Welcome to the Home Page for Einstein's Relativity
the class will run Monday, Wednesday and Friday 10:00 AM - 11:00 AM, in Room 184
and also a problem session on Monday, Physics 451-012 from 2:30 PM - 3:30 PM, in Room 184;
Office Hours: M, W, and F 11 AM - noon; Monday 1-2 PM
but I will always talk with a student in the class at any time I am found and free
(1879 - 1955)
for two black holes
colliding to become one
Einstein with Tagore
General Relativity is often described as "the most beautiful physical theory ever invented."
Therefore, consider enrolling in this course, and we will discuss at least some of the reasons why this comment is thought to be true.
This class will provide an overview of both of Einstein's theories of relativity:
This class will not completely prepare you for research
in this area: it will be an overview with insufficient depth for that purpose.
|| special relativity, which describes the requirements on elementary physics placed by the experimental proofs that we live in a universe where the notions of time (temporal progression) and (3-dimensional) space must be combined to describe observations of our 4-dimensional universe, which we call spacetime.
As well, it tells us that the distinction between these two notions is observer dependent!
|| and his theory of general relativity, which tells us how to understand the forces we describe as gravity, in our 4-dimensional spacetime|
We will concentrate on the details involving Einstein's equations, black holes, and the gravitational waves which have been recently observed. Perhaps there will be time for a little discussion of cosmology?
However, that is more likely than not exactly what you wanted anyway.
|| I assume you have a good foundation in standard undergraduate physics:
classical mechanics, electromagnetism, and the usual sophomore-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 studies on that before, except for linear algebra.
Our discussions on special relativity will take you from the material in introductory physics to becoming familiar with the use of 4-vectors in spacetime and the use of Minkowski diagrams to better visualize what is the underlying physics.
|| Undergraduates should register for Phys. 480-001, while graduate students should register for Physics 581-001.
Everyone will be involved in the same weekly lectures; however requirements for graduate students will be somewhat more demanding, particularly in that there will be some few special assignments only for them. |
It would be very helpful for everyone to also register for the 1-hour per week problem session, which is graded CR/NC, Physics 451-012.
Various sorts of things will happen there, varying from students working through problems, perhaps assigned in advance, to my trying to explain some things that have been unclear before.
Textbooks and Syllabus:This is the first time an attempt to teach all of Einstein's relativity theories to both undergraduate and graduate students, simultanously, has been tried at UNM. Therefore we all, together, will be making some attempts to create it as we go along,
with my own leadership!
I will be creating a general Syllabus for the course presented at this link, but it will be being posted as we go along. We will surely begin with the 4-dimensional approach to special relativity, and proceed onward.
I do believe that it is useful to have a printed textbook, and possibly several for this sort of a course that is actually an overview of several (related) subjects.
I have asked the bookstore to provide for sale Thomas Moore's book, A General Relativity Workbook, published by University Science Books.
I have studied this text, and believe that his approach might well be a good one to follow, although I will include some of my own pdf-handouts as well.
"I will indeed follow the order in this book, although perhaps loosely, including my own handouts as I deem useful. You will certainly need to acquire a copy and work through it thoughtfully, attempting to follow his approach as a work book. This definitely includes filling in the so-called "Boxes" that he provides liberally.
My own handouts will be accessible from this website.
Thomas Moore's book has an introduction, and 38 additional chapters. He suggests that one could consider those on
Gravitational Waves, Cosmology, and Spinning Black Holes as possibly optional, and says that he typically covers one chapter each lecture period. Of these last 3 optional ones, I have mentioned them just above in the order that I see as most important for our course; i.e., Grav. Waves first, Cosmology second, etc.
Given the recent observations of gravitational waves, I feel we should aim toward understanding them as much as, and as soon as, possible!
We will have 45 meetings; therefore I hope to follow Moore's approach, ordering of topics, and speed, although the speed will definitely depend on the students in the class.
We will often have discussions during the weekly problem session, as to how and where to go.
Lastly, there are a great many textbooks discussing both special and general relativity. Over many years I have personally used several different ones as textbooks for classes on this material. All are of course reasonably different from each other; therefore, I strongly encourage you to consider several of them, at least briefly, in an attempt to find a writer who presents things in a way consistent with your own way of thinking.
It is certainly important to be reading, and questioning, from more than one presentation of the material!
Since reading from many different points of view is a very good thing, I
have also appended two different lists.
- The first is a
listing of some books at a "popular" level, that concern ideas in general relativity.
