Transformation Vs. Physical Law

[universal_101]

No, it isn't. Remember we're talking about the rest frame of the particle: in that frame, the particles are at rest. So there is no "length" involved--only the particles' travel time (or, if you want to be really precise, since the particles are not moving but the source and detector are, in this frame: the time between when the source is co-located with the particles and when the detector is co-located with the particles, by the particles' clock).

Look at the formula DaleSpam posted, which explicitly uses the time in the particles' rest frame. Do you see any length in there?

I thought it would be simple to explain the necessity of the use of the transformation even in the rest frame of the particles.

In order to pinpoint, how does one calculate when was the particle at source and when at the detector? The simple equation would have been, contracting the distance between the source and detector and dividing it by the relative velocity.

But you never mentioned how are you going to calculate when the particle was at source and how much time it took to reach the detector.

 

[universal_101]

Yes.
I was tempted to add "of course", but obviously it's not obvious or you wouldn't be asking.

So here goes... You find yourself riding a relativistic particle down from the top of the atmosphere. You see the surface of the Earth rushing towards you at speed v=.999c, from a distance of 1 light-usec away. Note that neither this distance nor the speed came from any sort of transformation - you measured them directly.

Now, what is the probability that your relativistic but unstable steed will hit (be hit by) the surface of the Earth before it decays? Calculate the time the particle needs to live, by dividing the distance by the velocity, and plug it and lambda (the half-life of the particle expressed in terms of the particle's proper time, which is the time that you are measuring - see, still no transforms) into the formula... And out pops your answer.

You are using the increased half-life time of the particle, which is a transformation tool. Remember you would either use Time Dilation of half-life or the length contraction of the distance which the particle needs to travel. And these are not transformations but tools of it.

 

[Dale]

You are using the increased half-life time of the particle

Where, exactly? \lambda is the usual non-increased half life. It is not a function of speed.

 

[universal_101]

Where, exactly? \lambda is the usual non-increased half life. It is not a function of speed.

So that everyone can see it, does the differential ageing of the twins after the trip can be explained without using transformation tools ?

No it cannot, the same applies to the number of particles reaching Earth. That is, No matter what, in the end the ratio of the number of particles reaching Earth to the number of particles survived in the lab, is always a function of the speed of the particles which reach in higher quantity.

What you guys are missing is the point that, we need to use that same law for lab particles also.

Thanks

 

[Dale]

what I would really like to mention is that, to refute a theory we need just one experiment where as to give a theory the stature of fundamental fact there is NO limit on the Experiments.

This is true, which is why we continue to perform more precise and exact experiments in order to push the limits further and further.

This has nothing to do with the current thread, which is not an experimental challenge but a theoretical challenge. The challenge has been refuted. Your argument is unsound, based only on a flawed understanding. I have posted the law which pertains to radioactive decay. It is frame invariant, and no transformations are required to use it.

 

[Nugatory]

But at the first place, To calculate the number of unstable particles in any frame, we use the Lorentz transformation, don't we ?

No. See my and the other followup posts.


You may be confused by two ways in which we would use the Lorentz transforms:
- We might use the Lorentz transform in a classroom, just to demonstrate that the results of the calculation of unstable particles doesn't change from frame to frame. But there we've already done the calculation, and we're going through the exercise to demonstrate something about the Lorentz transform.

- Sometimes it is very difficult to get a time or distance measurement from the frame where we need it. For example, in a later post I have you, the observer, riding the unstable particle down from the sky as you do your measurements - easier said than done. In that case, we make the measurement in a more convenient frame, then Lorentz-transform it into the frame where we needed the measurement.

 

[Dale]

I note that you avoided answering my question about where exactly you think that we are bringing in a transformation into the law. Again, the law is invariant under the Lorentz transformation, but no transformation is needed in order to use the law.

So that everyone can see it, does the differential ageing of the twins after the trip can be explained without using transformation tools ?

