Cosimo StornaioloINFN-Sezione di Napoli

MG 12 Paris12-18 July 2009

SPACETIME METRIC DEFORMATIONS

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)

Papers

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

Introduction

Nature 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 conditions

The ignorance of the real structure of spacetime

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………

Deformation to parameterize the ignorance of the real spacetime geometry?

Let 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

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

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

But this is still an approximation

The 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 geoid 1. Ocean

2. Ellipsoid3. Local plumb4. Continent5. Geoid

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.

510hh

510TT

It 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

2

2 2 2 2 2 2 2 2

( )

( sin ) , ( sin )

in the case of the deformation of a sphere we haveg x g

g R d d g R d d

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

A generalization?

2n

In an n-dimensional manifold with metric

the metric has

degrees of freedom

Riemann theorem

( 1)2

n nf

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

2ab ab a bg h

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?

Deformations in more than three dimensions

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

Our definition of metric deformation

A Bab ABg e e ( ) ( )A C B D

ab AB C Dg x e x e

( ) ( )A BAB C D CDx x

( ) ( )A C B Dab AB C a D bg x e x e

The metric deformation

( ) ( )A AB Bx x

abg~ abg

are matrices of scalar fields in spacetime,

Deforming matrices are scalars with respect to coordinate transformations.

Properties of the deforming matrices

( )AB x

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.

Conformal transformations

AB

AB xx )()(

abDB

DCA

CABab gxeeg )(~ 2

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

Example of conformal transformations

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

M

M~

: A B A B C DAB AB C Dg e e g e e

M

M

M

They 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 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

Properties of the deforming matrices

Second deforming matrix

( )

( ) ( )

AB

A B C Dab AB C D

x

g x x e e G

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

C

AC B AB ABAB

AB AB AB AB

B B B BA A A A

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)

Expansion of the second deforming matrix

2 2 C D

AB AB AB CD A B

C D C D C DCD A B A B CD A B

G

Reconstructing 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

2ab ab abg g

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

Deforming the contravariant metric

abg 1 1C Dab AB a b

C DA Bg e e

We are now able to define the connections

where

and

is a tensor

Deformed connections

c c cab ab abC

2 2 1( ) 22c cd cd

ab d a b ab d a db b ad d abC g g g g

Finally we can define how the curvature tensors are deformed

Deformed Curvature tensors

aed

bce

bed

ace

bcd

aacd

bd

abcd

abc CCCCCCRR ~

aed

dbe

ded

abe

dbd

aabd

dabab CCCCCCRR ~

aeddbe

ded

abe

dbd

aabd

dab

abab

abab CCCCCCgRgRgR ~~~~~

The equations in the vacuum take the form

Einstein equations for the deformed spacetime in the vacuum

0

0

d dab ab d ab a db

e d e dab de db ae

R R C C

C C C C

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

The deformed Einstein equations in presence of deformed matter sources

d d e d e dab ab d ab a db ab de db aeR R C C C C C C

12ab ab abR T g T

1 12 2

d d e d e dd ab a db ab de db ae ab ab ab abC C C C C C T g T T Tg

Conformal transformations Kerr-Schild metricsMetric perturbations: cosmological

perturbations and gravitational waves

Some examples of spacetime metric deformations already present in literature

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;

Small deformations or gravitational perturbations

( ) ( )A B AB A Bx x ( ) 1A

B x

A B C Dab ab ab ab AB C D a bg g h g e e

Equations for small deformations

0~ aed

dbe

ded

abe

dbd

aabd

dabab CCCCCCRR

0 aed

dbe

ded

abe

dbd

aabd

d CCCCCC

0dbd

aabd

d CC

2 0a a da bc b c adh R h

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

2 0

0

0

d C d C C dd A CB A d CB A d CB

ABBA

d C CD d CA B CA d Be

The 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 deformations

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

Discussion and Conclusions