A new approach on Einstein equation in super spacetime.

In this paper, we introduce a new kind of super warped product spaces M ̄ (I) = I 1 | 0 × f M m | n , M ̄ (I I) = I 0 | 1 × f M m | n , and M ̄ (I I I) = I 1 | 1 × f M m | n , where M m | n is a supermanifold of dimension m | n , I δ | δ ′ is a superdomain with I = (0 , 1) and δ , δ ′ ∈ { 0 , 1 } , subject to the warp functions f (t) , f (t ̄) , and f (t , t ̄) , respectively. In each super warped product space, M ̄ (I) , M ̄ (I I) , and M ̄ (I I I) , it is shown that Einstein equations Ḡ A B = − Λ ̄ ḡ A B , with cosmological term Λ ̄ are reducible to the Einstein equations G α β = − Λ g α β on the super space M m | n with cosmological term Λ , where Λ ̄ and Λ are functions of f(t), f (t ̄) , and f (t , 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 spacetime. By using the generalized Robertson–Walker spacetime, as a super spacetime, and demanding for constancy of Λ ̄ , we can determine the warp functions and Λ which result in finding the solutions for Einstein equations Ḡ A B = − Λ ̄ ḡ A B and G α β = − Λ g α β . We have discussed the cosmological solutions, for each kind of super warped product space, in the special case of M 3 | 0 . [ABSTRACT FROM AUTHOR]