Yes, in fact, it is my preferred explanation since if you don't do a transformation you never even seem to get a paradox.

The age of each twin at the reunion is simply: \tau_P=\int_P d\tau

What you guys are missing is the point that, we need to use that same law for lab particles also.

No, what you are missing is that all physical laws are invariant under the transform whether it is the decay law or the age of the twins, but that doesn't mean that a transform needs to be used whenever you use a physical law.

 

[PeterDonis]

In order to pinpoint, how does one calculate when was the particle at source and when at the detector?

Um, remember, we're in the rest frame of the particles. That means we just watch when, by the particles' clock, the source passes, and when the detector passes. In the rest frame of the particles, these are direct measurements. No calculation is required.

The simple equation would have been, contracting the distance between the source and detector and dividing it by the relative velocity.

If you are not in the rest frame of the particles, then yes, you could do something like this. But this argument concedes the point--if you are not in the rest frame of the particles, then yes, you have to do a "transformation" to obtain the proper time in the rest frame of the particles. But this is irrelevant if you are in the rest frame of the particles.

Actually, it's even debatable whether you need to use a "transformation" in the case where we are not in the rest frame of the particles, as Russell E posted. Look at the formula I posted earlier, which is a modified version of DaleSpam's formula that uses values from the Earth frame to calculate the proper time in the particles' rest frame. Those values from the Earth frame are direct measurements, and you can plug them directly into my formula to get the answer. Whether or not this counts as doing a "transformation" is a question about words, not about physics.

But you never mentioned how are you going to calculate when the particle was at source and how much time it took to reach the detector.

If I am in the rest frame of the particles, I don't have to calculate this; I can measure it directly.

Basically, you are saying that we are not in the rest frame of the particles, so we have to calculate what the proper time is in that frame, since we can't measure it directly. Nobody is disputing this, nor is anybody disputing that you can use "transformations" to make the calculation. But if you want to talk about a hypothetical situation where we *can* measure directly the proper time in the rest frame of the particles (which is what "being in the rest frame of the particles" would mean), then we do *not* have to calculate the proper time in that frame, because we can measure it directly.

In any case, at this point we are talking more about how to interpret words, like "transformation" or what it means to be "in the rest frame of the particles", than about the physics, as I said above. I still don't see any substantive point about the physics in your posts that challenges the standard understanding.

 

[PeterDonis]

So that everyone can see it, does the differential ageing of the twins after the trip can be explained without using transformation tools ?

Sure it can; calculate everything in a single inertial frame, the frame of the non-traveling twin. Everything you need--the start time and end time in that frame, the distance the traveling twin goes before turning around, and the relative velocity--can be obtained by direct measurements in that frame. No transformations required.

Edit: I see DaleSpam posted the specific formula you would use.

 

[Mentz114]

So that everyone can see it, does the differential ageing of the twins after the trip can be explained without using transformation tools ?

This has been answered in the affirmative.

Given the equations for the worldlines of the twins in some inertial coordinates, we use only the Minkowski metric and those equations to calculate the times on the twins clocks. The answer will be the same whatever inertial coordinates are used, because the proper time is an invariant. No transformation or 'tools'.

[edit]Posted simultaneously with the above.

 

[Nugatory]

You are using the increased half-life time of the particle, which is a transformation tool. Remember you would either use Time Dilation of half-life or the length contraction of the distance which the particle needs to travel. And these are not transformations but tools of it.

I am not using the "increased half-length" of the particle (increased from what, for crissakes?). I'm using the half-life of the particle as I measured it, from other experiments in which I was riding around on other such particles, and it is the exact same value which an earthbound scientist sees and calculates when studying these particles at rest in his earthbound lab.

 

[Dale]

if you are not in the rest frame of the particles, then yes, you have to do a "transformation" to obtain the proper time in the rest frame of the particles.

I disagree with this slightly. The proper time is itself a frame invariant quantity. It can be calculated in any frame using only measurements and values relative to that frame. All frames will agree on the value.

