Table Of ContentTutorials, Schools, and Workshops in the
Mathematical Sciences
Sergio Cacciatori
Batu Güneysu
Stefano Pigola
Editors
Einstein Equations:
Physical and
Mathematical
Aspects of General
Relativity
Domoschool 2018
Tutorials, Schools, and Workshops
in the Mathematical Sciences
Thisserieswillserveasaresourceforthepublicationofresultsanddevelopments
presentedatsummerorwinterschools,workshops,tutorials,andseminars.Written
in an informal and accessible style, they present important and emerging topics
in scientific research for PhD students and researchers. Filling a gap between
traditional lecture notes, proceedings, and standard textbooks, the titles included
inTSWMSpresentmaterialfromtheforefrontofresearch.
Moreinformationaboutthisseriesathttp://www.springer.com/series/15641
¨
Sergio Cacciatori • Batu Guneysu • Stefano Pigola
Editors
Einstein Equations: Physical
and Mathematical Aspects
of General Relativity
Domoschool 2018
Editors
SergioCacciatori BatuGu¨neysu
DepartmentofScienceandHigh MathematischesInstitut
Technology Humboldt-Universita¨tzuBerlin
UniversityInsubrien Berlin,Germany
Como,Italy
StefanoPigola
DepartmentofScienceandHigh
Technology
UniversityInsubrien
Como,Italy
ISSN2522-0969 ISSN2522-0977 (electronic)
Tutorials,Schools,andWorkshopsintheMathematicalSciences
ISBN978-3-030-18060-7 ISBN978-3-030-18061-4 (eBook)
https://doi.org/10.1007/978-3-030-18061-4
MathematicsSubjectClassification(2010):83C05,83C15,83C20,83C35,83C40,83C47,83C57
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Preface
This book presents four lecture courses and, in addition, nine talks given at the
first edition of the Domoschool, the International Alpine School in Mathematics
andPhysics,heldinDomodossolainJuly2018withthetitle“EinsteinEquations:
PhysicalandMathematicalaspectsofGeneralRelativity.”
Domoschool is a 5-day summer school aiming to bring together young math-
ematicians and physicists working in the same field of physics. Four days are
devotedtofourlecturecourses(eachconsistingoffourlectures)givenbyprominent
experts. Despite its theoretical nature, the school also includes a short course on
experimentalaspects of the chosen topic, aiming to motivate a further theoretical
research.TheadditionaldayofDomoschoolisdevotedtosocialactivities(suchasa
hikingtourandadinner),aimingtoshowtheparticipantsthetownofDomodossola
and its surroundings. The school also includes a public meeting with the local
communityin orderto explain to the citizens of Domodossolathe contents of the
summer school and the scientific relevance. In 2018, the topic was chosen to be
GeneralRelativity.PubliclecturesweregivenbyDanielaBortoletto,LuigiBignami,
and Giancarlo Cella. Alessandro Carlotto gave an additional, spontaneous public
lecture,explainingthecontentsofthissummerschooltothebroaderaudience.
General Relativity is perhaps the most spectacular creation of Albert Einstein.
He started elaborating a new theory of gravitation immediately after his annus
mirabilis (1905). However, creating a relativistic theory of gravity took 10 long
yearsofhardandalmostunsuccessfulwork,andthecorrectformulationsuddenly
appearedinthelastfewmonthsofhisworks,writtenbetweentheendof1915and
the beginningof 1916. General Relativity immediately had many groundbreaking
consequences, even though the theory was not initially accepted by the scientific
community. For Einstein himself, the correct prediction of the secular precession
of Mercury’sperihelion was the smoking gun, but what broughtthe theory to the
forefrontof the worldwas the verification,in 1919,ofthe bendingof lightbythe
sun,predictedbythisnewtheory.Today,weknowhundredsoftestsagreeingwith
thepredictionsofGeneralRelativitywhilerulingoutseveralalternativetheoriesof
gravityandaddingverystrongconstraints.
v
vi Preface
However, the history of General Relativity is even more interesting. In 1917,
in a famous paper where he introduced the cosmological constant, Einstein gave
origintomoderncosmology,andin1918,hepredictedtheexistenceofgravitational
waves but considered it as just a theoretical speculation since the weakness of
the phenomenon appeared to make it undetectable. However, despite its exper-
imental verification before 1920, it is worth mentioning that General Relativity
did not receive much attention in the physics community for quite some time.
