Welcome to the Home Page for Physics 570
Monday and Wednesday, 5:30 - 7:00 PM , in Room 184
(1879 - 1955)
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
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.
| 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:
- 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.
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.
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.
- Introductory comments about tidal gravitational forces,
and geometry, 9 pages.
- A brief review of special relativity,
along with some notational conventions, 19 pages.
- A useful summary of the
Lorentz transformations of several useful physical quantities, 4 pages.
- Introduction and
Conventions on Vectors,
Tensors, and Matrices, 15 pages.
- Tangent Vectors and
Differential Forms over Manifolds 33 pages.
- Important notes on
Covariant Derivatives and Curvature; 72 pages.
- 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.
- 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.
- A Penrose conformal diagram for the Reissner-Nordström manifold.
- Discussion of observations
made by a uniformly accelerating observer; 15 pages.
- The Kerr metric, for
rotating stellar objects: some rather brief listings of properties
and equations; 4 pages.
- 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).
- Some notes on the Lorentz group and its subgroup,
rotations in 3-space 27 pages.
- A discussion of Lie
derivatives and Killing vectors; 15 pages
- Notes on Robertson-Walker Spacetimes: 7 pages.
- Discussions of Current State of Cosmology, by a practicing relativist: George Ellis:
- 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.
Usable Maple files are downloadable; they
require a right-click on the link, and then choosing "Save link as ...".
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 and Solutions are pdf-files, except when occasionally
there will be an html-file for a portion of the solutions.
be made available once the assignments have been turned in.
DUE at the beginning of the class period on the due date!
There are many modern software packages to perform tensor
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.
Worldwide Relativity Information Sites
- 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
- 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
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
They move fast; therefore, it is most interesting if you actually slowly "drag" the play button along
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.
Links to Exciting Astronomy News
Picture of the day
to the complete list of their pictures.
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
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.
Last updated/modified: 5 November, 2013