An apology to Androcles
in [Relativity]
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From: Daryl McCullough on 11 Aug 2010 14:48 ben6993 says... >The part about the effects on length of inter-molecular forces when >accelerating from speed 0 to v is still unclear, but to get from 0 to >v requires an acceleration, which is not part of SR. You seem to have a mistaken idea about what SR covers. SR can perfectly well describe accelerated motions. SR is a replacement for Newtonian mechanics, and the main point of Newtonian mechanics is to describe the trajectories of particles that are acted upon by forces. SR can do that just as well as Newtonian mechanics can. Instead of Newton's F = m d/dt v SR has a corresponding 4-dimensional equation: m d/dtau V^u = F^u where V^u is the velocity 4-vector, tau is proper time, and F^u is the 4-force. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 11 Aug 2010 15:07 ben6993 says... >I take this quote to mean that ... if L1 was the rest length of the >rod in frame K before the experiment began, then L1>L0. >That would mean that in frame K the rod length during the experiment >is L0/2 which is < L1/2. > >This seems nearly as bad as Androcles' length expansion! But it is >correct? I'm not sure if I understand what you are saying, but let me try to explain again, in detail, what I am saying: Suppose you have a rocket of length L at rest in some frame F. For definiteness, lets suppose that it is lined up along the x-axis. At the rear of the rocket is a powerful thruster. At time t=0 (as measured in frame F), I turn on the thruster. What happens immediately is that the rear of the rocket will start moving forward in the positive x direction. The front of the rocket will only start moving forward after a delay. That's because the information that the rocket has started moving has to propagate from the rear of the rocket to the front of the rocket. It propagates in the form of a compression wave that moves through the rocket at the speed of sound, which is much slower than the speed of light. By the time the compression wave reaches the front of the rocket, the rear of the rocket will have already moved a short distance. That means that the rocket will momentarily be shorter than it was before the thruster fired. (The rear has moved closer to the front, which means that the distance between front and rear is smaller). Now, once the compression wave reaches the front of the rocket, the front will start moving, too. As the rocket accelerates, the distance between the rear and the front will initially shrink, then will grow a little, and then it might shrink again. In general, there might be a complicated story as to how the length of the rocket changes with time. But eventually, if the rocket settles down into a constant speed, the rocket's length will settle down into some equilibrium length. If we actually knew all the forces at work on each little section of the rocket, then we could compute the equilibrium length. But if the forces holding the rocket together are Lorentz invariant, then we know that the length of the rocket, as measured in the original rest frame F, must be square-root(1-(v/c)^2) L, unless the acceleration has permanently stretched or compressed it. >Has L1/L0 for v=0.886c been estimated for a steel rod say? >Also, wouldn't a steel rod pushed out in front of an accelerating >spaceship be contracted due to forces inside the rod, while a rod >pulled behind the spaceship be stretched? A rod pushed in front of a rocket will *initially* compress to a length that is less than its equilibrium length, but then will expand to eventually settle down to its equilibrium length. A rod pulled behind a rocket will *initially* stretch to a length that is greater than its equilibrium length, but will then contract to its equilibrium length. -- Daryl McCullough Ithaca, NY
From: Edward Green on 11 Aug 2010 21:29 On Aug 6, 12:13 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > There are two different, but related, effects that could be called > "relativistic length contraction". First of all, if you set up > two standard inertial coordinate systems, then an object at rest > in one coordinate system will be measured to be contracted as > measured in the other coordinate system. This is a pure mathematical > derivation from the Lorentz transformations, which in turn are > derivable via Einstein's thought experiments. > > A second effect is that a rigid rod, when gently accelerated to reach > a constant velocity, and then allowed to reestablish its equilibrium > length, will be contracted relative to its original length, as measured > in its original rest frame. This is *not* a fact about coordinate > transformations, since it only mentions one coordinate system. It > is a fact about the nature of the forces inside the rod. The equilibrium > length of a solid object is a very complicated consequence of those > forces. > > These two different sorts of length contraction are related, of course. > Unless the forces inside a rod caused its length to be contracted > exactly the right amount by acceleration, the laws governing its > length could not be invariant under Lorentz transformations. > > There is similarly two different meanings of time dilation. > One meaning has to do with comparisons of time intervals in > two different inertial coordinate systems. That kind of time > dilation is a fact about Lorentz transformations. A second > type of time dilation is that a clock that is physically > accelerated and then decelerated will show less elapsed time > than a clock that had been at rest the whole while. Again, these > two types of time dilation are related, in that if clocks > *didn't* show the second kind of time dilation, then the > laws governing the behavior of clocks wouldn't be invariant > under Lorentz