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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 - PowerPoint PPT Presentation



Cosimo StornaioloINFN-Sezione di NapoliMG 12 Paris12-18 July 2009

SPACETIME METRIC DEFORMATIONS1D. Pugliese, Deformazioni di metriche spazio-temporali, tesi di laurea quadriennale, relatori S. Capozziello e C. StornaioloS. Capozziello e C. Stornaiolo, Space-Time deformations as extended conformal transformations International Journal of Geometric Methods in Modern Physics 5, 185-196 (2008)Papers2 In General Relativity, we usually deal with Exact solutions to describe cosmology, black holes, motions in the solar system, astrophysical objectsApproximate solutions to deal with gravitational waves, gravitomagnetic effects, cosmological perturbations, post-newtonian parametrizationNumerical simulationsIntroduction3Nature of dark matter and of dark energyUnknown distribution of (dark) matter and (dark) energyPioneer anomalyRelation of the geometries between the different scalesWhich theory describes best gravitation at different scalesAlternative theoriesApproximate symmetries or complete lack of symmetrySpacetime inhomogeneitiesBoundary conditionsInitial conditionsThe ignorance of the real structure of spacetime4Let us try to define a general covariant way to deal with the previous list of problems.In an arbitrary space time, an exact solution of Einstein equations, can only describe in an approximative way the real structure of the spacetime geometry. Our purpose is to encode our ignorance of the correct spacetime geometry by deforming the exact solution with scalar fields.Let us see howDeformation to parameterize the ignorance of the real spacetime geometry?5Let us considerthe surface of the earth, we can consider it as a (rotating) sphere

But this is an approximation

An example of geometrical deformations in 2D: the earth surface

6Measurments indicate that the earth surface is better described by an oblate ellipsoid

Mean radius 6,371.0km Equatorial radius 6,378.1km Polar radius 6,356.8km Flattening 0.0033528

But this is still an approximationThe Earth is an oblate ellipsoid


The shape of 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 geoid1. Ocean2. Ellipsoid3. Local plumb4. Continent5. Geoid8

An altimetric image of the geoid


Comparison between the deviations of the geoid from an idealized oblate ellipsoid and the deviations of the CMB from the homogeneity i.e. the departure of spacetime from homogeneity at the last scattering epoch.

10It is well known that all the two dimensional metrics are related by conformal transformations, and are all locally conform to the flat metric

Deformations in 2D

11The question is if there exists an intrinsic and covariant way to relate similarly metrics in dimensions A generalization?

12In an n-dimensional manifold with metric the metric has

degrees of freedom

Riemann theorem

13 In 2002 Coll, Llosa and Soler (General Relativity and Gravitation, Vol. 34, 269, 2002) showed that any metric in a 3D space(time) is related to a constant curvature metric by the following relation An attempt to define deformation in three dimensions

14Question: How can we generalize the conformal transformations in 2 dimensions an Coll and coworkers deformation in 3 dimensions to more than 3 dimension spacetimes?Deformations in more than three dimensions15Let us see if we can generalize the preceding result possibly expressing the deformations in terms of scalar fields as for conformal transformations. What do we mean by metric deformation? Let us first consider the decomposition of a metric in tetrad vectors Our definition of metric deformation


The metric deformation


Construction of a new metric

18The deformation matrices are the corrections of exact solutions to consider a realistic spacetime

They are the unknown of our problemThey can be obtained by phenomenolgical observationsOr even as solutions of a set of differential equations

Role of the deformation matrices19 are matrices of scalar fields in spacetime,

Deforming matrices are scalars with respect to coordinate transformations.Properties of the deforming matrices

20A particular class of deformations is given by

which represent the conformal transformations

This is one of the first examples of deformations known from literature. For this reason we can consider deformations as an extension of conformal transformations.

Conformal transformations

21Conformal transformations given by a change of coordinates flat (and open) Friedmann metrics and Einsteiin (static) universe

Conformal transformation that cannot be obtained by a change of coordinates closed Friedmann metric

Example of conformal transformations

22 If the metric tensors of two spaces and are related by the relation

we say that is the deformation of (cfr. L.P. Eisenhart, Riemannian Geometry, pag. 89)

More precise definition of deformation

23They are not necessarily realThey are not necessarily continuos (so that we may associate spacetimes with different topologies)They are not coordinate transformations (one should transform correspondingly all the covariant and contravariant tensors), i.e. they are not diffeomorphisms of a spacetime M to itselfThey can be composed to give successive deformationsThey may be singular in some point, if we expect to construct a solution of the Einstein equations from a Minkowski spacetime

Properties of the deforming matrices24 Second deforming matrix

25By lowering its index with a Minkowski matrix we can decompose the first deforming matrix

26Substituting in the expression for deformationthe second deforming matrix takes the form

Inserting the tetrad vectors to obtain the metric it follows that (next slide)Expansion of the second deforming matrix

27Reconstructing a deformed metric leads to

This is the most general relation between two metrics. This is the third way to define a deformation

Tensorial definition of the deformations

28To complete the definition of a deformation we need to define the deformation of the corresponding contravariant tensor Deforming the contravariant metric

29We are now able to define the connections



is a tensor

Deformed connections

30Finally we can define how the curvature tensors are deformed

Deformed Curvature tensors

31The equations in the vacuum take the formEinstein equations for the deformed spacetime in the vacuum

32In presence of matter sources the equations for the deformed metric are of the form

The deformed Einstein equations in presence of deformed matter sources

33Conformal transformations Kerr-Schild metricsMetric perturbations: cosmological perturbations and gravitational wavesSome examples of spacetime metric deformations already present in literature34In our approach the approximation is given by the conditions

This conditions are covariant for three reasons: 1) we are using scalar fields; 2) this objects are adimensional; 3) they are subject only to Lorentz transformations;

Small deformations or gravitational perturbations


Equations for small deformations


The corresponding equations and gauge conditions for the scalar potentials (in a flat spacetime)

37The gauge conditions are no more coordinate conditions. instead they are restrictions in the choice of the perturbing scalars.The meaning of gauge conditions for the deformations38We have presented the deformation of spacetime metrics as the corrections one has to introduce in the metric in order to deal with our ignorance of the fine spacetime structure.The use of scalars to define deformation can simplify many conceptual issuesAs a result we showed that we can consider a theory of no more than six scalar potentials given in a background geometryWe showed that this scalar field are suitable for a covariant definition for the cosmological perturbations and also for gravitational wavesUsing deformations we can study covariantly the Inhomogeneity and backreaction problemsCosmological perturbations but alsoGravitational waves in arbitrary spacetimesSymmetry and approximate symmetric properties of spacetimeThe boundary and initial conditionsThe relation between GR and alternative gravitational theories

Discussion and Conclusions39