Geometric Algebra Techniques for General Relativity

TL;DR

This paper introduces a geometric algebra framework for general relativity, simplifying the formulation and derivation of key solutions like Schwarzschild and Kerr, and analyzing the Weyl tensor's Petrov types.

Contribution

It develops a geometric algebra approach to general relativity, combining differential forms and coordinate methods for more efficient calculations.

Findings

Derived Schwarzschild and Kerr solutions using geometric algebra.

Provided a detailed classification of Weyl tensor Petrov types.

Demonstrated the effectiveness of geometric algebra in gravitational physics.

Abstract

Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.