- The second is a listing of technical books, that are certainly useful, but, as already stated, appeal to various different modes of thinking about the material. All would require some work to use as a text. Some I may occasionally refer to for additional thoughts and insight on a subject.
As already stated, I will be following the text and also the
handouts of my own creation, presented as pdf-files on this website, which should be read, at least more or less, in the order listed below.
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.
- A summary of special relativity, i.e., properties of 4-dimensional spacetime,
along with a very useful summary of many conventions about notation that will be used in class, 25 pages.
- Minkowski diagrams, some help and examples.
- Some notes on the Lorentz group and its subgroup,
rotations in 3-space 28 pages.
- A useful summary of the
Lorentz transformations of several useful physical quantities, 6 pages.
- While I have discussed some notations for vector spaces and matrices in the review of special relativity, it is hoped that
a complete summary of such things, including comments about the Levi-Civita symbol and determinants of matrices, will be useful and so is given here: Introduction and
Conventions on Vectors, Tensors, and Matrices, 23 pages.
- Introductory comments about tidal gravitational forces,
and geometry, 11 pages.
- In some sense all of the above handouts have dealt with review, or physical motivation. At this
point we begin considering the mathematical needs for general relativity!
Tangent Vectors and
Differential Forms over Manifolds 33 pages.
- Important notes on
Covariant Derivatives and Curvature; 73 pages.
This contains the physical interpretations for the ideas named in the title, and is extremely important.
It also has the clearest discussion of (non-holonomic) tetrad basis sets, best for physical interpretations of components of vectors.
- Important notes on
Structure of the Riemann Curvature Tensor; 14 pages.
- The Lie Derivative on Manifolds, and symmetries of the manifold as generated by Killing Vectors; 16 pages.
- Various very recent papers concerning the January, 2016 observations by LIGO of gravitational waves
emitted as two black holes merged:
From here on, the handouts consider various specific applications to physical systems.
- 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.
- An application for the "Guess" method for calculating affine connections from the orthonormal bases for 1-forms,
which shows how it does NOT work for the Kerr metric.
- 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).
- A discussion of Lie
derivatives and Killing vectors; 15 pages
- Beginnings of calculations for Weak Gravitational Fields
- Basic Reviews of Gravitational Waves, by Eanna Flannagan.
- Notes on Robertson-Walker Spacetimes: 9 pages.
- Recent Discussions of Current State of Cosmology, by a practicing relativist: George Ellis:
Exams and Homework Assignments: There will be two
In addition, there will be (more or less) weekly homework assignments,
with solutions posted after they have been turned in.
The grader is
Jaksa Osinski, who can be
emailed by clicking on his name. He will be at our weekly problem sessions, for help,
and will have an office hour once per week, on Mondays, from 11-12 AM, in the department lobby.
The problem sessions are very useful to acquire a complete understanding of material for this course.
We usually work out problems, at the blackboard, that are helpful. A listing of those is given here, after they are completed in the sessions:
- First actual session, Mon., 27 Aug.: No. 1
- Second session, Mon. 10 Sept.: No. 2
- Third session, Mon. 17 Sept.: No. 3
- Fourth session, Mon. 24 Sept.: No. 4
- Fifth session, Mon. 1 Oct.: No. 5
- Sixth session, Mon. 8 Oct.: No. 6
- Seventh session, Mon. 15 Oct.: No. 7
- Eighth session, Mon. 22 Oct.: No. 8
- Ninth session, Mon. 29 Oct.: No. 9
- Tenth session, Mon. 5 Nov.: No. 10
- Eleventh session, Mon. 12 Nov.: No. 11
- Twelfth session, Mon. 19 Nov.: No. 12
Worldwide Relativity Information Sites
- The Albert Einstein Institute, at the
Max-Planck Institut in Potsdam, Germany.
- the most important journal in the field is
Classical and Quantum Gravity, published by the Institute of Physics in Great Britain.
- 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
black hole can be found on the
website, maintained by Andrew Hamilton (Univ. Colo.). It also
has many other interesting links about special and general relativity.
- 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.
- 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
- Freely Falling, or Supported
Charges Do Not Radiate in a Gravitational Field: this does not violate the
This paper seems to provide a correct answer
to an old and somewhat controversial question.
Some earlier responses to this question are in
- 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.
- Very interesting
history of Gravity Probe B, As well, there is. a review article from IOP Science
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
at Livingston, LA, and Hanford, WA.
Last updated/modified: 5 December, 2018