Bryn Mawr College, Department of Physics
Physics 325: General Relativity, Spring 2012
Michael Schulz
Office: 340 Park Science Center
E-mail: mbschulz at brynmawr dot edu
Office phone: (610) 526 - 5367
Lecture (Park Science Center, Room TBD):
Tuesday, 11:15–12:45pm
Thursday, 11:15–12:45pm
Recitation section (Park Science Center, Room 336):
Thursday, 9:10–10:00am
Office hours:
Monday, 11:10–12noon
Tuesday, 6:10–7pm
Wednesday, 12:00–12:30pm, 1:30–2pm
Thursday, 12:45–1:45pm
Course webpage:
Course description:
A tentative schedule can be found on the calendar and assignments webpage.
There is a timeless beauty in Einstein's theories of special and general relativity. In special relativity, we start with two simple, empirically based assumptions: (i) the speed of light c is the same in all inertial frames, and (ii) the laws of physics are the same in all inertial frames. From these assumptions, we unify matter and energy, space and time, electromagnetism and mechanics, all in one fell swoop.
General relativity adds new depth to this beauty. Having achieved a peaceful coexistence of mechanics and electromagnetism, the next task is to incorporate gravity. Newtonian gravity, with its instantaneous "action at a distance" is incompatible with special relativity, and the solution is general relativity: We ask that the laws of physics be written in a form that is the same not only in inertial frames, but in all reference frames, including accelerating ones. In the course of this rewriting, we deduce an amazing thing — gravity can be reinterpreted as the curvature of spacetime. And this spacetime is a dynamical fabric, not a fixed backdrop. It warps and bends and evolves in time in response to the matter moving through it. The experimental predictions are equally spectacular: gravitational redshift, gravitational deflection of light, precession of planetary orbits, black holes, frame-dragging, big-bang cosmology, and gravity waves, to name a few. In addition, general relativity is well tested. It is used every day in the functioning of GPS, and is essential for understanding accretion disks, pulsars, and a host of other astrophysical phenomena.
In Physics 325, we will investigate the theory and experimental status of general relativity, adopting an "explore first, derive later" approach. This approach allows us to see the most interesting aspects of the theory as soon as possible.
Part I: Geometry, Newtonian Mechanics, Special Relativity and the Equivalence Principle. To set the stage for things to come, we'll start with the following topics, at least some of which should be familiar to you: Newtonian mechanics, Newtonian gravity, the variational principle, Galilean relativity, special relavity, spacetime diagrams, puzzles, paradoxes and gedanken experiments. We'll also talk a bit about what we mean by geometry, and what different kinds of geometries there are. As a bridge to general relativity proper, we'll then turn to the equivalence principle and see how to rescast Newtonian gravity in spacetime terms, which is a limiting case of general relativity.
Part II. The Curved Spacetimes of General Relativity. Solving the Einstein Equation — the defining equation of general relativity — is exceedingly difficult, but fortunately, most of our understanding of general relativity comes from handful of important solutions. The Schwarzschild solution describes the geometry outside of a spherical star or black hole. The Friedmann-Robertson-Walker solutions describe expanding (or crunching) universes of maximal spatial symmetry, the basis for cosmology. Gravitational waves are a third class of solutions. We'll devote a fair amount of time to developing an intuition for these spacetimes and to the behavior of observers within them. This will sharpen our understanding of metrics, gravitational redshift, geodesics, horizons and causal stucture, as summarized in Penrose diagrams. We'll also have a chance to relate the Schwarzschild solution to tests and applications of general relativity, including the two that provided its initial experimental confirmation: the deflection of startlight by the Sun and the procession of the perihelion of Mercury's orbit.
Part III. The Einstein Equation. Having explored the important spacetimes, it's time to finally gear up with some mathematical tools and study the equations of which these spacetimes are solutions. We will study the curvature tensor, geodesic deviation and vacuum Einstein Equation, which we will solve to derive the Schwarzschild solution.
Part IV. Additional topics. There are a few possibilities for the final part of the course. We can decide among them based on time and class interest: (1) We might want to extend our study of black holes to include the Kerr rotating black hole, along with its ergosphere and Penrose processes, and at a more qualitative level, charged black holes. Or we could explore black hole thermodynamics — for example the various approaches to Hawking radiation and Bekenstein-Hawking entropy. (2) Alternatively, we could extend our study of Einstein's equations to include sources, and then apply what we've learned to either of the following: (a) We might delve deeper into gravitational waves, studying their emission, for example, from binary stars. (b) Or, we could pursue any number of applications to cosmology, for example, a simple model of inflation.
Textbook:
James B. Hartle, Gravity: An Introduction to Einstein's General Relativity Addison Wesley (2003), ISBN 0-8053-8662-9. This textbook has a companion website containing Mathematica notebooks, web supplements, color images, and useful links, as well as a list of errata.
On reserve in Collier Library:
Bernard F. Schutz, A First Course in General Relativity
Sean Carroll, Spacetime and Geometry: an Introduction to General Relativity
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, GravitationThe first is an introductory book at a similar or slightly higher level than Hartle. It's the recommended first place to look if you want an alternative point of view. The second is a modern textbook at an intermediate level (graduate or advanced undergraduate). It's based on a set of free lecture notes available here. It does a good job of maintaining a readable, conversational tone, while treating the major topics and keeping the page count down. The third, weighing in at 1300 pages, is the bible. Published in 1973 and still a classic, it is a comprehensive yet accessible account written by three masters of the field and infused with their unique creative imprint.
Format:
We'll adopt the philosophy that class time is best spent bringing the material in your textbook to life through discussion and problem solving. I'll try to restrict lecturing to hitting the highlights of your reading assignments and to "big picture" ideas. For that reason, it will be very important for you to stay on top of the reading. Most of the class time will be devoted to working through problems.
Grading:
The grade is entirely determined by the weekly problem sets. There are no exams and there is no final project. However, there will be three review problem sets. These three problem sets will be shorter than the others, but will cover a broad range of topics and will count twice as much as the other problem sets. In the event that you are on the border between two grades, good class participation and other indications of effort throughout the course will help to justify the higher grade.
I hope that there will be much discussion both inside and outside of class. You are allowed (and encouraged!) to work on the problem sets together and to form study groups. The solutions you submit must of course be prepared yourself and not be reproductions of other people's work.
Accommodations:
Students who think they may need accommodations in this course because of the impact of a learning, physical, or psychological disability are encouraged to meet with me privately early in the semester to discuss their concerns. Students should also contact Stephanie Bell, Coordinator of Access Services (610-526-7351 or sbell@brynmawr.edu), as soon as possible, to verify their eligibility for reasonable academic accommodations. Early contact will help to avoid unnecessary inconvenience and delays.