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 linearly independent vector fields called a tetrad.[1]

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.) 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[2], a tetrad basis is chosen: a set of four independent vector fields

e_{a}=e_{a}{}^{\mu }\partial _{\mu }

for $a=1,\ldots ,4$ that together span the 4D vector tangent space at each point in spacetime. Dually, a tetrad determines (and is determined by) a dual co-tetrad—a set of four independent covectors (1-forms)

e^{a}=e^{a}{}_{\mu }dx^{\mu }

such that

e^{a}(e_{b})=e^{a}{}_{\mu }e^{\mu }{}_{b}=\delta _{b}^{a},

where $\delta _{b}^{a}$ is the Kronecker delta. A tetrad is usually specified by its coefficients $e^{\mu }{}_{a}$ with respect to a coordinate basis, despite the choice of a set of (local) coordinates $x^{\mu }$ being unnecessary for the specification of a tetrad. Each covector is a solder form.

From the point of view of the differential geometry of fiber bundles, the four vector fields $\{e_{a}\}_{a=1\dots 4}$ define a section of the frame bundle i.e. a parallelization of $M$ which is equivalent to an isomorphism $TM\cong M\times {\mathbb {R} ^{4}}$ . Since not every manifold is parallelizable, a tetrad can generally only be chosen locally.

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 tensor 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 four null 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 $\{\partial _{\mu }\}$ whereas the dual cotetrad is denoted $\{dx^{\mu }\}$ . These tangent vectors are usually defined as directional derivative operators: given a chart ${\varphi =(\varphi ^{1},\ldots ,\varphi ^{n})}$ which maps a subset of the manifold into coordinate space $\mathbb {R} ^{n}$ , and any scalar field $f$ , the coordinate vectors are such that:

\partial _{\mu }[f]\equiv {\frac {\partial (f\circ \varphi ^{-1})}{\partial x^{\mu }}}.

The definition of the cotetrad uses the usual abuse of notation $dx^{\mu }=d\varphi ^{\mu }$ to define covectors (1-forms) on $M$ . The involvement of the coordinate tetrad is not usually made explicit in the standard formalism. In the tetrad formalism, instead of writing tensor equations out fully (including tetrad elements and tensor products $\otimes$ as above) only components of the tensors are mentioned. For example, the metric is written as " $g_{ab}$ ". 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 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 $\mathbf {g}$ can be expressed as:

\mathbf {g} =g_{\mu \nu }dx^{\mu }dx^{\nu }\qquad {\text{where}}~g_{\mu \nu }=\mathbf {g} (\partial _{\mu },\partial _{\nu }).

(Here we use the Einstein summation convention). Likewise, the metric can be expressed with respect to an arbitrary (co)tetrad as

\mathbf {g} =g_{ab}e^{a}e^{b}\qquad {\text{where}}~g_{ab}=\mathbf {g} \left(e_{a},e_{b}\right).

Here, we use choice of alphabet (Latin and Greek) for the index variables to distinguish the applicable basis.

We can translate from a general co-tetrad to the coordinate co-tetrad by expanding the covector $e^{a}=e^{a}{}_{\mu }dx^{\mu }$ . We then get

\mathbf {g} =g_{ab}e^{a}e^{b}=g_{ab}e^{a}{}_{\mu }e^{b}{}_{\nu }dx^{\mu }dx^{\nu }=g_{\mu \nu }dx^{\mu }dx^{\nu }

from which it follows that $g_{\mu \nu }=g_{ab}e^{a}{}_{\mu }e^{b}{}_{\nu }$ . Likewise expanding $dx^{\mu }=e^{\mu }{}_{a}e^{a}$ with respect to the general tetrad, we get

\mathbf {g} =g_{\mu \nu }dx^{\mu }dx^{\nu }=g_{\mu \nu }e^{\mu }{}_{a}e^{\nu }{}_{b}e^{a}e^{b}=g_{ab}e^{a}e^{b}

which shows that $g_{ab}=g_{\mu \nu }e^{\mu }{}_{a}e^{\nu }{}_{b}$ .

Manipulation of indices

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: $\partial _{\mu }\partial _{\nu }=\partial _{\nu }\partial _{\mu }$ ), 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: $[e_{a},e_{b}]\neq 0$ . Thus, it is sometimes said that tetrad coordinates provide a non-holonomic basis.

For example, the Riemann curvature tensor is defined for general vector fields $X,Y$ by

R(X,Y)=\left(\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{[X,Y]}\right)

.

In a coordinate tetrad this gives tensor coefficients

R_{\ \nu \sigma \tau }^{\mu }=dx^{\mu }\left((\nabla _{\sigma }\nabla _{\tau }-\nabla _{\tau }\nabla _{\sigma })\partial _{\nu }\right).

The naive "Greek to Latin" substitution of the latter expression

R_{\ bcd}^{a}=e^{a}\left((\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c})e_{b}\right)\qquad {\text{(wrong!)}}

is incorrect because for fixed c and d, $\left(\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c}\right)$ is, in general, a first order differential operator rather than a zeroth order operator which defines a tensor coefficient. Substituting a general tetrad basis in the abstract formula we find the proper definition of the curvature in abstract index notation, however:

R_{\ bcd}^{a}=e^{a}\left((\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c}-f_{cd}{}^{e}\nabla _{e})e_{b}\right)

where $[e_{a},e_{b}]=f_{ab}{}^{c}e_{c}$ . Note that the expression $\left(\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c}-f_{cd}{}^{e}\nabla _{e}\right)$ is indeed a zeroth order operator, hence (the (c d)-component of) a tensor. Since it agrees with the coordinate expression for the curvature when specialised to a coordinate tetrad it is clear, even without using the abstract definition of the curvature, that it defines the same tensor as the coordinate basis expression.

Notes

De Felice, F.; Clarke, C.J.S. (1990), Relativity on Curved Manifolds, p. 133
Tohru Eguchi, Peter B. Gilkey and Andrew J. Hanson, "Gravitation, Gauge Theories and Differential Geometry", Physics Reports 66 (1980) pp 213-393.

References

De Felice, F.; Clarke, C.J.S. (1990), Relativity on Curved Manifolds (first published 1990 ed.), Cambridge University Press, ISBN 0-521-26639-4
Benn, I.M.; Tucker, R.W. (1987), An introduction to Spinors and Geometry with Applications in Physics (first published 1987 ed.), Adam Hilger, ISBN 0-85274-169-3

External links

General Relativity with Tetrads

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[1] De Felice, F.; Clarke, C.J.S. (1990), Relativity on Curved Manifolds, p. 133

[eguchi-2] Tohru Eguchi, Peter B. Gilkey and Andrew J. Hanson, "Gravitation, Gauge Theories and Differential Geometry", Physics Reports 66 (1980) pp 213-393.