This course discusses the elements of the classical general theory of relativity. It is expected that the students taking this course are already familiar with the language of the special theory of relativity, classical mechanics, and classical electromagnetism. The mathematical prerequisites include linear algebra, differential and integral calculus, vector calculus, and partial differential equations. In addition, do anticipate some messy calculations along the way.
Contents:
(Not necessarily to be strictly followed.)

1.Review of special relativity:
-A fun tour
-The Lorentz transformation
-Thomas precession as a manifestation of length contraction
-Spacetime diagram and 4-vectors
-The metric
-Geometrization of the law of conservation of energy-momentum
-Symmetry can be a severe constraint put in by hand
-Conservation law is a local thing

2.Uniformly accelerated observer, and his neighbors:
-Uniform acceleration is meaningful only to oneself
-Neighbor remaining close downstairs is not experiencing the same acceleration
-An event horizon and particle trajectories
-Nothing strange happens when falling into the event horizon
-The equivalence principle
-The gravitational redshift

3.Curvature:
-Curvature of a spatial curve
-Curvature of a curve lying on a surface
-Detecting the curvature without leaving the surface
-Concept of “connection”
-Parallel transport
-Foucault’s pendulum as a manifestation of parallel transport
-Thomas precession as a manifestation of parallel transport
-An aside on dual space, tensors, and all that
-Strain tensor, stress tensor, and stress-energy tensor
-The Riemann curvature tensor
-Geodesic deviation

4.Einstein’s field equation:
-From Newton’s law of gravitation to its relativistic generalization
-The weak field approximation and interpretation
-Gravitational waves
-The variational formulation

5.The Schwarzschild solution:
-The metric
-Precession of Mercury about the sun
-Bending of starlight near the sun
-The gravitational redshift, again
-The event horizon
-Nothing strange happens when falling into the event horizon
-Black holes

6.Cosmology:
-The Friedman-Robertson-Walker metric
-The expanding universe