General Relativity
Math 651B & Phys 640
Fall 2020

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 new theorems and mathematical ideas.  General relativity (GR) is also one of the two great pillars of modern physics (along with quantum theory). It has passed all experimental tests to date and it has been 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 about the 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 universe. 

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 space and time, even raising the possibility of existence of wormholes and other universes.

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 the essential 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 time permits.

Instructor David Klein
Santa Susana Hall,  Room 127  
Phone: (818) 677-7792
email:, web page:

Office Hours: MW 4:39 to 5 p.m. & by appointment (via zoom)

Class Meetings              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.

For Math Students: at least Math 462, Math 450a, but also recommended are any of the courses Math 450b, Math 501, Math 570, Math 550, Math 552.

For Physics Students: at least Phys 402, Phys 410, but also recommended are as many math courses as possible
Grading 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. 

Homework 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 class lectures.

Additional References

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, Ta-Pei Cheng

General Relativity, N.M.J. Woodhouse


An introduction to general relativity: spacetime and geometry, by Sean Carroll

Gravitation, by Charles Misner, Kip Thorne, and John Wheeler

General Relativity, by Robert Wald

Gravitation and Cosmology, Steven Weinberg

Relativistic Cosmology, G. Ellis, R. Maartens, and M. MacCallum

The Geometry of Minkowski Spacetime, Gregory Naber

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 and Toplogy

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 2

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 Wednesday October 21 by 5 p.m.

Click here for assignment 4 problems.

Assignment 5 Due Wednesday November 4 by 5 p.m.

Problems 7.6, 7.7, 7.13, 7.19 pg 172

Midterm 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.

Click here for assignment 7 problems.

Reference article, including historical background, for gravitational waves:  The 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

Evan Randles, Spacelike foliations of Robertson-Walker spacetime by Fermi space slices, Masters Thesis in Mathematics, May 2011

Fermi coordinates, simultaneity, and expanding space in Robertson-Walker cosmologies (with D. Klein), Annales Henri Poincaré, Vol. 12, p. 303-328 (2011),  DOI: 10.1007/s00023-011-0080-9. ArXiv:

Sam Havens, Fermi coordinates and relative motion in inflationary power law cosmologies, Masters Thesis in Mathematics, January 2013

Relative velocities, geometry, and expansion of space (with V.J. Bolós and D. Klein), Recent Advances in Cosmology, NOVA Science Publishers, Inc. (2013). ArXiv: arXive:1210.3161

James Kentosh, GPS test of the local position invariance of Planck's constant, Masters Thesis in Physics, June 2013

Global Positioning System Test of the Local Position Invariance of Planck’s Constant (with M. Mohageg), Physical Review Letters 108:110801, 2012. arXiv:1203.0102

Jake Reschke, Geometric extensions of Robertson-Walker spacetimes, Masters Thesis in Mathematics, June 2016

Pre-big bang geometric extensions of inflationary cosmologies (with D. Klein), Annales Henri Poincaré, Vol. 19, p. 565-606 (2018), DOI: 10.1007/s00023-017-0634-6, SpringerNature Open Access.

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.