Announcements:
New homework set due next Thursday
Legendre polynomials
Context:
Einstein began by realizing that the speed of light is constant.
This is a symmetry.
What does it mean that the speed of light is constant? Can I make the same statement about the speed of a train?
The distinction between the two is that both c and v_{train} are invariant if we rotate but only c is invariant if we boost.
The symmetry needs to be defined with respect to a set of transformations.
[A sphere rotated about its center remains unchanged. A sphere translated in some direction has changed. I will be able to tell that you moved the baseball but unable to tell if you rotate it (without watching). Sphere possesses symmetry wrt rotation but not translation.]
What is a set of transformations?
Consider rotations. There is no need to use mathematics to understand that there are such things as rotations. We understand that I can rotate a book. We can take these abstract elements and describe some general properties. For example,
We can represent all of the members of the rotation group as 3x3 matrices. Get this clear the matrices have all of the properties of the abstract rotations. We will want to examine how to represent physical states and the associated transformations on these states.
Einstein then went a step further with General Relativity to describe transformations of a local nature.
Weyl extended Einsteins notion by reducing the requirements and imagining that lengths could vary in a restricted but more general way than Einstein suggested.
Weyl was wrong
Consider a general charge distribution
It is very difficult, in general, to find the electric and magnetic fields a point _{} due to a charge distribution _{}
The potential at _{}for a point charge located at _{}is
_{}
For a charge distribution defined
by primed coordinates
charge distribution |
_{} |
field point |
_{} |
_{}
_{}
Usually the student is introduced to this procedure in E&M based on the above problem but expressed in Cartesian coordinates. The more revealing expansion is done in terms of spherical coordinates because the moments can be classified by their transformation properties under rotations.
How would one describe the monopole moment in terms of rotational properties?
{independent è a rotation will not change the monopole moment}
How about the dipole?
{vector è a rotation will mix the three components}
Originally a vector was introduced as a quantity with magnitude and direction. This type of entity was important because so many physics concepts force, position … possess this structure. However another natural way to introduce a vector is to characterize by its transformation characteristics under rotations. There are objects that transform as scalers, vectors, tensors under rotation.
To return to the problem of the charge distribution we discovered that we could break a general charge distribution into parts:
Under rotations ??????