Bevezetés az általános relativitáselméletbe I.

Előadásjegyzetek (előadás dátumával) – 2020/21 őszi félév

• 1. előadás: Topology (2020. szeptember 23.)
• 2. előadás: Differentiable manifolds (2020. szeptember 30.)
• 3. előadás: Tangent and dual spaces (2020. október 7.)
• 4. előadás: Tensors (2020. október 14.)
• 5. előadás: Covariant derivative (2020. október 21.)
• 6. előadás: Curvature (2020. november 4.)
• 7. előadás: Covariant derivative (2020. november 11.)
• 8. előadás: Maps of manifolds, Lie derivative and Killing fields (2020. november 25.)
• 9. előadás: Differentia forms, integration and the Frobenius theorem (2020. december 2.)

Szóbeli vizsga tételeinek listája

1. Topology: Topological spaces; Homeomorphisms; Closed sets; Hausdorff Topological spaces; notion of paracompactness;
2. Differentiable Manifolds: Class C^r; n-dimensional differentiable manifolds; Class C^r mappings of differentiable manifolds (curves, functions); Class C^r diffeomorphisms
3. Vectors: Directional derivative; Tangent space V_p,around point p; Vector fields; Parallel transport; Dual spaces, dual basis and one-forms;
4. Tensors: Tensors of type (k,l), operations, properties; Metric tensor, coordinate transformations;
5. Covariant derivative I: Covariant derivative; The partial derivative as local covariant derivative; The difference of covariant derivatives on different tensor fields;
6. Covariant derivative II: Christoffel symbols; Parallel transport; Covariant derivative associated with the metric;
7. Curvature tensor: Geometric interpretation, components from parallel transport; Ricci tensor and Ricci scalar; Properties, symmetries, decomposition (Weyl tensor, contracted Bianchi identity);
8. Geodesics: normal coordinates; affine parameters; spacelike, timelike or null curves; line element and proper time; properties of geodesics, extremization condition.
9. Maps of manifolds: Differentiable mappings (“pull back”, “carries along”); Symmetry transformation;
10. Lie derivative: Definition and properties of Lie derivative; adapted coordinates; Lie derivative in adapted coordinates;
11. Killing fields: Killing’s equation; The Riemann tensor; Conserved quantities;
12. Differential forms: Definition of p-forms; Outer product; Differentiation on p-forms;
13. Integration on manifolds: Orientability; Integrability of continuous n-form field α over n-dimensional orientable manifold; “Well behaved” surface; Stokes’ theorem; Gauss’s law form of Stokes’ theorem (volume element)
14. Frobenius’ theorem