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MANY - BODY PROBLEM in KALUZA-KLEIN MODELS with TOROIDAL COMPACTIFICATION

MANY - BODY PROBLEM in KALUZA-KLEIN MODELS with TOROIDAL COMPACTIFICATION. A lexander Zhuk , Alexey Chopovsky and Maxim Eingorn. Astronomical Observatory and Department of Theoretica l Physics , Odessa National University , Odessa , Ukraine a nd

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MANY - BODY PROBLEM in KALUZA-KLEIN MODELS with TOROIDAL COMPACTIFICATION

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  1. MANY-BODY PROBLEMinKALUZA-KLEIN MODELS with TOROIDAL COMPACTIFICATION AlexanderZhuk, AlexeyChopovskyand Maxim Eingorn AstronomicalObservatoryandDepartmentofTheoretical Physics, OdessaNationalUniversity,Odessa, Ukraine and North Carolina Central University, CREST and NASA Research Centers, Durham, USA

  2. 2 Evident statements: For a massive body, any gravitational theory should possess solutions which generalize the Schwarzschild solution of General Relativity. 2. These solutions must satisfy the gravitational experiments (the perihelion shift, the deflection of light, the time delay of radar echoes)at the same level of accuracy as General Relativity. What about multidimensional Kaluza-Klein models ?

  3. 3 Multidimensional KK models with toroidal compactification: Our external space-time (asymptotically flat) Compact internal space (mathematical tori) Class. Quant. Grav. 27 (2010) 205014, Phys. Rev. D83 (2011) 044005, Phys. Rev. D84 (2011) 024031, Phys. Lett. B 713 (2012) 154 : To satisfy the gravitational tests, a gravitating mass should have tension(negative pressure) in the internal space. E.g. black strings/branes have EoS . In this case, the variations of the internal space volume is absent. If , such variations result in fifth forth  contradiction with experiments. Can we construct a viable theory for a many-body system ?

  4. 4 Gravitational field of the many-body system Metrics: No matter sources -> Minkowskispacetime: Weak-field perturbations in the presence of gravitating masses:

  5. 5 Energy-momentum tensor of the system of Ngravitating masses: (D+1)-velocity Gravitating bodies are pressureless in the external space : They have arbitrary EoS in the internal spaces: where

  6. 6 Multidimensional Einstein equation: 3D radius-vector 3D velocity Solution:

  7. 7 Non-relativistic gravitational potential: Newton gravitational constant: periods of the tori

  8. 8 Gauge conditions and smearing To get the solutions (*), we used the standard gauge condition: This condition is satisfied: identically; up to ; -- is not of interest. The gravitatingmassesshouldbeuniformlysmearedover theextradimensions. Excited KK modes are absent !!!

  9. 9 Lagrangefunctionfor a many-bodysystem TheLagrangefunctionof a particlewiththemassin thegravitationalfieldcreatedbytheotherbodiesis given by: (*)

  10. 10 Two-body system The Lagrange function for the particle “1”:

  11. 11 L.D. Landauand E.M. Lifshitz, TheClassicalTheoryofFields, § 106: ThetotalLagrangefunctionofthe two-bodysystemshouldbe constructedsothatitleadstothecorrectvaluesoftheforces actingoneachofthebodiesforgivenmotion oftheothers. Toachieveit, we, first, willdifferentiatewithrespectto, settingafterthat. Then, weshouldintegratethisexpression withrespectto.

  12. 12 The two-bodyLagrangefunctionfromtheLagrangefunction fortheparticle“1": The two-bodyLagrangefunctionfromtheLagrangefunction fortheparticle "2": if

  13. 13 andshouldbesymmetricwithrespecttopermutations ofparticles 1 and 2 andshouldcoincidewitheachother is satisfied identically for any values of We construct the two-body Lagrange function for any value of the parameters of the equation of state in the extra dimensions.

  14. 14 Gravitationaltests Itcanbeeasilyseenthatthecomponentsofthemetricscoefficients intheexternal/ourspaceaswellasthe two-bodyLagrangefunctions exactlycoincidewithGeneralRelativityfor The latent soliton value. E.g. black strings/branes with How big can a deviation be from this value? ?

  15. 15 1. PPN parameters Eqs. (*): as in GR ! Shapiro time-delay experiment (Cassini spacecraft):

  16. 16 2. PerihelionshiftoftheMercury For a testbodyorbitingaroundthegravitatingmass, theperihelionshiftforoneperiodis In GR, apredictedrelativisticadvance agreeswiththeobservationstoabout 0.1%

  17. 3. Periastronshiftoftherelativistic binarypulsar PSR B1913+16 17 Two-body Lagrange function: Forthepulsar PSR B1913+16 theshiftis degreeperyear Much bigger than for the Mercury and with extremely high accuracy! Unfortunayely, masses are calculated from GR! Infuture, independentmeasurementsof these masseswill allowustoobtain a highaccuracyrestrictiononparameter.

  18. 18 Conclusion: WeconstructedtheLagrangefunctionofamany-bodysystem foranyvalueof inthe case of Kaluza-Klein models with toroidalcompactification of the internal spaces. 2. For , the external metric coefficients and the Lagrange function coincide exactly with GR expressions. 3. Thegravitationaltests (PPN parameters, perihelionandperiastron advances) requirenegligibledeviationfromthevalue. 4. Thepresenceofpressure/tensionintheinternalspaceresults necessarilyinthesmearingofthegravitatingmassesoverthe internalspaceandintheabsenceofthe KK modes. Thislooksveryunnaturalfromthepointofquantumphysics!!! Abigdisadvantageofthe Kaluza-Klein models withthe toroidal compactification.

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