{"title":"The Dirac equation in general relativity and the 3+1 formalism","authors":"Miguel Alcubierre","doi":"10.1007/s10714-025-03460-8","DOIUrl":null,"url":null,"abstract":"<div><p>I present a review of the Dirac equation in general relativity. Although the generalization of the Dirac equation to a curved spacetime is well known, it is not usually part of the standard toolkit of techniques known to people working on classical general relativity. Recently, there has been some renewed interest in studying solutions of the Einstein–Dirac system of equations, particularly in the context of the so-called “Dirac stars”. Motivated by this, here I present a review of the Dirac equation in general relativity, starting from Minkowski spacetime, and then considering the Lorentz group and the tetrad formalism in order to generalize this equation to the case of a curved spacetime. I also derive the form of the Dirac equation and its associated stress–energy tensor for the case of the 3+1 formalism of general relativity, which can be useful for the study of the evolution of the Dirac field in a dynamical spacetime.</p></div>","PeriodicalId":578,"journal":{"name":"General Relativity and Gravitation","volume":"57 8","pages":""},"PeriodicalIF":2.8000,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10714-025-03460-8.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"General Relativity and Gravitation","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10714-025-03460-8","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ASTRONOMY & ASTROPHYSICS","Score":null,"Total":0}
引用次数: 0
Abstract
I present a review of the Dirac equation in general relativity. Although the generalization of the Dirac equation to a curved spacetime is well known, it is not usually part of the standard toolkit of techniques known to people working on classical general relativity. Recently, there has been some renewed interest in studying solutions of the Einstein–Dirac system of equations, particularly in the context of the so-called “Dirac stars”. Motivated by this, here I present a review of the Dirac equation in general relativity, starting from Minkowski spacetime, and then considering the Lorentz group and the tetrad formalism in order to generalize this equation to the case of a curved spacetime. I also derive the form of the Dirac equation and its associated stress–energy tensor for the case of the 3+1 formalism of general relativity, which can be useful for the study of the evolution of the Dirac field in a dynamical spacetime.
期刊介绍：
General Relativity and Gravitation is a journal devoted to all aspects of modern gravitational science, and published under the auspices of the International Society on General Relativity and Gravitation.
It welcomes in particular original articles on the following topics of current research:
Analytical general relativity, including its interface with geometrical analysis
Numerical relativity
Theoretical and observational cosmology
Relativistic astrophysics
Gravitational waves: data analysis, astrophysical sources and detector science
Extensions of general relativity
Supergravity
Gravitational aspects of string theory and its extensions
Quantum gravity: canonical approaches, in particular loop quantum gravity, and path integral approaches, in particular spin foams, Regge calculus and dynamical triangulations
Quantum field theory in curved spacetime
Non-commutative geometry and gravitation
Experimental gravity, in particular tests of general relativity
The journal publishes articles on all theoretical and experimental aspects of modern general relativity and gravitation, as well as book reviews and historical articles of special interest.
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