Consistency of the speed of light

[Hans de Vries]

You start by showing the Lorentz transform (Eq. 3.4 from M&S-I p. 501) for any example you choose (M&S restricts their examples to motion along the x-axis, so we will need to agree to do the same), and I will show the corresponding LET transform (Eq. 3.6 from M&S-I p. 502). "This transform is--as far as the prediction of experimental results is concerned--completely equivalent to (3.4)". -- M&S-I p. 502.

Eq. (3.4)
$$t=(1-v^2)^{1/2}T-vx$$
$$x=(X-vT)/(1-v^2)^{1/2}$$


Eq. (3.6)
$$t=(1-v^2)^{1/2}T$$
$$x=(X-vT)/(1-v^2)^{1/2}$$


where we have chosen to readjust our clocks according to
$$f(x,v)=-vx$$

"We thus arrive at the remarkable result that a theory maintaining absolute simultaneity is equivalent to special relativity." -- M&S-I p. 503

Equivalent? Only in the preferred frame.

Note that 3.6 makes both length and speed anisotropic in any other
reference frame besides the preferred frame. Round wheels would be
only really round in the preferred frame.

We at Earth would see al our wheels changing shape in a 24 hours cycle
corresponding with the rotation of the earth.


Regards, Hans

 

[Aether]

That should clear up a lot of confusion. OK. Let's see you compute the one-way speed of light.

Here's the experiment. In an arbitrary frame of reference in the aether model, start with two synchronized clocks side-by-side and slowly transport one of them to any convenient distance D. Then compute D/(t2-t1). The answer better be c. (t1 is the time on the stationary clock when the light pulse is sent. t2 is the time when the light arrives as measured by the slowly transported clock. Take the limit of ultraslow transport for a perfect answer of c).

Lorentz transformation:
$$t_1=(1-v^2/c_0^2)^{1/2}T_1-vx_1/c_0^2$$
$$x_1=(X_1-vT_1)/(1-v^2/c_0^2)^{1/2}$$

$$t_2=(1-v^2/c_0^2)^{1/2}T_2-vx_2/c_0^2$$
$$x_2=(X_2-vT_2)/(1-v^2/c_0^2)^{1/2}$$

$$D/(t_2-t_1)=(x_2-x_1)/(t_2-t_1)=c_0$$

$$c(v,\theta)=c_0$$

M&S-I p.511 Eq. (6.16) gives the following result for first order effects when transport synchronization of clocks is used with the Lorentz transformation: $$c(\theta)=1-v(1+2\alpha)cos\theta$$ where $$\alpha =-1/2$$ corresponds to perfect Lorentz symmetry.

"In discussing the experiments we need the inverse velocity of light to second order in $$v/c$$..." M&S-III p. 810 Eq. (2.1) - $$1/c(\theta)=1+(\beta+\delta-1/2)v^2sin^2\theta+(\alpha-\beta+1)v^2$$

LET transformation:
$$t_1=(1-v^2/c_0^2)^{1/2}T_1$$
$$x_1=(X_1-vT_1)/(1-v^2/c_0^2)^{1/2}$$

$$t_2=(1-v^2/c_0^2)^{1/2}T_2$$
$$x_2=(X_2-vT_2)/(1-v^2/c_0^2)^{1/2}$$

$$D/(t_2-t_1)=(x_2-x_1)/(t_2-t_1)=c_0^2/(c_0+v)$$

$$c(v,\theta)=c_0^2/(c_0+v \cdot cos(\theta))$$

Hint: The speed of light $$c_0$$ is isotropic in the ether frame: $$(X_2-X_1)/(T_2-T_1)=c_0$$.

Assuming perfect Lorentz symmetry and using the LET transformation I get:
$$c_0/c(v,\theta)=1+(v/c_0) \cdot cos(\theta)$$. This is a dimensionless ratio, and as such it is a measurable (e.g., physical) quantity. However, you must synchronize two clocks to make this measurement; and exactly how you choose to do that determines whether the Lorentz transformation or the LET transformation should be applied.


