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Tetrads in General Relativity

III. Geometric Algebra

Geometric Product Rule

   With the definitions of the inner product (3.1) and outer product (3.2), the multiplicative properties of the tetrad base $\{ {{\mathbf{e}}_a}\} $ can then be summarized in the geometric product rule

3.3

\[ {{\mathbf{e}}_a}{{\mathbf{e}}_b} = {{\mathbf{e}}_a} \cdot {{\mathbf{e}}_b} + {{\mathbf{e}}_a} \wedge {{\mathbf{e}}_b} \]

An essential feature of the geometric product is that it is invertible, that is, the reciprocal basis $\{ {{\mathbf{e}}^a}(x)\} $ may be obtained algebraically:

3.4

\[  {{\mathbf{e}}^a}: = {({{\mathbf{e}}_a})^{ - 1}}, {\quad}{{\mathbf{e}}^a}{{\mathbf{e}}_b} = {{\mathbf{e}}^a} \cdot {{\mathbf{e}}_b} = \delta _b^a  \]

This is an equally valid basis for vectors and tensors in the geometric tangent space ${\mathcal{G}_4}$.

   Invertibility (3.4) relies on the fact that the newly defined geometric product (3.3) has the property of being associative, i.e. ${\mathbf{a}}({\mathbf{bc}}) = ({\mathbf{ab}}){\mathbf{c}}$ , in contrast to the inner product (3.1) and the outer product (3.2), separately. This also is key to defining more complex products such as ${\mathbf{a}} \cdot ({\mathbf{b}} \wedge {\mathbf{c}})$ . Useful identities valid in the GA formalism are

3.5

\[  {\mathbf{a}}({\mathbf{b}} \wedge {\mathbf{c}}) = 2({\mathbf{a}} \cdot {\mathbf{b}}){\mathbf{c}} - 2({\mathbf{a}} \cdot {\mathbf{c}}) {\mathbf{b}}\]

3.6

\[  {\mathbf{a}} \cdot ({\mathbf{b}} \wedge {\mathbf{c}}) = ({\mathbf{a}} \cdot {\mathbf{b}}){\mathbf{c}} - ({\mathbf{a}} \cdot {\mathbf{c}}){\mathbf{b}}  \]

3.7

\[  ({\mathbf{a}} \wedge {\mathbf{b}}) \cdot ({\mathbf{c}} \wedge {\mathbf{d}}) = ({\mathbf{a}} \cdot {\mathbf{d}}) ({\mathbf{b}} \cdot {\mathbf{c}}) - ({\mathbf{a}} \cdot {\mathbf{c}}) ({\mathbf{b}} \cdot {\mathbf{d}})  \]

By successively multiplying together vectors the complete algebra may be generated. Elements of this algebra are called multivectors. Proofs may be found in the references below.