Consistency of the speed of light

[clj4]

Again. What do you feel I have said that is incorrect regarding the field tensor? What do you feel I have misunderstood?

Are you denying that the form of Maxwell's equations and the Lorentz force law change form in GGT depending on whether you choose the contravarient or the covarient field tensor to define the electromagnetic fields in the GGT frame? The form of maxwell's equations requires this. And also, you'd be contradicting Gagnon's ref 9.Sure I have. I'll post it again. Follow through the proof yourself:

I found the form of physics in GGT frames not worth the effort. So instead I chose to do the calculations in a "lorentz frame", transform to some arbitrary "special frame" (where GGT and SR are defined to agree), then transform back to the "lab GGT frame". Because GGT and SR have identical metrics in the special frame, and have identical definitions of proper time (invarient interval $$ds^2=c^2 dt^2$$ is always true in the clock's rest/"proper" frame according to both SR and GGT), the frequency measured in a GGT frame agrees with the SR value (independent of the choice of "special frame").

Which statement do you deny?

Ah, now the "gregory" personality re-emerges (and the same prose, with no math).
Which reminds me , how did you convince "gregory" that he was wrong about the boundary conditions? You never produced the mathematical proof. Knowing "gregory", this wasn't such an easy task. Can you show us?

 

[NotForYou]

Ah, now the "gregory" personality re-emerges (and the same prose, with no math).
Which reminds me , how did you convince "gregory" that he was wrong about the boundary conditions? You never produced the mathematical proof. Knowing "gregory", this wasn't such an easy task. Can you show us?

I AM NOT GREGORY! You have been warned repeatedly to stick to the physics, and stop making accusations. Please do so.

He's my roommate. We can sit down and discuss the tensor equations. Two other friends have worked through the equations as well. Since the result was that I don't believe Gagnon made an error by assuming the boundary conditions were the same using his definition, there is no point in showing all the calculations (since everyone agrees with that conclusion).



As for the proof, why prose and no math? Because the result is basically by definition, so there is nothing really to show. Here, let me show you step by step. Tell me which step you disagree with.

1] DEFINITION: GGT and SR agree on the physical laws in one "special frame".

2] DEFINITION: let w = cutoff frequency of waveguide according to SR in the lab frame. This is measured by two events on a stationary clock, the time between two peaks T = 2 pi / w.

3] Transform into the "special frame". The proper time of the clock is invarient. So if we did ALL the calculations according to the physical laws in this moving frame ... while more complicated ... we know that the result is that SR will predict that the clock will still measure T.

4] GGT and SR agree on the laws of physics in this special frame. So doing ALL the calculations according to SR in this "special" moving frame is equivalent to doing the calculations for GGT in this frame.

5] Thus GGT also predicts that the clock will measure T.

6] Thus GGT and SR agree on the cutoff frequency of the waveguide, independent of the choice of the "special frame".


Which step do you deny?

 

[clj4]

I AM NOT GREGORY! You have been warned repeatedly to stick to the physics, and stop making accusations. Please do so.

He's my roommate. We can sit down and discuss the tensor equations. Two other friends have worked through the equations as well. Since the result was that I don't believe Gagnon made an error by assuming the boundary conditions were the same using his definition, there is no point in showing all the calculations (since everyone agrees with that conclusion).
As for the proof, why prose and no math? Because the result is basically by definition, so there is nothing really to show. Here, let me show you step by step. Tell me which step you disagree with.

1] DEFINITION: GGT and SR agree on the physical laws in one "special frame".

2] DEFINITION: let w = cutoff frequency of waveguide according to SR in the lab frame. This is measured by two events on a stationary clock, the time between two peaks T = 2 pi / w.

3] Transform into the "special frame". The proper time of the clock is invarient. So if we did ALL the calculations according to SR in this moving frame ... while more complicated ... we know that the result is that SR will predict that the clock will still measure T.

4] GGT and SR agree on the laws of physics in this special frame. So doing ALL the calculations according to SR in this "special" moving frame is equivalent to doing the calculations for GGT in this frame.

5] Thus GGT also predicts that the clock will measure T.

6] Thus GGT and SR agree on the cutoff frequency of the waveguide.Which step to you deny?

It doesn't work this way: you do the math.
Besides, you miss the point: Gagnon measured a difference in phase
$$\phi$$. Your task is to disprove (9). It has always been , but you keep trying to expedite it with prose or to do it by disproving other things. Remember, Gagnon sets to measure $$\phi$$.
$$\omega_c$$ is not relevant to the discussion. It is just a byproduct of the derivation of (7). You can view it as a value very close to $$\omega_1_0$$.
Think about it: what would you need to write to Phys Rev to refute the Gagnon paper? For sure, the prose above wouldn't wash.

Let's see the equations that disprove Gagnon (9).
While you are at it, I still want to see what you wrote to convince "gregory" of his mistake.

