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Tetrads in General Relativity

III. Geometric Algebra

Vector Coderivative

   In geometric algebra, the notion of a coderivative (covariant derivative operator) can be extended to a multivector directional derivative (MDD), i.e. an operator which takes the derivative of a multivector field in the direction of a vector. It can be shown that such an operator exists for every metric-compatible connection. 

   The basic properties of such a directional coderivative ${D_i}$ with respect to an arbitrary frame field $\{ {{\mathbf{e}}_j}\}$ may be summarized as follows:

Directional Coderivative ${D_i}$

  1. maps scalars into scalars: ${D_i}\phi = {\partial _i}\phi $; partial derivatives operate only on scalar components relative to the basis $\{ {{\mathbf{e}}_j}\} $;
  2. maps vectors into vectors. In particular, ${D_i}$ maps ${{\mathbf{e}}_j}$ into a vector, which can be expressed as a linear combination:
    \[  {D_i}{{\mathbf{e}}_j} = \Gamma _{ij}^k{{\mathbf{e}}_k}{\quad} \Gamma _{ij}^k = \left( {{D_i}{{\mathbf{e}}_j}} \right) \cdot {{\mathbf{e}}^k}  \]
    This defines the connection coefficients $\Gamma _{ij}^k$ for the frame $\{ {{\mathbf{e}}_j}\} $ , which can be arbitrary, save for the ‘metric compatibility’ condition (1.28);
  3. obeys the Leibniz rule. Thus, for any two multivector fields $A,B$,
    \[  {D_i}(AB) = ({D_i}A)B + A({D_i}B) \]
  4. is distributive with respect to addition,
    \[  {D_i}(A + B) = {D_i}A + {D_i}B  \]
  5. is related to operators ${D_m} = h_m^i{D_i}$ corresponding to different frames ${\text{\{ }}{{\mathbf{e}}^m}\} $, if the frame ${\text{\{ }}{{\mathbf{e}}^i}\}$ is related to the different frames by ${{\mathbf{e}}^i} = h_m^i{{\mathbf{e}}^m}$.

   The vector coderivative  is defined as $D: = {{\mathbf{e}}^i}{D_i}$, with properties directly derived from the multivector directional coderivative defined above. This includes the important property that the vector coderivative may be decomposed into the sum

3.10

\[DA = D \cdot A + D \wedge A\]

of the covariant divergence $D \cdot A: = {{\mathbf{e}}^i} \cdot {D_i}A$ and the covariant curl $D \wedge A: = {{\mathbf{e}}^i} \wedge {D_i}A$. The vector coderivative is a coordinate-free object that can be decomposed in any system of coordinates, e.g. $D = {{\mathbf{g}}^\mu }{D_\mu }$. Algebraically it may be treated as a vector, to be used as such in geometric, inner and wedge products.