& Phys 640
Einstein's general theory of relativity is both an elegant field
of mathematics and a practical branch of modern physics.
From a mathematical and aesthetic perspective, it is a gold standard
for what a physical theory should be, and it has generated troves of
theorems and mathematical ideas. General relativity (GR) is also
the two great pillars of modern physics (along with quantum theory). It
has passed all
experimental tests to date and it has
successful in a wide range of applications, including precise
calculations for time dilation (the slowing down of time) required for
Global Positioning Systems, predicting the bending of light for
gravitational optics, and calculating the precession of Mercury's
orbit. At the frontiers of theoretical physics, general
relativity also explains much of what we know
universe as a whole, establishing the fundamental principles of modern
cosmology – the branch of physics devoted to the study of the
universe, including the "big bang theory" and the expansion of the
In this theory, gravity is not conceptualized as a force, but rather is
identified as curvature of a four dimensional spacetime manifold.
GR incorporates Newton's universal
law of gravitation as a special case, but it also explains and predicts
a vastly wider array of physical phenomena than Newtonian physics,
including the existence of black holes and gravitational effects on
time, even raising the possibility of existence of wormholes and other
This introductory interdisciplinary course is designed for both math
and physics graduate
students. The course is largely self-contained with prerequisties
in math and physics developed as they are needed. We will begin
with a brief
introduction to special relativity, then develop some results of
mathematical machinery of differential geometry. Topics include
the Einstein field equations, spacetime
geometry of Schwarzschild and Kerr black holes
and stars, an introduction to cosmology, and gravitational waves as
Santa Susana Hall, Room 127
web page: www.csun.edu/~vcmth00m
Office Hours: MW 4:39 to
5 p.m. & by appointment (via zoom)
||Meetings, including office
hours, will be online via zoom due to the Covid
pandemic. For this reason attendance
during the class zoom meetings is mandatory and your video camera
should be on (with only rare exceptions) so that I can see you
for the duration of the class meetings.
Math Students: at least Math 462, Math 450a, but also
are any of the courses Math 450b, Math 501, Math 570, Math 550, Math
For Physics Students: at
least Phys 402,
Phys 410, but also recommended are as many math courses as possible
||There will be a midterm and a
final exam, each contributing one-third to the total grade for the
course, with homework assignments contituting the remaining third. Plus
grades (+) and minus grades (–) will be assigned.
||Student collaboration is
encouraged, but you
should understand everything you turn in.
General relativity: an
introduction for physicists, by Hobson, Estanthiou, and
Lasenby. Additional Mathematical development will be presented in
There are many excellent books on general relativity and cosmology that
can be used
as references or for supplemental reading. Among these are:
The Geometry of Spacetime: An
Introduction to Special and General Relativity, by James Callahan
A short course in general relativity,
by J. Foster and J. D. Nightingale
A first course in general relativity,
by Bernard Schutz
Introducing Einstein's Relativity,
by Ray D'Inverno
Relativity, Gravitation and Cosmology,
General Relativity, N.M.J.
An introduction to general
relativity: spacetime and geometry, by Sean Carroll
Gravitation, by Charles
Kip Thorne, and John Wheeler
General Relativity, by Robert
Gravitation and Cosmology,
Relativistic Cosmology, G.
R. Maartens, and M. MacCallum
The Geometry of Minkowski Spacetime,
Exact Space-Times in Einstein's
General Relativity, J. Griffiths and J. Podolsky
Gauge Fields, Knots, and Gravity,
J. Baez & J.P. Muniain
The large scale structure of
space-time, by Stephen Hawking and George Ellis
Semi-Riemannian geometry with
applications to relativity, by Barrett O'Niell
General relativity for mathematicians,
by R. K. Sachs and H. Wu
Cosmology, Steven Weinberg
Optional but recommended first assignment (ungraded). Watch some
or all of the following non-technical videos created by Sean Carroll
for a lay audience from his series "The Biggest Ideas in the
Universe." (I recommend that you watch them in the order listed
below. Don't worry if you don't understand everything in them.)
The Biggest Ideas in the Universe 13. Geometry
The Biggest Ideas in the Universe 13. Geometry and Topology Q&A
The Biggest Ideas in the Universe 16. Gravity
The Biggest Ideas in the Universe 16. Gravity Q&A
Assignment 1 (Special
Relativity). Due Monday, Sept 14 by 5 p.m.
A) Frames S and S' are in standard configuration and S' has velocity v
relative to S. Frames S' and S'' are in standard configuration
and S'' moves with velocity u relative to S'. Derive the addition
of velocity formula for w, the velocity of S'' relative to S, using
only hyperbolic rotation matrices and hyperbolic trig identities.
Express your answer in terms of u, v, and the speed of light c only.
B) Using the notation of part A), prove that w < c, provided u <
c and v < c.
C) Solve problems 1.2, 1.8, and 1.14 in the textbook.
Assignment 2 Due Wednesday Sept 30 by 5 p.m.
Problem 1.3 pg 24 (Special Relativity)
Problems 2.2, 2.7 pg 50
Problem 3.3 pg 88
Click here for addtional problems for
Assignment 3 Due Monday
October 12 by 5 p.m.
Problems 3.6, 3.14, 3.19 pg 88
Problems 4.2, 4.8 pg 108
Assignment 4 Due date
October 21 by 5 p.m.
Click here for assignment
Assignment 5 Due Wednesday
November 4 by 5 p.m.
Problems 7.6, 7.7, 7.13, 7.19 pg 172
to be sent Wednesday November 11, 6:30 p.m.; returned by Friday
November 13, 7 p.m.
Assignment 6 Due Wednesday
November 25 by 5 p.m.
Problems 9.25 on pg 227, 11.1, 11.3 on pg 283, and 13.7 on pg 350
Assignment 7 Due date
Wednesday December 2 by 5 p.m.
for assignment 7 problems.
Reference article, including
historical background, for gravitational
Mathematics of Gravitational Waves
Final Exam to be sent out via email Monday December
7, 6:30 p.m.; returned by Wednesday December 9, 7:30 p.m.
CSUN Masters Theses and student
publications on GR & Cosmology
foliations of Robertson-Walker spacetime by Fermi space slices,
Masters Thesis in Mathematics, May 2011
Sam Havens, Fermi
coordinates and relative motion in inflationary power law cosmologies,
Masters Thesis in Mathematics, January 2013
James Kentosh, GPS test of
the local position invariance of Planck's constant, Masters Thesis
Positioning System Test of the Local Position Invariance of Planck’s
(with M. Mohageg), Physical Review Letters 108:110801
Jake Reschke, Geometric
extensions of Robertson-Walker spacetimes, Masters Thesis in
Mathematics, June 2016
bang geometric extensions of
(with D. Klein), Annales Henri
Poincaré, Vol. 19, p. 565-606 (2018), DOI: 10.1007/s00023-017-0634-6
Astronomy Picture of the Day Archive
Note: This syllabus is subject to
change during the semester. Any changes will be posted here and
announced in class.