Consistency of the speed of light

[clj4]

Yes, that's what I mean, that's Eq. (7) applied to the experiment.

Interesting...I tried to do the numerical evaluation and I stumbled on the fact that the expression:

[\omega_2^2[1-\frac{v_x^2}{c^2}]-\omega^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}

happens to be imaginary.

 

[Aether]

Interesting...I tried to do the numerical evaluation and I stumbled on the fact that the expression:

[\omega_2^2[1-\frac{v_x^2}{c^2}]-\omega^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}

happens to be imaginary.

No wait, you have edited that equation. That isn't what I wrote in post #126.

 

[clj4]

No wait, you have edited that equation. That isn't what I wrote in post #126.

You are right.

 

[Aether]

You have fed us so much BS in the past that I am sorry to say but I don't believe that you are giving us the output of (7)...

You are right.

No wonder.

 

[clj4]

No wonder.

Not so fast again. Plugging in your omegas from post 144 in (9a) gives a much bigger number than the one you claim. Would you care to show how you get 0.0347?

 

[Aether]

Not so fast again. Plugging in your omegas in (9a) gives a much bigger number than the one you claim. Would you care to show how you get 0.0347?

OK.

This is how I got 0.0347 rad from the calculations in post #126. I'll re-do these calculations in a later post using L_1=L_2=2.4384 \meters, and plot Eq. (9) next to \Delta \phi(\theta)=k_2(\theta)L_2-k_1(\theta)L_1 over a 90-degree range of theta.

Here's a Kennedy-Thorndike type analysis (taking into account the Earth's rotation only):
\Delta \phi=k_2L_2-k_1L_1=\frac{L_2}{c}[\omega^2[1-\frac{v_x^2}{c^2}]-\omega_2^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}-\frac{L_1}{c}[\omega^2[1-\frac{v_x^2}{c^2}]-\omega_1^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}.

L_1=L_2=2.5 \ meters
\omega=2.52333\times 10^{11} \ rad/sec
\omega_1=2.52019\times 10^{11} \ rad/sec
\omega_2=1.39487\times 10^{11} \ rad/sec
c=2.99792458\times 10^8 \ m/s

For v_x=400 \ km/s and v_z=0:
\Delta \phi=1753.498118-104.942309=1648.555808 \ rad

For v_x=0 and v_z=400 \ km/s:
\Delta \phi=1753.500365-104.979859=1648.520506 \ rad

Apply Eq. (7) to a WR-28 waveguide to get a first guide wave number k_1, and multiply that by the length of the first waveguide L_1 to get a first phase; then apply Eq. (7) to a near-cutoff waveguide to get a second guide wave number k_2, and multiply that by the length of the second waveguide L_2 to get a second phase; subtract the first phase from the second phase to get \Delta \phi.

In order to more plainly show how different measured output phase differentials are compared, let's label the various \Delta \phi and k functions with their absolute velocity components from now on:

For v_x=400 \ km/s and v_z=0:
\Delta \phi (400000,0,0)=1753.498118-104.942309=1648.555808 \ rad

For v_x=0 and v_z=400 \ km/s:
\Delta \phi (0,0,400000)=1753.500365-104.979859=1648.520506 \ rad

(k_1(400000,0,0)+\frac{\omega}{c}\frac{v_z}{c})L_1=\frac{L_1}{c}[\omega^2[1-\frac{v_x^2}{c^2}]-\omega_1^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}=104.942309 \ rad

(k_2(400000,0,0)+\frac{\omega}{c}\frac{v_z}{c})L_2=\frac{L_2}{c}[\omega^2[1-\frac{v_x^2}{c^2}]-\omega_2^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}=1753.498118 \ rad

\Delta \phi(400000,0,0)=(k_2(400000,0,0)+\frac{\omega}{c}\frac{v_z}{c})L_2-(k_1(400000,0,0)+\frac{\omega}{c}\frac{v_z}{c})L_1=1648.555808 \ rad

\Delta \phi(0,0,400000)=(k_2(0,0,400000)+\frac{\omega}{c}\frac{v_z}{c})L_2-(k_1(0,0,400000)+\frac{\omega}{c}\frac{v_z}{c})L_1=1648.520506 \ rad

\Delta \phi(400000,0,0)-\Delta \phi(0,0,400000)=1648.555808-1648.520506=0.035302 \ rad

This is the change in \Delta \phi predicted for a 90-degree rotation of the apparatus. This number will be slightly lower when re-calculated below using L_1=L_2=2.4384 \meters.

 

[clj4]

OK.(k_1+\frac{\omega}{c}\frac{v_z}{c})L_1=\frac{L_1}{c}[\omega^2[1-\frac{v_x^2}{c^2}]-\omega_1^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}=104.942309 \rad

(k_2+\frac{\omega}{c}\frac{\v_z}{c})L_2=\frac{L_2}{c}[\omega^2[1-\frac{v_x^2}{c^2}]-\omega_2^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}=1753.498118 \ rad

\Delta \phi=(k_2+\frac{v_z}{c}\frac{v_z}{c})L_2-(k_1+\frac{v_z}{c}\frac{v_z}{c})L_1=1648.555808 \ rad

Do you want me to show more detail for any of these terms?

