Lorentz Contraction Circular Motion

[bcrowell]

*If* you regard a ruler as measuring the distance between worldlines, I believe you can get a well -defined answer for the circumference of a rotating disk. (You have to make some basic assumptions that the distance between worldlines is the shortest worldline connecting them, and that this distance is static because the geometry is static, and that you take the limit for closely space worldlines).

Can you explain in a little more detail what you mean by this? It seems to me that given any two world-lines A and B, you could have other world-lines connecting them that would have any length you want. Say you're using a +--- metric. Then to get a world-line C with big positive length that connects A and B, you could start at a point on A, maintain a coordinate velocity of 0, wait until B is about to hit you, then run back toward A at the speed of light. Lather, rinse, repeat. World-line D could be one that goes from A to B at the speed of light, giving a length of 0. World-line E, which can't be physically realized, runs back and forth between A and B at speeds that are always greater than c; it gives as big a negative length as you like.

 

[pervect]

Can you explain in a little more detail what you mean by this? It seems to me that given any two world-lines A and B, you could have other world-lines connecting them that would have any length you want. Say you're using a +--- metric. Then to get a world-line C with big positive length that connects A and B, you could start at a point on A, maintain a coordinate velocity of 0, wait until B is about to hit you, then run back toward A at the speed of light. Lather, rinse, repeat. World-line D could be one that goes from A to B at the speed of light, giving a length of 0. World-line E, which can't be physically realized, runs back and forth between A and B at speeds that are always greater than c; it gives as big a negative length as you like.

I think I got the sign wrong. But it's probably safer to say "extremizes" the distance, that way the sign doesn't matter :-). And I did mean to say that path along which we measure the distance was to be a straight line in SR (or in general a geodesic if space-time is curved) even if I forgot to specify it in my last post. We don't need to define how to synch clocks to define the notion of a straight line either - it's coordinate independent.

suppose we have two curves C1 and C2

C1 is (x=0, t=lambda)
C2 is (x=1, t=lambda)

The distance between C1 and C2 will in general depend where on C1 (or C2) we are. Let's say we want the distance between C1 and C2 at the point (0,0) on C1

Lets pick a point on C2, (0,tau). Then the square of the straight line distance will be the loretnz interval between them

d^2 = (1-0)^2 - (0-tau)^2 = 1 - tau^2

this is extremized when tau=0, making the distance one.

 

[Fredrik]

Hmm...so are you saying that Rindler, Grøn, and Dieks are all wrong? If so, then it would be interesting to see what you think is the error in their treatments.

I'm just saying that the closed continuous curve in Minkowski spacetime that satisfies $t=0, x^2+y^2=r^2, z=0$ in an inertial frame has proper length $2\pi r$, and that proper length is a coordinate independent property of a spacelike curve. I don't think any of those guys would disagree with that.

Regarding what they're doing, I just don't see the point. I don't see why they feel their calculations are worth doing, or why they insist on using terms like "spatial geometry". I also don't see a way to write down a rule that describes how to make length measurements using a radar device that's in an arbitrary state of motion. If the purpose of these calculations is to find such a rule, then I think it's worth doing, but I didn't get that impression.

 

[pervect]

The standard treatment is more abstract, but it should be equivalent to taking the radar distances between nearby points on the circumference and summing them.

 

[heldervelez]

CMB photons as standards clocks and rulers

Here the setup:

- The mother ship is moving inertially and not rotating, as verified by accelerometers and Sagnac interferometers.

- The mother ship sends out 3 space ships, at 120°-step angles.

- All 3 ships run the same acceleration program, that brings and keeps them in an orbit around the mother ship. Now all 3 are at rest in the same rotating frame.

- Each of the 3 ships is observing the other 2 ships through telescopes and measuring the apparent angle between them.

We can use CMB photons as standards clocks and rulers, when at rest in the mother ship.
CMB provides a commom reference.

Suppose we dispose radially in the circular disk a set of equal surface (when at rest) photodetectors.
For calibration purposes, when at rest, we measure the count of incoming photons in each device. The count must be equal in each other.
When moving the circle the count in each device will differ as the device is more inner or outer.
The number counts at each can be used to infer the equivalent area of each device or the geometry?

 

[bcrowell]

I also don't see a way to write down a rule that describes how to make length measurements using a radar device that's in an arbitrary state of motion.

See pp. 6-8 of Dieks, http://www.phys.uu.nl/igg/dieks/rotation.pdf

The standard treatment is more abstract, but it should be equivalent to taking the radar distances between nearby points on the circumference and summing them.

I don't know if one treatment is more standard than another. Dieks uses radar distances between nearby points. Rindler defines what he calls the "first standard form" of a stationary metric (section 9.3), and then interprets it as being equivalent to radar distance and ruler distance (first page of section 9.4).

 

[bcrowell]

Thanks, pervect, for the explanation in #72. I think I follow you now.

A.T., I've been thinking some more about the three-spaceships idea, and I've come to the conclusion that the ground rules we've set for that are not likely to be fruitful. The general idea here has been that there are issues with global clock synchronization and issues with dynamics of rigid rulers, so we want to find a purely static method of measurement that doesn't require clocks or rigid rulers. This has the flavor of the classic geometrical constructions, like compass and straightedge constructions. There are other families of constructions like these in plane geometry. There are constructions that can be done with a compass and no straightedge. There are others that can't be done with compass and straightedge, but can be done with origami. In general, such a system of constructions corresponds to a certain geometry with constructive axioms. Compass and straightedge are the instruments that are explicitly referred to in Euclid's axioms. If you restrict yourself to a system where there is only a straightedge and a method for drawing parallels, then you have affine geometry.

What we've been doing is essentially trying to perform a certain construction in a system where there is angular measure but no distance measure. That doesn't really work. There is no interesting geometrical system that is like this. Euclidean geometry has both angular and distance measure. Affine geometry has distance measure but no angular measure (and distances along non-parallel lines aren't comparable). But there is no system that has angular measure without distance measure, and that's because if you have angular measure, then your system is at least as strong as affine geometry (which only defines parallelism), and affine geometry allows a distance scale to be constructed using a ladder construction.

