The shape of multidimensional gravity

TL;DR

This paper derives a generalized gravitational potential for a space with one extra compact dimension, analyzing its behavior and deviations from Newtonian gravity, and discusses experimental constraints on extra dimensions.

Contribution

The paper presents a closed-form formula for gravitational potential in a space with topology  rac{a}{2\

Findings

The potential reduces to Newtonian form at large distances.

Deviations from Newtonian gravity are very small at observable scales.

Experimental data can limit the number of extra dimensions.

Abstract

In the case of one extra dimension, well known Newton's potential $\phi (r_3)=-G_N m/r_3$ is generalized to compact and elegant formula $\phi(r_3,\xi)=-(G_N m/r_3)\sinh(2\pi r_3/a)[\cosh(2\pi r_3/a)-\cos(2\pi\xi/a)]^{-1}$ if four-dimensional space has topology $\mathbb{R}^3\times T$. Here, $r_3$ is magnitude of three-dimensional radius vector, $\xi$ is extra dimension and $a$ is a period of a torus $T$. This formula is valid for full range of variables $r_3 \in [0,+\infty)$ and $\xi\in [0,a]$ and has known asymptotic behavior: $\phi \sim 1/r_3$ for $r_3>>a$ and $\phi \sim 1/r_4^2$ for $r_4=\sqrt{r_3^2+\xi^2}<<a$. Obtained formula is applied to an infinitesimally thin shell, a shell, a sphere and two spheres to show deviations from the newtonian expressions. Usually, these corrections are very small to observe at experiments. Nevertheless, in the case of spatial topology $\mathbb{R}^3\times T^{d}$, experimental data can provide us with a limitation on maximal number of extra dimensions.