Newton's law of universal gravitation.- The force law of gravitation.-
Newton's law of gravitation in local form.- Tidal forces.- The principle
of equivalence.- The general principle of relativity.- The covariance
principle.- Mach's principle.- The special theory of relativity.-
Coordinate systems and Minkowski diagrams.- Synchronization of clocks.-
The Doppler effect.- Relativistic time-dilation.- The relativity of
simultaneity.- The Lorentz contraction.- The Lorentz transformation.-
The Lorentz invariant interval.- The twin paradox.- Hyperbolic motion.-
Energy and mass.- Relativistic increase of mass.- Tachyons.- Magnetism
as a relativistic second order effect.- Vectors, tensors and forms.-
Vectors.- Four-vectors.- Tangent vector fields and coordinate vectors.-
Coordinate transformations.- Structure coefficients.- Tensors.-
Transformation of tensor components.- Transformation of basis 1-forms.-
The metric tensor.- Forms.- Rotating and accelerated reference frames.-
Rotating reference frames.- The spatial metric tensor.- Angular
acceleration of the rotating frame.- Gravitational time dilation.- Path
of photons emitted from the axis in a rotating frame.- The Sagnac
effect.- Uniformly accelerated reference frames.- Covariant
differentiation.- Differentiation of forms.- Exterior differentiation.-
Covariant derivative.- The Christoffel symbols.- Geodetic curves.- The
covariant Euler-Lagrange equations.- Application of the Lagrange
formalism to free particles.- Equation of motion from Lagrange's
equations.- Geodesic worldliness in spacetime.- Gravitational Doppler
effect.- The Koszul connection.- Connection coefficients and structure
coefficients in a Riemannian (torsion free) space.- Covariant
differentiation of vectors, forms and tensors.- Covariant
differentiation of a vector field in an arbitrary basis.- Covariant
differentiation of forms.- Generalization for tensors of higher rank.-
The Cartan connection.- Curvature.- The Riemann curvature tensor.-
Differential geometry of surfaces.- Surface curvature using the Cartan
formalism.- The Ricci identity.- Bianchi's 1st identity.- Bianchi's 2nd
identity.- Einstein's field equations.- Energy-momentum conservation.-
Newtonian fluid.- Perfect fluids.- Einstein's curvature tensor.-
Einstein's field equations.- The 'geodesic postulate' as a consequence
of the field equations.- The Schwarschild spacetime.- Schwarzschild's
exterior solution.- Radial free fall in Schwarzschild spacetime.- Light
cones in Schwarzschild spacetime.- Analytical extension of the
Schwarzschild coordinates.- Embedding of the Schwarzschild metric.-
Deceleration of light.- Particle trajectories in Schwarzschild 3-space.-
Motion in the equatorial plane.- Classical tests of Einstein's general
theory of relativity.- The Hafele-Keating experiment.- Mercury's
perihelion precession.- Deflection of light.- Black holes.- 'Surface
gravity': gravitational acceleration on the horizon of a black hole.-
Hawking radiation: radiation from a black hole.- Rotating black holes:
The Kerr metric.- Zero-angular-momentum-observers.- Does the Kerr space
have a horizon?.- Schwarzschild's interior solution.- Newtonian
incompressible star.- The pressure contribution to the gravitational
mass of a static, spherically symmetric system.- The
Tolman-Oppenheimer-Volkov equation.- An exact solution for
incompressible stars - Schwarzschild's interior solution.- Cosmology.-
Comoving coordinate system.- Curvature isotropy - the Robertson-Walker
metric.- Cosmic dynamics.- Hubble's law.- Cosmological redshift of
light.- Cosmic fluids.- Isotropic and homogeneous universe models.- Some
cosmological models.- Radiation dominated model.- Dust dominated model.-
Transition from radiation to matter dominated universe.-
Friegmann-Lemaître model.- Inflationary cosmology.- Problems with the
Big Bang models.- Cosmic inflation.
