The shape of multidimensional gravity
Abstract
Description
In the case of one extra dimension, well known Newton's potential $ϕ(r_3)=-G_N m/r_3$ is generalized to compact and elegant formula $ϕ(r_3,ξ)=-(G_N m/r_3)\sinh(2πr_3/a)[\cosh(2πr_3/a)-\cos(2πξ/a)]^{-1}$ if four-dimensional space has topology $\mathbb{R}^3\times T$. Here, $r_3$ is magnitude of three-dimensional radius vector, $ξ$ 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 $ξ\in [0,a]$ and has known asymptotic behavior: $ϕ\sim 1/r_3$ for $r_3>>a$ and $ϕ\sim 1/r_4^2$ for $r_4=\sqrt{r_3^2+ξ^2}<<a$. 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.
4 pages of Revtex4, 2 eps figures
4 pages of Revtex4, 2 eps figures