I Relativistic centrifugal force

[PAllen]

I'm not quite sure I understand procedure 2, I'd suggest the following simple one for measuring the circumference of the disk.

Make up a large number of short rods. You could replace them with taunt tape measures, as well, or little radar sets. It's simpler with rods, though. For more simplicity, make all of the rods of uniform, length.

Count how many of these little rods you can fit around the circumference of the disk.

Multiply the length of each rods by the number of rods. In the limit as the rods become shorter and shorter, the result is the circumference of the disk.

I am not concerned with measuring the circumference. I am talking about measuring the distance between two arbitrary particles of the disc via a taught tape measure (which should ideally be a geodesic of the quotient manifold geometry . I thought that was clear, since I never mentioned circumference.

 

[A.T.]

... but those individual spacelike curves cannot be "assembled" into a single closed spacelike curve that describes "the circumference of the disk at an instant of time"...

If the geometry of the disc doesn't change over time, that "at an instant of time" part might be irrelevant for practical purposes.

So I will agree there is some physical procedure that corresponds, over finite distances. to quotient space metric computations.

If you wanted to build an already rotating truss structure from pre-fabricated short beams (not spin one up), the quotient space metric would tell you how many of these short beams you need to order.

 

[PeterDonis]

If the geometry of the disc doesn't change over time, that "at an instant of time" part might be irrelevant for practical purposes.

This is what the quotient space construction does: it makes use of the fact that the congruence of worldlines describing the disk is stationary (which is what "doesn't change over time" means in the math) to derive an abstract 3-dimensional space in which each point corresponds to a worldline. This abstract space has a geometry and a metric that can be viewed as "the geometry of the disk". However, this abstract space does not correspond to any 3-dimensional subspace of the 4-dimensional spacetime, and that means it violates some intuitions about what "the geometry of the disk" means physically.

 

[pervect]

I am not concerned with measuring the circumference. I am talking about measuring the distance between two arbitrary particles of the disc via a taught tape measure (which should ideally be a geodesic of the quotient manifold geometry . I thought that was clear, since I never mentioned circumference.

OK, let's consider that case. I would say that we have a bucket of rods of uniform, very small, length, and we ask the question "what's the least number of rods can we use to go from one mark on the surface of the disk to another". Technically, we are taking the limit as the length of the rods grows shorter and shorter.

Do you think your idea of pulling a tape measure taut gives the same answer as this approach, or a different answer? If you think it's different, we might have to delve into the mathematical representation of the tape measure some more.

You claim that there is some technique that gives a different answer for the distance, but I don't quite follow what it is. Let's focus on that, the technique you claim gives a different answer.

1) Local to one observer, mark lines on tape, then extend it to some other disc observer and have them pull it taught. Have them mark where they are, and communicate the result to the other end. I claim this is what would normally be thought of as using a tape measure, and this will not match the quotient space metric.

I've been trying to imagine what you are saying here, an failing. I think of a "mark on the tape", and also "a mark on the disk", that I used earlier, as necessarily being some worldline in the congruence of worldlines that represents the spinning disk.

As long as the tape is static on the disk (not vibrating), marks on the disk should be the same as marks on the tape. And both are represented by worldlines that are in the congruence of worldlines that represent the spinning disk.

You seem to be claiming that we cannot use the quotient manifold to intepret the idea of the distance between worldlines as a distance between points in the quotient space. But I don't see why you are claiming this. Basically, to my mind, marks on the disk, marks on the tape (when the tape is at rest on the disk), and points in the quotient manifold all represent the same thing in different words.

 

[PeterDonis]

As long as the tape is static on the disk

But the tape can't stay static on the disk during the process of it being pulled taut after marks have been made on it. So if the marks are made on the tape before it is pulled taut, while the tape is restricted to a single local region of the disk, as @PAllen describes, then the relationship between marks on the tape and marks on the disk will change when the tape is pulled taut between points that are not restricted to a single local region of the disk.