The speed of light is generally anisotropic in LET (e.g., except for within the ether frame), absolute simultaneity is maintained, and this is empirically equivalent to SR. The trick is that the $$-vx/c_0^2$$ term that is used to maintain a constant speed of light in the Lorentz transformation is used instead to maintain absolute simultaneity in the LET transformation. Both ways are equally valid.

 

[Aether]

Equivalent? Only in the preferred frame.

Empirically equivalent, yes; independent of the frame. These transforms operate on abstract coordinates, and not on physical objects.

Note that 3.6 makes both length and speed anisotropic in any other reference frame besides the preferred frame. Round wheels would be only really round in the preferred frame.

We at Earth would see al our wheels changing shape in a 24 hours cycle
corresponding with the rotation of the earth.

Maybe so, but only in the same sense as the arctic circle looks inflated on a Mercator projection. We are talking about mere coordinate systems here, aren't we? My agrument is with taking the attributes of a coordinate system (e.g., constancy of the speed of light) and claiming that it is an emprically proven fact. I do not dispute that for whatever reason one may prefer the Lorentz transform over the LET transform for whatever purpose, but how is this any different than expressing a preference to use spherical polar coordinates over rectangular coordinates in some cases or vice versa? In both cases the two coordinate systems transform into one another quite freely.

 

[Aether]

Different test theories differ in their assumptions about what form the transform equations could reasonably take. There are at present four test theories of SR:
Robertson,Rev. of Mod. Phys. 21, p378 (1949).
Edwards, Am. J. Phys. 31 (1963), p482.
Mansouri and Sexl, Gen. Rel. Grav. 8 (1977), p497, p515, p809.
Zhang, Special Relativity and its Experimental Foundations.
Zhang discusses their interrelationships and presents a unified test theory encompassing the other three, but with a better and more interpretable parameterization. His discussion implies that there will be no more test theories of SR that are not reducible to one of the first three.

I have received my copy of Zhang's book "Special Relativity and its Experimental Foundations", (1997). In the preface he says this: "The key point in Einstein's theory is the postulate concerning the constancy of the (one-way) velocity of light, which contradicts the classical (nonrelativistic) addition law of velocities. The postulate is needed only for constructing well-defined inertial frames of reference or, in other words, only for synchronizing clocks (i.e., defining simultaneity). It is not possible to test the one-way velocity of light because another independent method of clock synchronization has not yet been found...Of course one could use the experiments to yield limits on the parameters in Robertson's transformations but not on the directional parameter q in Edwards' and MS' theories."

On the back cover Zhang says this: "...In particular, the discussions indicate that the one-way speed of light is not observable in the present laboratories...In the third part, variant types of experiments performed up to now are analyzed and compared to the predictions of special relativity. The analyses show that the experiments are tests of the two-way velocity, but not of the one-way velocity, of light."

On page 10 he says: "We have known that there is no instantaneous signal in nature and, therefore, the absolute simultaneity cannot be realized in any laboratory...It is well known that one always use a light signal for the clock synchronization in a laboratory. Therefore Einstein's simultaneity can be directly realized in experiments. We want to stress here that only the two-way speed, but not the one-way speed, of light has been already measured in the experimental measurements, and hence the isotropy of the one-way velocity of light is just a postulate...We shall see from Chap. 6 that a more general postulate, a choice of the anisotropy of the one-way velocity of light, together with the principle of relativity, would give the same physical predictions as Einstein's theory of special relativity."

 

[cincirob]

Einstein's second postulate states that the speed of light is constant as viewed from any frame of reference. Most of the books on relativity that I have been reading usually ask the reader to accept that fact because proving it is behind the scope of the book. Can anyone help me understand the actual reason behind the second postulate?

Hi, I'm new to the forum. I didn't read all the responses to your question, but it looks like it wandered away from your original question so I'd like to add a couple of thoughts.

First, if you search the net for "On the Electrodynamics of Moving Bodies" (OEMB for short) you can find Einstein's 1905 paper and read his explanation for the second postulate. Briefly, it can be interpretted this way: When Maxwell's equations (circa 1850) convinced people that light was a wave phenomenon they assumed that the wave must be the motion of some medium. For instance the medium for sound is air. The medium for ocean waves is water, etc.

