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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 ′′→′′
XV
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
здесьмоглабыбытьвашареклама
TH
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курениевреднодлявашегоздоровья
XVII
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