I came up with a somewhat more explicit way of showing this in the context of the spaceship experiment you've been talking about. The objection I raised earlier is that you can't do much of a quantitative test of GR unless you can measure r, since GR's only testable prediction about the spatial geometry is
$$\lim\epsilon/A = -\frac{3\omega^2}{1-2\omega^2r^2+\omega^4r^4}$$
for the angular deficit of a triangle per unit area. You can sort of get around this by the following trick. Start with a single spaceship in some randomly chosen state of inertial motion, which means that with unit probability it's rotating. The ship has two identical rocket engines. Fire one engine with constant thrust, construct a triangle in a lab inside the ship, and measure $\epsilon$ using protractors. During this measurement, there exists some inertial frame such that the ship is moving in a circle, but you can't directly measure what the circle's radius is. Now fire both engines simultaneously, and measure $\epsilon$ again. The circle now has some other radius. Although you don't know r, GR does predict that $\lim\epsilon/A=f(n)$, where n is the number of engines firing, and f is some function, which has adjustable parameters in it. By doing measurements with enough values of n, you can determine the adjustable parameters. It now seems like you have a clearcut test of GR, because you can measure f, which is essentially a test of the radial dependence of the Gaussian curvature. The problem with all this is that you don't know A, and A isn't constant. (To see that A can't be constant, note that $\epsilon/A$ diverges as n approaches infinity, whereas $-\pi < \epsilon \le 0$). To measure A, you need either clocks+radar, or some other kind of ruler. I think this is a symptom of the fact that, for the reasons I outlined above, the particular set of measuring instruments we've been discussing is not a fruitful one to talk about. Having a protractor is actually equivalent to having a ruler (by the affine-geometry ladder construction), so you might as well just use radar rulers.

Another issue that occurred to me is that geodesics of the spatial metric are not the same as geodesics of the spacetime metric, so you can't use laser beams to define the sides of your triangle. For example, a laser beam going outward from the axis is actually a curve in the rotating frame, and it's not a geodesic of the spatial metric.

 

[yuiop]

Proper length is a coordinate independent property of the curve, so there's nothing to work out. It's 2$\pi$r.

I haven't seen any comments about an issue that I raised on page 1. We all agree that it's not possible to get a disc spinning without stretching the material, right? When it's streched, the sum of the internal forces on any atom should be towards the center, while the centrifugal force is in the opposite direction. Has anyone worked out which one of these forces "wins"? Does the radius of the disc get larger or smaller when we give it a spin?

I now see that your assertion that the proper length of the circumference is 2*pi*r is conditional upon the outcome of the question that follows it, which perhaps can be rephrased as "What will be the proper length of the circumference, if the disc is slowly and carefully slowed down so that it is no longer rotating?"?

My answer to that question, is that the proper circumference of the disc when brought to rest is numerically the same as the result obtained by 2*pi*r*gamma when the disc was rotating, or put another way the radius gets smaller when the disc is given a spin.

Of course, in normal circumstances, a disc is torn apart by "centrifugal force" long before its its peripheral velocity reaches a tiny fraction of the speed of light, as can be seen by the difficulty in designing high speed, energy storage flywheels. To be able to spin a disc up in a way that demonstrates the radius gets smaller as it spins faster, an idealised form of angular acceleration would have to be set up that is similar to the ideal of linear Born rigid acceleration. For example a number of stations could be placed on spokes radiating from a common hub. The stations are equally spaced and equidistant from the hub and there are enough stations to aproximate the circumference of a disc. The stations are free to slide along the spokes if required and each station has its own rocket thrusters to maintain position. This is similar to the ideal of a Born rigid rocket having a micro thruster attached to every single atom of a linearly accelerating rocket. Now as the disc is spun up, the controllers of the stations are instructed to operate their outward thrusters so that they maintain constant radar distance with their immediate neighbours. Under these conditions, the final radius of the disc will be less than the initial radius and proper circumference of the disc will remain constant as measured by local radar measurements (or by ideal co-moving rulers as specified by A.T.) for any angular velocity.

I hope by the above argument, you will concede that the coordinate independent proper circumference of the disc is 2*pi*r*gamma where r is the final radius of the rotating disc and gamma is a function of the final peripheral velocity of the disc and this distance is distance measured by local rulers that are at rest with the rotating circumference as described by A.T. (i.e. the location of the end points of the rulers do not change over time according to the reference frame of the accelerating rotating disc riders). This circumference is also the distance that would be obtained by the sum of chained local radar distance measurements. Put yet another way, the proper circumference is 2*pi*r/sqrt(1-v^2/c^2) for any angular velocity, where r is the instantaneous radius (measured in the non rotating frame) and v is the instaneous peripheral velocity at any given time and this length value remains constant at all times, under the form of Born rigid angular acceleration I have described.

 

[Fredrik]

I now see that your assertion that the proper length of the circumference is 2*pi*r is conditional upon the outcome of the question that follows it, which perhaps can be rephrased as "What will be the proper length of the circumference, if the disc is slowly and carefully slowed down so that it is no longer rotating?"?

It isn't. It's only based on the fact that we have specified that the distance from the center to the edge in the center-of-mass inertial frame is r. It doesn't matter what the radius was before we gave the disc a spin.

The gamma in the claim that the circumference is 2*pi*gamma*r "in the rotating frame" comes from the fact that if the circumference is measured by a large number of co-moving inertial observers, the result is 2*pi*gamma*r. (Each inertial observer measures the length of a short segment that he's approximately co-moving with, and then you add up the results. It's approximately 2*pi*gamma*r, because their measuring devices are Lorentz contracted by a factor of gamma, and the approximation becomes exact in the limit of infinitely many co-moving observers). Why anyone would call this a measurement of the circumference in the rotating frame is beyond me.

The stations are free to slide along the spokes if required and each station has its own rocket thrusters to maintain position. This is similar to the ideal of a Born rigid rocket having a micro thruster attached to every single atom of a linearly accelerating rocket. Now as the disc is spun up, the controllers of the stations are instructed to operate their outward thrusters so that they maintain constant radar distance with their immediate neighbours. Under these conditions, the final radius of the disc will be less than the initial radius

You're describing a scenario where we can exactly cancel the centrifugal force on each component part, but you have also removed the internal forces between the component parts. The way I see it, they are the reason why the disc would get smaller when you give it a spin (and magically compensate for the centrifugal force).