The name given to the medium for light was aether. A number of experiments were performed over the next half century to observe the properties of aether, the most famous being the Michelson-Morley experiment (you can find Michelson's paper ont he net also). In OEMB Eisntein notes that all the experiments to find the nature or effects of aether failed. He then states his postulate that light propagates in "empty space" at a constant velocity c. Although he doesn't come right out and say it, he is proclaiming that aether doesn't exist (or at least is of no consequence regarding light).

Many people think that this postulate is where he declared that the speed of light is the same for all inertial observers. Actually it is the first postulate that declares this phenomenon. In the first postulate he says that the laws of electrodynamics and optics hold for all inertial observers (inertial means non-accelerating). This means that Maxwell's laws hold. And Maxwell's laws show that the speed of light c is a function of two properites of empty space, the permeability and permitivity of empty space. Since these two properties are constants, then c must be a constant.

In summary then, Einstein made postulates out of what observations of physics seemed to imply; that is, that aether didn't seem to have any measurable effects and that the motion of an observer didn't seem to change the outcome of various physical experiments.

 

[Jimmy Snyder]

Although he doesn't come right out and say it, he is proclaiming that aether doesn't exist (or at least is of no consequence regarding light).

I just popped into nit pick. In fact he does come right out and say it.

The introduction of a "luminiferous ether'' will prove to be superfluous

http://www.fourmilab.ch/etexts/einstein/specrel/www/

 

[cincirob]

Quote:
Originally Posted by cincirob
Although he doesn't come right out and say it, he is proclaiming that aether doesn't exist (or at least is of no consequence regarding light).

I just popped into nit pick. In fact he does come right out and say it.


Quote:
Originally Posted by A. Einstein
The introduction of a "luminiferous ether'' will prove to be superfluous
=======================================

You are correct, he does make the comment above. But my comments were directed specifically to the postulates where you must draw the inferrence.

 

[cincirob]

I have seen some comments in this thread about the Robertson-Mansouri-Sexl framework. I'm not very familiar with the work, but i found this reference to it.

I see that the graphics don't copy so here is the site : http://qom.physik.hu-berlin.de/prl_91_020401_2003.pdf

Modern Michelson-Morley Experiment using Cryogenic Optical Resonators
Holger Mu¨ller,1,2,* Sven Herrmann,1,2 Claus Braxmaier,2 Stephan Schiller,3 and Achim Peters1,†,‡
1Institut fu¨ r Physik, Humboldt-Universita¨t zu Berlin, Hausvogteiplatz 5-7, 10117 Berlin, Germany
2Fachbereich Physik, Universita¨t Konstanz, 78457 Konstanz, Germany
3Institut fu¨ r Experimentalphysik, Heinrich-Heine-Universita¨t Du¨sseldorf, 40225 Du¨sseldorf, Germany
(Received 27 January 2003; published 10 July 2003)
We report on a new test of Lorentz invariance performed by comparing the resonance frequencies of
two orthogonal cryogenic optical resonators subject to Earth’s rotation over
1 yr. For a possible
anisotropy of the speed of light c, we obtain c=c0  2:6  1:7  1015. Within the Robertson-
Mansouri-Sexl (RMS) test theory, this implies an isotropy violation parameter     12
 2:2  1:5  109, about 3 times lower than the best previous result. Within the general extension of the
standard model of particle physics, we extract limits on seven parameters at accuracies down to 1015,
improving the best previous result by about 2 orders of magnitude.

 

[metrictensor]

It may turn out that there is an new theory that explains why the speed of light is the same in all IRF's. This would lead to the same type of question to that theory's foundations. If you understand this you understand the nature of science. It causes one to ask what do I mean by 'why' when I ask a question of science. What am I looking for? What do I expect?

