The course addresses three general topics (see Figure 4):
differential geometry, special relativity, and general relativity.
Topics covered in differential geometry are:
- curves,
- curvature of curves,
- surfaces,
- first fundamental (metric) form of a surface,
- Einstein summation notation,
- Frenet frame,
- second fundamental form of a surface,
- curvature of a surface,
- geodesic of a surface,
- Christoffel symbols,
- Gauss' Theorema Egregium,
- Riemann-Christoffel curvature tensor,
- manifolds,
- tangent vector to a manifold, and
- Riemann metric geodesic of a manifold.
Notice that some standard topics from differential geometry (such as
differentiable forms) are omitted. We concentrate on
topics which we will need in the development of general relativity.
Topics covered in special relativity include:
- Newton's Laws of Motion,
- inertial frames,
- Michelson-Morley experiment,
- stellar aberration,
- Postulates of Special Relativity,
- simultaneity,
- proper time,
- timelike interval,
- spacelike interval,
- lightlike interval,
- Lorentz transformation,
- spacetime diagrams,
- Minkowski space, and
- causality.
Classical examples are
presented such as the detection of pions at Earth's surface and "putting
a 10m pole in a 5m barn."
Topics covered in general
relativity include:
- Principle of Equivalence,
- Ricci tensor,
- Einstein's field equations, and
- the Schwarzschild solution.
Though the motivation for Einstein's
choice for the field equations is weak, the remaining topics are well
motivated and the solutions are mathematically sound (though approximations
are sometimes necessary). In my opinion, the major accomplishment of
the general relativity topics is to explain exactly what Einstein's field
equations are (namely, 16 second order PDE's in 16 unknown functions)
and what it means to "solve" them (which is explicitly illustrated
in the Schwarzschild solution).