You are missing the 2\pi factor again (see your other formula).
But the main point is how do you arrive from thousands of radians to 0.0347?

 

[clj4]

OK.Let's assume that L_1=L_2=L

(k_1+\frac{\omega}{c}\frac{v_z}{c})L=\frac{L}{c}[\omega^2[1-\frac{v_x^2}{c^2}]-\omega_1^2[1-\frac{v_x^2}{c^2}-\frac{v_z^2}{c^2}]]^{1/2}=104.942309 \rad

Where is this coming from? You have never shown this before. I thought that we were talking about (9a)

 

[Aether]

You are missing the 2\pi factor again (see your other formula).

OK, give me a few minutes to finish constructing that post. The 2\pi factor was there originally because I was taking the omegas to be frequencies (as seemed to be implied throughout Gagnon), but I removed it when I changed the units on the omegas to radians.

But the main point is how do you arrive from thousands of radians to 0.0347?

That is how Kennedy-Thordike experiments work. You record two different \Delta \phi measurements (usually at different times of the year, but it can also be done at different times of the day). The 0.0347 radians is the difference between the two \Delta \phi measurements, one with v_x=400 \ km/s and the other with v_z=400 \ km/sec.

 

[clj4]

\Delta \phi=(k_2+\frac{\omega}{c}\frac{v_z}{c})L-(k_1+\frac{\omega}{c}\frac{v_z}{c})L_1=1648.555808 \ rad

This is new as well. How did you arrive to this new defiinition of

\Delta \phi

 

[Aether]

This is new as well. How did you arrive to this defiinition of

\Delta \phi

Let me finish constructing that post first. The "new" terms will cancel out, I'm just showing them here because you wanted to see my intermediate steps.

 

[clj4]

OK, give me a few minutes to finish constructing that post. The 2\pi factor was there originally because I was taking the omegas to be frequencies (as seemed to be implied throughout Gagnon), but I removed it when I changed the units on the omegas to radians.

That is how Kennedy-Thordike experiments work. You record two different \Delta \phi measurements (usually at different times of the year, but it can also be done at different times of the day). The 0.0347 radians is the difference between the two \Delta \phi measurements, one with v_x=400 \ km/s and the other with v_z=400 \ km/sec.

Would you care to show the complete calculations? This is getting to look like a mess.

 

[Aether]

Would you care to show the complete calculations? This is getting to look like a mess.

Sure, I've got to run as soon as I repair a few things in post #156. I'll return later to fill-in the intermediate details.

 

[Aether]

Would you care to show the complete calculations? This is getting to look like a mess.

I have attached a text file containing the output (and source code) of a high-precision (e.g., 1000 digit computation(s) with output truncated at 9 decimal places) numerical comparison of the output of Eq. (9) of Gagnon et al. (described as an “approximation” by the authors) and my Eq. (9a) which applies Eq. (7) to the experiment through a 90-degree rotation in the x-z plane. Eq. (9) appears to be reasonably approximating the experimental hypothesis of Gagnon et al.. The textual claim appearing in (Gagnon et al., 1988) that a “peak-to-peak phase shift of at least 19-degrees is predicted as the apparatus turns in the laboratory” is apparently over-estimated by an order of magnitude (e.g., a peak-to-peak phase shift of 1.9-degrees seems to be more consistent with Eq. (9) of the paper).

The key functions implemented are:

Eq. (7):
k_1(v_x,v_y,v_z)=-\frac{\omega}{c_0}\frac{v_z}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_x^2}{c_0^2})-\omega_1^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2}

k_2(v_x,v_y,v_z)=-\frac{\omega}{c_0}\frac{v_z}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_x^2}{c_0^2})-\omega_2^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2}

Eq. (9):
\Delta \phi_9(v_x,v_y,v_z)=\phi_0+\frac{\omega_1L}{2c_0}[\frac{\omega_1}{2\delta}]^{1/2}\frac{v^2}{c_0^2}sin^2(\theta)

Eq. (9a):
\Delta \phi_{9a}(v_x,v_y,v_z)=k_2(v_x,v_y,v_z)L_2-k_1(v_x,v_y,v_z)L_1

Within this analysis I have arbitrarily defined \phi_0 as:
\phi_0=\Delta \phi_{9a}(v,0,0)-\Delta \phi_9(v,0,0)

This should allow us to focus on Eqs. (5) through (8) of Gagnon now. We may also return to Eq. (9a) later in order to apply length-contractions to L_1, and L_2.

 

[clj4]

So,

1. At 0 AM

\theta=0 degrees , v_x=0 , v_z=390km/sec

2. At 6 AM
\theta=90 degrees , v_z=0 , v_x=390km/sec

Now, if we work the peak to peak difference between the phase difference at 0AM and 6AM respectively we should get 2 degrees whether we use the author's formula (9) or the formulas that you derived (9a).

 

[Aether]

So,

1. At 0 AM

\theta=0 degrees , v_x=0 , v_z=390km/sec

2. At 6 AM
\theta=90 degrees , v_z=0 , v_x=390km/sec

Now, if we work the peak to peak difference between the phase difference at 0AM and 6AM respectively we should get 2 degrees whether we use the author's formula (9) or the formulas that you derived (9a).