 

[clj4]

6] Thus GGT and SR agree on the cutoff frequency of the waveguide.

Ok, and the impact on (9) is? Can we have the mathematical disproof of (9)?

 

[NotForYou]

$$\omega_c$$ is not relevant to the discussion.

It is. Because I can show that GGT predicts the same $$\omega_c$$ as SR. Therefore Gagnon is wrong (eq. 8).

If you disagree with me, go back to my previous post and tell me which step you disagree with and why.

 

[clj4]

It is. Because I can show that GGT predicts the same $$\omega_c$$ as SR. Therefore Gagnon is wrong (eq. 8).

If you disagree with me, go back to my previous post and tell me which step you disagree with and why.

The item in discussion is k (eq. 7).
You just rederived k in agreement with Gagnon.
K intervenes in calculating (9) , the main equation of the Gagnon paper. So, your task is, and has always been, to disprove (9). Ank k looks different in GGT.
You will need to write all the math that supports steps [1]-[6] above and to rederive (9), the same way you did it with (7,8). When we agree that the derivation is correct, we are done. Until then, you have proven nothing.

And , please, show us how you proved "gregory" wrong. After all , I posted my disproof, it is only fair that you post yours.

 

[NotForYou]

And , please, show us how you proved "gregory" wrong. After all , I posted my disproof, it is only fair that you post yours.

I refuse to waste my time typing up a result that you already agree with. That is the end of it.

Besides, I explained what I calculated. You can reproduce it yourself if you really wish.

Until then, you have proven nothing.

Sorry, it doesn't work that way.
If you do a complicated calculation regarding 100 colliding electrons, and find that the total momentum is not conserved ... if I can prove that momentum is conserved, I can prove that your calculation is wrong ... without ever looking at it.

I have proven Gagnon's equation 8 is wrong.

Yes, in the future I'd like to find where Gagnon's error occurs, but I don't need to do that to prove he is wrong.


Again, if you disagree with my proof, then you must state which step you disagree with and why.

 

[NotForYou]

Enough bickering. I'd like to continue the search for Gagnon's error. This requires us first to agree on his calculations.


I want us to 100% agree on Gagnon's calculation of the electromagnetic wave, because if I show in the future where their error is I don't want to deal with any "backpeddling" trying to claim they didn't make an error. Because after finding their error, I'm not going to bother trying to correct their calculations. As I showed, there is a much easier way to find the predictions. (Please take good note of what I just said: I never intend to try to correct their calculations, as that is a waste of time. Do not expect this. Do not ask for this.)

So, the real question is:

1] Do we agree the electromagnetic wave that I calculated:
a) satisfies the wave equation (Gagnon eq 5)
b) satisfies the $$E_\parallel=0$$ boundary condition
c) agrees with Gagnon's dispersion relation (eq 7)
d) agrees with Gagnon's cutoff frequency (eq 8)

2] Do you agree this electromagnetic wave is the same as what Gagnon calculated?

 

[clj4]

As long as you agree to:

1. Prove Gagnon (9) wrong directly and mathematically (not through prose as in your chain of statements like [1]-[6])

2. You remember your derivation:

Gagnon defined the cutoff frequency as that which made k=0.
$$k=0= -\frac{v_z}{c}\frac{\omega_c}{c}+\frac{1}{c}\sqrt{(1-\frac{v_x^2}{c^2})\omega_c^2 - \omega_{10}^2}$$
$$\frac{v_z^2}{c^2}\frac{\omega_c^2}{c^2}=(1-\frac{v_x^2}{c^2})\frac{\omega_c^2}{c^2} - \frac{\omega_{10}^2}{c^2}$$
$$\omega_c = \omega_{10} (1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2})^{-1/2}$$

which is a DEFINITION of $$\omega_c$$ as the pulsation that "makes k=0". Nothing more nothing less.A DEFINITION. So you will not try again to use (8) as a criterion of demonstrating that Gagnon paper is invalid (see details below).

I'm gone skiing for the day. "See" you in the evening.

 

[clj4]

I have proven Gagnon's equation 8 is wrong.

No, for the last time, you haven't. A simple way of showing that is the following:

1. You DEFINED $$\omega_1_0=c*\frac{\pi}{a}$$

2. You further derived that $$\omega_c$$ as the pulsation that "makes k=0". So , this is nothing more nor less than the solution of an equation.See, I can write prose like you do :-)
So, please drop your claims that you found an error in Gagnon (8)

Yes, in the future I'd like to find where Gagnon's error occurs, but I don't need to do that to prove he is wrong.

And declare victory without mathematical proof? Give us some more prose? No way.

Again, if you disagree with my proof, then you must state which step you disagree with and why.

See above.

 

[clj4]

I refuse to waste my time typing up a result that you already agree with. That is the end of it.

Well, you called mine wrong (it turns out that it isn't). It is only fair to show how you did it. Knowing how difficult (if not impossible) it is to prove "gregory" wrong, this must be a real work of art, so I would like to see it and learn from it.

