Introduction To General Relativity - G. T.Hooft

Introduction To General Relativity - G. T.Hooft -...

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INTRODUCTION TO GENERAL RELATIVITY G. ’t Hooft CAPUTCOLLEGE 1998 Institute for Theoretical Physics Utrecht University, Princetonplein 5, 3584 CC Utrecht, the Netherlands version 30/1/98
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PROLOGUE General relativity is a beautiful scheme for describing the gravitational Feld and the equations it obeys. Nowadays this theory is often used as a prototype for other, more intricate constructions to describe forces between elementary particles or other branches of fundamental physics. This is why in an introduction to general relativity it is of importance to separate as clearly as possible the various ingredients that together give shape to this paradigm. After explaining the physical motivations we Frst introduce curved coordinates, then add to this the notion of an affine connection Feld and only as a later step add to that the metric Feld. One then sees clearly how space and time get more and more structure, until Fnally all we have to do is deduce Einstein’s Feld equations. As for applications of the theory, the usual ones such as the gravitational red shift, the Schwarzschild metric, the perihelion shift and light deflection are pretty standard. They can be found in the cited literature if one wants any further details. I do pay some extra attention to an application that may well become important in the near future: gravitational radiation. The derivations given are often tedious, but they can be produced rather elegantly using standard Lagrangian methods from Feld theory, which is what will be demonstrated in these notes. LITERATURE C.W. Misner, K.S. Thorne and J.A. Wheeler, “Gravitation”, W.H. ±reeman and Comp., San ±rancisco 1973, ISBN 0-7167-0344-0. R. Adler, M. Bazin, M. Schi²er, “Introduction to General Relativity”, Mc.Graw-Hill 1965. R. M. Wald, “General Relativity”, Univ. of Chicago Press 1984. P.A.M. Dirac, “General Theory of Relativity”, Wiley Interscience 1975. S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity”, J. Wiley & Sons. year ??? S.W. Hawking, G.±.R. Ellis, “The large scale structure of space-time”, Cambridge Univ. Press 1973. S. Chandrasekhar, “The Mathematical Theory of Black Holes”, Clarendon Press, Oxford Univ. Press, 1983 Dr. A.D. ±okker, “Relativiteitstheorie”, P. Noordho², Groningen, 1929. 1
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J.A. Wheeler, “A Journey into Gravity and Spacetime, Scientifc American Library, New York, 1990, distr. by W.H. Freeman & Co, New York. CONTENTS Prologue 1 literature 1 1. Summary o± the theory o± Special Relativity. Notations. 3 2. The E¨ otv¨ os experiments and the equaivalence principle. 7 3. The constantly accelerated elevator. Rindler space. 9 4. Curved coordinates. 13 5. The affine connection. Riemann curvature. 19 6. The metric tensor. 25 7. The perturbative expansion and Einstein’s law o± gravity. 30 8. The action principle. 35 9. Spacial coordinates. 39 10. Electromagnetism. 43 11. The Schwarzschild solution. 45 12. Mercury and light rays in the Schwarzschild metric. 50 13. Generalizations o± the Schwarzschild solution. 55 14. The Robertson-Walker metric. 58 15. Gravitational radiation. 62 2
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1. SUMMARY OF THE THEORY OF SPECIAL RELATIVITY. NOTATIONS.
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Introduction To General Relativity - G. T.Hooft -...

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