What you need a transformation for is to find the coordinate time in the rest frame of the particle, not the proper time. At any event on the worldline of the particle the proper time is equal to the coordinate time in the rest frame, but the coordinate time is also defined at events which are not on the worldline of the particle, so they are different things.

 

[GeorgeDishman]

The above mentioned law is well known, but there is NO law which explain the how many number of particles will reach the Earth.

Yes there is, you even say it is well known and DaleSpam gave you it mathematically.

Because, currently we use the part of a transformation to explain this effect.

No we don't, my point was that the law applies equally well in both the Earth frame and the particle frame. The value of the half-life obtained in the lab is in the particle's rest frame while we usually measure the thickness of the atmosphere in the Earth frame. It is inherent in the question you asked that that those are not the same hence applying the transform is one way to get both to the same frame. However, that isn't the only way. If you want to know the value of the particle half-life in the Earth frame, you must apply the time dilation factor but that can be obtained from many experiments, that of Ives and Stilwell for example, you don't need to use the Lorentz Transforms.

The problem is, you just used a transformation to explain a physical effect, which should be governed by a physical law, including, which is today known as Time Dilation of unstable particles due to motion.

The Lorentz Transforms can be used to convert between the frames to check for consistency but they aren't needed to predict the particle numbers, both length contraction and time dilation can be obtained empirically from experiment as independent laws without using the transforms.

 

[PeterDonis]

I disagree with this slightly.

Actually, I kind of did too, which is why I added the paragraph on it being debatable whether a "transformation" is actually needed.

What you need a transformation for is to find the coordinate time in the rest frame of the particle, not the proper time. At any event on the worldline of the particle the proper time is equal to the coordinate time in the rest frame, but the coordinate time is also defined at events which are not on the worldline of the particle, so they are different things.

Good point, I hadn't taken this into account.

 

[universal_101]

I think there is a huge misunderstanding or differences in the definition of the term use of transformation.

Because what I'm calling a use of transformation is the dependence of a property on the relative velocity.

What you guys are suggesting is that we don't need any kind of transformation to conclude the results in the rest frame of the particle. This is perfectly fine, but the whole point of the debate was,

1.) number of particles decay differently depending on their relative motion, now this difference is a function of relative velocity. is it not ?

2.) if this difference is a function of the relative velocity, then we must get those relative velocity terms from somewhere, and it is this somewhere which I'm suggesting comes only from the Lorentz transformation.

Whereas, it is perfectly OK to not have any transformation use if we are not comparing the results of different frames. It is the difference in the results which depends on relative velocity, and it is this relative velocity dependence which is concluded using Lorentz transformation.

Now, do you people agree that in order to understand the difference in ageing of the Twins, we use their relative velocity to calculate their difference in age. Which I thought was a use of transformation.

 

[GeorgeDishman]

I think there is a huge misunderstanding or differences in the definition of the term use of transformation.

Because what I'm calling a use of transformation is the dependence of a property on the relative velocity.

That would certainly cause confusion. What I am calling a transformation is a set of equations which allow coordinates stated in one reference frame to be translated into an equivalent set of numbers that represent the same events but with values stated in a different reference frame. I believe that is the standard meaning of the term.

The analogy I use is to draw dots on a blank sheet of paper and place a grid printed on a transparent sheet over the top. You can then read off coordinates for the dots. If you then rotate the grid sheet slightly, you get different coordinates for the same dots. A transformation would then allow you to calculate one set of coordinates from the other.

What you guys are suggesting is that we don't need any kind of transformation to conclude the results in the rest frame of the particle. This is perfectly fine, but the whole point of the debate was,

1.) number of particles decay differently depending on their relative motion, now this difference is a function of relative velocity. is it not ?

The half-life expressed in terms of proper time for a specific type of particle is independent of motion. The number of particles that decays in any specific amount of coordinate time depends on how you rotate the grid you put over the events.

