The Geometry of the Universe

274 Pages·2021·33.801 MB·English

by  Colin Rourke

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Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Series on Knots and Everything — Vol. 71
THE GEOMETRY OF THE UNIVERSE
Copyright © 2021 by Colin Rourke
All rights reserved.
ISBN 978-981-123-386-9 (hardcover)
ISBN 978-981-123-387-6 (ebook for institutions)
ISBN 978-981-123-388-3 (ebook for individuals)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/12195#t=suppl
Desk Editor: Yumeng Liu
Printed in Singapore
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Foreword
Theuniverseinwhichweliveisawonderful,amazingplace. Weknow
agreatdealaboutitthroughaseriesoffantasticobservationsusing
groundandspacebasedtelescopesandotherequipment, including
recently, equipment to detect gravitational waves. Cosmology, the
study of the universe, arouses a great deal of public interest, with
seriousarticlesbothinthescientificpressandinmajornewspapers,
withmanyofthetheoriesandconcepts(egthe“BigBang”and“black
holes”)discussed,ofteningreatdepth. Theobservationsthatsupport
thesediscussionsusesophisticatedandexpensivemachinerybothon
theearth(egtheLIGOequipmentusedtodetectgravitationalwaves)
andinorbitaroundtheearth(egtheHubbletelescope). Thisisvery
bigsciencewithbudgetstorivalthoseofwholecountries.
There is a consensus for the theoretical framework supporting and
interpreting these observations, which is known as the “standard
model”. Thisstartswithasingulareventknownasthe“BigBang”
andexpandsfromtheretofilltheentirevisibleuniverse. Theimage
presented is of a complete and full theory that (with a few minor
unsolvedproblems)explainsalltheobservationsthatwehave.
Thisimageis(likemanyimages)completelyfalse. Therearehuge
problemswiththestandardmodelandistheaimofthisbooktopresent
analternativemodel,whichavoidsthemostglaringoftheseproblems.
Thenewmodelisbasedaroundageometricdescriptionoftheuniverse
andhencethetitle. InthisnewmodelthereisnoBigBang—indeed
themodelhaswhatisknownasthePerfectCopernicanPrinciplethat
thereshouldbenospecialtimesorplaces—andnodarkmatter,which
isreplacedbyasimplegeometrichypothesis. Cosmologyisahighly
technicalsubjectinvolvingsomedeepanddifficultmathematicsand
presentsachallengetoanyauthorattemptingtocommunicateinaway
that is accessible to a non-technical audience. Nevertheless I shall
attempttodothis. ButIwillalsoallowmyselftheluxuryofgiving
fulltechnicaldetailsforreaderswhowishtoexplorethem.
Accordinglythebookisdividedintothreeparts:
v
vi Foreword
Part1 Thewholestorypresentedasfaraspossibleforanontechnical
reader
Part2 Thesamestory,toldagainbutforareaderwithsometechnical
knowledge
Part3 Appendiceswithfulltechnicaldetailsofseveraloftheimpor-
tanttopicscovered.
Part 1 should be readable (and understandable) by anyone with a
noddingacquaintancewiththebasiclanguageofcosmology: events,
lightspaths,galaxies,blackholesandsoon. Itcoversthewholestory
ofthebookinasuntechnicalwayaspossiblegiventhescopeofthe
topicscovered.
Part2coversthesamegroundagainbutwithenoughtechnicaldetailsto
satisfyareaderwithbasicknowledgeofmathematicsand/orphysics.
Part3consistsofappendiceswhicharereferredtointheotherparts
and which also contain the highly technical material omitted from
Section2.
SeveralpartsofthisbookarebasedonjointworkwithRobertMacKay
andIthankhimforallowingmetousethismaterialandalsofora
thoughtfulcriticalread. IwouldalsoliketothankRosembergToala
Enriquesfortheuseofthematerialinthedraftthreeauthorpaper[88]
onquasars. SpecialthanksareduetoRobertMacKay,IanStewartand
RobKirbyforunfailingsupportthroughthediscouragingprocessof
attemptingtopublishthisworkinseriousscientificjournals.
Anearlyversionofthisbookwiththetitle“Anewparadigmforthe
