Tetrad formalism

The tetrad formalism is a transformed coordinate approach to General Relativity.

In the tetrad formalism, a tetrad basis is chosen -- a set of four independent vector fields $\{e\_\{(a)\}^\{mu\}\}\_\{a=1dots4\}$ that together span the 4D vector space at each point in spacetime. All tensors of the theory are then expressed in this vector basis, by projecting them against members of the tetrad. For example, the spacetime metric itself can be transformed from a fixed coordinate basis to the tetrad basis:

:$g\_\{mu\; u\}\; ightarrow\; g\_\{(a)(b)\}\; =\; g\_\{mu\; u\}\; e\_\{(a)\}^\{mu\}\; e\_\{(b)\}^\{\; u\}$

The advantage of the tetrad formalism over the standard coordinate-based approach to GR lies in the ability to choose the tetrad basis to reflect important physical aspects of the spacetime.

Popular tetrad bases include orthonormal tetrads and null tetrads. The latter are used frequently in problems dealing with radiation, and are the basis of the Newman-Penrose formalism.