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 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.
Prerequisites:	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:

• The general Syllabus for the course will be 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.
• I have chosen to use,

  as the principal text Einstein's General Theory of Relativity, with Modern Applications in Cosmology,
  by Øyvind Grøn and Sigbjørn Hervik, Springer, 2007.

  This text proceeds somewhat more slowly than some I have used in the last several years, but, nonetheless, seems quite capable of arriving at the same places in the end. I believe this will be a good change for our class, and I have agreed with myself to try much harder to follow it in more detail than I have sometimes done in the past.

  I have also asked the bookstore to stock several copies of an optional text, Gravitation and Spacetime, 3rd Edition,
  by Hans C. Ohanian and Remo Ruffini, Cambridge, 2013.

  This text takes a very different approach, preferring to put in a lot of details concerning the history of experimental results that have led up to what we now believe to be true. This may well be interesting for some of you to read.

  There are actually a few other, new textbooks in this area, which I found it difficult to decide against. Therefore I have provided a list of those other plausible textbooks below, where three different sorts of books for other reading are presented.

  I will doubtless omit some few parts of the book by Grøn and Hervik, and will also append some extra thoughts, where I wish they had been presented somewhat differently.

  These things will have a written reference made available to you,
  usually with a specific reference to some pages of one of the other texts, or
  To one of the handouts of my own creation, presented as pdf-files, which are listed below.

  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 brief listing of those few other texts that I considered, before I settled on this text by Grøn & Hervik.
  • The third is a listing of technical books, mostly not quite so recently-published, that are certainly still useful.

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.
    • Penrose conformal diagram for maximal extension of the Kerr manifold
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:
    • 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.
    • References for some Work on Cosmologies alternative to the FLRW (isotropic and homogeneous) model.
17. Some older notes on Spinors.

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.

• 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.
• 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, were made by Leo Brewin at Monash University in Australia. 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.
  They move fast; therefore, it is most interesting if you actually slowly "drag" the play button along while watching.
• 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.
  • 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.
    However, the new book by Griffiths and Podolsky, listed in my online list of other books, is much more complete on this question.
  • 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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