## Video Lectures on the General Theory of Relativity

### Long and shorter videos on selected topics, problems, and computer exercises.

Virtual lectures by the instructor on selected topics, problems and computer exercises, supplementing the material presented in class. This is an ongoing effort, and you should expect the list to grow in time. Next to the video links, you will find links to transparencies, Mathematica & Maxima notebooks and other relevant material used in the videos.

You can also find the videos in a youtube video list. A separate list with computer exercises using Mathematica, can be found here, and a list of videos and material introducing xAct for general relativity here. The list below contains all the videos in those lists, together with some metadata.

- Unit 1: Differentiable Manifolds
- Manifolds: Differential Manifolds, topological spaces, charts, transition functions, atlases. (Slides)
- Vectors: Vectors as tangent to curves, tangent space, coordinate basis, component transformations, vector fields, integral curves, Lie bracket, Lie derivative. (Slides)
- One forms and tensors: One forms as linear maps on TM, cotagent space, gradient of a function, coordinate bases, tensors, tensor product, contractions, (anti)symmetrization. (Slides)
- Differential forms: Differential forms and form fields, wedge product, exterior derivative, interior product, Levi-Civita tensor, duality, Hodge-star operator. (Slides)
- Maps: Maps between manifolds, and pullback/pushforward of tensors. (Slides)
- Diffeomorphisms: Diffeomorphic manifolds, pullback/pushforward of tensors, Lie Derivative, components of the Lie derivative in a coordinate basis. (Slides)
- Lie Dierivatives: Proof of \(\mathcal{L}_V W=[V,W] \), \(\mathcal{L}_V f=V(f) \). Computation of \( \mathcal{L}_V\omega\). Geometric interpretation of \(\mathcal{L}_V W\) and \( [V,W] \). (Slides)
- Integration: Integration on manifolds, orientability, partitions of unity, integration on manifolds with a metric. (Slides)
- Stokes Theorem: Manifolds with boundary, induced orientation, restriction of differential forms on boundary, Stokes theorem, examples. (Slides)
- Computer Lab: Embedding surfaces in \( \mathbb{R}^3\). Plotting surfaces, curves on surfaces and tangent/normal vectors using parametric plots. Torus, cylinder, sphere, Klein Bottle, Möbius strip. (Math nb, Math pdf)
- Computer Lab: Embedding of \(S^2\), \(T^2\), \(S^1\times\mathbb{R}\) in \( \mathbb{R}^3\). Embedding of \( dS_4\) and \( AdS_4\) in \( \mathbb{R}^5\). Computation of the pullback/pushforward operator and of the induced metrics. (Slides)
- Computer Lab: Permutation of indices, sign of permutation, (anti)symmetrization of tensors, Levi-Civita symbol. (Slides, Math nb, Math pdf)
- Computer Lab: Differential Forms' algebra using Mathematica's intrinsic functions. Computation of wedge products and duals of forms. (Math nb, Math pdf)
- Computer Lab: xTerior differential forms' computer algebra package (part of xAct). Wedge product, exterior derivative, coordinate transformations. (Math nb, Math pdf)

- Unit 2: The Mathematics of Gravity (some of...)
- The Metric: Pseudo-Riemmannian metrics, abstract index notation, index raising/lowering, otrhonormal bases, observers, intertial frames, causal structure, examples (closed timelike curves, wormhole, flat-space cosmology). (Slides)
- Affine Connections: Covariant derivative operator, relation between different operators, torsion, Christoffel symbols, metric compatibility, divergence of vector field, directional covariant derivative, parallel transport, covariant derivative defined from parallel transport. (Slides)
- Geodesics: Definition, uniqueness, affine parameter, reparametrization, extremization of length/proper time, Riemann normal coordinates, parallel transport defined from geodesics (Shild's ladder), examples (Slides)
- Curvature: Riemann tensor, symmetries, contractions (Ricci tensor+scalar, Weyl tensor, Einstein tensor), parallel transport along closed curve, geodesic deviation. Exercise: proof of Riemann's symmetries and the 2nd Bianchi identity. (Slides)
- Killing vectors: Isometries, Killing vector field, \( {\cal L}_\xi g_{\mu\nu} = \nabla_{(\mu}\xi_{\nu)}=0 \), \( \partial_\sigma g_{\mu\nu} =0 \Rightarrow \partial_\sigma\) Killing v.f., conservation of \(\xi_\mu U^\mu\) along a geodesic, conservation of \(J^\mu = \xi_\nu T^{\mu\nu} \), \( \exists\, \xi\) timelike \(\Rightarrow\) conserved \(E\). Examples: \(S^2, \mathbb{R}^2, S^1\times\mathbb{R} \), Schwarzschild. Exercise: \(\nabla_\mu\nabla_\nu\,\xi_\lambda=R^\rho{}_{\mu\lambda\nu} \xi_\rho \). (Slides)
- Computer Lab: Use of Maxima and the ctensor package to compute \(\Gamma^\mu_{\nu\rho}, R^\mu{}_{\nu\rho\sigma}, R_{\mu\nu}, G_{\mu\nu}, R, R^2,\ldots\) (slides, wxMaxima notebook, wxMaxima notebook pdf)
- Computer Lab: Use of Mathematica to compute \(\Gamma^\mu{}_{\nu\rho}, R^\mu{}_{\nu\rho\sigma}, R_{\mu\nu}, G_{\mu\nu}, R, R^2,\ldots\) (slides, Math nb, Math pdf, Template notebook for fast calculations, Template .m file for fast calculations - can be used with the cli wolframscript or for copy/paste in a Wolfram cloud noteboook)
- Computer Lab: Introduction to xTensor for symbolic calculations in General Relativity. Downloading, installing, finding documentation. Defining a Manifold with metric and its Christoffel connection. Defining tensors. Tensorial expressions and contractions. \(\delta_\mu^\nu, \delta_{\mu_1\ldots\mu_n}^{\nu_1,\ldots,\nu_n}, \epsilon_{\mu_1,\ldots,\mu_n}\). Covariant derivative \(\nabla_\mu \) and \(\partial_\mu\). Lie derivatives and Lie brackets. Exercise: Prove that \({\cal L}_{[u,v]} w = [{\cal L}_u,{\cal L}_v]w \). (Math nb, Math pdf)
- Computer Lab: Using xTensor for symbolic computations involving the Riemann tensor and its contractions. Symmetries of \( R^\mu{}_{\nu\rho\sigma}, R_{\mu\nu}, G_{\mu\nu},\ldots \). Manipulations using RiemannToChristoffel, EinsteinToRicci, WeylToRiemann, TFRicciToRicci. Commutator \([\nabla_\mu,\nabla_\nu] \) and the functions SortCovDs, CommuteCovDs. Rules and substitutions: MakeRule, IndexSet, IndexRule. Exercises: Bianchi identities, geodesic deviation equation, \(\nabla_\mu\nabla_\nu \xi_\lambda = R_{\lambda\nu\mu\rho} \xi^\rho \) (Math nb, Math pdf)
- Computer Lab: Introduction to xCoba. Use of the xCoba Mathematica package to do tensor computations with components in General Relativity. (Math nb, Math pdf)
- Computer Lab: Use of the xCoba Mathematica package to compute the components of curvature tensors, Christoffel symbols, and the geodesic equations. Download the template notebook. (Friedmann [pdf], Schwarzschild [pdf], Rindler [pdf])