FLRW spaces as submanifolds of $\mathbb{R}^6$: restriction to the Klein-Gordon operator

Abstract

The FLRW spacetimes can be realized as submanifolds of $\mathbb{R}^6$. In this paper we relate the Laplace-Beltrami operator for an homogeneous scalar field $ϕ$ of $\mathbb{R}^6$ to its explicit restriction on FLRW spacetimes. We then make the link between the homogeneous solutions of the equation $\square_6 ϕ= 0$ in $\mathbb{R}^6$ and those of the Klein-Gordon equation $(\square_{f} - ξR^f + m^2)ϕ^f=0 $ for the free field $ϕ^f$ in the FLRW spacetime. We obtain as a byproduct a formula for the Ricci scalar of the FRLW spacetime in terms of the function $f$ defining this spacetime in $\mathbb{R}^6$.

7 pages, Revtex 4.1,