A new approach on Einstein equation in super spacetime

Abstract

In this paper, we introduce a new kind of super warped product spaces [Formula: see text], [Formula: see text], and [Formula: see text], where [Formula: see text] is a supermanifold of dimension [Formula: see text], [Formula: see text] is a superdomain with [Formula: see text] and [Formula: see text], subject to the warp functions [Formula: see text], [Formula: see text], and [Formula: see text], respectively. In each super warped product space, [Formula: see text], [Formula: see text], and [Formula: see text], it is shown that Einstein equations [Formula: see text], with cosmological term [Formula: see text] are reducible to the Einstein equations [Formula: see text] on the super space [Formula: see text] with cosmological term [Formula: see text], where [Formula: see text] and [Formula: see text] are functions of f(t), [Formula: see text], and [Formula: see text], 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 spacetime. By using the generalized Robertson–Walker spacetime, as a super spacetime, and demanding for constancy of [Formula: see text], we can determine the warp functions and [Formula: see text] which result in finding the solutions for Einstein equations [Formula: see text] and [Formula: see text]. We have discussed the cosmological solutions, for each kind of super warped product space, in the special case of [Formula: see text].