ah, ok!

But then, when both time AND space contract, then this would also apply to when OB measures B.

So then it would mean that both light beams, A and B, have the same speed in an *absolute* sense (like Lorentz said), as in the Lorentzian Relativity.

Then it works out: both OA and OB measure c both times, and relative to each other!

Chantal

What's the problem in scenario A? The rods, if alonged with the radial spokes, will not contract. If they are tangential to the radial lines, they will.

What's the problem?

the rods are parts of the outer part of the wheel -> tangential.

Chantal

In that case, they do contract, since they are always moving along their length. I've seen a number of pictures in books showing someone on one of those spinning fairground rides with a ruler measuring the radial direction and the tangential direction. It's a common example to show contraction happens in the direction of motion, just in all directions.

I believe the problem he is describing is something like this.

Assume a wheel designed like a wagon wheel, with spokes. Also assume measuring sticks that are along the outside of the wheel, oriented so that the center of the length of each measuring stick is touching the wheel, and the measuring stick is tangential to the wheel, oriented in the direction of its spin.

If the wheel spins very rapidly, then the measuring sticks contract. If they contract, then this would mean that the outer rim of the wheel contracts as well. However, if it does so, then this means that the circumference of the wheel is contracting. If the circumference contracts, then so must the diameter.

If this is correct, then as a wheel turns, it appears that the spokes must shrink in length, even though the spoke's length is perpendicular to the high-velocity motion.

I believe that is what is meant. Did I get it right?

In that case, here's what I believe would happen, approximately. (Disclaimer: I am a physics dilettante, years out of date, with a liking for relativity.) I will give it from the point of view of a spectator on the hub, one on the outer wheel, and one on one of the spokes part way out.

The guy at the center of the hub would see the measuring sticks on the outer rim get shorter, but the outer rim would stay right where it is. The contraction of the wheel would be balanced by the material of the wheel "stretching" to match.

The guy on the outer rim would see the wheel material stretch, but to him it wouldn't be stretching to stay the same, it would be stretching to expand. The wheel would get bigger from his point of view, as he would be measuring against objects that would be contracting, while the wheel would be prevented from contracting.

A guy on the spokes would see an in-between situation. If near the hub, he would see what the observer on the hub saw. The further out he got, the more it would appear to him that the spokes had stretched. As he approached the outer edge of the wheel, his observations would come to match those of the outer wheel observer.

I believe that is approximately correct. Does that help?

Hi korosten, Farsight, AlphaNumeric, Filksinger et al,

Information travels at the speed of light so all observations are "very local" in the situations supplied. You must be very specific about the frame of the observer and the frame of what is being observed and their relationship. Though Special Relativity (SR) was derived as a "Global Theory" in any real situation we have the appearance of mass which means that it does not obey the global requirements of SR but a restricted version which is actually more "interesting".

Near the speed of light the propagation time of light becomes a very significant factor in what is actually "seen" of the observee by the observer and must be accounted for. Special Relativity is not usually about what an observer actually sees but what is actually able to be measured. The theory "should" be interpreted through the concept of "observers" and "all" valid observations of systems should be made "very locally" to what is being observed and then corrected for the propagation time to the "casual observer". SR is really about how we compare measurements between two inertial frames at rest and how we should transfer those measurements "between" the two frames. So you can well understand what we "see" being a very personal and cosmocentric view of everything is not quite the same as having a large number of non-local observers "measuring" things all over the Universe from the one distributed inertial frame of reference (our frame) each with an ideal synchronized clock to our clock back at "base". Unfortunately we really do not have this ideal set of observers strung up throughout space watching and measuring things for us as the "local observers" or "Johnny on the spot" to all relativistic phenomena that we are mostly measuring out there in space.

1. Cart Wheel

This old chestnut has an easy and a hard solution. The easy solution is to consider the wheel being "massless" and you mentally attach points of reference to the rim and along one of the spokes to indicate what is going to occur.

