Canonical Gravity and Applications: Cosmology, Black Holes, and Quantum Gravity

313 Pages·2011·2.12 MB·English

by  Martin Bojowald

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CANONICAL GRAVITY AND APPLICATIONS
Canonical methods are a powerful mathematical tool within the field of gravitational
research, both theoretical and observational, and have contributed to a number of recent
developments in physics. Providing mathematical foundations as well as physical appli-
cations,thisisthefirstsystematicexplanationofcanonicalmethodsingravity.Thebook
discussesthemathematicalandgeometricalnotionsunderlyingcanonicaltools,highlight-
ingtheirapplicationsinallaspectsofgravitationalresearch,fromadvancedmathematical
foundationstomodernapplicationsincosmologyandblack-holephysics.Themaincanon-
icalformulations,includingtheArnowitt–Deser–Misner(ADM)formalismandAshtekar
variables,arederivedanddiscussed.
Ideal for both graduate students and researchers, this book provides a link between
standardintroductionstogeneralrelativityandadvancedexpositionsofblackholephysics,
theoreticalcosmology,orquantumgravity.
MartinBojowaldisanAssociateProfessorattheInstituteforGravitationandtheCosmos,
PennsylvaniaStateUniversity.Hepioneeredloopquantumcosmology,afieldinwhichhis
researchcontinuestofocus.

