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

29 Nov 2017  ·  Zapata J. P. Arias, Belokogne A., Huguet E., Queva J., Renaud J. ·

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 $\mathbb{R}^6$. read more

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