On a disk that was not rotating, it would be possible to pull the tape taut in a way that minimized distortion of the tape (and in the limit eliminated it) so that there would be no change in the relationship between marks on the tape and marks on the disk. But this is not possible for a rotating disk. This is one of the counterintuitive consequences of the congruence of worldlines that describes the disk not being hypersurface orthogonal.

You seem to be claiming that we cannot use the quotient manifold to intepret the idea of the distance between worldlines as a distance between points in the quotient space.

That's not the issue; distances in the quotient space can indeed be interpreted as one possible meaning of "distance between worldlines". But this meaning of "distance between worldlines" does not correspond to other possible meanings (in terms of various possible physical realizations) in the way that it would if the disk were not rotating. Basically, the rotation of the disk makes different interpretations of "distance" give different answers, which would give the same answers for a non-rotating disk.

 

[pervect]

That's not the issue; distances in the quotient space can indeed be interpreted as one possible meaning of "distance between worldlines". But this meaning of "distance between worldlines" does not correspond to other possible meanings (in terms of various possible physical realizations) in the way that it would if the disk were not rotating. Basically, the rotation of the disk makes different interpretations of "distance" give different answers, which would give the same answers for a non-rotating disk.

It sounds like we agree that the quotient manifold represents the space of the disk.

Do we agree that in the limit as the distance between two points on the disk (two worldlines in 4d spacetime) approaches zero, i.e. for very close points / worldlines, that physically significant notion of distance must agree with the radar notion of distance?

If so, then it follows we agree that the quotient space represents the "space" of the disk, and we agree on the metric associated with the quotient space. Now, while I could imagine using some connection other than the Levi-Civita connection of the metric to define distances (by measuring distances along geodesics defined by this alternative connection), I can't see any motivation for doing such a thing.

 

[PeterDonis]

It sounds like we agree that the quotient manifold represents the space of the disk.

For a suitable interpretation of "the space of the disk", yes. One of the points I'm trying to make is that the ordinary language phrase "the space of the disk" does not have a single unique interpretation. It just so happens that in simpler cases like a non-rotating disk, the different possible interpretations end up leading to the same result. But in the case of the rotating disk, they don't.

Do we agree that in the limit as the distance between two points on the disk (two worldlines in 4d spacetime) approaches zero, i.e. for very close points / worldlines, that physically significant notion of distance must agree with the radar notion of distance?

In the limit of close enough points/worldlines, all of the different possible notions of distance agree.

If so, then it follows we agree that the quotient space represents the "space" of the disk

No, it doesn't, because "the space of the disk" is a global concept, not a local concept. As long as we're clear about exactly which global concept we're talking about, there's no problem. But just saying "the space of the disk" doesn't pick out a single unique global concept. That's why terms like "quotient space", which is unambiguous, are preferred.

 

[A.T.]

However, this abstract space does not correspond to any 3-dimensional subspace of the 4-dimensional spacetime, and that means it violates some intuitions about what "the geometry of the disk" means physically.

To me personally, measuring a stationary geometry by laying out rulers on it seems quite intuitive and physical, while a 3-dimensional subspace of the 4-dimensional spacetime is a more abstract idea.

 

[PeterDonis]

measuring a stationary geometry by laying out rulers on it seems quite intuitive and physical

"Laying out rulers on it" is ambiguous. If it means the particular procedure that has been described, where each ruler only covers an infinitesimal distance and you have to do a careful calculation to combine all the infinitesimal ruler measurements into a total distance between two points that are not infinitesimally close, then I'm not sure that would seem "quite intuitive and physical" to everyone. What I think is "quite intuitive in physical" is the idea of just laying a single ruler between two points, but on the rotating disk you simply can't do that the way you can on a non-rotating disk.

 

[A.T.]

If it means the particular procedure that has been described, where each ruler only covers an infinitesimal distance and you have to do a careful calculation to combine all the infinitesimal ruler measurements into a total distance between two points that are not infinitesimally close, then I'm not sure that would seem "quite intuitive and physical" to everyone.

This is basically how humans have been surveying the curved surface of the Earth for centuries. To me it seems less abstract than slices through space-time.