Table Of ContentNOETHER SYMMETRIES IN THEORIES
OF GRAVITY
ThisvolumesummarizesthemanyalternativesandextensionstoEinstein’sGen-
eral Theory of Relativity and shows how symmetry principles can be applied to
identifyphysicallyviablemodels.Thefirstpartofthebookestablishesthefoun-
dations of classical field theory, providing an introduction to symmetry groups
and the Noether theorems. A quick overview of general relativity is provided,
including discussion of its successes and shortcomings; then several theories of
gravity are presented, and their main features are summarized. In the second
part, the “Noether Symmetry Approach” is applied to theories of gravity to
identify those which contain symmetries. In the third part of the book, these
selected models are tested through comparison with the latest experiments and
observations. This constrains the free parameters in the selected models to fit
the current data, demonstrating a useful approach that will allow researchers to
construct and constrain modified gravity models for further applications.
francesco bajardi is a researcher at Scuola Superiore Meridionale di
Napoli. His primary areas of research are general relativity, theoretical cosmol-
ogy, and theories of gravity. He graduated at the University of Palermo and
obtained his PhD at the Università di Napoli “Federico II.”
salvatore capozziello is Full Professor in Astronomy and Astrophysics
at the Università di Napoli “Federico II,” and former President of the Italian
Society for General Relativity and Gravitation. His primary areas of research
are general relativity, cosmology, and relativistic astrophysics. He has authored
more than 600 scientific publications with more than 34,000 citations.
Published online by Cambridge University Press
CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS
S.J.AarsethGravitationalN-BodySimulations:ToolsandAlgorithms†
D.AhluwaliaMassDimensionOneFermions
J.Ambjørn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheory
Approach†
A.M.AnileRelativisticFluidsandMagneto-fluids:WithApplicationsinAstrophysicsand
PlasmaPhysics
J.A.deAzcárragaandJ.M.IzquierdoLieGroups,LieAlgebras,CohomologyandSome
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O.Babelon,D.BernardandM.TalonIntroductiontoClassicalIntegrableSystems†
F.BajardiandS.CapozzielloNoetherSymmetriesinTheoriesofGravity
F.BastianelliandP.vanNieuwenhuizenPathIntegralsandAnomaliesinCurvedSpace†
D.BaumannandL.McAllisterInflationandStringTheory†
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N.D.BirrellandP.C.W.DaviesQuantumFieldsinCurvedSpace†
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andtheOperator-ProductExpansion†
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M.CreutzQuarks,GluonsandLattices†
P.D.D’EathSupersymmetricQuantumCosmology†
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Y.FrishmanandJ.SonnenscheinNon-PerturbativeFieldTheory:FromTwoDimensional
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J.A.FuchsAffineLieAlgebrasandQuantumGroups:AnIntroduction,withApplicationsin
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Y.FujiiandK.MaedaTheScalar-TensorTheoryofGravitation†
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R.GambiniandJ.PullinLoops,Knots,GaugeTheoriesandQuantumGravity†
T.GannonMoonshinebeyondtheMonster:TheBridgeConnectingAlgebra,ModularFormsand
Physics†
A.García-DíazExactSolutionsinThree-DimensionalGravity
M.GöckelerandT.SchückerDifferentialGeometry,GaugeTheories,andGravity†
C.Gómez,M.Ruiz-AltabaandG.SierraQuantumGroupsinTwo-DimensionalPhysics†
M.B.Green,J.H.SchwarzandE.WittenSuperstringTheoryVolume1:Introduction
M.B.Green,J.H.SchwarzandE.WittenSuperstringTheoryVolume2:LoopAmplitudes,
AnomaliesandPhenomenology
V.N.GribovTheTheoryofComplexAngularMomenta:GribovLecturesonTheoreticalPhysics†
J.B.GriffithsandJ.PodolskýExactSpace-TimesinEinstein’sGeneralRelativity†
T.HarkoandF.LoboExtensionsoff(R)Gravity:Curvature-MatterCouplingsandHybrid
Metric-PalatiniGravity
S.W.HawkingandG.F.R.EllisTheLargeScaleStructureofSpace-Time†
J.HarnadandF.BaloghTauFunctionsandTheirApplications
B.B.HuandE.VerdaguerSemiclassicalandStochasticGravity
F.IachelloandA.ArimaTheInteractingBosonModel†
F.IachelloandP.vanIsackerTheInteractingBoson-FermionModel†
C.ItzyksonandJ.M.DrouffeStatisticalFieldTheoryVolume1:FromBrownianMotionto
RenormalizationandLatticeGaugeTheory†
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C.ItzyksonandJ.M.DrouffeStatisticalFieldTheoryVolume2:StrongCoupling,MonteCarlo
Methods,ConformalFieldTheoryandRandomSystems†
G.JaroszkiewiczPrinciplesofDiscreteTimeMechanics
G.JaroszkiewiczQuantizedDetectorNetworks
