Weak gravitation from a small extra 2D sphere

TL;DR

This paper proposes a 6-dimensional model with a small extra 2D sphere to explain the weakness of gravity in our universe, deriving the 4D Newton constant and force coefficients from the model.

Contribution

It introduces a novel 6D framework with a warp factor that naturally explains weak gravity and derives the effective 4D gravitational constant and force coefficients.

Findings

The 4D Newton constant is proportional to the 6D constant times a small parameter to the tenth power.

The model predicts specific values for fifth force coefficients based on the small extra dimension.

The warp factor effectively localizes gravity and matter in our 4D universe.

Abstract

In order to explain weak gravitation in our 4-dimensional universe, a 6-dimensional model with a small extra 2D sphere is proposed. The traceless energy-momentum tensor is quite naturally appeared in our 6-dimensional model. The warp factor is given by $\phi (\theta ) = \epsilon + \sin{\theta }$, where $\epsilon $ plays a role of killing the singular point $\phi (\theta )=0$, and is assumed $0 < \epsilon \ll 1$. Any massive particle is rolling down into points along this geodesic line. The light ray can be shown to stay in our 4-dimensional universe. This suggest us that our 4-dimensional world can be located at $\theta =0 $ and/or $\theta = \pi $, its background metric being $\epsilon ^2 \eta _{\mu \nu }$. As a result, we have the 4-dimensional Newton constant, which is given by $G_N \simeq G_6 \epsilon ^{10}$ and the fifth force coefficients appeared here are $\alpha _i\simeq \epsilon ^{2(i-4)}$, $i=1, 2, 3$. Here $G_6$ is the gravitational constant in 6-dimensional spacetime. If we take $\epsilon = 10^{-3.8}$ against $G_6\sim 1($GeV$)^{-2}$, we get $G_N\sim 10^{-38}($Gev$)^{-2}$, the present time gravitational constant.