Homepage for Math 497C - Geometry and Relativity

This course is part of the MASS Program at Penn State







Lecture notes courtesy of Bob Yuncken.






Course Description:   Classical gravity.  Differential geometry of surfaces and higher-dimensional manifolds in space.  Gauss' theorema egregium, Gauss-Bonnet theorem.  Minkowski geometry, the hyperbolic plane, special relativity.  Riemann curvature, the Bianchi identities, general relativity Weak-field approximation, gravitational redshift, relativistic effects in the Global Positioning System.  The Schwarzchild solution, black holes, other topics as time allows.

Prerequisites:  The main prerequisite for this class is a knowledge ofAdvanced Calculus, especially techniques for manipulating partial derivatives(such as the multivariable Chain Rule), vector calculus (div, grad and curl), and integral theorems (Gauss, Green, Stokes etc).

Instructor:  John Roe, Professor of Mathematics, with the assistance of Bob Yuncken.

Meeting Times:  The class meets four times a week, on Mondays, Wednesdays, Thursdays, and Fridays from 1.25 to 2.15 p.m.   in 103 McAllister.  Mondays, Wednesdays and Fridays will be devoted to lectures, and Thursdays to a review session.

Office hours will take place Wednesday 2.15 -- 3.00 (right after class, as soon as I get back to my office) and Friday 10.15 -- 11.00.
Students are strongly encouraged to make use of available office hours to discuss any questions or problems that they may have about the course or about mathematics more generally .

Textbook: Our `textbook' will be the Dover reprint of Einstein, Minkowski and Lorenz' papers, The Principle of Relativity (ISBN 0-486-60081-5).  My aim is that by the end of the course we are able to read and to some extent understand papers III and VII   in that book - Einstein'sElectrodynamics of Moving Bodies and Foundations of the General Theory of Relativity - two of the most important physics papers ever published.

Bob Yuncken is maintaining notes from the course.  Contact him for information.

You may find the following books useful for supplemental reading, but they are not required:
 
 
Frank Morgan Riemannian Geometry: A Beginner's Guide
James Callahan The Geometry of Spacetime
Timothy Gowers Mathematics: A Very Short Introduction
Manfredo P. do Carmo Differential Geometry of Curves and Surfaces
Paul Dirac General Theory of Relativity
Arthur Eddington Space, Time and Gravitation
Edwin Taylor and John Wheeler Exploring Black Holes: Introduction to General Relativity.

Grading   Your grade will be computed on the basis of the following factors:

Course Schedule:  I have divided the course material into twelve units of about a week each.  These are listed below.  It is very likely that this plan may be modified as the course progresses.
 
 
Unit 1 Newtonian Gravity
Unit 2 Groups and Geometry
Unit 3 Surface Geometry
Unit 4 Gauss' Remarkable Theorem
Unit 5 The Gauss-Bonnet Theorem
Unit 6 Minkowski Space and Hyperbolic Geometry
Unit 7 Special Relativity
Unit 8 Higher Dimensional Manifolds
Unit 9 Curvature and the Bianchi Identities
Unit 10 General Relativity
Unit 11  The weak field approximation and the Schwarzchild solution
Unit 12 Applications of General Relativity

Academic Integrity Statement All Penn State policies regarding ethics and honorable behavior apply to this course. Academic integrity is the pursuit of scholarly activity free from fraud and deception and is an educational objective of this institution.  Academic dishonesty includes, but is not limited to, cheating, plagiarizing, fabricating of information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. For any material or ideas obtained from other sources, such as the text or things you see on the web, in the library, etc., a source reference must be given. Direct quotes from any source must be identified as such. All exam answers must be your own, and you must not provide any assistance to other students during exams. Any instances of academic dishonesty will be pursued under the University and Eberly College of Science regulations concerning academic integrity.

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