Alternate derivation of Lorentz Trans.

[timetraveldude]

Has anybody come up with a way to derive the LT not based on the constantcy of the speed of light in all inertial reference frames?

 

[HallsofIvy]

WHY would one want to? The constancy of the speed of light was the experimental data that led to the Lorenz transformation. If the speed of light was not constant why would one want or need the Lorenz transformation?

 

[Fredrik]

I'm not sure what you (timetraveldude) have in mind here. The Lorentz tranformation is equivalent to the statement that the speed of light is the same to all inertial observers. I don't think the question really makes sense. Any postulate that you can use as a starting point for a derivation of the Lorentz tranformation will include that stuff about the speed or light, whether it's apparent or not.

 

[arildno]

I think Lorentz himself had another way...
If I remember correctly, he made some ad-hoc assumption that objects contracted in the direction of motion.
This is of course only of historical interest; I think Lorentz attempted to account for the Michelson-Morley experiment while simultaneously saving the ether hypothesis.

Einstein, of course, presented a far more elegant, and convincing, chain of argument for the necessity of the Lorentz transformation.

 

[timetraveldude]

WHY would one want to? The constancy of the speed of light was the experimental data that led to the Lorenz transformation. If the speed of light was not constant why would one want or need the Lorenz transformation?

I am not sure if you are correct. As I understand it, Maxwell's equations were not invariant under the Galilean transformations. Einstein felt that the laws are physics should be the same in all inertial reference frames. So either the Galilean transformations were wrong or Maxwell's equations.

 

[timetraveldude]

Simultaneity post

Maybe you should see my other post in this forum. The relativity of simultaneity argument is based on light but if you use sound instead of light all inertial observers will agree whether two events were simultaneous. On the other hand if you can derive the Lorentz transformations without using the 2nd postulate (i.e. through a different means) then the relativity of simultaneity is preserved.

 

[timetraveldude]

I'm not sure what you (timetraveldude) have in mind here. The Lorentz tranformation is equivalent to the statement that the speed of light is the same to all inertial observers. I don't think the question really makes sense. Any postulate that you can use as a starting point for a derivation of the Lorentz tranformation will include that stuff about the speed or light, whether it's apparent or not.

This is not true. The Lorentz transformations only describe distance and time transformations between coordinate systems if there is a universal speed limit.

The impression I get from the people here is that they have memorized the details but do not know how to think critically. I have proved that you can derive the Lorentz transformations without utilizing the 2nd postulate of SR. Am I the only one? There was a paper in 1972 that did this exact thing but using a different method from mine. The idea is basically that if you are teaching a mechanics course and want to incorporate SR without reference to electro-dynamics you need a different way of introducing the Lorentz transformations.

 

[jcsd]

The problem is for you argument to have any validity you need something that travels on a null worldline, light does whereas sound does not.

 

[Fredrik]

This is not true. The Lorentz transformations only describe distance and time transformations between coordinate systems if there is a universal speed limit.

OK. When I think of "the speed of light" I don't even think of light. To me "the speed of light" is just a name that represents the universal speed. That's why I thought your suggestion sounded so strange. But OK, you don't want to do a derivation that doesn't involve a universal speed, you want to do a derivation that doesn't involve light (or anything else from the classical or quantum theory of electrodynamics). That's a different story.

I've seen a derivation like that once, or at least a part of it. Unfortunately I don't remember who wrote it. Their idea was to assume nothing at all about the properties of space, except rotational and translational invariance, and try to determine the most general rule for addition of velocities. The result they got was of course not the non-relativistic "u+v" but a relativistic-looking formula (u+v)/(1+Kuv), where K was a non-negative real number that couldn't be determined from the postulates they had started with.

The constant K can of course be identified with 1/c², but there's no need to do that just yet. Instead, we can use the velocity addition formula as the starting point of a derivation of the rest of special relativity, including the Lorentz transformation.

 

[timetraveldude]

The problem is for you argument to have any validity you need something that travels on a null worldline, light does whereas sound does not.

My argument is perfectly valid. Again you are using as evidence what I am questioning. If you want to remain in the realm of logical thinkers you need to understand this is not acceptable.

