Welcome to the Home Page for Einstein's Relativity

Fall, 2018
the class will run Monday, Wednesday and Friday 10:00 AM - 11:00 AM, in Room 184
and also a problem session on Monday, Physics 451-012 from 2:30 PM - 3:30 PM, in Room 184;
Office Hours: M, W, and F 11 AM - noon; Monday 1-2 PM
but I will always talk with a student in the class at any time I am found and free

Albert Einstein (1879 - 1955)

spacetime diagram
for two black holes
colliding to become one

Einstein with Tagore

General Relativity is often described as "the most beautiful physical theory ever invented."
Therefore, consider enrolling in this course, and we will discuss at least some of the reasons why this comment is thought to be true.

General Introduction

This class will provide an overview of both of Einstein's theories of relativity:
    special relativity, which describes the requirements on elementary physics placed by the experimental proofs that we live in a universe where the notions of time (temporal progression) and (3-dimensional) space must be combined to describe observations of our 4-dimensional universe, which we call spacetime. As well, it tells us that the distinction between these two notions is observer dependent!   
    and his theory of general relativity, which tells us how to understand the forces we describe as gravity, in our 4-dimensional spacetime
We will concentrate on the details involving Einstein's equations, black holes, and the gravitational waves which have been recently observed. Perhaps there will be time for a little discussion of cosmology?
This class will not completely prepare you for research in this area: it will be an overview with insufficient depth for that purpose.
     However, that is more likely than not exactly what you wanted anyway.

I assume you have a good foundation in standard undergraduate physics: classical mechanics, electromagnetism, and the usual sophomore-level special relativity.
Also you should have a mathematics background in calculus, differential equations, and linear algebra.
The mathematics of general relativity is differential geometry, but I am not assuming you have had any studies on that before, except for linear algebra.
Our discussions on special relativity will take you from the material in introductory physics to becoming familiar with the use of 4-vectors in spacetime and the use of Minkowski diagrams to better visualize what is the underlying physics.
Undergraduates should register for Phys. 480-001, while graduate students should register for Physics 581-001. Everyone will be involved in the same weekly lectures; however requirements for graduate students will be somewhat more demanding, particularly in that there will be some few special assignments only for them.
It would be very helpful for everyone to also register for the 1-hour per week problem session, which is graded CR/NC, Physics 451-012.
Various sorts of things will happen there, varying from students working through problems, perhaps assigned in advance, to my trying to explain some things that have been unclear before.

Textbooks and Syllabus:

  • This is the first time an attempt to teach all of Einstein's relativity theories to both undergraduate and graduate students, simultanously, has been tried at UNM. Therefore we all, together, will be making some attempts to create it as we go along, with my own leadership!
    I will be creating a general Syllabus for the course presented at this link, but it will be being posted as we go along. We will surely begin with the 4-dimensional approach to special relativity, and proceed onward.
  • I do believe that it is useful to have a printed textbook, and possibly several for this sort of a course that is actually an overview of several (related) subjects.
    Therefore, I have asked the bookstore to provide for sale Thomas Moore's book, A General Relativity Workbook, published by University Science Books.
    I have studied this text, and believe that his approach might well be a good one to follow, although I will include some of my own pdf-handouts as well.
    "I will indeed follow the order in this book, although perhaps loosely, including my own handouts as I deem useful. You will certainly need to acquire a copy and work through it thoughtfully, attempting to follow his approach as a work book. This definitely includes filling in the so-called "Boxes" that he provides liberally.
    My own handouts will be accessible from this website.
  • Thomas Moore's book has an introduction, and 38 additional chapters. He suggests that one could consider those on Gravitational Waves, Cosmology, and Spinning Black Holes as possibly optional, and says that he typically covers one chapter each lecture period. Of these last 3 optional ones, I have mentioned them just above in the order that I see as most important for our course; i.e., Grav. Waves first, Cosmology second, etc.
    Given the recent observations of gravitational waves, I feel we should aim toward understanding them as much as, and as soon as, possible!
    We will have 45 meetings; therefore I hope to follow Moore's approach, ordering of topics, and speed, although the speed will definitely depend on the students in the class.
    We will often have discussions during the weekly problem session, as to how and where to go.
  • Lastly, there are a great many textbooks discussing both special and general relativity. Over many years I have personally used several different ones as textbooks for classes on this material. All are of course reasonably different from each other; therefore, I strongly encourage you to consider several of them, at least briefly, in an attempt to find a writer who presents things in a way consistent with your own way of thinking.
    It is certainly important to be reading, and questioning, from more than one presentation of the material!
  • Since reading from many different points of view is a very good thing, I have also appended two different lists.
  • As already stated, I will be following the text and also the handouts of my own creation, presented as pdf-files on this website, which should be read, at least more or less, in the order listed below.
  • Handouts to supplement the texts: parts of the course will follow these closely.
    They are Acrobat-readable (*.pdf) files that you should print out, at appropriate times during the course of the class.
    1. A summary of special relativity, i.e., properties of 4-dimensional spacetime, along with a very useful summary of many conventions about notation that will be used in class, 25 pages.
    2. Minkowski diagrams, some help and examples.
    3. Some notes on the Lorentz group and its subgroup, rotations in 3-space   28 pages.
    4. A useful summary of the Lorentz transformations of several useful physical quantities, 6 pages.
    5. While I have discussed some notations for vector spaces and matrices in the review of special relativity, it is hoped that a complete summary of such things, including comments about the Levi-Civita symbol and determinants of matrices, will be useful and so is given here: Introduction and Conventions on Vectors, Tensors, and Matrices,     23 pages.
    6. Introductory comments about tidal gravitational forces, and geometry, 11 pages.
    7. In some sense all of the above handouts have dealt with review, or physical motivation. At this point we begin considering the mathematical needs for general relativity!
      Tangent Vectors and Differential Forms over Manifolds     33 pages.
    8. Important notes on Covariant Derivatives and Curvature; 73 pages.
      This contains the physical interpretations for the ideas named in the title, and is extremely important.
      It also has the clearest discussion of (non-holonomic) tetrad basis sets, best for physical interpretations of components of vectors.
    9. Important notes on Structure of the Riemann Curvature Tensor; 14 pages.
    10. The Lie Derivative on Manifolds, and symmetries of the manifold as generated by Killing Vectors; 16 pages.
    11. Various very recent papers concerning the January, 2016 observations by LIGO of gravitational waves emitted as two black holes merged:

