Semi-Riemann Geometry and General Relativity Shlomo Sternberg September 24, 2003 2 0.1 Introduction Thisbookrepresentscoursenotesforaonesemestercourseattheundergraduate 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 whatareknownastheprincipalcurvatures. Theprecisedefinitionsaregivenin the text.The chapter culminates with Gauss’ Theorema egregium which asserts thatifwethickenatwodimensionalsurfaceevenlyonbothsides,thenthethese integrandsdependonlyontheintrinsicgeometryofthesurface,andnotonhow the surface is embedded. We give two proofs of this important theorem. (We giveseveralmorelaterinthebook.) Thefirstproofmakesuseof“normalcoor- 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. ChapterIIis arapidreview ofthe differentialandintegralcalculus onman- 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”. ChapterIIIdiscussesthefundamentalnotionsoflinearconnectionsandtheir curvatures,andalsoCartan’smethodofcalculatingcurvatureusingframefields anddifferentialforms. WeshowthatthegeodesicsonaLiegroupequippedwith abi-invariantmetricarethetranslatesoftheoneparametersubgroups. Ashort 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. WediscussEinstein’sgeneraltheoryofrelativityfromthepointofviewof theEinstein-Hilbertfunctional. InfactweborrowthetitleofHilbert’spaperfor theChapterheading. Wealsointroducetheprincipleofgeneralcovariance,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 perfectedbyBarrettO’Neill. ItisthenaturalsettingforthegeneralizedGauss- Codazzi type equations. Although technically somewhat demanding at the be- ginning, the range of applications justifies the effort in setting up the theory. Applicationsrangefromcurvaturecomputationsforhomogeneousspacestocos- mogeny and eschatology in Friedman type models. ChapterXdiscussesthePetrovclassification,usingcomplexgeometry,ofthe 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 isessentiallyindependentoftheentirecourse,butIthoughtitusefultoinclude, asitwouldbeofinterestinanymoreadvancedtreatmentoftopicsinthecourse. 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. . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2 Normal coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.3 The Euler field E and its image P. . . . . . . . . . . . . . . . . . 127 6.4 The normal frame field. . . . . . . . . . . . . . . . . . . . . . . . 128 6.5 Gauss’ lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.6 Minimization of arc length. . . . . . . . . . . . . . . . . . . . . . 131 7 Special relativity 133 7.1 Two dimensional Lorentz transformations. . . . . . . . . . . . . . 133 7.1.1 Addition law for velocities. . . . . . . . . . . . . . . . . . 135 7.1.2 Hyperbolic angle. . . . . . . . . . . . . . . . . . . . . . . . 135 7.1.3 Proper time. . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.1.4 Time dilatation. . . . . . . . . . . . . . . . . . . . . . . . 137 7.1.5 Lorentz-Fitzgerald contraction. . . . . . . . . . . . . . . . 137 7.1.6 The reverse triangle inequality. . . . . . . . . . . . . . . . 138 7.1.7 Physical significance of the Minkowski distance. . . . . . . 138 7.1.8 Energy-momentum . . . . . . . . . . . . . . . . . . . . . . 139 7.1.9 Psychological units. . . . . . . . . . . . . . . . . . . . . . 140 7.1.10 The Galilean limit. . . . . . . . . . . . . . . . . . . . . . . 142 7.2 Minkowski space. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.2.1 The Compton effect. . . . . . . . . . . . . . . . . . . . . . 143 7.2.2 Natural Units. . . . . . . . . . . . . . . . . . . . . . . . . 146 7.2.3 Two-particle invariants. . . . . . . . . . . . . . . . . . . . 147 8 CONTENTS 7.2.4 Mandlestam variables. . . . . . . . . . . . . . . . . . . . . 150 7.3 Scattering cross-section and mutual flux. . . . . . . . . . . . . . . 154 8 Die Grundlagen der Physik. 157 8.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.1.1 Densities and divergences. . . . . . . . . . . . . . . . . . . 157 8.1.2 Divergenceofavectorfieldonasemi-Riemannianmanifold.160 8.1.3 The Lie derivative of of a semi-Riemann metric. . . . . . 162 8.1.4 The covariant divergence of a symmetric tensor field. . . . 