Tetrad formalism

The tetrad formalism is an approach to general relativity that replaces the choice of a coordinate basis by the less restrictive choice of a local basis for the tangent bundle, i.e. a locally defined set of four linearly independent vector fields called a tetrad.

In the tetrad formalism all tensors are represented in terms of a chosen basis. (When generalised to other than four dimensions this approach is given other names, see Cartan formalism.) As a formalism rather than a theory, it does not make different predictions but does allow the relevant equations to be expressed differently.

The advantage of the tetrad formalism over the standard coordinate-based approach to general relativity lies in the ability to choose the tetrad basis to reflect important physical aspects of the spacetime. The abstract index notation denotes tensors as if they were represented by their coefficients with respect to a fixed local tetrad. Compared to a completely coordinate free notation, which is often conceptually clearer, it allows an easy and computationally explicit way to denote contractions.

In the tetrad formalism, a tetrad basis is chosen: a set of four independent vector fields ${\displaystyle \scriptstyle \{e_{(a)}=e_{(a)}^{\mu }\partial _{\mu }\}_{a=1\dots 4}}$ that together span the 4D vector tangent space at each point in spacetime. Dually, a tetrad determines (and is determined by) a dual co-tetrad—a set of four independent covectors (1-forms) ${\displaystyle \scriptstyle \{e^{(a)}=e_{\mu }^{(a)}dx^{\mu }\}_{a=1\dots 4}}$ such that

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