As korosten has correctly observed length contraction should occur in the direction of motion of the rim and it you have set up an orthogonal set of axes with the X axis pointing along the direction of motion then this length should not only "appear" to external observers to "contract" and Time for the observed phenomenon to dilate. This axis can only "instantaneously" be considered to be pointing along the tangential direction since the Special Theory is all about a preferred direction (the X axis) and this is changing from instant to instant as the point on the rim precesses around the center (relative to a hypothetical observer at the center). If you ignore this vectorial acceleration and concentrate only on speed then the outer circumference of the Cartwheel will contract relative to the radius. To an observer sited at the axis of the Cartwheel no length contraction occurs in the direction transverse to motion so the distance to the rim apparently remains constant while the circumference shrinks. Near the edge of the Cartwheel in that radial direction lesser length contractions will occur as we move incrementally along one of the spokes from the rim towards the hub. As V -> C of the rim the outer circumference of the Cartwheel shrinks towards zero length. This breaks the relationship...
Circumference = 2πR
R (which is measured from the center to the rim) remains "constant" while the Circumference shrinks to zero. So the geometry of the space is modified for a moving observer on the circumference of the wheel. For a massless wheel and massless observer on the rim of the wheel as the velocity of the "observer" tends toward the speed of light... time dramatically slows down and the velocity of the rim observer "rest" frame tends toward infinity relative to the observer at the center. Yes.... I said infinity... because what SR removes in "appreciable time" is returned as a calculation of increased velocity because velocity is s/t as t -> 0. This is relative to the "external observer".

Time for the moving observer in the moving observer frame is "relatively" slowing down and in the limit as V(tangential) -> C, will effectively stop if viewed by external rest frame observers. From the moving observer frame (who cannot notice this effect) the "apparent" distance traveled (which is in the rest frames of the external observer) seems to allow that speed to increase without limit (not even limited by the speed of light). In one sense this would give the "Star Trek" motion of the starry background where stars can seem to zip bye as if at "warp speed". The downside of all this is our traveler is being "projected" on a one way trip into the external observer's distant future and a few moments of the relatively high velocity reference frame will result in potentially millions of years of the alternative frames time. The frames are not equivalent since one is accelerated and the other remains unaccelerated and thus distinguishable. There are other effects... optical effects... such as stellar aberration but that is another story.

This is not a special function of simply "traveling in circles" but occurs for all travelers and it is usually referred to as the "Twin Paradox" since this is the result of time slowing down, in a relative sense, to those external observers. Any traveler whose rest frame closes on the speed of light (causes an external inertial frame of reference to have a relative velocity near the speed of light by virtue of relative motion) will experience this effect and it is an inseparable dual conjugate phenomena of space contraction and time dilation and stretches the external "relative" life expectancy of any ongoing process accordingly. A human being could travel a hundred billion light years distance in a matter of minutes of his rest frame time (according to clocks carried by him personally)... appearing to have no upper bound on velocity. At the "same time" external clocks in other frames of rest in relative motion will "appear" to do two different things relative to the high speed observer... when our observer approaches them they seem to speed up and when he recedes from them they slow down (an optical Doppler Effect accompanying these phenomena is inextricably linked where a blue and red shift occur respectively). What will also happen is the speeding up "bits" exceed the slowing down "bits" such that the overall effect is a nett speeding up of all external clocks. All the while the traveler's internal clock "appears" to maintain a constant rate according to all internal reckoning.

Above all else the meaning of Special Relativity is saying that for all observers all observations in their own rest frames the velocity of the speed of light is constant. Interpreted accurately means that light flashes from all sources moving relatively or otherwise "appear" to expand on the surfaces of perfect spheres in the vacuum of space at EXACTLY the Speed of Light regardless of the state of motion of source or observer. Relative to the observer these flashes reach the observer after a temporal delay period equal to CT seconds where T is the period of time elapsed since the flash was launched. This indicates to an observer where the position of the flash event "was" T seconds ago but since we observe in the "now" and have no way to immediately correct for propagation time (internally) this flash can be seen to indicate a contemporaneous event relative to a very local one and thus swap the roles of temporal causality. This is not an unusual feature since it occurs even in your bedrooms because a flash of light from across the room can appear to precede or lag a similar event closer or further away respectively simply because of the different propagation times. At the low velocity things are moving in my room (if you get my drift) you hardly notice this phenomena if at all. Where it becomes very important is when those flashes of light are coming from an object whose velocity is a significant proportion of the propagating velocity itself. Unless we have a way of correcting the times of events the information reaching our eyes is meaningless. Thus we have Special Relativity.

