Introduction to Einstein's Theory of Relativity

From Newton's Attractive Gravity to the Repulsive Gravity of Vacuum Energy

The revised and updated 2nd edition of this established textbook provides a self-contained introduction to the general theory of relativity, describing not only the physical principles and applications of the theory, but also the mathematics needed, in particular the calculus of differential forms. Les mer
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The revised and updated 2nd edition of this established textbook provides a self-contained introduction to the general theory of relativity, describing not only the physical principles and applications of the theory, but also the mathematics needed, in particular the calculus of differential forms.Updated throughout, the book contains more detailed explanations and extended discussions of several conceptual points, and strengthened mathematical deductions where required. It includes examples of work conducted in the ten years since the first edition of the book was published, for example the pedagogically helpful concept of a "river of space" and a more detailed discussion of how far the principle of relativity is contained in the general theory of relativity. Also presented is a discussion of the concept of the 'gravitational field' in Einstein's theory, and some new material concerning the 'twin paradox' in the theory of relativity.




Finally, the book contains a new section about gravitational waves, exploring the dramatic progress in this field following the LIGO observations.



Based on a long-established masters course, the book serves advanced undergraduate and graduate level students, and also provides a useful reference for researchers.

Fakta

Innholdsfortegnelse

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.-

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