Transformation Vs. Physical Law

[universal_101]

Most of the people here, who responded to the last thread posted by me, may think of me as someone who does not want to understand Relativity, and instead is just barking at the wrong tree. But I'm posting the same Logical contradiction of using Lorentz Transformation to conclude Time Dilation of unstable moving particles with the definition of physical law.

But first, let me make sure that people here understand the basic nature of the problem I'm encountering with the relativity of transformations and Physical laws.

As the topic suggests, the problem starts with the definition of physical laws and transformation itself. Let me make it more clear by using an example and the respective definitions.

A physical law must be invariant under a transformation from one observer to another. In other words, it is independent of who is observing it. the conclusion of using a physical law for a physical process must be same for all observers(inertial).

Whereas, a transformation, let's consider a co-ordinate transform in geometry first, then we can simply extend the concept for the Lorentz Transformation. In geometry the shape of any object(circle, parabola, line) does not depend on the position of the origin of the co-ordinate system, even though the co-ordinates(x,y,z) of these objects can change.

The same applies to the Lorentz transformation, the outcome of a physical law cannot change under transformation, even though the parameters of the equation governing the physical law changes after the transformation.


Both of these(LT and Physical law), can be analogously visualized in the following example.

Consider a live play in a large auditorium, Now the parts of the play that shows what happens to the characters in the play, can be considered as a physical law(for example, a characters death). Whereas, the observation from different positions of the auditorium can be calculated as the transformation of the events in play for different observers. That is, everybody sees the death of the character but their view can be different depending on their positions.

Now, coming back to my original question,

If the number of unstable particles reaching the Earth is invariant under Lorentz transformation. Then this phenomena must be explained by a physical law and not by the transformation itself. Since, a transformation cannot keep the numbers invariant if this phenomena were to be actually explained by the transformation of observers. But as we all know, the transformation around this phenomena does keep the numbers invariant must imply that this phenomena is governed by a physical law and not by the transformation.

So,What is wrong with the above Logical argument?

Thanks,

 

[Mentz114]

If the number of unstable particles reaching the Earth is invariant under Lorentz transformation.

It is. This is a necessary property of the transformation

Then this phenomenon must be explained by a physical law and not by the transformation itself. Since, a transformation cannot keep the numbers invariant if this phenomena were to be actually explained by the transformation of observers. But as we all know, the transformation around this phenomena does keep the numbers invariant must imply that this phenomena is governed by a physical law and not by the transformation.

Everything is governed by physical law, and this is in no way challenged or altered by the invariance of laws under LT.

The invariance gives the prediction that different observers see the same outcome of physical phenomena. Which is what we want, is it not ?

For instance, the Lagrangian that governs electrodynamics is Lorentz invariant. So a Lorentz transformation will not predict that different observers see different outcomes to electrodynamic phenomena.

 

[vin300]

Whereas, a transformation, let's consider a co-ordinate transform in geometry first, then we can simply extend the concept for the Lorentz Transformation. In geometry the shape of any object(circle, parabola, line) does not depend on the position of the origin of the co-ordinate system, even though the co-ordinates(x,y,z) of these objects can change.

The same applies to the Lorentz transformation, the outcome of a physical law cannot change under transformation, even though the parameters of the equation governing the physical law changes after the transformation.

The shape of a two dimensional object does change as seen from different frames. A circle becomes an ellipse, an ellipse becomes an ellipse of different eccentricity or a circle(a circle is an ellipse with eccentricity 0), a parabola also changes its focal parameter.

 

[universal_101]

Thanks for your reply,

It is. This is a necessary property of the transformation

Of-course it is, but again it means that it must be a physical law behind the phenomena.

Everything is governed by physical law, and this is in no way challenged or altered by the invariance of laws under LT.

Yes, everything is governed by physical laws, but there is none for Time Dilation of unstable particles.

We are using a transformation in place of a physical law to explain a physical process.

The invariance gives the prediction that different observers see the same outcome of physical phenomena. Which is what we want, is it not ?

For instance, the Lagrangian that governs electrodynamics is Lorentz invariant. So a Lorentz transformation will not predict that different observers see different outcomes to electrodynamic phenomena.

Agreed , and I'm also not suggesting that the number of particles should depend on transformation. What I'm suggesting is, it must be governed by a physical law instead of a transformation that which predicts how many particles should reach a particular destination.

 

[universal_101]

The shape of a two dimensional object does change as seen from different frames. A circle becomes an ellipse, an ellipse becomes an ellipse of different eccentricity or a circle(a circle is an ellipse with eccentricity 0), a parabola also changes its focal parameter.

Thanks for the reply,

But I was suggesting that it is the transformation of the equations of the shapes while shifting origin which does not change the shapes of the objects.

What you are explaining is the Lorentz transformation of these shapes, which do changes with different observer speeds. Yes.

