This book has resulted from a course in the general theory of relativity at the University of Oslo where the author has lectured for more than twenty years. Although the text is designed for master students, it is rather self-contained. Since mathematics courses on differential geometry and tensor calculus usually employ a rather abstract notation different from the component notation used in physical applications, the book introduces not only an introduction to the physical principles of the theory and physical applications of the theory, but also introduces the mathematics which is needed, in particular the calculus of differential forms. Detailed calculations are given of the bending of light, the perihelion precession of Mercury and the predictions for the Hafele-Keating experiment. The Tolman-Oppenheimer-Volkoff equation is deduced and solved for an incompressible fluid to give the internal Schwarzschild solution. Rotating black holes are discussed. The Friedmann-Robertson-Walker universe models are deduced. Also the reader will become familiar with the Universe model which is now considered as the standard model of the universe; a flat model filled with vacuum energy and cold matter. The inflationary era at the first moment of the history of our universe is also discussed.

Klappentext

This book collects lectures on the general theory of relativity given by Dr. Øyvind Grøn at the University of Oslo, Norway. This accessible text allows students to follow the deductions all the way throughout the book.

Inhalt

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.