Einstein equations with cosmological constant in Super Space-Time

TL;DR

This paper introduces super warped product spaces and demonstrates how Einstein equations with a cosmological constant reduce to equations on the super space, revealing the warp functions' role in cosmological terms.

Contribution

It develops a new framework for super warped product spaces and derives Einstein equations with cosmological constants within this structure, linking warp functions to cosmological terms.

Findings

Einstein equations reduce to super space equations with cosmological terms.

Warp functions determine the cosmological constant's behavior.

Solutions are found for specific super space-time configurations.

Abstract

We introduce a new kind of super warped product spaces $\bar{M}_{_{(I)}}=\textbf{I}^{1|0}\times_f M^{m|n}$, $\bar{M}_{_{(II)}}=\textbf{I}^{0|1}\times_{f} M^{m|n}$, and $\bar{M}_{_{(III)}}=\textbf{I}^{1|1}\times_{f} M^{m|n}$, where $M^{m|n}$ is a supermanifold of dimension $m|n$, $\textbf{I}^{\delta|\delta'}$ is standard superdomain with $\textbf{I}=(0,1)$ and $\delta,\delta' \in \{0,1\}$, subject to the warp functions $f(t)$, $f(\bar t)$, and $f(t, \bar t)$, respectively. In each super warped product space, $\bar{M}_{_{(I)}}$, $\bar{M}_{_{(II)}}$, and $\bar{M}_{_{(III)}}$, it is shown that Einstein equations $\bar{G}_{AB}=-\bar{\Lambda}\bar{g}_{AB}$, with cosmological term $\bar{\Lambda}$ are reducible to the Einstein equations $G_{\alpha\beta} = -\Lambda g_{\alpha\beta}$ on the super space $M^{m|n}$ with cosmological term ${\Lambda}$, where $\bar{\Lambda}$ and ${\Lambda}$ are functions of $f(t)$, $f(\bar t)$, and $f(t, \bar t)$, as well as ($m$, $n$). This dependence points to the origin of cosmological terms which turn out to be within the warped structure of the super space-time. By using the Generalized Robertson-Walker space-time, as a super space-time, and demanding for constancy of $\bar{\Lambda}$ we can determine the warp functions and $\Lambda$ which result in finding the solutions for Einstein equations $\bar{G}_{AB}=-\bar{\Lambda}\bar{g}_{AB}$ and $G_{\alpha\beta} = -\Lambda g_{\alpha\beta}$. We have discussed the cosmological solutions, for each kind of super warped product space, in the special case of $M^{3|0}$.