Preferred Frame Theory indistinguishable from SR
in [Math]
|
From: Paul Stowe on 6 Jul 2010 20:56 On Jul 6, 12:42 am, harald <h...(a)swissonline.ch> wrote: > On Jul 5, 8:26 pm, PaulStowe<theaether...(a)gmail.com> wrote: > > > > > > > On Jul 5, 10:00 am, harald <h...(a)swissonline.ch> wrote: > > > > On Jul 5, 2:46 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > harald says... > > > > > >Acceleration effects are not identified as gravitational fields in > > > > >Newtonian physics (which, as you now know, you didn't know); and > > > > >neither is that the case in SRT. In those theories acceleration is > > > > >"absolute", and no gravitational fields are caused by acceleration.. > > > > > This is a topic for another discussion, but I'm talking about > > > > "pseudo-gravitational" fields, which crop up in both SR and Newtonian > > > > physics if you use accelerated coordinates. > > > > We agree on "pseudo", while Einstein rejected that. > > > > > >> I think it is because you have not made it very well. I still > > > > >> have no idea what your point is. > > > > > >Just study Einstein's paper carefully, > > > > > I want to know what *YOUR* point is. State it in your own words. > > > > I did, also in the part of my sentence that you exactly here snipped: > > > > *you'll know what theory the paradox challenges*. > > > > THAT (and only that) was my point: the clock paradox challenges the > > > General PoR. > > > > You certainly are aware that, despite Einstein's *suggestion* to that > > > effect in the introduction of his 1905 paper, SRT is *not* based on > > > the General PoR. > > > Strictly speaking the 'domain' of the 'special' theory of relativity > > is limited to inertial states. The original principle of relativity > > as expressed in Einstein's 1905 work covered only that domain. > > PS I overlooked the error you made here - a confusion that is often > seen. As Eric points out, SRT is *not* (never was!) limited to > inertial states. It's the same as for Newtonian mechanics. SRT uses > Poincare's PoR, which refers to Newtonian reference systems; and we > can switch reference system whenever we like, using the LT. Thus the > twin problem is a trivial exercise in SRT. The twin paradox however > was aiming Einstein's *General* PoR. > > Harald 'I' never said Lorentzian Relativity (LR) or it renormalized cousin (SR) is 'limited' to inertial frames. I said that, traditionally, the concept commonly called 'special' relativity is such (special) because it is considered the subset of general relativity limited to unaccelerated frames of reference. Nature places no barrier or boundary on relativity, human concepts do! This is why I refused to call LR 'LET'. LR is no more limited to such artificial boundaries than SR... Paul Stowe
From: harald on 7 Jul 2010 05:29 On Jul 7, 2:47 am, Paul Stowe <theaether...(a)gmail.com> wrote: > On Jul 6, 12:25 am, harald <h...(a)swissonline.ch> wrote: > > > > > On Jul 5, 8:26 pm, PaulStowe<theaether...(a)gmail.com> wrote: > > > > On Jul 5, 10:00 am, harald <h...(a)swissonline.ch> wrote: > > > > > On Jul 5, 2:46 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > > harald says... > > > > > > >Acceleration effects are not identified as gravitational fields in > > > > > >Newtonian physics (which, as you now know, you didn't know); and > > > > > >neither is that the case in SRT. In those theories acceleration is > > > > > >"absolute", and no gravitational fields are caused by acceleration. > > > > > > This is a topic for another discussion, but I'm talking about > > > > > "pseudo-gravitational" fields, which crop up in both SR and Newtonian > > > > > physics if you use accelerated coordinates. > > > > > We agree on "pseudo", while Einstein rejected that. > > > > > > >> I think it is because you have not made it very well. I still > > > > > >> have no idea what your point is. > > > > > > >Just study Einstein's paper carefully, > > > > > > I want to know what *YOUR* point is. State it in your own words. > > > > > I did, also in the part of my sentence that you exactly here snipped: > > > > > *you'll know what theory the paradox challenges*. > > > > > THAT (and only that) was my point: the clock paradox challenges the > > > > General PoR. > > > > > You certainly are aware that, despite Einstein's *suggestion* to that > > > > effect in the introduction of his 1905 paper, SRT is *not* based on > > > > the General PoR. > > > > Strictly speaking the 'domain' of the 'special' theory of relativity > > > is limited to inertial states. The original principle of relativity > > > as expressed in Einstein's 1905 work covered only that domain. So > > > Harald is correct, the paradox is confined to the situation where, you > > > have identical twins one remain in the original inertial frame, the > > > other accelerated rapidly (nearly instantaneously) to speed ~c travels > > > for x time wrt the original FOR, reverses comes to an equally rapid > > > stop (wrt the original FOR) then returns the same way. Since SRT is > > > based upon v^2 effects (second order quantities) the directionality of > > > any asymmetry is lost in the expressions that quantify changes. > > > However, there is NO! paradox, either in nature, or SRT, once one > > > understands that limitation. The traveling twin, not the stay at home > > > twin will be physically younger. On a one-way trip however, we can't > > > say which one would be for an equal physical duration. That would > > > depend upon the speeds of both FOR relative to the CMBR... > > > Directionality does matter. > > > I don't follow your last sentence. Perhaps you mean, as Langevin put > > it, that a change of direction of speed does matter for the asymmetry. > > > Regards, > > Harald > > According to LR the rate at which time passes is related to the > absolute speed wrt to the aether frame. OK > For any round trip direction > is irrelevant, the total travel necessary to complete the circuit will > guarantee the total time will be relative to the delta velocity > between the systems. However!, if, for example you are traveling at > 600,000 Kps in some direction as measured by the CMBR Doppler OK, let's assume that the CMBR is at rest in the ether > and accelerate in a direction as to bring your speed to zero wrt to the > CMBR, LR predicts that your rate of time passage is now proceeding > faster than your stay at home twin. Yes, as measured with a truly stationary reference system (which thus records "absolute time"), the stay-at-home clock is slowed down by its motion at 600 Mm/s while your clock is in rest. > That twin is now receding from > you at 600,000 Kps and to EVER! hope to get back to him you must catch > up to him. Doing so requires you to go faster, longer, than ANY > outbound track, making the total elapsed time still less when you get > back than his. Right. > However, if you don't go back, LR say your time passes > faster. Note however that for the outbound one-way trip your clock > rate depends upon whether you're increasing or decreasing your speed > relative to the CMBR. Definitely direction dependent. Yes indeed, your "absolute" speed depends of course on the direction of your speed relative to a moving system. All clear now. ;-) Regards, Harald
From: harald on 7 Jul 2010 06:10 On Jul 7, 2:56 am, Paul Stowe <theaether...(a)gmail.com> wrote: > On Jul 6, 12:42 am, harald <h...(a)swissonline.ch> wrote: > > > > > On Jul 5, 8:26 pm, PaulStowe<theaether...(a)gmail.com> wrote: > > > > On Jul 5, 10:00 am, harald <h...(a)swissonline.ch> wrote: > > > > > On Jul 5, 2:46 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > > harald says... > > > > > > >Acceleration effects are not identified as gravitational fields in > > > > > >Newtonian physics (which, as you now know, you didn't know); and > > > > > >neither is that the case in SRT. In those theories acceleration is > > > > > >"absolute", and no gravitational fields are caused by acceleration. > > > > > > This is a topic for another discussion, but I'm talking about > > > > > "pseudo-gravitational" fields, which crop up in both SR and Newtonian > > > > > physics if you use accelerated coordinates. > > > > > We agree on "pseudo", while Einstein rejected that. > > > > > > >> I think it is because you have not made it very well. I still > > > > > >> have no idea what your point is. > > > > > > >Just study Einstein's paper carefully, > > > > > > I want to know what *YOUR* point is. State it in your own words. > > > > > I did, also in the part of my sentence that you exactly here snipped: > > > > > *you'll know what theory the paradox challenges*. > > > > > THAT (and only that) was my point: the clock paradox challenges the > > > > General PoR. > > > > > You certainly are aware that, despite Einstein's *suggestion* to that > > > > effect in the introduction of his 1905 paper, SRT is *not* based on > > > > the General PoR. > > > > Strictly speaking the 'domain' of the 'special' theory of relativity > > > is limited to inertial states. The original principle of relativity > > > as expressed in Einstein's 1905 work covered only that domain. > > > PS I overlooked the error you made here - a confusion that is often > > seen. As Eric points out, SRT is *not* (never was!) limited to > > inertial states. It's the same as for