The Symmetric Twin Paradox
in [Relativity]
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From: Sue... on 21 Jun 2010 00:19 On Jun 20, 11:26 pm, colp <c...(a)solder.ath.cx> wrote: > On Jun 21, 12:24 pm, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: > > > > > On Jun 20, 8:03 pm, colp <c...(a)solder.ath.cx> wrote: > > > > On Jun 21, 5:52 am, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: > > > > > On Jun 20, 2:27 am, colp <c...(a)solder.ath.cx> wrote:> On Jun 20, 11:35 am, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: > > > > > > > On Jun 19, 7:17 pm, colp <c...(a)solder.ath.cx> wrote: > > > > > > > 1. SR predicts that each twin observes the other twin to age more > > > > > > > slowly both on the outgoing leg and the return leg. > > > > > > > No... > > > > > ================= > > > > > > How does a four dimensional model of spacetime provide for an > > > > > alternative interpretation of the symmetric twin thought experiment? > > > > > It contributes a bit of mathematical rigour. > > > > Additional rigor does not invalidate the usual interpretation of SR. > > > It certainly does. Apply rigorous maths > > to an optical illusion and your interpretation > > will change. ==========> > How do you think that works in the case of the symmetric twin paradox? It works with everything. You don't get answers 'till you formalize your question. > > > > > > In the symmetric twin paradox, do you deny that SR predicts that each > > > twin observes the other twin to age more slowly in the outgoing leg? > > > I suppose you refer to the absurdity in the > > 1905 paper pointed out by Paul Langevin, > > among others. > > No, that isn't what I'm referring to. Will you please answer my > question? You haven't asked a question. You formed your scenario around what some think relativity says, rather that what it actually says. Here it is again, word for word: << the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude sqrt(-1) ct proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same rôle as the three space co-ordinates. >> http://www.bartleby.com/173/17.html Where is there room for paradox? Symmetric or otherwise? Sue...
From: colp on 21 Jun 2010 00:25 On Jun 21, 3:25 pm, "Inertial" <relativ...(a)rest.com> wrote: > "colp" <c...(a)solder.ath.cx> wrote in message > > news:572cf302-7007-41ba-a08d-77cf2dde07a7(a)40g2000pry.googlegroups.com... > > > > > On Jun 21, 12:10 pm, "Inertial" <relativ...(a)rest.com> wrote: > >> "colp" <c...(a)solder.ath.cx> wrote in message > > >>news:73c42da8-03e8-4f07-acbf-92c78718d7ba(a)j36g2000prj.googlegroups.com.... > > >> > On Jun 20, 9:14 pm, "Inertial" <relativ...(a)rest.com> wrote: > >> >> "colp" <c...(a)solder.ath.cx> wrote in message > > >> >> > What do you think that SR actually says about the symmetric twin > >> >> > thought experiment? > > >> >> You are the one making claims .. you'd been asked repeatedly to show > >> >> the > >> >> math backing up your claim. you refuse to do so. Until you do, you > >> >> cannot > >> >> be taken seriously > > >> > I have already shown the math, and I've also reposted it in response > >> > to an earlier post of yours. > > >> I've shown you are wrong > > > According to your logic you cannot be taken seriously. > > Of course I can .. by my own, and any reasonable logic. Wrong. You said: "You are the one making claims .. you'd been asked repeatedly to show the math backing up your claim. you refuse to do so. Until you do, you cannot be taken seriously" Do you think that I should live up to standards that you yourself cannot live up to? You haven't shown the math for the turnaround, all you have done is to falsely claim that you have. It would be easy for you to prove me wrong simply pasting in what you previously wrote, had you actually shown the math.
