Non-coordinates basis in General Relativity and Cartan's structure equations

TL;DR

This paper introduces non-coordinate bases and Cartan's structure equations in General Relativity, emphasizing their utility in simplifying curvature calculations using tetrads or vierbeins.

Contribution

It provides a detailed introduction to non-coordinate bases and Cartan's structure equations, enhancing understanding and calculation methods in Riemannian geometry within General Relativity.

Findings

Clarifies the use of tetrads and vierbeins in local frames.

Demonstrates how Cartan's structure equations simplify curvature computations.

Provides detailed notation and conceptual explanations.

Abstract

The basic and fundamental aspects of General Relativity are in general analysed in mathematical level of coordinate basis or holonomic frame by several authors in the literature. However, for many purposes it is more convenient to use a general basis, often called in four dimensions, a tetrad or vierbein, very useful in a local frame with orthonormal basis or pseudo-orthonormal basis. This text presents an introduction to non-coordinate basis and the two Cartan's structure equations that are mathematical implements in Riemannian geometry that facilitate the calculation of curvature tensors. The purpose of this text is to approach the language and the notation of tetrad field or vierbein with conceptual and calculational details.