2.) if this difference is a function of the relative velocity, then we must get those relative velocity terms from somewhere, and it is this somewhere which I'm suggesting comes only from the Lorentz transformation.

You can get the time dilation factor (which accounts for the rotation of the x,t coordinate grid) from the Lorentz Transforms but my point was that that is not the only method, you can also get it directly from empirical measurement.

Now, do you people agree that in order to understand the difference in ageing of the Twins, we use their relative velocity to calculate their difference in age.

Yes, I agree, if you don't know the velocity, you cannot calculate the difference in their ages (obviously).

Which I thought was a use of transformation.

No, I disagree, you can calculate how much each twin ages in the stay-at-home twin's frame throughout. You need the time dilation factor of course but that, as I have said before, can be obtained empirically from the Ives-Stilwell experiment without the use of a transform. In that experiment, all measurements are made in the lab frame.

 

[Dale]

Because what I'm calling a use of transformation is the dependence of a property on the relative velocity.

Then you should say "dependence on velocity" instead of "use of transformation".

There is nothing wrong or illogical or unphysical or circular about things depending on velocity.

1.) number of particles decay differently depending on their relative motion, now this difference is a function of relative velocity. is it not ?

This is factually incorrect. The number of particles at any given event is frame invariant, as can be clearly seen in the equation I posted.

2.) if this difference is a function of the relative velocity, then we must get those relative velocity terms from somewhere, and it is this somewhere which I'm suggesting comes only from the Lorentz transformation.

There is no difference, so this question is somewhat moot.

However, there are other quantities which do depend on the velocity, such as the rate of decay wrt coordinate time. The velocity has to be measured in some frame. If it is measured in the frame of interest then no transformation is required. If it is measured in some other frame then it must be transformed into the frame of interest. The same applies to any frame-variant quantity.

It is the difference in the results which depends on relative velocity, and it is this relative velocity dependence which is concluded using Lorentz transformation.

Not in general, no.

Now, do you people agree that in order to understand the difference in ageing of the Twins, we use their relative velocity to calculate their difference in age. Which I thought was a use of transformation.

You thought incorrectly. Either twin can measure the relative velocity of the other directly, without using any transformation.

 

[Mentz114]

Now, do you people agree that in order to understand the difference in ageing of the Twins, we use their relative velocity to calculate their difference in age.

No, I don't agree.

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.

 

[ghwellsjr]

So that everyone can see it, does the differential ageing of the twins after the trip can be explained without using transformation tools ?

Now, do you people agree that in order to understand the difference in ageing of the Twins, we use their relative velocity to calculate their difference in age. Which I thought was a use of transformation.

"The differential ageing of the twins after the trip" can be explained by Relativisic Doppler analysis without using transformation tools, frames, relative velocity or anything else derived from or dependent on Special Relativity. All that is required is the Principle of Relativity, and the experimental evidence that the one way speed of light is independent of the motion of the source of the light and that the traveler's speed is the same in both directions but you don't have to know what that speed is or how it relates to time dilation or to the Doppler factor. I outlined the details here:

Normal Doppler, where there is a medium such as air for sound, is not relativistic, meaning that two observers don't hear the same thing coming from the other one because we have to take into account their relative speed in the medium.

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. It can be easily demonstrated that two inertial observer with a relative motion between them, traveling along the same line will see a Doppler factor while approaching that is the reciprocal of the Doppler factor while receding. This by itself is all that is necessary to show that if one of those twins remains inertial while the other one travels away at some constant speed creating a constant Doppler factor less than one and then turns around and approaches at that same constant speed, he will observe a Doppler factor that is the inverse of the first one, a number greater than one. Let's say that the return Doppler factor is N, a number greater than one and the departing Doppler factor is 1/N. We take the average of these two numbers to get the total ratio of their accumulated ages. This number will always be greater than one and in fact is equal to gamma. Speedo will watch Goslo's clock constantly advancing during his entire trip, first slower then his own and then at turn-around faster than his own and when they meet, Goslo's clock will have advance gamma times the amount his own clock has advanced. This is exactly addressing the question that Michio Cuckoo asked in his first post.

