\( \newcommand{\bs}{\boldsymbol} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \newcommand{\third}{{\scriptstyle \frac{1}{3}}} \newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter] \newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)

Tetrads in General Relativity

Appendix C

Duality Operation

   The forming of the product of the grade-$n$ pseudoscalar (volume element) ${\operatorname{I} _n}$ of a geometric vector space $\mathcal{G}{_n}$, with a grade-$p$ multivector ${{\mathbf{A}}_p}$, is called a duality operation:

C.1

\[{{\mathbf{A}}_p} \to {*\mathbf{A}}_p := {\operatorname{I} _n}{{\mathbf{A}}_p}\]

If ${{\mathbf{A}}_p}$ is a blade, ${\operatorname{I} _n}{{\mathbf{A}}_p}$ returns a blade of grade $r=n-p \leqslant n$, embedded in the subspace of vectors not contained in ${{\mathbf{A}}_p}$. This blade ${*\mathbf{A}}_p$ is called the dual of ${{\mathbf{A}}_p}$ or its orthogonal complement. The operation can be reversed by multiplication with the inverse $\operatorname{I} ^n$. Both are duality operations equivalent (up to a sign) to the Hodge Star operation in the theory of differential forms.

   Since the duality operation is linear, it is sufficient to consider how the volume element $\operatorname{I} _n$ acts on orthonormal $p$-blades. For $1 \leqslant p \leqslant n$ it follows with (12.3):

C.2

\[{\operatorname{I} _n}{{\mathbf{e}}^{{a_1}}} \wedge ... \wedge {{\mathbf{e}}^{{a_p}}} = (-1)^s \frac{{{{( - 1)}^{r(r - 1)/2}}}}{{r!}} {\varepsilon ^{{a_1}.......{a_p}{a_{p + 1}}...{a_n}}}{{\mathbf{e}}_{{a_{p + 1}}}} \wedge ... \wedge {{\mathbf{e}}_{{a_n}}}\]

The minus sign depends on the number of permutations that must be performed to bring both sets of indices $\{p+1,...,n\}$ at the right-hand side in ascending numerical order.

   Given a geometric vector space ${\mathcal{G}_n}$ with $\{ {{\mathbf{e}}_i};i = 1,..,n\}$ an orthonormal base. The set of all possible blades $\{ 1,\,{{\mathbf{e}}_i},\,{{\mathbf{e}}_i} \wedge {{\mathbf{e}}_j}, \,{{\mathbf{e}}_i} \wedge {{\mathbf{e}}_j} \wedge {{\mathbf{e}}_k},...\}$ with $i < j < k < ..$, provides a basis for the entire algebra which has the dimension $\dim ({\mathcal{G}_n}) = {2^n}$. The fact that the basis vectors anti-commute ensures that each product in the basis set is totally antisymmetric. 

   In particular, the space ${\mathcal{G}_4}$ has ${2^4} = 16$ linearly independent elements: one scalar, four vectors, six bivectors, four trivectors and one speudoscalar. Applying a dual operation, meaning in this case $\{{\mathbf{e}}\} \to {\text{I}}^4\{{\mathbf{e}}\}$, one derives:

C.3

Trautman Forms

  1. $\quad {\mathbf{\eta }}\quad \;\, := {\text{I}}^4$
  2. $\quad {{\boldsymbol{\eta }}_a}\quad := {\text{I}}^4{{\mathbf{e}}_a} =\frac{1}{3}{{\mathbf{e}}^b} \wedge {{\bs{\eta }}_{ab}}= - \frac{1}{{3!}}{\varepsilon _{abcd}}{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}$
  3. $ \quad {{\boldsymbol{\eta }}_{ab}}\,\,\,\,: = \operatorname{I} ^4{{\mathbf{e}}_a} \wedge {{\mathbf{e}}_b} = - \frac{1}{2}{{\mathbf{e}}^c} \wedge {{\bs{\eta }}_{abc}} = - \frac{1}{2}{\varepsilon _{abcd}} {{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}$
  4. $\quad {{\boldsymbol{\eta }}_{abc}}\,\,:=\operatorname{I}^4{{\mathbf{e}}_a} \wedge {{\mathbf{e}}_b} \wedge {{\mathbf{e}}_c} = {\varepsilon _{abcd}}{{\mathbf{e}}^d}$
  5. $\quad {{\mathbf{\eta}}_{abcd}}:= \operatorname{I}^4{{\mathbf{e}}_a} \wedge {{\mathbf{e}}_b} \wedge {{\mathbf{e}}_c} \wedge {{\mathbf{e}}_d} ={\varepsilon _{abcd}}$

These $\boldsymbol{\eta}$-forms, first introduced by Andrzej Trautman (1973), constitute a dual basis for the algebra of the geometric space $\mathcal{G}_4$.