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General Relativity PX436: General Relativity Gareth P. Alexander University of Warwick email: [email protected] office: D1.09, Zeeman building http://go.warwick.ac.uk/px436 Wednesday 6th December, 2017 Contents Preface iii Books and other reading iv 1 Gravity and Relativity2 1.1 Gravity........................................2 1.2 Special Relativity...................................6 1.2.1 Notation....................................8 1.3 Geometry of Minkowski Space-Time........................ 10 1.4 Particle Motion.................................... 12 1.5 The Stress-Energy-Momentum Tensor....................... 14 1.6 Electromagnetism................................... 15 1.6.1 The Transformation of Electric and Magnetic Fields........... 17 1.6.2 The Dynamical Maxwell Equations..................... 17 1.6.3 The Stress-Energy-Momentum Tensor................... 20 Problems.......................................... 23 2 Differential Geometry 27 2.1 Manifolds....................................... 27 2.2 The Metric...................................... 29 2.2.1 Lorentzian Metrics.............................. 31 2.3 Geodesics....................................... 32 2.4 Angles, Areas, Volumes, etc.............................. 35 2.5 Vectors, 1-Forms, Tensors.............................. 36 2.6 Curvature....................................... 39 2.7 Covariant Derivative................................. 46 2.7.1 Continuity, Conservation and Divergence................. 47 2.8 Summary....................................... 49 Problems.......................................... 50 3 Einstein's General Theory of Relativity 55 4 The Schwarzschild Solution 60 4.1 Derivation of the Metric............................... 60 4.2 The Event Horizon.................................. 62 4.3 Kruskal-Szekeres Coordinates............................ 64 4.4 The Classical Tests of General Relativity...................... 66 4.4.1 Precession of Perihelia............................ 67 4.4.2 Gravitational Lensing............................ 68 Problems.......................................... 72 i 5 Gravitational Collapse and Black Holes 77 5.1 Gravitational Collapse................................ 77 5.2 The Kerr Black Hole................................. 79 5.2.1 Geodesics of the Kerr Metric........................ 80 5.2.2 Event Horizon and Ergoregion....................... 81 5.3 Observational Evidence for Black Holes...................... 82 Problems.......................................... 84 6 Gravitational Waves 87 6.1 Linearised General Relativity............................ 87 6.2 The Graviton is a Spin 2 Particle.......................... 89 6.3 Gravitational Radiation............................... 91 6.4 Inspiral { the fate of binary systems........................ 93 6.5 Observations of Gravitational Waves........................ 94 6.5.1 Hulse-Taylor Binary............................. 94 6.5.2 LIGO and VIRGO Collaborations..................... 95 Problems.......................................... 97 7 Cosmology 100 7.1 de Sitter Universe and Cosmological Constant................... 100 7.2 The Friedmann-Lema^ıtre-Robertson-Walker Metric................ 101 7.3 Dark Energy and Λ-CDM.............................. 105 Problems.......................................... 107 Solutions to selected problems 109 ii Preface The aim of this course is to give an account of gravity and general relativity, as I understand it. I hope that you will find the subject as fascinating as I did when I was an undergraduate. The first four weeks of the course will be devoted to an exposition of the theoretical foun- dations of general relativity leading to a derivation of the Einstein equations. This will be divided in equal parts into a discussion of the stress-energy-momentum tensor { the source of gravitation { and curvature { its influence in the gravitational field. For the remaining six weeks we will study the predictions and implications of the Einstein equations, covering the classical tests { the motion of celestial bodies and gravitational lensing {, stellar collapse and black holes, gravitational waves and cosmology. The topics chosen, and emphasis of presenta- tion, reflect my own knowledge and expertise and lie very much in the tradition of theoretical physics. At the same time, I have tried to promote and give preference to those parts of the theory that either have been tested against experiment and observation, or are the focus of current experiments. General relativity is a theory of the structure of space and time and as such makes consid- erable use of (pseudo-)Riemannian geometry. It is lamentable that though geometry is taught at school before one learns calculus, it is essentially