MANY-BODY PROBLEM in KALUZA-KLEIN MODELS with TOROIDAL COMPACTIFICATION Astronomical Observatory and Department of Theoretical Physics, Odessa National University,Odessa, Ukraine and North Carolina Central University, CREST and NASA Research Centers, Durham, USA Alexander Zhuk, Alexey Chopovsky and Maxim Eingorn

1.For a massive body, any gravitational theory should possess solutions which generalize the Schwarzschild solution of General Relativity. Evident statements: 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 ? 2

Multidimensional KK models with toroidal compactification: Our external space-time (asymptotically flat) Compact internal space (mathematical tori) Class. Quant. Grav. 27 (2010) , Phys. Rev. D83 (2011) , Phys. Rev. D84 (2011) , 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 ? 3

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

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

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

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

Gauge conditions and smearing To get the solutions (*), we used the standard gauge condition : This condition is satisfied: up to ;identically; -- is not of interest. The gravitating masses should be uniformly smeared over the extra dimensions. Excited KK modes are absent !!! 8

Lagrange function for a many-body system The Lagrange function of a particle with the mass in the gravitational field created by the other bodies is given by: (*) 9

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

L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, § 106: The total Lagrange function of the two-body system should be constructed so that it leads to the correct values of the forces acting on each of the bodies for given motion of the others. To achieve it, we, first, will differentiate with respect to, setting after that. Then, we should integrate this expression with respect to. 11

The two-body Lagrange function from the Lagrange function for the particle "2": The two-body Lagrange function from the Lagrange function for the particle “1": 12 if

and should be symmetric with respect to permutations of particles 1 and 2 and should coincide with each other is satisfied identically for any values of 13 We construct the two-body Lagrange function for any value of the parameters of the equation of state in the extra dimensions.

Gravitational tests It can be easily seen that the components of the metrics coefficients in the external/our space as well as the two-body Lagrange functions exactly coincide with General Relativity for The latent soliton value. E.g. black strings/branes with How big can a deviation be from this value? ? 14

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

2. Perihelion shift of the Mercury 16 For a test body orbiting around the gravitating mass, the perihelion shift for one period is In GR, a predicted relativistic advance agrees with the observations to about 0.1%

3. Periastron shift of the relativistic binary pulsar PSR B Two-body Lagrange function: For the pulsar PSR B the shift is degree per year Much bigger than for the Mercury and with extremely high accuracy! Unfortunayely, masses are calculated from GR! In future, independent measurements of these masses will allow us to obtain a high accuracy restriction on parameter. 17

Conclusion: 1.We constructed the Lagrange function of a many-body system for any value of in the case of Kaluza-Klein models with toroidal compactification of the internal spaces For, the external metric coefficients and the Lagrange function coincide exactly with GR expressions. 3. The gravitational tests (PPN parameters, perihelion and periastron advances) require negligible deviation from the value. 4. The presence of pressure/tension in the internal space results necessarily in the smearing of the gravitating masses over the internal space and in the absence of the KK modes. This looks very unnatural from the point of quantum physics!!! A big disadvantage of the Kaluza-Klein models with the toroidal compactification.