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Relativity Theory

Log for lectures (Edwin)


Relativity, VT09

[C]=Ta-Pei Cheng, Relativity, Gravitation and Cosmology, Oxford 2006.
[D]=Lecture notes by P. Dunsby
  • Lecture 1: Introduction and overview. Chapter 1 in [C] (complimentary: Chapter 1.1-1.3 in [D]). Remark: I strongly recommend that you prepare for the lecture by reading the corresponding chapers in [C] before and after the lectures and check your understanding by trying to answer the review questions. Reference [D] provides a somewhat more concise and informal discussion of much of the course material which you might find helpful.
  • Lecture 2: Invariance of Newton's equations under Galileo transformations: what it means  in terms of formulas and intuitively. Example rotations and boosts (Sect. 2.1.1-2.1.2 in [C]).  Lorentz symmetry (Sect. 2.1.3-2.1.4). Implications of Lorentz symmetry (Sections 2.2  and 2.3.4 in [C]). Remark:  If you have not seen SR relativity before you might want to study other sources on this, e.g. the article on Wikipedia or Chapter 1 on Dunsby's web course (containing much of the material in [D]).
  • Lecture 3: Invariance of ds²=-c²dt²+dx² under Lorentz transformation (2.2.3 in [C]). Math background needed to understand Minkowski's interpretation of this: general coordinates and metric tensor (2.3.1 in [C]; I described 2.3.2 in [C] and recommend very much that you study that in detail, convincing yourselves that all claims made there are correct). Introduction to tensors in SR (parts of 10.2 in [C]). Recommended complementary reading: Chapter 2 in [D]. Remarks: i) I strongly recommend that you carefully study the material covered in todays lecture by reading it again in [C], checking all details. I also recommend you study 10.1 in [C]. 2) ii)  I think formula (2.26) in [C] should be corrected: I got just one factor gamma (and not gamma-squared).
  • Lecture 4: (Sections 2 and 10 in [C]) Vectors and tensors in Minkowski space. Proper time, 4-velocity, 4-momentum and 4-acceleratrion and their physical interpretations. Relativistic form of Newton's equation. relativistic form of Maxwell's equations. Remarks: (i) Please study on your own Box 10.3 and Section 10.4 in [C] - two homework problems tomorrow are based in this. (ii) Advise on how to prepare for KS1: I suggest that you first concentrate on the problems which I just marked in bold face on the course plan.
  • Lecture 5: Equivalence principle, its significance and implications: bending of light rays by gravity, gravitational redshift and time dilation (Chapter 3 in [C]).  Remarks: (i) I recommend that you also read the section on todays topic in Dunsby's lecture notes.  (ii) I recommend you take a look at the Wikipedia article on the EP (many interesting details on history,  discussion of various experimental tests etc.). (iii) Please note that the midterm test tomorrow will be in FB52.
  • Lecture 6: Introduction to the mathematical description of curved spaces: curved surfaces embedded in 3D Euclidean spaces. Intrinsic coordinates, metric tensor, general coordinate transformations, geodesic, curvature, spaces with constant curvature, Riemann's generalization to higher dimensions (a glimpse) (most of Chapter 4 in [C]). Remarks: (i) I recommend you fresh up your knowledge on variational calculus, checking out some more details on what is shortly described in Eqs. (4.23)-(4.27) in [C]. You can e.g. take a look in Sects. 1 and 2.1 in the following notes:  Var.pdf  (Sofia will discuss this next time) (ii) I recommend you take a look at the Wikipedia article on Gaussian curvature and explore the links from there. (iii) The chapter discussed in todays class is the key to understanding the math of GR. I recommend you study this carefully in the book and check all computations there in detail.
  • Lecture 7: Details on the geodesic equation (I plan to put out some notes on this here). Curvature K of curved surfaces ("Gauss' beautiful theorem") and examples. 2D and 3D spaces with constant curvature ([C] 4.3). GR as geometric theory of gravity: introductions to [C] Section 5.
  • Lecture 8: GR as geometric theory of gravity. Action of a particle in Newton's physics, SR and GR. Newtonian limit of geodesic equation. Tidal forces. ([C] Section 5).