"See" you in the evening.

 

[NotForYou]

Well, you called mine wrong (it turns out that it isn't). It is only fair to show how you did it.

It IS wrong, and I already told you why several times.
You defined one of maxwell's equations to be invarient that Gagnon did not. You are disagreeing directly with the results of gagnon ref 9.

No, for the last time, you haven't. A simple way of showing that is the following:

1. You DEFINED $$\omega_1_0=c*\frac{\pi}{a}$$

2. You further derived that $$\omega_c$$ as the pulsation that "makes k=0". So , this is nothing more nor less than the solution of an equation.

1. Yes. Because $$\omega_1_0$$ is DEFINED by gagnon as the "usual value" of the mode frequency (ie the value according to SR, or equivalently, the value if the waveguide rest frame was the "absolute frame").

2. No. You're relying on Gagnon's dispersion relation to be correct. You are trying to use Gagnon's incorrect results to prove itself. This is not a valid proof. (Besides, as I showed, his dispersion relation (eq 7) is wrong because it predicts the wrong cutoff frequency.)


So please, try again. Tell me which step in my Gagnon disproof you believe is wrong.

So, please drop your claims that you found an error in Gagnon (8)

No. The proof is straight forward and correct. Because of this, I don't even need to look at Gagnon's paper to know that it is wrong. We are only looking to satisfy our curiousity.

--------------------------------------------

As long as you agree to:
1. Prove Gagnon (9) wrong directly and mathematically (not through prose as in your chain of statements like [1]-[6])

I told you that I do not intend to fix their calculations. I believe they made a big error somewhere that invalidates their derivation of eq 7 ... immediately invalidating all further work. So I only intend to find where their derivation of eq 7 is wrong. That is where I will stop. Because to do anything more would require me to start their work over from scratch ... which isn't "correcting" their paper anymore, it is just redoing it. I told you I am not willing to waste my time on that.

So I only intend to show that their derivation of eq 7 is wrong. That is where I will stop.



So again - I want us to 100% agree on Gagnon's calculation of the electromagnetic wave, because if I show in the future where their error is I don't want to deal with any "backpeddling" trying to claim they didn't make an error.

So to make 100% sure, let me know if you agree on the following:

1] Do we agree the electromagnetic wave that I calculated:
a) satisfies the wave equation (Gagnon eq 5)
b) satisfies the boundary condition $$E_\parallel=0$$
c) agrees with Gagnon's dispersion relation (eq 7)
d) agrees with Gagnon's cutoff frequency (eq 8)

2] Do you agree this electromagnetic wave is the same as what Gagnon calculated?

 

[clj4]

I believe they made a big error somewhere that invalidates their derivation of eq 7 ...

Go to post 391.

 

[NotForYou]

I believe they made a big error somewhere that invalidates their derivation of eq 7 ...

Go to post 391.

I told you, I will not take the time to fix their errors. I will however, take the time to FIND their errors. Given time, I am confident that I can find the error in their derivation.


So, I'd like to continue the search for Gagnon's error. This requires us first to agree on his derivation.


I want us to 100% agree on Gagnon's calculation of the electromagnetic wave, because if I show in the future where their error is I don't want to deal with any "backpeddling" trying to claim they didn't make an error.

So, the question is:

1] Do we agree the electromagnetic wave that I calculated:
a) satisfies the wave equation (Gagnon eq 5)
b) satisfies the $$E_\parallel=0$$ boundary condition
c) agrees with Gagnon's dispersion relation (eq 7)
d) agrees with Gagnon's cutoff frequency (eq 8)

2] Do you agree this electromagnetic wave is the same as what Gagnon calculated?

If we agree on these, then we can move on ... and I will have to find Gagnon's error.

 

[clj4]

Given time, I am confident that I can find the error in their derivation.So, I'd like to continue the search for Gagnon's error. This requires us first to agree on his derivation.I want us to 100% agree on Gagnon's calculation of the electromagnetic wave, because if I show in the future where their error is I don't want to deal with any "backpeddling" trying to claim they didn't make an error.

So, the question is:

1] Do we agree the electromagnetic wave that I calculated:
a) satisfies the wave equation (Gagnon eq 5)
b) satisfies the $$E_\parallel=0$$ boundary condition
c) agrees with Gagnon's dispersion relation (eq 7)
d) agrees with Gagnon's cutoff frequency (eq 8)

2] Do you agree this electromagnetic wave is the same as what Gagnon calculated?

If we agree on these, then we can move on ... and I will have to find Gagnon's error.

Please, continue looking for the error. But try not to be tricky, the way you set your questions would make sure you that you have no work to do since you already "divined" the fact that (8) is wrong.Do you think you are dealing with a bunch of diletantes? So, I agree ONLY that:

1] Do we agree the electromagnetic wave that I calculated:
a) satisfies the wave equation (Gagnon eq 5)
b) satisfies the $$E_\parallel=0$$ boundary condition

A) I also agree that you used a correct (albeit very restrictive) mathematical method in order to derive (7,8) from (5) such that (7,8) look exactly as in the paper.