163 8.2 Varying the metric and the connection. . . . . . . . . . . . . . . 167 8.3 The structure of physical laws. . . . . . . . . . . . . . . . . . . . 169 8.3.1 The Legendre transformation. . . . . . . . . . . . . . . . . 169 8.3.2 The passive equations. . . . . . . . . . . . . . . . . . . . . 172 8.4 The Hilbert “function”. . . . . . . . . . . . . . . . . . . . . . . . 173 8.5 Schrodinger’s equation as a passive equation. . . . . . . . . . . . 175 8.6 Harmonic maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9 Submersions. 179 9.1 Submersions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 9.2 The fundamental tensors of a submersion. . . . . . . . . . . . . . 181 9.2.1 The tensor T. . . . . . . . . . . . . . . . . . . . . . . . . . 181 9.2.2 The tensor A. . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.2.3 Covariant derivatives of T and A.. . . . . . . . . . . . . . 183 9.2.4 The fundamental tensors for a warped product. . . . . . . 185 9.3 Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.3.1 Curvature for warped products. . . . . . . . . . . . . . . . 190 9.3.2 Sectional curvature. . . . . . . . . . . . . . . . . . . . . . 193 9.4 Reductive homogeneous spaces. . . . . . . . . . . . . . . . . . . . 194 9.4.1 Bi-invariant metrics on a Lie group. . . . . . . . . . . . . 194 9.4.2 Homogeneous spaces.. . . . . . . . . . . . . . . . . . . . . 197 9.4.3 Normal symmetric spaces. . . . . . . . . . . . . . . . . . . 197 9.4.4 Orthogonal groups. . . . . . . . . . . . . . . . . . . . . . . 198 9.4.5 Dual Grassmannians. . . . . . . . . . . . . . . . . . . . . 200 9.5 Schwarzschild as a warped product.. . . . . . . . . . . . . . . . . 202 9.5.1 Surfaces with orthogonal coordinates. . . . . . . . . . . . 203 9.5.2 The Schwarzschild plane. . . . . . . . . . . . . . . . . . . 204 9.5.3 Covariant derivatives. . . . . . . . . . . . . . . . . . . . . 205 9.5.4 Schwarzschild curvature. . . . . . . . . . . . . . . . . . . . 206 9.5.5 Cartan computation. . . . . . . . . . . . . . . . . . . . . . 207 9.5.6 Petrov type.. . . . . . . . . . . . . . . . . . . . . . . . . . 209 9.5.7 Kerr-Schild form. . . . . . . . . . . . . . . . . . . . . . . . 210 9.5.8 Isometries. . . . . . . . . . . . . . . . . . . . . . . . . . . 211 9.6 Robertson Walker metrics. . . . . . . . . . . . . . . . . . . . . . . 214 9.6.1 Cosmogeny and eschatology.. . . . . . . . . . . . . . . . . 216 CONTENTS 9 10 Petrov types. 217 10.1 Algebraic properties of the curvature tensor . . . . . . . . . . . . 217 10.2 Linear and antilinear maps. . . . . . . . . . . . . . . . . . . . . . 219 10.3 Complex conjugation and real forms. . . . . . . . . . . . . . . . . 221 10.4 Structures on tensor products. . . . . . . . . . . . . . . . . . . . 223 10.5 Spinors and Minkowski space. . . . . . . . . . . . . . . . . . . . . 224 10.6 Traceless curvatures. . . . . . . . . . . . . . . . . . . . . . . . . . 225 10.7 The polynomial algebra. . . . . . . . . . . . . . . . . . . . . . . . 225 10.8 Petrov types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 10.9 Principal null directions. . . . . . . . . . . . . . . . . . . . . . . . 227 10.10Kerr-Schild metrics. . . . . . . . . . . . . . . . . . . . . . . . . . 230 11 Star. 233 11.1 Definition of the star operator. . . . . . . . . . . . . . . . . . . . 233 11.2 Does ?:∧kV →∧n−kV determine the metric? . . . . . . . . . . 235 11.3 The star operator on forms. . . . . . . . . . . . . . . . . . . . . . 240 11.3.1 For R2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 11.3.2 For R3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 11.3.3 For R1,3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 11.4 Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . 243 11.4.1 Electrostatics.. . . . . . . . . . . . . . . . . . . . . . . . . 243 11.4.2 Magnetoquasistatics. . . . . . . . . . . . . . . . . . . . . . 244 11.4.3 The London equations.. . . . . . . . . . . . . . . . . . . . 246 11.4.4 The London equations in relativistic form. . . . . . . . . . 248 11.4.5 Maxwell’s equations. . . . . . . . . . . . . . . . . . . . . . 249 11.4.6 Comparing Maxwell and London. . . . . . . . . . . . . . . 249 10 CONTENTS