  • Lecture 9: Schwarzschild spacetime and important experimental tests of GR: gravitational lensing, precession pf Mercury's perihelion, Black holes ([C] Section 6). Remarks: (i) I only described qualitatively how to solve the geodesic equation for the Schwarzschild metric - details can be found in [C] or in the last chapter of [D] (the latter I find actually somewhat more transparent). (ii) I only described qualitatively how the Schwarzschild solution is obtained - some more details will provided later in the course. If you want to see more details already now I recommend you take a look at Wikipedia's article on deriving the Schwarzschild solution. You might also want to take a look on Wikipedia's articles on the Schwarzschild metric and black holes (nice figures and historic information!). (iii) Advise on how to prepare for KS2: I suggest that you first concentrate on the problems which I just marked in bold face on the course plan. (iv) Note on how to obtain Newton's equations from the geodesic eq. using the Lagrange formalism can (soon) be found here.
  • Lecture 10: Introduction to black hole physics ([C] Section 6). Introduction to cosmology: what it is about, short review on experimental facts, short introduction to Big Bang model (Robertson-Walker metric and Friedman equation) etc.  (a more detailed discussion of all this can be found in [C], Sections 7 and 8) Remarks:  A lot of interesting material is available on www. If you want to know more about this I recommend you explore the Wikipedia pages on Black holes, the RW metric and cosmology.
  • Lecture 11: Tensors on curved spaces: covariant and contravariant vectors;  how to rise and lower indices;  coordinate transformations of vectors and tensors; covariant derivative vector fields; example: polar coordinates in the plane ([C] 10.1, 11.1.1-2; problem (11.2) (a) and (b) in [C]).
  • Lecture 12: Doppler effect in SR. Energy momentum tensor ([C] Box 10.1 and 10.4). Parallel transport. Riemann curvature tensor. (C [11.2] and [11.3]). Comments: (i) A good symmary of tensor calculus can be found in Einstein's original paper available from here. (ii) Since this is an introductory course I try to keep the math on a basic and somewhat oldfashioned level (pretty much in the spririt of Einstein's original paper).  A modern approach is based on differential geometry. If you want to get a flavor of this I suggest you take a look in Carroll's lecture notes available from here. (iii) Carroll's lecture notes are somewhat more advanded than our course book. However, I hope that you will enjoy reading them after the course.
  • Lecture 13:  Flatness theorem: significance and proof ([C] 4.2.2. and Box 11.1); Riemann curvature tensor, Ricci tensor and Ricci scalar: definition, what they are, their properties; why they are important in GR ([C] 11.3). Einstein's field equations ([C] 12.2.1). Relativistic form of Maxwell's equations, 4-vector potential, gauge invariance (10.3). Remark: if you want to read more on gauge theories I suggest you take a look on Wikipedia. Advise on how to prepare for KS3: I suggest that you first concentrate on the problems which I just marked in bold face on the course plan.
  • Lecture 15:  Sketch of how to derive Friedmann's eqs. from GR ([C] 12.4.2). Introduction to gravitational waves ([C] Section 13.1, very sketchy overview over 13.2-4). Remarks on various more advanced topics not covered in the course: 1) Using variational principles to derive GR (see appendix of Einstein's original paper). 2) Remarks on differential geometry (there is gentle introduction to this on Wikipedia). Remarks:  (i) Advise on how to prepare for the exam: as I mentioned on the course-PM, the exam will consist on two parts. Part one will be essentially on the same material as the midterm tests, and part two will be on more challenging problems and questions based in the material from the course book which we covered in the course. As I stressed several times during the course, the exam now will be quite different from old exams on the course homepage (they focused much more on computations and on SR that we did during this course).  (ii) A  concise summary of the course can be found in notes by Carroll. I also recommend the lecture notes of Carroll available from the course literature page which is on a somewhat more advanced level than the course book but which you might now appreciate. (iii) We will answer questions you might have about problems etc. in the second hour in Friday's exercise class.
 
Thank you for a pleasant course.

Edwin



Last updated: Mar 4, 2009