The shape of multidimensional gravity

In the case of one extra dimension, well known Newton’s potential $\phi (r_3)=-G_N m/r_3$isgeneralizedtocompactandelegantformula$\phi(r_3,\xi)=-(G_N m/r_3)\sinh(2\pi r_3/a)[\cosh(2\pi r_3/a)-\cos(2\pi\xi/a)]^{-1}$iffour-dimensionalspacehastopology$$\mathbb{R}$^3\times T$\mathrm{.}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 $r_3\in [0,+\mathrm{\infty })$ and $\xi\in [0,a]$andhasknownasymptoticbehavior:$\phi \sim 1/r_3$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.