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Tetrads in General Relativity

I. Metric and Connection

Differential Forms

   In the case of a metric space, there is an isomorphism between the reciprocal basis $\{ {{\mathbf{g}}^\mu }\} $ as defined above in (1.2) and the differential forms $\{ {\text{d}}{x^\mu }\}$, called 1-forms in the theory of differential forms pioneered by the mathematician Élie Cartan. For example, in the theory of differential forms the exterior derivative of a scalar function (zero-form) $f(x)$ is defined as the gradient 1-form

1.6

\[{\text{d}}f(x): = {({\partial _\mu }f)_x}{\text{d}}{x^\mu } = {({\partial _\mu }f)_x}{{\mathbf{g}}^\mu }\]

In particular, it follows that $\text{d}{x^\nu} = {\partial _\mu }{x^\nu }\text{d}{x^\mu}  = {\text{d}}{x^\nu }$. Thus, it is seen that the action of the exterior differential is the same as the action of the gradient $\nabla = {{\mathbf{g}}^\mu }{\partial _\mu }$.

   A general 1-form is a linear, real-valued function ${\mathbf{A}} = {A_\mu }{\text{d}}{x^\mu }$ with respect to the basis differentials $\text{d}{x^\mu }$. The exterior derivative ${\text{d}}$ acts as the curl  $\nabla \wedge $ on the 1-form ${\mathbf{A}}$ raising the degree to give the 2-form

1.7

\[ {\text{d}}{\mathbf{A}}: = {\partial _\mu }{A_\nu }\text{d}{x^\mu } \wedge \text{d}{x^\nu } = {\partial _\mu }{A_\nu }{{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }\]

This defines an anti-symmetric covariant tensor ${F_{\mu \nu }} = {F_{[\mu \nu ]}}$ with the differential 2-form

1.8

\[ {\mathbf{F}}: = \frac{1}{2}{F_{\mu \nu }} \text{d}{x^\mu } \wedge \text{d}{x^\nu } = {\partial _{[\mu }}{A_{\nu ]}} {{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }\]

The bracket, like the wedge product, indicates anti-symmetrization of the indices with an implicit factor 1/2.

   Differential forms are in one-to-one correspondence with totally antisymmetric tensor fields. These objects are particularly suited for analyzing fields on manifolds and vector spaces in GRT, providing a more flexible and powerful framework than traditional tensor calculus. For example, compared with the tensor formalism, differential forms together with Cartan’s structure equations (see section IV and section X) offer an efficient way of obtaining the connection elements. In section III the theory of forms is extended to become a Geometric Algebra (GA), also known as a Clifford Algebra, or SpaceTime Algebra (STA).