B) So, if there is any error in their thinking it has to be in deriving (5).

C) We have already excluded the initial conditions of (5) since you "proved" "gregory" wrong.

Out of curiosity, I wonder what are you hoping to achieve? Suppose that by absurd you manage to prove Gagnon wrong, you will still have to deal with C.M.Will's paper that shows clearly that the M-S theory is not indistinguishable from SR. It even tells you exactly why (and so do M-S themselves) The same goes for the Krisher-C.M.Will paper. So why drag this agony for another 200 posts?

 

[NotForYou]

Here is the issue. I believe that Gagnon made a mistake in their derivation. I believe the electromagnetic wave that they get is incorrect. However they don't show their solution of the wave equation ... so we need to agree on what their solution of the wave equation is, so that I may continue looking for their specific error.

Again, to do this, we need to agree on what their solution of the wave equation is.

#1] Do we agree on what their solution of the wave equation is?

So, I agree ONLY that:

A) I also agree that you used a correct (albeit very restrictive) mathematical method in order to derive (7,8) from (5) such that (7,8) look exactly as in the paper.

Yes, I believe that I have recreated their solution to the wave equation.

You say my math is "correct" but "very restrictive". Do you believe that I have not recreated their solution? If so, say so now. I don't want to have to return to this issue.

Once we agree here, then it will be upon my shoulders to show exactly what their error was. Let me know if we agree and I'll get to work...

 

[clj4]

Find Gagnon's error. The only thing left for you to work on is the partial differential equation (5).

 

[NotForYou]

Find Gagnon's error. The only thing left for you to work on is the partial differential equation (5).

Here is the issue. I believe the electromagnetic wave that they get is incorrect. However they don't show their solution of the wave equation ... so we need to agree on what their solution of the wave equation is, so that I may continue looking for their specific error.

Again, to do this, we need to agree on what their solution of the wave equation is.

#1] Do we agree on what their solution of the wave equation is?

All they state is that
- they started with the wave equation (eq 5)
- looked for a TE mode
- assumed the z,t dependence to be exp(ikz - iwt)
- applied the boundary condition $E_\parallel=0$
- their solution has a dispersion relation of eq 7
- their solution has a frequency for k=0 according to eq 8

I did their steps and got a solution that agrees with eq 7,8.

$$E_z = 0$$ (TE mode)
$$E_x = \exp(-yf/2)[\exp(+y\sqrt{f^2/4-g}) - \exp(-y\sqrt{f^2/4-g})]\exp(ikz-i\omega t)$$
where
$$f= - 2\frac{v_y}{c}\frac{i\omega}{c}$$
$$g=-k^2 +2k\frac{v_z}{c}\frac{\omega}{c} + (1-\frac{v^2}{c^2})\frac{\omega^2}{c^2}$$

Do we agree that this is Gagnon's solution?


If we don't agree, let's work on that. If we do agree, then I can continue my search.

 

[clj4]

This is what you got at post 361 for $$E_y$$, you applied the zero boundary condition to it and you got k. All the calculations seemed correct, we double checked them together and they reproduced Gagnon (7,8) perfectly.
It may, or may not be the correct procedure since the correct way of dealing with waveguides is to use $$E(x,y)= E_x(x)*E_y(y)$$ and to separate the original equation into two equations, one in x and the other one in y.

So where is this thing going? Looks like you are going in circles. You need to either prove wrong:

1. Eq(5) (the partial differential equation that is at the origin of it all

or

2. Eq(9) , i.e. the expression that is 0 for SR and non-zero for GGT

So why do you keep coming back to (7,8)?

 

[NotForYou]

This is what you got at post 361 for $$E_y$$, you applied the zero boundary condition to it and you got k. All the calculations seemed correct, we double checked them together and they reproduced Gagnon (7,8) perfectly.

I assume you mean $$E_x$$, but yes.

It may, or may not be the correct procedure since the correct way of dealing with waveguides is to use $$E(x,y)= E_x(x)*E_y(y)$$ and to separate the original equation into two equations, one in x and the other one in y.

Whoa... hold on. We appear to be using different notations. (maybe that has lead to some confusion) I have used $$E$$ to refer to the electric field (a vector field). And $$E_x,E_y,E_z$$ to refer to the components of that vector. However, you seem to take E to just be a component and then further, the subscript to just refer to the separable function of just that subscript.

I hope this hasn't been the cause of some of the confusion.
Different notation is fine as long as all parties are aware of it (but it is easier if we just use one).

So where is this thing going? Looks like you are going in circles. You need to either prove wrong:

1. Eq(5) (the partial differential equation that is at the origin of it all
or
2. Eq(9) , i.e. the expression that is 0 for SR and non-zero for GGT

So why do you keep coming back to (7,8)?

