# General Relativity - a one year course

Summer semester 2011
Lectures: Paweł Nurowski, Tuesdays 16:15, KMMF, Vth floor, Hoża 74.
Tutorials: Paweł Nurowski, Tuesdays 18:15, KMMF, Vth floor, Hoża 74.

Exam: A student atending the lectures and the tutorials should not have any trouble in obtaining a very good mark at the exam. The dates of the exam will be established later.

Syllabus and the scans of instructor's notes from lectures/tutorials (Click on the blue-highlited symbols):

 Date Subject of the lecture/tutorial Remarks 17.02 (0) Models of space time - an overview; (1) Notations and basic notions from differential geometry: 1) Differentiable manifold. 2) Differentiable maps, diffeomorphisms. 3) Cartesian product of manifolds. 4) Submanifolds. 5) Vector fields. 6) Commutator of vector fields. 7) Tangent vectors and tangent space. 8) 1-parameter groups of transformations. This lecture is an overview of a basic udergraduate course on differential geometry on `naked' manifolds. Read it only if you really need it. 22.02 9) Local 1-parameter groups of transformations. 10) Integral curves of a vector field. 11) Properties of the commutator. 12)-14) Vector distributions. 15) Integral manifolds. 16) Frobenius theorem. 17) Commuting vector fields and coordinate systems; For those that know a basic differential geometry course, only the point 17), including an important corollary from the Frobenius theorem, with its proof included at the begining of the next scan, can be interesting. 1.03 (2) Tensors: 1) Tensors as multilinear maps. 2) Vector space and its dual as tensor spaces. 3) Dimensions of tensor spaces. 4) Change of basis. 5) Transformation of the components. 6) Tensors as equivalence classes; (3) Cartan's formalism of vector valued forms: 1) An action of a group on a set. 2) Transitive actions. 3) An orbit. 4)-5) Effectiveness and freeness of the action. 6) An action of GL(n,R) on the space of bases. 7) GL(n,R) acting on a Cartesian product of the space of bases and a (given) representation space of GL(n,R): tensors (revisited), tensor densities of a given weight (Levi-Civita symbol, determinant of the metric), pseudotensors. The begining of this lecture/tutorial is standard. A new, and very important material, starts at point (3). It is the point (3) of the scans, where the material required at the exam starts. 8.03 8) Covariant tensor fields. 9) Antisymmetric covariant tensor fields. 10) Exterior differential. 11) Interior product (hook operator). 12) Derivations of Z-graded algebras. 13) Lie derivativve of tensor fields. 14) Algebraic formula for the exterior derivative. 15) Local anholonomic frames and the Frobenius theorem revisited. Recommended reading: a) A. Trautman's article on how the notion of Lie derivative evolved. b) This could be followed by the article of Frölicher and Nijenhuis, about the structure of the algebra of derivations of the Cartan algebra. Both Trautman's and FN's articles show that there are nontrivial derivations of degree p>1. c) A student interested in forms of negative degree can look here, or in my article with D. Robinson here. d) A question: if there are forms of negative degree, what happens with the argument that there are no derivations of the Cartan algebra of degree smaller than -1? 15.03 15) Local anholonomic frames, Maurer-Cartan theorem, Frobenius theorem again, theorems of Pfaff and Darboux. 16) An example of Exterior Differential System, its integration. 17) Liouville's equation and its integration. 22.03 19) Coordinate (holonomic) frames. 20) Representation-valued forms. 21) Examples of such forms. (4) Covariant differentiation. 1) How to differentiate a vector field? 2) Linear connections and their curvatures. 3) Covariant exterior differential. Torsion and Ricci formula. 4) Bianchi identities. 5) Cartan's structure equations and their integrability conditions. 6) Covariant derivative. 29.03 7) Covariant differentiation; Koszul notation and axioms. 8) Torsion tensor. 9) Curvature tensor. 10) Parallel transport. 11) Selfparallels. (5) Riemannian manifolds. 1) Geodesics. 5.04 2) Christoffel symbols and their properties 3) Levi-Civita connection. Determination of a connection in terms of the torsion and nonmetricity. 4) Riemann tensor and its symmetries. 5) Riemann tensor symmetry implied by the IInd Bianchi identity. 6) Metricity of the connection and parallel transport. A particular thing should be remembered: when calculating the curvature of a given metric, it is good to chose the most convenient frame. Not always the coordinate frame is the most easy to handle with. In many cases the use of an orthonormal (or a null) frame is more convenient. In these frames the metric coefficients are constants. In particular, the counterparts of the Christoffel symbols in such frames are called Ricci rotation coefficients . 12.04 7) Vanishing of the Riemann tensor. 8) Riemann tensor and isometries. 9) Decomposition of the Riemann tensor onto irreducible parts. 10) Weyl tensor and conformal invariance. 11) Contractes second Bianchi identity. Einstein tensor. (6) Formulation of General Relativity. 1) What ingredients every relativistic theory of gravity should have? 2) The model. 3) Movement of free particles. 4) Newtonian limit for the equation of a free paticle. It is very important that students read a book of W. Kopczynski and A. Trautman `Spacetime and gravitation', especially its parts devoted to pre-Einsteinian theories of space-times, as well as the chapter summerizing Einstein's path to his field equations. 19.04 5) Einstein's field equations. 6) The Einstein Universe. 7) Linearization of Einstein's equations. 8) Newtonian limit of Einstein's equations. 10.05 9) Tidal forces - Jacobi equation. 10) How the field equations determine the motion of matter? 11) Continuity equation in GR. 12) Gravitational waves in linearized theory. 13) Evolution of a ball of dust in the field of monochromatic gravitational wave. (7) Symmetries. 1) Transformation groups. 2) G-invariant tensors. 3) Killing fields and the Killing equation. 4) Maximal number of symmetries. Spaces of constant curvature. 5) DeSitter and anti-DeSitter spaces. 17.05 6) Stationary and static gravitational fields. 7) Spherical symmetry. 9) Integration of the spherically symmetric Ric(g)=0 equations. Schwarzschild metric. 24.05 10) Maximal analytic extension of the Schwarzschild metric. Kruskal diagram. 11) Radial free fall in Schwarzschild spacetime. (8) Basic physical properties of the Schwarzschild spacetime. 1) Equations of motion of a test particle in Schwarzschild. 2) Perihelion advance. 3) Deflection of light. (9) Spherically symmetric relativistic stars. 1) Matching conditions in GR. 2) Spherically symmetric, static distribution of mass. 3) Relativistic equations for stars. Oppenheimer-Volkoff equation. 4) Example: a star with constant density. 31.05 (10) Cosmology 1) Cosmological principles. 2) Mathematical formulation of spatial homogeneity. 3) Bianchi models. 4) Bianchi classification of 3-dimensional Lie algebras. 5) Mathematical formulation of spatial isotropy. 6) Tensors invariant with respect to the isotropy group. 7) Robertson-Walker metric. 8) Friedman-Lemaitre-Robertson-Walker (FLRW) cosmologies. 9) Friedman models and topology. 10) Measurable cosmological parameters. Questions for the exam. ??? To be continued... I thank all the students for their patience, dilligence and active participation in the lectures!