Complex Spacetimes and the Newman-Janis trick

0.68 MB·

by  Del Rajan

Complex Spacetimes and the
Newman-Janis Trick
Deloshan Nawarajan
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1 School of Mathematics and Statistics
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Athesis
submittedtotheVictoriaUniversityofWellington
infulfilmentoftherequirementsforthedegreeof
MasterofScience
inMathematics.
VictoriaUniversityofWellington
2015

Abstract
In this thesis, we explore the subject of complex spacetimes, in which the math-
ematical theory of complex manifolds gets modified for application to General
Relativity. We will also explore the mysterious Newman-Janis trick, which is
an elementary and quite short method to obtain the Kerr black hole from the
Schwarzschild black hole through the use of complex variables. This exposition
will cover variations of the Newman-Janis trick, partial explanations, as well as
originalcontributions.

Acknowledgements
I want to thank my supervisor Professor Matt Visser for many things, but three
things in particular. First, I want to thank him for taking me on board as his
research student and providing me with an opportunity, when it was not a trivial
decision. Iamforevergratefulforthat.
IalsowanttothankMattforhisamazingsupportasasupervisorforthisresearch
project. This includes his time spent on this project, as well as teaching me on
other current issues of theoretical physics and shaping my understanding of the
Universe. Icouldn’thaveaskedforabettermentor.
Last but not least, I feel absolutely lucky that I had a supervisor who has the
personalcharacteristicsofbeingverykindandbeingverysupportive.
I want to thank my fellow graduate students in the School of Mathematics &
Statistics. Your company and our time spent together is a highlight of 2015 for
me.
I also want to thank Baktash and Padideh Fazelzadeh, Gayathara De Silva, Ash-
wyn Sathanantham and Chirag Ahuja for 2014. Without you, this would not be
possible.

Dedicatedtomyfamily.
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Contents
1 Introduction 1
2 ReviewofGeneralRelativity 3
2.1 Lorentziangeometry . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Vacuumspacetimes . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 NullTetrads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 ComplexManifoldTheory 11
3.1 ComplexLinearAlgebra . . . . . . . . . . . . . . . . . . . . . . 12
3.2 ComplexManifolds . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 FunctionsonaComplexManifold . . . . . . . . . . . . . . . . . 20
3.4 VectorsonaComplexManifold . . . . . . . . . . . . . . . . . . 21
3.5 TensorsonaComplexManifold . . . . . . . . . . . . . . . . . . 25
3.6 AlmostComplexManifolds . . . . . . . . . . . . . . . . . . . . 29
3.7 HermitianManifolds . . . . . . . . . . . . . . . . . . . . . . . . 32
3.8 Ka¨hlerManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 ComplexSpacetimes 41
4.1 ModifyingLorentziansignaturemetrics . . . . . . . . . . . . . . 42
4.2 ModifyingComplexManifoldTheory . . . . . . . . . . . . . . . 48
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viii CONTENTS
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 TheNewman-Janistrick 59
5.1 Newman-Janistrick . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 ExtensionsandApplications . . . . . . . . . . . . . . . . . . . . 64
5.3 Giampieri’smethod . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Analternativeversion . . . . . . . . . . . . . . . . . . . . . . . . 67
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6 Variousexplanations 71
6.1 Kerr-Talbotexplanation . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Newman’sexplanation . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Flaherty’sexplanation . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 Schifferetal. explanation . . . . . . . . . . . . . . . . . . . . . . 80
6.5 Drake-Szekeres’explanation . . . . . . . . . . . . . . . . . . . . 80
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7 Originalcontribution 83
7.1 Newman-JanisversusGiampieri . . . . . . . . . . . . . . . . . . 83
7.2 Non-holomorphicproblems . . . . . . . . . . . . . . . . . . . . . 90
7.3 Anewapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.5 ResemblancebetweenSchwarzschildandKerr . . . . . . . . . . 117
8 SummaryandConclusions 121