Welcome to the Home Page for Physics 570

Albert Einstein (1879 - 1955)

spacetime diagram for two black holes colliding to become one

Einstein with Tagore

First, an Advertisement for General Relativity:

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.

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, although unlikely since it is only a one-semester course. The first half will focus primarily on the mathematics and basic structure of the theory, with relevant physical motivation and insight thrown in along the way, but the main applications in the second half.
Prerequisites: I assume you have a good foundation in undergraduate physics, particularly in classical mechanics, electromagnetism, and special relativity, and a mathematics background in advanced 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 half learning the relevant differential geometry.

Textbooks and Syllabus:

• A Prospective Syllabus is given at this link. It proceeds week by week, with references to the texts mentioned below. It should be conceived as an ordered listing of the majority of things we want to talk about; however, you will see it has two different weeks scheduled as "catch-up" if and when we have fallen behind in the ordering.
• The principal text is Spacetime and Geometry: An Introduction to General Relativity, by Sean M. Carroll, which is modern and rather good. The text has its own website, where you may find various useful things, including lists of Errata, relevant since this is its first edition. As well, you may go to Sean Carroll's own website, to see other things in which he is interested.
  For my lengthier discussion of this text, please go to my listing of more detailed comments on our texts. However, I note here that we will doubtless omit some few parts of Carroll's book, and will append some few extra thoughts, where I wish they were done somewhat differently. There will be handouts of my own creation on those things, presented as pdf-files.
• An optional book available at the bookstore is A first course in general relativity, by Bernard F. Schutz. It is a very well-written, advanced-undergraduate level textbook, that goes too slowly for our purposes, but therefore is an excellent reference for more understanding of some of the details that we may have gone over "too fast."
• Since reading from many different points of view is a very good thing, I have also appended two different lists.
  • The first one is a listing of some books at a "popular" level, that concern ideas in general relativity.
  • The second is a listing of technical books, mostly not quite so recently-published, that are certainly still useful.
  • A website maintained by John Baez has a nicely-written and ordered reading list of material on relativity for undergraduates (created and maintained by Chris Hilman).

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.

0. A useful summary of the Lorentz transformations of several useful physical quantities.
1. Introduction and Conventions on Vectors, Tensors, and Matrices,     26 pages.
2. Tangent Vectors and Differential Forms     23 pages.
3. For the Sphere as a Manifold: Good Coordinate Charts and their use to understand the Magnetic Vector Potential for a Magnetic Monopole.
   Then some by C.N. Yang on the Dirac constraint because of the existence of magnetic monopoles.
4. Some (older) Notes on Rotations in 3-space   17 pages.
5. Important notes on Covariant Derivatives and Curvature; 25 pages.
6. A discussion of Lie derivatives and Killing vectors; 10 pages. It has been modified, yet again, on 23 Febr., 2007, hopefully to make things more clear.
7. A summary of local properties of spherically symmetric, static spacetimes; 8 pages;
   and also some notes on the Kruskal extensions.
8. Discussion of observations made by a uniformly accelerating observer; 15 pages.
9. The Kerr metric, for rotating stellar objects: some rather brief listings of properties and equations; 4 pages.
10. 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).
11. Notes on Spinors and their use to study Minkowski-signature manifolds: 33 pages
12. Notes on Robertson-Walker Spacetimes: 7 pages.
    • Explanation of Input Data for the Maple files below, used to integrate the elliptic equations.
    • Maple file on Calculations for FRW Cosmologies with k=+1
    • Maple file on Calculations for FRW Cosmologies with k=-1
13. Collapsing Dust Metric, from Tolman [3 pages]
14. Recent Discussions of Current State of Cosmology, by a practicing relativist: George Ellis:
    • Pedagogical discussion of horizons in FRW Cosmologies, written in the American Journal of Physics,, 1993.
    • Excellent review article on the 83 years of General Relativity and Cosmology, 1999: 39 pages.
    • More recent, quite brief summary, 2001---posing questions to be answered
    • A particular explicit model for inflation: the emergent universe, 2003.

Exams and Homework Assignments: There will be two "mid-term examinations" during the course of the semester, 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 Steve Flammia, who can be emailed by clicking on the link just previous, and who will also have some office hours for the class.

• The Albert Einstein Institute, at the Max-Planck Institut in Potsdam, Germany.
• the most important journal in the field, Classical and Quantum Gravity.
• Astrophysical Gravitational Wave Sources: NASA Data Archive, and information about LISA, the projected, orbiting gravitational wave telescope.
• Historical Exhibit on Albert Einstein, from the American Institute of Physics.
• The latest Newsletter, Gravlens, of the General Relativity Division of the American Physical Society.
• Very interesting discussion and movies of both orbiting around and falling into a vacuum, Schwarzschild black hole. Done by Andrew Hamilton, whose main home page has many other interesting links about special and general relativity and interesting things in the sky.
• Some interesting movies showing the fact that when one uses light rays to view very-fast-moving, 3-dimensional objects they appear to rotate, by Leo Brewin at Monash University in Australia, can be found at this webpage.
  I have copied two of the movies that I liked the best, which may be found here:
  • for a fast-moving Rubik's Cube moving toward you, and
  • for a fast-moving steam locomotive moving past you.
• An interesting history of the ideas in general relativity, beginning with Aristotle and Copernicus, along with many further links to biographical information on the researchers involved, can be found at this link, created by people at St. Andrews University in Scotland.
• A year-long course on General Relativity is taught every year at Cal. Tech. This link takes you to the webpage for that class, created by Marc Kamionkowski.
• Finley's page of Other Interesting Links for Relativity
• From time to time, a student asks a question which is too complicated to fully discuss in class.
  If possible I will then create a webpage with a listing of articles appropriate to answering that question.
  At the beginning of the semester there were two such listings here. Several more have been added today, 24 April, 2007.
  • Questions about "warp drives";
  • Questions concerning Reality and Stability of Black-Hole Interiors
  • Freely Falling, or Supported Charges Do Not Radiate in a Gravitational Field: this does not violate the Equivalence Principle.
    This paper seems to provide a correct answer to an old and somewhat controversial question.
    Some earlier responses to this question are in this paper.
  • Questions somewhat related to the radiation of charges are those concerning the backreaction of particles in a gravitational field. They are discussed very carefully and rigorously in this paper of Eric Poisson.
  • Is the concept of a graviton---as a spin 2, massless object "something like" a photon" really an idea consistent with full, nonlinear general relativity?
    This paper says no, although this is still a very controversial issue.
  • deSitter and anti-deSitter spacetimes are described in these 11 pages from Hawking and Ellis' book.
    This put here since there were quite a few questions about it at the last class, and there wasn't really time to discuss it more.
  • Here is a link to the webpage describing current theoretical research studies in cosmology at the University of Capetown.
  • Very interesting 14 April history of Gravity Probe B, and associated comments through the American Institute of Physics concerning what it's doing.

• Astronomy Picture of the day
  • Index to the complete list of their pictures.
• COBE satellite data on cosmic microwave background
• Very interesting background primer on Cosmology and the astronomical measurements that allow us to make inferences about it.
• The official webpage for of the LIGO observatory for gravitational waves, otherwise known as the Laser Interferometer Gravitational Wave Observatory, built and in the final testing stages now at Livingston, LA, and Hanford, WA.

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