Perfect Fluid FRW with Time-varying Constants Revisited

In this paper we revise a perfect fluid FRW model with time-varying constants \textquotedblleft but\textquotedblright taking into account the effects of a \textquotedblleft$c$-variable\textquotedblright into the curvature tensor. We study the model under the following assumptions, $div(T)=0$ and $div(T)\neq0,$ and in each case the flat and the non-flat cases are studied. Once we have outlined the new field equations, it is showed in the flat case i.e. K=0, that there is a non-trivial homothetic vector field i.e. that this case is self-similar. In this way, we find that there is only one symmetry, the scaling one, which induces the same solution that the obtained one in our previous model. At the same time we find that \textquotedblleft constants" $G$ and $c$ must verify, as integration condition of the field equations, the relationship $G/c^{2}=const.$ in spite of that both \textquotedblleft constants" vary. We also find that there is a narrow relationship between the equation of state and the behavior of the time functions $G,c$ and the sign of $\Lambda$ in such a way that these functions may be growing as well as decreasing functions on time $t,$ while $\Lambda$ may be a positive or negative decreasing function on time $t.$ In the non-flat case it will be showed that there is not any symmetry. For the case $div(T)\neq0,$ it will be studied again the flat and the non-flat cases. In order to solve this case it is necessary to make some assumptions on the behavior of the time functions $G,c$ and $\Lambda.$ We also find the flat case with $div(T)=0,$ is a particular solution of the general case $div(T)\neq0.$