Too big to revolve like an atom?

Question

At what point do objects naturally start revolving as a disk? From an atom, with which the electron revolves as a sphere, to a galaxy, that revolves in the same direction? I herd about rigid rotation and regular rotation. Either answer is ok. Thanks

Answer 1

This question leads to some subtleties. There are at least two distinct notions of "revolution" that could be meaningful in physics. Namely, "to revolve" can mean:

1. To have angular momentum;
2. To transform by a particular kind of Euclidean isometry (a rotation) (or, to be broader and more technical, a representation of that Euclidean isometry).

As far as we can tell, objects of all sizes can have nonzero angular momentum. The spin of an electron, for example, is responsible for its magnetic moment. Likewise, galaxies and clusters of galaxies can have angular momentum as well. This is quite aside from whether they rotate rigidly or whether there is a nonlinear rotation curve (a linear one would imply a rigid rotation) as in a galaxy or the rings of Saturn. For really big objects (galactic clusters and beyond), however, angular momentum becomes a less useful notion, because it is not in general conserved in general relativity (there's no consistent way to define AM of the gravitational field).

The second notion, however, puts definite size / angular velocity limits on the objects it can be applied to. If a body has too big a radius angular velocity product, rigid rotation would imply some parts of the body's moving at greater than $c$ relative to other parts. Indeed, the notion of rigid transformations is not meaningful in special and general relativity: you cannot accelerate a rod by a local shove at one end for this would imply the "shove signal" reaching the other end instantaneously (look up the Barn Pole paradox). So in general we can say that $\omega\,r \ll c$ for the second notion to be valid. When one has $\omega\,r \sim c$ then we have the situation described by the Ehrenfest Paradox. The notion of Born Rigidity must replace the more everyday notion.

At the other end of the length scale, transformation by rotation is only meaningful if the object concerned is not invariant with respect to rotation. In other words, we need to be able to sight different parts of the body and track their individual movements. This definitely applies to objects approaching the hundreds of nanometer scale and probably less: an cell nucleus irradiated with a vortex mode (a particular laser mode with angular momentum) under a microscope is seen to "spin" under the microscope. But the notion becomes more and more problematic as we approach the atomic domain. It's not meaningful to talk about rotation of a hydrogen atom in its ground state: the object is mixture delocalized quantum fields in their non ground state and it doesn't really make sense to ascribe different parts of the spherically symmetric atom to different polar azimuth angles, just as it is not meaningful to distinguish "different" subatomic elements. We as physicists don't believe the notion of electrons orbitting a nucleus like little planets: this notion has been outmoded since the 1930s. That doesn't mean, however, that the hydrogen atom cannot have angular momentum: its constituent electrons and quarks have intrinsic spin angular momentum. A notion of rotation, however, is meaningful in molecules where we can meaningfully talk about relative orientations of different bonds.

So "rotation" in the everyday sense applies roughly to objects the size of molecules up to the size * angular velocity limit $\omega\,r \ll c$.

Answer 2

Angular momentum, or a measure of rotation, in very large astrophysical or cosmological bodies and energy can become relativistic, and must be treated using General Relativity. In General Relativity (GR) angular momentum is not too different from momentum and energy. Different quantities, the last two are contained in the stress energy momentum tensor and the angular momentum is defined by a 3 tensor formed by the exterior product of position and the stress energy momentum tensor.

All three quantities are conserved only for certain conditions in GR. But those can be very interesting and useful cases such as in our astrophysical observations. And all have no limits on how big they can be. This is contrary to what @RodVance wrote. This is explained below.

The conditions for conservation are that some symmetries exist (similar to classical mechanics and QM), except in GR the symmetries don't happen in every spacetime. The symmetries in GR are defined by Killing vector fields (basically the flows of the symmetries, along which the metric does not change). Energy is from a timelike Killing vector, momentum a space like one, and angular momentum from an axial or tangential Killing vector. Those entities are also conserved in the large (eg, total energy lot and and total energy flowing out asymptotically out at spatial or null infinity) for spacetimes that are asymptotically flat. These were derived by ADM for spatial infinity and by BMS for null infinity.

An example where angular momentum is conserved in GR is for rotating stationary (i.e., equilibrium) black holes (BH), a Kerr metric. It has an axial Killing vector, so angular momentum is conserved. It is conserved when one extracts energy from them using the Penrose process, and the angular momentum the BH looses is carried away by the same particles that carry away the energy. Not too similarly, but equally conservative is the energy and angular momentum at infinity carried away by gravitational waves from the energy and the angular momentum (both intrinsic and orbital) of two merging BHs. That was what we detected from the two BH mergers observed by LIGO.

The concept that GR does not generally conserve those quantities is true, but you can find many cases and conditions in which they are conserved in GR, and in which it is useful to know that. For LiGO the conservation of energy allows them to estimate luminosity distance from the strain (and thus energy) of the detected gravitational waves.

Finally, the limits. Yes, there can be limits for the angular momentum (J) of BHs. If J is too large, equal to M^2 in terms of natural units and M the mass of the BH, it is called extreme. Any more and the horizon would not exist. It is conjectured that there can not be singularities without horizons, and none except for some special cases with certain fields have been able to be constructed mathematically in GR.

Large bodies of mass and energy can be rotating and have a large J. No limit is known except for that BH limit. J can increase without limit with nothing going faster than c. Just the rotation is not rigid, but so what. It is rotation and it is angular momentum. I do not know what the largest 'known' J in astrophysics is, but there is absolutely nothing that says that J is limited. As for rotating, again, as long as the object going around is not going faster than c it's ok. Born rigidity or any such may apply to rigid bodies but not astrophysical bodies. Those are not rods. And wr is for special relativity, not GR where you have to take the metric into account.

One more item, in the nonrelativistic limit. Spiral galaxies tend to have disks because of the way the gas that formed the stars gets attracted and the particles collide with each other, with the angular momentum conservation also entering in. See the very good explanation of that at Why the galaxies form 2D planes (or spiral-like) instead of 3D balls (or spherical-like)?