An apology to Androcles
in [Relativity]
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From: harald on 12 Aug 2010 05:44 On Aug 12, 3:42 am, Edward Green <spamspamsp...(a)netzero.com> wrote: > 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. That still works perfectly, as I learned in the "old" days (1980's) from Alonso&Finn, Fundamental University Physics (apparently it's even in reprint). As it happens to be on my shelf, I'll quickly put it here, with my comments in []: F = dP/dt = d(mv)/dt ; F and v are vectors and m=gamma*m_0 , so that: F = d/dt[m_0*v/sqr(1-v^2/c^2)] - For straight line motion: [I'll put v^2/c^2 = B] F = d/dt[m_0*v/(1-B)^1/2] = m_0[(dv/dt)/(1-B)]^3/2 = m/(1-B) dv/dt [By the way, you can also write this as: F = gamma^2 m a = gamma^3 m_0 a] - For circular motion: F = m_0/(1-B)^1/2 dv/dt = m dv/dt [= gamma m_0 dv/dt] As abs(dv/dt) = v^2/R, [no vector], the centripetal Force: F_N = m_0/(1-B)^1/2 v^2/R = m v^2/R = pv/R [scalar] - In general: a_T = dv/dt and a_N = v^2/R, so that: F_T = m/1-v^2/c^2 = k^2 m a_T [they use k for gamma; thus F_T = gamma^3 m_0 a_T] F_N = m a_N [thus F_N = gamma m_0 a_N] Cheers, Harald
From: Daryl McCullough on 12 Aug 2010 09:08 Edward Green says... > >On Aug 11, 2:48=A0pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> 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. Well, to rewrite it in terms of coordinate time, rather than proper time, you use the relationship between tau and t: d tau = 1/gamma dt Then V^u = d/dtau x^u = gamma d/dt x^u d/dtau V^u = gamma d/dt (gamma d/dt x^u) So in terms of coordinates, with x^0 = ct, we have: m gamma d/dt (gamma) = F^0 m gamma d/dt (gamma v^j) = F^j or d/dt (gamma mc^2) = 1/gamma c F^0 d/dt (gamma mv^j) = 1/gamma F^j So if we define E = mc^2 p^j = gamma m v^j P = 1/gamma c F^0 (the "power", or rate of transference of energy) f^j = 1/gamma F^j then we recover the 4 equations in 3+1 formalism: d/dt E = P d/dt p^j = f^j But note that f^j is not simply a spatial component of the 4-force F^j. They differ by a factor of gamma. In the same way, v^j is not just the spatial component of the 4-velocity V^j, they also differ by a factor of gamma. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 09:37 Edward Green says... >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. I know that people always hate my Euclidean geometry analogies, but they are actually *very* close. There is a similar effect in good old Newtonian physics when it comes to rotations. Take a yardstick that has dimensions 36 inches long by 1 inch wide. Initially, it is oriented along the x-axis. Let's define the y-extent of the yardstick to be the distance between a point on one edge of the yardstick to the point on the other edge vertically above the first point. Initially, then,the y-extent of the yardstick is delta-y = 1 inch. Now, rotate the yardstick 45 degrees. Since nothing is perfectly rigid, the yardstick's shape will distort, it will twist a little, but eventually it will settle down. At that time, the y-extent will be given by delta-y' = delta-y/cos(45) (You have to draw a picture to see this, but the reason that delta-y' is greater than delta-y is that the vertical line through the yardstick now cuts at an angle across the yardstick. If you turn the yardstick through 90 degrees, then the vertical line will run the entire length of the yardstick, 36 inches). Now, the shape and size of the yardstick is determined by intermolecular forces. Those forces have to act in exactly the right way in order to satisfy the above formula. But if they *didn't* act in exactly that way, then the laws of physics governing yardsticks would not be rotationally invariant. (You can imagine laws that are not rotationally invariant, so that rotating a yardstick changes its length, but normal physical forces don't work that way). >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. Sure, that's a legitimate way of thinking about it, just as it is legitimate (although a little strange) to think of rotating yardsticks in terms of a physical expansion of the y-extent of the yardstick. Viewing it geometrically has a certain simplicity--the yardstick is unchanged, it's just rotated, so its projection along the x-axis and the y-axis are changed. But it's actually a fact about the forces making up the yardstick that make the geometric interpretation possible. If forces were not invariant under rotations, then the yardstick *wouldn't* be unchanged by rotation. >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. Yeah, there is nothing wrong with the view that starts with a description in a particular frame, and expresses everything in terms that are meaningful to that frame. However, if the laws of physics possess a certain kind of symmetry, then there is nothing special about that frame. -- Daryl McCullough Ithaca, NY