 

[jcsd]

My argument is perfectly valid. Again you are using as evidence what I am questioning. If you want to remain in the realm of logical thinkers you need to understand this is not acceptable.

The problem is that sound is totally irrelevant to considerations of simultaneity in relativity. Special relativity is inertanally self-consistent so any questioning of well-know results such the fialure of simulatenity at distance cannot come from poniticating it must come from experimental evidmnece, yet you offer none.

 

[cragwolf]

To answer the OP, yes, there are many alternative derivations of the Lorentz Transformations, some of which do not assume the constancy of the speed of light. I'll just list a few that don't assume the constancy of the speed of light:

Y.P.Terletskii, "Paradoxes in the Theory of Relativity", Plenum Press, New York, 1968, P17
R.Weinstock, "New Approach to Special Relativity", Am. J. Phys. 33 640-645 (1965)
A.R.Lee and T.M.Kalotas, "Lorentz Transformation from the First Postulate", Am. J. Phys. 43 434-437 (1975)
J.M.Levy-Leblond, "One more Derivation of the Lorentz Transformation", Am. J. Phys. 44 271-277 (1976)
A.Sen, "How Galileo could have derived the Special Theory of Relativity", Am. J. Phys. 62 157-162 (1994)
J.H.Field, "Space-Time Exchange Invariance: Special Relativity as a Symmetry Principle", [http://arxiv.org/physics/0012011 ]

 

[timetraveldude]

To answer the OP, yes, there are many alternative derivations of the Lorentz Transformations, some of which do not assume the constancy of the speed of light. I'll just list a few that don't assume the constancy of the speed of light:

Y.P.Terletskii, "Paradoxes in the Theory of Relativity", Plenum Press, New York, 1968, P17
R.Weinstock, "New Approach to Special Relativity", Am. J. Phys. 33 640-645 (1965)
A.R.Lee and T.M.Kalotas, "Lorentz Transformation from the First Postulate", Am. J. Phys. 43 434-437 (1975)
J.M.Levy-Leblond, "One more Derivation of the Lorentz Transformation", Am. J. Phys. 44 271-277 (1976)
A.Sen, "How Galileo could have derived the Special Theory of Relativity", Am. J. Phys. 62 157-162 (1994)
J.H.Field, "Space-Time Exchange Invariance: Special Relativity as a Symmetry Principle", [http://arxiv.org/physics/0012011 ]

Thank you. You are the first person I have met in this thread who actually thinks.

 

[Garth]

Have you tried K calculus? Developed by Milne in his Kinematic cosmology in the 1930's and used by d'Inverno in "Introducing Einstein's Relativity".
Garth

 

[timetraveldude]

Have you tried K calculus? Developed by Milne in his Kinematic cosmology in the 1930's and used by d'Inverno in "Introducing Einstein's Relativity".
Garth

Thanks. WOW! Two useful posts in a row. This is a violation of statistics.

It is amazing that the people who make the most useless posts are the so called mentors.

 

[Hurkyl]

Just to make sure you realize, these approaches will derive the constancy of the speed of light.

 

[quantumdude]

Fredrik: The Lorentz tranformation is equivalent to the statement that the speed of light is the same to all inertial observers.

timetraveldude: This is not true.

Yes, it is true.

Einstein started with the constant speed of light postulate and Maxwell's equations. Requiring the latter to be covariant, he derived the Lorentz transformation. But you could just as easily start from the Lorentz transformation and derive from that the speed of light postulate, and of course the covariance of Maxwell's equations.

It is amazing that the people who make the most useless posts are the so called mentors.

Lose the attitude.

 

[quantumdude]

Never mind folks. Timetraveldude is just another alias for our beloved protonman and tenzin.

He won't be joining us in this thread anymore.

 

[fixizrox]

Yes, it is true.

Einstein started with the constant speed of light postulate and Maxwell's equations. Requiring the latter to be covariant, he derived the Lorentz transformation. But you could just as easily start from the Lorentz transformation and derive from that the speed of light postulate, and of course the covariance of Maxwell's equations.

