Maxim Eingorn & Alexander Zhuk

WEAK-FIELD LIMITIN F(R)-THEORIES OF GRAVITY

WITH 3 AND MORE SPATIAL DIMENSIONS

Department of TheoreticalPhysics & Astronomical

Observatory

Odessa National Universitynamed after I.I.Mechnikov

I

Multidimensional R-theoryof gravity (multidimensional

generalization of GR)

( )∫ Ω−⇒ dgRS Dg 1

+compactification

of all extra dimensions=

(linear) Kaluza-Klein models

Toroidal compactification=>

contradiction with allclassical gravitational tests,except the frequency shift

Multidimensionalf(R)-theories of gravity

( ) ( )∫ Ω−⇒ dgRfS Dg 1

+ compactificationof all extra dimensions=

(nonlinear) Kaluza-Klein models

II

The natural question: do nonlinear Kaluza-Klein modelscontradict the same classical gravitational tests? Yes.

I. Flat Minkowskispacetime background

III

– expansion in a Taylor series

0=Λ=>

=>

( )[ ] 1lim0

=→

RRfR

IV

V

– the gauge condition

=>

VI

– the D-dimensionalLaplace operator

=> =>

VII

– the D-dimensionalPoisson equation

– the relationship between 2D-dimensional gravitationalconstants

– the rest mass density

– the D-dimensionalradius-vector

– the boundary condition

– the 3-dimensionalSchwarzschild radius

– the Newtoniangravitational constant

VIII the (D = d + 3)-dimensional space

the flat external space

the internalspace (tori)

– the internal space volume – periodsof tori

– the relationship betweenmultidimensional and Newtoniangravitational constants

=> – the zero Kaluza-Kleinmode

IX

– a mass for the additionalscalar degree of freedom

=>

( ) ( ) diaRR iii ,...,2,1,,, 33 =+= ξξ rr

– periodic boundary conditions

X

( ) 0lim 3...13

=+∞→

rdkk

rf

– the boundary condition=>

=>the zero mode

XI

the nonrelativisticgravitational potential=>

Inverse square law experiments=>

=>

XII

Solar system (classicalgravitational tests) =>

=>

=> contradiction for D>3

II. Curved de Sitterspacetime background

=>

– the cosmological constant

– background metriccoefficients

– background curvature

ikg0

0R

XIII

– the differences between the total perturbed valuesand the corresponding background values

The case Without loss of generality

=> =>

=>

=>

XIV

ikη mnη=> the same equation as in the case I of flat Minkowski spacetime background, but

( ) 20Rfa ′′=

=>

&

– the mass squared of the scalar degree of freedom

=> the same equation as in the case I, but ( ) ( )00 Rff ′′→′′

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Conclusions

Nonlinear Kaluza-Klein models contradict all classical gravitational tests, except the

frequency shift.

здесь

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Based on

M. Eingorn and A. ZhukWeak-field limit of f(R)-gravity in

3 and more spatial dimensionsPhys. Rev. D 84, 024023 (2011)

arXiv:gr-qc/1104.1456

XVI

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курениевреднодлявашегоздоровья

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