Multidimensional gravity in non-relativistic limit

TL;DR

This paper derives an exact multidimensional gravitational potential solution, showing how it transitions from Newtonian to higher-dimensional behavior, and explores implications for experiments and models addressing the hierarchy problem.

Contribution

It provides an exact solution for the gravitational potential in multidimensional space with compact extra dimensions, including special cases and implications for experimental measurements and hierarchy problem solutions.

Findings

Exact solution for Poisson equation in $M_{3+d}$ space.

Potential transitions from Newtonian to multidimensional behavior.

Models with smeared extra dimensions can solve hierarchy problem.

Abstract

It is found the exact solution of the Poisson equation for the multidimensional space with topology $M_{3+d}=\mathbb{R}^3\times T^d$. This solution describes smooth transition from the newtonian behavior $1/r_3$ for distances bigger than periods of tori (the extra dimension sizes) to multidimensional behavior $1/r^{1+d}_{3+d}$ in opposite limit. In the case of one extra dimension $d=1$, the gravitational potential is expressed via compact and elegant formula. 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 $G_N$. 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 $M_{Pl(10)}$. 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.