# Relativistic Merry-go-round

*Most recent answer: 10/22/2007*

Q:

I’m not sure if you received this question.
A ring has cirumfurance = diameter * 3.14.
Now, if this ring, layed flat atop a smooth surface, rotates with linear speed close to the speed of light, its cirumfurance shrinks to close to zero. However, since there is no motion in the diametric direction, its diameter remains unchanged.
How could a ring, on a flat surface, have cirumfurance less than diameter * 3.14 ?
To better visualize this situation, we can start by drawing a circle on the surface right below the ring before the rotation starts. Now the question becomes: How could the rotating ring have the same diameter as the circle but less circumfurance?

- Mehran (age 53)

Lisle, Illinois

- Mehran (age 53)

Lisle, Illinois

A:

Mehran- This is a familiar example, used by Einstein to help introduce
General Relativity. Let’s look at this rotating disk from two points of
view- the point of view of some standing on the ground and that of
someone on the disk. We’ll assume that the geometry from the ground
point of view has all the normal geometrical properties that we’re used
to. Now if by ’circumference’ we mean the length that the ground
observer traces out in the dirt directly below the rim of the spinning
disk, it is obviously pi* diameter, where diameter is the distance
across that circle. You may wonder how that can be since the moving
parts of the disk are Lorentz contracted, but there are all sorts of
other stresses etc in a spinning disk, so it stretches some, and we
simply know that Euclid’s geometry works well to describe any figures
under ordinary circumstances in our standard frames. Remember, we don’t
have to worry anout how fast the disk is spinning because we’re making
measurements on the part of the ground that almost touches the disk.

Now how do things look for someone making measurements on the disk? As you say, if he measures the diameter by laying out meter sticks in a standard way, he’ll get the same length as we get, because those meter sticks are not moving lengthwise with respect to us and hence are not Lorentz contracted lengthwise. However, the meter sticks used to measure the circumference, along the rim, ARE Lorentz contracted, so it takes MORE than pi*diameter of them to cover the rim. So unless you were to for some reason pick a different, longer, path to measure the dimeter, you end up with the circumference/diameter ration being MORE than pi in the frame of the spinning disk. Of course, you might choose some other path, like that followed by a light ray, and get a longer diameter.

So the answer is that if you pick odd frames like that of the disk, with different parts accelerating different ways, you can’t use the standard Euclidean geometry of space-time. The weird thing is that if gravity is present, it mimics non-uniform acceleration. So, strictly speaking, the simple frames we started with don’t exist, but the world can come close to behaving that way. For the Earth, the circumference is short of what we would expect for Euclidean geometry by only about an inch.

Mike W.

The observer on the rim of the disk can get a different answer depending on how the circumference of the disk is measured. If the whirling observer on a disk whose rim is traveling close to the speed of light observes how fast the dirt is going by underneath, using his own meter stick and clock, he will get the answer that he is going at close to the speed of light. If he measures the time on his clock it takes to make one revolution (by looking at a mark in the dirt, say), and he multiplies that by his speed, he will get an answer that is less than pi*diameter. This is a weird quantity because it is not measured in a single frame of refernce. The whirling observer accelerates constantly, and this is the sum of lots of little pieces measured in a succession of uniformly moving frames of reference.

This situation has practical consequences. Storing a large number of bunches of charged particles in a circular storage ring and then accelerating them to high energies involves this effect. Typically, the rings of magnets in a modern synchrotron are fixed in radius and the radio-frequency cavities are fixed in frequency and their spacing. The charged particles travel at nearly the speed of light all the time, so their travel times do not change much as the energy is raised from immense to really immense. Nor does the spacing of the bunches around the ring. What changes though, is that in one moving bunch’s frame, the neighboring bunches get farther apart as the energy is increased. This has an effect on the electrostatic force one bunch exerts on another as the energy increases (they go down. Real accelerators have more troubles with residual electromagnetic fields oscillating in the metal beampipe). In the frame of one of the bunches, the distance to the next has increased, but the same number of bunches stay equally spaced around the ring, so the whirling observer thinks the circumference has increased. But, paradoxically, it takes less of his time to go around that circle at approximately the same speed. (this is observed when putting particles with known lifetimes, such as muons, into these storage rings -- they make more turns around the ring on average before decaying).

Tom

Now how do things look for someone making measurements on the disk? As you say, if he measures the diameter by laying out meter sticks in a standard way, he’ll get the same length as we get, because those meter sticks are not moving lengthwise with respect to us and hence are not Lorentz contracted lengthwise. However, the meter sticks used to measure the circumference, along the rim, ARE Lorentz contracted, so it takes MORE than pi*diameter of them to cover the rim. So unless you were to for some reason pick a different, longer, path to measure the dimeter, you end up with the circumference/diameter ration being MORE than pi in the frame of the spinning disk. Of course, you might choose some other path, like that followed by a light ray, and get a longer diameter.

So the answer is that if you pick odd frames like that of the disk, with different parts accelerating different ways, you can’t use the standard Euclidean geometry of space-time. The weird thing is that if gravity is present, it mimics non-uniform acceleration. So, strictly speaking, the simple frames we started with don’t exist, but the world can come close to behaving that way. For the Earth, the circumference is short of what we would expect for Euclidean geometry by only about an inch.

Mike W.

The observer on the rim of the disk can get a different answer depending on how the circumference of the disk is measured. If the whirling observer on a disk whose rim is traveling close to the speed of light observes how fast the dirt is going by underneath, using his own meter stick and clock, he will get the answer that he is going at close to the speed of light. If he measures the time on his clock it takes to make one revolution (by looking at a mark in the dirt, say), and he multiplies that by his speed, he will get an answer that is less than pi*diameter. This is a weird quantity because it is not measured in a single frame of refernce. The whirling observer accelerates constantly, and this is the sum of lots of little pieces measured in a succession of uniformly moving frames of reference.

This situation has practical consequences. Storing a large number of bunches of charged particles in a circular storage ring and then accelerating them to high energies involves this effect. Typically, the rings of magnets in a modern synchrotron are fixed in radius and the radio-frequency cavities are fixed in frequency and their spacing. The charged particles travel at nearly the speed of light all the time, so their travel times do not change much as the energy is raised from immense to really immense. Nor does the spacing of the bunches around the ring. What changes though, is that in one moving bunch’s frame, the neighboring bunches get farther apart as the energy is increased. This has an effect on the electrostatic force one bunch exerts on another as the energy increases (they go down. Real accelerators have more troubles with residual electromagnetic fields oscillating in the metal beampipe). In the frame of one of the bunches, the distance to the next has increased, but the same number of bunches stay equally spaced around the ring, so the whirling observer thinks the circumference has increased. But, paradoxically, it takes less of his time to go around that circle at approximately the same speed. (this is observed when putting particles with known lifetimes, such as muons, into these storage rings -- they make more turns around the ring on average before decaying).

Tom

*(published on 10/22/2007)*