 

[Mentz114]

Yes, everything is governed by physical laws, but there is none for Time Dilation of unstable particles.

Time dilation appears as part of the transformation between frames.

We are using a transformation in place of a physical law to explain a physical process.

This is wrong.
The process is governed by the laws. Observations of the process from different frames is governed by the transformation.

I have to say I admire your gall. You don't understand this stuff, which has been around for decades and examined by the best minds of our time - and still you think you've found a paradox.

 

[Dale]

If the number of unstable particles reaching the Earth is invariant under Lorentz transformation. Then this phenomena must be explained by a physical law and not by the transformation itself. Since, a transformation cannot keep the numbers invariant if this phenomena were to be actually explained by the transformation of observers. But as we all know, the transformation around this phenomena does keep the numbers invariant must imply that this phenomena is governed by a physical law and not by the transformation.

The phenomenon is explained by a physical law. The law is invariant under the Lorentz transformation. Is that clear enough?

 

[Dale]

Yes, everything is governed by physical laws, but there is none for Time Dilation of unstable particles.

Nonsense. Of course there is a physical law that exhibits time dilation of unstable particles. I mentioned it in the last thread.

 

[Austin0]

Most of the people here, who responded to the last thread posted by me, may think of me as someone who does not want to understand Relativity, and instead is just barking at the wrong tree. But I'm posting the same Logical contradiction of using Lorentz Transformation to conclude Time Dilation of unstable moving particles with the definition of physical law.

But first, let me make sure that people here understand the basic nature of the problem I'm encountering with the relativity of transformations and Physical laws.

As the topic suggests, the problem starts with the definition of physical laws and transformation itself. Let me make it more clear by using an example and the respective definitions.

A physical law must be invariant under a transformation from one observer to another. In other words, it is independent of who is observing it. the conclusion of using a physical law for a physical process must be same for all observers(inertial).

Whereas, a transformation, let's consider a co-ordinate transform in geometry first, then we can simply extend the concept for the Lorentz Transformation. In geometry the shape of any object(circle, parabola, line) does not depend on the position of the origin of the co-ordinate system, even though the co-ordinates(x,y,z) of these objects can change.

The same applies to the Lorentz transformation, the outcome of a physical law cannot change under transformation, even though the parameters of the equation governing the physical law changes after the transformation.


Both of these(LT and Physical law), can be analogously visualized in the following example.

Consider a live play in a large auditorium, Now the parts of the play that shows what happens to the characters in the play, can be considered as a physical law(for example, a characters death). Whereas, the observation from different positions of the auditorium can be calculated as the transformation of the events in play for different observers. That is, everybody sees the death of the character but their view can be different depending on their positions.

Now, coming back to my original question,

If the number of unstable particles reaching the Earth is invariant under Lorentz transformation. Then this phenomena must be explained by a physical law and not by the transformation itself. Since, a transformation cannot keep the numbers invariant if this phenomena were to be actually explained by the transformation of observers. But as we all know, the transformation around this phenomena does keep the numbers invariant must imply that this phenomena is governed by a physical law and not by the transformation.

So,What is wrong with the above Logical argument?

Thanks,

I think you can consider the Lorentz math itsself, physical law . Unlike the Galilean transform that described no physics itself but was entirely a simple transformation..It is an elvolution of Newtonian mechanics which tells us how much energy it will take to accelerate an electron etc.,etc.
Since these aspects of physics affect the instruments of physics themselves ,clocks ,rulers etc. it is natural to encorporate them directly into the coordinate
system as part of the transformation. I.e. An addition to the Galilean transform.
This is just my view of course.

 

[Nugatory]

If the number of unstable particles reaching the Earth is invariant under Lorentz transformation. Then this phenomena must be explained by a physical law and not by the transformation itself. Since, a transformation cannot keep the numbers invariant if this phenomena were to be actually explained by the transformation of observers. But as we all know, the transformation around this phenomena does keep the numbers invariant must imply that this phenomena is governed by a physical law and not by the transformation.

So,What is wrong with the above Logical argument?

The phrase that I've emphasized in bold... It's not necessarily true.

Let's start with a more precise definition of what we're measuring: the number of particles that are detected between two events in spacetime (for example, "I turned the detector on and started counting" and "I turned the detector off and checked the counts"). There is no time or distance involved here, so the results are (unsurprisingly) the same for all observers regardless of relative motion, time dilation, and the like. If we have a sufficiently complete specification of the initial conditions, we can predict this value from a frame-independent physical law that gives the decay time of the particles as a function of the proper time experienced by the particle itself.

Now, different observers may find different rates of arrival at the detector. This also isn't surprising, because the rate of arrival is found by dividing the number of arrivals by the time that the detector is on - and the different observers are measuring time differently so they're dividing by different values, so getting different rates. Different observers may also calculate different particle lifetimes as measured by their different clocks - but again, these are different clocks so there's no surprise there.