C.V.JohnsonD-Branes†
P.S.JoshiGravitationalCollapseandSpacetimeSingularities†
J.I.KapustaandC.GaleFinite-TemperatureFieldTheory:PrinciplesandApplications,2nd
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V.E.Korepin,N.M.BogoliubovandA.G.IzerginQuantumInverseScatteringMethodand
CorrelationFunctions†
K.KrasnovFormulationsofGeneralRelativity
J.KroonConformalMethodsinGeneralRelativity
M.LeBellacThermalFieldTheory†
L.LusannaNon-InertialFramesandDiracObservablesinRelativity
Y.MakeenkoMethodsofContemporaryGaugeTheory†
S.MallikandS.SarkarHadronsatFiniteTemperature
A.MalyarenkoandM.Ostoja-StarzewskiTensor-ValuedRandomFieldsforContinuumPhysics
N.MantonandP.SutcliffeTopologicalSolitons†
N.H.MarchLiquidMetals:ConceptsandTheory†
I.MontvayandG.MünsterQuantumFieldsonaLattice†
P.NathSupersymmetry,Supergravity,andUnification
L.O’RaifeartaighGroupStructureofGaugeTheories†
T.OrtínGravityandStrings,2ndedition
A.M.OzoriodeAlmeidaHamiltonianSystems:ChaosandQuantization†
M.ParanjapeTheTheoryandApplicationsofInstantonCalculations
L.ParkerandD.TomsQuantumFieldTheoryinCurvedSpacetime:QuantizedFieldsand
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V.N.PopovFunctionalIntegralsandCollectiveExcitations†
L.V.ProkhorovandS.V.ShabanovHamiltonianMechanicsofGaugeSystems
S.RaychaudhuriandK.SridharParticlePhysicsofBraneWorldsandExtraDimensions
A.RecknagelandV.SchiomerusBoundaryConformalFieldTheoryandtheWorldsheet
ApproachtoD-Branes
M.ReuterandF.SaueressigQuantumGravityandtheFunctionalRenormalizationGroup
R.J.RiversPathIntegralMethodsinQuantumFieldTheory†
R.G.RobertsTheStructureoftheProton:DeepInelasticScattering†
P.RomatschkeandU.RomatschkeRelativisticFluidDynamicsInandOutofEquilibrium:And
ApplicationstoRelativisticNuclearCollisions
C.RovelliQuantumGravity†
W.C.SaslawGravitationalPhysicsofStellarandGalacticSystems†
R.N.SenCausality,MeasurementTheoryandtheDifferentiableStructureofSpace-Time
M.ShifmanandA.YungSupersymmetricSolitons
Y.M.ShnirTopologicalandNon-TopologicalSolitonsinScalarFieldTheories
H.Stephani,D.Kramer,M.MacCallum,C.HoenselaersandE.HerltExactSolutionsof
Einstein’sFieldEquations,2ndedition†
J.StewartAdvancedGeneralRelativity†
J.C.TaylorGaugeTheoriesofWeakInteractions†
T.ThiemannModernCanonicalQuantumGeneralRelativity†
D.J.TomsTheSchwingerActionPrincipleandEffectiveAction†
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E.J.WeinbergClassicalSolutionsinQuantumFieldTheory:SolitonsandInstantonsinHigh
EnergyPhysics†
J.R.WilsonandG.J.MathewsRelativisticNumericalHydrodynamics†
† Availableinpaperback
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Published online by Cambridge University Press
Noether Symmetries in Theories
of Gravity
With Applications to Astrophysics and Cosmology
FRANCESCO BAJARDI
Scuola Superiore Meridionale di Napoli
SALVATORE CAPOZZIELLO
Università degli Studi di Napoli “Federico II”
Published online by Cambridge University Press
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education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781009208741
DOI:10.1017/9781009208727
©FrancescoBajardiandSalvatoreCapozziello2023
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress&Assessment.
Firstpublished2023
A catalogue record for this publication is available from the British Library.
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Congress.
ISBN978-1-009-20874-1Hardback
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Published online by Cambridge University Press
Dedicated to all women who,
like Emmy Noether,
are fighting for Science and Knowledge.
Published online by Cambridge University Press
Published online by Cambridge University Press
Contents
Preface page xv
Acknowledgments xviii
Amalie Emmy Noether: A Life for Mathematics xix
Notation xxii
Acronyms xxiv
Part I PRELIMINARIES 1
1 The Concept of Symmetry 3
1.1 Symmetries in Physics 5
1.1.1 The Unitary Group 11
1.1.2 The Translation Group 12
1.1.3 The Rotation Group 12
1.1.4 The Lorentz Group 13
1.1.5 The Poincaré Group 15
2 The Two Noether Theorems 16
2.1 Noether’s First Theorem 17
2.1.1 First Demonstration 17
2.1.2 Second Demonstration 19
2.1.3 Internal Symmetries 21
2.2 Noether’s Second Theorem 23
2.3 The Lie Derivative: Applications and Beyond 25
2.4 The Noether–Bessel-Hagen Theorem: Symmetries of Equations
of Motion 28
3 Applications of Noether’s First Theorem to Fields
and Particles 34
3.1 The Free Scalar Field: The U(1) Gauge Invariance 34
3.2 Invariance under Translations and Rotations 36
3.2.1 Space-Time Translations 36
3.2.2 Rotations 38
3.3 The Lorentz Invariance 40
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