Multiply-warped product metrics and reduction of Einstein equations

TL;DR

This paper demonstrates how Einstein equations for multiply warped product metrics in higher dimensions can be reduced to simpler Einstein equations on lower-dimensional submanifolds, with related cosmological constants.

Contribution

It provides a method to reduce multidimensional Einstein equations for multiply warped metrics to lower-dimensional equations with explicit relations for cosmological constants.

Findings

Einstein equations for multiply warped metrics can be decomposed into submanifold equations.

The cosmological constants are functions of warp functions and submanifold dimensions.

The approach simplifies analyzing higher-dimensional Einstein equations in warped geometries.

Abstract

It is shown that for every multidimensional metric in the multiply warped product form $\bar{M} = K\times_{f_1} M_1\times_{f_2}M_2$ with warp functions $f_1$, $f_2$, associated to the submanifolds $M_1$, $M_2$ of dimensions $n_1$, $n_2$ respectively, one can find the corresponding Einstein equations $\bar{G}_{AB}=-\bar{\Lambda}\bar{g}_{AB}$, with cosmological constant $\bar{\Lambda}$, which are reducible to the Einstein equations $G_{\alpha\beta} = -\Lambda_1 g_{\alpha\beta}$ and $G_{ij} =-\Lambda_2 h_{ij}$ on the submanifolds $M_1$, $M_2$, with cosmological constants ${\Lambda_1}$ and ${\Lambda_2}$, respectively, where $\bar{\Lambda}$, ${\Lambda_1}$ and ${\Lambda_2}$ are functions of ${f_1}$, ${f_2}$ and $n_1$, $n_2$.