Newtonian mechanics. SRT uses > > Poincare's PoR, which refers to Newtonian reference systems; and we > > can switch reference system whenever we like, using the LT. Thus the > > twin problem is a trivial exercise in SRT. The twin paradox however > > was aiming Einstein's *General* PoR. > > > Harald > > 'I' never said Lorentzian Relativity (LR) or it renormalized cousin > (SR) is 'limited' to inertial frames. I said that, traditionally, the > concept commonly called 'special' relativity is such (special) because > it is considered the subset of general relativity limited to > unaccelerated frames of reference. Nature places no barrier or > boundary on relativity, human concepts do! This is why I refused to > call LR 'LET'. LR is no more limited to such artificial boundaries > than SR... > > Paul Stowe Hi Paul, Eric pointed at "the 'domain' of the 'special' theory of relativity is limited to inertial states", while its domain includes every kind of motion. Although he may be nitpicking, it's important to be sure that you didn't mean what you wrote - completely erroneous concepts have been built on subtle misunderstandings. Regards, Harald
From: Daryl McCullough on 7 Jul 2010 07:46 harald says... > >On Jul 6, 5:18=A0pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> harald says... >> >> >The twin scenario was presented by Langevin in 1911 to show that >> >physical acceleration is "absolute", even more so with SRT than with >> >Newton's mechanics. >> >> What does that mean? As I said, proper acceleration (as measured by >> an accelerometer) is absolute, but coordinate acceleration is certainly >> not. > >It means that you agree on that point with Langevin. Well, it's hard for me to believe that Einstein was unaware of the fact that an accelerometer can measure accelerations. >> Here's an analogy: A flat Euclidean plane has no notion of a preferred >> direction. Any direction is as good as any other. But it certainly has >> a notion of a *change* of direction. If you draw a path on the Euclidean >> plane, then you can unambiguously determine whether the line is >> straight or curved, because a straight line connecting two points is >> shorter than any curved line connecting the same two points. If you >> measure the lengths of two curves, you can determine which one is >> straight. > >Sorry but I can't resist pointing out the error of the above: a >straight trajectory relatively to an Euclidean plane is *only* >measured to be "straight" if that plane is part of what Einstein >called the "privileged" group of inertial "spaces". I think you are confused about this point. Euclidean space has an associated metric, which determines the lengths of paths. A straight line is defined relative to that metric as the path that minimizes the length between two points. It has nothing to do with any "privileged space". Having said that, we can define a special group of coordinate systems for the Euclidean plane---the Cartesian coordinate systems, via the requirement: A line is straight <=> It can be parametrized so that (d/ds)^2 x = (d/ds)^2 y = 0. Being a straight line is independent of coordinate system. The associated coordinate acceleration being zero is dependent on a choice of a special coordinate system. >> >"The laws of physics must be of such a nature that they apply to >> >systems of reference in any kind of motion". >> >As a result, physical acceleration is, according to Einstein's GRT, >> >*relative* - which is just the contrary of what Langevin argued based >> >on his "twins" example of SRT. >> >> As I said, proper acceleration is definitely *not* relative, but >> coordinate acceleration trivially *is*. But proper acceleration is >> measuring acceleration relative to *freefall*. > >Then we both disagree with Einstein; That's ridiculous. Einstein certainly knew that an accelerated observer feels "inertial forces", and an unaccelerated observer does not. Whatever was meant by his generalized principle of relativity, he certainly did *not* mean that what is now known as proper acceleration is undetectable. The modern way of looking at it is that "inertial forces" are felt whenever the observer is accelerating *relative* to freefall. Einstein originally thought of the equivalence principle differently: He thought that an object accelerating in a gravitational field felt two different kinds of forces: (1) inertial forces due to acceleration, and (2) gravitational forces. These two forces canceled in the case of freefall. >and this was the central point of the twin's paradox, >as criticism against Einstein's theory. I don't think you are correct. The twin paradox is not a