- #31
Karl Coryat
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So I must be wrong in my assumption that such equidistance is a condition of the detectors clicking together.PeterDonis said:
I The Simultaneity Paradox: Investigating Net Rotation of a Balanced Beam
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The discussion centers on the simultaneity paradox involving a balanced beam struck by lightning bolts at both ends. An observer at rest sees the strikes as simultaneous, resulting in no net rotation, while a moving observer perceives a time difference between the strikes. Despite this difference, both observers agree that the beam does not tilt due to the simultaneous arrival of shock waves at the fulcrum. The beam flexes but remains in equilibrium because the effects of the impacts cancel each other out. Ultimately, the concept of balanced torques is frame-dependent, highlighting the complexities of relativistic mechanics.
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Then you just have a variation of Einstein's train experiment. Light hits the mirrors. In the frame in which the mirrors are at rest, the light leaves the Mirrors at c, relative to the mirrors in both directions and meet at the center. In the frame where the Mirrors of moving to the right, the light travels at c relative to this frame, and hits/leaves the left mirror first. If the mirrors are moving at 0.5c, then the left mirror's light has a speed of 0.5c with respect to the mirrors and the right mirror's light has a speed of 1.5c with respect to the Mirror.Karl Coryat said:
Same result. Light leave the left mirror first and takes longer to "catch up" to the center than the later leaving right mirror light, and they meet at the midpoint between mirrors.
- #33
Karl Coryat
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I guess my faulty assumption was that 0.5c and 1.5c (with respect to moving objects) were not things that could exist in this universe. But, if I were moving along with these mirrors, my understanding is that I would not measure the speed of light at those values, is that correct?
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PeterDonis
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Karl Coryat said:
More precisely, you were wrong in your assumption that "equidistance" is a frame-invariant condition. It isn't. You would need to specify the condition as equidistance in the cable's rest frame.
- #35
PeterDonis
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Janus said:
This is a very misleading way of stating it; the "speeds" you describe here are not speeds in the usual sense of the term and are not subject to the rule that speeds cannot exceed the speed of light. Nor do they obey the relativistic velocity addition rule. Nor are they what an observer, in any frame, would actually measure as the speeds of the light beams--those would all be measured to move at c.
I don't know if there is a standard term in relativity for the things you are calling "speeds", but I would call them "rate differences"--they are what you get if you just subtract the mirror speed from the light speed, paying appropriate attention to signs, and now using "speed" in its correct relativistic sense, as the speeds that would actually be measured. So the left mirror and the light are moving in the same direction, thus their speeds have the same sign and the rate difference is ##1 - 0.5 = 0.5##, while the right mirror and the light are moving in opposite directions, thus their speeds have opposite signs and the rate difference is ##1 - (- 0.5) = 1.5##.
Karl Coryat said:
The things @Janus called "speeds" are not properly called that. See above.
- #36
Karl Coryat
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Thanks, guys. That was very enlightening.
- #37
Ibix
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Not to scale, but here's a quick sketch of @Karl Coryat's two fibers connected to a black box detector, made at three different times in a frame where the apparatus is moving. The fibers are of equal length.
In the top diagram a pulse of light (coloured red) enters one end of the apparatus. In the second diagram the red pulse has traveled some way along the fiber, and the other pulse (coloured blue) enters the other end of the apparatus. In the final diagram both pulses arrive at the detector simultaneously. I've added vertical grey lines showing the points in space where the light pulses entered the fibers and where they meet at the detector. You can see that the distances are very different, although the distances from detector to fiber end are equal at all times. The point is that, in a frame where the apparatus is moving, "where the light entered the fiber" and "where the fiber end is now" are two different things in general.
Since this isn't drawn to scale, the same diagram works just as well for the "light moving in free space" variant.
In the top diagram a pulse of light (coloured red) enters one end of the apparatus. In the second diagram the red pulse has traveled some way along the fiber, and the other pulse (coloured blue) enters the other end of the apparatus. In the final diagram both pulses arrive at the detector simultaneously. I've added vertical grey lines showing the points in space where the light pulses entered the fibers and where they meet at the detector. You can see that the distances are very different, although the distances from detector to fiber end are equal at all times. The point is that, in a frame where the apparatus is moving, "where the light entered the fiber" and "where the fiber end is now" are two different things in general.
Since this isn't drawn to scale, the same diagram works just as well for the "light moving in free space" variant.