If you plug Eq. (7) into Eq. (9a), then that is correct. However, I will now argue that Eq. (6) is invariant over rotations in the x-y plane, but Eqs. (7) and (8) (which were derived using Eq. (6)) are not, and therefore Eqs. (7) and (8) are not generally valid (e.g., they are only valid, if at all, when v_y=0).

For a waveguide lying along the z direction of the laboratory-coordinate system:
Eq. (6):
E(x,y,z)=E(x,y)exp(ikz-i\omega t)

Eq. (7):
k_g(v_x,v_y,v_z)=-\frac{\omega}{c_0}\frac{v_z}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_x^2}{c_0^2})-\omega_c^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2}

Eq. (8):
\omega_c(v_x,v_y,v_z)=\omega_{mn}[1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2}]^{-1/2}

Therefore, I propose (provisionally) the following two new versions of these equations which are both invariant over rotations in the x-y plane:

Eq. (7a):
k_z(v_x,v_y,v_z)=-\frac{\omega}{c_0}\frac{v_z}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_x^2}{c_0^2}-\frac{v_y^2}{c_0^2})-\omega_z^2(1-\frac{v_x^2}{c_0^2}-\frac{v_y^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2}

Eq. (8a):
\omega_i(v_x,v_y,v_z)=\omega_{mni}[1-\frac{v_x^2}{c_0^2}-\frac{v_y^2}{c_0^2}-\frac{v_z^2}{c_0^2}]^{-1/2}=\gamma (v_x,v_y,v_z)\omega_{mni}

If this is acceptable, then I will also propose (provisionally) four new equations (6a), (7b), (6b), and (7c) for waveguides lying along the x and y directions of the laboratory-coordinate system respectively (see post #124):

For a waveguide lying along the x direction of the laboratory-coordinate system:
Eq. (6a):
E(x,y,z)=E(y,z)exp(ikx-i\omega t)

Eq. (7b):
k_x(v_x,v_y,v_z)=-\frac{\omega}{c_0}\frac{v_x}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_y^2}{c_0^2}-\frac{v_z^2}{c_0^2})-\omega_z^2(1-\frac{v_x^2}{c_0^2}-\frac{v_y^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2}

For a waveguide lying along the y direction of the laboratory-coordinate system:
Eq. (6b):
E(x,y,z)=E(x,z)exp(iky-i\omega t)

Eq. (7c):
k_y(v_x,v_y,v_z)=-\frac{\omega}{c_0}\frac{v_y}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})-\omega_z^2(1-\frac{v_x^2}{c_0^2}-\frac{v_y^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2}

If this is acceptable, then it is apparent that we could use either (7b), (7c), or (7a) to complete a k_i-invariant rotation of the apparatus about any of the three respective spatial axes. However, Eq. (7) leads us to conclude that we can't (in general) complete a k_z-invariant rotation of the apparatus about the y axis. Eq. (7c) generally contradicts both Eq. (7) and Eq. (7a) for rotations in the x-z plane.

 

[clj4]

Why would we complicate things this way? The lab (the Earth) is assumed to rotate in the x-z plane (y is assumed to be the axis going through poles, v_y=0).So the equations given in the paper (and your equivalent (9a)) are necessary and sufficient in describing the experiment.

Eq. (7c) generally contradicts both Eq. (7) and Eq. (7a) for rotations in the x-z plane.

Not so.

(7a),(8a) are nothing but (7) and (8) for v_y=0 which is the case of the experiment. The experiment clearly assumes v_y=0. There is no need for the newly introduced (7a),(7b),(7c).

 

[Aether]

Why would we complicate things this way? The lab (the Earth) is assumed to rotate in the x-z plane (y is assumed to be the axis going through poles, v_y=0).So the equations given in the paper (and your equivalent (9a)) are necessary and sufficient in describing the experiment.

Eq. (7) does not appear (to me) to be generally valid over a rotation in the x-z plane. If this is true, then the equations given are not necessary and sufficient for correctly describing the experiment.

Not so.

(7a),(8a) are nothing but (7) and (8) for v_y=0 which is the case of the experiment. The experiment clearly assumes v_y=0. There is no need for the newly introduced (7a),(7b),(7c).

I have introduced them to show that Eq. (7) does not appear (to me) to be generally valid over a rotation in the x-z plane.

 

[clj4]

Eq. (7) does not appear (to me) to be generally valid over a rotation in the x-z plane. If this is true, then the equations given are not necessary and sufficient for correctly describing the experiment.

I have introduced them to show that Eq. (7) does not appear (to me) to be generally valid over a rotation in the x-z plane.

I disagree. Your equation is nothing but (7) with v_y=0 which also happens to be the case in the actual experiment. Looks like a last ditch diversion.

 

[Aether]

I disagree. Your equation is nothing but (7) with v_y=0 which also happens to be the case in the actual experiment.

Eq. (7a) helps establish the pattern that I used to generate Eq. (7c), and Eq. (7c) generally contradicts Eq. (7). I am still working toward this, to "try to give a generalized expression for the guide wave number that is valid (or at least consistent with Gagnon's Eq. (6)) for any angle between the absolute velocity vector and the waveguide." I don't think that Eq. (7) does that, not even for rotations in the x-z plane.