Remember, Einstein's second postulate is that the propagation of light is c, meaning that the unmeasurable one-way time is equal to one-half of the round-trip time and the Doppler analysis does not require that or depend on that in any way. In fact it works the same in any frame even in those for which the two one-way times for a round trip propagation of light are not equal. In other words, it is making no statement about the synchronization of the clocks of the two twins while they are separated, only the final outcome of the time difference when they reunite.

 

[Samshorn]

"The differential ageing of the twins after the trip" can be explained by Relativisic Doppler analysis without using ... anything else derived from or dependent on Special Relativity. All that is required is the Principle of Relativity, and the experimental evidence that the one way speed of light is independent of the motion of the source of the light and that the traveler's speed is the same in both directions...

Not true, as explained in detail in the other thread where you made that claim. By the way, the principle of relativity is founded on experimental evidence, just as much as is the invariance of light speed in terms of standard inertial coordinates, so it makes no sense to take one as a principle and the other as an "experimental" proposition. They are both experimentally founded propositions that we adopt as principles. Now, as to your specific claim, the independence of light speed on the motion of the source is necessary but not sufficient to derive relativistic Doppler, because it doesn't rule out directional dependence. You need, in addition to the principle of relativity, the full principle of lightspeed invariance, including isotropy of light speed (in terms of standard inertial coordinates). And of course you need to specify the numerical value of this invariant speed in order to quantify the relativistic effects (such as asymmetric aging). Taken together, these are sufficient to derive all of special relativity, including (but not limited to) relativistic Doppler.

 

[ghwellsjr]

Not true, as explained in detail in the other thread where you made that claim. First, the principle of relativity is founded on experimental evidence, just as much as is the invariance of light speed in terms of standard inertial coordinates, so it makes no sense to take one as a principle and the other as an "experimental" proposition. They are both experimentally founded propositions that we adopt as principles. Second, the independence of light speed on the motion of the source is necessary but not sufficient to derive relativistic Doppler, because it doesn't rule out directional dependence. You need, in addition to the principle of relativity, the full principle of lightspeed invariance, including isotropy of light speed (in terms of standard inertial coordinates). And taken together, these are sufficient to derive all of special relativity, including (but not limited to) relativistic Doppler.

I'm not trying to derive relativistic Doppler. I'm saying that since two inertial observers can experimentally determine that the Doppler based on light is the same for both of them as they approach each other and that it is the same for both of them as they recede away from each other and that these two Doppler factors are reciprocals of each other, then that is all they need to know to predict that if they enact the twin scenario where they depart from each other and remain inertial for a while and then one of them accelerates back toward the other one with the recprocal Doppler, their accumulated age ratio can be calculated from the Doppler factor.

I'm also saying that this analysis does not require any synchronization of remote clocks by any method or the establishment or definition of any frame of reference or coordinate system or any theory about transforming coordinates between different coordinate systems, which is what universal_101 is contending.

 

[Samshorn]

I'm saying that since two inertial observers can experimentally determine that the Doppler based on light is the same for both of them as they approach each other and that it is the same for both of them as they recede away from each other and that these two Doppler factors are reciprocals of each other, then that is all they need to know to predict ... their accumulated age ratio ... from the Doppler factor. I'm also saying that this analysis does not require ... the establishment or definition of any frame of reference or coordinate system...

Still not true. Your premise is that we can experimentally determine that the Doppler shift when receding at a certain speed is the reciprocal of the Doppler shift when approaching at the same speed. The problem is that you haven't thought about how they would deterime that they are approaching each other at the same speed that they were formerly receding from each other. They obviously can't use the Doppler shift, because that would be circular and devoid of physical content. In other words, they can't simply define their approach speed to be equal to their receed speed when the Doppler shifts are reciprocal. For that proposition to have physical meaning, they need some independent measure of speed, which comes from the systems of coordinates in which the homogeneous and isotropic equations of mechanics hold good. There is simply no way of getting the effects of special relativity without establishing the correlation (implicitly or explicitly) with inertia.