From: Inertial on 21 Jun 2010 00:26 "Sue..." <suzysewnshow(a)yahoo.com.au> wrote in message news:cf792cf0-49ba-481e-ae7c-99bb64bff6da(a)w12g2000yqj.googlegroups.com... > On Jun 20, 11:26 pm, colp <c...(a)solder.ath.cx> wrote: >> On Jun 21, 12:24 pm, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: >> >> >> >> > On Jun 20, 8:03 pm, colp <c...(a)solder.ath.cx> wrote: >> >> > > On Jun 21, 5:52 am, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: >> >> > > > On Jun 20, 2:27 am, colp <c...(a)solder.ath.cx> wrote:> On Jun 20, >> > > > 11:35 am, "Sue..." <suzysewns...(a)yahoo.com.au> wrote: >> >> > > > > > On Jun 19, 7:17 pm, colp <c...(a)solder.ath.cx> wrote: >> > > > > > > 1. SR predicts that each twin observes the other twin to age >> > > > > > > more >> > > > > > > slowly both on the outgoing leg and the return leg. >> >> > > > > > No... >> >> > > > ================= >> >> > > > > How does a four dimensional model of spacetime provide for an >> > > > > alternative interpretation of the symmetric twin thought >> > > > > experiment? >> >> > > > It contributes a bit of mathematical rigour. >> >> > > Additional rigor does not invalidate the usual interpretation of SR. >> >> > It certainly does. Apply rigorous maths >> > to an optical illusion and your interpretation >> > will change. > > ==========> > >> How do you think that works in the case of the symmetric twin paradox? > > It works with everything. You don't > get answers 'till you formalize your > question. True >> > > In the symmetric twin paradox, do you deny that SR predicts that each >> > > twin observes the other twin to age more slowly in the outgoing leg? >> >> > I suppose you refer to the absurdity in the >> > 1905 paper pointed out by Paul Langevin, >> > among others. >> >> No, that isn't what I'm referring to. Will you please answer my >> question? > > You haven't asked a question. You formed > your scenario around what some think relativity > says, rather that what it actually says. True > Here it is again, word for word: Sue now provides a quote which is talking ABOUT SR, and how laws are reformulated to be 'relativistic' (eg think of doppler) .. forms that degenerate to approximately the same as the familiar forms when v <<c, but what Sue shows isn't all of 'what SR says'. So they following is NOT the theory of specila relativity, nor is it what SR says about the particular scenario. As usual, Sue's quotes are not relevant .. regardless of whether what she is saying (if anything) is correct or not. > << the four-dimensional space-time continuum of the > theory of relativity, in its most essential formal > properties, shows a pronounced relationship to the > three-dimensional continuum of Euclidean geometrical space. > In order to give due prominence to this relationship, > however, we must replace the usual time co-ordinate t by > an imaginary magnitude > > sqrt(-1) > > ct proportional to it. Under these conditions, the > natural laws satisfying the demands of the (special) > theory of relativity assume mathematical forms, in which > the time co-ordinate plays exactly the same r�le as > the three space co-ordinates. >> > http://www.bartleby.com/173/17.html > > Where is there room for paradox? > Symmetric or otherwise? There isn't. SR is proven mathematically to be self-consistent .. so there is no paradox. All one can do it perform experiments that can possibly show a result different to what is predicted
From: Inertial on 21 Jun 2010 00:30 "colp" <colp(a)solder.ath.cx> wrote in message news:bd2683bc-e843-41a1-acc3-91fd70137ffd(a)h37g2000pra.googlegroups.com... > On Jun 21, 3:25 pm, "Inertial" <relativ...(a)rest.com> wrote: >> "colp" <c...(a)solder.ath.cx> wrote in message >> >> news:572cf302-7007-41ba-a08d-77cf2dde07a7(a)40g2000pry.googlegroups.com... >> >> >> >> > On Jun 21, 12:10 pm, "Inertial" <relativ...(a)rest.com> wrote: >> >> "colp" <c...(a)solder.ath.cx> wrote in message >> >> >>news:73c42da8-03e8-4f07-acbf-92c78718d7ba(a)j36g2000prj.googlegroups.com... >> >> >> > On Jun 20, 9:14 pm, "Inertial" <relativ...(a)rest.com> wrote: >> >> >> "colp" <c...(a)solder.ath.cx> wrote in message >> >> >> >> > What do you think that SR actually says about the symmetric twin >> >> >> > thought experiment? >> >> >> >> You are the one making claims .. you'd been asked repeatedly to >> >> >> show >> >> >> the >> >> >> math backing up your claim. you refuse to do so. Until you do, >> >> >> you >> >> >> cannot >> >> >> be taken seriously >> >> >> > I have already shown the math, and I've also reposted it in response >> >> > to an earlier post of yours. >> >> >> I've shown you are wrong >> >> > According to your logic you cannot be taken seriously. >> >> Of course I can .. by my own, and any reasonable logic. > > Wrong. You said: "You are the one making claims .. you'd been asked > repeatedly to show the > math backing up your claim. you refuse to do so. Until you do, you > cannot be taken seriously" Yeup. > Do you think that I should live up to standards that you yourself > cannot live up to? I am not making the claims against SR. If you make such claims, you should provide the proof. I shouldn't have to do it for you. Once you do, i can show you where you are wrong (as I did). > You haven't shown the math for the turnaround, all you have done is to > falsely claim that you have. The burden of proof that there is a contradiction or paradox in SR is yours. If you really want to see the analysis done, I'll do it for you. Do you then promise to admit you were wrong and go away? > It would be easy for you to prove me wrong I already pointed out your errors ... job done. You need to correct for those errors and try again. > simply pasting in what you > previously wrote, had you actually shown the math.