 

[Integral]

In the 1860's Maxwell discovered that he could cast the equations, which bear his name, in the form of a standard wave equation. When he did that, he found a combination of physical constants expressed the velocity of these electromagnetic waves. That constant was $\frac 1 { \sqrt{ \mu_0 \epsilon_0}}$. It is said that he was surprised to find that expression evaluated to a number which was equal to the then accepted value for the speed of light.
So this was a theoretical prediction that the speed of light was a physical constant. The meaning of this was hotly debated for the rest of the century, it implied that electromagnetism behaved differently from massive bodies. As mentioned Michelson and Morley preformed an experiment in an effort to detect the motion of light through the aether. They failed to detect any medium through which light was propagating.
Einstein's postulate that the speed of light was constant to all observers was a result of the failure, over the previous 50 yrs, of physicist trying to prove that it was not. The constancy was predicted theoretically and verified experimentally. Therefore when A.E. wrote down that postulate, it was accepted by the physics community without debate, because it was common knowledge. Einstein tied together the physics of Newton and Electo Magnetism with a simple and very straight forward derivation.

 

[clj4]

Lorentz transformation:
$$t_1=(1-v^2/c_0^2)^{1/2}T_1-vx_1/c_0^2$$
$$x_1=(X_1-vT_1)/(1-v^2/c_0^2)^{1/2}$$

$$t_2=(1-v^2/c_0^2)^{1/2}T_2-vx_2/c_0^2$$
$$x_2=(X_2-vT_2)/(1-v^2/c_0^2)^{1/2}$$

$$D/(t_2-t_1)=(x_2-x_1)/(t_2-t_1)=c_0$$

$$c(v,\theta)=c_0$$

M&S-I p.511 Eq. (6.16) gives the following result for first order effects when transport synchronization of clocks is used with the Lorentz transformation: $$c(\theta)=1-v(1+2\alpha)cos\theta$$ where $$\alpha =-1/2$$ corresponds to perfect Lorentz symmetry.

"In discussing the experiments we need the inverse velocity of light to second order in $$v/c$$..." M&S-III p. 810 Eq. (2.1) - $$1/c(\theta)=1+(\beta+\delta-1/2)v^2sin^2\theta+(\alpha-\beta+1)v^2$$

LET transformation:
$$t_1=(1-v^2/c_0^2)^{1/2}T_1$$
$$x_1=(X_1-vT_1)/(1-v^2/c_0^2)^{1/2}$$

$$t_2=(1-v^2/c_0^2)^{1/2}T_2$$
$$x_2=(X_2-vT_2)/(1-v^2/c_0^2)^{1/2}$$

$$D/(t_2-t_1)=(x_2-x_1)/(t_2-t_1)=c_0^2/(c_0+v)$$

$$c(v,\theta)=c_0^2/(c_0+v \cdot cos(\theta))$$

Hint: The speed of light $$c_0$$ is isotropic in the ether frame: $$(X_2-X_1)/(T_2-T_1)=c_0$$.

Assuming perfect Lorentz symmetry and using the LET transformation I get:
$$c_0/c(v,\theta)=1+(v/c_0) \cdot cos(\theta)$$. This is a dimensionless ratio, and as such it is a measurable (e.g., physical) quantity. However, you must synchronize two clocks to make this measurement; and exactly how you choose to do that determines whether the Lorentz transformation or the LET transformation should be applied.


The speed of light is generally anisotropic in LET (e.g., except for within the ether frame), absolute simultaneity is maintained, and this is empirically equivalent to SR. The trick is that the $$-vx/c_0^2$$ term that is used to maintain a constant speed of light in the Lorentz transformation is used instead to maintain absolute simultaneity in the LET transformation. Both ways are equally valid.

Hmmm,

It is clear how you got the second formula but it is not clear at all how you got the first formula $$c(v,\theta)=c_0$$. Would you care to show the intermediate steps?

 

[clj4]

:
If the two arms of a round-trip interferometer are parallel, then these two terms cancel on subtraction because the two $$(\delta-\beta+1/2)*[(v/c)sin(\theta)]^2$$ terms are both proportional to $$sin^2(\theta)$$. These two terms do not cancel in Michelson-Morley or Kennedy-Thorndike experiments because one of the terms is proportional to $$sin^2(\theta)$$ while the other term is proportional to $$cos^2(\theta)$$ (when the two arms of the interferometer are orthogonal... but they would cancel if the two arms of the interferometer were made parallel, and that is why they are not made parallel). Gagnon's interferometer is not round-trip so I'm not completely sure that this cancellation of terms applies, but he hasn't explained exactly how he got to Eq. (9)

According to (9) in Gagnon the terms do not cancel.
This has been your challenge from the beginning, to prove that expression (9) is zero. Your explanations have been all over the map, the previous one had to do with delta/beta being zero, the one before had to do with some impossible to correlate transformation thru the Lorentz transforms, most of them had to do with the confusion about the framework (which was incorrectly taken to be SR).