A new golden era started in the 1960s when black hole physics attracted the
attentionofmathematiciansandphysicists,likeRogerPenrose,StephenHawking,
SubrahmanyanChandrasekhar,etc., even though the first black hole solution was
already known in 1917, thanks to the works of Karl Schwarzschild. In the same
years, the first observations of neutron stars, pulsars, and very compact objects,
whose theoretical description required the use of a relativistic theory of gravity,
contributed heavily to the acceptance of General Relativity as the true theory of
gravity. In particular, in 1974, Russell Alan Hulse and Joseph Hooton Taylor Jr.
observedthe orbitofa pulsararounda neutronstar measuringa decreasingofthe
orbiting period in perfectagreement with the prediction of the Einstein equations
about the emission of gravitational waves. The direct detection of gravitational
waves, however,was notbefore the twenty-first century.Starting from September
2015 up to August 2017, six events have been detected by LIGO and Virgo labs:
GW150914, GW151226, GW170104, GW170608, GW170814, and GW170817.
The first five events correspondedto the catastrophic merging of black hole pairs
after tremendouslyviolent dances. All of them agreed with the predictionsof the
EinsteinequationsascomputedbythenewmethodsintroducedbyThibaultDamour
andcollaborators.The sixtheventwas evenmoreformidable:itwas generatedby
the merging of two neutron stars, giving rise not only to gravitational waves but
also to electromagnetic waves, allowing for comparisonsand several new tests of
thetheories.Thisrepresentedthebeginningofanewera!Mergingofneutronstars,
calledmultimessengers,isanewinstrumentforlookingatouruniverseaswehave
neverseenitbefore.
But the surprisesarising from the Einstein equationsare notonly at the exper-
imental level. The Cauchy problem for the Einstein equations can be formulated
by specifying, on a suitable spacelike surface, initial data that have to satisfy
certain constraintequations.In their study of precisely these constraintequations,
AlessandroCarlottoandRichardSchoendiscoveredcertainlocalizationphenomena
having many consequences. Among them, as Piotr Chrusciel pointed out, there is
a gravitation screening phenomenon,very different from the one that can happen
inNewtoniangravity.Afteracentury,Einstein’sequationscontinuetoprovidenew
surprises!Thesearethereasonsunderlyingourchoicetoorganizeaschoolonboth
theoreticalandexperimentalrecentachievementsinGeneralRelativity.
Thebookisdividedintotwoparts.Thefirstpartisdevotedtothecoursesgiven
attheschoolbyleadinginternationalexperts.
The first course, given by Alessandro Carlotto, had the primary scope of
introducinggraduatestudentstothemathematicsofgravitationalisolatedsystems.
It was divided into four lectures. The first lecture provided a nice introduction
Preface vii
to Einstein equations expressed in terms of constraint equations for the Cauchy
problem,startingfromNewtoniangravityandpassingthroughLorentziangeometry.
The second lecture was devoted to the positive Riemannian mass theorem, by
Richard Schoen and Shing-TungYau, which states the positivity of the Arnowitt-
Deser-Misner mass for asymptotically flat manifoldswith Euclidean signature. In
particular,hepresentedtheveryrecentproofvalidfordimensionsD ≥8.Inlecture
three, it was shown that a massless asymptoticallyflat Riemannian manifoldwith
nonnegativescalar curvatureis necessarily a flat Euclideanspace. Italso included
an interesting discussion of similar rigidityphenomena.Finally,the fourthlecture
coursewasdevotedtothegluingmethodsusedbyAlessandroCarlottoandRichard
Schoeninordertoconstructnewlocalizedsolutions,whichareasymptoticallyflat
initialdatasets,withpositiveADMmassthatarepreciselytrivialoutsideaconeof
agivenangle.
