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
(published on 10/22/2007)
