The Geometric Representations of the BLL Family of Symmetry Groups

[Alfred Einstead]

I decided to give scribd out a try and (if it works out okay) to place
all my notes, expositories, some of my unpublished research, etc. on
there. The content spans a large number of fields of mathematics,
physics, computer science, logic and other fields in addition.

The following has been uploaded to Scribd:
http://www.scribd.com/doc/143478572/Geometric-Representation-of-the-BLL-Fam=
ily-of-Groups

Geometric Representation of the BLL Family of Groups
These are some notes I=92ve written up on the BLL family and their
geometric representations.

We will derive 4 and 5 dimensional coordinate representations of the
Bacry/L=E9vy-Leblond (BLL) family of groups, both with and without
central extensions. An appendix is also attached providing a
derivation of the symmetry group SL (n +1) for the law of inertia in n
dimensions.

The BLL family includes the (anti-)deSitter group, Bargmann & Galilei
groups, Poincare' group, Riemannian (4-D spherical) group,
Lobachevskian group (for 4-D negative curvature hyperbolic geometry),
Carroll group (for c =3D 0), Static group (the "non-relativistic"
version of Carroll =3D the group is associated with Hellenistic era
spacetime geometry), the Newton-Hooke (and anti-Newton-Hooke group),
Para-Poincare' (and anti Para-Poincare', sometimes also called Hooke-
Newton) and para-Galilei. I think that's all of them.

This is meant to be part of a larger series that will develop all the
major results of kinematics and dynamics in the generalized framework
of kinematics governed by BLL symmetry (e.g. a non-relativistic or
Euclidean version of the Dirac equation; fluid dynamics; QFT, etc.)

[Alfred Einstead]

On May 24, 4:52 pm, Alfred Einstead <federation2...@netzero.com>
wrote:
> I decided to give scribd out a try...
> The following has been uploaded to Scribd...

What I *really* wanted to see is whether it's one of those sites what
assume what you put on there is "ta da! All done. Lookey here!" or more
along the lines of the Web, itself, "this is where we're at right now.
Future changes/additions pending." That's something I'll check on
hopefully soon.

A second expository (and actually a redistribution of something I
posted a now-invalid link to a while back):

http://www.scribd.com/doc/144951260/Tetrad-Formulation-of-the-Einstein-Fiel=
d-Equations-The-Newman-Penrose-Equations

Tetrad Formulation of the Einstein Field Equations: The Newman-Penrose
Equations
Summary:
The Einstein field equations can be formulated in terms of null tetrads.
In turn, this sets the stage for the SL(2,C) theoretic representation of
gravitational dynamics, presented by Carmeli (discussed in Chapter 10 of
his 1983 Classical Fields treatise), as well as other related
formalisms, like the Newman- Penrose equations or Ashtekar
representation. The following will introduce the requisite background.

Sections:
1. The Null Tetrad and Spinor Frame
2. The Connection Coefficients
3. The Structure Coefficients
4. The Newman-Penrose Equations
5. Tetrad Components
6. The Covariant Exterior Derivatives and Optical Scalars
7. The Electromagnetic and Gauge Fields

What follows here is something almost never seen in the literature --- a
presentation of the spinor calculus in invariant, algebraic form that is
significantly simplified relative to the index- based notation seen in
the current literature, and more intuitive geometrically (as well as
more robust; for instance, we go far beyond the well-known decomposition
of 2-forms while also explaining its origin). This is applied to the end
of not just providing a more transparent and geometrically intuitive
development of the Newman-Penrose formalism; but going beyond it,
exposing its oversights and showing how one might generalize to
Riemann-Cartan geometry, which is the geometry in which spinor-valued
quantities actually reside.

This is a legacy article from my archive coming from several years back.
Time permitting, it will be reformatted, and redone, with some of the
material covered by the original Newman- Penrose papers added in.