Zafar Ahsan The Potential of Fields in Einstein’s Theory of Gravitation ’ The Potential of Fields in Einstein s Theory of Gravitation Zafar Ahsan The Potential of Fields ’ in Einstein s Theory of Gravitation 123 Zafar Ahsan Department ofMathematics AligarhMuslim University Aligarh, Uttar Pradesh, India ISBN978-981-13-8975-7 ISBN978-981-13-8976-4 (eBook) https://doi.org/10.1007/978-981-13-8976-4 ©SpringerNatureSingaporePteLtd.2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface The theory of relativity has developed in two phases—special theory of relativity and general theory of relativity. Special theory of relativity adapted the concept of inertial frametothebasiclawofconstancyofthevelocityoflightdispensingwith theconceptofabsolutespaceandtimeofGalilean–Newtonianmechanics,whilethe generaltheoryofrelativitycameintoexistenceasanextensionofthespecialtheory of relativity. WiththetoolsofRiemanniangeometry,Einsteinwasabletoformulateatheory that predicts the behaviour of objects in the presence of gravitational, electro- magnetic and other forces. Through his general theory of relativity, Einstein redefined gravity. From the classical point of view, gravity is the attractive force between massive objects in three-dimensional space. In general relativity, gravity manifests as a curvature offour-dimensional spacetime. Conversely, curved space andtimegenerateseffectsthatareequivalenttogravitationaleffects.J.A.Wheelar has described the results as ‘matter tells spacetime how to bend and spacetime returnsthecomplementbytellingmatterhowtomove’.Onotherhand,cosmology is the science of the universe as a whole, and the study of cosmology requires several kinds of physics. Since the dominant force on the cosmic scale is gravita- tion,thisisthebasicingredientthatistakencareofbyEinstein’sgeneraltheoryof relativity. The matter distribution of fluids, gases, fields, etc., in the spacetime is given by the Einstein field equations. A cosmological model is a model of our universe that predicts the observed properties of the universe and explains the phenomena of the early universe. In a more restricted sense, cosmological models are the exact solutions of the Einstein field equations for a perfect fluid. The global geometry of the spacetime is determined by the Riemann curvature tensor, which can be decomposed in terms of the Weyl conformal tensor, Ricci tensorandmetrictensor.Thisdecompositioninvolvescertainirreducibletensors.In empty spacetime, the pure gravitational radiation field is described by the Weyl conformal tensor. However, when gravitational waves propagate through matter, the Weyl conformal tensor is still pertinent. In 1962, C. Lanczos thought that the Weylconformaltensorcanalsobederivedfromasimplertensorfield.Moreover,it is known that an electromagnetic field is generated through the covariant v vi Preface differentiationofavectorfield.Thequestionnowis:Whetheritispossibleornotto generate the gravitational field through a potential? The answer is yes; one can generatethegravitationalfieldthroughthecovariantdifferentiationofatensorfield. Thistensorfieldthatcanactasapotentialtothegravitationalfieldisnowknownas the Lanczos potential. Further, certain physical problems in general relativity are often conveniently described using a tetrad formalism adapted to the geometry of the particular situation. ThepresentbookdealswithadetailedstudyoftheLanczospotentialingeneral relativity and comprises eight chapters. Chapters 1–3 deal with a detailed study of tetrad formalism and its important examples—Newman–Penrose and Geroch– Held–Penrose formalisms. These discussions will then be used in the study of the Lanczos potential. Chapter 4 defines the Lanczos potential, and the equation by meansofwhichthegravitationalfieldiscreatedhasbeenderived.Suchequationis called Weyl–Lanczos equation, and this equation and other related results are expressed in terms of Newman–Penrose and Geroch–Held–Penrose formalisms. Chapter 5 gives a general prescription on how to generate a gravitational field of algebraically special fields, which is supported by a number of examples, while Chap.6dealswithyetanothermethodtoobtaintheLanczospotentialforaperfect fluidspacetime,andtheseresultsarethenusedtogeneratethegravitationalfieldof some cosmological models. Chapter 7 defines the Lanczos potentials for some well-known solutions of Einstein field equations, which have been obtained using tetrad formalisms. Apart from tetrad formalism, there are also some other methods to obtain the