The Dirac equation in general relativity A guide for calculations Peter Collas1 and David Klein2 (September 7, 2018) Abstract In these informal lecture notes we outline different approaches used in doing calculations involving the Dirac equation in curved spacetime. We have tried to clarify the subject by carefully pointing out the various con- ventions used and by including several examples from textbooks and the existing literature. In addition some basic material has been included in the appendices. It is our hope that graduate students and other re- searchers will find these notes useful. arXiv:1809.02764v1 [gr-qc] 8 Sep 2018 1Department of Physics and Astronomy, California State University, Northridge, Northridge, CA 91330-8268. Email:
[email protected]. 2Department of Mathematics and Interdisciplinary Research Institute for the Sci- ences, California State University, Northridge, Northridge, CA 91330-8313. Email:
[email protected]. Contents 1 The spinorial covariant derivative3 1.1 The Fock-Ivanenko coefficients . .3 1.2 The Ricci rotation coefficient approach . .8 1.3 The electromagnetic interaction . .9 1.4 The Newman-Penrose formalism . 10 2 Examples 14 2.1 Schwarzschild spacetime N-P . 14 2.2 Schwarzschild spacetime F-I . 16 2.3 Nonfactorizable metric . 18 2.4 de Sitter spacetime, Fermi coordinates . 19 3 The Dirac equation in (1+1) GR 23 3.1 Introduction to (1+1) . 23 3.2 The Dirac equation in the Milne universe . 24 4 Scalar product 28 4.1 Conservation of j in SR . 28 4.2 The current density in GR . 29 4.3 The Scalar product in SR .