Let's try again:

1. How did you arrive to the expressions
$$\phi_1(t)=\phi_1(0)+\omega_1*(2L_1/c)(1/\eta_1+(\delta-\beta+1/2)*[(v/c)sin(\theta)]^2$$

$$\phi_2(t)=\phi_2(0)+\omega_2*(2L_1/c)(1/\eta_2+(\delta-\beta+1/2)*[(v/c)sin(\theta)]^2$$

for the two wave guides?

2. Are you aware that the parenses don't match, therefore the expressions are incomprehensible? You will need to correct that.

Look again at $$\phi_1(t)=\phi_1(0)+\omega_1*(2L_1/c)(1/\eta_1+(\delta-\beta+1/2)*[(v/c)sin(\theta)]^2$$

3. How does the term $$\1\/\eta_1$$ "disappear" from your considerations?

4. Where in your calculations do you factor in that the cutoff frequencies for the two waveguides are arranged to be very different? Gagnon seems to make a great deal of the fact that expression (9) is obtained by driving the two waveguides one just above the cutofff and the second one way above the cutoff frequency.

 

[Aether]

Hmmm,

It is clear how you got the second formula but it is not clear at all how you got the first formula $$c(v,\theta)=c_0$$. Would you care to show the intermediate steps?

This is just a statement that in SR one-way light speed is isotropic, but I will examine it anyway.

SR, RMS (Robertson-Mansouri-Sexl text theory), and Lorentz ether theory are all in agreement that the speed of light $$c_0$$ is defined to be isotropic with respect to at least one particular inertial frame (e.g., within the ether frame $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ -- actually, all velocites are isotropic within this frame, so this equation really relates just to this example, I suppose that a more general equation like this $$\frac{X_2-X_1}{T_2-T_1}=v_0$$ is always valid). In SR however, the Lorentz transform relates this particular (arbitrary) inertial frame with every other inertial frame while explicitly preserving light speed, and therefore the speed of light is further defined (within SR) to be isotropic with respect to every other inertial frame as well.

For example, these are (two sets of) the Lorentz transforms for motion along the x-axis:

$$t_1=((1-v^2/c_0^2)^{1/2}T_1-vx_1/c_0^2)$$,
$$x_1=((X_1-vT_1)/(1-v^2/c_0^2)^{1/2})$$,
$$t_2=((1-v^2/c_0^2)^{1/2}T_2-vx_2/c_0^2)$$,
$$x_2=((X_2-vT_2)/(1-v^2/c_0^2)^{1/2})$$.

Rather than defining light-speed isotropy for all (inertial) frames, I will justify this equation $$c(v,\theta)=c_0$$ by defining light-speed isotropy only in the ether frame (e.g., $$\frac{X_2-X_1}{T_2-T_1}=c_0$$) and then combine this definition with the Lorentz transformations to show that light speed is therefore isotropic in all (inertial) frames $$\frac{x_2-x_1}{t_2-t_1}=c_0$$; such a showing is ultimately coordinate-system dependent, and it is definitely not something that could ever be proven by an experiment.

So, to verify that this equation $$\frac{x_2-x_1}{t_2-t_1}=c_0$$ is true (when $$\frac{X_2-X_1}{T_2-T_1}=c_0$$) then simply transform the four coordinates from any arbitrary inertial frame (e.g., use any v you like) into the ether frame using the Lorentz transform equations given above and verify that you always arrive back at this equation $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ which we have defined above to be true.

