An apology to Androcles
in [Relativity]
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From: Daryl McCullough on 6 Aug 2010 12:13 ben6993 says... >http://en.wikipedia.org/wiki/Bell's_spaceship_paradox: >Extract: "Bell pointed out that length contraction of objects as well >as the lack of length contraction between objects in frame S can be >explained physically, using Maxwell's laws. The distorted >intermolecular fields cause moving objects to contract =97 or to become >stressed if hindered from doing so. In contrast, no such forces act in >the space between rockets." > >I don't know about the supposed length contraction of objects and the >suposed lack of length contraction between objects noted in the above >extract. That seems to be a different thing from SR. In the >derivation of the SR length contraction you could have two moving rods >and the light emitted from the far end of the first rod and reflected >back off the near end of the far rod after traveling through the gap. >So L0 would instead now represent the gap between the two rods. SR >should show that the gap contracted, just as much as a rod would have >done. > >If there was a Bell-type contraction of a rod coupled with an SR >contraction of the rod then wouldn't the effects be multiplicative and >you would get a resultant L0/4 instead of L0/2? But why are the >intermolecular fields distorted by constant relative velocity. I >know that there are acceleration in Bell's spaceship paradox, maybe >the answer lies there. I need to read the paradox again. 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. -- Daryl McCullough Ithaca, NY
From: ben6993 on 7 Aug 2010 19:36 On Aug 6, 5:13 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > ben6993 says... > > > > > > >http://en.wikipedia.org/wiki/Bell's_spaceship_paradox: > >Extract: "Bell pointed out that length contraction of objects as well > >as the lack of length contraction between objects in frame S can be > >explained physically, using Maxwell's laws. The distorted > >intermolecular fields cause moving objects to contract =97 or to become > >stressed if hindered from doing so. In contrast, no such forces act in > >the space between rockets." > > >I don't know about the supposed length contraction of objects and the > >suposed lack of length contraction between objects noted in the above > >extract. That seems to be a different thing from SR. In the > >derivation of the SR length contraction you could have two moving rods > >and the light emitted from the far end of the first rod and reflected > >back off the near end of the far rod after traveling through the gap. > >So L0 would instead now represent the gap between the two rods. SR > >should show that the gap contracted, just as much as a rod would have > >done. > > >If there was a Bell-type contraction of a rod coupled with an SR > >contraction of the rod then wouldn't the effects be multiplicative and > >you would get a resultant L0/4 instead of L0/2? But why are the > >intermolecular fields distorted by constant relative velocity. I > >know that there are acceleration in Bell's spaceship paradox, maybe > >the answer lies there. I need to read the paradox again. > > 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. > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text - Thanks for your explanations, Daryl and Harald. Yes, I was overlooking that the measuring rod would be contracted too in frame k, as well as the rod. So the rest length in k is always L0, no matter what is the speed v. This implies that although the rod will always measure L0 in frame k, and for v=0.886c, then the rod will always appear as L0/2 in frame K. "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." (Quoted from Daryl) 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? 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?
From: harald on 8 Aug 2010 08:23 On Aug 8, 1:36 am, ben6993 <ben6...(a)hotmail.com> wrote: > On Aug 6, 5:13 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > ben6993 says... > > > >http://en.wikipedia.org/wiki/Bell's_spaceship_paradox: > > >Extract: "Bell pointed out that length contraction of objects as well > > >as the lack of length contraction between objects in frame S can be > > >explained physically, using Maxwell's laws. The distorted > > >intermolecular fields cause moving objects to contract =97 or to become > > >stressed if hindered from doing so. In contrast, no such forces act in > > >the space between rockets." > > > >I don't know about the supposed length contraction of objects and the > > >suposed lack of length contraction between objects noted in the above > > >extract. That seems to be a different thing from SR. In the > > >derivation of the SR length contraction you could have two moving rods > > >and the light emitted from the far end of the first rod and reflected > > >back off the near end of the far rod after traveling through the gap. > > >So L0 would instead now represent the gap between the two rods. SR > > >should show that the gap contracted, just as much as a rod would have > > >done. > > > >If there was a Bell-type contraction of a rod coupled with an SR > > >contraction of the rod then wouldn't the effects be multiplicative and > > >you would get a resultant L0/4 instead of L0/2? But why are the > > >intermolecular fields distorted by constant relative velocity. I > > >know that there are acceleration in Bell's spaceship paradox, maybe > > >the answer lies there. I need to read the paradox again. > > > 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. > > > -- > > Daryl McCullough > > Ithaca, NY- Hide quoted text - > > > - Show quoted text - > > Thanks for your explanations, Daryl and Harald. > > Yes, I was overlooking that the measuring rod would be contracted too > in frame k, as well as the rod. So the rest length in k is always L0, > no matter what is the speed v. This implies that although the rod will > always measure L0 in frame k, and for v=0.886c, then the rod will > always appear as L0/2 in frame K. > > "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." (Quoted from Daryl) > > 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. Certainly not; you correctly understood it at first. Rephrasing your quote of Daryl: Suppose that a rigid rod at rest in K is gently accelerated to reach a constant velocity and allowed to obtain equilibrium. According to measurements in K it will then be contracted relative to its original length because its equilibrium length is now less than before. Thus, as you wrote, "for v=0.886c, then the rod will [..] appear as L0/2 in frame K." Is that clearer? Harald > 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? > > 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?