Lanczos potential. Such methods have been discussed in this chapter andappliedtofindtheLanczospotentialforGödelcosmologicalmodel.Chapter8, containssomemoreapplicationsoftheNewman–Penroseformalism.Amethodfor finding thesolution ofEinstein–Maxwell equations,usingNPformalism,hasbeen discussedindetail.Theinteraction between aPetrov typeNgravitational fieldand null electromagnetic field has been considered, and a metric describing this situa- tion has been obtained. A systematic and detailed study of symmetries of the spacetime (which are also known as collineations) has also been made in this chapter.Eachchapterofthisbookendswithalistofreferenceswhichbynomeans is a complete bibliography of the Lanczos potential and tetrad formalism; only the work referred to in this book has been included in the list. Some portions of this book were completed at Universiti Sains Islam Malaysia (USIM),Nilai,Malaysia,duringmystayasVisitingProfessor.Iamhighlythankful to Prof. Musa Ahmad, Vice Chancellor of the university, and Dr. Nurul Sima, Head, Department of Mathematics, for their excellent support and encouragement. Thanks are also due to the learned referee for his valuable suggestions and com- ments. I am also grateful to my publisher for having faith in me and specially to Mr. Shamim Ahmad for introducing me to the world of Springerand his guidance during the preparation of the manuscript. Aligarh, India Zafar Ahsan April 2019 Contents 1 The Tetrad Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Tetrad Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Directional Derivative and Ricci Rotation Coefficient . . . . . . . . . . 3 1.4 The Commutation Relation and the Structure Constants . . . . . . . . 6 1.5 The Ricci and the Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 The Newman–Penrose Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Complex Null Tetrad and the Spin-Coefficients . . . . . . . . . . . . . . 12 2.3 The Riemann, Ricci and Weyl Tensors . . . . . . . . . . . . . . . . . . . . 14 2.4 The Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 The Ricci Identities (NP Field Equations) . . . . . . . . . . . . . . . . . . 18 2.6 The Physical and Geometrical Meanings of the Spin-Coefficients and the Optical Scalars . . . . . . . . . . . . . . . . . . . 20 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 The Geroch–Held–Penrose Formalism . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Spacetime Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 GHP Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 The Geometry of the Null Congruences. . . . . . . . . . . . . . . . . . . . 33 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 Lanczos Potential and Tetrad Formalism . . . . . . . . . . . . . . . . . . . . . 41 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Lanczos Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Weyl–Lanczos Equation and Tetrad Formalism . . . . . . . . . . . . . . 47 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 vii viii Contents 5 Lanczos Potential for Algebraically Special Spacetimes . . . . . . . . . . 51 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Lanczos Potential for Petrov Type II Spacetimes . . . . . . . . . . . . . 51 5.3 Lanczos Potential for Petrov Type D Spacetimes . . . . . . . . . . . . . 54 5.4 Lanczos Potential for Petrov Type III Spacetimes. . . . . . . . . . . . . 56 5.4.1 Kaigorodov Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4.2 A Type III Solution with Twist . . . . . . . . . . . . . . . . . . . . 57 5.5 Lanczos Potential for Petrov Type N Spacetimes . . . . . . . . . . . . . 59 5.5.1 Generalized pp-Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.5.2 Pure Radiation Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.5.3 Kaigorodov Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.5.4 Twist-Free Gravitational Field . . . . . . . . . . . . . . . . . . . . . 63 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6 Lanczos Potential and Perfect Fluid Spacetimes . . . . . . . . . . . . . . . . 67 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2 General Observers and NP Formalism . . . . . . . . . . . . . . . . . . . . . 67 6.3 Perfect Fluid Spacetimes and Lanczos Potential . . . . . . . . . . . . . . 73 6.3.1 Shear-Free and Irrotational Spacetimes . . . . . . . . . . . . . . . 73 6.3.2 Bianchi Type I Universes. . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.3 Gödel Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7 Lanczos Potential For the Spacetime Solutions. . . . . . . . . . . . . . . . . 