Tetrad formalism

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.^[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, 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 $\scriptstyle\{e_{(a)} = e_{(a)}^{\mu} \partial_\mu\}_{a=1\dots4}$ 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) $\scriptstyle\{e^{(a)} = e^{(a)}_{\mu} dx^\mu\}_{a=1\dots4}$ such that

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

where $\delta^{(a)}_{(b)}$ is the Kronecker delta. A tetrad is usually specified by its coefficients $e_{(a)}^{\mu}$ with respect to a coordinate basis, despite the fact that the choice of a tetrad does not actually require the additional choice of a set of (local) coordinates $x^\mu$.

From a mathematical point of view, the four vector fields $\scriptstyle\{e_{(a)}\}_{a=1\dots4}$ define a section of the frame bundle i.e. a parallelization of $\scriptstyle M$ which is equivalent to an isomorphism $\scriptstyle TM \cong M\times {\mathbb R^4}$. 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 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 $\scriptstyle\{\partial_\mu\}$ whereas the dual cotetrad is denoted $\{d x^\mu\}$. These tangent vectors are usually defined as directional derivative operators: given a chart $\scriptstyle\varphi = (\varphi^1, \ldots, \varphi^n)$ which maps a subset of the manifold into coordinate space $\scriptstyle\mathbb R^n$, and any scalar field $\scriptstyle f$, 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 $\scriptstyle\otimes$ as above) only components of the tensors are mentioned. For example, the metric is written as "$\scriptstyle g_{ab}$". 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 $\bold g$ can be expressed as:

$\bold g = g_{\mu\nu}dx^\mu dx^\nu~~~~~~~~~~~\text{where}~g_{\mu\nu} = \bold g(\partial_\mu,\partial_\nu)$

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

$\bold g = g_{ab}e^{(a)}e^{(b)}~~~~~~~~~~~\text{where}~g_{ab} =\bold g(e_{(a)},e_{(b)})$

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

$\bold g = g_{ab}e^{(a)}e^{(b)} = g_{ab}e^{(a)}_\mu e^{(b)}_\nu dx^\mu dx^\nu = g_{\mu\nu}dx^{\mu}dx^{\nu}$

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

$\bold g = g_{\mu\nu}dx^{\mu}dx^{\nu} = g_{\mu \nu} e_{(a)}^{\mu} e_{(b)}^{\nu} e^{(a)} e^{(b)} = g_{ab}e^{(a)}e^{(b)}$

which shows that $g_{ab} = g_{\mu\nu}e_{(a)}^\mu e_{(b)}^\nu$. For notational simplicity one usually drops the round brackets around the indices, recognizing that they can both label a set of (co)vectors and tensor components with respect to the (co)tetrad defined by these (co)vectors.

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: $\partial_\mu\partial_\nu = \partial_\nu\partial_\mu$), 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 $[e_a, e_b] = e_a e_b - e_b e_a \ne 0$. For example, the Riemann curvature tensor is defined for general vectorfields $X, Y$ by

$R(X,Y) = (\nabla_X \nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})$.

In a coordinate tetrad this gives tensor coefficients

$R^\mu_{\ \nu\sigma\tau} = dx^\mu((\nabla_\sigma\nabla_\tau - \nabla_\tau\nabla_\sigma)\partial_\nu).$

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

$R^a_{\ bcd} = e^a((\nabla_c\nabla_d - \nabla_d\nabla_c)e_b) ~~~~~~~~~\text{(wrong!)}$

is incorrect because for fixed c and d, $(\nabla_c\nabla_d - \nabla_d\nabla_c)$ is, in general, a first order differential operator rather than a zero'th 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:

$R^a_{\ bcd}= e^a((\nabla_c\nabla_d - \nabla_d\nabla_c - f^e_{cd}\nabla_e)e_b)$

where $[e_a, e_b] = f^c_{ab}e_c$. Note that the expression $(\nabla_c\nabla_d - \nabla_d\nabla_c - f^e_{cd}\nabla_e)$ 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.

See also

• Frame bundle
• Orthonormal frame bundle
• Principal bundle
• Spin bundle
• Connection (mathematics)
• G-structure
• Spin manifold
• Spin structure
• Dirac equation in curved spacetime

Notes

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

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