Welcome to the Home Page for Physics 570

Spring, 2014
Monday and Wednesday, 5:30 - 7:00 PM , in Room 184

Albert Einstein (1879 - 1955)
spacetime diagram
for two black holes
colliding to become one
Einstein with Tagore

First, an Advertisement for General Relativity:

Einstein's theory of general relativity is a classic example of a field theory:
    a theory describing the behavior of a field that exists at every point and every time, and its interactions.
General relativity can lay claim to (at least) three differences from most other field theories:
it is unique in that the equations of motion of particles through the field may be derived directly from the theory itself;
the field is in fact the curvature itself of the very points and times at which it is defined---via their tidal variations;
the field interacts with itself. [This is not quite unique since there are other (quantum) fields that also do this: Yang-Mills theories.]

Some reasonable understanding of this subject should actually be a part of the education of any professional physicist!
In addition, you can hardly even keep up with the Science pages of the New York Times if you don't understand the underpinnings of modern cosmological research.

This course will certainly 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.

General Introduction

The purpose of this class:
will be to learn the theory of general relativity, Einstein's theory of relativistic gravity, as well as some basic applications, including at least solar-system tests of gravitational theories, black holes, gravitational waves, and cosmology, with others possible if they can be fitted into a one-semester course.
  The first third to half of the course will focus primarily on the basic structure of the theory, with relevant physical motivation and insight thrown in along the way, and also provide a reasonable introduction to the needed mathematics. You do NOT need to already know more physics and mathematics than is described in the Prerequisite section just below. The major applications will come after that, although perhaps some discussion of motions around spherical stars, and weak gravitational waves will come in the earlier sections.
I assume you have a good foundation in standard undergraduate physics: classical mechanics, electromagnetism, and the usual junior-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: we will spend a good fraction of the first portion of the course learning the relevant differential geometry.
An extended/advanced course in special relativity is NOT necessary. Only the basic ideas of spacetime, 4-vectors, Minkowski diagrams, etc. are needed from special relativity; our time will mostly be concerned with questions involving gravitational fields in 4-dimensional spacetime.

Textbooks and Syllabus:

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. Introductory comments about tidal gravitational forces, and geometry, 9 pages.
  2. A brief review of special relativity, along with some notational conventions, 19 pages.
  3. A useful summary of the Lorentz transformations of several useful physical quantities, 4 pages.
  4. Introduction and Conventions on Vectors, Tensors, and Matrices,     15 pages.
  5. Tangent Vectors and Differential Forms over Manifolds     33 pages.
  6. Important notes on Covariant Derivatives and Curvature; 72 pages.
  7. A summary of all the different parts of the curvature tensor, and efficient modes of presentation of them is given in this link to an Acrobat file.
  8. 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.
  9. A Penrose conformal diagram for the Reissner-Nordström manifold.
  10. Discussion of observations made by a uniformly accelerating observer; 15 pages.
  11. The Kerr metric, for rotating stellar objects: some rather brief listings of properties and equations; 4 pages.
  12. 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).
  13. Some notes on the Lorentz group and its subgroup, rotations in 3-space   27 pages.
  14. A discussion of Lie derivatives and Killing vectors; 15 pages
  15. Notes on Robertson-Walker Spacetimes: 7 pages.
  16. Discussions of Current State of Cosmology, by a practicing relativist: George Ellis:

Exams and Homework Assignments: There will be a "mid-term examination" sometime soon after Spring Break, but no final examination.
In addition, there will be (more or less) weekly homework assignments, with solutions posted after they have been turned in.
The grader for the course is Ninnat Dangniam who may be found in class, if you need to set up an appointment to talk with him about grading questions.

Homework Assignments Due Date Homework Solutions
HW #1, due 29 Jan. Solutions for HW #1
HW #2, due 5 Febr. Solutions for HW #2
HW #3, due 12 Febr. Solutions for HW #3
HW #4 due 19 Febr. Solutions for HW #4
HW #5 due 26 Febr. Solutions for HW #5
HW #6 due 5 March Solutions for HW #6
HW #7 due 12 March. Solutions for HW #7
An Exam on Wednesday, 26 March, 2014 Solutions will be available here.
HW #8 due 2 April. Solutions for HW #8
HW #9 due 9 April. Solutions for HW #9
HW #10 due 16 April. Solutions for HW #10
HW #11 due 23 April. Solutions for HW #11
Usable Maple files are downloadable; they require a right-click on the link, and then choosing "Save link as ...".

Homework assignments and Solutions are pdf-files, except when occasionally there will be an html-file for a portion of the solutions.
Solutions will be made available once the assignments have been turned in.
Homework is DUE at the beginning of the class period on the due date!

There are many modern software packages to perform tensor calculations.
I prefer the program grtensor, which is described in more detail in this linked webpage.
After you have a reasonably-good understanding of how the process works, I see no reason why you shouldn't have an algebraic computing system do the work for you.

Links to Worldwide Relativity Information Sites

Links to Exciting Astronomy News

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Last updated/modified: 5 November, 2013