This is what I'm thinking: Suppose that the radius at rest is s. If we could add an inward force that exactly cancels the centrifugal force on each atom, we would have r<s, because the forceful stretching of the circumference that occurs when we increase the angular velocity will create internal forces in the inward direction. The shrinking of the disc will produce an outward force that grows as the disc get smaller (and the pressure increases), and at some point an equilibrium is reached. This must happen before the radius has shrunk to s/gamma, because that's what we would expect the radius to become if we neglect that compressing the disc will increase the pressure in the material. So the r is going to be somewhere between s and s/gamma (probably much closer to s than s/gamma).

and proper circumference of the disc will remain constant as measured by local radar measurements (or by ideal co-moving rulers as specified by A.T.) for any angular velocity.

Here you seem to be neglecting that the co-moving rulers get Lorentz contracted.

I hope by the above argument, you will concede that the coordinate independent proper circumference of the disc is 2*pi*r*gamma where r is the final radius of the rotating disc and gamma is a function of the final peripheral velocity of the disc and this distance is distance measured by local rulers that are at rest with the rotating circumference as described by A.T.

No, the rulers would measure 2*pi*r*gamma, because they're Lorentz contracted by a factor of gamma, so you need more of them to fill up the entire circumference. The coordinate independent proper length of the circumference is 2*pi*r, because that's what it is in the center-of-mass inertial frame. The curve that has coordinate-independent proper length 2*pi*gamma*r is a spiral in spacetime, not a circle in space. (It's not even a circle in the spiral-shaped hypersurface that Grøn calls "rest space", since the endpoint isn't the same as the starting point).

 

[Ich]

Why anyone would call this a measurement of the circumference in the rotating frame is beyond me.

Because you can actually and comfortably stuff 8 rulers in the circumference of a 1 ruler radius. For an indefinitely long time, with anyone on board having time to accurately measure length. No synchronization issues there.

If I somehow lost track of what this discussion is about, my apologies.

 

[A.T.]

No, the rulers would measure 2*pi*r*gamma, because they're Lorentz contracted by a factor of gamma, so you need more of them to fill up the entire circumference.

But in the rotating frame the rulers are not contracted, yet you still need more of them to fill up the entire circumference. So the circumference is more than 2*pi*r in the rotating frame.

The curve that has coordinate-independent proper length 2*pi*gamma*r is a spiral in spacetime, not a circle in space.

So a spiral in spacetime cannot be a circle when projected onto space? And why should I even care how the circular ruler is represented in spacetime, if the ruler is at rest in my rotating frame, and I read off what it measures.

 

[Fredrik]

But in the rotating frame the rulers are not contracted, yet you still need more of them to fill up the entire circumference.

How do you make sense of that statement? I assume the rulers are supposed to be "at rest" in the rotating frame. What can that possibly mean other than that they're at rest in a sequence of co-moving inertial frames? If that's what they are, then they're Lorentz contracted. If the rulers are instead being held in place by a circular rim, then they will get bent and squeezed in addition to being Lorenz contracted.

So a spiral in spacetime cannot be a circle when projected onto space?

Of course it can, but who's talking about projections? The projection onto space has circumference 2*pi*r, not 2*pi*gamma*r. Your guys defined "rest space" to be that spiral surface just to get a larger circumference, but the price we pay is a) that we're not even talking about a closed curve anymore (so why call it "circumference"?), and b) that we're not talking about a set of simultaneous events (so why call it "space"?)

And why should I even care how the circular ruler is represented in spacetime,

One reason why you should care about these things is that they're needed to justify the terminology. This isn't just "semantics". I still feel like these calculations that "prove the non-euclidean geometry of the rotating frame" are the equivalent of claiming that pigs can fly and then redefining "fly" until the statement is true.

A circular ruler is by the way, at any time in the center-of-mass inertial frame, represented by a circle, which has a coordinate independent proper length 2*pi*r.

if the ruler is at rest in my rotating frame, and I read off what it measures.

Is there any reason why you wouldn't describe this result as obtaining the wrong result because the rulers have been deformed from their rest shapes?

 

[yuiop]

It might help to return to the numerical example to clarify a few points:

Experiment:
Disk radius = 1 light second in the non rotating frame.
Instantaneous velocity of a point of the rim of the rotating disk is 0.8c clockwise, relative to an observer just outside the disk in the non rotating frame.
Gamma = 1/0.6 = 1.666666

Disk rotating with a rim velocity of 0.8c:

Circumference = 2*pi*r = 2*pi*1.00 = 6.28 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Observer on the disc.)

Disk not rotating.

Circumference = 2*pi*r = 2*pi*1.67 = 10.47 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.67*1.00 = 10.47 lightseconds. (Observer on the disc.)

Note that the observer on the disc always measures the circumference as 10.47 lightseconds. This is the proper length of the circumference whether measured by rulers or local radar measurements. When the disc is spinning the non rotating observer sees the disc circumference length contracted to 6.28 lightseconds. I am assuming the Born rigid rotation method I described in post #78.

Here you seem to be neglecting that the co-moving rulers get Lorentz contracted...

No, the assumption of length contraction was built in.

I understand your concern that the location of two ends of the tape measure wrapped around the disc do not seem to be measured simultaneously because of the spiral path that a point on the rim of the disc takes through spacetime. However, this an artifact of the Einstein synchronisation method, It is easy to establish that spatial distance around the rim of the disc does in fact have a dt of zero by placing a single clock at the location where the two ends of the tape measure overlap each other. Since the two ends are at the same location in space and time there should be no question that the two ends are in fact measured simultaneously. This makes more sense when the clocks on the rim are synchronised by a single common signal from the centre of the disc.

 

[atyy]

I still feel like these calculations that "prove the non-euclidean geometry of the rotating frame" are the equivalent of claiming that pigs can fly and then redefining "fly" until the statement is true.

