The shape of multidimensional gravity

Abstract

In the case of one extra dimension, well known Newton's potential $\varphi (r_3)=-G_{N}m/r_3$ is generalized to compact and elegant formula $\varphi (r_3,\xi )=-(G_{N}m/r_3)\sinh (2\pi r_3/a)[\cosh (2\pi r_3/a)-\cos (2\pi \xi /a){]}^{-1}$ 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: $\varphi \sim 1/r_3$ for $r_3>>a$ and $\varphi \sim 1/r_4^2$ 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.

Comments: 4 pages of Revtex4, 2 eps figures