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OK, say I have a very long , straight, rigid bar/rod type object. At one end, attached perpendicular to 1st is some type of motor to spin the rod, much like the hand of a clock. What is there to prevent the *linear* velocity at the very opposite opposite end of this bar to exceed the speed of light, if it was long enough?

The constancy of the speed of light also implies that there are no rigid bodies, therefore the bar will not move as a straight rod - it will bend by the sufficient amount and move at a velocity less than `c' What happens is this : the far end does not know what you have done to this end before a time l/c has passed, where l is the length of the rod. It will start moving only after this time. If you have already turned it by a large angle at the other end then you will end up winding it like a spring. I cannot think of an easier way of putting it.

jasjeet s bagla

In my opinion the rigidity of the stick is not relevant. Special relativity as noted above limits the rate at which variation of the force on the near end of the stick can be transmitted through the stick to cause variation of the acceleration (what is sometimes called "jerk") of the far end. The material properties of the stick (what it is made of) impose, incidentally, a stricter limit: a limit on the force that can be transmitted by the stick, and hence a limit on the acceleration of the far end. But neither of these limits by themselves prevents the velocity of the far end from exceeding the speed of light c: you can always accelerate the stick slowly enough to avoid both. The reason the far end of the stick cannot exceed c going in a circle is just the same reason it cannot exceed c going in a straight line: the inertia (resistance to further increase in velocity) grows without limit as its velocity closes in on c, and becomes infinite at c. Thus there is no force sufficient, no matter how long applied, and whether applied directly or through the lever of the stick, to accelerate the stick end past c.

An interesting variation on this questions is: what if I wave back and forth a flashlight pointed at the sky? Is not the light beam at some distance from me moving sideways faster than c? Interestingly, the answer to this question is "yes." Special relativity is not an issue because the interpretation of the light beam as one object moving sideways is entirely in our minds. In fact the "moving" beam consists of trains of photons heading straight outward always but in different directions: nothing real is moving sideways, so there is no limit on the sideways "velocity" of the beam. This is not an entirely academic question, either. A neutron star radiates most strongly at its magnetic poles, throwing out jets of radiation as it rotates like a water sprinkler throws out jets of water. If the sideways "velocity" of these beams were limited when they got out to Earth by the speed of light we could tell nothing from it. But actually the sideways "velocity" is just determined by the rotation speed of the neutron star, and by measuring it (e.g. measuring the time it takes the beam to sweep once around the sky and come back to pointing at Earth) we can determine the rotation period of the neutron star. From knowing the mass (by other means) this places upper limits on the size of the star, hence lower limits on its density, and can corroborate the thesis that the star is a hyperdense object.

christopher grayce

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