Metrics
The metric used in the 4D rigid geometric algebra over 3D Euclidean space is the $$4 \times 4$$ matrix $$\mathfrak g$$ given by
- $$\mathfrak g = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\\end{bmatrix}$$ .
The metric exomorphism matrix $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below.
The metric antiexomorphism matrix $$\mathbb G$$, often called the "antimetric", corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below.
The product of the metric exomorphism matrix $$\mathbf G$$ and metric antiexomorphism matrix $$\mathbb G$$ for any metric $$\mathfrak g$$ is always equal to the $$16 \times 16$$ identity matrix times the determinant of $$\mathfrak g$$. That is, $$\mathbf G \mathbb G = \det(\mathfrak g) \mathbf I$$.
The metric and antimetric determine bulk and weight, duals, dot products, and geometric products.
In the Book
- The metric and antimetric are introduced in Sections 2.8.1 and 2.8.2.