ignored in university level physics courses until one meets general relativity. This is to the diminishment of both subjects and makes the presentation of both in a course such as this difficult. It is my conviction that the essential ideas can be gained from the example of surfaces, of widespread relevance to physics in their own right, which have the benefit of being directly visualisable. In addition to emphasising this, I have also tried to adopt a more modern approach than in many textbooks, using intrinsic descriptions in preference to the old-fashioned tensor calculus. These lecture notes have been written in an informal and colloquial style. At times the presentation is quite detailed; at others I have taken liberties. As a general rule, there is more material in these notes than I will be able to cover in the lectures, although not too much more. What may appear in the exam should be inferred from the official syllabus and what I say in lectures rather than the fullness of these lecture notes. In preparing these notes I discovered that images taken by NASA are generally not copyrighted and can be used freely for personal or educational purposes. This is a most wonderful thing and I am grateful to be able to use their inspiring images to illustrate these lecture notes. Their image galleries can be found at this url http://www.nasa.gov/multimedia/imagegallery/index.html. I am grateful to the students who took the course in previous years for their feedback and suggestions, and to my colleagues who read my notes and spotted mistakes or points for improvement, especially Nick d'Ambrumenil and George Rowlands. I shamelessly copy Newton in his Principia (1687): I heartily beg that what I have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and investigated by new endeavours of my readers. iii Books and other reading There are lots of good books. You are encouraged to read as many as you can. Some with particular relevance to the course are listed below. 1. A. Einstein, The Meaning of Relativity, Princeton University Press, Princeton, 1945. In 1921 Einstein visited the United States and gave a series of lectures at Princeton on his theory of relativity. This text is a reissue of those lectures. Einstein's theory of relativity is universally viewed as amongst the most beautiful and remarkable scientific achievements in recorded history. It is a joy to be able to read his own presentation of it. An electronic copy of the book is freely available at this url https://www.gutenberg.org/files/36276/ 36276-pdf.pdf. 2. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields: Volume 2 of the Course of Theoretical Physics, 4th edition, Butterworth-Heinemann, Oxford, 1975. A wonderful account with the highest standard of theoretical physics. It is moderately dated and gives a traditional `tensor calculus' approach to the differential geometry, but thereafter the treatment of physical consequences of the theory is excellent. If you consider yourself a theorist, then this is the book I would like to recommend. 3. M.P. Hobson, G.P. Efstathiou, and A.N. Lasenby, General Relativity: An Introduction for Physicists, Cambridge University Press, Cambridge, 2006. This was the previous lecturer, Elizabeth Stanway's, recommended textbook. Being written relatively re- cently it is modern in style and up to date. It should go without saying that it can still be recommended highly. 4. J.B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Addison Wesley, 2002. Hartle's text is one of the more recent ones and it shows in the style of presentation and how up to date the applications and examples are. I am sure some of you will really like it. 5. B. Schutz, A First Course in General Relativity, 2nd edition, Cambridge University Press, Cambridge, 2009. A standard reference for undergraduate courses in general relativity that is often enthusiastically recom- mended. 6. C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation, W.H. Freeman, 1973. The big black coffee table, as it is affectionately known, is perhaps the most widely recommended text at advanced level. Written by three who did so much to advance general relativity, and geometrical thinking in particular, it is a classic. Its major drawback, however, is its length; 1279 pages. 7. R.M. Wald, General Relativity, Chicago University Press, Chicago, 1984. One of the most widely recommended texts for graduate courses in the US, this is the book that I first read when learning general relativity. It has a more modern style than most, which is what attracted me, adopting an `abstract index notation' that I believe was advocated by Penrose. iv 8. W. Rindler, Relativity: Special, General, and Cosmological, 2nd edition, Oxford University Press,
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