General Relativity
(Math 527 and Physics 622)
Spring 2022
Instructor:
Paul Aspinwall
Credits: 1.0, Hours: 3.0
Time: TuTh 10:15AM - 11:30AM
Location: Physics 235
Description Introduction to the basic concepts and techniques
of General Relativity. The course will cover the fundamentals of
tensor calculus, Riemannian geometry, and Einstein's equations, as
well as applications to cosmology and black holes.
This is a core course for students who want to work in general relativity,
cosmology, gravitational lensing, theoretical astrophysics, string theory,
or related subjects.
Homework
Prerequisites A sound knowledge of multivariable calculus (at
least Math 212) and linear algebra (at least Math 218). A basic
knowledge of classical mechanics and electromagnetism is desirable
but the course will endevour to be self-contained.
Exams
A take-home final will be
given due back at noon Wednesday, April 27.
Synopsis
A rough outline is as follows.
0. Special Relativity
- Minkowski Space
- Lorentz Transformations
I. Manifolds and Tensors
- Tangent vectors and differentiable maps
- Curves, vector fields, and one-forms
- Tensor fields and the abstract index notation
II. Riemannian Geometry
- Covariant derivatives and parallel transport
- Curvature and geodesics
- Computing curvature
III. The Einstein Field Equations
- General and special covariance
- Einstein's equation
- The weak-field limit
IV. Applications
- Cosmology
- Robertson-Walker universes
- The cosmological constant (“dark energy”)
- The Schwarzschild solution
- Gravitational red shift
- Black holes
- Perihelion precession and bending of light
- The Kruskal extension
- Further Analysis of Black Holes
- The Reissner-Nordström Solution
- The Kerr Solution
- The Ergosphere
- Black Hole Thermodynamics
- Gravitational Waves
- Inflation
- Slow Roll Inflation
- Eternal Inflation and the Multiverse
Textbooks
The course will be based on the text:
- Robert M. Wald,
General Relativity, University of Chicago Press, Chicago, 1984.
See also
- Sean M. Carroll, Spacetime and Geometry, An Introduction to
General Relativity, Addison Wesley 2004. (See also
gr-qc/9712019
for what might be considered to be an earlier form of this book online.)
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