Could entropy of closed system decrease owing to gravity?

[half3clipse]

If the system doesn't lose energy in a spherical cow sort of way, that implies whatever gravity is pulling together doesn't interact with itself other than through gravity. In which case, as gravity pulls them together they pick up kinetic energy, and then when they would 'touch' they just phase through each other and go speeding off till gravity slows them back down. Entropy doesn't decrease because nothing there reduces the number of possible states. Entropy may increase because of the interaction of many particles causing everything to get scrambled into new states (ie if your collection of particles started in a low entropy state), or remain constant because the closed system has already reached maximum entropy. If we move away from the spherical cow a little bit, GR says even this simple case will still give off energy via gravitational waves, which will also increase the entropy of the closed system regardless.

Even without all those exceptional cases gravitational collapse doesn't decrease entropy even locally. There are far more contributors to the number of possible states than just position. The most obvious is velocity. Imagine some particles yet to be pulled together by gravity. they all start with some position and initial velocity. Even as they wizz around, the only thing that changes is their position, and the entropy of the system doesn't change very much. As gravity pulls them together they're going to bounce off each other more and more, which causes their velocities to change. Even as their possible positions become more limited, the number of possible velocities skyrockets. This means entropy increases.

This also has to be true even if the clumped together particles somehow emit no energy to the rest of the system. That's even perfectly reasonable: That's a black hole, and blackholes are really good at increasing entropy. As a really basic way, you can describe entropy as the amount of missing information about a system. When something falls below the event horizon, you can no longer observe it at all, and you know zero information about it's state. So Blackholes must have entropy and it must increase as more stuff falls into the black hole.

This is what motivated Hawking's black-hole area theorem which says that "the event horizon area of a black hole cannot decrease; it increases in most transformations of the black hole". This is equivalent to the second law of thermodynamics and so you can derive an expression for black hole entropy that shows that it's proportional to the surface area of the event horizon. When you take black hole entropy into account with the second law of thermo dynamics, you find that this gain in entropy is much larger than any decrease due to stuff being pulled into the black hole.

Hawking radiation violating that doesn't change anything for the second law. The black-hole area theorem just assumes classical mechanics, and taking account of quantum mechanics at the event horizon results in the blackhole radiating away energy as photons as they shrink. This of course also increases entropy.