However the observers do agree about how their respective clocks are related so after they've made all their measurements they can go back and compare notes. When they do, they'll find that there is no paradox - all of their measurements are consistent with the observation itself, and with the expected particle lifetimes as a function of the passage of time in the particles proper time.

 

[universal_101]

The phenomenon is explained by a physical law. The law is invariant under the Lorentz transformation. Is that clear enough?

The above statement is clear as anything.

But which physical law is there at work ? but remember, it should not involve any kind of transformation, if it has to be a physical law !

 

[GeorgeDishman]

If the number of unstable particles reaching the Earth is invariant under Lorentz transformation. Then this phenomena must be explained by a physical law and not by the transformation itself.

The physical law relates to the probability of decay in a given time. For large numbers, we quantify that as the half-life of the particle.

Since, a transformation cannot keep the numbers invariant if this phenomena were to be actually explained by the transformation of observers. But as we all know, the transformation around this phenomena does keep the numbers invariant must imply that this phenomena is governed by a physical law and not by the transformation.

In the rest frame of the particle, the our atmosphere is thin (due to 'length contraction'), it takes a short time to pass through as the Earth rushes into meet the particle, so fewer particles decay than if they were moving slowly.

Transformed to the Earth frame, the atmosphere is thicker but the particles suffer 'time dilation' which extends their half-life so the number reaching the ground is the same.

Where do you see the problem?

 

[Dale]

The above statement is clear as anything.

But which physical law is there at work ? but remember, it should not involve any kind of transformation, if it has to be a physical law !

The usual decay law: \frac{dn}{d\tau}=-\lambda n which has the solution n=n_0 e^{-\lambda \tau}

 

[universal_101]

I think you can consider the Lorentz math itsself, physical law . Unlike the Galilean transform that described no physics itself but was entirely a simple transformation..It is an elvolution of Newtonian mechanics which tells us how much energy it will take to accelerate an electron etc.,etc.
Since these aspects of physics affect the instruments of physics themselves ,clocks ,rulers etc. it is natural to encorporate them directly into the coordinate
system as part of the transformation. I.e. An addition to the Galilean transform.
This is just my view of course.

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.

 

[universal_101]

The physical law relates to the probability of decay in a given time. For large numbers, we quantify that as the half-life of the particle.

The above mentioned law is well known, but there is NO law which explain the how many number of particles will reach the Earth. Because, currently we use the part of a transformation to explain this effect.

In the rest frame of the particle, the our atmosphere is thin (due to 'length contraction'), it takes a short time to pass through as the Earth rushes into meet the particle, so fewer particles decay than if they were moving slowly.

Transformed to the Earth frame, the atmosphere is thicker but the particles suffer 'time dilation' which extends their half-life so the number reaching the ground is the same.

Where do you see the problem?

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.

Thanks

 

[universal_101]

The usual decay law: \frac{dn}{d\tau}=-\lambda n which has the solution n=n_0 e^{-\lambda \tau}

Does this law explain or account for the number of particles reaching the Earth, without using any transformation.

 

[universal_101]

The phrase that I've emphasized in bold... It's not necessarily true.

Let's start with a more precise definition of what we're measuring: the number of particles that are detected between two events in spacetime (for example, "I turned the detector on and started counting" and "I turned the detector off and checked the counts"). There is no time or distance involved here, so the results are (unsurprisingly) the same for all observers regardless of relative motion, time dilation, and the like. If we have a sufficiently complete specification of the initial conditions, we can predict this value from a frame-independent physical law that gives the decay time of the particles as a function of the proper time experienced by the particle itself.

Now, different observers may find different rates of arrival at the detector. This also isn't surprising, because the rate of arrival is found by dividing the number of arrivals by the time that the detector is on - and the different observers are measuring time differently so they're dividing by different values, so getting different rates. Different observers may also calculate different particle lifetimes as measured by their different clocks - but again, these are different clocks so there's no surprise there.

However the observers do agree about how their respective clocks are related so after they've made all their measurements they can go back and compare notes. When they do, they'll find that there is no paradox - all of their measurements are consistent with the observation itself, and with the expected particle lifetimes as a function of the passage of time in the particles proper time.

Thanks for your view,

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

Since we use the Lorentz transformation, it cannot be a physical law as argued in the original post.

 

[Mentz114]

Agreed , and I'm also not suggesting that the number of particles should depend on transformation.

Good, because it is invariant.

What I'm suggesting is, it must be governed by a physical law instead of a transformation that which predicts how many particles should reach a particular destination.

The transformation does not predict how many many particles should reach a particular destination. The transformation changes the observers coordinates.

There is a physical law that decides the number, which law happens to be invariant under transformation .

