Physics 495

Special relativity takes seriously the concept of spacetime, a (4-dimensional) union of the usual three dimensions of space and one of time.
We will first use kinematical problems to develop our intuition to work well in spacetime. This will include creating an ability to use Minkowski diagrams, 4-vectors, differential forms, Lorentz boosts (between inertial reference frames), and both 4x4 (real) and 2x2 (complex) matrices, which will allow an introduction to simple tensor analysis.
Then we will apply that intuition to various interesting problems, including at least electromagnetism and perfect fluids, and perhaps the relativistic behavior of spin.

Some important highlights might include the following:

• Coordinates and vectors are quite different objects, and need to be distinguished carefully; 1-forms are another type of vector---the objects that appear "under integral signs." This introduces covariant and contravariant vectors.
• Velocity, momentum, acceleration, force and others are 4-vectors, a concept that is common and important in physics.
  However, some physical quantities are more complicated: energy density, the magnetic field, electric and magnetic dipoles, stress and strain for fluids, all of which involve consideration of second-rank tensors, which may be viewed as matrices, tensors, or even 2-forms.
• The generalization of the usual (3-dimensional) notions of gradient, curl, and divergence to 4 dimensions involves 2-forms and duality (first invented by physicists in the 1890's).
• Interesting consequences follow when we consider particles which have a constant, non-zero acceleration, as measured in the inertial frame in which they are momentarily at rest.
• The generalization of angular momentum to 4 dimensions is the basic structure underlying the Lorentz transformations that convert measurements from one inertial reference frame to another.
• By better understanding the Lorentz group, we can learn new things about the usual notions of spin. We can also "derive" Dirac's equation for neutrinos and electrons.
• Magnetic monopoles are much more interesting than are electric monopoles (charges), at least if they were to exist.

As discussed in more detail elsewhere, I will be using "Introduction to Special Relativity," by Wolfgang Rindler as a text. However, various handouts will be made available on many of these ideas as we proceed along.