The second course was given by Felix Finster and concernsthe linear stability
fornon-extremalKerrblackholesunderperturbations.Thecourseintroducedtoan
advancedtopicinaverydidacticway.Theauthorstartedwiththedescriptionofthe
Kerrsolution,examiningitssymmetriesandtheassociatedconservedquantities.He
then discussed the main instability processes, like superradiance and the Penrose
process for extracting energy, and the long-time behavior of the Dirac field. It
was then shown that the Teukolsky equation, arising in the decoupling of waves
of different spin, can always be fully separated. This allowed him to analyze the
stabilityofthenon-extremalKerrspacetimeafterexpressingtheprobleminterms
of the decay of solutions of the Cauchy problem for the Teukolsky equation. An
outlineoftheproofofthestabilitywasfinallygiven.
The third course was given by Alexander Kamenshchik and was devoted to
cosmology.This course was rather atypicalfor cosmology in the sense that when
solely looking at some symmetries of the Einstein equation completely, new
phenomena can arise. The course started with a discussion of the symmetries of
theBianchiclassificationofthree-dimensionalLiealgebrasand,inparticular,ofthe
homogeneouscosmologies.The author then took a closer look at the solutionsof
the Einstein equationsfor Bianchi-Iand Bianchi-IIuniversesand the geometryof
Bianchi-IXuniverses.HealsoexplainedtheoscillatorymethodintroducedbyV.A.
Belinski, I.M. Khalatnikov, and E.M. Lifshitz for approaching the cosmological
singularityanditsHamiltoniandescription,theMixmasteruniverse,introducedby
C. Misner. Finally, it was shown how chaosenters in cosmologyand how infinite
dimensionalLiealgebrasmightplayacrucialroleinthiscontext.
ThefourthcoursewasgivenbyGiancarloCella.Thiscoursewasverydifferent
from the other ones in the sense that it was devoted to experimental physics: the
authorexplainedtheexperimentsoftheLIGO-Virgolabs,whichdetectedthegravi-
tationalwaveeventsmentionedabove.Thelecturesgavequiteacompletepanorama
on the phenomenological and experimental aspects and showed the magnificent
predictive power of General Relativity. After a historical introduction, the basic
principles of the gravitational detectors were discussed. Then, the fundamental
sources of noise were analyzed: seismic, thermal, and quantum noise, parametric
instabilities, and gravity fluctuations. Next, the detections were described, both
viii Preface
for the black holes coalescences and for the multimessenger event GW170817.
The potential of multimessenger detection was also discussed. Finally, the author
presentedproposalsforotherpossiblesourcesbeyondcoalescences,likecontinuous
sources,burstsignals,andstochasticbackgrounddetections.
Theschoolalsoincludedtwoothercoursesnottransposedinthepresentbook.A
coursegivenbyPiotrChrusciel,“IntroductiontotheGeneralRelativisticConstraint
Equations,” was not included here because the course aimed to introduce the
physicsstudentswhoattendedDomoschooltothemathematicalconceptsofGeneral
Relativity. The presented material is available in the literature and can be found,
forexample,onhispersonalhomepage:homepage.univie.ac.at/piotr.chrusciel.The
other course not included here was a course given by Eric Poisson, “Advanced
NewtonianGravity,”whose lectureshave beenbased on his book[E. Poisson and
Clifford M. Will, Gravity: Newtonian, Post-Newtonian, Relativistic, Cambridge
UniversityPress(2014)Cambridge,UK].
Although Domoschool is a school aiming to offer PhD students and young
researchers high-level courses given by experts, it also provides space for young
researcherstopresenttheirresearchresultsinshorttalks.Reflectingthesetalks,the
secondpartofthebookpresentsthesetalksinaconferenceproceeding-typemanner.
