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 will
focus primarily on the mathematics and basic structure of the theory, with relevant physical
motivation and insight thrown
in along the way, with the major applications coming after that.
|| 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 is 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 a principal text, the (paperback) book, Relativity,
by Hans Stephani,
- who was, until his recent death, an important name in the understanding of
the conclusions arrived at concerning all parts of the universe, via Einstein's field equations.
In addition, I have asked the bookstore to provide for those who might want it, an optional,
additional text, A first course in general relativity, 2nd Edition, 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."
- An additional optional book that might have been of use to some is
Spacetime and Geometry: An Introduction to
General Relativity, by Sean M. Carroll.
- This book is modern and good, but has quite a different
"attitude" than our designated text, as the author comes from a background in theoretical particle
physics. It is, however, now out of print; nonetheless, it does have
its own website,
where you may find an earlier, online, free version.
doubtless omit some few parts of Stephani's book, and will also append some extra
thoughts, where I wish they had been presented somewhat differently.
- These things will have
handouts of my own creation, presented as pdf-files, which are listed somewhere below.
- A particular concept that Stephani more or less leaves out is the subject of
the vector spaces of differential forms, also referred to as p-forms.
I will use this
idea extensively; there is a handout on "vectors" which discusses it, and it is also
discussed in some detail in Schutz' book.
- 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.
0. A useful summary of the
Lorentz transformations of several useful physical quantities, 4 pages.
- Introduction and
Conventions on Vectors,
Tensors, and Matrices, 25 pages.
- Tangent Vectors and
Differential Forms 32 pages.
- Important notes on
Covariant Derivatives and Curvature; 40 pages.
- For the Sphere as a Manifold:
Good Coordinate Charts, 5 pages, with figure,
and their use to understand
the Magnetic Vector
Potential for a Magnetic Monopole, 12 pages, including the above on projective coordinates for the sphere.
by C.N. Yang on the Dirac constraint because of the existence of magnetic monopoles, 27 pages.
- Some notes on the Lorentz group and its subgroup,
rotations in 3-space 27 pages.
- A discussion of Lie
derivatives and Killing vectors; 11 pages.
- A summary of local properties of
spherically symmetric, static
spacetimes; 9 pages;
and also some notes on the Kruskal extensions.
- 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).
- Notes on Spinors and
their use to study Minkowski-signature manifolds: 33 pages
- An invited paper on colliding(plane)
gravitational waves by Valería Ferrari, presented at the 1989 conference of the
Society for General Relativity and Gravitation (GRG), in Boulder, Colorado.
- Notes on Robertson-Walker Spacetimes: 7 pages.
- Collapsing Dust Metric, from Tolman [3 pages]
- Recent Discussions of Current State of Cosmology, by a practicing relativist: George Ellis:
Exams and Homework Assignments: There will be two
"mid-term examinations" during
the course of the semester, 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 Jonathan Allen 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: 4 January, 2010