From: harald on 12 Aug 2010 15:33 On Aug 12, 3:37 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Edward Green says... > > >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. > > I know that people always hate my Euclidean geometry analogies, > but they are actually *very* close. > > There is a similar effect in good old Newtonian physics when it > comes to rotations. Take a yardstick that has dimensions 36 inches > long by 1 inch wide. Initially, it is oriented along the x-axis. > Let's define the y-extent of the yardstick to be the distance > between a point on one edge of the yardstick to the point on the > other edge vertically above the first point. Initially, then,the > y-extent of the yardstick is delta-y = 1 inch. Now, rotate the yardstick > 45 degrees. Since nothing is perfectly rigid, the yardstick's > shape will distort, it will twist a little, but eventually it will > settle down. At that time, the y-extent will be given by > > delta-y' = delta-y/cos(45) > > (You have to draw a picture to see this, but the reason that > delta-y' is greater than delta-y is that the vertical line > through the yardstick now cuts at an angle across the yardstick. > If you turn the yardstick through 90 degrees, then the vertical > line will run the entire length of the yardstick, 36 inches). > > Now, the shape and size of the yardstick is determined by > intermolecular forces. Those forces have to act in exactly > the right way in order to satisfy the above formula. But if > they *didn't* act in exactly that way, then the laws of > physics governing yardsticks would not be rotationally > invariant. > > (You can imagine laws that are not rotationally invariant, > so that rotating a yardstick changes its length, but normal > physical forces don't work that way). > > >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. > > Sure, that's a legitimate way of thinking about it, just as > it is legitimate (although a little strange) to think of rotating > yardsticks in terms of a physical expansion of the y-extent of > the yardstick. > > Viewing it geometrically has a certain simplicity--the yardstick > is unchanged, it's just rotated, so its projection along the x-axis > and the y-axis are changed. But it's actually a fact about the > forces making up the yardstick that make the geometric interpretation > possible. If forces were not invariant under rotations, then the > yardstick *wouldn't* be unchanged by rotation. > > >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. > > Yeah, there is nothing wrong with the view that starts with > a description in a particular frame, and expresses everything > in terms that are meaningful to that frame. However, if the > laws of physics possess a certain kind of symmetry, then there > is nothing special about that frame. > > -- > Daryl McCullough > Ithaca, NY Nice, everything you write looks quite perfect - until the logical glitch right at the end. I would improve it as follows (which is certainly not by chance closer to the way Einstein put it!): "If the laws of physics possess a certain kind of symmetry, then there is nothing special about that frame for the laws of physics." A minor change? Not really! Compare: "If a picture is a perfect copy of a painting, then there is nothing special about the painting." with: "If a picture is a perfect copy of a painting, then there is nothing special about the painting for imaging purposes." Cheers, Harald
From: Edward Green on 13 Aug 2010 16:22
On Aug 12, 3:33 pm, harald <h...(a)swissonline.ch> wrote: > On Aug 12, 3:37 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: <...> > Nice, everything you write looks quite perfect - until the logical > glitch right at the end. > > I would improve it as follows (which is certainly not by chance closer > to the way Einstein put it!): > > "If the laws of physics possess a certain kind of symmetry, then there > is nothing special about that frame for the laws of physics." > > A minor change? Not really! Compare: > > "If a picture is a perfect copy of a painting, then there is nothing > special about the painting." > > with: > > "If a picture is a perfect copy of a painting, then there is nothing > special about the painting for imaging purposes." I don't really understand your... watchamacallit: simile? analogy? Anyway, even if there is nothing special about a certain frame, we can always construct a complete description of events from that frame, which can be a very powerful point of view, I think. As Bell would have said: in eliminating certain modes of thinking which were wrong, we also forgot some which are right. |