transformations. Ah, you seem to be coming around to my point of view. There _must_ be something in the Lorentzian point of view, for otherwise, how are we to explain the contracting rod, except by the feeble "it is now at rest in a different frame of reference". There must be some detailed atomic physics behind the contraction too, as seen from a fixed reference frame. Soon you will be admitting that it is a consistent point of view to say that the accelerated clock has, on average, been running slower, because (in a fixed frame of reference) it has on average been moving faster -- and there is in turn a detailed atomic explanation about processes in the moving clock as seen in that reference frame. The Mechanical Universe has a very nice animation of a "light clock", consisting of a pulse of light bouncing back and forth transversely to the line of relative motion, showing just how this works out in a particularly simple case. I think they still manage to poo-poo the whole idea that Lorentz was up to something. I think the "different path lengths in spacetime" POV and the Lorentzian POV are complimentary, not contradictory. Ed "marveling how well his new Google Groups killfile is working out" Green
From: Edward Green on 11 Aug 2010 21:42 On Aug 11, 2:48 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > ben6993 says... > > >The part about the effects on length of inter-molecular forces when > >accelerating from speed 0 to v is still unclear, but to get from 0 to > >v requires an acceleration, which is not part of SR. > > You seem to have a mistaken idea about what SR covers. SR can perfectly > well describe accelerated motions. SR is a replacement for Newtonian > mechanics, and the main point of Newtonian mechanics is to describe > the trajectories of particles that are acted upon by forces. SR > can do that just as well as Newtonian mechanics can. > > Instead of Newton's > > F = m d/dt v > > SR has a corresponding 4-dimensional equation: > > m d/dtau V^u = F^u > > where V^u is the velocity 4-vector, tau is proper time, and F^u > is the 4-force. Hmm... if I can come down off my hobby horse, and admit ignorance for a second, how would that work out with 3-momenta? I rather had the idea that dP/dt = F still worked in three space.
From: Androcles on 11 Aug 2010 23:09
"Edward Green" <spamspamspam3(a)netzero.com> wrote in message news:d34c72f4-47fc-4423-83c8-d24104045b19(a)l20g2000yqm.googlegroups.com... | On Aug 6, 12:13 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) | wrote: | | > There are two different, but related, effects that could be called | > "relativistic length contraction". First of all, if you set up | > two standard inertial coordinate systems, then an object at rest | > in one coordinate system will be measured to be contracted as | > measured in the other coordinate system. This is a pure mathematical | > derivation from the Lorentz transformations, which in turn are | > derivable via Einstein's thought experiments. | > | > A second effect is that a rigid rod, when gently accelerated to reach | > a constant velocity, and then allowed to reestablish its equilibrium | > length, will be contracted relative to its original length, as measured | > in its original rest frame. This is *not* a fact about coordinate | > transformations, since it only mentions one coordinate system. It | > is a fact about the nature of the forces inside the rod. The equilibrium | > length of a solid object is a very complicated consequence of those | > forces. | > | > These two different sorts of length contraction are related, of course. | > Unless the forces inside a rod caused its length to be contracted | > exactly the right amount by acceleration, the laws governing its | > length could not be invariant under Lorentz transformations. | > | > There is similarly two different meanings of time dilation. | > One meaning has to do with comparisons of time intervals in | > two different inertial coordinate systems. That kind of time | > dilation is a fact about Lorentz transformations. A second | > type of time dilation is that a clock that is physically | > accelerated and then decelerated will show less elapsed time | > than a clock that had been at rest the whole while. Again, these | > two types of time dilation are related, in that if clocks | > *didn't* show the second kind of time dilation, then the | > laws governing the behavior of clocks wouldn't be invariant | > under Lorentz transformations. | | Ah, you seem to be coming around to my point of view. There _must_ be | something in the Lorentzian point of view, for otherwise, how are we | to explain the contracting rod, except by the feeble "it is now at | rest in a different frame of reference". There must be some detailed | atomic physics behind the contraction too, as seen from a fixed | reference frame. | | Soon you will be admitting that it is a consistent point of view to | say that the accelerated clock has, on average, been running slower, | because (in a fixed frame of reference) it has on average been moving | faster -- and there is in turn a detailed atomic explanation about | processes in the moving clock as seen in that reference frame. | | The Mechanical Universe has a very nice animation of a "light clock", | consisting of a pulse of light bouncing back and forth transversely to | the line of relative motion, showing just how this works out in a | particularly simple case. I think Do you have any evidence to support that ludicrous assertion? | they still manage to poo-poo the | whole idea that Lorentz was up to something. I think Do you have any evidence to support that ludicrous assertion? | the "different | path lengths in spacetime" POV and the Lorentzian POV are | complimentary, not contradictory. | | Ed "marveling how well his new Google Groups killfile is working out" | Green |