Introduction to general relativity I

We are pleased to offer an introductory elective course (i.e. not a core-program course) on General Relativity for domestic and international students currently completing an undergraduate or a graduate-degree Physics program in Hungary. The course is entirely conducted either in Hungarian or in English, depending on the students request. The live lectures are normaly delivered entirely on-campus at Eötvös Lóránd University, however, in response to the COVID-19 pandemic we had to shift to online delivery and the course will be running remotely over the semester.

Lecture notes and course materials (with date of lecture) – Fall Semester 2020/21

• Lecture 1: Topology (September 23, 2020)
• Lecture 2: Differentiable manifolds (September 30, 2020)
• Lecture 3: Tangent and dual spaces (October 7, 2020)
• Lecture 4: Tensors (October 14, 2020)
• Lecture 5: Covariant derivative (October 21, 2020)
• Lecture 6: Curvature (November 4, 2020)
• Lecture 7: Covariant derivative (November 11, 2020)
• Lecture 8: Maps of manifolds, Lie derivative and Killing fields (November 25, 2020)
• Lecture 9: Differentia forms, integration and the Frobenius theorem (December 2, 2020)

List of topics for the oral exam

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; 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