Some exact inhomogeneous solutions of Einstein’s equations with symmetries on the hypersurfaces t=const

The solution of Einstein’s field equations is studied for a metric written in the form (δ≠γ)ds2=−α2(t,r,θ,φ)dt2 +e2β(t,r) dr2+e2γ(t,r) dθ2 +e2δ(t,r)M2(θ)dφ2. A perfect fluid, which flows orthogonally to the hypersurfaces t=const is considered as matter content. These hypersurfaces admit a translational Killing vector, which will not be, in general, a Killing vector of the whole space‐time. All the possible solutions are obtained when α depends on the variable φ. These solutions represent either a perfect fluid without expansion or vacuum with a cosmological constant Λ0. Some particular inhomogeneous solutions are obtained for α independently of φ. These solutions are physical, the fluid obeys an equation of state p=ρ (stiff matter), and the space‐time admits, apparently, only a group G2 of isometries. A vacuum family is also obtained in this case.