No, it's like redefining "pig" until the statement is true

 

[Fredrik]

No, it's like redefining "pig" until the statement is true

I approve of this post.

Disk rotating with a rim velocity of 0.8c:

Circumference = 2*pi*r = 2*pi*1.00 = 6.28 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Observer on the disc.)

I'd rather say it like this:

Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Lots of observers on the disc who add their results)

Disk not rotating.

Circumference = 2*pi*r = 2*pi*1.67 = 10.47 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.67*1.00 = 10.47 lightseconds. (Observer on the disc.)

Did you really mean to have the radius shrink by a factor of gamma when the disc is given a spin (from being initially at rest)?

I am assuming the Born rigid rotation method I described in post #78.

Ah, but this has very little to do with an actual solid disc, and I don't think the term "Born rigid" is appropriate here (but I'd have to think that through to be sure).

I understand your concern that the location of two ends of the tape measure wrapped around the disc do not seem to be measured simultaneously because of the spiral path that a point on the rim of the disc takes through spacetime.

That isn't my concern at all.

It is easy to establish that spatial distance around the rim of the disc does in fact have a dt of zero

Sounds like you're considering a circle of radius r in the hypersurface of constant time coordinate. That's what I'm doing. That's the curve that has coordinate independend proper length 2*pi*r, which is not equal to the sum of the results of lots of distance measurements by co-moving rulers on the edge.

 

[yuiop]

Did you really mean to have the radius shrink by a factor of gamma when the disc is given a spin (from being initially at rest)?

Yes, that is what I was hinting at when I earlier said "put another way the radius gets smaller when the disc is given a spin".

I agree the Born rigid rotation method is artificial, and the shrinking of the radius will not happen spontaneously or naturally. It is a bit like accelerating a rocket from rest to a new relative velocity. The clocks onboard the rocket will not stay synchronised naturally and we have to articificially resync the clocks again. If we desire that the speed of light should remain constant and isotropic to observers onboard the rocket we have to fiddle with the clocks. If we desire the proper circumference to remain constant on the disc we have to fiddle with the radius. Nature does not care about our desires.

Ah, but this has very little to do with an actual solid disc, and I don't think the term "Born rigid" is appropriate here (but I'd have to think that through to be sure).

Even better, consider a solid rotating cylinder. Spinning a solid cylider up while trying to maintain a constant radius will produce enormous stress in the surface of the cylinder even if centrifugal/centripetal forces did not exist and the cylinder will inevitably tear itself apart as its circumference tries to length contract. The nearest example in nature is a spinning neutron star which is held together at very high rotational velocities by gravity, but the gravity makes that example difficult to analyse and distill out the purely kinematic aspects.

If you insist on the radius remaining constant when the solid cylinder is spun up, then if the circumference was 10.47 lightseconds when the cylinder was not rotating, then the circumference would be measured as 2*pi*r*gamma = 2*pi*1.67*1.67 = 17.52 light seconds by observers on the surface of the cylinder, when the cylinder is rotating with a rim velocity of 0.8c.

That isn't my concern at all.

Well I am now not sure what your concerns are. Perhaps the best way forward would be for you to describe what practical method, the observers onboard the disc, would use to measure the circumference as being simply 2*pi*r.

I think most people on this forum would agree that science is more about what you would measure than about what is "really" happening. I have described several measurement methods, but none of them come up with a circumference of 2*pi*r for the proper circumference of a rotating disc.

I'd rather say it like this:
"Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Lots of observers on the disc who add their results)"

Or (One observer on the disc with one long tape measure.)


Pigs fed and watered and limbering up ready for takeoff.

 

[A.T.]

I assume the rulers are supposed to be "at rest" in the rotating frame.

That is the point.

What can that possibly mean other than that they're at rest in a sequence of co-moving inertial frames?

Still true, but irrelevant, because I want to consider just one frame: the rotating one

If that's what they are, then they're Lorentz contracted.

In which frame are they Lorentz contracted? Obviously not in the rotating frame, in which they are at rest. Objects at rest are not Lorentz contracted.

Is there any reason why you wouldn't describe this result as obtaining the wrong result because the rulers have been deformed from their rest shapes?

The rulers are at rest and so they preserve their rest lengths.

One reason why you should care about these things is that they're needed to justify the terminology.

The justification for calculating this non-Euclidean geometry in a rotating frame and calling it "spatial" is the same justification as for calculating the Centrifugal/Coriolis-forces and calling them "forces", in classical mechanics:

I want to do calculations in the rotating frame, but use the physics laws that were designed for inertial frames.

 

[Fredrik]

OK. After thinking some more, I agree that it makes sense to say that the circumference is measured to be 2*pi*gamma*r in the rotating frame. I'll try to explain why. A classical theory is a set of statements that makes predictions about results of experiments. A mathematical structure (like Minkowski spacetime) can never define a theory by itself. The theory is defined by a set of axioms that tells us how to interpret the mathematics as predictions about results of experiments. This means that even if we have an operational procedure (like using a tape measure) that associates a number with dimensions of length with the circumference of a disc, it isn't possible to relate this to a mathematical quantity in the theory unless there's an axiom that describes how to do that. The theory may not consider what we have done to be a measurement. Because of that, it doesn't make much sense to discuss these things without properly defining the theory first. The theory I have in mind when I use the term "special relativity" is defined by the following three axioms:

1. Physical events are represented by points in Minkowski spacetime. (A consequence of this is that motion is represented by curves, and this suggests the definition of a "particle" as a system the motion of which can be represented by exactly one curve).
2. A clock measures the proper time of the curve in Minkowski spacetime that represents its motion.
3. A radar device measures infinitesimal lengths in the following way: If the roundtrip time is T, then cT/2 is the approximate proper length of the spacelike geodesic from the midpoint of the timelike geodesic through the emission event and the detection event to the reflection event. The approximation becomes exact in the limit T→0. (I haven't found a way to say this that isn't really awkward).

Actually these just define a framework in which we can define classical special relativistic theories of matter and interaction (in several different ways). I won't go into details about those things here. Note that I could have chosen to define 3 in a different way:

3'. A radar device moving as represented by a timelike geodesic measures lengths in the following way: If the roundtrip time is T, then cT/2 is the proper length of the spacelike geodesic from the midpoint of the worldline between the emission event and the detection event to the reflection event.

