Plebanski Formulation of General Relativity: A Practical Introduction

TL;DR

This paper provides a clear, pedagogical introduction to Plebanski's self-dual two-forms formulation of general relativity, demonstrating its practical application to key solutions and its advantages over traditional methods.

Contribution

It offers an explicit, accessible presentation of Plebanski's formulation, illustrating how to derive important solutions and limits of GR using this alternative formalism.

Findings

Derivation of Schwarzschild solution within Plebanski formalism

Application to Newtonian limit and gravitational waves

Comparison showing efficiency of Plebanski approach

Abstract

We give a pedagogical introduction into an old, but unfortunately not commonly known formulation of GR in terms of self-dual two-forms due to in particular Jerzy Plebanski. Our presentation is rather explicit in that we show how the familiar textbook solutions: Schwarzschild, Volkoff-Oppenheimer, as well as those describing the Newtonian limit, a gravitational wave and the homogeneous isotropic Universe can be obtained within this formalism. Our description shows how Plebanski formulation gives quite an economical alternative to the usual metric and frame-based schemes for deriving Einstein equations.