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Tetrads in General Relativity

II. Tetrad and Vierbein

Vierbein Field

   Associated with the tetrad frame at each point is a set of local coordinates ${x^a}$, $a \in \left\{ {0,1,2,3} \right\}$. Unlike the coordinates ${x^\mu }$ of the background geometry, the local coordinates ${x^a}$ do not extend beyond the local frame at each point. In these coordinates the scalar spacetime distance is $d{s^2} = {\eta _{ab}}d{x^a}d{x^b}$. Writing $d{x^a} = {e_\mu }^ad{x^\mu }$ one derives the relation:

2.4

\[  {g_{\mu \nu }}(x) = {e_\mu }^a(x) {e_\nu }^b(x){\eta _{ab}} \]

expressing ${g_{\mu \nu }}(x)$ in terms of Einstein's vierbein fields  ${e_\mu }^a(x)$ and the flat metric. This way, the spacetime metric may be seen as a deformation of the Minkowskian (tangent space) metric, the product of vierbein fields carrying the metric information. The first-order vierbein ${e_\mu }^a \simeq {\partial _\mu }{x^a} $ is called trivial. Except for some special cases e.g. a freely falling system, vierbeins in curved space are non-trivial: ${\partial _{[\mu }}{e_{\nu ]}}^a \ne 0$; see section Structure Coefficients.

   Being a symmetric rank-2 tensor in $d = 4$ dimensions, the metric tensor at left-hand side of (2.4) has $d(d + 1)/2$ independent components. On the other hand, since they have no particular symmetries, the number of independent components of the vierbeins is ${d^2}$. This means that the choice of the vierbeins is not unique. The difference in independent components is $d(d - 1)/2$, which matches precisely with the number of generators of the local Lorentz group in $d$-dimensions. Therefore, all the equivalent choices of the vierbein are related by local Lorentz transformations (2.3).

   If the local coordinates ${x^a}$ are kept fixed at each physical point $x$, the vierbeins ${e_\mu }^a$ change under manifold diffeomorphism, i.e. an invertible and differential map ${x^\mu } \to {x'^\mu }$, according to the rule

2.5

\[{e_\mu }^a(x) \to {e'_\mu }{_{}^a}(x') = \frac{{\partial {x^\nu }}}{{\partial {{x'}^\mu }}}{e_\nu }^a(x)\]

Thus, the vierbein field ${e_\mu }^a(x)$ may be thought of as forming four covariant tetrad fields ${{\mathbf{e}}^a}: = {e_\mu }^a{{\mathbf{g}}^\mu }$, one for each value of the upper index, with ${e_\mu }^a(x)$ the components with respect to the coordinate basis.