gustavolacerda writes,
if physics-entropy is subjective, wouldn’t you get some
serious paradoxes?

My initial reaction was, “Absolutely! This theory of mine makes no sense.” Consider a black hole. It’s surface area is proportional to its entropy, so if entropy was subjective, then two people would claim the same black hole had different sizes.

Let us clearly show that the notion of subjective entropy is nonsense. Consider two observers, Alice and Bob. Both observe a black hole and agree it has the same size and entropy. Together they throw a box of gas towards the black hole. The difference between Alice and Bob is that Alice knows more about the gas than Bob does. Alice thinks the box contains a bunch of cold atoms in a perfectly known state. Bob thinks that the box contains a bunch of hot atoms in an uncertain state.

So Alice and Bob watch the box drift towards the black hole. As the box falls in it starts radiating its energy away. At a certain point Bob says, “See there, the box as been sucked into the black hole!”. The box has gotten close enough to the black hole that it has done its fancy gravitational well interacting thing. The event horizon has expanded, the entropy has been added to the black hole.

But Alice says, “No. The box hasn’t entered the black hole yet. See it is still there.” Alice points vaguely at the dark object. “Where?”, asks Bob. Alice and Bob see a photon flash. “There!”, exclaims Alice, “That was light from one of the atoms giving off energy as it falls closer to the event horizon.” Bob disagrees, “No, that is just Hawking radiation.”

After a long wait, and many more photons, Bob has seen enough Hawking radiation that the black hole has shrunk back to its original size. “The box is now gone,” says Alice, “All of its mass has been converted to radiation so there is nothing left to enter the black hole.” Before returning home, Alice and Bob stare at the black hole a moment. Both of them agree on its state once again.

Oops, I set out to show that my subjective entropy is crap, but I ended up showing how two people could observe the same events happening with a black hole, even if they disagree about the entropy and size of the black hole. One person sees a large black hole with hawking radiation. The other person see a small black hole with a pile of junk near its event horizon, and the junk radiating energy as it falls in.

I think that subjective entropy is a broken theory, but I am still looking for a paradox.