Start with:
$$\frac{x_2-x_1}{t_2-t_1}=c_0$$

Transform the four space-time coordinates (by substitution) using the Lorentz transforms:
$$\frac{((X_2-vT_2)/(1-v^2/c_0^2)^{1/2})-((X_1-vT_1)/(1-v^2/c_0^2)^{1/2})}{((1-v^2/c_0^2)^{1/2}T_2-vx_2/c_0^2)-((1-v^2/c_0^2)^{1/2}T_1-vx_1/c_0^2)}=c_0$$




Now, simply reduce this equation to $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ to show that $$c(v,\theta)=c_0$$ is true (this doesn't actually prove that it is true for all $$\theta$$ because we're only looking at motion along the x-axis).

$$\frac{X_2-vT_2-X_1+vT_1}{(1-v^2/c_0^2)T_2-\frac{v(X_2-vT_2)}{c_0^2}-(1-v^2/c_0^2)T_1-\frac{v(X_1-vT_1)}{c_0^2}}=c_0$$

$$\frac{(X_2-vT_2-X_1+vT_1)c_0^2}{(1-v^2/c_0^2)T_2c_0^2-v(X_2-vT_2)-(1-v^2/c_0^2)T_1c_0^2-v(X_1-vT_1)}=c_0$$

$$\frac{(X_2-vT_2-X_1+vT_1)c_0^2}{( c_0^2-v^2)(T_2-T_1)-v(X_2-vT_2-X_1+vT_1)}=c_0$$

$$(X_2-vT_2-X_1+vT_1)c_0^2=c_0(( c_0^2-v^2)(T_2-T_1)-v(X_2-vT_2-X_1+vT_1))$$

$$(X_2-vT_2-X_1+vT_1)c_0^2=c_0( c_0^2-v^2)(T_2-T_1)-v c_0(X_2-vT_2-X_1+vT_1)$$

$$(X_2-vT_2-X_1+vT_1)(c_0^2+v c_0)=c_0( c_0^2-v^2)(T_2-T_1)$$

$$(X_2-vT_2-X_1+vT_1)c_0(c_0+v)=c_0( c_0+v)( c_0-v)(T_2-T_1)$$

$$(X_2-vT_2-X_1+vT_1)c_0=c_0( c_0-v)(T_2-T_1)$$

$$X_2-vT_2-X_1+vT_1=c_0(T_2-T_1)-v(T_2-T_1)$$

$$X_2-vT_2-X_1+vT_1+vT_2-vT_1=c_0(T_2-T_1)$$

$$X_2-X_1=c_0(T_2-T_1)$$

$$\frac{X_2-X_1}{T_2-T_1}=c_0$$

 

[clj4]

This is just a statement that in SR one-way light speed is isotropic, but I will examine it anyway.

SR, RMS (Robertson-Mansouri-Sexl text theory), and Lorentz ether theory are all in agreement that the speed of light $$c_0$$ is defined to be isotropic with respect to at least one particular inertial frame (e.g., within the ether frame $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ -- actually, all velocites are isotropic within this frame, so this equation really relates just to this example, I suppose that a more general equation like this $$\frac{X_2-X_1}{T_2-T_1}=v_0$$ is always valid). In SR however, the Lorentz transform relates this particular (arbitrary) inertial frame with every other inertial frame while explicitly preserving light speed, and therefore the speed of light is further defined (within SR) to be isotropic with respect to every other inertial frame as well.

For example, these are (two sets of) the Lorentz transforms for motion along the x-axis:

$$t_1=((1-v^2/c_0^2)^{1/2}T_1-vx_1/c_0^2)$$,
$$x_1=((X_1-vT_1)/(1-v^2/c_0^2)^{1/2})$$,
$$t_2=((1-v^2/c_0^2)^{1/2}T_2-vx_2/c_0^2)$$,
$$x_2=((X_2-vT_2)/(1-v^2/c_0^2)^{1/2})$$.

Rather than defining light-speed isotropy for all (inertial) frames, I will justify this equation $$c(v,\theta)=c_0$$ by defining light-speed isotropy only in the ether frame (e.g., $$\frac{X_2-X_1}{T_2-T_1}=c_0$$) and then combine this definition with the Lorentz transformations to show that light speed is therefore isotropic in all (inertial) frames $$\frac{x_2-x_1}{t_2-t_1}=c_0$$; such a showing is ultimately coordinate-system dependent, and it is definitely not something that could ever be proven by an experiment.

