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## Transformation Vs. Physical Law

 Quote by universal_101 This must be a new physics, since what you are suggesting is that, difference in the ages of the Twins after the trip, is independent of their relative velocity during the trip.
No, he is suggesting (correctly) that the age of each twin is independent of their relative velocity. I could shoot one of the twins dead while the rocket is in flight and the aging of the other twin would be not be affected in the least.

Each twin's age depends only on the path that twin takes through space-time. So I compute the age of twin one at the reunion by looking at twin one's path through space-time; twin two and the relative velocity don't enter into this computation at all. Then I compute the age of twin two at the reunion by looking at twin two's path through space-time; twin one and the relative velocity don't enter into this calculation at all.

And now that I know their ages at the moment of reunion... I know what the difference in their ages is.

 Quote by Nugatory Each twin's age depends only on the path that twin takes through space-time.
That path is calculated/based/defined by their relative velocity.

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 Quote by universal_101 That path is calculated/based/defined by their relative velocity.
It is not.

If it were, I wouldn't be able to calculate the aging of anyone who didn't have a twin... Surely you aren't suggesting that an only child suspended in the midst of empty space (so that's there's no relative velocity because there's nothing to be relative to) won't age, or that I can't calculate the aging? All we need to do is to look at his wristwatch.

But if I can calculate the aging of the isolated only child without considering his velocity relative to a twin that he doesn't even have.... Clearly I can do the same for either twin, just by ignoring the other twin and the relative velocity between them.

 Quote by atyy Exactly. However, if one is able to specify a coordinate system, then one can use the coordinates to describe events. In special relativity as in geometry, both the coordinate-system invariant and the coordinate-system descriptions are useful, with the proviso that when using the latter the coordinate system must be specified.
Agreed, but the invariant events, cannot depend on which coordinate system we use, and it is this use of the coordinate system which I'm questioning.

That is, how many events(differential age of the twins) would take place is concluded by using the gamma factor of a transformation.

Now, it is this use of the gamma factor to produce difference in the ages of the Twins, make it necessary to have real Length contraction phenomena, to which we don't have any experimental support.

Instead, it is the Time Dilation of unstable particles(using LT) which is directly challenged by the recent new findings, of dependence of the nuclear decay rates on the Earth-Sun distance, which includes beta decaying particles(Muons).

 Quote by Nugatory It is not. If it were, I wouldn't be able to calculate the aging of anyone who didn't have a twin... Surely you aren't suggesting that an only child suspended in the midst of empty space (so that's there's no relative velocity because there's nothing to be relative to) won't age, or that I can't calculate the aging? All we need to do is to look at his wristwatch. But if I can calculate the aging of the isolated only child without considering his velocity relative to a twin that he doesn't even have.... Clearly I can do the same for either twin, just by ignoring the other twin and the relative velocity between them.
To calculate the difference in the age of two twins, we need two twins to compare

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 Quote by universal_101 Agreed, but the invariant events, cannot depend on which coordinate system we use, and it is this use of the coordinate system which I'm questioning. That is, how many events(differential age of the twins) would take place is concluded by using the gamma factor of a transformation. Now, it is this use of the gamma factor to produce difference in the ages of the Twins, make it necessary to have real Length contraction phenomena, to which we don't have any experimental support. Instead, it is the Time Dilation of unstable particles(using LT) which is directly challenged by the recent new findings, of dependence of the nuclear decay rates on the Earth-Sun distance, which includes beta decaying particles(Muons).
Do you believe in energy?

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Quote by universal_101
 Quote by mentz114 The clock time of a twin depends only on their *own* worldline. It is completely irrelevant what the other twin is doing. Relative velocity does not come into it, except implicitly when we choose a frame in which to do the calculation. This does not have to be one of the twins frames. The difference in age is the only time both twins come into the calculation.
This must be a new physics, since what you are suggesting is that, difference in the ages of the Twins after the trip, is independent of their relative velocity during the trip.
But I've said we only need both twins if we want to compare their worldlines. The proper length of a worldline depends only on that worldline - not on a relative velocity.

You keep moving the argument around. Do you still think there is a paradox buried in all this ?

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Quote by universal_101
 Quote by ghwellsjr If we assume the Principle of Relativity for light, we are assuming that what each twin sees of the other one is symmetrical and not dependent on their relative speed in any medium.
This is incorrect, 2 and 1/2, 3 and 1/3, or any other form like x and 1/x are inversely symmetrical, but saying that these values, for example, 2,3 and x is independent of the relative velocity makes them arbitrary. I mean if they does not depend on the relative velocity, then how come you choose one over the other and say they are different, since 2 and 3 are obviously different.
I didn't say that the Doppler factor is not dependent of the relative speed between the twins, I said it's not dependent on each twin's relative speed in any medium. I'm also saying that the speed is not important to being able to derive the difference in aging. All we need is the knowledge that the two Doppler factors are reciprocals, and that the traveling twin spends the same amount of time traveling away as he does toward the other twin based on his own clock.

For example, with Dopplers of 2 and 1/2, the average of them is 1.25 which means that as the traveling twin kept his eye on the stationary twin's clock through the entire trip, he first saw it ticking at 1/2 the rate of his own, then for the return trip, he watched it tick twice as fast as his own. You can confirm that at a relative speed of 0.6c, the relativistic Doppler factors are 2 and 1/2 and that gamma equals 1.25.

Another example, with Dopplers of 3 and 1/3, the average is 5/3 or 1.667, and this occurs with a relative speed of 0.8c which produces a gamma of 1.667.

The point is that we don't need to know the value of the speed in order to calculate the age difference which happens to be equal to gamma.

