# Does entropy decrease in the late universe? [duplicate]

The early universe had an entropy of $$10^{88} k_B$$ . This entropy does mostly come from photons, quarks and leptons and stayed constant over time, due to adiabatic expansion. Sagittarius A* has an entropy of $$10^{91} k_B$$ , all black holes combined determine the entropy of the whole universe today (roughly $$10^{103} k_B$$ ). If the whole universe was one black hole it would have $$10^{123} k_B$$ worth of entropy.

But since Hawking we know that black holes dissipate energy and get therefore smaller which in return means we lose entropy, since $$S_{BH} = \frac{k_B \cdot A}{4 l_{p}^2}$$ . Wouldn't this mean that in the far far state of the universe we would have a decrease in entropy?

Your question seems to assume that when a black hole evaporates, it gets smaller - so its entropy gets smaller - but that's it. But as the black hole evaporates, something new is created ("Hawking Radiation"). Hawking Radiation is produced around the black hole's event horizon (not strictly "from" the horizon itself) during the entire lifetime of the black hole, and has entropy. In particular, it is possible for it to have more entropy than the black hole - it's not obvious it must have less, and definitely it isn't zero.

But does it really have more entropy than the black hole, and how is that entropy calculated? Unfortunately, I'm not really an expert in black hole thermodynamics. One 1982 paper I found seems to claim that

the entropy of the radiation evaporated by an uncharged, nonrotating black hole into vacuum in the course of its lifetime is approximately 4/3 times the initial entropy of this black hole.

Another 2006 paper says that

The generalised second law guarantees the increase of the total entropy of the whole system which consists of the black hole and the Hawking field. ... The rate of entropy increase of the Hawking field (the entropy emission rate by the black hole) grows faster than the rate of entropy decrease of the black hole along the black hole evaporation in the empty space. The origin of the entropy increase of the Hawking field is the increase of the black hole temperature. Hence an understanding of the generalised second law in the context of the nonequilibrium thermodynamics is suggested; even if the self-relaxation of the Hawking field does not take place, the temperature increase of the black hole during the evaporation process causes the entropy increase of the Hawking field to result in the increase of the total entropy.

I think my question could be answered by this post:

https://physics.stackexchange.com/a/100586/355021

"...generalised second law of black hole thermodynamics, which broadly states that the entropy of the black hole plus the Hawking radiation cannot decrease."