serious criticism against anything that Einstein believed. >According to his theory, we are entitled to say that such an object >is *not* (properly) accelerating but that instead a "real" >gravitational field is induced through the universe which accelerates >all the *other* objects. I think you are confusing the physical content of Einstein's theory with the way he chose to describe it. Saying that fictitious forces due to acceleration are equivalent to gravitational forces doesn't mean that they are both real, it means that they are both *fictitious*. The modern view is that gravitational forces *are* inertial forces due to acceleration relative to freefall. Einstein specifically declined, in the dialog you pointed to, to make the distinction between "real" and "fictitious". He wrote: "In the first place I must point out that the distinction real - unreal is hardly helpful. In relation to K' the gravitational field "exists" in the same sense as any other physical entity that can only be defined with reference to a coordinate system, even though it is not present in relation to the system K." What I interpreted him to be saying is that the fictitious gravitational fields due to acceleration are real in the sense that they enter into the equations of motion in the accelerated coordinate system in exactly the same way that "real" gravitational forces do. The modern preference is to say that *neither* is real---the only real effects are ones that can be expressed in a coordinate-free manner. >> >It should not be surprising that this was not only very confusing for >> >bystanders (who already hardly understood the difference between the >> >two theories), but that it even looked like a contradiction >> >> I would like to hear any coherent explanation of why it looks like >> a contradiction. > >You spotted it yourself here above; but evidently, you refuse to >believe it I don't think you have correctly characterized what Einstein believed. He certainly did not believe that what is now called proper acceleration is undetectable. >just as you refuse to believe your own eyes when you >replace Einstein's "real" by "pseudo", assuming that it was just a >glitch. It wasn't. A lot of the confusion in physics discussions are because people are caught up in interpreting *words*, as if we are analyzing some holy text. I don't *care* what words Einstein, or anyone else, uses. His theories have physical content that are independent of the words used to describe them. Einstein certainly struggled over how best to convey his theories to the layman, and I can't say that he necessarily succeeded very well. >> The bare statement "The laws of physics must be of >> such a nature that they apply to systems of reference in any kind of >> motion" is not a contradiction---on the contrary, it is nearly a >> tautology. You can always write the laws of physics so that you >> can use an arbitrary coordinate system. > >If you think that a postulate of physics is a tautology, then probably >you misinterpret its meaning. Einstein didn't *realize* it was a tautology. He thought that the requirement that a physical theory be written in a way that had the same form in all coordinate systems would uniquely pin down the theory, or at least eliminate some candidate theories. He was wrong about that; the requirement of general covariance doesn't actually eliminate any candidate theories, since they can always be rewritten so as to be generally covariant. However, the principle of general covariance *does* serve as a heuristic in developing theories, if one attempts to come up with theories that are *simple* when written in a coordinate-independent way. General Relativity *is* simple for a theory of gravity when written in a coordinate-independent way, while Newtonian gravity is not. Newtonian gravity looks simple in inertial coordinates, but not in more generalized coordinate systems. >> To derive a paradox from the twin thought experiment, you >> need to reason something like this: >> >> 1. There exists two coordinate systems, C1 and C2, such that >> the path of the traveling twin, as described in C1, is the >> same as the path of the stay-at-home twin, as described in C2. >> >> 2. Therefore, the predicted age of the traveling twin, computed >> using C1, must be the same as the predicted age of the stay-at-home >> twin, computed using C2. > >No, you are thinking "inside the box" of SRT while this has nothing to >do with such an SRT problem. It is strongly related to your own >objection. I have read many of your posts, and I have *yet* to see you explain in what sense you think that the twin paradox is a