I'm also saying that this analysis does not require any ... theory about transforming coordinates between different coordinate systems, which is what universal_101 is contending.

Well, it obviously doesn't require any transforming of coordinates, but it does imply Lorentz invariance, which entails the covariance of the physical parameters under a certain class of transformations.

The answer to the OP is that the physical law describing the half-life of a sub-atomic particle moving in the x, y, and z directions by the amounts dx, dy, dz in the time dt is purely a function of the quantity sqrt[dt^2 - dx^2 - dy^2 - dz^2] where x,y,z,t are any single system of inertial coordinates. No transformation is involved. (But of course x,y,z,t do have to be coordinates in terms of which the laws of mechanics hold good.)

In fact, we find that every physical process and phenomenon (not just the half-lives of sub-atomic particles) has this same form, in the sense that the physical laws don't depend on the absolute values of x,y,z,t, nor even on the absolute values of dx,dy,dz,dt or their ratios, but only on the quantity dt^2 - dx^2 - dy^2 - dz^2. The fact that these physical laws work equally well in terms of any standard system of inertial spacetime coordinates implies that this quadratic quantity is the same in all of them. After noticing this, and then seeing it confirmed over and over again for all known physical laws, we begin to expect it to be true, even when trying to formulate the laws governing previously unknown phenomena. This property, called Lorentz invariance, is not itself a physical law, it is an attribute of all known physical laws.

It's useful to know about Lorentz invariance because it enables us to compute things very easily by taking a short cut. If we already know that a certain physical law (such as the law for the half-life of a particle) is Lorentz invariant, we know that we can compute things in any convenient system of standard inertial coordinates, and then very simply express the results in terms of any other system of coordinates using the Lorentz transformation (which happens to be the transformation that preserves that quadratic quantity appearing in the physical laws). But this is just a computational shortcut, used by people who know what they're doing. If it confuses the OP, he can just go ahead and do things the more laborious (and less insightful) way.

 

[Austin0]

Thanks for the view,

I agree that Lorentz transformation is more than just a transformation in modern physics. It is exactly what I'm questioning. It seems as if the transformation is multipurpose, it can be a physical law at times and also can be a transformation at other.

Do you see this contradiction of basic physics concept.

Actually, no, I don't see any contradiction.

Do you doubt the validity of time dilation as a function of relative motion as it is described in the Lorentz math??

If you don't then I don't understand why you think there is a problem. Is it the semantic question of whether time dilation is called a law or a transformation? You seemed to agree that it could be both so I am confused as to your point here.

 

[Austin0]

Still not true. Your premise is that we can experimentally determine that the Doppler shift when receding at a certain speed is the reciprocal of the Doppler shift when approaching at the same speed. The problem is that you haven't thought about how they would deterime that they are approaching each other at the same speed that they were formerly receding from each other. They obviously can't use the Doppler shift, because that would be circular and devoid of physical content. In other words, they can't simply define their approach speed to be equal to their receed speed when the Doppler shifts are reciprocal. For that proposition to have physical meaning, they need some independent measure of speed, which comes from the systems of coordinates in which the homogeneous and isotropic equations of mechanics hold good. There is simply no way of getting the effects of special relativity without establishing the correlation (implicitly or explicitly) with inertia.



Well, it obviously doesn't require any transforming of coordinates, but it does imply Lorentz invariance, which entails the covariance of the physical parameters under a certain class of transformations.

The answer to the OP is that the physical law describing the half-life of a sub-atomic particle moving in the x, y, and z directions by the amounts dx, dy, dz in the time dt is purely a function of the quantity sqrt[dt^2 - dx^2 - dy^2 - dz^2] where x,y,z,t are any single system of inertial coordinates. No transformation is involved. (But of course x,y,z,t do have to be coordinates in terms of which the laws of mechanics hold good.)