With 3', we have a theory that's at least as worthy of the name "special relativity" as anything Einstein could have written down in 1905, but it doesn't make any prediction at all about the circumference of the disc in the rotating frame. This theory simply doesn't tell us how to make measurements with non-inertial measuring devices. This is of course exactly why we should prefer 3 over 3'. If we put lots of tiny radar devices along the edge of the rotating disc, have them measure the distance to the next device, and then add up the results, the total will clearly be 2*pi*gamma*r (in the limit of infinitely many radar devices).

My first thought was that it doesn't make sense to call the result obtained this way "the measured circumference in the rotating frame". I thought that it made no sense to describe the sum of many measurements made by measuring devices in different states of motion as the result of a single measurement in a frame where all devices have constant spatial coordinates. But then I realized that this is exactly what we do when we claim to have used axiom 3 to measure something (non-infinitesimal) in an inertial frame. All the measuring devices have the same velocity, but not the same world lines, so we're definitely adding up results from measuring devices in different states of motion. If we allow ourselves to say that we have measured a non-infinitesimal length in an inertial frame (using axiom 3 rather than 3'), then we have no reason not to allow ourselves to say that we have measured non-infinitesimal lengths in the rotating frame.

What I did before is the equivalent of using 3' for measurements in inertial frames, and 3 only for measurements in non-inertial frames. But if we include 3 in the definition of the theory, we don't need 3'.

I still disagree that it makes any sense to interpret this as a non-euclidean spatial geometry, because the term "spatial geometry" can only refer to the geometry of a hypersurface of points that are all assigned the same time coordinate. Such a hypersurface is flat, and the circumference of the disc in that hypersurface is 2*pi*r. This is a coordinate independent proper length of a closed curve.

 

[Fredrik]

In which frame are they Lorentz contracted?

In the center of mass inertial frame. When you describe these things from that frame, the reason why the result is 2*pi*gamma*r is that the rulers are Lorentz contracted, and therefore give you a result that isn't the proper length of the closed curve that defines the circumference of the disc in "space".

The rulers are at rest and so they preserve their rest lengths.

You keep talking about the rulers being at rest in the rotating frame, but you also keep ignoring that they would be crushed by the centrifugal force. (The whole disc would of course be messed up, but the deformations of the measuring devices are more important since they're what we use to measure things). You also haven't said anything that indicates that any of your ideas are based on an actual definition of the theory. To avoid the centrifugal forces, we're going to have to make the rulers inertial, and it's very far from clear that measurements made using inertial rulers can be called a measurement in the rotating frame. This is something that would have to be derived from the axioms of the theory (and first you would have to specify what the axioms are).

The justification for calculating this non-Euclidean geometry in a rotating frame and calling it "spatial" is the same justification as for calculating the Centrifugal/Coriolis-forces and calling them "forces", in classical mechanics:

I don't follow you at all here, but I haven't ruled out that it's because I'm tired and need to get some sleep.

 

[A.T.]

To avoid the centrifugal forces, we're going to have to make the rulers inertial,

Fine, for practical purposes let's make them inertial and let them meet at the circumference for a moment. In the rotating frame they will be at rest in that moment, so they will measure the correct circumference in the in the rotating frame.

The justification for calculating this non-Euclidean geometry in a rotating frame and calling it "spatial" is the same justification as for calculating the Centrifugal/Coriolis-forces and calling them "forces", in classical mechanics:
I want to do calculations in the rotating frame, but use the physics laws that were designed for inertial frames.

I don't follow you at all here, but I haven't ruled out that it's because I'm tired and need to get some sleep.

It is very simple:

Let's take Newtons 1st & 2nd law. They are very useful and simple, but hold only in inertial frames and fail in rotating frames. So you have two choices:

1) Never use rotating frames for calculations

2) Assume Centrifugal/Coriolis-forces in the rotating frame and still use Newtons 1st & 2nd law.

That trick works fine as long as omega * r << c, otherwise you need to assume more things to keep your simple "inertial frame physics" working. One of these things is a non-Euclidean spatial geometry.

The question if the spatial geometry is really non-Euclidean here is analogous to the question if those inertial forces are really forces. Some say "No, it is just a math-trick", others say "If it walks like a duck and quacks like a duck, let's call it 'duck'". I personally don't really care.

 

[bcrowell]

I still disagree that it makes any sense to interpret this as a non-euclidean spatial geometry, because the term "spatial geometry" can only refer to the geometry of a hypersurface of points that are all assigned the same time coordinate.

This is what I originally thought about this example, but actually it's incorrect. Rindler has a nice description of this in ch. 9 of Relativity: Special, General, and Cosmological. You can define two properties: stationary and static. Stationary means there's a timelike Killing vector. Static means that in addition, there's no rotation (i.e., no Sagnac effect). In a stationary case, you have a preferred time coordinate, which is defined by placing a master clock at some point in space, sending out a carrier wave from that clock, and adjusting all other clocks at other points so as to eliminate Doppler shifts, i.e., calibrating them so that they agree with the master clock on the frequency of the sine wave. In the static case, you can also globally match the phase of the master clock, and you have Einstein synchronization, and this Einstein synchronization is independent of your choice of where to place the master clock. Our example is stationary but not static. Because of the symmetry of the problem, you probably want to put the master clock at the center. Then the moving clocks will all be running at different rates. Clocks at the same theta are Einstein-synchronized, but clocks at different thetas aren't.

If you do all this, a surface of simultaneity is simply a light-cone centered on an event at the axis. A cone has zero intrinsic curvature.

However, the spatial geometry determined by obervers using co-moving radar rulers is not the same as the (flat) spatial geometry obtained by restricting to that surface of simultaneity. The reason is that if you orient a radar-ruler in the azimuthal direction, you are measuring the spatial separation between events that are Einstein-synchronized, whereas the surface of simultaneity isn't Einstein-synchronized azimuthally.

Such a hypersurface is flat, and the circumference of the disc in that hypersurface is 2*pi*r. This is a coordinate independent proper length of a closed curve.