consistency problem for any position that Einstein is likely to have believed. He certainly did *not* believe that (proper) acceleration is undetectable, so your interpretation of what he meant by the generalized principle of relativity is almost certainly wrong. If Einstein really believed what you seem to think he believed, then he certainly made a huge conceptual blunder. The dialog that Einstein makes it clear that he was aware of the differences between the two twins. The one twin's description of the journey has no "gravitational fields", while the other twin's description *does* have such fields. Clearly, their situations are not the same, and clearly Einstein understood the differences. His dialog, far from showing that his generalized principle of relativity is contradictory, shows that both twins compute the same answer for the question: How old is each twin when they get back together? The dialog as presented is a demonstration of the *consistency* of Einstein's theory, it is not a demonstration of inconsistency. I think you are very confused. -- Daryl McCullough Ithaca, NY
From: harald on 7 Jul 2010 09:45
On Jul 7, 1:46 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > harald says... > > > > >On Jul 6, 5:18=A0pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> harald says... > > >> >The twin scenario was presented by Langevin in 1911 to show that > >> >physical acceleration is "absolute", even more so with SRT than with > >> >Newton's mechanics. > > >> What does that mean? As I said, proper acceleration (as measured by > >> an accelerometer) is absolute, but coordinate acceleration is certainly > >> not. > > >It means that you agree on that point with Langevin. > > Well, it's hard for me to believe that Einstein was unaware of the > fact that an accelerometer can measure accelerations. Einstein was as aware as most physicists that an accelerometer does not distinguish between an acceleration and a gravitational field; however, he pushed that idea to the extreme. > >> Here's an analogy: A flat Euclidean plane has no notion of a preferred > >> direction. Any direction is as good as any other. But it certainly has > >> a notion of a *change* of direction. If you draw a path on the Euclidean > >> plane, then you can unambiguously determine whether the line is > >> straight or curved, because a straight line connecting two points is > >> shorter than any curved line connecting the same two points. If you > >> measure the lengths of two curves, you can determine which one is > >> straight. > > >Sorry but I can't resist pointing out the error of the above: a > >straight trajectory relatively to an Euclidean plane is *only* > >measured to be "straight" if that plane is part of what Einstein > >called the "privileged" group of inertial "spaces". > > I think you are confused about this point. Euclidean space has > an associated metric, which determines the lengths of paths. > A straight line is defined relative to that metric as the path > that minimizes the length between two points. It has nothing to > do with any "privileged space". I agree that a straight line relative to Euclidean space can be clearly defined. Perhaps I misunderstood that you meant with straight "path" a straight trajectory as defined in Newtonian mechanics and SRT. If you did not mean that, I don't know what you tried to say. > Having said that, we can define a special group of coordinate > systems for the Euclidean plane---the Cartesian coordinate systems, > via the requirement: > > A line is straight > <=> > It can be parametrized so that (d/ds)^2 x = (d/ds)^2 y = 0. > > Being a straight line is independent of coordinate system. With that claim I wonder if I truly misunderstood you; for in Newtonian mechanics as well as SRT, a path is very much dependent of the kind of coordinate system that you use. > The associated coordinate acceleration being zero is dependent > on a choice of a special coordinate system. Yes. But what was your point? > >> >"The laws of physics must be of such a nature that they apply to > >> >systems of reference in any kind of motion". > >> >As a result, physical acceleration is, according to Einstein's GRT, > >> >*relative* - which is just the contrary of what Langevin argued based > >> >on his "twins" example of SRT. > > >> As I said, proper acceleration is definitely *not* relative, but > >> coordinate acceleration trivially *is*. But proper acceleration is > >> measuring acceleration relative to *freefall*. > > >Then we both disagree with Einstein; > > That's ridiculous. Einstein certainly knew that an accelerated > observer feels "inertial forces", and an