Weak gravitation from a small extra 2D sphere

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 $\varphi (\theta )=\epsilon +\sin \theta$, where $\epsilon$ plays a role of killing the singular point $\varphi (\theta )=0$, 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.

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• Gravity in a warped 6D world with an extra 2D sphere