Tetrad formalism

The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four^[a] linearly independent vector fields called a tetrad or vierbein.^[1] It is a special case of the more general idea of a vielbein formalism, which is set in (pseudo-)Riemannian geometry. This article as currently written makes frequent mention of general relativity; however, almost everything it says is equally applicable to (pseudo-)Riemannian manifolds in general, and even to spin manifolds. Most statements hold simply by substituting arbitrary ${\displaystyle n}$ for ${\displaystyle n=4}$. In German, "vier" translates to "four", and "viel" to "many".

The general idea is to write the metric tensor as the product of two vielbeins, one on the left, and one on the right. The effect of the vielbeins is to change the coordinate system used on the tangent manifold to one that is simpler or more suitable for calculations. It is frequently the case that the vielbein coordinate system is orthonormal, as that is generally the easiest to use. Most tensors become simple or even trivial in this coordinate system; thus the complexity of most expressions is revealed to be an artifact of the choice of coordinates, rather than a innate property or physical effect^{[citation needed]}. That is, as a formalism, it does not alter predictions; it is rather a calculational technique.

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.

The significance of the tetradic formalism appear in the Einstein–Cartan formulation of general relativity. The tetradic formalism of the theory is more fundamental than its metric formulation as one can not convert between the tetradic and metric formulations of the fermionic actions despite this being possible for bosonic actions ^{[citation needed]}. This is effectively because Weyl spinors can be very naturally defined on a Riemannian manifold^{[2] [citation needed]} and their natural setting leads to the spin connection. Those spinors take form in the vielbein coordinate system, and not in the manifold coordinate system.

The privileged tetradic formalism also appears in the deconstruction of higher dimensional Kaluza–Klein gravity theories^[3] and massive gravity theories, in which the extra-dimension(s) is/are replaced by series of N lattice sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.^[4] Vielbeins commonly appear in other general settings in physics and mathematics. Vielbeins can be understood as solder forms.

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