This is correct, but restriction to the hypersurface doesn't correspond to the geometry measured by co-moving observes with radar rulers.

 

[yuiop]

It occurs to me that we can carry out an adaptation of A.T's angle measurement on a more local scale on the disc. This is the set up:

Two points A and B are locations on the rim of the disc and the distance from A to B is a small fraction of the total circumference of the disc. Two theodolites are placed at A and B and lined up to view each other and measure the angles. The actual angles are not important here. Once the theodolites are lined up accurately, they are welded so that they can no longer be adjusted. The theodolites are then swapped with each other and it will be noticed that they no longer line up with each other. By such a method the inhabitants of the inside of a cylinder will be able to establish that light paths are not isotropic in their world and if they are clever enough, maybe even figure out they are rotating, even if they do not have any view outside their cylindrical world and even if they do not have access to the entire inner circumference. The main cause of the difference in the angular measurements is abberation. They could also shine laser beems at each other. It would be noticed that they can block one beem at a time conclusively proving the two light paths do not follow the same path.

 

[bcrowell]

By such a method the inhabitants of the inside of a cylinder will be able to establish that light paths are not isotropic in their world and if they are clever enough, maybe even figure out they are rotating, even if they do not have any view outside their cylindrical world and even if they do not have access to the entire inner circumference.

This is basically a measurement of the Sagnac effect, which is certainly a method that works for detecting by local measurements whether you're in a rotating frame.

The restriction to angle measurements, rather than distance measurements, is not a useful one, as I pointed out in #77: Any method of angular measurement can also be used to determine distances. Macroscopic, solid rulers will not work (because Born rigidity is kinematically impossible), but radar rulers will.

 

[Fredrik]

I read a couple of pages of Rindler's book (the stuff surrounding (9.26) (which is the metric I'm including below)). He talks about a rotating lattice, rather than a rotating disc. The substitution $\phi\rightarrow\phi-\omega t$ puts the metric in the form

$$ds^2=(1-r^2\omega^2)\bigg(dt-\frac{r^2\omega}{1-r^2\omega^2}d\phi\bigg)^2-dr^2-\frac{r^2}{1-r^2\omega^2}d\phi^2-dz^2$$

This is still just the metric of Minkowski spacetime expressed in a funny way, but then he says that "the metric of the lattice is the negative of the last three terms". This is an interesting comment. The only way I can make sense of it is this: Instead of considering Minkowski spacetime, he decides to consider another manifold. Its underlying topological space is $\mathbb R\times[0,\infty)\times[0,2\pi)\times\mathbb R$ with the topology induced by the topology on $\mathbb R^4$. Note that the simplest definition of Minkowski spacetime uses $\mathbb R^4$ as the underlying topological space, so we can say that we're dealing with a proper subset of Minkowski spacetime. The coordinate systems are defined to be the coordinate systems of Minkowski spacetime restricted to that proper subset. The metric is defined to have components

$$g_{00}=1-r^2\omega^2,\ g_{11}=1,\ g_{22}=\frac{r^2}{1-r^2\omega^2},\ g_{33}=1$$

$$g_{\mu\nu}=0 \mbox{ when }\mu\neq\nu$$

in the coordinate system defined by the inclusion map for that subspace. ($I:M'\rightarrow\mathbb R^4,\ I(x)=x,\ \forall x\in M'$)

I don't have a problem with saying that "space" in this spacetime has a non-euclidean geometry. It just seems so incredibly pointless to introduce a different spacetime when the one we had was just fine. Think of the definition of SR in my post #88. SR is a theory about one specific spacetime, Minkowski spacetime. What Rindler is really saying here (without actually saying it) is that there's another theory that can reproduce the predictions of SR when we're dealing with uniform rotation.

I don't think this is very interesting. Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR, but it isn't even SR.

 

[DrGreg]

I read a couple of pages of Rindler's book (the stuff surrounding (9.26) (which is the metric I'm including below)). He talks about a rotating lattice, rather than a rotating disc. The substitution $\phi\rightarrow\phi-\omega t$ puts the metric in the form

$$ds^2=(1-r^2\omega^2)\bigg(dt-\frac{r^2\omega}{1-r^2\omega^2}d\phi\bigg)^2-dr^2-\frac{r^2}{1-r^2\omega^2}d\phi^2-dz^2$$

This is still just the metric of Minkowski space expressed in a funny way, but then he says that "the metric of the lattice is the negative of the last three terms".

I'm struggling to make sense of this myself, but you need to read what Rindler says before this, at the start of section 9.6. From what I can gather, the first term in the above metric takes account of the moving lattice's simultaneity (which isn't dt=0). Or, to put it another way, the radar distance measurement of infinitesimally close lattice points implies a pair of events for which the first term vanishes, I think (?).

 

[A.T.]

SR is a theory about one specific spacetime, Minkowski spacetime.

I rather see it the other way around: Minkowski spacetime is one possible geometrical interpretation of SR.

What Rindler is really saying here (without actually saying it) is that there's another theory that can reproduce the predictions of SR, when we're dealing with uniform rotation

Another theory would imply different predictions. But so, it is at most a different geometrical interpretation.

Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR, but it isn't even SR.

Most sources rather indicate that historically this problem motivated the development of GR. So it might be correct that this is not SR anymore.

 

[atyy]

I don't think this is very interesting. Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR, but it isn't even SR.

To mix metaphors, aren't there many ways to slice a pig, or parts of a pig? It's just some noninertial coordinates covering a piece of Minkowski spacetime.

Edit: Actually, I'm a bit puzzled on reading Wiki's http://en.wikipedia.org/wiki/Born_coordinates which says that there Langevin observers are not hypersurface orthogonal, so what is the hypersurface that Rindler has defined?

Edit: I took a look at Poisson's http://www.physics.uoguelph.ca/poisson/research/agr.pdf . It's not exactly relevant, since it deals with timelike geodesics (section 2.3), and my guess is the Langevin observers are not geodesic. However, he says that although there are congruences that are not hypersurface orthogonal (section 2.3.3), these congruences still have a "transverse metric" which is purely "spatial" in the limited sense that it is "locally spatial" (section 2.3.1, that last term is mine, see his notes for the real maths).

