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Tetrads in General Relativity

III. Geometric Algebra

Vector Derivative

   The main derivative operator in GA is the vector derivative $\nabla = {{\mathbf{e}}^j}{\partial _j}$, with $\{ {{\mathbf{e}}^j}\}$ the reciprocal base with respect to an arbitrary frame field $\left\{ {{{\mathbf{e}}_j}} \right\}$.

Vector Derivative $\nabla$

  1. yields the gradient when operating on a scalar field $\phi = \phi (x)$
    \[  \nabla \phi (x) = {{\mathbf{e}}^j}{\,\partial _j}\,\phi (x)  \]
  2. yields the geometric product when operating on a vector field
    \[  \nabla {\mathbf{a}} = \nabla \cdot {\mathbf{a}} + \nabla \wedge {\mathbf{a}} \]
  3. yields the divergence through the inner product
    \[\nabla \cdot {\mathbf{a}} = {{\mathbf{e}}^j} \cdot {\partial _j}{\,\mathbf{a}} = {\partial _j}{\,a^j}{\qquad}{a^j}: = {{\mathbf{e}}^j} \cdot {\mathbf{a}}\]
  4. yields the curl through the wedge product
    \[  \nabla \wedge {\mathbf{a}} = {{\mathbf{e}}^i} \wedge {\,\partial _i}{\,\mathbf{a}}= {\partial _i}{\, a^j}{{\,\mathbf{e}}^i} \wedge {{\mathbf{e}}_j} \]

   The operator $\nabla $ has all the algebraic properties of a vector and can act over any multivector with the inner, outer or geometric product. For example, if $\phi $ is a scalar field and ${\mathbf{a}}$ and ${\mathbf{b}}$ two vector fields, then

3.8

\[ \nabla (\phi {\mathbf{a}}) = (\nabla \phi ){\mathbf{a}} + \phi (\nabla {\mathbf{a}})  \]

3.9

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

The commutativity or anticommutativity ${( - 1)^p}$ depends on the degree/rank $p$ of the object that $\nabla$ is acting upon. For a summary of properties see references below.