unaccelerated observer > does not. Sure he did. :-) > Whatever was meant by his generalized principle of relativity, You mean that you really did not know, and that you still don't - even after reading all his explanations?! > he certainly did *not* mean that what is now known > as proper acceleration is undetectable. Indeed. Perhaps it helps to say it in other words than he did: he meant *indistinguishable* from gravitation. > The modern way of looking at it is that "inertial forces" are > felt whenever the observer is accelerating *relative* to freefall. > Einstein originally thought of the equivalence principle differently: > He thought that an object accelerating in a gravitational field felt > two different kinds of forces: (1) inertial forces due to acceleration, > and (2) gravitational forces. These two forces canceled in the case > of freefall. ??? I strongly doubt that. Reference please! > >and this was the central point of the twin's paradox, > >as criticism against Einstein's theory. > > I don't think you are correct. The twin paradox is not a serious > criticism against anything that Einstein believed. So far it is not clear to you what the twin paradox criticized, and still you claim that it wasn't serious criticism against anything that Einstein believed. > >According to his theory, we are entitled to say that such an object > >is *not* (properly) accelerating but that instead a "real" > >gravitational field is induced through the universe which accelerates > >all the *other* objects. > > I think you are confusing the physical content of Einstein's theory > with the way he chose to describe it. The purpose with which you and I try to describe things here is to make the physical content of what think clear to the other. Do you seriously believe that Einstein tried to do the opposite, to hide the meaning of his words? > Saying that fictitious forces > due to acceleration are equivalent to gravitational forces doesn't > mean that they are both real, it means that they are both *fictitious*. Yup. > The modern view is that gravitational forces *are* inertial forces > due to acceleration relative to freefall. > > Einstein specifically declined, in the dialog you pointed to, to make > the distinction between "real" and "fictitious". He wrote: > > "In the first place I must point out that the distinction real - unreal is > hardly helpful. In relation to K' the gravitational field "exists" in the same > sense as any other physical entity that can only be defined with reference to a > coordinate system, even though it is not present in relation to the system K." > > What I interpreted him to be saying is that the fictitious gravitational > fields due to acceleration are real in the sense that they enter into > the equations of motion in the accelerated coordinate system in exactly > the same way that "real" gravitational forces do. The modern preference > is to say that *neither* is real---the only real effects are ones that > can be expressed in a coordinate-free manner. Good, we are making progress. :-) Einstein held that, as he put it, acceleration is "relative": according to his theory we may just as well claim that the traveler is *not* physically accelerated, contrary to Langevin's and your claim. He thought to solve the problem by saying that at the turnaround (according to the stay-at-home), the traveler may consider himself as remaining in place against an induced gravitational field that appears. > >> >It should not be surprising that this was not only very confusing for > >> >bystanders (who already hardly understood the difference between the > >> >two theories), but that it even looked like a contradiction > > >> I would like to hear any coherent explanation of why it looks like > >> a contradiction. > > >You spotted it yourself here above; but evidently, you refuse to > >believe it > > I don't think you have correctly characterized what Einstein > believed. He certainly did not believe that what is now called > proper acceleration is undetectable. He did not believe that no effect is detected, nor did he or I suggest that. > >just as you refuse to believe your own eyes when you > >replace Einstein's "real" by "pseudo", assuming that it was just a > >glitch. It wasn't. > > A lot of the confusion in physics discussions are because people are > caught up in interpreting *words*, as if we are analyzing some holy > text. I don't *care* what words Einstein, or anyone else, uses. In that case we have nothing to discuss, nor can you really discuss the clock paradox: it is foremost concerned with physical concepts that had been expressed with words as well