 

[bcrowell]

I don't think this is very interesting.

Nobody can force you to be interested. Einstein, historically, thought it was interesting -- in fact, he considered it a crucial example that helped him to develop GR. Ehrenfest thought it was interesting. Gron and Rindler thought it was interesting enough to put it in their textbooks. Dieks thought it was interesting enough to write a paper about. Rizzi and Ruggiero thought it was interesting enough to edit an entire anthology on the topic. I think it's interesting, and so do A.T., kev, and others who have kept this thread going for nearly 100 posts. But if you don't think it's interesting, that's up to you. Diff'rent strokes.

Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR,

I don't think anybody said it was "the" correct way to treat rotation in SR. I'm sure there are many different ways of looking at it. As far as correctness, please feel free to point out anything you think is correct in the treatments by Rindler, Dieks, and Gron -- but the last time I asked you this, you said it was correct but not interesting.

but it isn't even SR.

Historically, Einstein considered it interesting as a bridge from SR to GR. Whether a treatment like Rindler's or Dieks' is SR or GR depends on your point of view, and on how you define SR. From the perspective of the year 2010, my opinion is that the most reasonable way to define SR is that it deals with 3+1 spacetime that's flat. By that definition, this example is SR, because the underlying 3+1 spacetime is flat. Historically, there was some doubt about whether accelerating observers could be incorporated into SR, but I think the clear answer from a modern point of view is that they can.

The coordinate systems are defined to be the coordinate systems of Minkowski space restricted to that proper subset.

No, this is incorrect. See #91.

I'm struggling to make sense of this myself, but you need to read what Rindler says before this, at the start of section 9.6. From what I can gather, the first term in the above metric takes account of the moving lattice's simultaneity (which isn't dt=0). Or, to put it another way, the radar distance measurement of infinitesimally close lattice points implies a pair of events for which the first term vanishes, I think (?).

Yeah, this is pretty much right. The point about the moving lattice's simultaneity is subtle, since there's no way of doing an Einstein synchronization over any finite area. However, all you need for a radar-ruler measurement of distance is Einstein synchronization on a line segment, and since that doesn't enclose any area, you're OK. I personally think Dieks' treatment is a lot more transparent than Rindler's, and I would suggest starting there. Rindler's is more general and abstract. The thing that confused me at first was that I thought there was a surface of simultaneity, which would appear as an equation of constraint that would eliminate one variable from the metric. That's incorrect, for the reasons outlined in #91. If you read Dieks' treatment, I think it's more clear what's going on. There's no variable whose differential is dt in the rotating frame. You just substitute in for dt, which mathematically represents doing a local Einstein synchronization between the two ends of a radar-ruler of length dl. So this is the only thing that I think might not be quite right in your quote above. In the last sentence, it's not that you're making the time term vanish, it's that you're eliminating the variable dt.

 

[Fredrik]

I rather see it the other way around: Minkowski spacetime is one possible geometrical interpretation of SR.

If we use my definitions of the terms "theory" and "special relativity", those statements don't make much sense. So your definitions must be different than mine.

Another theory would imply different predictions.

Since I define a theory as a set of statements that makes predictions about results of experiments*, I have to consider two different sets of statements that lead to the same predictions different but equivalent theories.

*) That's just a part of it actually. I'll post the full definition if someone requests it.

From the perspective of the year 2010, my opinion is that the most reasonable way to define SR is that it deals with 3+1 spacetime that's flat.

Yes, I agree. (See #88 for a more complete definition).

By that definition, this example is SR, because the underlying 3+1 spacetime is flat.

As I said in my previous post, I can't interpret Rindler's statement that way. Gron appears to be considering a hypersurface in Minkowski space that consists of a bunch of spirals, but Rindler appears to be considering a different spacetime altogether. I think the "space" part of the spacetime Rindler is describing must be isomorphic to the submanifold of Minkowski space that Gron is considering. So it appears that Gron is doing SR, but ends up using a very weird terminology, like using the term "circumference" about the proper length of a spiral and the term "space" about a hypersurface of events that aren't assigned the same time coordinate by the coordinate system we're using. Rindler is not doing SR, but avoids the terminology issues.

Historically, there was some doubt about whether accelerating observers could be incorporated into SR, but I think the clear answer from a modern point of view is that they can.

The real issue is whether we can describe a way to perform measurements of proper length using non-inertial measuring devices. Axiom 3 in #88 takes care of that, so we didn't really need to study the rotating disc problem to understand accelerating "observers".

No, this is incorrect. See #91.

I don't think you understood what I meant. In order to define a manifold, you have to specify a set, a topology on that set, and all the coordinate systems that we're allowed to use. That's what I was doing. #91 is about something else entirely.

OK. After thinking some more, I agree that it makes sense to say that the circumference is measured to be 2*pi*gamma*r in the rotating frame.

I think I may have to retract this comment, or at least elaborate a bit. In order to get this to be true, we either have to use the term "circumference" about the proper length of a spiral (or an even uglier curve), or define the "rotating frame" as the spatial part of an entirely different spacetime. (See my previous two posts). It "makes sense" to redefine "fly" or "pig" to make the statement "pigs can fly" true, but that doesn't mean that we should. SR allows us to do these things, but doesn't give us a reason why we should.

 

[bcrowell]

The coordinate systems are defined to be the coordinate systems of Minkowski space restricted to that proper subset.

I don't think you understood what I meant. In order to define a manifold, you have to specify a set, a topology on that set, and all the coordinate systems that we're allowed to use. That's what I was doing. #91 is about something else entirely.

I think I did understand what you meant, and what you meant was incorrect. In the first quote, you're defining a space consisting of a subset of Minkowski space. That isn't what you want to do in this example, because there is no global time synchronization, and therefore no natural way to pick such a subset, and it doesn't define the metric you want by the usual process of restricting a space to a lower-dimensional subspace. To define the spatial manifold successfully, you just want to define the set as the set of coordinates $(r,\theta)$, the same way you would do if you were just going to talk about ordinary plane polar coordinates. The reason I can tell that you're not understanding this correctly is that you keep on talking about spirals. The only reason to talk about spirals is if you're imagining this as a process where you induce a new metric by restricting the parent space to a surface of simultaneity. All of your previous objections (that the subspace is flat) make perfect sense with this approach. That's why it's not a fruitful approach.

