Syllabus and the scans of instructor's notes from lectures/tutorials (Click on the blue-highlited symbols):
| Date | Subject of the lecture | Remarks |
| 13.04 | 1) Models of space time without gravity: a) manifold, b) affine space as a manifold, c) lines and affine transformations, d) repere | This lecture is based on Chapter 3 of the book Spacetime and gravitation by W. Kopczyński, A. Trautman. The book can be downloaded from here: Trautman & Kopczyński Link to the video of the lecture |
| 17.04 | 2) Models of spacetimes without gravity: a) Galilean spacetime, b) Aristotelian spacetime, c) Minkowski spacetime | Also this lecture is based on the book Spacetime and gravitation by W. Kopczyński, A. Trautman: Trautman & Kopczyński ; recomended chapters to extend the material from the lecture are: Chapters 3,4,5,6,7. Link to the video of the lecture |
| 20.04 | 3) Tensors: a) tensors as multilinear maps, b) setup - an associated bundle over a point, c) examples: tensors, tensor densities, pseudotensors | Link to the video of the lecture |
| 24.04 | 4) Tensor fields on manifolds: a) vector fields, b) covariant tensor fields, c) k-forms, d) Cartan algebra, e) derivations: exterior derivative, hook operator, Lie derivative, f) nonholonomic frames, g) Maurer-Cartan theorem | In three lecture notes related to Lecture 3), 4), and 5), as well as during the Lectures 3,4,5, I kept making an annoying error telling that the dimension of the space of tensors of type (r,s) over an n-dimensional vector space is N=n(r+s). This is obviously WRONG. I am very sorry for this lapsus. The CORRECT dimension is of course N=n(r+s). Please correct this everywhere in your notes. Link to the video of the lecture |
| 27.04 | 5) Connections, curvature and torsion: a) k-forms of a given representation type, b) exterior covariant derivative, c) curvature as an obstruction to gauging connection to zero everywhere, d) torsion of a connection, e) covariant derivative | Link to the video of the lecture |
| 8.05 | 6) Newton's spacetime: a) Autoparallels, b) affine parametrization, c) inertial and gravitational mass, d) free falls as autoparallels, e) Poisson field equation as curvature condition for Newton's connection. | This lecture is based on Chapter 10 of the book Spacetime and gravitation by W. Kopczyński, A. Trautman: Trautman & Kopczyński . Much detailed presentation of the subject can be found in an article of A. Trautman, Comparison of Newtonian and Relativistic theories of spacetime, which can be downloaded from here, and an earlier one, Sur la theorie newtonienne de la gravitation, which can be downloaded from here . Link to the video of the lecture |
| 11.05 | 7) Riemannian manifolds: a) Geometric formulation of Newton's gravitation (summary of lecture 6), b) geodesics, c) Levi-Civita connection, d) torsion and metricity determin connection. | Link to the video of the lecture |
| 15.05 | 8) Einstein's spacetime: a) Formulation of Einstein's theory, b) correspondence principle, c) Newton's potential as a metric component in weak field approximation, d) energy momentum tensor, e) Ricci identities, f) Bianchi identities, g) symmetries of Riemann tensor, h) Ricci tensor, i) contracted second Bianchi identity and Eisntein's tensor, j) Einstein's field equations. | Read Chapter 11 and 12 of the book Spacetime and gravitation by W. Kopczyński, A. Trautman: Trautman & Kopczyński . Especially Chapter 12 with a story describing struggles of Einstein to get his field equations is interesting Link to the video of the lecture |
| 22.05 | 9) Einstein Universe, linearization of Einstein's equations: a) Cosmological constant to save Einstein's early view on the Universe, b) Einstein's equations in linear regime, c) interpretation of the constant next to the energy-momentum tensor in Eisntein's equations. | Link to the video of the lecture |
| 25.05 | 10) Predictions of linearized Einstein's gravity: a) Tidal forces and Jacobi equation (geodesic deviation equation), b) gravitational waves in linearized theory, c) evolution of a ball of dust in the field of of monochromatic wave. | Link to the video of the lecture |
| 29.05 | 11) Symmetries and deSitter and anti-deSitter solutions: a) Lie derivative of tensor fields, b) Killing equations, c) maximal number of symmetries on a Riemannian manifold, d) decomposition of the Riemann tensor onto irreducible components: Weyl and Ricci tensors, Ricci scalar , e) deSitter and anti-deSitter spacetimes. | Link to the video of the lecture |
| 1.06 | 12) Schwarzschild solution and its friends: a) Stationary and static gravitational fields, b) spherical symmetry, c) Schwarzschild metric as the unique Ricci flat metric with spherical symmetry, d) coupled Einstein-Maxwell system aqnd spherical symmetry, e) Reisner-Nordstroem solution with cosmological constant. | Link to the video of the lecture |
| 5.06 | 13) Maximal analytic extension of the Schwarzschild metric: a) Kruskal coordinates, b) Kruskal diagram, c) radial free fall on the Schwarzschild black hole, d) comparison of times of a free fall as measured by a falling observer and a and observer far a way from the hole. | Link to the video of the lecture |
| 8.06 | 14) Analysis of orbits in Schwarzschild spacetime: a) energy functional, b) stable orbits, c) perihelion advance, d) deflection of light, e) how field equations determine particles movement?. | Link to the video of the lecture |
| 26.06 | 15) Sperically symmetric stars: a) equations of motions from the field equations, b) matching conditions in GR, c) spherically symmetric static distributions of mass, d) Oppenheimer-Volkoff equations, e) Kerr's metric and its basic properties. | Link to the video of the lecture |
| 29.06 | 16) Basic cosmological models: a) cosmological principles, b) mathematical formulation of spatial homogeneity, c) Bianchi models, d) mathematical formulation of spatial isotropy, e) Friedmann-Lemaitre-Robertson-Walker cosmologies, f) Friedman-Lemaitre equations, g) basis measurable cosmological parameters, h) critical density and Omega. | Link to the video of the lecture I thank all the students for their patience, dilligence and active participation in the lectures! |