The references refer to the books in the list of text books linked from the sidebar. From lecture 16 onwards the precise matching of material to lecture is schematic.
GENERAL RELATIVITY
Lecture 1: Units, c=G=hbar=1, Minkowski metric, principal of general covariance, transformation property of the metric tensor. Isometries, the example of Lorentz-Fitzgerald transformations, the isometries of the Minkowski metric. (P 227-232, d'I 58ff)
Tensors
Lecture 2: Summary of previous lecture, definition of scalars, covariant and contravariant vectors and tensors, new tensors from old: linearity, tensor product, contraction, raising and lowering. Symmetrization and anti-symmetrization. (d'I 60-64).
Lecture 3: The covariant derivative of a contavariant vector, the connection. The covariant derivative of a general tensor using Leibnitz rule. (d'I 72-74).
Lecture 4: Aside making connection with the more fundemental definition of tensors used in differential geometry. Torsion, a tensor, set to zero. (d'I 74). Explicit formula for a torsion free metric connection. (d'I 82-85).
Geodesics, the Riemann tenson, curvature
Lecture 5: Parallel transport, the geodesic equations. Geodesics minimize distance. (d'I 74-76).
Lecture 6: Definition of the Riemann tensor. (d'I 77, but note that the indices are in a different order). Explicit formula for Riemann tensor (d'I 86, note 1). Symmetries of the Riemann tensor (d'I 86). Number of components of the Riemann tensor, (W 142-146, MTW 326, given as an exercise, see also 334 and 343, note 2).
Lecture 7: Summary of previous lecture, outline description of the Weyl
tensor (d'I 87-88). Geometric interpretation of the Riemann tensor:
parallel transport around a closed loop. (W 135-136, note 3 to
appear). There is another geometric interpretation of the Riemann
tensor: geodesic deviation, I haven't covered this, but if you are
interested it is treated in a usefully off-hand way in (W 148-149).
Lecture 8 Curvature on the surface of a
two-sphere. Introduction to the geodesic coordinates (note 4 to
appear).
Lecture 9 More on geodesic coordinates (d'I
77-78). The Riemann tensor in geodesic coordinates: if the Riemann
tensor in non-zero then any coordinate choice making the metric
Minkowskian at a point must have second order corrections around that point. Note that
the symmetries of the Riemann tensor are obvious in geodesic
coordinates. The Bianchi identity (d'I 87).
The Einstein equations
Lecture 10 The
Bianchi identity. Two contractions of the Bianchi identity. The
Einstein tensor. Some physics, geodesic motion, some more physics, the
Einstein equation. (d'I 142-143).
Lecture 11 Einstein's equation
again. Discussion of the cosmological constant. The Newtonian limit,
first half of the calculation (W, P2 268-272).
Lecture 12 The Newtonian limit, the whole calculation.
Lecture 13 Review of the Newtonian limit, some comments on the cosmological constant. Killing's equation and the Killing vector. (d'I 102-103, but note that he uses the Lie derivative, which we haven't covered. W 375ff).
Schwarzschild solution and corrections to Newtonian gravity
Lecture 14 Schwarzschild solution (d'I 180ff, W 179-182).
Lecture 15 Discussion of the Schwarzschild radius (MTW part VII). The Schwarzschild solution in isotropic coordinates (d'I 189). Deviation of light by the sun (W 188ff, d'I 199-201).
Lecture 16 Further discussion of geodesic deviation of
light. The anomalous perihelion of mercury, beginning of
calculation.
Lecture 17 The anomalous perihelion of mercury.
Field theory approach
Lecture 18 Integration: the measure in curved space.
Lecture 19 More on the measure, the divergence theorem in curved space. The square root of g multiplied by the covariant derivative is the derivative multiplied by the sqyare root of g.
Lecture 20 Cancelled.
Lecture 21 Revision of Lagrangians in flat space, first for
finite dimensional systems and then for fields. The Klein-Gordon example and the Maxwell example.
Lecture 22 The Einstein-Hilbert actions for the Einstein tensor: varying g, the variation of R is a total derivative, the Palantini identity from geodesic coordinates.
Lecture 23 Further discussion of the Einstein-Hilbert action, inclusion of matter.
Lecture 24 Inclusion of matter, deriving Einstein equations and
field equations from a single Lagrangian. The Klein-Gordon example.
Problem classes
Lecture 25 Problem sheet 1.
Lecture 26 Problem sheet 1.
Lecture 27 Cancelled.
COSMOLOGY
Lecture 28 Cancelled.
Lecture 29 Contents of the universe (L, chapter 1), Obler's paradox "why is the sky dark (at night)" (L 3).
Lecture 30 Cancelled.
Roberson-Walker metric, RW-Freidman equations
Lecture 31 Isotropy and spatial homogeneity, three-spaces with
constant curvature, three possibilities, flat space, the three-sphere,
the three-hyperboloid (L 39-40). The Roberson-Walker metric.
Lecture 32 Energy-momentum for macroscopic matter, applying the
Einstein equation to the Roberson-Walker metric; three equations: the Friedman equation, the accelleration equation and the fluid equation.
Lecture 33 Physical parameters, the Hubble parameter and
Hubble's law, the decelleration parameter, redshift and the density
parameter. (L chapter 6).
Lecture 34 Solving the equations for a dust universe and the different values of k. (L 31).
Lecture 35 The closed universe and its cyclic behaviour, calculating the maximum size of a closed universe.
Lecture 36 The radiation universe. Mixed universe, eventual dust domination of the universe.
Lecture 37 The age of the Universe. The flatness problem.
Non-zero cosmological constant
Lecture 38 Working out the equations for non-zero cosmological
constant. The Einstein static universe. The de-Sitter space.
Lecture 39 More on the de-Sitter space.
Lecture 40 Relating the redshift and the decelleration parameter, the correction to Hubble's law.
Lecture 41 Discussion of the Hubble's law calculation. The age
of the universe with non-zero cosmological constant.
Lecture 42 Recap of the age of the universe. Some remarks on the history of the early universe.
Inflation
Lecture 43 The history of the universe, the cosmic microwave background.
Lecture 44 Cancelled.
Lecture 45 Inflation, solving the equations, the slow roll (M chaper 9, K&T chapter 8).
Lecture 46 Inflation, discussion, the fluctuation scale.
Problem classes
Lecture 47 Problem sheet 2.
Lecture 48 Problem sheet 2.
Lecture 49 Problem sheet 2.
Lecture 50 Problem sheet 3.
Lecture 51 Problem sheet 3.
SPECIAL TOPICS
Gravitational Radiation
Lecture 52 Introduction to gravitational radiation, the linear Einstein equations and the harmonic gauge.
Lecture 53 Obtaining the harmonic gauge. Start of plane wave solutions.
Lecture 54 Plane wave solutions, solving the equation, satisfying the harmonic gauge.
Lecture 55 The number of polarizations, residual invariance, transverse, tracefree polarization.
Lecture 56 Generation of graviational waves, solving the linearized Einstein equations.
Lecture 57 Approximate solutions for the generation of gravitational waves; couples to the quadrapole.
Kaluza-Klein theory
Lecture 58 Introduction to Kaluza-Klein theory. The Kaluza-Klein spacetime, the Killing vector, the Einstein-Hilbert action.
Lecture 59 Recovering four-dimensional relativity and electromagnetism. The scalar field.
Lecture 60 The geodesic equation, momentum as charge. Vacuum solutions, the Kaluza-Klein monopole.