8 Bi-invariant forms from representations. 76 3.14.9 The Weinberg angle. 78 3.15 Frame fields. 78 3.16 Curvature tensors in a frame field. 79 3.17 Frame fields and curvature forms. 79 3.18 Cartan’s lemma. 82 3.19 Orthogonal coordinates on a surface. 83 3.20 The curvature of the Schwartzschild metric . 84 3.21 Geodesics of the Schwartzschild metric. 85 3.21.1 Massive particles. 88 3.21.2 Massless particles. 93 4 The bundle of frames. 95 4.1 Connection and curvature forms in a frame field. 95 4.2 Change of frame field. 96 4.3 The bundle of frames. 98 4.3.1 The form ϑ........................... 99 4.3.2 The form ϑ in terms of a frame field. 99 4.3.3 The definition of ω. ..................... 99 4.4 The connection form in a frame field as a pull-back. 100 4.5 Gauss’ theorems. 103 4.5.1 Equations of structure of Euclidean space. 103 CONTENTS 7 4.5.2 Equations of structure of a surface in R3. 104 4.5.3 Theorema egregium. 104 4.5.4 Holonomy. 104 4.5.5 Gauss-Bonnet. 105 5 Connections on principal bundles. 107 5.1 Submersions, fibrations, and connections. 107 5.2 Principal bundles and invariant connections. 111 5.2.1 Principal bundles. 111 5.2.2 Connections on principal bundles. 113 5.2.3 Associated bundles. 115 5.2.4 Sections of associated bundles. 116 5.2.5 Associated vector bundles. 117 5.2.6 Exterior products of vector valued forms. 119 5.3 Covariant differentials and covariant derivatives. 121 5.3.1 The horizontal projection of forms. 121 5.3.2 The covariant differential of forms on P . 122 5.3.3 A formula for the covariant differential of basic forms. 122 5.3.4 The curvature is dω. 123 5.3.5 Bianchi’s identity. 123 5.3.6 The curvature and d2. 123 6 Gauss’s lemma. 125 6.1 The exponential map.
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