 

[yuiop]

I have found another worm to add to the can. To a non rotating observer the radius of the disc is r or 1 lightsecond in the numerical example. To an observer on the rotating disc, the radar measured radius (R) is r/gamma = 0.6 lightseconds. This makes the situation even less Euclidean to the disc observer, because by his measurements of circumference and radius using radar measurements, the circumference is 2*pi*R*gamma^2.

Fredrik, you have still failed to provide a practical method by which an observer on the disc will measure the proper circumference as being simply 2*pi*r and when you do find a method, I predict it will not have the desirable properties of transitive time synchronisation, isotropic speed of light, same distance measured in either direction, a local speed of light of c, etc, etc.

 

[atyy]

Edit: I took a look at Poisson's http://www.physics.uoguelph.ca/poisson/research/agr.pdf . It's not exactly relevant, since it deals with timelike geodesics (section 2.3), and my guess is the Langevin observers are not geodesic. However, he says that although there are congruences that are not hypersurface orthogonal (section 2.3.3), these congruences still have a "transverse metric" which is purely "spatial" in the limited sense that it is "locally spatial" (section 2.3.1, that last term is mine, see his notes for the real maths).

Yeah, this is pretty much right. The point about the moving lattice's simultaneity is subtle, since there's no way of doing an Einstein synchronization over any finite area. However, all you need for a radar-ruler measurement of distance is Einstein synchronization on a line segment, and since that doesn't enclose any area, you're OK. I personally think Dieks' treatment is a lot more transparent than Rindler's, and I would suggest starting there. Rindler's is more general and abstract. The thing that confused me at first was that I thought there was a surface of simultaneity, which would appear as an equation of constraint that would eliminate one variable from the metric. That's incorrect, for the reasons outlined in #91. If you read Dieks' treatment, I think it's more clear what's going on. There's no variable whose differential is dt in the rotating frame. You just substitute in for dt, which mathematically represents doing a local Einstein synchronization between the two ends of a radar-ruler of length dl. So this is the only thing that I think might not be quite right in your quote above. In the last sentence, it's not that you're making the time term vanish, it's that you're eliminating the variable dt.

I just read Gron's treatment, which is identical to Rindler's as far as I can tell, except he gives a picture of the surface, and also states in the preceding chapter "... there does not exist a single space of simultaneity encompassing the "rest spaces" of all observers in an arbitrary reference frame. In this sense the 3-dimensional space described by the spatial metrical tensor is local." So I suppose this is all very opaque from the SR point of view, but apparently is a useful precursor to thinking about things in GR, just like Fredrik's favourite index notation for tensors in SR

 

[Fredrik]

I think I did understand what you meant, and what you meant was incorrect. In the first quote, you're defining a space consisting of a subset of Minkowski space. That isn't what you want to do in this example, because there is no global time synchronization, and therefore no natural way to pick such a subset, and it doesn't define the metric you want by the usual process of restricting a space to a lower-dimensional subspace.

I was really confused by the first comments, but the last one shows that you have definitely misunderstood what I'm doing. I'll try to explain it in a different way. Minkowski spacetime can be defined in the following way:

* Choose the set $\mathbb R^4$.
* Choose the standard topology.
* Choose the standard coordinate systems (i.e. all $C^\infty$ injective functions from $\mathbb R^4$ into $\mathbb R^4$)
* Choose the metric to have components $\eta_{\mu\nu}$ in the coordinate system I, where I is the identity map on $\mathbb R^4$.

What I'm doing is to define a different spacetime manifold M':

* Choose the set $\mathbb R\times [0,\infty)\times [0,2\pi)\times\mathbb R$, i.e. the range of the specific coordinate system on Minkowski spacetime that we've been considering.
* Choose the topology inherited from $\mathbb R^4$.
* Choose the coordinate systems to be the same as the ones used in the construction of Minkowski spacetime, but restricted to this smaller subset of $\mathbb R^4$.
* Do not choose the metric that would be inherited from Minowski spacetime if we thought of this as a submanifold (it would have the same components as the Minkowski metric), but instead choose the metric that has components

$$g_{00}=1-r^2\omega^2,\ g_{11}=1,\ g_{22}=\frac{r^2}{1-r^2\omega^2},\ g_{33}=1$$

$$g_{\mu\nu}=0 \mbox{ when }\mu\neq\nu$$

in the coordinate system I, where I is now the restriction of the identity map of $\mathbb R^4$ to M'.

Edit: This may not be enough. Maybe we should use $[0,2\pi]$ instead of $[0,2\pi)$, and then identify the line $\phi=0$ with the line $\phi=2\pi$.

The reason I can tell that you're not understanding this correctly is that you keep on talking about spirals.

Not sure if that means that I've misunderstood something, or if it means that you have.

The only reason to talk about spirals is if you're imagining this as a process where you induce a new metric by restricting the parent space to a surface of simultaneity. All of your previous objections (that the subspace is flat) make perfect sense with this approach. That's why it's not a fruitful approach.

Then we agree about that. I thought this was the approach taken by Grøn & Hervik in their book. Are you saying it isn't?

I thought I made it clear that this is how I interpreted what they were doing, and no one has objected against that until now. A.T. even defended calling the spiraling hypersurface "space".

As I've been saying, Rindler appears to be doing something different. To be more precise, he appears to be doing what I described for the second time earlier in this post.

 

[atyy]

Concerning the curved "spatial" hypersurface, are these statements true?
(i) it is everywhere locally orthogonal to the rotating observers
(ii) it is not a spacelike hypersurface, which is what disqualifies the rotating observers from being "hypersurface orthogonal" in the technical sense

 

[Fredrik]

If we're talking about the "spiral" hypersurface of Minkowski spacetime, then (i) is true and (ii) is not. (That's actually implied by (i)). I think it's also true for the alternative approach that I've been describing (as my interpretation of what Rindler is doing), but I'm not 100% sure about that one.