Looks like a last ditch diversion.

I called this shot in post #124.

 

[clj4]

Eq. (7a) helps establish the pattern that I used to generate Eq. (7c), and Eq. (7c) generally contradicts Eq. (7). I am still working toward this, to "try to give a generalized expression for the guide wave number that is valid (or at least consistent with Gagnon's Eq. (6)) for any angle between the absolute velocity vector and the waveguide." I don't think that Eq. (7) does that, not even for rotations in the x-z plane.

I called this shot in post #124.

Make v_y=0 , (7) and (7a) are identical .
Now, if you want something really general, you shouldn't simply permute the indexes but you should derive (7) under the circumstance that the two waveguides make an angle \theta with the vector v. I think that this is exactly what (9) is. I don't think that your simple permutation of indexes produces what you are after.

If you wanted to do this, I think that you would need a better definition of the "field of the waveguide" (6):

\exp(i<k,\frac{r}{|r|}>-i\omega*t) instead of

\exp(ikz-i\omega*t)

where <*,*> is the dot product between the unit vector k and the unit positional vector \frac{r}{|r|}

Transform Gagnon equation (5) into polar coordinates in the x-z plane and work things from base principles
For example:
Set
z=r*cos(\theta)
x=r*sin(\theta)
with \theta varying from 0 to \pi/2
I am willing to bet that the final outcome will be a much more ellegant derivation of Gagnon (9)

I am willing to work with you on this.

 

[Aether]

Make v_y=0 , (7) and (7a) are identical .
Now, if you want something really general, you shouldn't simply permute the indexes but you should derive (7) under the circumstance that the two waveguides make an angle \theta with the vector v. I think that this is exactly what (9) is. I don't think that your simple permutation of indexes produces what you are after.

I only showed that to prove that Eq. (7) is not generally valid. It still remains to be seen exactly what is generally valid.

If you wanted to do this, I think that you would need a better definition of the "field of the waveguide" (6):

\exp(i&lt;k,\frac{r}{|r|}&gt;-i\omega*t) instead of

\exp(ikz-i\omega*t)

where <*,*> is the dot product between the unit vector k and the unit positional vector \frac{r}{|r|}

Something like that, yes.

Transform Gagnon equation (5) into polar coordinates in the x-z plane and work things from base principles
For example:
Set
z=r*cos(\theta)
x=r*sin(\theta)
with \theta varying from 0 to \pi/2
I am willing to bet that the final outcome will be a much more ellegant derivation of Gagnon (9)

If so, then we look at Eq. (5). If not, and Gagnon falls, then we may still want to look at Eq. (5) and ref (9) anyway. We'll need to see full 3D rotations before moving on in any case.

I am willing to work with you on this.

Thank-you. I am studying ref (10) at the moment.

 

[clj4]

I only showed that to prove that Eq. (7) is not generally valid. It still remains to be seen exactly what is generally valid.

You haven't proved that "(7) is not generally valid". As you can see I rejected all the "made up" formulas (7b),(7c) as being baseless. This is also why I suggested that you worked from base principles. Actually you haven't proved anything yet. Until you do, (7) stands.

 

[Aether]

Immediately following Eq. (5), Gagnon et al. state the following: “Substituting the usual solution for a plane wave traveling in the z direction, i.e., E(z)=Eexp(ikz-i\omega t)} and solving for k, we find that the wave number corresponds to a wave with phase velocity c+v, to the first order of approximation. The classical velocity addition is thus obtained for electromagnetic waves moving in a reference frame.” Eq. (7) is given soon thereafter as:

k_g=-\frac{\omega}{c_0}\frac{v_z}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_x^2}{c_0^2})-\omega_c^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2} (Eq. (7)).​

This term in Eq. (7):

\omega_c^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})​

is invariant over rotations in the x-z plane, so we may simplify our analysis by applying Eq. (7) to the case of an unguided electromagnetic wave traveling in a vacuum along the z-direction of the laboratory-coordinate system (e.g., where \omega_c=0):

k_0(v_x,0,v_z)= \omega \frac{(1-\frac{v_x^2}{c_0^2})^{1/2}-\frac{v_z}{c_0}}{c_0} (Eq. (7d)).​

The vacuum wave number for an unguided electromagnetic wave in an isotropic coordinate system (such as SR, or the \Sigma frame of RMS) is:

k_0=\frac{\omega}{c_0} (Eq. (7e)),​

and I will now argue that the vacuum wave number for an unguided electromagnetic wave in an anisotropic coordinate system such as RMS is:

k_0(v_z)=\frac{\omega}{c_z} (Eq. (7f)​

where c_z is the one-way speed of light in the longitudinal z-direction of propagation…Note: the round-trip speed of light is isotropic in RMS as well as SR, so the average speed of light in the transverse direction is always c_0...see post #92).

