# Topics in General Relativity

I have created a one-semester course in Advanced GR and another one-semester course in Advanced GR + Cosmology. The materials of these two courses will eventually be merged into a free book. For now, the lecture notes in their present form are available here.

### Topics in advanced General Relativity

In this course, I explain and use only coordinate-free differential geometry in the index-free notation. There are no tensor indices, Christoffel symbols or other non-tensors, coordinate transformations, or special reference systems chosen to simplify calculations. This course was given in Munich in the Fall 2005.

Topics include: Asymptotic structure of spacetime, conformal diagrams, null surfaces, Raychaudhury equation, black holes, the holographic principle, singularity theorems, Einstein-Hilbert action, Newtonian limit of Einsten's equations, redshift, Noether's theorem, tetrad (vierbein) formalism, spinor fields in curved spacetime, Hamiltonian formulation of GR, basics of quantum cosmology.

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Source in LaTeX is given as well.

1. -- Preface -- Suggested literature
1. 1 Calculus in curved space
2. -- 1.1 Summary -- 1.1.1 Index-free notation -- 1.1.2 Sample practice problems -- 1.2 Basic notions: Manifolds and vector fields -- 1.2.1 Definitions -- 1.2.2 Manifolds and coordinates -- 1.2.3 Manifolds: intrinsic picture -- 1.2.4 Tangent spaces -- 1.2.5 Tangent vectors as short curve segments -- 1.2.6 *Tangent space as space of derivations -- 1.2.7 Vector fields and flows -- 1.2.8 *Tangent bundle -- 1.2.9 Tensor fields -- 1.2.10 Commutator of vector fields -- 1.2.11 Connecting vectors -- 1.3 Lie derivative -- 1.3.1 Commutator as Lie derivative -- 1.3.2 Lie derivative of tensors -- 1.3.3 Geometric interpretation -- 1.4 Calculus of differential forms -- 1.4.1 Volume as antisymmetric tensor -- 1.4.2 Motivation for differential forms -- 1.4.3 Antisymmetric tensors -- 1.4.4 *Oriented volume and n-vectors -- 1.4.5 Determinants -- 1.4.6 Differential forms -- 1.4.7 *Canonical decomposition of 1-forms and 2-forms -- 1.4.8 The Poincaré lemma -- 1.4.9 Integration of forms -- 1.5 Metric -- 1.5.1 Motivation: metric on surfaces -- 1.5.2 Definition -- 1.5.3 Examples of metrics -- 1.5.4 Orthonormal frames -- 1.5.5 Correspondence of vectors and covectors -- 1.5.6 The Levi-Civita tensor .oldsymbol{varepsilon} -- 1.6 Affine connection -- 1.6.1 Motivation -- 1.6.2 General properties of connections -- 1.6.3 The "coordinate derivative" connection -- 1.6.4 Compatibility with the metric -- 1.6.5 Torsion and torsion-freeness -- 1.6.6 Levi-Civita connection -- First examples -- Derivation of the Levi-Civita connection -- 1.6.7 Killing vectors -- 1.6.8 *Koszul formula and the Lie derivative -- 1.6.9 Divergence of a vector field -- 1.7 Calculations in index-free notation -- 1.7.1 Abstract index notation -- 1.7.2 Converting expressions into index-free notation -- 1.7.3 Index-free computations of trace -- 1.7.4 Summary of calculation rules -- 1.8 Curvature -- 1.8.1 Curvature of a connection -- 1.8.2 Bianchi identities -- 1.8.3 Ricci tensor and scalar -- 1.8.4 Calculations with the curvature tensor -- 1.9 Geodesic curves, geodesic vector fields -- 1.9.1 Parallel transport of vectors -- 1.9.2 Geodesics -- 1.9.3 Geodesics extremize proper length -- 1.9.4 *Motion under external forces -- 1.9.5 Deviation of geodesics -- 1.10 Example: hypersurface of constant curvature -- 1.10.1 Tangent bundle and induced metric -- 1.10.2 Induced connection -- 1.10.3 Riemann tensor within the hypersurface
