Violation of the weak energy condition: Is it generic of spontaneous scalarization?

It was recently shown by Whinnett and Torres [A. W. Whinnett and D. F. Torres, Astrophys. J. 603, L133 (2004).] that the phenomenon of spontaneous scalarization in compact objects (polytropes) was accompanied also by a spontaneous violation of the weak energy condition (WEC). Notably, by the encounter of negative-energy densities as measured by a static observer at several points of the star. Here we argue that such a situation is not generic of scalar-tensor theories of gravity (STT). We support this conclusion by numerical results within a class of STT and by using three realistic models of dense matter. However, we show that the 'angular parts' of the additional conditions needed for the WEC to hold ρeff+Tii(eff)≥0 tend to be 'slightly violated' at the outskirts of the star