Now, let’s consider two special cases of Eq. (7d): 1) Absolute motion is in the longitudinal direction of wave propagation:

k_0(0,0,v_z)= \omega \frac{(1-\frac{v_z}{c_0})}{c_0} (Eq. (7g));​

and 2) Absolute motion is transverse to the direction of wave propagation:

k_0(v_x,0,0)= \omega \frac{(1-\frac{v_x^2}{c_0^2})^{1/2}}{c_0} (Eq. (7h)).​

Eq. (7g) closely approximates Eq. (7f) if:

\frac{1}{c_z}=\frac{1}{c_0+v_z},​

and Eq. (7h) closely approximates (to second order) Eq. (7f) if:

\frac{1}{c_z}=\frac{\frac{1}{c_0+v_x}+\frac{1}{c_0-v_x}}{2}​

(round trip speed of light is always c_0 in SR and RMS, but not in Galileian relativity). However, in the RMS coordinate system Eq. (7f) evaluates to:

k_0(0,0,v_z)=\frac{\omega}{c_z}=\omega \frac{c_0+v_z}{c_0^2} ,​

and for absolute motion normal to the z-direction of wave propagation:

k_0(v_x,0,0)=\omega (\frac{\frac{c_0+v_x}{c_0^2}+\frac{c_0-v_x}{c_0^2}}{2})=\frac{\omega}{c_0}.​

Gagnon et al. appear to have based Eq. (7) on a first order approximation to a wave’s phase velocity (e.g., the Galilean transform) rather than the GGT/RMS transform, and their experimental hypothesis appears to be directed ((un)intentionally?) toward a refutation of Galilean relativity as opposed to RMS.

 

[clj4]

The above is unreadable, could you reformat it a little? Thank you

 

[clj4]

Immediately following Eq. (5), Gagnon et al. state the following: “Substituting the usual solution for a plane wave traveling in the z direction, i.e., E(z)=Eexp(ikz-i\omega t)} and solving for k, we find that the wave number corresponds to a wave with phase velocity c+v, to the first order of approximation. The classical velocity addition is thus obtained for electromagnetic waves moving in a reference frame.” Eq. (7) is given soon thereafter as:

k_g=-\frac{\omega}{c_0}\frac{v_z}{c_0}+\frac{1}{c_0}[\omega^2(1-\frac{v_x^2}{c_0^2})-\omega_c^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})]^{1/2} (Eq. (7)).​

This term in Eq. (7):

\omega_c^2(1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2})​

is invariant over rotations in the x-z plane, so we may simplify our analysis by applying Eq. (7) to the case of an unguided electromagnetic wave traveling in a vacuum along the z-direction of the laboratory-coordinate system (e.g., where \omega_c=0):

Thank you, this is much better.

But \omega_c is clearly not 0 in the experiment, so I cannot accept (7d). I thought that we were done with the ad-hoc invention of formulas, that you were going to rederive things from base principles. You come back with the same thing. can you solve partial differential equations?
K is the solution of the partiial differential equation (5), you cannot keep coming with cooked up formulas. If you want to refute the paper you need to solve (5) from base principles. I gave you a tool, the conversion to polar coordinates. On a different issue, k and \omega are variables that get tied together by (5), so this should be your starting point, not all the different speculations as to how to connect the two, They are connected by equation (5).

 

[Aether]

Thank you, this is much better.

But \omega_c is clearly not 0 in the experiment, so I cannot accept (7d). I thought that we were done with the ad-hoc invention of formulas, that you were going to rederive things from base principles. You come back with the same thing. can you solve partial differential equations?
K is the solution of the partiial differential equation (5), you cannot keep coming with cooked up formulas. If you want to refute the paper you need to solve (5) from base principles.

OK, I'm working up to that.

I gave you a tool, the conversion to polar coordinates.

You said that k was a "unit vector", but it is defined on p. 212 of ref (10) as: k^2=(\frac{\pi p}{a})^2+(\frac{\pi q}{b})^2 where p=1, q=0, (in this experiment) and (a,b) are the interior cross-sectional dimensions of a waveguide. Apparently, how the waveguide coordinates (x,y,z)=(a,b,L) transform over a rotation of the apparatus (combined with the one-way speed of light) is what ultimately determines the outcome of this experiment/analysis. We can expect that any anisotropy in the one-way speed of light due to RMS is going to be exactly offset by how these spatial coordinates transform; not so in Galileian relativity however.

On a different issue, k and \omega are variables that get tied together by (5), so this should be your starting point, not all the different speculations as to how to connect the two, They are connected by equation (5).

This is how k (in addition to the equation that I just gave) and \omega are defined:

...bearing in mind that k is independent of \omega and \beta--it is a constant depending only on the mode concerned and the geometry of the waveguide cross-section.

The "mode concerned" is/are an integer(s), so k varies with the geometry of the waveguide cross-section only.

Since it is phase locked to a reference oscillator which is at rest in the laboratory, the output frequency of the klystron [f=\frac{\omega}{2\pi}] is unaffected by rotation of the apparatus.

 

[777tiM777]

prior to einsteins special theory of relativity maxwell proved (theoretically) that the propogation of electromagnetic waves is 3.00*10^8 m/s (c = 1/sqrt(e0*u0)), all einstein did essentially in his special theory of relativity was expand galilean relativity(which stated something along the lines that the velocity viewed from different frames of reference will not be the same, but the laws such as the law of conservation of eneregy will still hold true in any inertial frame of reference) to say that all laws of physics hold true in all inertial frames of reference, and since maxwell's equations showed that the speed of light is constant it fell under the "all laws of physics"...im really sorry if someone else posted this argument before me, i didnt look through all the posts... hope this still helps answer your question

 

[Aether]

You come back with the same thing. can you solve partial differential equations? K is the solution of the partiial differential equation (5), you cannot keep coming with cooked up formulas. If you want to refute the paper you need to solve (5) from base principles.