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

Not necessarily. We can use the rest frame of the particle. We only use the LT when we want to see what happens in a different frame.

 

[Nugatory]

Thanks for the view,
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.

No contradiction that I see... The transform describes certain aspects of physical law, namely how observations of time and space differ between observers in relative motion. It's very convenient to describe these differences in terms of coordinate transforms because we generally state our observations of time and space in terms of coordinate systems.

 

[PeterDonis]

Since, a transformation cannot keep the numbers invariant if this phenomena were to be actually explained by the transformation of observers. But as we all know, the transformation around this phenomena does keep the numbers invariant must imply that this phenomena is governed by a physical law and not by the transformation.

The transformation tells how different observers view the invariant event--the death of a character in the play, or the arrival of a given number of unstable particles at a given detector. So just as we expect the "transformation" from one audience viewpoint to another to keep invariant the death of the character in the play (while changing observer-dependent details such as the exact angle at which the character's face is viewed), we expect the transformation from one observer's viewpoint to another to keep invariant the number of unstable particles arriving at the detector (while changing observer-dependent details such as the time, according to that observer, that the particles take to travel from source to detector, or the distance between the two). As of course it does.

So your own analogy perfectly supports the facts of the Lorentz transformation; yet you talk as if you are somehow pointing out a problem. What problem?

 

[universal_101]

I have to say I admire your gall. You don't understand this stuff, which has been around for decades and examined by the best minds of our time - and still you think you've found a paradox.

If something is there for decades and so many people admire it, does not make that something correct or does it. I don't want to include history, which says otherwise.

But 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.

And No, I'm not looking for paradoxes, instead I'm looking for solutions.

Thanks

 

[universal_101]

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

Not necessarily. We can use the rest frame of the particle. We only use the LT when we want to see what happens in a different frame.

I don't think even using the rest frame of the particle, you can calculate the number of particles reaching Earth without using any kind of transformation of any property what so ever.

 

[Mentz114]

If something is there for decades and so many people admire it, does not make that something correct or does it. I don't want to include history, which says otherwise.

Which something are you talking about ? Special relativity ?

But 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.

Sure. What experimental evidence have got ?

And No, I'm not looking for paradoxes, instead I'm looking for solutions.

In the first post of this thread you asked for an explanation of a paradox

... instead I'm looking for solutions.

Solutions to what problem ? The fact that physical laws must be Lorentz invariant is not a problem.

 

[PeterDonis]

I don't think even using the rest frame of the particle, you can calculate the number of particles reaching Earth without using any kind of transformation of any property what so ever.

If you know the time in the rest frame of the particle, what could you possibly need to transform? That's the only variable in the formula DaleSpam posted (everything else is a physical constant or a known initial condition).

 

[universal_101]

The transformation tells how different observers view the invariant event--the death of a character in the play, or the arrival of a given number of unstable particles at a given detector. So just as we expect the "transformation" from one audience viewpoint to another to keep invariant the death of the character in the play (while changing observer-dependent details such as the exact angle at which the character's face is viewed), we expect the transformation from one observer's viewpoint to another to keep invariant the number of unstable particles arriving at the detector (while changing observer-dependent details such as the time, according to that observer, that the particles take to travel from source to detector, or the distance between the two). As of course it does.

So your own analogy perfectly supports the facts of the Lorentz transformation; yet you talk as if you are somehow pointing out a problem. What problem?

The problem is, it is the audience viewpoint of one special position in audience, which is utilized in determining how many characters will die in a certain play.

That is, the number of particles reaching Earth are determined by the tools of transformation. This is a big problem, at-least to my understanding.

 

[Dale]

Does this law explain or account for the number of particles reaching the Earth, without using any transformation.

Yes.

Since it is a law of physics and since all laws of physics are diffeomorphism invariant, we know that it is invariant under the Lorentz transform. But no transform is required in order to use it.

 

[universal_101]

If you know the time in the rest frame of the particle, what could you possibly need to transform? That's the only variable in the formula DaleSpam posted (everything else is a physical constant or a known initial condition).

I think you left the Length part, since in order to calculate how many particles reached, we must know how much they traveled.

And this length is to be transformed.

 

[Mentz114]

I don't think even using the rest frame of the particle, you can calculate the number of particles reaching Earth without using any kind of transformation of any property what so ever.

If you do the calculation in the rest frame of the particle, only the coordinates of that frame are used. No transformation is used. As others have said above.

I think you left the Length part, since in order to calculate how many particles reached, we must know how much they traveled.

Yes, measured in rest frame coordinates.

 

[PeterDonis]

I think you left the Length part, since in order to calculate how many particles reached, we must know how much they traveled.

And this length is to be transformed.

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?

 

[Nugatory]

Does this law explain or account for the number of particles reaching the Earth, without using any transformation.

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.

 

[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.