Relative Rotation: Physics Approach for Newton's Bucket in Empty Universe

[Alfredo Tifi]

Is there any physics approach useful to establish what should happen to a Newton's water bucketful rotating in a hypothetically empty universe (but for the bucket)? What the bucket would be rotating respect to in that one-bucket-universe? Everything rotating seems to have absolute (not relative) effects in our actual Universe...

 

[Dale]

Is there any physics approach useful to establish what should happen to a Newton's water bucketful rotating in a hypothetically empty universe (but for the bucket)?

Physics was designed to describe this universe. We can speculate about other universes, but such speculations cannot be tested and are therefore philosophical not scientific

 

[PeterDonis]

In relativity there are basically two ways to approach this.

The first and simplest way is to model spacetime as flat, i.e., to assume that the bucket's stress-energy is negligible. Then, in that flat spacetime, you can define a 4-momentum vector and a 4-angular momentum tensor for the bucket. If the 4-angular momentum tensor is nonzero, the bucket is rotating. This basically defines "rotating" to mean "rotating relative to an inertial frame at rest relative to the bucket's center of mass"; in other words, you are using the geometry of flat spacetime to define what "rotating" means.

The more sophisticated way would be to take the bucket's own stress-energy into account, and compute its effect on the spacetime geometry using the Einstein Field Equation. Doing this would show that a non-rotating bucket gives rise to a different spacetime geometry than a rotating bucket (heuristically, approximately Schwarzschild vs. approximately Kerr). These different spacetime geometries have measurable effects; for example, a rotating bucket would produce a nonzero Lense-Thirring precession in a satellite orbiting it (this is the effect that was recently measured for the Earth by Gravity Probe B), whereas a non-rotating bucket would not. This approach basically defines "rotating" to mean that the spacetime geometry effects associated with rotation are present, i.e., it is using the difference between two curved spacetime geometries to define what "rotating" means.

 

[Alfredo Tifi]

Thank you PeterDonis. This is the kind of answer I hoped. Rotational effects are not due to far Universe-stars. It is a local effect. Thus, what holds for the true Universe is presumably or reasonably valid for a flat & empty universe, in contrast with Mach's principle. Furthermore rotation and space-time curvature are absolute (they can't be canceled in any frame, inertial or not).

 

[PeterDonis]

Rotational effects are not due to far Universe-stars. It is a local effect.

From the standpoint of immediate causes, yes. However, you've now opened up a new can of worms. See below.

what holds for the true Universe is presumably or reasonably valid for a flat & empty universe, in contrast with Mach's principle.

In terms of mathematical models, yes, that's true; we can construct a mathematical model of a flat spacetime containing a bucket, and it says what I described in my previous post. Or we can construct a mathematical model of an asymptotically flat curved spacetime containing a bucket with stress-energy, and tell whether it is rotating or not by looking at the spacetime geometry (whether effects like Lense-Thirring are present).

However, whether or not these mathematical models are actually physically reasonable, if we consider them as models of an entire universe, rather than just an isolated system within a larger universe, is a very different question. I think most physicists would say that they're not: that there is no physically reasonable way for a flat spacetime containing nothing but a bucket (or any other isolated object) to exist, and similarly for an asymptotically flat spacetime containing a single isolated object (whether it's rotating or not). In order to have a physically reasonable model of an entire universe, you have to work, at least on a large scale, with something like the FRW spacetimes used in cosmology, and those aren't flat or even asymptotically flat.

In order for models like those I described in my previous post to make sense in the context of a model of the universe as a whole, we have to consider them as small patches of spacetime, describing some particular isolated object, inside a larger FRW spacetime that describes the rest of the universe. Fortunately, there is a way to do this: it turns out that if we have an isolated region that is surrounded by a spherically symmetric distribution of matter, the matter outside the region produces zero spacetime curvature inside the region. So, for example, we could consider a patch of flat spacetime, containing a single bucket, whose extent is very large compared with the size of the bucket, but very small compared with the universe as a whole. The rest of the matter in the universe will be spherically symmetric about the patch of flat spacetime, at least to a very, very good approximation, simply because the density of matter in the rest of the universe is, to a good approximation, uniform on large enough scales. So, inside the patch, things work just like they would if the bucket and the flat spacetime were the only objects in the universe: we can still define a 4-momentum and 4-angular momentum for the bucket, relative to the flat spacetime in the patch, and tell whether the bucket is rotating by whether its 4-angular momentum is nonzero. Similar remarks apply to the curved spacetime models I described, since they are asymptotically flat, so as long as we consider a patch large enough compared to the size of the isolated object, spacetime near the boundary of the patch will be flat.

Viewing these isolated models in the fashion I've just described amounts to using Mach's Principle to explain why the patches of spacetime around the isolated objects are flat (or asymptotically flat), by appealing to the distribution of matter in the rest of the universe. This is why, for example, "rotating" in the sense of my previous post (in either case I described) turns out to be the same, at least to a very good approximation, as "rotating" in the sense of rotation with respect to the distant stars. So from a physical standpoint, I think Mach's Principle is a good way to describe why the models I described in my previous post work as well as they do, even though our actual universe contains a lot of other stuff besides one bucket or one isolated object.

rotation and space-time curvature are absolute (they can't be canceled in any frame, inertial or not).

In both senses of "rotation" from my previous post, yes, that's true: the angular momentum tensor is a tensor and can't be made to vanish in any frame if it is non-vanishing in one frame; and effects like the Lense-Thirring effect are independent of any choice of coordinates, since they're directly observable.

 

[pervect]

As I understand the math, if you have a bucket in a universe empty of matter, you still haven't defined quite enough to solve Einstein's field equations. Like the case in electromagnetism, you need to specify some boundary conditions , this is a general mathematical requirement for solving partial differential equations (PDE's), which is what the underlying math is at the most basic level.

So for starters, you assume the universe contains no gravitational waves.

I don't think this is quite enough assumptions yet to solve the Einstein's field equations (EFE's). I believe you also need some assumptions about topology. I suspect that it's sufficient to say that the universe is unbounded (not compact). But actually, this is a good question for a mathematician and something more rigorous than an off-the-cuff post.

But if you do have a solution to EFE's, and it is infinite and asymptotically flat, you can tell from the metric of space-time for this asymptotically flat solution whether or not that universe (which has only the bucket in it) is rotating or not. Of course, to have a metric, you do need to assume that the space-time exists without other observers in it. But this is rather necessary to apply the EFE's at all, so I'd argue this is part of a starting assumption that GR holds. WHich I made, but not explicitly until now.

If you don't assume GR holds,, all bets are off, but the question becomes too vague for me to say much other than you can probably do whatever you like, as long as you're self-consistent.