I ordered a couple of advanced calculus textbooks to study partial differential equations in more depth; it will be at least a few weeks before I solve (5) for k. I think that I'll probably also need that to derive an anisotropic RMS transformation tensor for use with Eq. (6) from ref (9). I'll use that to transform both k and Eq. (8b) independent of Eq. (5).

In ref (10) the mode of the waveguide is identified by (p,q) rather than (m,n) as is used in Gagnon et al. (I will use (p,q) to identify the mode and reserve (m,n) for a different purpose), and on pp. 212-213 the cutoff angular frequency is defined as:

\omega_{pq}=[p^2\pi^2\epsilon \mu \frac{c_0^2}{a_0^2}+q^2\pi^2\epsilon \mu \frac{c_0^2}{b_0^2}]^{1/2} Eq. (8b),​

where \epsilon \mu=1 for a vacuum “filled” waveguide.

According to Gagnon et al., Eq. (8) is supposed to transform a cutoff angular frequency \omega_{mn} in the absolute frame into a cutoff angular frequency \omega_c in a moving frame:

\omega_c=\omega_{mn}[1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2}]^{-1/2} Eq. (8);​

however, this seems to be inconsistent with the RMS transforms (their Eq. (1), also see post #92):

x=\gamma(x_0-vt_0),\ t=\gamma^{-1},\ y=y_0,\ z=z_0 Eq. (1).​

For example, with (p,q)=(1,0), \epsilon \mu =1, where a_0 is a distance along the x-axis and absolute motion is along the x-axis, then Eq. (1) transforms Eq. (8b) into Eq. (8c) (note that transverse wave motion is round-trip averaged):

\omega_c=\frac{2\pi c_0}{(a_0-vt_0)+(a_0+vt_0)}[1-\frac{v_x^2}{c_0^2}]^{1/2}=\frac{\pi c_0}{a_0}[1-\frac{v_x^2}{c_0^2}]^{1/2} Eq. (8c);​

but Eq. (8) transforms Eq. (8b) into Eq. (8d) (the exponent in (8d) is of the opposite sign as in (8c)):

\omega_c=\pi \frac{c_0}{a_0}[1-\frac{v_x^2}{c_0^2}]^{-1/2} Eq. (8d);​

 

[clj4]

I ordered a couple of advanced calculus textbooks to study partial differential equations in more depth; it will be at least a few weeks before I solve (5) for k. I think that I'll probably also need that to derive an anisotropic RMS transformation tensor for use with Eq. (6) from ref (9). I'll use that to transform both k and Eq. (8b) independent of Eq. (5).

In ref (10) the mode of the waveguide is identified by (p,q) rather than (m,n) as is used in Gagnon et al. (I will use (p,q) to identify the mode and reserve (m,n) for a different purpose), and on pp. 212-213 the cutoff angular frequency is defined as:

\omega_{pq}=[p^2\pi^2\epsilon \mu \frac{c_0^2}{a_0^2}+q^2\pi^2\epsilon \mu \frac{c_0^2}{b_0^2}]^{1/2} Eq. (8b),​

where \epsilon \mu=1 for a vacuum “filled” waveguide.

According to Gagnon et al., Eq. (8) is supposed to transform a cutoff angular frequency \omega_{mn} in the absolute frame into a cutoff angular frequency \omega_c in a moving frame:

\omega_c=\omega_{mn}[1-\frac{v_x^2}{c_0^2}-\frac{v_z^2}{c_0^2}]^{-1/2} Eq. (8);​

however, this seems to be inconsistent with their Eq. (1):

x=\gamma(x_0-vt_0),\ t=\gamma^{-1},\ y=y_0,\ z=z_0 Eq. (1).​

For example, with (p,q)=(1,0), \epsilon \mu =1, where a_0 is a distance along the x-axis and absolute motion is along the x-axis, then Eq. (1) transforms Eq. (8b) into Eq. (8c) (note that transverse wave motion is round-trip averaged):

\omega_c=\frac{2\pi c_0}{(a_0-vt_0)+(a_0+vt_0)}[1-\frac{v_x^2}{c_0^2}]^{1/2}=\frac{\pi c_0}{a_0}[1-\frac{v_x^2}{c_0^2}]^{1/2} Eq. (8c);​

but Eq. (8) transforms Eq. (8b) into Eq. (8d) (the exponent in (8d) is of the opposite sign as in (8c)):

\omega_c=\pi \frac{c_0}{a_0}[1-\frac{v_x^2}{c_0^2}]^{-1/2} Eq. (8d);​

You are inverting (8b), right? I would expect to see a term in \omega_{pq} in (8c). I am not seeing it. Could you show the steps, at a superficial view, if I look at (8b) and at (1), i would expect to see the exponent {-1/2} in (8c)
Who are a0 and b0?

 

[Aether]

You are inverting (8b), right?

No.

I would expect to see a term in \omega_{pq} in (8c). I am not seeing it.