      From here on, the handouts consider various specific applications to physical systems.

    12. A summary of local properties of spherically symmetric, static spacetimes; 9 pages;
      and also some notes on the Kruskal extensions.
      and some figures showing light cones along radial, inward trajectories in both Schwarzschild and Kruskal coordinates,
      as well as a Maple file that can be downloaded and run, showing radial, inward timelike trajectories in considerable detail.
    13. A Penrose conformal diagram for the Reissner-Nordström manifold.
    14. Discussion of observations made by a uniformly accelerating observer; 15 pages.
    15. The Kerr metric, for rotating stellar objects: some rather brief listings of properties and equations; 4 pages.
    16. An application for the "Guess" method for calculating affine connections from the orthonormal bases for 1-forms,
      which shows how it does NOT work for the Kerr metric.
    17. the important, original paper on rotating black holes:
      Rotating Black Holes: Locally Nonrotating frames, energy extraction, and scalar synchrotron radiation
      by James M. Bardeen, William H. Press, and Saul A. Teukolsky, The Astrophysical Journal, 178, 347-369 (1972).
    18. A discussion of Lie derivatives and Killing vectors; 15 pages
    19. Beginnings of calculations for Weak Gravitational Fields
    20. Basic Reviews of Gravitational Waves, by Eanna Flannagan.
    21. Notes on Robertson-Walker Spacetimes: 9 pages.
    22. Recent Discussions of Current State of Cosmology, by a practicing relativist: George Ellis:

    Exams and Homework Assignments: There will be two examinations,
    In addition, there will be (more or less) weekly homework assignments, with solutions posted after they have been turned in.
    The grader is Jaksa Osinski, who can be emailed by clicking on his name. He will be at our weekly problem sessions, for help, and will have an office hour once per week, on Mondays, from 11-12 AM, in the department lobby.

    Homework Assignments Due Date Homework Solutions
    HW #1, due Monday, 27 August Solutions for HW #1
    HW #2, due Friday, 31 August Solutions for HW #2
    HW #3, due Friday, 7 September. Solutions for HW #3
    HW #4, due (Wednesday) 12 September. Solutions for HW #4
    HW #5, due (Monday) 17 September. Solutions for HW #5
    Exam 1, (Friday) 21 September. Solutions for Exam 1
    HW #6 due (Wednesday) 26 September Solutions for HW #6
    HW #7 due (Monday) 1 October Solutions for HW #7
    HW #8 due (Monday) 8 October Solutions for HW #8
    HW #9 due (Monday) 15 October Solutions for HW #9
    HW #10 due (Friday) 19 October Solutions for HW #10
    HW #11 due (Wednesday) 24 October Solutions for HW #11
    HW #12 due (Wednesday) 31 October Solutions for HW #12
    HW #13 due (Wednesday) 7 November Solutions for HW #13
    HW #14 due (Wednesday) 14 November Solutions for HW #14
    HW #15 due (Wednesday) 21 November Solutions for HW #15
    Exam 2 will be (Wednesday) 28 November Solutions for Exam 2
    Last Day of Class 7 December

    The problem sessions are very useful to acquire a complete understanding of material for this course.
    We usually work out problems, at the blackboard, that are helpful. A listing of those is given here, after they are completed in the sessions:

    1. First actual session, Mon., 27 Aug.: No. 1
    2. Second session, Mon. 10 Sept.: No. 2
    3. Third session, Mon. 17 Sept.: No. 3
    4. Fourth session, Mon. 24 Sept.: No. 4
    5. Fifth session, Mon. 1 Oct.: No. 5
    6. Sixth session, Mon. 8 Oct.: No. 6
    7. Seventh session, Mon. 15 Oct.: No. 7
    8. Eighth session, Mon. 22 Oct.: No. 8
    9. Ninth session, Mon. 29 Oct.: No. 9
    10. Tenth session, Mon. 5 Nov.: No. 10
    11. Eleventh session, Mon. 12 Nov.: No. 11
    12. Twelfth session, Mon. 19 Nov.: No. 12

      Links to Worldwide Relativity Information Sites

    Links to Exciting Astronomy News

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    Last updated/modified: 5 December, 2018