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

TL;DR

This paper explores how FLRW spacetimes can be embedded in six-dimensional space and relates the Laplace-Beltrami operator in this setting to the Klein-Gordon operator on these spacetimes, providing new insights into scalar fields and curvature.

Contribution

It establishes a connection between the Laplace-Beltrami operator in six-dimensional space and the Klein-Gordon operator on FLRW spacetimes, including a formula for the Ricci scalar.

Findings

Derived a formula for the Ricci scalar in terms of the embedding function

Linked solutions of the 6D wave equation to Klein-Gordon solutions in FLRW

Provided a geometric interpretation of scalar fields in FLRW as submanifolds

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 $\phi$ of $\mathbb{R}^6$ to its explicit restriction on FLRW spacetimes. We then make the link between the homogeneous solutions of the equation $\square_6 \phi = 0$ in $\mathbb{R}^6$ and those of the Klein-Gordon equation $(\square_{f} - \xi R^f + m^2)\phi^f=0 $ for the free field $\phi^f$ 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 $\mathbb{R}^6$.