CANONICAL GRAVITY AND
APPLICATIONS
Cosmology, black holes, quantum gravity
MARTIN BOJOWALD
InstituteforGravitationandtheCosmos
ThePennsylvaniaStateUniversity
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
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Firstpublished2011
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LibraryofCongressCataloguinginPublicationdata
Bojowald,Martin.
Canonicalgravityandapplications:cosmology,blackholes,andquantumgravity/MartinBojowald.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-19575-1(hardback)
1.Quantumgravity. 2.Generalrelativity(Physics) 3.Cosmology. I.Title.
QC178.B625 2010
530.14(cid:2)3–dc22 2010038771
ISBN978-0-521-19575-1Hardback
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Contents
1 Introduction page1
2 Isotropiccosmology:aprelude 4
2.1 Equationsofmotion 5
2.2 Matterparameters 8
2.3 Energyconditions 10
2.4 Singularities 11
2.5 Linearperturbations 12
3 Hamiltonianformulationofgeneralrelativity 17
3.1 Constrainedsystems 18
3.2 Geometryofhypersurfaces 40
3.3 ADMformulationofgeneralrelativity 50
3.4 Initial-valueproblem 72
3.5 First-orderformulationsandAshtekarvariables 82
3.6 Canonicalmattersystems 98
4 Modelsystemsandperturbations 113
4.1 Bianchimodels 113
4.2 Symmetry 129
4.3 Sphericalsymmetry 147
4.4 Linearizedgravity 163
5 Globalandasymptoticproperties 184
5.1 Geodesiccongruences 185
5.2 Trappedsurfaces 198
5.3 Asymptoticinfinity 203
5.4 Matchingofsolutions 217
5.5 Horizons 235
6 Quantumgravity 248
6.1 Constrainedquantizationandbackground
independence 249
6.2 Quantumcosmology 258
v
vi Contents
6.3 Quantumblackholes 271
6.4 Thestatusofcanonicalquantumgravity 273
AppendixA:Somemathematicalmethods 274
References 289
Index 300
1
Introduction
Einstein’sequation
G =8πGT (1.1)
ab ab
presents a complicated system of non-linear partial differential equations of up to
second order for the space-time metric g . As a tensorial equation, it determines the
ab
structureofspace-timeinacovariantandcoordinate-independentway.Nevertheless,coor-
dinatesareoftenchosentoarriveatspecificsolutions,andtheEinsteintensorissplitinto
its components in the process. In component form, one then notices that some of the
equations are of first order only; they do not appear as evolution equations but rather as
constraints on the initial values that can be posed for the second-order part of Einstein’s
equation.Moreover,somecomponentsofthemetricdonotappearassecond-orderderiva-
tivesatall.
Physically, all these properties taken together capture the self-interacting nature of the
gravitationalfieldanditsintimaterelationshipwiththestructureofspace-time.Einstein’s
equationisnottobesolvedonagivenbackgroundspace-time,itssolutionsratherdetermine
how space-time itself evolves starting with the structure of an initial spatial manifold.
Generalcovarianceallowsonetoexpresssolutionsinanycoordinatesystemandtorelate
solutionsbasedonlyondifferentchoicesofcoordinatesinconsistentways.Consistencyis
ensuredbypropertiesofthefirst-orderpartoftheequation,andcoordinateredundancyby
thedifferentbehaviorsofmetriccomponents.Allthesepropertiesarethuscrucial,butthey
makethetheoryratherdifficulttoanalyzeandtounderstand.
Instead of solving Einstein’s equation just as one set of coupled partial differential
equations, the use of geometry provides important additional insights by which much
information can be gained in an elegant and systematic way. There is, first, space-time
itself which is equipped with a Riemannian structure and thus encodes the gravitational
field in a geometrical way. Geometry allows many identifications of observable space-
time quantities, and it provides means to understand space-time globally and to arrive at
general theorems, for instance regarding singularities. These structures can be analyzed
with differential geometry, which is provided in most introductory textbooks on general
relativity and will be assumed at least as basic knowledge in this book. (More advanced
1
2 Introduction
geometricaltopicsareprovidedintheAppendix.)Wewillbeassumingfamiliaritywiththe
firstpartofthebookbyWald(1984),andusesimilarnotations.
In addition to space-time, also the solution space to Einstein’s equation, just like the
solutionspaceofanyfieldtheory,isequippedwithaspecialkindofgeometry:symplectic
or Poisson geometry as the basis of canonical methods. General properties of solution
spaces regarding gauge freedom, as originally analyzed by Dirac, are best seen in such
a setting. In this book, the traditional treatment of systems with constraints following
Dirac’s classification will be accompanied by a mathematical discussion of geometrical
propertiesofthesolutionspacesinvolved.Withthiscombination,amorepenetratingview
canbedeveloped,showinghownaturalseveralofthedistinctionsmadebyDiracarefrom
a mathematical perspective. In gravity, these techniques become especially important for
understanding the solutions of Einstein’s equation and their relationships to each other
and to observables. They provide exactly the systematic tools required to understand the
evolution problem and consistency of Einstein’s equation and the meaning of the way
in which space-time structure is described, but they are certainly not confined to this
purpose. Canonical techniques are relevant for many applications, including cosmology
of homogeneous models and perturbations around them, and collapse models of matter
distributionsintoblackholes.Regardingobservationalaspectsofcosmology,forinstance,
canonicalmethodsprovidesystematictoolstoderivegauge-invariantobservablesandtheir
evolution.Finally,canonicalmethodsareimportantwhenthetheoryistobequantizedto
obtainquantumgravity.
Wewillfirstillustratetheappearanceandapplicationofcanonicaltechniquesingravity
by the example of isotropic cosmology. What we learn in this context will be applied to
generalrelativityinChapter3,inwhichthemainversionsofcanonicalformulations—those
duetoArnowitt,DeserandMisner(ADM)(2008)andareformulationintermsofAshtekar
variables — are derived. At the same time, mathematical techniques of symplectic and
Poissongeometrywillbedeveloped.Applicationsatthisgenerallevelincludeadiscussion
oftheinitial-valueproblemaswellasanexhibitionofcanonicalmethodsandtheirresults
innumericalrelativity.Canonicalmattersystemswillalsobediscussedinthischapter.
JustasoneoftensolvesEinstein’sequationinasymmetriccontext,symmetry-reduced
modelsprovideinterestingapplicationsofthecanonicalequations.Classesofthesemodels,
generalissuesofsymmetryreduction,andperturbationsaroundsymmetricmodelsarethe
topicofChapter4.Themaincosmologicalimplicationsofgeneralrelativitywillbetouched
upon in the process. From the mathematical side, the general theory of connections and
fiberbundleswillbedevelopedinthischapter.Sphericallysymmetricmodels,then,donot
onlyprovideinsightsaboutblackholes,butalsoillustratethesymmetrystructuresbehind
thecanonicalformulationofgeneralrelativity(intermsofLiealgebroids).
Chapter 5 does not introduce new canonical techniques, but rather, shows how they
are interlinked with other, differential geometric methods often used to analyze global
propertiesofsolutionsofgeneralrelativity.Theseincludegeodesiccongruences,singularity
theorems, the structure of horizons, and matching techniques to construct complicated
solutionsfromsimplerones.Theclassofphysicalapplicationsinthischapterwillmainly