3. 2 Geometry of null surfaces
4. -- 2.1 Null vectors -- 2.1.1 Orthogonal complement spaces -- 2.1.2 Divergence of a null vector field -- 2.2 Null surfaces -- 2.2.1 Three-dimensional hypersurfaces -- 2.2.2 Integrable vector fields -- 2.2.3 Frobenius theorem -- 2.2.4 Null surfaces -- 2.2.5 Examples of null surfaces -- 2.2.6 Lightcones are null surfaces -- 2.2.7 Null functions -- 2.2.8 Null functions generate null geodesics -- 2.2.9 Every lightray comes from null functions -- 2.2.10 Conformal invariance -- 2.3 Raychaudhuri equation -- 2.3.1 Distortion tensor -- 2.3.2 Rotation -- 2.3.3 Introducing Raychaudhuri equation -- 2.3.4 Shear for timelike congruences -- 2.3.5 Shear for null congruences -- 2.4 Applications of Raychaudhuri equation -- 2.4.1 Energy conditions -- 2.4.2 Focusing of timelike geodesics -- 2.4.3 Repulsive gravity -- 2.4.4 Focusing of null geodesics -- 2.5 Null tetrad formalism
5. 3 Asymptotically flat spacetimes
6. -- 3.1 Stationary spacetimes -- 3.1.1 Newtonian limit -- Example: Schwarzschild spacetime -- Example: de Sitter spacetime -- 3.1.2 Redshift -- 3.1.3 Conformal Killing vectors -- 3.1.4 Gravitational potential -- 3.1.5 Energy -- 3.2 Conformal infinity -- 3.2.1 Conformal infinity for Minkowski spacetime -- 3.2.2 Conformal diagrams -- Standard procedure -- Method of lightrays -- New definition of conformal diagrams -- Minkowski spacetime -- Cauchy surfaces and artificial boundaries -- 3.2.3 How to draw conformal diagrams -- Further examples -- 3.3 Asymptotic flatness -- 3.4 Conformal radiation fields -- 3.4.1 Scalar field in 1+1 dimensions -- 3.4.2 Scalar field in 3+1 dimensions -- 3.4.3 Electromagnetic field -- 3.4.4 Gravitational radiation field -- 3.4.5 Asymptotic behavior of radiation
7. 4 Global techniques
8. -- 4.1 Singularity theorems -- 4.1.1 Singularities and geodesic incompleteness -- 4.1.2 Past-incompleteness of inflation -- 4.1.3 Conjugate points on geodesics -- 4.1.4 Second variation of proper length -- 4.1.5 Singularity in collapsing or expanding universe -- 4.1.6 Singularity in a closed universe -- 4.1.7 Singularity in gravitational collapse -- 4.2 Hawking's area theorem -- 4.3 Holographic principle
9. 5 Variational principle
10. -- 5.1 Lagrangian formulation -- 5.1.1 Classical field theory -- Covariant volume element -- Minimal coupling to gravity -- Gauss's law with covariant derivatives -- 5.1.2 Einstein-Hilbert action -- 5.1.3 Nonlinear f(R) gravity -- 5.1.4 Energy-momentum tensor -- 5.1.5 General covariance -- 5.1.6 Symmetries and Noether theorems -- Translational symmetry -- Internal symmetry -- Infinite-dimensional (gauge) symmetry -- 5.2 Hamiltonian formulation -- 5.2.1 Electrodynamics in Hamiltonian formulation -- 5.2.2 Hamiltonian mechanics of constrained systems -- 5.2.3 Gauss-Codazzi equation -- 5.2.4 Boundary term in Einstein-Hilbert action -- 5.2.5 The Hamiltonian for pure gravity -- Derivation of Eq. ([eq:R4 through R3]) -- 5.2.6 Constraints in General Relativity -- 5.3 Quantum cosmology -- 5.3.1 Wave function of the universe -- 5.3.2 Wheeler-DeWitt equation -- 5.3.3 Interpretation of the wave function -- 5.3.4 "Minisuperspace"
