# Tetrad Fields and Spacetime

A spacetime is often described in terms of a tetrad field, that is, by giving a set of basis vectors at each point. Let the vectors of the tetrad be denoted by e_{a}^{μ}, where μ is a tensor index and where a is a tetrad index that serves to number the vectors from 1 to 4.

The inner prioducts of the vectors, e_{aμ} e_{b}^{μ} = η_{ab}, are assumed to be constants given a priori. For orthonormal tetrad fields, η_{ab} is just the Minkowski matrix, Diag(1, -1, -1, -1). We may also use its inverse matrix to define quantities η^{ab} = (η_{ab})^{-1}. The fact that the vectors are linearly independent is expressed by the completeness relation g_{μν} = e^{a}_{μ} e^{b}_{ν} η_{ab}.

Then the following convention is natural and convenient: g_{μν}, e_{a}^{μ} and η_{ab} are all “fundamental” tensors, that is, tensors whose application does not change the object itself, but only the nature of an index. Just as g_{μν} raises and lowers indices and changes them from contravariant to covariant and vice versa, η_{ab} is used to raise and lower tetrad indices, while e_{a}^{μ} can be used to change an index from a tensor index to a tetrad one and vice versa. Thus

T_{μν}

T_{aν} = e_{a}^{μ} T_{μν}

T_{μb} = e_{b}^{ν} T_{μν}

T_{ab} = e_{a}^{μ} e_{b}^{ν} T_{μν}

are understood to be manifestations of the “same” tensor T. Thanks to orthonormality and completeness we may convert back and forth between the different descriptions without any loss of information. In what follows, we will freely make use of this property without explicitly calling attention to it.

The processes of changing from a tensor index to a tetrad index and back have been called “strangulation” and “resurrection” respectively.

Derivatives

Associated with tetrad fields are several different types of derivative.

1) **The Covariant Derivative**

A semicolon will be used to denote the usual covariant derivative which ignores tetrad indices, and for example treats e_{a}^{μ} as simply a set of vectors.

Define

A^{μ}_{aν} = e_{a}^{μ}_{;ν}

This field is known as the Ricci rotation coefficient. It plays a central role in what follows. The strangled components A_{baν} are antisymmetric in a and b, since

A_{baν} = e_{b}^{μ} e_{aμ;ν} = (e_{b}^{μ} e_{a}_{μ})_{;ν} – e_{b}^{μ}_{;ν} e_{a}^{μ} = η_{ab;ν} – e_{a}^{μ} e_{bμ;ν} = – A_{abν}

Our convention automatically guarantees that the unstrangled components of an object have the same symmetry properties as the strangled ones. Therefore A_{μνσ} is also antisymmetric in μ and ν. Furthermore, any tensor A_{μνσ} antisymmetric in μ and ν satisfies identically

A_{μνσ} = A_{μ[νσ]} + A_{ν[σμ]} + A_{σ[νμ]}

The importance of this is that A_{μ[νσ]} = e^{b}_{μ} e_{b[ν,σ]} involves only partial derivatives, indicating that the Ricci rotation coefficients may be calculated directly without knowledge of the Christoffel symbols.

To get an intuitive feel for the Ricci rotation coefficients, let us calculate the change in the tetrad under an infinitesimal displacement. Since orthonormality is perserved, this change can only be a Lorentz rotation, Ω_{ab}

δe_{a}^{μ} = Ω_{ab} e^{bμ} = e_{a}^{μ}_{;ν} δx^{ν}

showing that Ω_{ab} = A_{baν} δx^{ν}. Thus A_{baν} measures the rotation of the tetrad in the ab plane when we make a small step in the ν direction.

2)** The Intrinsic Derivative**

The intrinsic derivative of any quantity with respect to a given tetrad, denoted by a stroke, is obtained by strangling completely, taking the covariant derivative of the resulting scalar (or equivalently its partial derivative), and then resurrecting. For example,

T_{aμ|ν} = (T_{aσ} e_{b}^{σ})_{;ν} e^{b}_{μ}

According to this definition, intrinsic differentiation commutes with strangulation and resurrection. Equivalently, e_{b}^{μ}_{|ν} = 0. For example:

e_{b}^{μ} T^{a}_{μ|ν} = T^{a}_{b|ν}

With the help of the Ricci rotation coefficient we can write the relationship between the intrinsic derivative and the covariant derivative, with a correction term appearing for each tensor index. For example:

T^{a}_{μ|ν} = T^{a}_{μ;ν} + T^{aσ} A_{σμν}

3) **The Invariant Derivative**

The third type of derivative is the invariant derivative, denoted by a dot, in which we first resurrect any tetrad indices that might be present, take the covariant derivative, then strangle back again. That is,

T_{aμ.ν} = (T^{b}_{μ} e_{b}^{σ})_{;ν} e_{a}^{σ}

As before, invariant differentiation commutes with strangulation, and equivalently e_{a}^{μ}_{.ν} = 0. For an object with only tensor indices, the invariant derivative is identical to its covariant derivative. For this reason the invariant derivative is the natural generalization of the covariant derivative to tetrad analysis. The relationship between the covariant and invariant derivative can be written using the Ricci rotation coefficient:

T_{a}^{μ}_{.ν} = T_{a}^{μ}_{;ν} – T_{b}^{μ} A^{b}_{aν}

with a similar term required for each tetrad index.

**The Riemann Tensor**

For any vector T,

T_{a.ν} = T_{a;ν} – T^{b} A_{baν}

T_{a.νσ} = T_{a;νσ} – T^{b}_{;ν} A_{baσ} – T^{b}_{;σ} _{Abaν} – T^{b} A_{baν.σ}

Antisymmetrizing on ν and σ, the first three terms drop out:

T_{a.[νσ]} = – T^{b} A_{ba[ν.σ]}

By comparison,

T_{a.[νσ]} = (T_{μ} e_{a}^{μ})_{.[νσ]} = e_{a}^{μ} T_{μ;[νσ]} = -1/2 e_{a}^{μ} T^{τ} R_{τμνσ} = -1/2 T^{b} R_{baνσ}

Since this must hold identically for all T, we have a simple expression for the Riemann tensor:

R_{τμνσ} = 2 A_{τμ[ν.σ]}

— This article was originally part of Physics Forums member Bill_K‘s PF blog. He may not respond to comments.

## Leave a Reply

Want to join the discussion?Feel free to contribute!