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Tetrads in General Relativity

II. Tetrad and Vierbein fields

Tetrad Field

   A tetrad  (Greek foursome) is a set of axes $\{ {{\mathbf{e}}_a}(x);a = 0,1,2,3\}$ at a point $x = \{ {x^\mu }\} $ of the spacetime ${\mathcal{M}_4}$ that span the tangent space ${V_4}(x) = {T_{x}}\mathcal{M}_4 $. A common choice is an orthonormal tetrad, where the axes form a local frame at each point, so that the scalar products of the axes constitute the Minkowskian metric ${\eta _{ab}} = (1, - 1, - 1, - 1)$:

2.1

\[  {{\mathbf{e}}_a}(x) \cdot {{\mathbf{e}}_b}(x) = {\eta _{ab}}  \]

By using the matrix inverse of the metric, a reciprocal orthonormal tetrad basis may be defined by

2.2

\[  {{\mathbf{e}}^a}(x) \cdot {{\mathbf{e}}_b}(x) = \delta _b^a  \]

This construction exists at every point of ${\mathcal{M}_4}$ , independent of the coordinate basis $\{ {{\mathbf{g}}_\mu },{{\mathbf{g}}^\nu }\} $.

   The inner product (2.1) is invariant under local Lorentz transformations (LLT’s)

2.3

\[{{\mathbf{e}}_a}(x) \to {{\mathbf{e'}}_a}(x) = {\Lambda _a}^b(x){{\mathbf{e}}_b}(x)\]

The matrices ${\Lambda _a}^b(x)$ represent position-dependent inverse Lorentz transformations which operate on the basis vectors. Thus, the tetrad indices $a,b,c,...$ may be understood as forming the vector representation of the Lorentz group $\text{SO}(1,3)$ with six independent degrees of freedom, three degrees of freedom in spatial rotations, and three more in Lorentz boosts. Tetrad transformations rotate the tetrad axes at each point, while leaving the back-ground coordinates $\{x^\mu \}$ unchanged. So, in the context of GRT the Lorentz group is the symmetry group of local tetrad rotations and boosts; see section VI, Gauge Principle.

   The tetrad base at each point of spacetime defines a local Lorentz frame with a Minkowski metric, representing the proper rest frame of a local (idealized) observer. On assumption of the local nature of physics, the tetrad formalism offers a physical interpretation of GRT in terms of concepts of special relativity; see section VI, Postulates.