The Shape of Multidimensional Gravity: Non-relativistic Limit

Abstract. It is found the exact solution of the Poisson equation for the multidimensional space with topology M3+d = R3 ×T d . This solution describes smooth transition from the Newtonian behavior 1/r3 for distances bigger than periods of tori (the extra dimension sizes) to multidimensional behavior 1/r1+d 3+d in opposite limit. In the case of one extra dimension d = 1, the gravitational potential is expressed via compact and elegant formula. Obtained formula is applied to an infinitesimally thin shell, a shell, a sphere and two spheres to show deviations from the Newtonian expressions. It is shown that the corrections to the gravitational constant in the Cavendish-type experiment can be within the measurement accuracy of Newton’s gravitational constant GN . It is proposed models where the test masses are smeared over some (or all) extra dimensions. In 10-dimensional spacetime with 3 smeared extra dimensions, it is shown that the size of 3 rest extra dimensions can be enlarged up to submillimeter for the case of 1TeV fundamental Planck scale MPl(10). In the models where all extra dimensions are smeared, the gravitational potential exactly coincides with the Newtonian one. Nevertheless, the hierarchy problem can be solved in these models.