8.962, GENERAL RELATIVITY
Spring, 2003
Introduction to the basic concepts and techniques of General Relativity.
This is a core course for students interested
in cosmology, string theory, gravitational lensing, theoretical astrophysics,
or related subjects.
Prerequisites:
a basic facility with multivariable calculus
and linear algebra.
Texts:
-
R. M. Wald,
General Relativity
(University of Chicago Press, Chicago, 1984).
-
A. O. Petters, H. Levine, and J. Wambsganss,
Singularity Theory and Gravitational Lensing
(Birkhauser, Boston, 2001).
Supplemental References:
-
E. Bertschinger,
Notes on General Relativity
-
S. Carroll,
Notes on General Relativity
-
S. Chandrasekhar,
The Mathematical Theory of Black Holes
(Clarendon Press, Oxford, 1992).
-
J. Hartle,
Gravity: An Introduction to Einstein's General Relativity
(Addison Wesley, San Francisco, 2003).
-
C. Misner, K. Thorne, and J. A. Wheeler,
Gravitation
(W. H. Freeman and Company, San Francisco, 1973).
-
B. Schutz,
A First Course in General Relativity
(Cambridge University Press, Cambridge, 1985).
-
S. Weinberg,
Gravitation and Cosmology
(John Wiley and Sons, New York, 1972).
Instructor:
Arlie O. PETTERS:
- Office: 37-664C
- Phone: (617) 452-4640
- Email:
petters@space.mit.edu
Class Times: Tuesdays and Thursdays: 9:30 a.m. - 11:00 a.m. in
Room 4-163.
Recitation Time: Tuesdays: 1:00 p.m.
in Room 4-231 (Jung-Tsung Shen, jushen@mit.edu).
COURSE OUTLINE (tentative):
1. Introduction
- Newtonian gravity
- Special relativity
- General relativity
2. Manifolds and Tensor Fields
- Topological spaces and manifolds
- Tangent vectors, tangent maps, and curves
- Vector fields and One-Forms
- Tensors: multilinearity, fields, and coordinate transformations
- Tensor contractions
3. Spacetimes
- Metric tensors and isometries
- Causality and time orientation
- Mathematical model of spacetime
- Examples of spacetimes
4. Spacetime Geometry
- Minkowski geometry
- Levi-Civita covariant derivative
- Parallel transport, geodesics, and Killing vector fields
- Exponential map and normal coordinates
- Type changing, metric contraction, and divergence
- Riemann, Ricci, and scalar curvatures
5. Spacetime Matter Content
- Particles and energy-momentum
- Fluids: particle density and particle flux
- Dust flows and perfect fluids
- Stress-energy tensor
- Local conservation of stress-energy
6. The Einstein Equation
- The Principle of Equivalence
- Summary of general relativity's core assumptions
- Einstein's equation and the cosmological constant
- Analogies between general relativity, classical mechanics, and electromagnetism
- The Newtonian limit
7. Application I: Big Bang Cosmology
- Modeling the cosmos under spatial homogeneity and isotropy
- Einstein's equations for a Robertson-Walker spacetime
- Equations of state: matter, radiation, and the vacuum
- Dynamical equations and cosmological parameters
- Dynamics of a Robertson-Walker universe
8. Application II: Gravitational Lensing
- Spacetime geometry of a gravitational lens system
- Gravitational lenses as density perturbations
- Fermat's principle and time delay functions
- Lensed images, magnification, and flux conservation
- Critical curves and caustics
- Lensing observables
- Applications: dark matter, Hubble's constant, extra-solar planets, etc.
9. Application III: Schwarzschild Black Hole
- Schwarzschild metric
- Gravitational redshift
- Energy diagrams and classification of geodesic orbits
- Perihelion precession and Mercury
- Relativistic gravitational lensing
- The Galactic Black Hole (guest lecturer: Fred Baganoff)
10. Application IV: Gravitational Waves
- Linearized Einstein equations
- Plane gravitational waves
- Gravitational wave action on freely falling particles
- Gravitational wave polarizations
- Detection of gravitational waves: LIGO (guest lecturer: Nergis Mavalvala)
Grade:
based on problem sets (75% from regular problem sets; 25% from final
comprehensive problem set).