\( \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \newcommand{\third}{{\scriptstyle \frac{1}{3}}} \newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter] \newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)

Tetrads in General Relativity

Preface

Summary

The essence of General Relativity not necessarily consists in the application of Riemannian geometry to physical space and time. In my view, it is much more useful to regard General Relativity above all as a theory of gravitation, whose connection with geometry arises from the peculiar empirical properties of gravitation, properties summarized by Einstein’s Principle of the Equivalence of Gravitation and Inertia.
Steven Weinberg, 1972

   The conventional formulation of General Relativity is based on a Riemannian geometry of a torsion-free spacetime with a covariantly conserved metric. The tetrad formalism, introduced by Albert Einstein in 1928 and independently developed by Hermann Weyl in 1929, generalizes the choice of basis for the tangent space 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 known as tetrads or vierbeins. This geometric structure exists independently of any prior gravity-model assumptions, facilitating a local description through inertial frames at each point.

   Physically, these local inertial frames represent the standard laboratory setups employed by observers conducting measurements in space and time. From a given local frame, different classes of observer frames may be obtained by performing local (point-dependent) Lorentz transformations. These include transformations that boost away local gravity. The statement that these different tetrad bases are equivalent, may be viewed as the gauge principle of local Lorentz symmetry. It encompasses the essence of Einstein’s equivalence principle that gravity may be annulled by acceleration.

   In the further development of general relativity, the mathematical insights of Élie Cartan (1869-1951) had a major influence, in particular, his theory of differential forms  (1899) and the concept of the Cartan connection (1926) with its corresponding curvature. By allowing torsion, Einstein’s vierbein formulation becomes a Poincaré gauge theory of gravitation in a Riemann-Cartan geometry.

   In 1913 Cartan introduced spinors in geometry before their physical relevance was discovered by Paul Dirac. In the context of General Relativity, and its extension to the Einstein-Cartan Theory, the tetrad formalism proves especially advantageous when working with spinors. It offers a natural and coherent framework for describing spinor fields and their interactions with gravity, seamlessly linking with the Dirac equation in curved spacetime.

   Mathematically the tetrad formalism is augmented by elevating the tangent spaces of the spacetime manifold to a Geometric Algebra (GA). This framework naturally accommodates arbitrary bases and encompasses the theory of differential forms. Additionally, calculations using GA are often more straightforward than those in traditional approaches. The GA formalism does not alter the underlying physics; it serves as an effective computational tool in this guide.