as with equations. > His theories have physical content that are independent of the words used > to describe them. Without definitions of the variables and their fields of application, there is just mathematics without physical meaning. > Einstein certainly struggled over how best to convey his theories > to the layman, and I can't say that he necessarily succeeded very > well. His papers such as this one were conveyed to scientists - only they read physics journals. His popular 1920 book however was directed at laymen, and I now think that he did a reasonable job there (I changed my mind on that). > >> The bare statement "The laws of physics must be of > >> such a nature that they apply to systems of reference in any kind of > >> motion" is not a contradiction---on the contrary, it is nearly a > >> tautology. You can always write the laws of physics so that you > >> can use an arbitrary coordinate system. > > >If you think that a postulate of physics is a tautology, then probably > >you misinterpret its meaning. > > Einstein didn't *realize* it was a tautology. He thought that the > requirement that a physical theory be written in a way that had the > same form in all coordinate systems would uniquely pin down the theory, > or at least eliminate some candidate theories. There is also the key thought "simplest form" which is not written there but implied: it is included in his special relativity definition, of which the GPoR is an extension. Anyway, a theory is that what its author says it is. Otherwise for example, I could claim that Ken Seto's Mechanics is right, and that he doesn't know his theory well enough yet, but we will do that for him! > He was wrong about that; > the requirement of general covariance doesn't actually eliminate any > candidate theories, since they can always be rewritten so as to be > generally covariant. > > However, the principle of general covariance *does* serve as a heuristic > in developing theories, if one attempts to come up with theories that are > *simple* when written in a coordinate-independent way. General Relativity > *is* simple for a theory of gravity when written in a coordinate-independent > way, while Newtonian gravity is not. Newtonian gravity looks simple in > inertial coordinates, but not in more generalized coordinate systems. Good! > >> To derive a paradox from the twin thought experiment, you > >> need to reason something like this: > > >> 1. There exists two coordinate systems, C1 and C2, such that > >> the path of the traveling twin, as described in C1, is the > >> same as the path of the stay-at-home twin, as described in C2. > > >> 2. Therefore, the predicted age of the traveling twin, computed > >> using C1, must be the same as the predicted age of the stay-at-home > >> twin, computed using C2. > > >No, you are thinking "inside the box" of SRT while this has nothing to > >do with such an SRT problem. It is strongly related to your own > >objection. > > I have read many of your posts, and I have *yet* to see you explain > in what sense you think that the twin paradox is a consistency problem > for any position that Einstein is likely to have believed. Einstein explains the cause of the distrust or criticism clearly enough; nothing that I add can make clearer what the issue was. Now, he believed (or pretended) to have solved the paradox to satisfaction (which doesn't UN-make it a paradox, as you seem to think). However, his solution was never published in another physics journal. Dingle rejected it, but didn't manage to clearly express why. The physics FAQ rejects it, but without presenting strong arguments. Nevertheless his solution is (was) known to be wrong, as I discussed here: http://groups.google.com/group/sci.physics.foundations/msg/68cd1c181f8191d2 [..] > The dialog that Einstein makes it clear that he was aware of the differences > between the two twins. The one twin's description of the journey has no > "gravitational fields", while the other twin's description *does* have > such fields. Clearly, their situations are not the same, and clearly > Einstein understood the differences. Yes - at first sight he really "solved the paradox". :-) > His dialog, far from showing that > his generalized principle of relativity is contradictory, shows that both > twins compute the same answer for the question: How old is each twin when > they get back together? > > The dialog as presented is a demonstration of the *consistency* of > Einstein's theory, it is not a demonstration of inconsistency. That was its purpose... > I think you are very confused. I think that you forgot to check the meaning of "paradox". Harald |