SPACETIME METRIC DEFORMATIONS Cosimo Stornaiolo INFN-Sezione di Napoli MG 12 Paris 12-18 July 2009
Papers D. Pugliese, Deformazioni di metriche spazio-temporali, tesi di laurea quadriennale, relatori S. Capozziello e C. Stornaiolo S. Capozziello e C. Stornaiolo, Space-Timedeformationsasextendedconformaltransformations International Journal ofGeometricMethods in ModernPhysics 5, 185-196 (2008)
Introduction In GeneralRelativity, weusually deal with • Exactsolutionstodescribecosmology, blackholes, motions in the solar system, astrophysicalobjects • Approximatesolutionsto deal withgravitationalwaves, gravitomagneticeffects, cosmologicalperturbations, post-newtonianparametrization • Numericalsimulations
The ignoranceof the realstructureofspacetime • Nature of dark matter and of dark energy • Unknowndistributionof (dark) matter and (dark) energy • Pioneer anomaly • Relation of the geometriesbetween the differentscales • Whichtheorydescribes best gravitation at differentscales • Alternative theories • Approximatesymmetries or complete lackofsymmetry • Spacetimeinhomogeneities • Boundaryconditions • Initialconditions
Deformationtoparameterize the ignoranceof the realspacetimegeometry? Letustrytodefine a generalcovariant way to deal with the previouslistofproblems. In anarbitraryspacetime, anexactsolutionof Einstein equations, can onlydescribe in anapproximative way the realstructureof the spacetimegeometry. Ourpurposeistoencodeourignoranceof the correctspacetimegeometrybydeforming the exactsolutionwith scalar fields. Letusseehow………
An exampleofgeometricaldeformations in 2D: the earthsurface • Letusconsiderthesurfaceof the earth, we can consider it as a (rotating) sphere Butthisisanapproximation
The Earthisan oblate ellipsoid • Measurments indicate that the earthsurfaceisbetterdescribedbyan oblate ellipsoid Meanradius 6,371.0 km Equatorialradius 6,378.1 km Polarradius 6,356.8 km Flattening 0.0033528 • Butthisisstillanapproximation
The shapeof the Geoid We can improve our measurements an find out that the shape of the earth is not exactly described by any regular solid. We call the geometrical solid representing the Earth a geoid 1. Ocean2. Ellipsoid3. Local plumb4. Continent5. Geoid
An altimetricimageof the geoid
Comparisonbetween the deviationsof the geoidfromanidealized oblate ellipsoid and the deviationsof the CMB from the homogeneityi.e. the departureofspacetimefromhomogeneity at the last scatteringepoch.
Deformations in 2D • Itiswellknownthatall the twodimensionalmetrics are relatedbyconformaltransformations, and are alllocallyconformto the flatmetric
A generalization? • The questionisifthereexistsanintrinsicand covariant way torelatesimilarlymetrics in dimensions
Riemanntheorem • In ann-dimensionalmanifoldwithmetric the metrichas degreesoffreedom
An attempttodefinedeformation in threedimensions In 2002 Coll, Llosa and Soler (General Relativity and Gravitation, Vol. 34, 269, 2002) showedthatanymetric in a 3D space(time) isrelatedto a constant curvature metricby the following relation
Deformations in more thanthreedimensions • Question: How can we generalize the conformal transformations in 2 dimensions an Coll and coworkers deformation in 3 dimensions to more than 3 dimension spacetimes?
Ourdefinitionofmetricdeformation Letusseeifwe can generalize the precedingresultpossiblyexpressing the deformations in termsof scalar fieldsasforconformaltransformations. What do wemeanbymetricdeformation? Letus first consider the decompositionof a metric in tetradvectors
The metricdeformation
Propertiesof the deformingmatrices • are matricesof scalar fields in spacetime, • Deformingmatrices are scalarswithrespectto coordinate transformations.
Conformaltransformations A particularclassofdeformationsisgivenby whichrepresent the conformaltransformations Thisisoneof the first examplesofdeformationsknownfromliterature. Forthisreasonwe can considerdeformationsasanextensionofconformaltransformations.
Exampleofconformaltransformations • Conformaltransformationsgivenby a changeofcoordinatesflat (and open) Friedmannmetrics and Einsteiin (static) universe • Conformaltransformationthatcannotbeobtainedby a changeofcoordinatesclosedFriedmannmetric
More precise definitionofdeformation If the metrictensorsoftwospaces and are relatedby the relation wesaythatis the deformationof (cfr. L.P.Eisenhart, RiemannianGeometry, pag. 89)
Propertiesof the deformingmatrices • They are notnecessarilyreal • They are notnecessarilycontinuos (so thatwemay associate spacetimeswithdifferenttopologies) • They are not coordinate transformations (oneshouldtransformcorrespondinglyall the covariant and contravarianttensors), i.e. they are notdiffeomorphismsof a spacetimeM toitself • They can becomposedtogive successive deformations • Theymaybesingular in some point, ifweexpecttoconstruct a solutionof the Einstein equationsfrom a Minkowskispacetime
Seconddeformingmatrix
Byloweringitsindexwith a Minkowskimatrixwe can decompose the first deformingmatrix
Expansionof the seconddeformingmatrix Substituting in the expressionfordeformation the seconddeformingmatrixtakes the form Inserting the tetradvectorstoobtain the metricitfollowsthat (next slide)
Tensorialdefinitionof the deformations Reconstructing a deformedmetricleadsto Thisis the mostgeneralrelation betweentwometrics. Thisis the third way todefine a deformation
Deforming the contravariantmetric To complete the definitionof a deformationweneedtodefine the deformationof the correspondingcontravarianttensor
Deformedconnections We are nowabletodefine the connections where and is a tensor
Deformed Curvature tensors Finallywe can definehow the curvature tensors are deformed
Einstein equationsfor the deformedspacetime in the vacuum The equations in the vacuum take the form
The deformed Einstein equations in presenceofdeformedmattersources In presenceofmattersources the equationsfor the deformedmetric are of the form
Some examplesofspacetimemetricdeformationsalreadypresent in literature • Conformaltransformations • Kerr-Schildmetrics • Metricperturbations: cosmologicalperturbations and gravitationalwaves
Smalldeformations or gravitationalperturbations • In ourapproach the approximationisgivenby the conditions • Thisconditions are covariantforthreereasons: • 1) we are using scalar fields; • 2) thisobjects are adimensional; • 3) they are subjectonlytoLorentztransformations;
Equationsforsmalldeformations
The correspondingequations and gaugeconditionsfor the scalar potentials (in a flatspacetime)
The meaningofgaugeconditionsfor the deformations • The gaugeconditions are no more coordinate conditions. insteadthey are restrictions in the choiceof the perturbingscalars.
Discussion and Conclusions • Wehavepresented the deformationofspacetimemetricsas the correctionsonehasto introduce in the metric in orderto deal withourignoranceof the fine spacetimestructure. • The useofscalarstodefinedeformation can simplifymanyconceptualissues • As a resultweshowedthatwe can consider a theoryof no more thansix scalar potentialsgiven in a background geometry • Weshowedthatthis scalar field are suitablefor a covariantdefinitionfor the cosmologicalperturbations and alsoforgravitationalwaves • Usingdeformationswe can studycovariantly the • Inhomogeneity and backreactionproblems • Cosmologicalperturbations butalso • Gravitationalwaves in arbitraryspacetimes • Symmetry and approximatesymmetricpropertiesofspacetime • The boundary and initialconditions • The relation between GR and alternative gravitationaltheories