The lorentz transformations say nothing about the speed of light being the same in all inertial reference frames. I derived the LT without any reference at all to light.

 

[quantumdude]

The lorentz transformations say nothing about the speed of light being the same in all inertial reference frames.

Of course they do. As has been said repeatedly, you can derive the speed of light postulate from the LT.

I derived the LT without any reference at all to light.

Good for you. Now use it to derive the relativistic velocity addition law, and then then Einstein's speed of light postulate.

 

[quantumdude]

Has anybody come up with a way to derive the LT not based on the constantcy of the speed of light in all inertial reference frames?

I derived the LT without any reference at all to light.

Looks like you had the answer to your own question all along.

It makes one wonder why you bothered to ask it here.

 

[Fredrik]

Of course they do. As has been said repeatedly, you can derive the speed of light postulate from the LT.

I think timetraveldude knows that. I think he understands that the Lorentz transformations imply the existence of a velocity that's the same to all inertial observers. What he's trying to make a big deal of here, is that there's nothing in the Lorentz transformations that explicitly mentions light. Sure, they mention the speed of light (the universal velocity), but they don't say that this is the same thing as the speed of light (photons/electromagnetic waves). That's why he won't accept that the Lorentz transformation is equivalent to the speed of light postulate.

 

[Fredrik]

Timetraveldude, if you have derived the Lorentz transformations without using light, that's not really a big deal. As I mentioned before, the most general velocity addition law that's consistent with rotational and translational invariance has been shown to be (u+v)/(1+Kuv), where K is just a constant to be determined later.

If K is not 0, we can define a constant c that has dimensions of velocity: c²=1/K. The velocity addition formula can be used to derive the Lorentz tranformations, and this will lead us to the idea of Minkowski space. Now, if we try to construct a theory of light that's consistent with the idea that Minkowski space is an accurate representation of space and time, we will eventually end up with QED, Maxwell's equations, and the identification c = the speed of light.

 

[Haelfix]

Umm, lorentz invariance is *required* for Maxwells equations to be self consistent. Now, the identification is that the universal speed is precisely C, the somewhat arbitrary constant that appears in those equations.

You are free of course to pick a higher value.. call it vmax, but then you contradict experiment. So experiment ultimately fixes the consistency of the theory.

 

[Fredrik]

I agree. I just think it's interesting that even if you've never heard of Maxwell's equations, and have no idea what the speed of light is, it's still possible to realize that SR (with some universal velocity) is at least a possibility.

 

[Stingray]

I agree. I just think it's interesting that even if you've never heard of Maxwell's equations, and have no idea what the speed of light is, it's still possible to realize that SR (with some universal velocity) is at least a possibility.

Interestingly, you can go even further in that "what if" scenario. If you write down Newtonian physics (including Newtonian gravity, but not electromagnetism) in a spacetime-like geometric form, you find that it is very natural to add a 1-parameter extension to the theory. If that parameter is set equal to 1/c^2, you have general relativity!

 

[fizixrox]

Timetraveldude, if you have derived the Lorentz transformations without using light, that's not really a big deal. As I mentioned before, the most general velocity addition law that's consistent with rotational and translational invariance has been shown to be (u+v)/(1+Kuv), where K is just a constant to be determined later.

If K is not 0, we can define a constant c that has dimensions of velocity: c²=1/K. The velocity addition formula can be used to derive the Lorentz tranformations, and this will lead us to the idea of Minkowski space. Now, if we try to construct a theory of light that's consistent with the idea that Minkowski space is an accurate representation of space and time, we will eventually end up with QED, Maxwell's equations, and the identification c = the speed of light.

Fredrik,

I assumed that the idea that you can derive an equation for a universal speed limit independent of the constantcy of the speed of light was not original. But I teach SR and have come to know it a bit. It often occurred to me to think of the 'c' in the Lorentz transformations as a universal speed limit and no t the speed of light. This was the result of contemplating some of the pedagogical derivations of time dilation and Lorentz contraction. They were not rigorous enough to be conclusive.