From: ben6993 on 10 Aug 2010 14:54 On Aug 8, 1:23 pm, harald <h...(a)swissonline.ch> wrote: > On Aug 8, 1:36 am, ben6993 <ben6...(a)hotmail.com> wrote: > > > > > > > On Aug 6, 5:13 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > ben6993 says... > > > > >http://en.wikipedia.org/wiki/Bell's_spaceship_paradox: > > > >Extract: "Bell pointed out that length contraction of objects as well > > > >as the lack of length contraction between objects in frame S can be > > > >explained physically, using Maxwell's laws. The distorted > > > >intermolecular fields cause moving objects to contract =97 or to become > > > >stressed if hindered from doing so. In contrast, no such forces act in > > > >the space between rockets." > > > > >I don't know about the supposed length contraction of objects and the > > > >suposed lack of length contraction between objects noted in the above > > > >extract. That seems to be a different thing from SR. In the > > > >derivation of the SR length contraction you could have two moving rods > > > >and the light emitted from the far end of the first rod and reflected > > > >back off the near end of the far rod after traveling through the gap.. > > > >So L0 would instead now represent the gap between the two rods. SR > > > >should show that the gap contracted, just as much as a rod would have > > > >done. > > > > >If there was a Bell-type contraction of a rod coupled with an SR > > > >contraction of the rod then wouldn't the effects be multiplicative and > > > >you would get a resultant L0/4 instead of L0/2? But why are the > > > >intermolecular fields distorted by constant relative velocity. I > > > >know that there are acceleration in Bell's spaceship paradox, maybe > > > >the answer lies there. I need to read the paradox again. > > > > 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. > > > > -- > > > Daryl McCullough > > > Ithaca, NY- Hide quoted text - > > > > - Show quoted text - > > > Thanks for your explanations, Daryl and Harald. > > > Yes, I was overlooking that the measuring rod would be contracted too > > in frame k, as well as the rod. So the rest length in k is always L0, > > no matter what is the speed v. This implies that although the rod will > > always measure L0 in frame k, and for v=0.886c, then the rod will > > always appear as L0/2 in frame K. > > > "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." (Quoted from Daryl) > > > 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. > > Certainly not; you correctly understood it at first. Rephrasing your > quote of Daryl: > > Suppose that a rigid rod at rest in K is gently accelerated to reach a > constant velocity and allowed to obtain equilibrium. According to > measurements in K it will then be contracted relative to its original > length because its equilibrium length is now less than before. > > Thus, as you wrote, "for v=0.886c, then the rod will [..] appear as > L0/2 in frame K." > > Is that clearer? > > Harald > > > > > 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? > > > 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?- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - Thank you. Yes, the main thing is it is very clear that in SR "for v=0.886c, then the rod will [..] appear as > L0/2 in frame K." 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. So I will think about that separately from SR.
From: harald on 11 Aug 2010 06:29
On Aug 10, 8:54 pm, ben6993 <ben6...(a)hotmail.com> wrote: > On Aug 8, 1:23 pm, harald <h...(a)swissonline.ch> wrote: [..] > > > 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. > > > Certainly not; you correctly understood it at first. Rephrasing your > > quote of Daryl: > > > Suppose that a rigid rod at rest in K is gently accelerated to reach a > > constant velocity and allowed to obtain equilibrium. According to > > measurements in K it will then be contracted relative to its original > > length because its equilibrium length is now less than before. > > > Thus, as you wrote, "for v=0.886c, then the rod will [..] appear as > > L0/2 in frame K." > > > Is that clearer? > > > Harald [..] > > Thank you. > > Yes, the main thing is it is very clear that in SR > "for v=0.886c, then the rod will [..] appear as > > L0/2 in frame K." > > 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. So I will think > about that separately from SR. That's wrong, SRT deals with accelerations the same as classical mechanics deals with them. Einstein introduced the discussion as follows (note that I shortened it a little): "We now imagine the rod lying in the stationary system, and a uniform motion is then imparted to the rod. We now inquire as to the length of the moving rod". I'll try to make it understandable by filling in a few more historical details that I'll put in chronological sequence, and slightly simplified. 1. The laws of Maxwell were intended for a system that is in "in rest" in the ether. 2. Heaviside elaborated on these laws and showed that the electric and magnetic fields of a moving charge will be contracted. 3. Others reasoned that if material objects are somehow bound by electric fields, then naturally these objects should also contract due to that motion; and it was speculated that even electrons would contract. 4. According to SRT the laws of Maxwell are valid wrt any standard inertial measurement system. We may thus pretend that our measurement system is "in rest" and apply Maxwell's laws. Therewith many formerly "absolute" expressions obtained with SRT a meaning that depends on one's perspective; in particular the word "contraction" became as relative as the expression "in rest". Thus: 5. According to SRT, an object that is accelerated from "rest" will become "contracted"; this contraction consists mainly of the contraction of its intermolecular fields. For that reason it is sometimes said that Maxwell's equations are "relativistic": they did not require modification. Cheers, Harald |