I'm trying to get us to agree on Gagnon's solution to the wave equation because that is where his error is.

--------------------
Where is their error?

I'll let Griffith's Introduction to electrodynamics (3rd ed.) do the talking:

$$\nabla^2\vec{E} = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \vec{E}, \ \ \nabla^2\vec{B} = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \vec{B}$$ (eq 9.41)

(pg 377) "Now the wave equations for E and B (Eq. 9.41) were derived from Maxwell's equations. However, whereas every solution to Maxwell's equations (in empty space) must obey the wave equation, the converse is not true; Maxwell's equations impose extra constrains on E and B."

This makes sense because Maxwell's equations are 4 equations (with E and B coupled), which we reduced to just 2 uncoupled equations. Here's an easy example:
$$\vec{B}=0, E_x=0,E_y=0, E_z = \cos(kx - kt/c)$$ is a solution to the wave equations (eq 9.41), but obviously aren't solutions to Maxwell's equations (because $$\frac{1}{c^2}\frac{\partial}{\partial t} \vec{E} \ne \nabla \times \vec{B} = 0$$).

That was about Maxwell's equations in a "normal Lorentz frame" and not a GGT frame, but the reasons behind it remain the same.

This is where Gagnon makes their error. Let me demonstrate.
$$\nabla \times E = -\frac{\partial}{\partial t}B$$ (still true in a GGT frame, according to Gagnon's choice ... see ref 9)
$$-\frac{\partial E_x}{\partial z} + \frac{\partial E_z}{\partial x} = -\frac{\partial B_y}{\partial t}$$
$$-ikE_x = -\frac{\partial B_y}{\partial t}$$

$$B_y = -\frac{k}{w} E_x + function(x,y,z)$$

So we are not free to just make B whatever we want and ignore other boundary conditions. Remember, from the boundary condition on a waveguide $$B_\perp=0$$ we know $$B_y(x=0)=B_y(x=b)=0$$, where b is the width of the waveguide in the x direction. Yet, $$B_y$$ cannot satisfy this. So their solution to the wave equation is not valid.

No where in the paper do they mention this boundary condition. Since they were solving for the electric field, I believe they just felt it was not relevant (as you yourself did when boundary conditions were first brought up). This is their error.

 

[clj4]

Whoa... hold on. We appear to be using different notations. (maybe that has lead to some confusion) I have used $$E$$ to refer to the electric field (a vector field). And $$E_x,E_y,E_z$$ to refer to the components of that vector. However, you seem to take E to just be a component and then further, the subscript to just refer to the separable function of just that subscript.

Yes, I used a different notation, let me use your exact notation from post 361 to make things clear.

$$E_x(x,y,z,t)=X(x)Y(y)exp(ikz-i\omega*t)$$

I think that the problem in your assumption that Gagnon made an error in deriving the solution to equation (5) stems from your simplifying hack X(x)=const which I criticised earlier. If you look up reference [1] (the book by Stuart Wentworth, p338-355) the solutions depend on both variables, i.e. BOTH x and y.
So , you should not end up with an ordinary differential equation as you did in post 361 but with a system of seprarable equations, in BOTH y AND x. This is YOUR mistake that you keep trying to attribute to Gagnon.

I'm trying to get us to agree on Gagnon's solution to the wave equation because that is where his error is.

--------------------
Where is their error?

I'll let Griffith's Introduction to electrodynamics (3rd ed.) do the talking:

$$\nabla^2\vec{E} = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \vec{E}, \ \ \nabla^2\vec{B} = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \vec{B}$$ (eq 9.41)

(pg 377) "Now the wave equations for E and B (Eq. 9.41) were derived from Maxwell's equations. However, whereas every solution to Maxwell's equations (in empty space) must obey the wave equation, the converse is not true; Maxwell's equations impose extra constrains on E and B."

This makes sense because Maxwell's equations are 4 equations (with E and B coupled), which we reduced to just 2 uncoupled equations. Here's an easy example:
$$\vec{B}=0, E_x=0,E_y=0, E_z = \cos(kx - kt/c)$$ is a solution to the wave equations (eq 9.41), but obviously aren't solutions to Maxwell's equations (because $$\frac{1}{c^2}\frac{\partial}{\partial t} \vec{E} \ne \nabla \times \vec{B} = 0$$).

That was about Maxwell's equations in a "normal Lorentz frame" and not a GGT frame, but the reasons behind it remain the same.