InDomoschool2018,nineoftheparticipantspresentedashorttalk:
Gabriele Bozzola and Vasileios Paschalidis (University of Arizona, Tucson,
USA):GenerationofInitialDataforGeneral-RelativisticSimulationsofCharged
BlackHoles
Federico Capone (University of Southampton, UK): BMS Symmetries and
Holography:AnIntroductoryOverview
Vittorio De Falco(Silesian Universityin Opava,CzechRepublic):Relativityof
ObserverSplittingFormalismandSomeAstrophysicalApplications
DavideFermi(UniversitàdegliStudidiMilano):SomeRemarksonaNewExotic
SpacetimeforTimeTravelbyFreeFall
Colin MacLaurin(UniversityofQueensland,Brisbane,Australia):Generalised
Gullstrand-PainleveSlicingsofSchwarzschild
JirˇíRyznerandMartinŽofka(CharlesUniversityinPrague):CrystalSpacetimes
withDiscreteTranslationalSymmetry
Jirˇí Veselý and Martin Žofka (Charles University in Prague): Electrogeodesics
andExtremalHorizonsinKerr–Newman–(anti-)deSitter
SebastianH.VölkelandKostasD.Kokkotas(UniversityofTübingen,Germany):
HearingtheNatureofCompactObjects
Adamantia Zampeli (Charles University in Prague): Minisuperspace Quantisa-
tionviaConditionalSymmetries
Domoschool was a project initiated by Sergio Cacciatori, but its realization
wasmadepossible,thankstotheopen-mindedpeopleandtheenthusiasticsupport
of the Municipality of Domodossola, which supported the school both financially
and concretely in its organization. For this, we are indebted to the Mayor Lucio
Pizzi; CouncilorforCulture DanieleFolino,who wasthe firstone welcomingour
proposal and connecting us to the right people; and, especially, Deputy Mayor
Angelo Tandurella, who was fully available to us, assisting step-by-step through
theevolution,thelogistics,thepublicevents,andsoon.
Preface ix
Additionally,wewanttoacknowledgethenoprofitassociationArs.Uni.VCO(Asso-
ciazione per lo Sviluppo della Cultura di Studi Universitari e della Ricerca nel
Verbano Cusio Ossola); its outgoing president and current vice president, Giulio
Gasparini,whowaspresentduringthefoundationoftheschool;andtheincoming
president, Stefania Cerutti, who inaugurated the school, together with Angelo
Tandurella. We are particularly indebted to the secretary, Andrea Cottini, and the
communicationmanager,FedericaFili,fortheircontinuoussupport,organizingthe
school, assisting all participants, and, in particular, together with the Rosminian
Fathers(especiallyPadre Fausto),makingit possible to gainCollegio Rosminias
thelocationoftheschool.
Furthermore, we are particularly grateful to the following speakers of
Domoschoolwhokindlyacceptedourinvitationsandwhose participationbrought
highprestigetotheschool:
AlessandroCarlottofromETH,Zurich,andIAS,Princeton
FelixFinsterfromUniversitätRegensburg,Germany
Alexander Kamenshchik from Università degli Studi di Bologna, Italy, and
LandauInstitute,Moscow
GiancarloCella,INFN,Pisa,Italy
PiotrChrusciel,fromtheUniversityofVienna,Vienna,Austria
EricPoisson,fromtheUniversityofGuelph,Guelph,Ontario,Canada
Weacknowledgethespeakerswhoacceptedourinvitationforthepubliclectures
givenatRoveretoSquarejustoutsidethecityhallofDomodossola.Thankstothem,
thepubliceventwasacompletesuccess.Theyareasfollows:
DanielaBortoletto,fromtheUniversityofOxford,Oxford,UKLuigiBignami,
Italianscientificjournalistandpopularizer
GiancarloCella,researcheratINFN
Alessandro Carlotto, who kindly spontaneously offered his contribution to the
publiclectures
Moreover, we extend our acknowledgments to the participants of the school,
includingtwohighschoolteachers,who,withtheiractiveparticipation,contributed
tomakingtheatmosphereoftheschoolparticularlypleasant.And,toothermembers
of the scientific board, Francesco Belgiorno, Simone Noja, Riccardo Re, Mauro
Giudici, and Pietro Antonio Grassi, as well as the members of the organizing
committee,AndreaCottiniandGiorgioMantica,wethankyouforyoursupport.
Last, but not least, we are grateful to the sponsors of Domoschool: Città
di Domodossola, GNFM-INDAM, Fondazione CRT, and Consiglio Regionale del
Piemonte.
Como,Italy SergioCacciatori
Berlin,Germany BatuGüneysu
Como,Italy StefanoPigola
February28,2019
