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Tetrads in General Relativity

III. Geometric Algebra

Integration Measure

   The line integral of a multivector field on a manifold ${\mathcal{M}_n}$ along a one-parameter curve $\bf{x}(\lambda)$ is defined by

3.11

\[\int {d\lambda } \frac{{\partial {\mathbf{x}}}}{{\partial \lambda }}{\mathbf{F}}(x): = \int {d{\mathbf{x}_\lambda}}\, {\mathbf{F}}(x)\]

The key concept here is the vector-valued measure

3.12

\[d{\mathbf{x}_\lambda} = d\lambda \frac{{\partial {\mathbf{x}}}}{{\partial \lambda }} = d\lambda \left| {\frac{{\partial {\mathbf{x}}}}{{\partial \lambda }}} \right|{\mathbf{I}}{\qquad}{\mathbf{I}}: = \frac{{\partial {\mathbf{x}}}}{{\partial \lambda }}{\left| {\frac{{\partial {\mathbf{x}}}}{{\partial \lambda }}} \right|^{ - 1}}\]

with ${\text{I}}$ the unit vector giving an orientation to the domain of integration.

   Definition (3.12) extends naturally to higher dimensional surfaces and volumes:

Directed Measure

  1. Let $\{ {{\mathbf{g}}_1}, \ldots ,{{\mathbf{g}}_p}\} $, $1 \leqslant p \leqslant n$,  be a subset of coordinate base vectors that span a $p$-dimensional subspace in an overall $n$-dimensional vector space $\mathcal{G}_n$.
  2. In GA the $p$-blade ${{\mathbf{g}}_1} \wedge {{\mathbf{g}}_2} \wedge \cdots \wedge {{\mathbf{g}}_p}$ is interpreted to be the oriented volume of the $p$-parallelotope subtended by these basis vectors. The oriented measure (volume element) is then constructed by defining
    \[{d}{\mathbf{x}_p} := d{x^1}...d{x^p}{{\mathbf{g}}_1} \wedge .... \wedge {{\mathbf{g}}_p}\]
    where the $d{x^i}$ are ordinary scalar coordinate integration elements.
  3. The normalized wedge product of the tangent vectors defines the unit volume for the subspace:
    \[{{\mathbf{I}}_p}(x): = {{\mathbf{g}}_1} \wedge ... \wedge {{\mathbf{g}}_p}/ \left| {{{\mathbf{g}}_1} \wedge ... \wedge {{\mathbf{g}}_p}} \right|\]
    The orientation is specified by the ordering of tangent vectors.
  4. The case $p=n=4$ is especially important; see Pseudoscalar Volume. For a metric space the denominator is the square root of the determinant $g := {\text{Det}({\mathbf{g}}_i} \cdot {{\mathbf{g}}_j)}$ of the metric tensor, leading to the signed scalar measure
    \[{d}{x}_4: = {d^4}x{{\mathbf{g}}_0} \wedge ... \wedge {{\mathbf{g}}_3} = {d^4}x\sqrt {\left| g \right|} {\operatorname{I} _4}\]
    5. This is the invariant volume measure as defined in general relativity, multiplied by the pseudoscalar unit volume $\operatorname{I} _4$ giving the measure an sign.