11. 6 Tetrad methods
12. -- 6.1 Tetrad formalism -- 6.1.1 Tetrads -- 6.1.2 Examples -- 6.1.3 Hodge duality -- 6.1.4 Levi-Civita connection -- 6.1.5 Connection as a set of 1-forms -- 6.1.6 *Solving equations for n-forms -- 6.2 Applications of tetrad formalism -- 6.2.1 Computing geodesic equations -- 6.2.2 Determining Killing vectors -- 6.2.3 Curvature as a set of 2-forms -- 6.2.4 Ricci tensor and Ricci scalar -- 6.2.5 Einstein-Hilbert action in tetrads -- 6.3 Connections on vector bundles -- 6.3.1 Vector bundles as generalization of tangent bundles -- 6.3.2 Examples of bundles -- 6.3.3 Covariant derivatives on vector bundles -- 6.3.4 Gauge theories and associated bundles -- 6.3.5 Tangent bundle as associated bundle
13. 7 Spinors
14. -- 7.1 Introducing spinors -- 7.1.1 Quaternions and rotations -- 7.1.2 The Lorentz group -- 7.1.3 Lorentz transformations of spinors -- 7.2 Spinor algebra -- 7.2.1 The fundamental 2-form -- 7.2.2 Relationship of spinors and vectors -- 7.2.3 Simplification of spinorial tensors -- 7.3 Equations for spinor fields -- 7.3.1 Spinors in curved spacetime -- 7.3.2 Covariant derivative on spinors -- 7.3.3 Maxwell equations -- 7.3.4 Dirac equation
15. A Elements of Special and General Relativity
16. -- A.1 Special Relativity -- A.1.1 Spacetime -- A.1.2 Motion of bodies in SR -- A.2 Index notation -- A.3 Transition to General Relativity -- A.4 Covariant derivative -- A.4.1 Curved coordinates -- A.4.2 Curved space and induced metric -- A.4.3 Covariant derivative -- A.4.4 Properties of covariant derivative -- A.4.5 ^{*}Choice of connection -- A.5 Curvature -- A.5.1 Parallel transport -- A.5.2 Riemann tensor -- A.5.3 ^{*}Expressing Riemann tensor through Gamma_{.lpha.eta}^{ambda} -- A.6 Covariant integration -- A.6.1 Determinant of the metric -- A.6.2 Covariant volume element -- A.6.3 Derivative of the determinant -- A.6.4 Covariant divergence -- A.6.5 Integration by parts -- A.7 Einstein's equation
17. B How not to learn tensor calculus
18. -- B.1 Tensor algebra -- B.2 Tensor calculus -- B.3 Hints
19. C Calculations and proofs
20. -- C.1 For Chapter [cha:Calculus-in-curved] -- C.2 For Chapter [cha:Asymptotically-flat-spacetimes] -- C.3 For Chapter [cha:Tetrad-methods]
21. D Comments on literature
22. -- D.1 Comments on Ludvigsen's General Relativity
23. E License for this text
24. -- E.1 Author's position on commercial publishing

### Advanced general relativity and cosmology

This course was given in Heidelberg in the Fall 2007. This course contains some of the material of the GR course as well as more modern topics, such as f(R) gravity, dynamics of inflation, and basics of inflationary perturbation theory. In particular, I attempted to derive, in a constructive way, the formula for the Mukhanov-Sasaki scalar field. I still use the index-free and coordinate-free approach to gravity for all calculations. I also introduce a simplified index-free notation for taking the trace (or contraction) of arbitrary tensors.

Only the lecture slides are available (but they are quite detailed, not the usual presentation-type slides).

File: lecture-drafts.pdf - version for on-screen viewing

File: lecture-drafts_2s.ps.gz - version for double-sided printing GR_course.pdf lecture_drafts.pdf WS06-T6-All_Problems_and_Solutions.pdf