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Tetrads in General Relativity

II. Tetrad and Vierbein

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 tetrad base ${{\mathbf{e}}_a}(x)$ at each point of spacetime defines a local Lorentz frame with a Minkowski metric. It may be imagined to be the proper rest frame of a local (idealized) observer able to measure physical quantities, space and time distances in particular. On assumption of the local nature of physics, the tetrad formalism thus offers an interpretation of GRT in terms of concepts of special relativity; see section Postulates.

   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 Gauge Principle.

   Tetrads are geometric objects defined independently of coordinates. Tetrad components of tensors therefore do not change when a coordinate transformation is applied. Quantities that are unchanged by a coordinate transformation are coordinate gauge invariant. Quantities that are unchanged under a tetrad transformation are tetrad gauge invariant. For example, tetrad tensors are coordinate gauge-invariant, while coordinate tensors are tetrad gauge-invariant.