In fact, we find that every physical process and phenomenon (not just the half-lives of sub-atomic particles) has this same form, in the sense that the physical laws don't depend on the absolute values of x,y,z,t, nor even on the absolute values of dx,dy,dz,dt or their ratios, but only on the quantity dt^2 - dx^2 - dy^2 - dz^2. The fact that these physical laws work equally well in terms of any standard system of inertial spacetime coordinates implies that this quadratic quantity is the same in all of them. After noticing this, and then seeing it confirmed over and over again for all known physical laws, we begin to expect it to be true, even when trying to formulate the laws governing previously unknown phenomena. This property, called Lorentz invariance, is not itself a physical law, it is an attribute of all known physical laws.

It's useful to know about Lorentz invariance because it enables us to compute things very easily by taking a short cut. If we already know that a certain physical law (such as the law for the half-life of a particle) is Lorentz invariant, we know that we can compute things in any convenient system of standard inertial coordinates, and then very simply express the results in terms of any other system of coordinates using the Lorentz transformation (which happens to be the transformation that preserves that quadratic quantity appearing in the physical laws). But this is just a computational shortcut, used by people who know what they're doing. If it confuses the OP, he can just go ahead and do things the more laborious (and less insightful) way.

Hi regarding relative velocities in these scenarios. Isn't that always problematic if we are considering hypothetical real world situations?? But normally in a case like ghwellsjr's it is assumed we extended a virtual frame all the way to the destination to measure velocity exactly , no? We do make the assumption that approach is equivalent to recession.

Regarding the invariant interval, I understood it was a direct derivation from the Lorentz math. Is this incorrect?

 

[Austin0]

I'm not trying to derive relativistic Doppler. I'm saying that since two inertial observers can experimentally determine that the Doppler based on light is the same for both of them as they approach each other and that it is the same for both of them as they recede away from each other and that these two Doppler factors are reciprocals of each other, then that is all they need to know to predict that if they enact the twin scenario where they depart from each other and remain inertial for a while and then one of them accelerates back toward the other one with the recprocal Doppler, their accumulated age ratio can be calculated from the Doppler factor.

I'm also saying that this analysis does not require any synchronization of remote clocks by any method or the establishment or definition of any frame of reference or coordinate system or any theory about transforming coordinates between different coordinate systems, which is what universal_101 is contending.

Hi Firstly I don't agree or in fact even understand the OP's point, but I have to mention that the Doppler shift equation is itself derived from and expressing the fundamental transformation isn't it? With classical Doppler it seems to me there would be no age differential, no?

 

[Dale]

Regarding the invariant interval, I understood it was a direct derivation from the Lorentz math. Is this incorrect?

If you start with the spacetime interval you can derive the Lorentz transform as a class of transforms that leaves the interval invariant. If you start with the transform you can derive the interval as a quantity that is invariant. It just depends what you want to consider a postulate and what you want to consider a derived result. The math doesn't care which direction you go.

I find a certain appeal to starting with the interval. After all, to me, the notion of distance seems more basic than the notion of coordinates.

 

[ghwellsjr]

I'm saying that since two inertial observers can experimentally determine that the Doppler based on light is the same for both of them as they approach each other and that it is the same for both of them as they recede away from each other and that these two Doppler factors are reciprocals of each other, then that is all they need to know to predict ... their accumulated age ratio ... from the Doppler factor. I'm also saying that this analysis does not require ... the establishment or definition of any frame of reference or coordinate system...

