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Plebański formulation of general relativity: a practical introduction

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 Plebański. 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 Plebański formulation gives quite an economical alternative to the usual metric and frame-based schemes for deriving Einstein equations.

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Acknowledgements

The author was supported by an EPSRC Advanced Fellowship. The author is grateful to an anonymous referee for suggesting a list of references that grew into the “historical remarks” subsection.

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Correspondence to Kirill Krasnov.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Krasnov, K. Plebański formulation of general relativity: a practical introduction. Gen Relativ Gravit 43, 1–15 (2011). https://doi.org/10.1007/s10714-010-1061-x

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Keyword

  • Self-dual formulation of general relativity