| That was excellent work by minutephysics. Your intuition about Boltzmann entropy (disguised as the probability of finding a cloud with no net rotation in its bulk vs one with some) interested me. I think it highlights a "past hypothesis" problem in the wake of your link to the minutephysics video. Your description is pretty much entirely gravitational. When you add matter-matter collisions, especially inelastic ones, you get changes in internal energy of the grains, and radiation outwards, and that's what drives collapse of gas clouds in timescales shorter than the age of the universe. Changes in internal energy (e.g. internal rotational degrees of freedom in large dust grains, large being at least the size of a hydrogen molecule) alters the "bounce" from a collision making it easier for collisional grains to clump together. See <https://news.ycombinator.com/item?id=36418364> for a neat example involving partially filled water bottles being dropped to the floor. When the water is swirled, the bounce is reduced. Analogously, one can swirl or flex an isolated molecule around some axis. For much smaller grains (e.g. atomic hydrogen) that feel electromagnetism, the light emitted by collisions ("scattering") is even more important, as it carries away internal pressure from the cloud, rather than just shifting it around internally to the cloud. In other words, for each in a pair of collided grains, subsequent collisions will be at lower energies, and so have smaller recoils. Looking at this through the lens of General Relativity (it being a gravitational problem, after all), we want to consider a covariant angular momentum quantity. In suitable coordinates (~ Cartesian, with the origin at the centre of momentum of the cloud), this angular momentum is best associated with the off-diagonal spatial shear components of the stress-energy tensor. That's the upper blue triangle in this diagram: https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor#/... In the diagram the 0-3 are the four Lorentzian spacetime dimensions, and 0 is the timelike one. Roughly, for T^{mn}, m is the (signed) "goesinto" direction and n is the (signed) "goesoutof" one, or if you prefer, the flux of m-momentum in the n-direction. Essentially the goal in the early collapse of a cloud is to shuffle the nonzeros out of the pressure diagonal (in green) into the energy (T^00 in red, which dominates a tensor contraction to a Newton-like mass) and the components above the pressure diagonal, which one can think of as heat, angular momentum, and radiation. Matter-matter interactions enable this shuffling much faster than for matter that only interacts gravitationally. A cloud of cold collisionless, non-radiating, non-interacting (except via gravity) dust with few internal degrees of freedom has trouble collapsing gravitationally. "Their gravitational interaction will cause them to move towards each other" is true, but if they don't interact non-gravitationally they'll mostly just slide right past each other. We see this in galaxy-galaxy collisions (like in the famous Bullet cluster) where stars are spaced far enough apart that they're essentially as non-colliding as dark matter; it's the less-compacted interstellar gas and dust clouds which smack into each other and throw off lots of X-ray radiation, which helps the gas swirl around near the site of the collision and collapse into star forming regions. > [many] disorderly configurations ... with at least some angular momentum Sure, but the thrust of the minutephysics video (if not exactly answering the question your comment's parent asked) is evolving from a system with relatively little angular momentum to a system with a lot, and contracting from a blob to an arrangement with a clear axis and eventually towards a thin disc. I would think that initial conditions of condensed objects moving on roughly circular Keplerian orbits roughly constraned to a plane is less typical than initial conditions of a cloud with random internal motion. From a Boltzmann perspective, there are a lot more microstates which can describe the cloud-blob macrostate than the star system macrostate. But the "past hypothesis" problem is that if the latter evolves from the former, then surely the primordial gas-cloud must be of lower entropy than the nice orderly-looking star system, especially if one finds cats, cars, computers, and teapots in it? So the question from my second paragraph becomes: does a cloud with random internal motions and no bulk rotation have more entropy than a cloud that has a clear rotational axis? As short-duration snapshots, the answer appears to be yes, and for reasons very similar to the ones you gave in your comment. But because of the evolution of the former to the latter, which is evidently common (we see stars and galaxies everywhere in our sky, and have good models for star formation and ok ones for galaxy formation), the answer appears to be no. https://en.wikipedia.org/wiki/Past_hypothesis (2nd last paragraph) is a terse starting point if you want more. There are bound to be youtube videos about it too, hopefully at least as good as the minutephysics video at the top. PS: for experts who want to think about angular momentum differently, e.g. the J parameter in a Kerr black hole or some other vacuum spacetime, are directed to ยง5.11 of Misner Thorne & Wheeler vs e.g. the problems at the end of Wald's chapter on asymptotic flatness. |