This is where Gagnon makes their error. Let me demonstrate.
$$\nabla \times E = -\frac{\partial}{\partial t}B$$ (still true in a GGT frame, according to Gagnon's choice ... see ref 9)
$$-\frac{\partial E_x}{\partial z} + \frac{\partial E_z}{\partial x} = -\frac{\partial B_y}{\partial t}$$

Not to be nitpicky but you mean:

$$\frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x} = -\frac{\partial B_y}{\partial t}$$

Small elementary mistakes...

$$-ikE_x = -\frac{\partial B_y}{\partial t}$$

You mean:

$$ikE_x = -\frac{\partial B_y}{\partial t}$$ ?

$$B_y = -\frac{k}{w} E_x + function(x,y,z)$$

ahem, you mean:

$$B_y = \frac{k}{w} E_x + function(x,y,z)$$

Hmmm...In your rush to prove Gagnon wrong haven't you forgotten a second equation? (actually several of them)? :-)
The above "disproof" is pure conjecture, it is mathematically incorrect, it is incomplete and irrelevant.

There is no reason whatsoever to infer that Gagnon may have missed the correct application of boundary conditions either from the Gagnon paper or from the Chang paper.
Did you make up a strawman to defeat it?
What in the Gagnon paper let's you believe that what you wrote above about "Gagnon's mistake" is true? There is absolutely nothing in the paper that would give any credibility to your conjecture relative to boundary conditions. Actually , it looks like you pulled it all out of your b...

So we are not free to just make B whatever we want and ignore other boundary conditions. Remember, from the boundary condition on a waveguide $$B_\perp=0$$ we know $$B_y(x=0)=B_y(x=b)=0$$, where b is the width of the waveguide in the x direction. Yet, $$B_y$$ cannot satisfy this. So their solution to the wave equation is not valid.

Did it dawn on you that nowhere in the Gagnon paper there is any explicit solution to the wave guide equation? That all they show is the k expression? How can you infer anything about solutions, boundary conditions, etc? Your writeup is pure conjecture (and incorrect as well) that you are trying to pass as fact.Just because you cannot find the correct general solution of Gagnon eq (5) doesn't mean that Gagnon set the equation incorrectly. On the other hand it is clear that you need to go back to your post 361 and redo your calculations. The right way this time.

 

[clj4]

You can temporarily get Gagnon's ref (9) here:

Thank you, very nice of you!

 

[Aether]

Thank you, very nice of you!

My pleasure. Since this is my issue I'm happy to do as much of the foot-work as I can.

 

[NotForYou]

Let us step back a second. What IF Gagnon made an error solving the wave equation? He didn't explicitly show his solution to the wave equation. Does this mean it is impossible to prove him wrong?

No. One can do the calculations themselves and show what the correct answer is. However, clj4 refuses to let anyone progress in this manner - we "have to find Gagnon's specific error". This is not possible, because they don't explain their calculations explicitly enough, let alone show their solution to the wave equation.

I know clj4 cannot be advocating that someone who doesn't show their solution will always be right. So clj4, if we're not allowed to calculate the result ourself, how ARE we supposed to prove Gagnon wrong?

---------------------------

Here I will calculate the result myself.

1] DEFINITION: In SR, the invarient line element is $$ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2$$.

2] DEFINITION: GGT and SR agree on the physical laws in one chosen "special frame", and agree on the metric / invarient line element in this frame.

3] DEFINITION: "Generalized Galilean trasforms" are defined as a modification of Lorentz transformations by changing the simultaneity convention such that simultaneity is always agreed upon with the "special frame". (You can read what this definition makes the transformations look like mathematically in Gagnon ref 9, eq 2.)

4] DEFINITION: Proper time is the time a clock would measure between two events which are on its path and coincident with its position. That is $$\tau = \int_{e_1}^{e_2} dt$$ in the clock's rest frame (where $e_1,e_2$ are the two events).

5] From #4 given #1, the proper time in SR is $$\tau = \int_{e_1}^{e_2} dt = \frac{1}{c} \int_{e_1}^{e_2} \sqrt{-ds^2}$$ because the two events are coincident with the clocks position and the clock does not move in its rest frame, therefore $$dr=0$$ in the clocks rest frame.

6] Gagnon ref 9 shows that from #3, given #2 and #1, the GGT line element in an arbitrary GGT frame is:
$$ds^2 = (dr)^2 -c^{-2}(dr \cdot v)^2 + 2(v \cdot dr)dt - c^2 dt^2$$

7] From #4 given #6, the proper time in GGT is also $$\tau = \int_{e_1}^{e_2} dt = \frac{1}{c} \int_{e_1}^{e_2} \sqrt{-ds^2}$$ because the two events are coincident with the clocks position and the clock does not move in its rest frame, therefore $$dr=0$$ in the clocks rest frame.

8] From #2, if we do all calculations in the "special frame" both GGT and SR will agree on the path and two event coordinates cooresponding to a clock measuring time.

9] From #8 given #7 and #5, GGT and SR will always agree on the proper time, the time measured by a moving clock.

10] Given #9, therefore no experiment can distinguish between SR and GGT by measuring a time difference using a clock.


Since Gagnon's experiment is just measuring the phase difference, which is just a time measurement ($$\Delta \phi = \omega \Delta t$$), it can not distinguish between SR and GGT.

