The entropy growth in the Universe was studied by Berman [1]. Here, we re-address the problem. One of the reasons for such return to this subject is that in the latter paper, we considere the absolute temperature to be represented by the Machian temperature, the sort of thing that is problematic, so, now, we conform to standard treatises, where the absolute temperature of the Universe is identified by its radiation, rather than by its matter contents. The better result is that we find either for the Universe or for black holes, the same dependence of entropy, while in reference [1], the non-standard temperature definition, yielded, for the Universe, an entropic formula dependent on However, the main motivation of entropy study is the deficiency of Standard Cosmological theory, whereby the fluid is isentropic, nobody is willing to accept the sort of thing. The entropy of the Universe is a measure of the disorder which entails negative information, according to Claude Shannons in-terpretation in the theory of communication. Berman and Gomide [2] showed that the Universe resembled a Kerr white-hole. Here, we just show that the formula for entropy of the Universe is the same as for a white-hole, i.e., varies with the square of the radius. Then we delve into dark energy considerations showing its physical origin, as heat flow in the Universe.

Berman[3] has shown, by pseudotensors, that the total energy of the Universe, is constant zero.

From Thermodynamics, the entropy of a reversible process, is given by [4],

where and stand for absolute temperature, entropy, total energy, pressure and volume.

For Cosmology, one conveys the absolute temperature, as the one for radiation.So, we write, for cosmology,

while, for Black-Holes, and Cosmology, we take the perfect gas equation of state,

where, and stands for energy density.

According to George F.R. Ellis [5], many statements, about the nature of entropy, are wrong when gravity is dominant, as is the case in Cosmology. Such studies, he says, are ill-founded. We tried to get rid of such inconveniencies, by working directly upon first principles.

As the total energy of the Universe, is constant zero, we find,

On the other hand, Machian analyses make the energy density dependent on in accordance with Berman [6,7]. The Machian theory, includes the zero-total energy theorem, and includes Brans-Dicke relation, which we take as exact.

From the energy statement,

we find,

Then, the energy density, say, of the mass, is given by,

thus proving our previous assertion.

Then, we have,

where is a constant.

Still, the radiation law is,for a black body,

where is constant,in accordance with any Cosmology textbook [8].

Thus,

Then, we find,

or, upon integration,

with and constants.

Then, the Universal entropy varies with the square of the scale-factor.The Hawking formula for Black-Holes, is also dependent on the surface of the event horizon, ie, it is also R²-dependent [9].

In a very popular paper Berman [10], has shown that entropy growth is attained by a cosmological “constant”, that represents dark energy, and varies with cosmic time. His result, is given by formula,

where the constant arises from the three-volume of the Universe, when described by,

On the other hand, when one deals with Raychaudhuris equation,[11], and disconsider shear, and turn into a non-accelerating fluid, we find [12],

where stands for the heat or energy flow.

On combining all the above relations, one concludes that the dark energy, which causes the accelerating Universe, is due to heat flow.

We find that entropy grows and the Universe has a cosmological “constant” representing dark energy, which physically signifies that there is a positive heat flow in the Universe.

MSB thanks for the important incentive offerred by Miss Solange Lima Kaczyk,now a brand new advocate, continued during the last five years of his research in Cosmology.