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 $\varphi$ of ${\mathbb{R}}^6$ to its explicit restriction on FLRW spacetimes. We then make the link between the homogeneous solutions of the equation ${\square }_6\varphi =0$ in ${\mathbb{R}}^6$ and those of the Klein-Gordon equation $({\square }_f-\xi R^f+m^2){\varphi }^f=0$ for the free field ${\varphi }^f$ 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,

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Cited by in corpus (1)

• FLRW embeddings in ${\mathbb{R}}^{n+2}$, differential geometry and conformal photon propagator