universe”(suitableforreaderswithtechnicalknowledge)isstoredon
thearXivatastro-ph/0311033andcanalsobedownloadedfrom
theauthor’swebpage.
ColinRourke
October2020
MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UK
[email protected] http://msp.warwick.ac.uk/~cpr
AMSclassification 85A40;83C57,85A15,85A05,83F05
Abouttheauthor
ColinRourkeisaprofessoremeritusofmathematicsattheUniversityofWarwick,
andhasalsotaughtatthePrincetonInstituteforAdvancedStudy, QueenMary
CollegeLondon,theUniversityofWisconsinatMadison,andtheOpenUniversity,
wherehemastermindedrewritingthepuremathematicscourse; hehasrecently
retiredfromlecturingaftercompletingahalf-century(of50yearsoflecturing). He
haswrittenpapersinhigh-dimensionalPLtopology,low-dimensionaltopology,
combinatorialgrouptheoryanddifferentialtopology.
In1996,dissatisfiedwiththerapidlyrisingfeeschargedbythemajorpublishers
of mathematical research journals, Colin decided to start his own journal, and
wasablyassistedbyRobKirby,JohnJonesandBrianSanderson. Thatjournal
becameGeometry&Topology. UnderColin’sleadership,GThasbecomealeading
journalinitsfieldwhileremainingoneoftheleastexpensiveperpage. GTwas
joinedin1998byaproceedingsandmonographsseries,Geometry&Topology
Monographs,andin2000byasisterjournal,Algebraic&GeometricTopology.
Colinwrotethesoftwareandfullymanagedthesepublicationsuntilaround2005
whenhecofoundedMathematicalSciencesPublishers(withRobKirby)totake
overtherunning. MSPhasnowgrowntobecomeaformidableforceinacademic
publishing. WithhiswifeDaphne,herunsasmallholdinginNorthamptonshire,
wheretheyfarmHebrideansheepandAnguscattle.
In2000hestartedtakinganinterestincosmologyandpublishedhisfirstsubstantial
foray on the ArXiv preprint server in 2003. For the past twelve years he has
collaborated with Robert MacKay also of Warwick University with papers on
redshift,gammarayburstsandnaturalobserverfields. Henowfeelsthathehas
mastered the basis of a completely new geometrical way of understanding the
universe,withouteitherdarkmatteroraBigBang. Itisthisthatispresentedin
thisbook.
Photo:PipSheldon
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Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Abouttheauthor . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Tableofcontents . . . . . . . . . . . . . . . . . . . . . . . . . . ix
PART1 1
1 FromtheGreekstoEinstein 3
1.1 KeplerandNewton . . . . . . . . . . . . . . . . . . . . . . . . . 5
Kepler’slawsofplanetarymotion . . . . . . . . . . . . . . . . . 5
1.2 Newtonianphysics: gravitationanddynamics . . . . . . . . . . . 6
Newton’slawsofmotion . . . . . . . . . . . . . . . . . . . . . . 6
Newton’slawofgravitation . . . . . . . . . . . . . . . . . . . . . 7
1.3 DerivationofKepler’slaws2and3. . . . . . . . . . . . . . . . . 7
Angularmomentum . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 MaxwellandtheroadtoSpecialRelativity . . . . . . . . . . . . . 10
TheætherandtheMichelson–Morleyexperiments . . . . . . . . 10
2 Einstein,relativity,modelbuildinganddeSitter 13
2.1 SpecialRelativity . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 GeneralRelativity . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Modelbuilding . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Einstein’s“biggestblunder” . . . . . . . . . . . . . . . . . . . . 19
2.4 deSitterspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
deSitterspaceandMach’sprinciple . . . . . . . . . . . . . . . . 22
3 Thebiggestblunder,darkmatterandquasars 25
3.1 Thebiggestblunder . . . . . . . . . . . . . . . . . . . . . . . . . 25
Weyl’spostulate . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
DeSitterspaceasamodelfortheuniverse . . . . . . . . . . . . . 27
3.2 Thestandardmodel . . . . . . . . . . . . . . . . . . . . . . . . . 29
ix