Getting back to the Cartwheel.. as Filksinger has indicated the rim "foreshortens" and the "spokes" stay essentially unchanged so Filksinger has implied that there is physical distortion. Now I realize that in any system where mass is involved and we have rotating coordinate systems there are going to be "huge" stresses at relativistic velocities. I do not think this is the way to look at it and I will say that since relativity is a "system" which removes the discrimination for a preference for one coordinate system or another. The state of motion of a sysem in Special Relativity is not internally distinguishable... all inertial frames are equivalent. This can only occur if we use a "test mass" for the cartwheel... a mass so diminishingly small that no matter what the "centripetal acceleration" negligible forces are developed. In an ideal case this cartwheel should be composed of light to make the analogy perfect and then this "rim" is the wall of the light-cone. Nevertheless the effect is the geometry of the wheel will be curved in such a way that seen from the outside it's external radius or diameter (far field condition) tends toward zero whereas within the radius of the spokes (the near field condition) there is a space with a positive area tending to the area of...
A = πR².
This is a single planar spin. It is possible to think of such an object in higher dimensions and this may be "embedded in six extra spatial dimensions", this makes the object have a total of nine spatial dimensions (including the three spatial dimension "we" the external observers abide within. In such a circumstance it is possible to "visualize" the object having spins in all three orthogonal planes simultaneously and this would result in a spherical region whose external region would be the two dimensional surface of a sphere and it radius tends toward zero while the interior volume would be essentially unchanged.

This does not make a lot of sense unless you carefully examine the geometry of Special Relativity and length contraction can be considered as a rotation in space as V -> C through an angle arcsin (V/C) away from all observers both "internal and external". Clearly this is describing a relativistic cavity for a shell whose "dimensional walls" can never be reached by any material object. It has become an enclosed space. If this "shell" was composed of a six dimensional object capable of spinning in all three planes at once at the speed of light this would be capable of "nipping off an internal cavity" inside of our space where the internal volume could be significantly large while the external dimensions as seen by external observers would seem insignificantly small.

The "wall" of such a dimensional object would be practically impenetrable and "unreachable" from our three dimensional "spacetime"...

2) 2 light beams from left to right:

Easy question... Relativity ensures that all observers measure the speed of light as being a constant from all sources... always equal to the "speed of light". This is not a debating point it is a fact about measurements made on anythings of this nature. Once again you cannot measure light any distance away from from a measuring instrument and therefore you are always measuring "locally" and this has shown no variation in velocity. This is no surprise since all forces are mitigated by the electromagnetic force in one or other of it's many forms (or by gravitation which is really a pseudo-force) which all propagate at the speed of light... The photons are the "exchange particle" of force in our Universe, we could never notice anything varying from this because of this linking of "dimensions and time". This answers all those questions. The only factor being changed is the measured frequency of sources. As seen from an observer as mentioned before that receding sources will undergo a Doppler Red Shift in frequency and for all approaching sources no matter what the direction all sources are blue shifted. This ignores gravitational red shift and effects due to Hubble Expansion of the Universe where every point is space is moving without acceleration away from every other point (on average).

As an additional point... in the situation of photons themselves this space contraction and time dilation in the rest frame of the photon leads to the unusual conclusion that along the path of the "free photon" in its internal time frame the photon "connects" all points it "touches" with a zero time and zero distance metric. This is the most extreme form of the way light itself actually sees the Universe in which it is moving. It never ages and "instantly connects" events of emission and absorption. It is only in our "less relevant" frame we see the separation of these two events as being separate.... But I caution that is a purely chauvinistic view of the Universe and is not a view shared by photons whose existence can be quite a great deal different from what is perceived. Photons have an almost zero life expectancy... that is ignoring the emission and absorption epochs which are "near field".

Cheers

Hi GoodElf et al,

I would like to add a visual description to the discussion. Although SR deals with how to reconcile viewpoints of observers in relative motion, it is interesting to ask what the rotating wheel would actually LOOK LIKE from the point of view of an observer on the rim.

Consider a single spoke of the wheel, say the spoke that meets the observer's location on the rim. Various light rays leave different points on this spoke and all arrive at the observer's eye at the same time to form his image of the spoke. Of course, the rays must have left the spoke at different times since they must travel different distances. Since the spoke is rotating, rays that left the spoke near the hub must have left when it was in an earlier orientation. Therefore, the spoke will appear curved to the observer on the rim. The other spokes will appear curved (each somewhat differently) as well.

Now consider light rays emitted radially from the hub. Since the observer is moving along with the rim, these rays appear to be coming from a direction that is tilted slightly forward along the observer's line of travel. (This is the same effect as stellar aberration due to the earth's orbit, which in turn is basically the same as the effect that makes rain seem to fall at a slant along the side windows of a moving bus.) Therefore, the observer does not seem to be traveling in a direction perpendicular to the hub, nor does the hub appear to be in the center of the wheel. This means that the wheel doesn't even appear round to this observer, but is distorted and curved like a Salvadore Dali watch (well, with more mathematical regularity than that, but you get the picture, I hope).