I then did a literature review and found that this idea has been taken up and written about in academic journals.

I would be curious to know if you have any references to "the most general velocity addition law that's consistent with rotational and translational invariance has been shown to be (u+v)/(1+Kuv)" as you mentioned above.

Congrads on an excellent post.

 

[Project]

Has anybody come up with a way to derive the LT not based on the constantcy of the speed of light in all inertial reference frames?

There are many derivations of the Lorentz transformation that do not use Einstein's second postulate. See http://www.everythingimportant.org/relativity/special.pdf for example.

 

[Project]

There are many derivations of the Lorentz transformation that do not use Einstein's second postulate. See http://www.everythingimportant.org/relativity/special.pdf for example.

Also see http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000043000005000434000001

 

[zwicky]

Just to make sure you realize, these approaches will derive the constancy of the speed of light.

I think these approaches will derive the existence of a fundamental velocity scale instead. Experimentally equal to the speed of light in vacuo for not to short wave length.

 

[PatrickPowers]

This is not true. The Lorentz transformations only describe distance and time transformations between coordinate systems if there is a universal speed limit.

The impression I get from the people here is that they have memorized the details but do not know how to think critically. I have proved that you can derive the Lorentz transformations without utilizing the 2nd postulate of SR. Am I the only one? There was a paper in 1972 that did this exact thing but using a different method from mine. The idea is basically that if you are teaching a mechanics course and want to incorporate SR without reference to electro-dynamics you need a different way of introducing the Lorentz transformations.

Lorentz derived the Lorentz transforms without assuming the speed of light was the same in every frame of reference. He thought that there was one frame of reference where it was true and in every other frame there was distortion of space and time that gave a sort of delusion that the speed of light was the same in that frame.

I am told that the math is exactly the same as the Einstein math, so there is no way to distinguish these same two interpretations. Lorentz transforms work perfectly well when applied to sonons, wave packets that are limited to the speed of sound and really do have a preferred frame of reference.

The reason the Einstein interpretation is preferred is that there is no reason to prefer any particular frame over any other and the preferred frame is impossible to identify. So there is no reason to believe that it exists and it seems an unnecessary, arbitrary complication.

 

[harrylin]

[..]
The reason the Einstein interpretation is preferred is that there is no reason to prefer any particular frame over any other and the preferred frame is impossible to identify. So there is no reason to believe that it exists and it seems an unnecessary, arbitrary complication.

It's an alternative way to derive the invariance of the (measured) speed of light.
On a side note: I think that it's a misnomer to call such a presumed cause that corresponds to a frame that is not preferred over other frames for doing physics, a "preferred frame". And evidently Newton and Lorentz disagreed with such Machian (positivistic) reasoning... I read somewhere that following that same line of reasoning, Mach also held that there was no reason to believe that atoms exists because they could not directly be measured.
This could be a new discussion topic and I now found more about it here:
http://en.wikipedia.org/wiki/Ernst_Mach

 

[Fredrik]

This thread had been dead for more than 7 years. I wonder if it's a new necropost record.

 

[harrylin]

This thread had been dead for more than 7 years. I wonder if it's a new necropost record.

Wow! OK then I guess that this thread can go back to its old state.

 

[keithR]

Just been looking over some old physics books of my student days.
In Special Relativity, Rindler,W. 1960, section 3 describe how Lorentz and Fitzgerald derived L-F length Contraction formula based on an ether, which is familiar from Einstein's Relativity theory. The same section in Rindler also shows how Lorenz derived the time-dilation (the same one as in Relativity), based on either theory and the constancy of observed light speed. The derivations are easy...much easier than the derivations I have seen in Relativity theory.
Exercise 12 on p24 Rindler says "as far as it goes, the Lorentz theory (with ether) is parallel to the Einstein theory (with no ether, but with the relativity principle).

But, on further reading, I find I still do not understand the purported resolutions of the twin paradox.
The twin scenario is: (i) two twins move apart at a constant speed relative to each other.
(ii) By relativity they both observe, by em-signals, that the other seems to be ageing faster than themselves.
(iii) One of the twins misses the other, turns around, and returns to their twin at the same relative velocity.
When they meet again the one who turned around has aged less.
The turnaround, which we may assume instantaneous, is the only difference between the twins, since they are both in inertial frames during the rest of their separation.