From: colp on 21 Jun 2010 00:45
On Jun 21, 2:42 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > colp says... > > > > > > >On Jun 21, 1:10=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> What SR says for those sorts of thought experiments is that, as measured > >> in any inertial coordinate system, > > >> 1. Light travels at constant velocity, at speed c, in all directions, > >> independent of the motion of the source. > >> 2. An ideal clock traveling at speed v for time period t will show > >> an elapsed time of T =3D t square-root(1-(v/c)^2). > > >You are saying that T = t / gamma, where gamma = 1 / square-root(1-(v/ > >c)^2) > > >How do you reconcile this with your earlier statement? > > ><quote> > >If you look at the Lorentz transformation for time, you > >find: > > >t' = gamma (t - vx/c^2) > > These are talking about two different things: (1) the elapsed time on > a *single* clock, and (2) the time *coordinate*. They are related, > but are not the same thing. > > To set up a coordinate system, you can't just use a single clock, > you have to have a system of clocks, and you have to have a way > of synchronizing them. The Lorentz transformation for the time > coordinate takes into account two different things: (1) the relative > rate of one clock as measured in a coordinate system in which that > clock is not at rest, and (2) the fact that two different frames > will use different synchronizations for their clocks. > > Let's see how that works for just two clocks. > > Suppose you have two clocks, A and B, that are lined up > along the x-axis, and are at rest in some frame F'. Suppose > that the distance between the clocks is X' as measured > in frame F'. If you are an observer in frame F', then > how do you go about making sure that your clocks are > synchronized? Well, here's one way of doing it: You > know the distance between the clocks, X'. You know that > light has speed c. So it takes light a time X'/c to travel > from clock A to clock B. So what you do is set clock > A to show t'=0, and then immediately send a light signal > to clock B. When B receives the signal, that clock is > set to t'=X'/c. Voila! They are synchronized. > > Now, let's look at things from the point of view > of frame F, the frame in which the two clocks are > moving at speed v along the x-axis (with clock B > ahead of clock A). Let's assume that clock A started > at x=0 at time t=0, so the location of clock A > at later times is given by x = vt. > > As measured in frame F, the > distance between the clocks is not X', but is X'/gamma, > because of length contraction. So the location > of B is given by x = vt + X'/gamma. > > That's the first > complication. The second complication is that when > A sends a light signal to clock B, the light has > to travel further than just X'/gamma, because B > is moving away the whole time. > > To compute this effect, let t be the time it takes > for the light to travel from A to B. During that time, > light travels a distance ct, and clock B travels a > distance vt. So the light has to travel a distance > of X'/gamma + vt in time t. So we must have: > > ct = X'/gamma + vt > > or > > t = X'/gamma * 1/(c-v) > > Since 1/gamma = gamma * 1/gamma^2, and > 1/gamma^2 = 1-(v/c)^2, we can rewrite this as follows: > > t = X' gamma (1-(v/c)^2)/(c-v) > = X'/c^2 gamma (c^2 - v^2)/(c-v) > = X'/c^2 gamma (c+v) > > So, from the point of view of frame F, it takes time X'/c^2 gamma (c+v) > for the light signal to travel from A to B. During that time, clock > A advances by 1/gamma * (that time), because moving clocks are slowed. > So A advances by X'/c^2 (c+v) by the time the light signal reaches B. > But, the synchronization scheme involved setting clock B to time X'/c. > So as measured in frame F, the two clocks are out of synch; > the difference in their setting is X'/c^2 (c+v) - X'/c = X' v/c^2. > Clock B is behind clock A by an amount X' v/c^2. In addition, > clock B is running slow by a factor of gamma. So the time t' > shown on clock B is given by: > > t' = t/gamma + X' v/c^2 > > We said above that B's location is given by x = vt - X'/gamma. > So we can express X' in terms of x as: > > X' = gamma (x-vt). > > Putting this into the equation for t' gives: > > t' = t/gamma - gamma (x-vt) v/c^2 > = gamma (t/gamma^2 - xv/c^2 + v^2/c^2 t) > = gamma (t (1-(v/c)^2) - xv/c^2 + (v/c)^2 t) > = gamma (t - xv/c^2) > > So, in frame F, although each clock is running slower by > a factor of gamma, the effect of the synchronization causes > the coordinate t' to vary with x as predicted by the Lorentz > transformations. O.K. What remains from my previous post is the question of how you get from the original premises of SR to a determination on when you should apply the transformation for a single clock, and when you should apply the transformation for the time coordinate in the case of the twins. |