Non-relativistic limit of multidimensional gravity: exact solutions and applications

TL;DR

This paper derives exact solutions for the gravitational potential in multidimensional spaces with compact extra dimensions, revealing a smooth transition from Newtonian to higher-dimensional gravity and proposing models that address the hierarchy problem.

Contribution

It provides exact solutions for the Poisson equation in multidimensional spaces and explores models with smeared extra dimensions that preserve Newtonian gravity.

Findings

Exact solutions for gravitational potential in multidimensional space.

Potential transition from Newtonian to higher-dimensional behavior.

Models with smeared extra dimensions that solve the 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. These exact solutions are applied to some practical problems to get the gravitational potentials for considered configurations. Found potentials are used to calculate the acceleration for point masses and gravitational self-energy.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 regardless of size of the extra dimensions. Nevertheless, the hierarchy problem can be solved in these models.