The derivation of the age difference usually considers one twin stationary on earth, and the other moving away. In moving away, the L-transformations shorten lengths and dilate times of the mover. The time-space measurement coordinate frame of the mover tighten up on the null/light cone.
But, why is this reasonable, relativistically speaking? In all inertial frames the speed of light is the same, so why is one coordinate measurement frame more lightlike than any other?
Some (including Mach?) justify the asymmetry by invoking the distant universe towards which the mover moves, and the stayer does not. This seems to be invoking a kind of "ether" in terms of the distant stars. But we know they are not fixed, but moving and accelerating away! Not convincing.
Some explanations note the red or blue shift observed in light from the partner, and this does seem to correspond to differing relative rate of aging behaviour. But, why do such considerations overcome the inertial frame equivalence of the two twins on the bulk of their journeys?
Sorry if these are all well warn and ignorant considerations...as I am sure they are. A reference to a really clear and solid resolution of the twin paradox would be appreciated.

 

[ghwellsjr]

Just been looking over some old physics books of my student days.
In Special Relativity, Rindler,W. 1960, section 3 describe how Lorentz and Fitzgerald derived L-F length Contraction formula based on an ether, which is familiar from Einstein's Relativity theory. The same section in Rindler also shows how Lorenz derived the time-dilation (the same one as in Relativity), based on either theory and the constancy of observed light speed. The derivations are easy...much easier than the derivations I have seen in Relativity theory.
Exercise 12 on p24 Rindler says "as far as it goes, the Lorentz theory (with ether) is parallel to the Einstein theory (with no ether, but with the relativity principle).

Actually, both LET and SR are based on the same first postulate, the principle of relativity, but they have different second postulates. LET's second postulate is that light propagates at c only in a single frame, the rest state of the immovable ether, whereas SR's second postulate is that light propagates at c in any inertial frame you want to pick. Note that the first postulate concerns things that can be measured and observed, like measuring the round-trip speed of light, while the second postulate concerns that which we can neither measure nor observe, the propagation of light.

But, on further reading, I find I still do not understand the purported resolutions of the twin paradox.
The twin scenario is: (i) two twins move apart at a constant speed relative to each other.
(ii) By relativity they both observe, by em-signals, that the other seems to be ageing faster than themselves.

You've got this backwards, they each observe the other ageing more slowly than themselves while they are traveling away from each other.

(iii) One of the twins misses the other, turns around, and returns to their twin at the same relative velocity.

And as soon as he turns around and travels back he will observe the twin that remained stationary as ageing faster than himself and this will continue for the entire trip back. When he gets back he will see that the sum of the equal intervals of observed slow ageing and fast ageing adds up to the actual amount that the stationary twin aged during the trip.

Now what does the stationary twin observe of the traveling twin? Well, he's not going to see the traveling twin turn around until long after he actually turns around because he has to wait for the em (light) signal or image of the turn-around to propagate back to him and until it reaches him, he will continue to see the traveling twin ageing at a slower rate than himself. Eventually, he will see his twin turn around and at that point, he will see him age faster than himself. So because he observes the traveling twin aging at a slower rate during most of the trip, he agrees that the traveling twin actually did age less than himself at the end of the trip.

To summarize: the traveling twin sees the stationary twin age slow for 1/2 the time and fast for 1/2 the time while the stationary twin sees the traveling twin age slow for, say, 3/4 of the time and fast for 1/4 of the time and this is why there is an imbalance in their ages when they reunite.

When they meet again the one who turned around has aged less.

Yes, and hopefully it makes sense from the viewpoint of what each twin actually sees and observes but the acceleration at the turn around isn't what made it happen.

The turnaround, which we may assume instantaneous, is the only difference between the twins, since they are both in inertial frames during the rest of their separation.

