## Transformation Vs. Physical Law

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,

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 Quote by universal_101 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.

 Quote by universal_101 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.

## Transformation Vs. Physical Law

 Quote by Mentz114 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.

 Quote by Mentz114 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.
 Quote by Mentz114 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.

 Quote by vin300 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.

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 Quote by universal_101 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.

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 Quote by universal_101 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?

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 Quote by universal_101 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.

 Quote by 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,

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.

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 Quote by universal_101 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.

 Quote by DaleSpam 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 !

 Quote by universal_101 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 in to 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?

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 Quote by universal_101 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 $$n=n_0 e^{-\lambda \tau}$$

 Quote by Austin0 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.

 Quote by GeorgeDishman 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.

 Quote by GeorgeDishman 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 in to 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

 Quote by DaleSpam 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.

 Quote by Nugatory 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.