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General Relativity GENERAL RELATIVITY —LECTURE NOTES — PRELIMINARY VERSION FEBRUARY 4, 2020 PROF.DR.HAYE HINRICHSEN LEHRSTUHL FÜR THEORETISCHE PHYSIK III FAKULTÄT FÜR PHYSIK UND ASTRONOMIE UNIVERSITÄT WÜRZBURG,GERMANY SUMMER TERM 2018 Contents 1 Mathematical foundations1 1.1 Elements of group theory . .1 1.1.1 Groups . .1 1.1.2 Cosets . .2 1.1.3 Normal subgroups . .2 1.1.4 Quotient groups . .3 1.1.5 Homomorphisms . .3 1.1.6 Kernel, image, and the corresponding quotient groups . .4 1.2 Vector spaces . .5 1.2.1 Fields . .5 1.2.2 Vector space axioms . .5 1.2.3 Affine spaces . .6 1.2.4 Representation of vectors . .6 1.3 Linear maps . .8 1.3.1 Definition and properties . .8 1.3.2 Representation of linear maps . .9 1.3.3 Basis transformations . .9 1.4 Composite vector spaces . 11 1.4.1 Direct sum ⊕ .............................. 11 1.4.2 Representation of direct sums . 11 1.4.3 Tensor product ⊗ ............................ 12 1.4.4 Calculation rules for tensor products . 13 1.4.5 Representation of the tensor product . 14 1.4.6 Multiple tensor products . 16 1.5 Multilinear forms . 16 1.5.1 1-forms . 16 1.5.2 Representation of 1-forms . 17 1.5.3 How 1-forms change under basis transformations . 19 1.5.4 Tensors . 20 1.5.5 Representation of tensors . 21 1.5.6 Induced basis in the space of tensors N(q,p)V ............ 22 1.5.7 Tensors versus matrices . 22 1.5.8 Tensor product of tensors . 23 1.5.9 Representation of the tensor product . 23 1.5.10 Contraction . 24 1.5.11 Representation of a contraction . 25 1.5.12 Tensor algebra . 26 1.6 Metric . 26 1.6.1 Metric tensor and scalar product . 26 1.6.2 Representation of the metric tensor . 28 c HAYE HINRICHSEN —GENERAL RELATIVITY — CC BY-SA 4.0 iv Contents 1.6.3 Examples . 28 1.6.4 Musical isomorphism V $ V∗ .................... 29 1.6.5 Representation of [ and ]: Raising and lowering of indices . 30 1.6.6 Application of the musical operators to tensors . 32 1.6.7 Transformation behavior of the metric . 32 1.6.8 Determinant of the metric . 32 ij 1.6.9 Differentiating the determinant g with respect to g or gij..... 34 2 Differential forms 37 2.1 Exterior algebra . 37 2.1.1 Exterior product (wedge product) . 37 2.1.2 q-multivectors . 39 2.1.3 p-forms . 40 2.1.4 Exterior algebra . 41 2.1.5 Representation of p-forms . 42 2.1.6 Representation of the wedge product . 43 2.1.7 Visual interpretation of the exterior algebra . 44 2.1.8 The volume form w ........................... 45 2.1.9 Representation of the volume form . 47 2.1.10 Contraction i .............................. 48 2.1.11 Representation of the contraction i in the exterior algebra . 49 2.2 Hodge duality . 50 2.2.1 Illustrative description of the Hodge duality . 50 2.2.2 Induced scalar product on p-forms . 51 2.2.3 Representation of the generalized scalar product . 52 2.2.4 Hodge duality on the basis of the generalized scalar product . 52 2.2.5 Hodge-star operator? ......................... 54 2.2.6 Representation of the Hodge-star operator? ............ 54 2.2.7 Properties of the Hodge-star operator ? ............... 55 2.2.8 Hodge-? operator represented in an orthonormal basis . 55 2.2.9 Self-duality * . 56 2.3 Functions, coordinate systems and differential forms . 57 2.3.1 Scalar functions, curves and directional derivatives . 58 2.3.2 Differentials . 60 2.3.3 Coordinate systems . 61 2.3.4 Coordinate basis . 62 2.3.5 Representation of fields in coordinate systems . 63 2.3.6 Changing between different coordinate systems . 65 2.3.7 Degenerate differential forms and zero vector fields . 67 2.4 Derivatives . 67 2.4.1 Generalized differential . 67 2.4.2 Exterior derivative . 69 2.4.3 Representation of the exterior algebra . 70 2.4.4 The Poincaré lemma . 71 2.4.5 Relation to ordinary vector analysis . 71 2.4.6 The co-differential operator . 72 2.4.7 Lie bracket . 73 c HAYE HINRICHSEN —GENERAL RELATIVITY — CC BY-SA 4.0 Contents v 2.5 Integration of forms . 74 2.5.1 Special cases . 74 2.5.2 Generic integrals over p-forms . 76 2.5.3 Stokes theorem . 76 2.6 Tensor-valued forms * . 76 3 Elementary concepts of differential geometry 79 3.1 Manifolds . 79 3.1.1 Maps . 80 3.1.2 Changes between different maps . 81 3.1.3 Functions on manifolds . 82 3.2 Tangent space and cotangent space . 83 3.2.1 Directional derivatives and differentials: . 83 3.2.2 Tangent bundle and cotangent bundle . 84 3.2.3 Excursus: fiber bundles * . 84 3.2.4 Coordinate basis . 85 3.2.5 Structural coefficients . 86 3.3 Parallel transport . 87 3.3.1 Transport of geometric objects . 87 3.3.2 Parallel transport of tangent vectors . 88 3.3.3 Covariant derivative of vector fields . 89 3.3.4 Connections . 90 3.3.5 Representation of the connection . 91 3.3.6 Representation of the connection in the coordinate basis . 92 3.3.7 Covariant transformation behavior . 93 3.3.8 Geodesic lines . 93 3.3.9 How the connection is calculated . 94 3.3.10 Covariant derivative of arbitrary tensor fields . 97 3.3.11 Exterior derivative of the tensorial forms* . 99 3.4 Curvature . 100 3.4.1 Riemann curvature tensor . 100 3.4.2 Representation of the Riemannian curvature tensor . 101 3.4.3 Symmetries of the curvature tensor . 101 3.4.4 Ricci tensor . 102 3.4.5 Interpretation of curvature tensors . 103 4 Electrodynamics as a gauge theory 105 4.1 U(1) gauge theory . 105 4.1.1 Intrinsic degrees of freedom . 105 4.1.2 Representation of intrinsic degrees of freedom . 108 4.1.3 Gauge transformations . 109 4.1.4 Two-dimensional U(1) gauge theory . 110 4.1.5 Covariant derivative . 112 4.1.6 Intrinsic curvature: The electromagnetic field . 112 4.2 Electrodynamics in terms of differential forms . 113 4.2.1 The electromagnetic field as a differential form . 113 4.2.2 Equation of motions in differential forms . 114 4.2.3 Equations of motion in components . 115 c HAYE HINRICHSEN —GENERAL RELATIVITY — CC BY-SA 4.0 vi Contents 4.2.4 U(1) gauge symmetry . 116 4.2.5 Action . 116 4.2.6 Wave equation . 117 4.2.7 Representation of electrodynamics . 117 4.2.8 Charge conservation . 117 5 Field equations of general relativity 119 5.1 Concept of General Theory.
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