This isn't quite an accurate statement for two reasons:

1) Everybody and everything is in all inertial frames. Usually when we say that a twin is in his inertial frame, we mean he is at rest in that inertial frame, but his other twin is also in the inertial frame, it's just that he is moving.

2) The stationary twin is at rest in a single inertial frame during the entire scenario but this isn't true for the traveling twin. The traveling twin is at rest in one inertial frame during the outbound portion of the trip and then he is at rest in a different inertial frame during the inbound portion of the trip. So now we have three inertial frames to consider but we could also consider any other frame and they will all give the same answer as to amount that each twin aged during the scenario.

The derivation of the age difference usually considers one twin stationary on earth, and the other moving away. In moving away, the L-transformations shorten lengths and dilate times of the mover.

And this is the easiest frame in which to analyze the scenario: moving twin's clock runs slower, therefore he ages less than the stationary twin when they reunite. Problem solved. No need to consider any other frame.

But if you want, you can transform all the relevant events in this first frame into one of the other frames to see how it looks there. What you will find is that while the traveling twin is at rest during one half of the trip, the "stationary" twin ages less, but during the other half of the trip, the traveling twin has to travel at a higher speed than the "stationary" twin and so he experiences even more time dilation and ends up younger.

The time-space measurement coordinate frame of the mover tighten up on the null/light cone.
But, why is this reasonable, relativistically speaking? In all inertial frames the speed of light is the same, so why is one coordinate measurement frame more lightlike than any other?
Some (including Mach?) justify the asymmetry by invoking the distant universe towards which the mover moves, and the stayer does not. This seems to be invoking a kind of "ether" in terms of the distant stars. But we know they are not fixed, but moving and accelerating away! Not convincing.
Some explanations note the red or blue shift observed in light from the partner, and this does seem to correspond to differing relative rate of aging behaviour. But, why do such considerations overcome the inertial frame equivalence of the two twins on the bulk of their journeys?
Sorry if these are all well warn and ignorant considerations...as I am sure they are. A reference to a really clear and solid resolution of the twin paradox would be appreciated.

I cannot relate to these last comments of yours. Hopefully, I have steered you in the right direction to be able to understand the resolution of the twin paradox. If not, ask for clarification on any points that are still unclear.

 

[keithR]

Thanks ghwellsjr, for your careful explanations, relating to the twin paradox.
Almost all of what you have explained I accept, and in fact your explanations are the same as those I have read elsewhere as I have been trying to find clarification.
However, I still remain with the same unease that I don't really "see" the complete adequacy of the explanations in answering my specific misgivings. Some of these you have not commented on, and others you say you do not relate to, presumably because my words are non-sensical within the SR context. Let me focus on some of these specific misgivings.
Sorry if I just continue to grovel in my own mess.

1. First, I would prefer to set aside accounts of the twin aging process in terms of what how they see each other aging. I accept the correctness of the account, but would prefer to deal with how a person actually is aging, rather than how they are seen to be aging.
(i) possibly, you will be unhappy with this preference, since simultaneity in SR is defined in terms of such light/em signals.
(ii) but I do feel the "observer-based" account is misleading.
Reasons:
(a) It does not matter if the velocity is away or towards the "fixed" observer since the effects depend on v2. The time units, or the ticks of the clock in the moving inertial frame are longer than they are in the fixed frame by a factor of β(1+v2)1/2where β=(1-v^2)-1/2 and c=1.
(b) I get this from the usual twins diagram, with the "y" axis being t and the "x" axis being x for the stationary frame of reference. Hence (0,1) refers to the stationary x-origin, after unit time. (0',1') refers to the moving x'-origin, when t'=1. Calculate (x,t) corresponding to (0',1') using the inverse LT (i.e. t=β(t'+vx') , x=β(x'+vt') to get β(v,1) which defines the direction fo the time axis in terms of the stationary frame of ref. It is alond the direction of movemement.
Taking a numerical example which I have seen used in the twin discussion elsewhere, let v=3/5, so that the rocket travels out 3 light years during 5 years in the stationary time frame.
Then β=5/4, which is the time-dilation factor, yielding a moving flight time of 4 years.

