[1] It is a special case of the more general idea of a vielbein formalism, which is set in (pseudo-)Riemannian geometry.
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].
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].
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.
From the point of view of the differential geometry of fiber bundles, the n vector fields
Since not every manifold is parallelizable, a vielbein can generally only be chosen locally (i.e. only on a coordinate chart
These tangent vectors are usually defined as directional derivative operators: given a chart
, the coordinate vectors are such that: The definition of the cotetrad uses the usual abuse of notation
The involvement of the coordinate tetrad is not usually made explicit in the standard formalism.
When the tetrad is unspecified this becomes a matter of specifying the type of the tensor called abstract index notation.
It allows to easily specify contraction between tensors by repeating indices as in the Einstein summation convention.
Changing to and between general tetrads is much similar and equally necessary (except for parallelizable manifolds).
Likewise, the metric can be expressed with respect to an arbitrary (co)tetrad as Here, we use choice of alphabet (Latin and Greek) for the index variables to distinguish the applicable basis.
The manipulation with tetrad coefficients shows that abstract index formulas can, in principle, be obtained from tensor formulas with respect to a coordinate tetrad by "replacing greek by latin indices".
However care must be taken that a coordinate tetrad formula defines a genuine tensor when differentiation is involved.
Since the coordinate vector fields have vanishing Lie bracket (i.e. commute:
), naive substitutions of formulas that correctly compute tensor coefficients with respect to a coordinate tetrad may not correctly define a tensor with respect to a general tetrad because the Lie bracket is non-vanishing:
For example, the Riemann curvature tensor is defined for general vector fields
by In a coordinate tetrad this gives tensor coefficients The naive "Greek to Latin" substitution of the latter expression is incorrect because for fixed c and d,
, the parallel transport of a differential corresponds to The above can be readily verified simply by taking
on the Lie group is the Cartan metric, aka the Killing form.
[6] These vielbeins are used to perform calculations in sigma models, of which the supergravity theories are a special case.