81 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.2 Some Special Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.2.1 Charged Rotating Black Hole. . . . . . . . . . . . . . . . . . . . . . 82 7.2.2 Kerr Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.2.3 Reissner–Nördstrom Metric . . . . . . . . . . . . . . . . . . . . . . . 84 7.2.4 Schwarzschild Exterior Solution . . . . . . . . . . . . . . . . . . . . 84 7.2.5 Vaidya Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2.6 Kantowski–Sachs Solution . . . . . . . . . . . . . . . . . . . . . . . . 87 7.3 Change of Tetrad and Lanczos Potential . . . . . . . . . . . . . . . . . . . 88 7.3.1 Gödel Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3.2 Taub Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.3.3 Petrov Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.3.4 Kaigorodov Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.3.5 Kasner Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3.6 Schwarzschild Exterior Solution . . . . . . . . . . . . . . . . . . . . 94 7.3.7 C-Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.3.8 Siklos Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Contents ix 7.4 Some Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.4.1 Method of Local and Isometric Embedding. . . . . . . . . . . . 96 7.4.2 Method of the Lovelock’s Theorem . . . . . . . . . . . . . . . . . 98 7.4.3 Method of the Wave Equation . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8 Newman–Penrose Formalism, Solution of Einstein–Maxwell Equations and Symmetries of the Spacetime. . . . . . . . . . . . . . . . . . . 103 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8.2 A Solution of Einstein–Maxwell Equations . . . . . . . . . . . . . . . . . 103 8.2.1 Equations for PR Fields and Their Simplifications. . . . . . . 105 8.2.2 The Solution Describing the Interaction of PR Fields. . . . . 111 8.3 Symmetries of Type N Pure Radiation Fields. . . . . . . . . . . . . . . . 112 8.3.1 NP Formalism and Collineations for PR Fields . . . . . . . . . 113 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Concluding Remarks ... .... ..... .... .... .... .... .... ..... .... 123 About the Author Zafar Ahsan is a former Professor at the Department of Mathematics, Aligarh Muslim University, India, where he completed his Ph.D. in Mathematics in 1979. He has previously served as a Visiting Associate at the Inter-University Centre of Astronomy and Astrophysics (IUCAA), Pune (1996–2005); a UGC Visiting ProfessoratSardarPatelUniversity,VallabhVidyanagar,Anand(2004–2005);and as a Visiting Professor at Universiti Sains Islam Malaysia (October–December, 2016). At present, Prof. Ahsan is a Research Fellow at Institut Sains Islam, Malaysia. ProfessorAhsanisalifememberofseverallearnedsocietiesincludingtheIndian AssociationofGeneralRelativityandGravitation,theAstronomicalSocietyofIndia, International Electronic Engineering Mathematical Society, Indian Mathematical Society and the Tensor Society of India. Further, he has served as Editor-in-Chief oftheAligarhBulletinofMathematics(2012–2015),Editor-in-ChiefoftheJournal ofTensorSocietyofIndia(2010–2012)andManagingEditoroftheAligarhBulletin of Mathematics (1998–2012). He is currently Editor, Palestine Journal of MathematicsandJournalofInterpolation&ApproximationinScientificComputing; andamemberoftheeditorialboardsofseveraljournals,includingtheGlobalJournal ofAdvancedResearchonClassicalandModernGeometries,theIslamicUniversity ofGazaJournalofNaturalandEngineeringStudies,andtheBulletinoftheCalcutta Mathematical Society and the Journal of the Calcutta Mathematical Society. With over 40 years of teaching experience, he has published five books and over 100 researchpapersinseveralinternationaljournalsofrepute.Hisresearchinterestsarein gravitationalwaves,symmetriesofspace–time,exactsolutionsofEinsteinequations, tetrad formalisms and differential geometric structures in general relativity. His current research interest, apart from General Relativity and Gravitation, is in the Qur’an,ScienceandModernCosmology. ProfessorAhsanisthePresidentofAnjumanFaroogh-e-Science(Associationfor PromotionofScience),Aligarhbranch;andhasbeenaCouncilMemberoftheIndian AssociationforGeneralRelativityandGravitation(2002–2006).Heistherecipient of the Nishan-e-Azad award (2017) for the promotion of science in Urdu; the International Einstein Award for Scientific Achievement (2011); the Rashtriya xi