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: [tex]\frac{dn}{d\tau}=-\lambda n[/tex] which has the solution [tex]n=n_0 e^{-\lambda \tau}[/tex]

 

[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: [tex]\frac{dn}{d\tau}=-\lambda n[/tex] which has the solution [tex]n=n_0 e^{-\lambda \tau}[/tex]

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? [itex]\lambda[/itex] is the usual non-increased half life. It is not a function of speed.

 

[universal_101]

Where, exactly? [itex]\lambda[/itex] 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.