\omega_{pq}=[p^2\pi^2\epsilon \mu \frac{c_0^2}{a_0^2}+q^2\pi^2\epsilon \mu \frac{c_0^2}{b_0^2}]^{1/2} Eq. (8b),​

with (p,q)=(1,0), and \epsilon \mu =1:

\omega_{10}=\pi \frac{c_0}{a_0}.​

\omega_{10} is a cutoff angular frequency in the absolute frame, and \omega_c is that cutoff angular frequency in a moving frame:

\omega_c=\frac{\pi c_0}{a}.​

Could you show the steps, at a superficial view, if I look at (8b) and at (1), i would expect to see the exponent {-1/2} in (8c)

To transform \omega_{10} to \omega_c using Eq. (1) we need to suppose that a_0 is a distance along the x-axis, and absolute motion is along the x-axis (this is necessary for now because Eq. (1) can only be used for motion and distance along the x-axis):

\omega_{c+}=\frac{\pi c_0}{\gamma (a_0-vt_0)}.​

However, a_0 lays along the x-axis while the wave propagates along the z-axis, and E reciprocates in the \pm x-direction. So, this last equation only applies while the transverse wave motion is in the +x direction; while the transverse wave motion is in the -x direction this equation applies:

\omega_{c-}=\frac{\pi c_0}{\gamma (a_0+vt_0)}.​

Eq. (8c) is an attempt to compute the average of these two equations, and may change slightly if this average turns out not to be done quite right.

\omega_c=\frac{2\pi c_0}{\gamma((a_0-vt_0)+(a_0+vt_0))}=\frac{\pi c_0}{\gamma a_0};

\omega_c=\frac{2\pi c_0}{(a_0-vt_0)+(a_0+vt_0)}[1-\frac{v_x^2}{c_0^2}]^{1/2}=\frac{\pi c_0}{a_0}[1-\frac{v_x^2}{c_0^2}]^{1/2} Eq. (8c);​

Who are a0 and b0?

a_0 and b_0 are the dimensions (in the absolute frame) of the interior cross-section of the waveguide. a_0 is taken to be along the x-axis, b_0 is taken to be along the y-axis. The electromagnetic wave propagates longitudinally along the z-axis. In a moving frame, these two dimensions transform to a and b respectively.

It is possible to compute \omega_c and k for a waveguide in a moving frame simply by transforming these coordinates (a,b).

 

[clj4]

Thank you, I will check it out. At a quick glance the average doesn't look right, doesn't lok like an average. Have you considered checking ref 9, apparently they did all the support calculations there. Might save you a lot of time.

 

[Aether]

Have you considered checking ref 9, apparently they did all the support calculations there. Might save you a lot of time.

Ref (9) should be very useful later, but it doesn't have anything to say about waveguides, k, etc. per se.

 

[clj4]

I think I found the error. Omega transforms like 1/t so instead of 1/gama you should have gama in your formula

 

[Aether]

I think I found the error. Omega transforms like 1/t so instead of 1/gama you should have gama in your formula

\omega transforms like 1/t, but \omega_{pq} is defined by the spatial geometry of the waveguide (and the round-trip speed of light). It does seem strange, I also thought that it should transform like 1/t at first. We could leave that as an open question for now.

 

[clj4]

\omega transforms like 1/t, but \omega_{pq} is defined by the spatial geometry of the waveguide (and the round-trip speed of light). It does seem strange, I also thought that it should transform like 1/t at first. We could leave that as an open question for now.

No, we cannot leave it open, it is a clear error.

 

[Aether]

No, we cannot leave it open, it is a clear error.

It is a clear error on whose part? The boundary conditions on the waveguide "require the tangential component of the electric field in the laboratory frame to vanish at the waveguide walls", and this identifies

\omega_c=\omega_{10}=\frac{\pi c_0}{a}=\frac{2\pi c_0}{\lambda},​

so:

\frac{\omega_c}{c_0}= \frac{2\pi}{\lambda}=k^0,​

which corresponds to the timelike component of a wave 4-vector k^\mu and that transforms like t, right?

 

[Aether]

That turns the whole idea of physical science on its head. No experiment can be devised that would prove any theory.

Sure there are experiments that can be devised to prove many theories, but this is not possible in the case of the postulate of SR regarding the constancy of the one-way speed of light. That is because all measurements of speed are inherently coordinate-system dependent. Doppler shifts, for example, are measurable in a coordinate-system independent way (e.g., \frac{v}{c} is a coordinate-system independent dimensionless ratio), so you could actually prove something (within the limits of the precision of your measurements) by making such a measurement. The difference between coordinate-system dependent vs. independent measurements is what we're talking about here.

 

[Jimmy Snyder]

Sure there are experiments that can be devised to prove many theories.

I deleted my post because it was redundant. Name a theory and the experiment that proves it.

 

[Aether]

I deleted my post because it was redundant. Name a theory and the experiment that proves it.

Theory: The Sun is made up of mostly hydrogen and helium; Experiment that proves it: spectroscopic analysis of sunlight vs. ionized hydrogen and helium. Duh.

You deleted your original post. If you still don't want to have this conversation (at least not here and now), then delete your second post too and I'll delete my responses.

 

[Jimmy Snyder]

Duh.