 

[clj4]

So clj4, if we're not allowed to calculate the result ourself, how ARE we supposed to prove Gagnon wrong?

By being honest and by not trying to snow us at very turn. You have argued that:

1. The boundary conditions were not transformed correctly (as "gregory").

When this proved to be false you argued that:

2. Expression (8) is wrong

When this proved to be false (you proved it yourself in post 361) you argued that:

3. Gagnon did not solve his wave equation (5) correctly.

When this was proven wrong , you are coming back with some more stories (see above post). T.Chang, the author of ref. 9 clearly disagrees with your post when he proposes his paper as the starting point of differentiating between GGT and SR. Between you (who have a history of miscues, errors, irrelevant proofs) and Chang , Kirsher, C.M.Will, (the editors and the reviewers of Phys Rev.) etc, who claim exactly the opposite in peer refereed journals, one would always pick the latter.

We understand that you are an "aetherist" committed to proving the equivalence between the M-S theory and SR. But there is mounting eveidence that your claims are not correct (even M-S disagree).

I already told you how you can prove Gagnon wrong: by going back over your calculations at post 361 and doing them right. If you are willing to wait one day, I will even help you . I will post the clues as to how to solve Gagnon (5) correctly. And, BTW, with all the correct boundary conditions in place :-)

 

[NotForYou]

By being honest and by not trying to snow us at very turn.

This does not answer the question.

IF gagnon made a mistake in solving the wave equation, since they don't show their solution, how can one show WHERE their error is? You can't. You can only show that they are wrong. But you refuse to let us solve for the predictions of GGT ourselves to show that they are wrong.

You have argued that:
1. The boundary conditions were not transformed correctly (as "gregory").

You have serious issues. I am not gregory. As the moderator strongly told you: stop accusations and stick to the physics!

2. Expression (8) is wrong
When this proved to be false (you proved it yourself in post 361) you argued that:

3. Gagnon did not solve his wave equation (5) correctly.

No, I did not "prove myself wrong" in post 361. I reproduced Gagnon's results and showed that their solution was wrong (ie post 361 is wrong). That was my intention all along. I never claimed post 361 represented the correct calculations for the experiment, I only claimed that they were Gagnon's calculations for the experiment.

I already told you how you can prove Gagnon wrong: by going back over your calculations at post 361 and doing them right.

No. You have already shown that you will not accept anything that disagrees with Gagnon. Since working out the calculations correctly would disagree with Gagnon, you would then declare that the calculations must be wrong.

In other words, you have made it so that you REFUSE to even look at any calculations that show Gagnon is wrong. This is completely unscientific.

Read my proof above. It shows why Gagnon's experiment (and many others) CANNOT distinguish between GGT and SR.

I have backed up my statements. If you disagree, back up your statements by showing us where the "error" is in the proof above.

 

[clj4]

This does not answer the question.

IF gagnon made a mistake in solving the wave equation, since they don't show their solution, how can one show WHERE their error is? You can't. You can only show that they are wrong. But you refuse to let us do any calculations showing they are wrong.

It does answer the question in the sense that your "aetherist" religion has promped you to forsake the scientific truth, the whole ADDITIONAL body of evidence and has driven you, in front of hundreds of daily viwers to giving us phony proof after phony proof. I refused and I continue to refuse to let you do any PHONY calculation . I will help you (see above post) if you agree to your calculations properly and honestly. I am interested in the scientific truth but you can't snow me, nor the other thread viewers. If Gagnon is wrong, then so be it, but prove it mathematically, not thru statements.
I am perfectly confortable with the fact Gagnon might be wrong, after all , there are much stronger papers from Krisher , C.M.Will that prove that M-S and SR. are not equivalent.
I am willing to help you analyze Gagnon in a HONEST, MATHEMATICALLY CORRECT way. But I will not prostitute myself in the name of religion.

No, I did not "prove myself wrong" in post 361. I reproduced Gagnon's results and showed that their solution was wrong (ie post 361 is wrong). That was my intention all along. I never claimed post 361 represented the correct calculations for the experiment, I only claimed that they were Gagnon's calculations for the experiment.

No, the error is YOURS. You reduced the problem to a unidimensional one by using the gratuitous hack X(x)=constant.
Gagnon and his collaborators are much better than you want to make them look. Give them some respect.
As I mentioned, I can help you find the correct solution that shows the correct dependency on both y AND x. I believe that this is what Gagnon really did. This will lead you to figuring out k which is the key to Gagnon eq (9) which is the key to the whole experiment.

No. You have already shown that you will not accept anything that disagrees with Gagnon. Since working out the calculations correctly would disagree with Gagnon, you would then declare that the calculations must be wrong.

I don't "declare" your calculations wrong. I have proven them wrong, time and again. We could all see the holes in them. From 100miles. If you are willing to sit down and do your calculations correctly, then I will remove all my objections.