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Quote by Austin0
 Quote by ghwellsjr The story I am discussing now does not look at the relative length of time in each phase for both twins but only for the one that turns around. His two times are equal and knowing the Doppler factors are reciprocal allows him to derive the value of gamma without invoking any other considerations.
Perhaps you could explain this trick?
Reciprocity of Doppler by itself ,without the gamma factor , does not imply aging differential.

So you are assuming that factor behind the scene , applying that to Speedo's hypothetical
observations and then asserting that Speedo, if he were mathematically inclined, could derive the Lorentz transformation directly from these observations.

Are you really claiming that the gamma is not involved or necessary to the explanation?
I never said that it is possible to derive the Lorentz transformation. Gamma is not the Lorentz transformation, it just happens, among other things, to be equal to the ratio of the accumulated times for the two twins and it can be derived just from the reciprocal Doppler factors.

 Quote by atyy Do you believe in energy?
I believe in energy conservation.

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Quote by universal_101
 Quote by mentz114 The clock time of a twin depends only on their *own* worldline. It is completely irrelevant what the other twin is doing. Relative velocity does not come into it, except implicitly when we choose a frame in which to do the calculation. This does not have to be one of the twins frames. The difference in age is the only time both twins come into the calculation.
This must be a new physics, since what you are suggesting is that, difference in the ages of the Twins after the trip, is independent of their relative velocity during the trip.
What I'm trying to say is that we don't need the relative velocity between the twins explicitly in the calculation. We could choose any inertial frame to parameterize the worldlines and still get the correct result.

Suppose I concede the point that the calculation could be done in one of the twins frames - what conclusion would you draw from that ?

Are you still saying that using a transformation to change frames invalidates the laws of dynamics ?

 Quote by ghwellsjr I didn't say that the Doppler factor is not dependent of the relative speed between the twins, I said it's not dependent on each twin's relative speed in any medium. I'm also saying that the speed is not important to being able to derive the difference in aging. All we need is the knowledge that the two Doppler factors are reciprocals, and that the traveling twin spends the same amount of time traveling away as he does toward the other twin based on his own clock. For example, with Dopplers of 2 and 1/2, the average of them is 1.25 which means that as the traveling twin kept his eye on the stationary twin's clock through the entire trip, he first saw it ticking at 1/2 the rate of his own, then for the return trip, he watched it tick twice as fast as his own. You can confirm that at a relative speed of 0.6c, the relativistic Doppler factors are 2 and 1/2 and that gamma equals 1.25. Another example, with Dopplers of 3 and 1/3, the average is 5/3 or 1.667, and this occurs with a relative speed of 0.8c which produces a gamma of 1.667. The point is that we don't need to know the value of the speed in order to calculate the age difference which happens to be equal to gamma.
You inherently used the relative velocity, when you talk about the reciprocity of the Doppler values, i.e. 2 and 1/2 etc. It is very surprising that you and other people here are claiming that difference in the age is independent of relative velocity.

 Quote by Mentz114 What I'm trying to say is that we don't need the relative velocity between the twins explicitly in the calculation. We could choose any inertial frame to parameterize the worldlines and still get the correct result. Suppose I concede the point that the calculation could be done in one of the twins frames - what conclusion would you draw from that ? Are you still saying that using a transformation to change frames invalidates the laws of dynamics ?
If the difference in the age is independent of the relative velocity, why does Muons moving at different speeds decay at different rates.

Please, don't say that, they do so because their worldlines are different, but there is no relation to the relative velocity.

Because the concept of the world-lines is abstract, and even difference in the world-lines of the two objects in a frame, is known as Lorentz transformation, whereas, we are talking about the difference in age which is invariant.

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universal_101, how is the comparison of proper times, as it is done in the twins paradox relevant to the expression of physcal laws in one frame or another ?

In the case of the muon decay it has been pointed out that we can write the 'law' (equations ) governing detector counts in any frame with no inconsistency ?

 I just saw this

 If the difference in the age is independent of the relative velocity, why does Muons moving at different speeds decay at different rates.
I don't know what you mean by 'different rates'. The life-time of the muon is invariant. What changes between frames is the distance they travel as expressed in different coordinates.

 Quote by ghwellsjr I never said that it is possible to derive the Lorentz transformation. Gamma is not the Lorentz transformation, it just happens, among other things, to be equal to the ratio of the accumulated times for the two twins and it can be derived just from the reciprocal Doppler factors.
'Derived just from the reciprocal Doppler factors" because those factors are found through a Lorentzian transform (relativistic Doppler) derived from and inherently containing the gamma factor , Or do you think this is not so??
SO when you say one twin sees the time on the other twins clock you are calculating what he sees using the transformed version of classical Doppler. yes?

I think Gamma cannot be derived simply from reciprocity of classical Doppler. Do you think otherwise? Because that is what you seem to be saying.

 Quote by ghwellsjr I never said that it is possible to derive the Lorentz transformation. Gamma is not the Lorentz transformation, it just happens, among other things, to be equal to the ratio of the accumulated times for the two twins and it can be derived just from the reciprocal Doppler factors.
IT JUST HAPPENS !! No, just happens to be...... does not come under the domain of the classical physics. On the other note, I myself think that it just happens to be equal to the gamma factor from the Lorentz Transformations, but again as I suggested the problem vanishes if we conclude the difference in the nuclear decay rates of moving Muons by using some physical law.

 Quote by Mentz114 I don't know what you mean by 'different rates'. The life-time of the muon is invariant. What changes between frames is the distance they travel as expressed in different coordinates.
First of all, you don't have any experimental proof of that change in distance, whereas, I do have the experimental evidence of different rates.

So, you cannot refute the notion of different rates, since, experimental proof is needed to claim otherwise.

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