Re: Paradox in need of resolution.
- From: Tom Roberts <tjroberts@xxxxxxxxxx>
- Date: Wed, 28 Sep 2005 11:10:58 -0500
RP wrote:
[...]
Your whole description is overly complicated, and your analysis is woefully inadequate, in part because of the complexity of your description. Gravitation is COMPLICATED, so let's eliminate it; rods/cords/strings are also complicated, so let's omit them, too: Let me propose a different gedanken which exhibits the essential aspects of yours, but is MUCH easier to analyze:
This is purely SR: In Minkowski spacetime consider inertial frame A and inertial frame B moving rapidly along a common X/X' axis relative to A; the point X'=0 is at rest in B and moves to increasing values of X in frame A. Their Y and Z axes are parallel (everything happens in the plane Z=0).
Initially at rest in frame A are two identical rockets, R1 and R2, separated along X with R1 at X1 and R2 at X2 in frame A, X2>X1; both are located at Y=Z=0 in frame A, and both carry standard clocks that are initially synchronized with the (standard) coordinate clocks of frame A. At time T=0 in A both R1 and R2 blast off along the Y axis with identical and constant proper acceleration. At any time t>0 in frame A, both rockets have identical values of Y, and their clocks both display the same time (which is progressively less than the coordinate time of frame A at their current location). Both rockets asymptotically approach speed c relative to frame A, and their direction is always along the Y axis.
Observed from frame B, R1 is the lead/front rocket, and R2 is the lagging/rear rocket. R2 blasts off at a smaller value of t' than does R2. So at any coordinate time in B after R2's blastoff R2 will have a larger value of Y' than R1. This difference in Y' values will increase over time in B, approaching an asymptotic value (which is finite, and the value depends on lots of details). Both rockets asymptotically approach speed c relative to frame B, but their direction changes (initially along -X', asymptotically approaching Y') -- this is their direction of motion, not the direction in which they are pointing (which is always along Y').
Note, however, that the observer in frame B could observe the locations of the rockets at a given value of their onboard clocks (rather than at a common value of coordinate time t'); when this is done both rockets have the same value of Y' and both have the same speed and direction of motion relative to frame B.
You wanted to attach a rod/cord/string between the rockets. That has many difficulties, so instead let's line up a series of rockets between R1 and R2, and have them all blast off at t=0 in A, with the same proper acceleration as R1 and R2. Think of these rockets as short pieces of "string", but there's no "sagging" or tension, and the entire "string" accelerates along with the rockets R1 and R2. Clearly in frame A, at a given coordinate time t all the rockets (incl R1 and R2) have the same value of Y, and their onboard clocks all read identical values.
Observed from frame B after R2 blastoff, this row of rockets is not a straight line, but asymptotically approaches one. It is this complex shape that makes a simple string extremely difficult to analyze. Moreover, each of these rockets is moving differently relative to frame B, in a complicated and changing direction (initally along -X', but increasingly pointed toward Y', and asymptotically parallel to Y'; the different rockets approach the asymptote differently as a function of frame-B time t'). Note, however, that if observer B measures the locations of all the rockets at a given value of their individual onboard clocks, they all have the same value of Y' and all have the same speed and direction relative to frame B.
You want to discuss the "tension" in the "string" between R1 and R2, and made wildly-incorrect statements about it. This gedanken permits us to analyze it. You seem to think that because the path length of the "string" in frame B is clearly longer than a straight line, the observer in frame B must conclude there is tension in the string. That is not so. The deviation from a straight line is due to the difference in coordinate clock synchronization, and that quite clearly does not affect the string itself.
In any short region of a string, its molecules will arrange themselves so in ther own rest frame the inter-molecular bonds are as relaxed as possible. That's why I split the continuous string into a series of discrete rockets. The relationships between adjacent rockets are only simple at a common value of their proper times. Clearly this is frame independent, and the above construction implies that the relationship between adjacent rockets at common proper time (in their instantaneously-comoving rest frame) is EXACTLY the same as it was initially in frame A.
So by comparing this gedanken to yours, we can conclude there is no tension in your rod/cord/string (assuming there was no tension initially, which you did not specify but is clearly required). Clearly the wild rotations you imagined will not occur.
Tom Roberts tjroberts@xxxxxxxxxx .
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- Paradox in need of resolution.
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