Physical Sciences and Mathematics Commons^™

How To Create A Two-Component Spinor, Charles G. Torre

How to... in 10 minutes or less

Let (M, g) be a spacetime, i.e., a 4-dimensional manifold M and Lorentz signature metric g. The key ingredients needed for constructing spinor fields on the spacetime are: a complex vector bundle E -> M ; an orthonormal frame on TM ; and a solder form relating sections of E to sections of TM (and tensor products thereof). We show how to create a two-component spinor field on the Schwarzschild spacetime using the DifferentialGeometry package in Maple. PDF and Maple worksheets can be downloaded from the links below.

The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson

Research Vignettes

The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

Higher Order Symmetries Of The Kdv Equation, Ian M. Anderson

Research Vignettes

In this worksheet we symbolically construct the formal inverse of the total derivative operator and use it to construct the recursion operator for the higher-order symmetries of the KdV equation. Using this recursion operator we generate the first 5 generalized symmetries of the KdV equation and verify that they all commute.

PDF and Maple worksheets can be downloaded from the links below.

A Homogeneous Solution Of The Einstein-Maxwell Equations, Charles G. Torre

Research Vignettes

We exhibit and analyze a homogeneous spacetime whose source is a pure radiation electromagnetic field [1]. It was previously believed that this spacetime is the sole example of a homogeneous pure radiation solution of the Einstein equations which admits no electromagnetic field (see [2] and references therein). Here we correct this error in the literature by explicitly displaying the electromagnetic source. This result implies that all homogeneous pure radiation spacetimes satisfy the Einstein-Maxwell equations.

PDF and Maple worksheets can be downloaded from the links below.

How To Create A Lie Algebra, Ian M. Anderson

How to... in 10 minutes or less

We show how to create a Lie algebra in Maple using three of the most common approaches: matrices, vector fields and structure equations. PDF and Maple worksheets can be downloaded from the links below.