Tetrad Formalism: Ricci Coefficients & Riemann Tensor

Tetrads and Notation

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.

Orthonormality and completeness

The inner products 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.

Fundamental tensors and index conversion

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

Component transformations

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

Strangulation and resurrection

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.

Ricci rotation coefficient

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_aμ)_;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 preserved, 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

Derivation of Riemann via Ricci coefficients

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 A_baν – 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]

Author note

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

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