β(1+v^2)1/2=√34/4, and since the "length" of the space-time trajectory, in the stationary frame is √34. So again, it is confirmed that the time experienced on the outgoing rocket is 4 years.
The same applies on the return journey.
What is seem through mutual light based tv's, or the spectrum shifts involved, is interesting, but to my mind, confuses the issue rather.

Similarly to above, we find that the space-like (1',0') in the rocket, is β(1,v) in the stationary frame. This, and the β(v,1) for the rocket time-like (0',1') are what I meant when in my last paragraph previously, when I said the time and space basis vectors close up (symetrically) on the null cone as v→c.
One of my previous comments on this issue, which you did not relate to, was as to how this squeezing up of moving (space,time) frames to the light cone, with increasing speed, consistent with the relativity principle: in any frame, the speed of light is still c, so why is squeezing up with speed possible? This is anther form of the "twin-paradox" I think.
Penrose (fabric of Reality p409, in his discussion of the "paradox") mentions this squeezing up of the time-like and space-like (orthoganal) axis to the null cone as v→c.
But I don't think penrose's use of the triangle inequality really addresses the paradox, since he just says with the longest side of the space-time triangle (in proper time) is in the frame that is stationary. But how do we say what frame is stationary!?

(c) All the above, right or wrong, just says I do accept the correctness of the age reduction in the twin example, from the LT equations, and form space time diagrams.

Explaining the life shortening in another way does not relieve me of the feeling of paradox.

The paradox only arises when we realize that the results should be symmetric between the two frames, as inertial frames, during the both the steady motion periods between the departure and turn-round points.
If you do not see this is paradoxical, you do not see the twin paradox.
Rindler, page 30 says: "If the two clocks were the whole content of the universe this would indeed be a paradox", so, he sees the paradox. Do you?
Rindler then says, as you do, one of the clocks has jumped inertial frames, so there is a lack of symmetry between the two twins. But, what has this got to do with the issue, if we can say that we assume the turn-around can be considered immediate. Rindler mentions "preferred frames"!. This sounds like "Ether" in another form.
As I mentioned in my first post, Mach's answer to the paradox was the "distant fixed stars". Another illusionary preferred frame.

Sorry, I don't know to quote your message points, ghwellsjr.
Thanks for your stimulating discussion.

 

[ghwellsjr]

I'm not sure which paradox you are referring to.

1) Are you talking about the "paradox" of two observers traveling with respect to each other, neither one changing their speed or direction and they each see the other ones clock running slower than their own?

2) Or are you talking about the "Twin Paradox" where both twins always see the other ones clock running slower than their own and by the same amount, yet when they return, one of them has aged less than the other one?

Virtually all paradoxes in Special Relativity come about as a result of conflating the parameters determined by two different inertial Frames of Reference or by not correctly applying the Lorentz Transformation to get from one inertial Frame of Reference to another one. You said:

The paradox only arises when we realize that the results should be symmetric between the two frames, as inertial frames, during the both the steady motion periods between the departure and turn-round points.

But I can't tell whether the two frames you are talking about here are the rest frame of the stationary twin and the rest frame of the traveling twin during the outbound portion of the trip OR if you are talking only about the rest frame of the traveling twin during the outbound portion of the trip and the rest frame of the traveling twin during the inbound portion of the trip OR if you are talking about the traveling twin being in a single inertial rest frame during the entire trip (except for the negligible instant of turn-around) and the rest frame of the stationary twin.

Please tell me which of the these three options you are talking about in the above quote with regard to the two frames (or maybe some other option).

 

[harrylin]

[..] why do such considerations overcome the inertial frame equivalence of the two twins on the bulk of their journeys?
Sorry if these are all well warn and ignorant considerations...as I am sure they are. A reference to a really clear and solid resolution of the twin paradox would be appreciated.

Hi keith,

The first clear and solid resolution of the twin paradox was perhaps in the first discussion of that scenario (without any "paradox"), here:
http://en.wikisource.org/wiki/The_Evolution_of_Space_and_Time

It's very long-winding, but in this context mainly the sections of p.47 + p.50-52 are of interest.