I don't like the direction this discussion is taking.

All that your experiment proves is that there is something in or on the Sun that produces the same spectral lines as Hydrogen and Helium do on the Earth. An extraordinary coincidence indeed, but I'm afraid that does not prove your theory.

 

[Aether]

All that your experiment proves is that there is something in or on the Sun that produces the same spectral lines as Hydrogen and Helium do on the Earth. An extraordinary coincidence indeed, but I'm afraid that does not prove your theory.

OK, Theory #2: There is something in or on the Sun that produces the same spectral lines as Hydrogen and Helium do on the Earth; Experiment that proves it: spectroscopic analysis of sunlight vs. ionized hydrogen and helium.

 

[Jimmy Snyder]

OK, Theory #2: There is something in or on the Sun that produces the same spectral lines as Hydrogen and Helium do on the Earth; Experiment that proves it: spectroscopic analysis of sunlight vs. ionized hydrogen and helium.

The experiment that you cite cannot have taken place since light from the sun is slightly red-shifted. The lines coming from the sun do not exactly match those of Hydrogen and Helium on the earth.

 

[Aether]

The experiment that you cite cannot have taken place since light from the sun is slightly red-shifted. The lines coming from the sun do not exactly match those of Hydrogen and Helium on the earth.

The experiment has taken place, and as I said before "you could actually prove something (within the limits of the precision of your measurements) by making such a measurement".

 

[Jimmy Snyder]

The experiment has taken place.

But the results were not as you claim. The spectral lines were not the same, just similar.

No amount of experimentation can ever prove me right; a single experiment can prove me wrong. -- Albert Einstein

 

[masudr]

An experiment can change the status of an assertion from "hypothesis" to "likely to be true".

Note that this is in NO WAY WHATSOEVER a proof. Proof of some conjecture, in the mathematical (and the only relevant) sense, implies there is no way that some conjecture can ever be false.

Your statements about the spectral line merely say that at the time the experiment was done the measurements agreed with those that occurred on Earth (even assuming the light wasn't redshifted). However, to PROVE the conjecture, you would need to show that no matter when you did the experiment and under WHATEVER circumstance, you would still get the same results. There is no experiment that can ever do that to any theoretical conjecture or hypothesis. That is the status of theory or postulate in ANY empirical science.

 

[Jimmy Snyder]

An unexplained center-to-limb variation of solar wavelength has been known for 75 years.

http://www.Newtonphysics.on.ca/Chromosphere/CHROMOSPHERE.html

 

[Aether]

But the results were not as you claim. The spectral lines were not the same, just similar.

No amount of experimentation can ever prove me right; a single experiment can prove me wrong. -- Albert Einstein

An experiment can change the status of an assertion from "hypothesis" to "likely to be true".

Note that this is in NO WAY WHATSOEVER a proof. Proof of some conjecture, in the mathematical (and the only relevant) sense, implies there is no way that some conjecture can ever be false.

Your statements about the spectral line merely say that at the time the experiment was done the measurements agreed with those that occurred on Earth (even assuming the light wasn't redshifted). However, to PROVE the conjecture, you would need to show that no matter when you did the experiment and under WHATEVER circumstance, you would still get the same results. There is no experiment that can ever do that to any theoretical conjecture or hypothesis. That is the status of theory or postulate in ANY empirical science.

OK, there is a sense in which what you are saying makes sense: "Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, or conjectures" -- http://en.wikipedia.org/wiki/Theory. However, in the context of this thread and Einstein's quote that isn't what we're talking about at all. We are talking about something very specific, about whether it is possible (or not) to make a coordinate-independent measurement of the one-way speed of light. We all agree (at least for the purposes of this discussion) that we can measure the round-trip speed of light. If you want to argue that we really can't even measure the round-trip speed of light (or anything else for that matter), then please do that in another thread.

I interpret Einstein's quote as a warning that he is assuming that Lorentz symmetry is perfect, that no experiment can ever prove that it is perfect, but a single experiment could prove that it is not perfect. Not some general notion which would also apply to every other scientist in the world just as well.

 

[Jimmy Snyder]

If you want to argue that we really can't even measure the round-trip speed of light, then please do that in another thread.

I didn't say that you can't do experiments. I said that you can't prove theories. It was in response to the original post in this thread.

 

[gregory_]

Hello, I've been watching this topic off and on for quite awhile now and it appears to be getting off track. I'm sorry to finally speak up on a note like this... but if it is possible, can we please focus on the issue at hand again?

We are talking about something very specific, about whether it is possible (or not) to make a coordinate-independent measurement of the one-way speed of light.

It appears that this is what the topic started as. And it appears to be what you keep trying to return to. But I don't understand why all the calculations dealing with Gagnon's experiment are necessary for this topic.

You have argued your point well and clearly. I have seen no mathematically consistent argument against your statement that coordinate independent one-way velocity measurements do not exist.

So, to help clarify this long thread, has everyone come to an agreement on this main point and you have moved onto the specifics of one experiment?

Since it is easy to mathematically prove that coordinate independent one-way velocity measurements do not exist, if there is still a debate on this point, it would seem more appropriate to settle this simple question first instead of trying to do it with the messy details of just one particular experiment. Wouldn't you agree?