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.
Contents
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.
Mathematical formulation
In the tetrad formalism, a tetrad basis is chosen: a set of four independent vector fields
where
From a mathematical point of view, the four vector fields
All tensors of the theory can be expressed in the vector and covector basis, by expressing them as linear combinations of members of the (co)tetrad. For example, the spacetime metric itself can be transformed from a coordinate basis to the tetrad basis.
Popular tetrad bases include orthonormal tetrads and null tetrads. Null tetrads are composed of light cone vectors, so are used frequently in problems dealing with radiation, and are the basis of the Newman–Penrose formalism and the GHP formalism.
Relation to standard formalism
The standard formalism of differential geometry (and general relativity) consists simply of using the coordinate tetrad in the tetrad formalism. The coordinate tetrad is the canonical set of vectors associated with the coordinate chart. The coordinate tetrad is commonly denoted
The definition of the cotetrad uses the usual abuse of notation
Changing tetrads is a routine operation in the standard formalism, as it is involved in every coordinate transformation (i.e., changing from one coordinate tetrad basis to another). Switching between multiple coordinate charts is necessary because, except in trivial cases, it is not possible for a single coordinate chart to cover the entire manifold. Changing to and between general tetrads is much similar and equally necessary (except for parallelizable manifolds). Any tensor can locally be written in terms of this coordinate tetrad or a general (co)tetrad.
For example, the metric tensor
(here we use the Einstein summation convention). Likewise, the metric can be expressed with respect to an arbitrary (co)tetrad as
We can translate from a general co-tetrad to the coordinate co-tetrad by expanding the covector
from which it follows that
which shows that
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 vectorfields have vanishing Lie bracket (i.e. commute:
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,
where