So, to verify that this equation $$\frac{x_2-x_1}{t_2-t_1}=c_0$$ is true (when $$\frac{X_2-X_1}{T_2-T_1}=c_0$$) then simply transform the four coordinates from any arbitrary inertial frame (e.g., use any v you like) into the ether frame using the Lorentz transform equations given above and verify that you always arrive back at this equation $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ which we have defined above to be true.

Start with:
$$\frac{x_2-x_1}{t_2-t_1}=c_0$$

Transform the four space-time coordinates (by substitution) using the Lorentz transforms:
$$\frac{((X_2-vT_2)/(1-v^2/c_0^2)^{1/2})-((X_1-vT_1)/(1-v^2/c_0^2)^{1/2})}{((1-v^2/c_0^2)^{1/2}T_2-vx_2/c_0^2)-((1-v^2/c_0^2)^{1/2}T_1-vx_1/c_0^2)}=c_0$$

Now, simply reduce this equation to $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ to show that $$c(v,\theta)=c_0$$ is true (this doesn't actually prove that it is true for all $$\theta$$ because we're only looking at motion along the x-axis).

Well, not to be nitpicking but you wrote above
$$c(v,\theta)=c_0$$ for ANY $$\theta$$
The following derivation is true only for $$\theta=0$$

$$\frac{X_2-vT_2-X_1+vT_1}{(1-v^2/c_0^2)T_2-\frac{v(X_2-vT_2)}{c_0^2}-(1-v^2/c_0^2)T_1-\frac{v(X_1-vT_1)}{c_0^2}}=c_0$$

There are some sign errors in the first expression. The correct thing is:

$$\frac{X_2-vT_2-X_1+vT_1}{(1-v^2/c_0^2)T_2-\frac{v(X_2-vT_2)}{c_0^2}-(1-v^2/c_0^2)T_1+\frac{v(X_1-vT_1)}{c_0^2}}=c_0$$
You would then get:

$$\frac{X_2-vT_2-X_1+vT_1}{T_2-\frac{vX_2}{c_0^2}-T_1+\frac{vX_1}{c_0^2}}=c_0$$

If you divide both the numerator and the denominator by $$T_2-T_1$$ and you remember that $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ you indeed get your result.
Very ugly...

 

[Aether]

Well, not to be nitpicking but you wrote above
$$c(v,\theta)=c_0$$ for ANY $$\theta$$
The following derivation is trur only for $$\theta=0$$

That's right. Do you think that it's important to show a more general proof for motion in 3D?

There are some sign errors in the first expression. The correct thing is:

$$\frac{X_2-vT_2-X_1+vT_1}{(1-v^2/c_0^2)T_2-\frac{v(X_2-vT_2)}{c_0^2}-(1-v^2/c_0^2)T_1+\frac{v(X_1-vT_1)}{c_0^2}}=c_0$$
You would then get:

$$\frac{X_2-vT_2-X_1+vT_1}{T_2-\frac{vX_2}{c_0^2}-T_1+\frac{vX_1}{c_0^2}}=c_0$$

If you divide both the numerator and the denominator by $$T_2-T_1$$ and you remember that $$\frac{X_2-X_1}{T_2-T_1}=c_0$$ you indeed get your result.

There is a sign error in the first two equations that I gave, but the rest are OK. The correct equations are:

$$\frac{X_2-vT_2-X_1+vT_1}{(1-v^2/c_0^2)T_2-\frac{v(X_2-vT_2)}{c_0^2}-(1-v^2/c_0^2)T_1+\frac{v(X_1-vT_1)}{c_0^2}}=c_0$$
$$\frac{(X_2-vT_2-X_1+vT_1)c_0^2}{(1-v^2/c_0^2)c_0^2T_2-v(X_2-vT_2)-(1-v^2/c_0^2)c_0^2T_1+v(X_1-vT_1)}=c_0$$

Very ugly...

What do you mean by this?

 

[clj4]

That's right. Do you think that it's important to show a more general proof for motion in 3D?

That the reasoning is only valid for theta=0.