Still not true. Your premise is that we can experimentally determine that the Doppler shift when receding at a certain speed is the reciprocal of the Doppler shift when approaching at the same speed. The problem is that you haven't thought about how they would deterime that they are approaching each other at the same speed that they were formerly receding from each other. They obviously can't use the Doppler shift, because that would be circular and devoid of physical content. In other words, they can't simply define their approach speed to be equal to their receed speed when the Doppler shifts are reciprocal. For that proposition to have physical meaning, they need some independent measure of speed, which comes from the systems of coordinates in which the homogeneous and isotropic equations of mechanics hold good. There is simply no way of getting the effects of special relativity without establishing the correlation (implicitly or explicitly) with inertia.

I said if two inertial observers start off approaching each other (from far apart) and then pass each other so that they are then receding, they will continue at the same speed, won't they? I wasn't talking yet about the twin scenario.

But beyond that, I have thought about how we can demonstrate that the two Doppler factors (coming and going at the same speed) are reciprocals and I found the answer in Hermann Bondi's book, http://archive.org/details/RelativityCommonSense, pages 77 to 80. So we can figure it out either by experiment or by analysis.

I'm also saying that this analysis does not require any ... theory about transforming coordinates between different coordinate systems, which is what universal_101 is contending.

Well, it obviously doesn't require any transforming of coordinates, but it does imply Lorentz invariance, which entails the covariance of the physical parameters under a certain class of transformations.

The answer to the OP is that the physical law describing the half-life of a sub-atomic particle...

I wasn't addressing the initial issue (which was already thoroughly addressed and discarded by universal_101 in his other thread that got locked because it was going around in circles) but only his contention that transformation tools are required to explain the twin paradox, which I did in a way that I thought might make sense to him.

 

[ghwellsjr]

Hi Firstly I don't agree or in fact even understand the OP's point, but I have to mention that the Doppler shift equation is itself derived from and expressing the fundamental transformation isn't it? With classical Doppler it seems to me there would be no age differential, no?

I am not using the Doppler shift equation, if by that you mean the one that calculates the Doppler factor as a function of relative speed. I'm only saying that the approaching and receding Doppler factors are reciprocals for the same relative speed but we aren't concerned with what that relative speed is or how it relates to the Doppler factor. Of course, you can also confirm that this is true based on that Doppler shift equation, but that is immaterial for the analysis that I have given.

There are many ways to derive the equation but that is irrelevant to what I am saying. And yes, the classical Doppler shift equation won't work because it is not relativistic.

 

[Austin0]

Yes there is, you even say it is well known and DaleSpam gave you it mathematically.



No we don't, my point was that the law applies equally well in both the Earth frame and the particle frame. The value of the half-life obtained in the lab is in the particle's rest frame while we usually measure the thickness of the atmosphere in the Earth frame. It is inherent in the question you asked that that those are not the same hence applying the transform is one way to get both to the same frame. However, that isn't the only way. If you want to know the value of the particle half-life in the Earth frame, you must apply the time dilation factor but that can be obtained from many experiments, that of Ives and Stilwell for example, you don't need to use the Lorentz Transforms.



The Lorentz Transforms can be used to convert between the frames to check for consistency but they aren't needed to predict the particle numbers, both length contraction and time dilation can be obtained empirically from experiment as independent laws without using the transforms.

Hi
could you point me to the experimental tests revealing length contraction?
I have looked without coming across anything. Thanks

 

[Austin0]

I am not using the Doppler shift equation, if by that you mean the one that calculates the Doppler factor as a function of relative speed. I'm only saying that the approaching and receding Doppler factors are reciprocals for the same relative speed but we aren't concerned with what that relative speed is or how it relates to the Doppler factor. Of course, you can also confirm that this is true based on that Doppler shift equation, but that is immaterial for the analysis that I have given.

There are many ways to derive the equation but that is irrelevant to what I am saying. And yes, the classical Doppler shift equation won't work because it is not relativistic.

yes I understand your point regarding reciprocity and the relative length of time in each phase. And certainly agree.
But to suggest you can apply this principle to the twins question to explain the difference in final age, without invoking the gamma factor inherent in the relativistic Doppler equation, is a different story. Wouldn't you agree?