Cosimo Stornaiolo INFN-Sezione di Napoli MG 12 Paris 12-18 July 2009
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- Slide 1
- Cosimo Stornaiolo INFN-Sezione di Napoli MG 12 Paris 12-18 July
2009
- Slide 2
- D. Pugliese, Deformazioni di metriche spazio-temporali, tesi di
laurea quadriennale, relatori S. Capozziello e C. Stornaiolo S.
Capozziello e C. Stornaiolo, Space-Time deformations as extended
conformal transformations International Journal of Geometric
Methods in Modern Physics 5, 185-196 (2008)
- Slide 3
- In General Relativity, we usually deal with Exact solutions to
describe cosmology, black holes, motions in the solar system,
astrophysical objects Approximate solutions to deal with
gravitational waves, gravitomagnetic effects, cosmological
perturbations, post-newtonian parametrization Numerical simulations
In General Relativity, we usually deal with Exact solutions to
describe cosmology, black holes, motions in the solar system,
astrophysical objects Approximate solutions to deal with
gravitational waves, gravitomagnetic effects, cosmological
perturbations, post-newtonian parametrization Numerical
simulations
- Slide 4
- Nature of dark matter and of dark energy Unknown distribution
of (dark) matter and (dark) energy Pioneer anomaly Relation of the
geometries between the different scales Which theory describes best
gravitation at different scales Alternative theories Approximate
symmetries or complete lack of symmetry Spacetime inhomogeneities
Boundary conditions Initial conditions Nature of dark matter and of
dark energy Unknown distribution of (dark) matter and (dark) energy
Pioneer anomaly Relation of the geometries between the different
scales Which theory describes best gravitation at different scales
Alternative theories Approximate symmetries or complete lack of
symmetry Spacetime inhomogeneities Boundary conditions Initial
conditions
- Slide 5
- Let 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 how Let 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 how
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- Let us consider the surface of the earth, we can consider it as
a (rotating) sphere But this is an approximation Let us consider
the surface of the earth, we can consider it as a (rotating) sphere
But this is an approximation
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- Measurments indicate that the earth surface is better described
by an oblate ellipsoid Mean radius 6,371.0 km Equatorial radius
6,378.1 kmEquatorial Polar radius 6,356.8 kmPolar Flattening
0.0033528Flattening But this is still an approximation Measurments
indicate that the earth surface is better described by an oblate
ellipsoid Mean radius 6,371.0 km Equatorial radius 6,378.1
kmEquatorial Polar radius 6,356.8 kmPolar Flattening
0.0033528Flattening But this is still an approximation
- Slide 8
- 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. Ocean 2.
Ellipsoid 3. Local plumb 4. Continent 5. Geoid
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- It is well known that all the two dimensional metrics are
related by conformal transformations, and are all locally conform
to the flat metric
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- The question is if there exists an intrinsic and covariant way
to relate similarly metrics in dimensions
- Slide 13
- In an n-dimensional manifold with metric the metric has degrees
of freedom In an n-dimensional manifold with metric the metric has
degrees of freedom
- Slide 14
- 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
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- 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 ?
- Slide 16
- Let 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
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- The deformation matrices are the corrections of exact solutions
to consider a realistic spacetime They are the unknown of our
problem They can be obtained by phenomenolgical observations Or
even as solutions of a set of differential equations The
deformation matrices are the corrections of exact solutions to
consider a realistic spacetime They are the unknown of our problem
They can be obtained by phenomenolgical observations Or even as
solutions of a set of differential equations
- Slide 20
- are matrices of scalar fields in spacetime, Deforming matrices
are scalars with respect to coordinate transformations. are
matrices of scalar fields in spacetime, Deforming matrices are
scalars with respect to coordinate transformations.
- Slide 21
- A 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. A
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.
- Slide 22
- Conformal 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
- Slide 23
- 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) 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)
- Slide 24
- They are not necessarily real They 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
itself They can be composed to give successive deformations They
may be singular in some point, if we expect to construct a solution
of the Einstein equations from a Minkowski spacetime They are not
necessarily real They 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 itself They can be composed to
give successive deformations They may be singular in some point, if
we expect to construct a solution of the Einstein equations from a
Minkowski spacetime
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- By lowering its index with a Minkowski matrix we can decompose
the first deforming matrix
- Slide 27
- Substituting in the expression for deformation the second
deforming matrix takes the form Inserting the tetrad vectors to
obtain the metric it follows that (next slide) Substituting in the
expression for deformation the second deforming matrix takes the
form Inserting the tetrad vectors to obtain the metric it follows
that (next slide)
- Slide 28
- Reconstructing a deformed metric leads to This is the most
general relation between two metrics. This is the third way to
define a deformation
- Slide 29
- To complete the definition of a deformation we need to define
the deformation of the corresponding contravariant tensor
- Slide 30
- We are now able to define the connections where and is a tensor
We are now able to define the connections where and is a
tensor
- Slide 31
- Finally we can define how the curvature tensors are
deformed
- Slide 32
- The equations in the vacuum take the form
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- In presence of matter sources the equations for the deformed
metric are of the form
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- Conformal transformations Kerr-Schild metrics Metric
perturbations: cosmological perturbations and gravitational
waves
- Slide 35
- In 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; In 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;
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- The gauge conditions are no more coordinate conditions. instead
they are restrictions in the choice of the perturbing scalars.
- Slide 39
- We 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 issues
As a result we showed that we can consider a theory of no more than
six scalar potentials given in a background geometry We showed that
this scalar field are suitable for a covariant definition for the
cosmological perturbations and also for gravitational waves Using
deformations we can study covariantly the Inhomogeneity and
backreaction problems Cosmological perturbations but also
Gravitational waves in arbitrary spacetimes Symmetry and
approximate symmetric properties of spacetime The boundary and
initial conditions The relation between GR and alternative
gravitational theories