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 will include the following:

• Although not usually explained in elementary physics, coordinates and vectors are quite different objects---usually having different dimensions---and therefore need to be distinguished carefully. There are even two sorts of vectors, i.e., objects which behave in some linear way:
  1. the first sort is often referred to as tangent vectors, i.e., tangents to a curve
  2. the second sort are often referred to as co-vectors, or 1-forms, which are the objects that appear under integral signs, also a linear process.
  This approach allows us to introduce covariant and contravariant vectors in a useful way.
  Examples of the two kinds of vectors are velocity and momentum or the electric and magnetic fields.
• We will spend some time talking about the difference between what is measured, and what is "seen," using light rays, and about the unexpected consequences this can have for distant, large objects. There are some interesting movies that have been created for this as well.
• 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). This will also allow us to give useful, clear definitions for length, distance, area, volume, and (4-dimensional) hypervolume.
• Interesting consequences follow when we consider particles which have a constant but 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 will talk some about the vector spaces whose elements are called "spinors," and can use that to "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.

We will be using two different texts, for relatively different purposes.

I find that more advanced students have often gone rapidly through an introduction to the formulae involved with very fast motions, i.e., special relativity, without having spent, perhaps, sufficient time thinking about basic underlying questions involved. The text "Relativity: Special, General, and Cosmological" (2nd Edn.) by Wolfgang Rindler is very useful for spending additional time considering such questions.

The difference between tangent vectors and (differential) 1-forms is actually very important even though it is usually glossed over for undergraduate students. This and other interesting questions are considered in detail in our second text, "A First Course in General Relativity," (2nd Edn.) by Bernard Schutz. Despite the title, the first 4 1/2 chapters provide quite a useful introduction to mathematical techniques that are quite useful in special relativity.

Lastly, there will also be various handouts that will be made available on many of these ideas as we proceed along.

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Last updated/modified: 16 July, 2009