Einstein equations with cosmological constant in Super Space-Time

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 $M^{3|0}$.