An observer at the hub (and rotating along with it) would see the spokes all curve as well. Now this observer's point of view is particularly interesting, because this observer is stationary himself, but is using a rotating coordinate system. A rotating coordinate system is always an accelerated one, accelerated coordinate systems produce fictitious forces, and GR treats gravitational force as just such a fictitious force. Therefore, there should be a GR point of view on this which says that the hub observer is a free-fall observer in a rather odd gravitational field. Perhaps the curvature associated with this field accounts for the disparity between radius and circumference in this coordinate system. (I am not sure, as I haven't personally done the math for this. Perhaps I'm on the wrong track here.)

However, these examples of how things look to different observers bring up one of the biggest GOTCHAs in SR. What's wrong with the following "explanation" of the contraction of the circumference but not the radius?
From the hub observer's point of view, the spokes are curved, but the circumference still forms a circle. Therefore, the length of a spoke is now greater than the radius of the circle, so the ratio circumference/spoke_length is now less than 2pi. This is perfectly consistent, since the circumference has contracted but the spokes have not, so of course they must bend. The answer is pretty obvious once you see it, but I'll let folks think about it before I post it.

Cheers!

--Stuart Anderson

that was a lot to read :-)

Ok, to the wheel:
--------------------

Observer in the CENTER:
so I see the outer rim getting shorter. But the diameter should not change.

I can see how this should work if the wheel is no longer "flat" but bent in some way (looking like a rotating U).

Is this what happens?

To the light beams:
----------------------
=============
The problem is what is *meant* with c being *constant*.

Let's look at the 3 situations - this is what SHOULD be measured:
Please correct me if I'm wrong!

1.OA stationary relative to OB: (this one is obvious)
OA measures c for A
OB measures c for B
OB measures c for A

2.OA moving relative to OB:
OA measures c for A
OB measures c for B
OB measures c for A

3.OB moving relative to OA:
OA measures c for A
OB measures c for B
OB measures c for A

There are 2 interpretations:

Let's look at all cases in BOTH interpretations. I write after each case WHY c is measured (or why NOT)

A.) Constant in an *absolute* sense (assuming an absolute coordinate system OC that is true for both OA and OB, so there would be a "third" coordinate system underlying both, let's call it OC.
====================================
In this "interpretation", c is always c relative to OC, but it is different if OA or OB is moving.
But becasue of time dilations, ALL observers still see ALL light beams as c!

1.OA stationary relative to OB (and OC): (this one is obvious)

2.OA moving relative to OB (and OC):
OA measures A: A SLOWER than c rel. to OA, BUT because of time/space dilation relative to OA
=> still measures c! -> OK

OB measures B: OB not moving rel. to OA ->c => ok

OB measures A: because A=c for *OC*, OB also measures c => OK


3.OB moving relative to OA:
OA measures A: OA not moving rel. to OC/OB ->c => ok
OB measures B: B SLOWER than c for OB, BUT because of time/space dilation relative to OC
=> still measures c! -> OK

OB measures A: A appears SLOWER than c as seen from OA rel. to OB BUT because of time/space dilation relative to OC
=> still measures c! -> OK


So in summary, there is *NO CONTRADITION* if we view it that way!


B.) Constant in a *relative* sense (NO absolute coordinate system, all is relative to the observer and ONLY relative to the observer, equivalence of motion/rest)
====================================

1.OA stationary relative to OB (and OC): (this one is obvious)
all measure c.

2.OA moving relative to OB:

OA measures A: A = c, BUT because of time/space dilation, OA would measure >> c -> NOT OK

OB measures B: B = c , OB not moving rel. to OA ->c => ok

OB measures A: A >> c for OB! => NOT OK!

3.OB moving relative to OA: same problems

=============================
So the conclusion is, that c is constant in an ABSOLUTE sense.
Otherwise you get contradictions...

Chantal

Oops. Yes, I completely forgot to take into account the fact that to the observer on the outside of the wheel, the rest of the wheel is moving. Assuming the wheel's outer edge is doing .9 c, the other side of the wheel would appear to the observer to be moving at .994475 c. Thus, to him the far side of the wheel would appear to be shrinking, even more than he appeared to be shrinking from the point of view of someone on the hub, while his side remains the same. Similar but lesser effects would happen in relation to other points along the wheel.

Due to light propogation delays, the spokes would, the further towards the far side they got from the obeserver on the rim, the more bent they would appear to be. Again, assuming .9 c, the observer on the rim would, when looking at the point directly opposed to him, see a point that was, assuming I am not missing a relativistic distortion, .9 * pi/2 further ahead than he would see if the wheel was standing still.

I really should have thought my answer through better.

EDIT: Where I said "side", I meant "point on the wheel". I really need to pay better attention.