I have backed up my statements. If you disagree, back up your statements by showing us where the "error" is in the proof above.

Simple: rework the solution to Gagnon eq (5) on a correct basis, i.e. go thru the separation of variables. If you do this correctly after I show you how to approach the problem, I will have no objection. You will have to abandon your "aetherist" religion though in order to do that. At least for the duration of the calculations :-)

 

[gregory_]

No clj4. NotForYou and I are indeed different people. Please stick to the physics.

What in the Gagnon paper let's you believe that what you wrote above about "Gagnon's mistake" is true? There is absolutely nothing in the paper that would give any credibility to your conjecture relative to boundary conditions.

I worked with my roommate on this. We have stated our reaons for this:

#1) Their solution treats $v_x$ and $v_y$ vastly different. Due to the symmetry of the problem, this does not seem possible. So we need to make some assumptions to even account for that.

#2) Also, while I admit it is conjecture, there IS evidence to give credibility to our conjecture.
- They explicitly mention the $E_\parallel=0$ boundary condition. They do not mention the $B_\perp=0$ boundary condition.
- They say they state that their approach is to solve the wave equation. They mention applying a boundary condition as a constraint, but never mention going back and applying Maxwell's equations as further constraints (which is necessary).


So there IS evidence to give credibility to our conjecture.
But, it IS ultimately a conjecture. They do not show their solution to the wave equation. For this reason it is IMPOSSIBLE to show definitively where Gagnon's error is. This does not mean it is IMPOSSIBLE for Gagnon to be wrong (and I truly hope clj4 is not claiming this).

It means the only way to show Gagnon is wrong is to work out the predictions to this experiment ourselves, and show what the correct prediction is. Clj4, if you do not allow this ... you are making it so that Gagnon can't even be wrong in principle. You have trapped yourself in an unscientific circular logic.

So the only means to proceed is: work out the predictions to this experiment ourselves, and check if Gagnon is correct or incorrect.


I have shown before that this experiment would predict the same result in a GGT or SR frame. NotForYou has as well (albeit he was more specific).

Clj4, you can't just state that we are wrong without even considering our arguements. If you want to say we are wrong, back it up ... show us where you believe NotForYou's proof is wrong and why you believe that.

 

[clj4]

Sure, there is a way. Solve Gagnon (5). Go ahead.
I will even help you.

 

[NotForYou]

Sure, there is a way. Solve Gagnon (5). Go ahead.
I will even help you.

If we get a different answer than Gagnon, will you believe that Gagnon is wrong, or believe that we made a mistake?

Besides, we are starting all over to resolve it. So why not solve it the easier way? Do the calculations in the "special frame" where SR and GGT agree. It is not necessary to solve gagnon eq 5.

 

[clj4]

If we get a different answer than Gagnon, will you believe that Gagnon is wrong, or believe that we made a mistake?

Provided that you two don't try to cheat in your calculations, yes. Be aware that we will need to take the calculations a little beyond computing k, all the way to eq (9). Because if (9) still shows a second order effect, Gagnon is still right. This should be straightforward.
I only care about the scientific truth. If you manage to prove Gagnon wrong you get C.M.Will to refute next. This should be fun.

You game?

 

[NotForYou]

What is wrong with doing the calculations in the "special frame" where SR and GGT agree? That will be much easier.

 

[NotForYou]

I see you added this:

I only care about the scientific truth. If you manage to prove Gagnon wrong you get C.M.Will to refute next. This should be fun

This is the very reason that I want to discuss _general_ results. It will speed up this discussion tremendously. If you cared about scientific truth, you would consider my general proofs like above. So let us start there.

 

[gregory_]

IMPORTANT NOTE:
I don't know if it is a typo, or if it is just difficult to read, but eq 2 in Chang's paper (Gagnon ref 9) should read:

$$\bf{r} = \bf{r}_0 + \frac{\gamma -1}{v^2}(\bf{r}_0 \cdot \bf{v})\bf{v} - \gamma \bf{v}t_0, \ \ \ t=\gamma^{-1}t_0$$

In Aether's scans, the $$(\gamma-1)$$ looks almost like $$(r-1)$$. I thought I'd point that out to prevent any confusion.


EDIT: latex processing is apparently not working ... not my fault: https://www.physicsforums.com/showthread.php?t=116446

 

[gregory_]

I must admit, I am a naive sucker. No matter how many times clj4 claims he'll "believe it if you show proof" ... well, I keep hoping that this time will be the time where he actually steps back and discusses the material with the intent to learn. Please clj4, this has gone on for way too many posts now.


Since latex processing is down here (and it is annoying to type it in with forum tags anyway), I have written up the correct solution to the wave equation and attached it. There are probably some typos, let me know about them but don't miss the main point because of them.


I only went so far as obtaining omega_c. It is indeed the same as in SR (as has been proved before). Once these calculations are accepted, I would like to pause to see why we should have expected this all along (ie the simple proofs given by me and others).