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Tetrads in General Relativity

I. Metric and Connection

Reciprocal Base

   The inverse mapping of ${\mathbf{x = x}}(x)$ is the set of scalar-valued functions ${x^\mu } = {x^\mu }({\mathbf{x}})$. The gradients of these scalar fields are the vector fields

1.2

\[{{\mathbf{g}}^\mu }({\mathbf{x}}) : = \nabla {x^\mu }({\mathbf{x}})\]

Suppose $f({\mathbf{x}})$ is scalar field defined over some region of the manifold. Given a vector ${\mathbf{a}}$ in the tangent space, the vector-valued gradient $\nabla : = {\partial _{\mathbf{x}}}$ is defined though the directional derivative:

1.3

\[ {\mathbf{a}} \cdot {\partial _{\mathbf{x}}}f({\mathbf{x}}): = \mathop {\lim } \limits_{\varepsilon \to 0} \frac{{f({\mathbf{x}} + \varepsilon {\mathbf{a}}) - f({\mathbf{x}})}}{\varepsilon } \]

   Note that ${\mathbf{a}} \cdot {\partial _{\mathbf{x}}}{\mathbf{x}} = {\mathbf{a}}$ when $f({\mathbf{x}}) = {\mathbf{x}}$. Hence:

1.4

\[ {{\mathbf{g}}_\mu } \cdot {{\mathbf{g}}^\nu } = {{\mathbf{g}}_\mu } \cdot \nabla {x^\nu } = \partial {x^\nu }/\partial {x^\mu } = \delta _\mu ^\nu \]

This implies the gradient relations

1.5

\[ {{\mathbf{g}}_\mu } \cdot \nabla = {\partial _\mu }{\quad}\nabla = {{\mathbf{g}}^\mu }{\partial _\mu } \]

   The gradient (cotangent) vectors $\{ {{\mathbf{g}}^\mu }\} $ as defined in (1.2) form a reciprocal basis to the $\{ {{\mathbf{g}}_\mu }\} $, i.e., the reciprocal basis vectors can be written as a linear sum over the holonomic basis vectors, and vice versa. Both vector bases cover the same tangent space and are both referred to as the natural (holonomic) coordinate basis at point $\mathbf{x}$, or equivalently, $x = \{ {x^\mu }\} $; there is no need to introduce a dual space to define the inner product. This construction is only possible in metric spaces.