Null energy condition of dynamical black holes in spatially flat FLRW space-times

A new identity reported here for the first time is used for studying how the null energy condition can be applied to the models of dynamical black holes proposed recently [I. I. Cot\u aescu, Eur. Phys. J. C (2022) 82:86]. It is shown that this condition selects the models with physical meaning whose horizons evolve in accordance with the general black hole theory in frames with null coordinates [S. A. Hayward, Phys. Rev. D {\bf 49} (1994) 6467].