Bivectors in Newman-Penrose formalism in General Relativity -- from electromagnetism to Weyl curvature tensor

TL;DR

This paper reviews the use of bivectors within the Newman-Penrose formalism in General Relativity, focusing on electromagnetic fields and the Weyl curvature tensor, providing detailed calculations and conversions.

Contribution

It introduces the calculation methods for bivectors and complex Weyl tensor coefficients in Newman-Penrose formalism, linking electromagnetism to curvature analysis.

Findings

Derived complex coefficients of the Weyl tensor in Newman-Penrose formalism.

Converted electromagnetic fields from Minkowski to bivectors in null tetrad basis.

Provided detailed calculation procedures for bivectors in the formalism.

Abstract

The use of the bivectors in the General Relativity with Newman-Penrose formalism is important to the description of the exact solutions of the Einstein's field equations. This review is devoted to introduce the basic ideas with calculation details of the bivectors in Newman-Penrose formalism through conversion of the complex self-dual electromagnetic field from orthornormal Minkowski basis to bivectors in the complex null tetrad basis. Furthermore, in this context, it is obtained the complex coefficients of the Weyl tensor in Newman-Penrose formalism.