Asymptotes of solutions of a perfect fluid coupled with a cosmological constant in four-dimensional spacetime with toroidal symmetry

摘要

Asymptotes of solutions of a perfect fluid when coupled with a cosmological constant in four-dimensional spacetime with toroidal symmetry are studied. In particular, it is found that the problem of self-similar solutions of the first kind for a fluid with the equation of state, p = kρ, can be reduced to solving a master equation of the form, $$ 2 F(q, k)frac{q''(xi)}{q'(xi)} - G(q,k) q'(xi) = frac{4}{xi}. $$ For k = 0 and k = −1/3 the general solutions are obtained and their main local and global properties are studied in detail.