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Recently, it has been found conditions for the heat flux where its introduction into the energy-momentum tensor, given his no mechanical nature, has no problems [1]. This has been achieved by checking the validity of the second law of thermodynamics in a fluid that is boosted by a Lorentz transformation of a non-commoving frame [2]. It is known that the condition, that turns out to be the null energy condition, involves the presence of a set of reference frames where Landau-Lifshitz frame is one of them. Moreover the entropy production remains positive, implying that there are no problems or issues that violate the second law of thermodynamics. In the present contribution we review the above condition, and apply it for a self-gravitating relativistic fluid in a spherically symmetric distribution in pure local coordinates. We find that our condition influences Schwarzschild fields, relating the condition of slow evolution with thermal quantities of the fluid.

Vanishing condition for the heat flux and slow evolution of a spherically fluid distribution