There is a sign error in the first two equations that I gave, but the rest are OK. The correct equations are:

$$\frac{X_2-vT_2-X_1+vT_1}{(1-v^2/c_0^2)T_2-\frac{v(X_2-vT_2)}{c_0^2}-(1-v^2/c_0^2)T_1+\frac{v(X_1-vT_1)}{c_0^2}}=c_0$$
$$\frac{(X_2-vT_2-X_1+vT_1)c_0^2}{(1-v^2/c_0^2)c_0^2T_2-v(X_2-vT_2)-(1-v^2/c_0^2)c_0^2T_1+v(X_1-vT_1)}=c_0$$

Yes, I corrected it for you. I also cleaned up the proof a little.

What do you mean by this?

That the whole RMs theory is exceptionally ugly. But never mind this detour, let's return to my 4 questions on the Gagnon experiment.

 

[Aether]

But never mind this detour, let's return to my 4 questions on the Gagnon experiment.

clj4 and I are continuing a discussion on this topic that were were having here http://www.bautforum.com/showthread.php?t=38765 (see post #47) because Latex isn't supported by BAUT at this time.

The issue at hand is this paper [D.R. Gagnon et al., Guided-wave measurement of the one-way speed of light, Physical Review 38A(4), 1767 (1988); http://imaginary_nematode.home.comca...et_al_1988.pdf.] which seems to be claiming that RMS predicts a different outcome to their experiment than does SR (although even that isn't really made clear in the paper).

Now I've got to study some more about waveguides and try to reverse engineer Eq. (9) from (Gagnon et al., 1988) because it wasn't derived in the paper.

 

[clj4]

. No experiment has ever been able to distinguish between these two points of view, and if one ever does it could only favor the ether view.

What makes you think that? In light of the experimental proof that shows exactly the opposite...see our little discussion

 

[clj4]

Can you read Hans' version? If not, I'll post a pdf. I made a .pdf first, but it was very large so I wen't for a .doc; it wasn't any smaller though.

Suppose that LET turns out only to be useful as a waypoint (for some) on the philosophical journey to SR. Even so, what is the justification for tolerating false (however well intentioned) claims that the constancy of the speed of light is an empirically determined fact? Why not just state up front that partiality to SR is simply a matter of coordinate choice, and not an empirical necessity?

You know (by now) that the claims of one way light speed constancy are far from being false. You also know that there are several experiments proving it.

 

[clj4]

Lorentz transformations amount to an arbitrary choice of Einstein's clock synchronization convention. Lorentz symmetry is equally well represented by transformations in which absolute simultaneity is maintained.

Correct. The problem starts when the experiments come back and show that the Robertson-Mansoury-Sexl HYPOTHESIS of light speed anisotropy is wrong.

 

[clj4]

Their (M&S) general approach may also be useful in rotating coordinate systems, where the usual assumption of isotropy has issues. While one can always chose not to use rotating coordiantes, they are convenient enough that sometimes it's worth giving up the conveniences of isotropy for the convenience of using rotating coordinates.

It would be useful except that they derived it under very restrictive conditions (translation only).
The RMS (R stands for Robertson) is indeed a "test theory". Looks like there are quite a few one-way light speed experiments that refute the light speed anisotropy. A good thing because RMS is one of the ugliest theories ever.

 

[clj4]

This is misleading. The M&S paperer is referred to because of the
parameterization scheme for possible deviations of Special Relativity.
It's only you who uses it to promote your ether theory.

The math of M&S is correct in the preferred frame, not in any other.



Do you at all read my post? do you look at my examples. No you don't

SR is the ABC of physics. Something you have to understand pretty
well before you can start to learn some real physics. The examples I
gave are the simplest it gets in understanding the basic mechanisms in SR
and the simplest way to show that your Ether theory with absolute time
can never work.

Now try to do the math. Try to understand the physics. Don't just rely on
some statement you have found somewhere in a paper. It's now time for you
to prove your ether theory by actually showing how it can account for
these relativistic effects.

1) How can two observers both see the other in a Lorentz contracted state?

2) How can you rotate the deBroglie wavefronts of particles if you go from
one reference frame to another in order to keep them at right angles with
the direction of their speed? How can a single transformation rotate these
wavefront at all kinds of different angles depending on the speed of the
particles?

Let's see if you can do that without non simultaneity.

People here are willing to help others to get ahead. That's why it's called a
Physics Help and Math Help forum. But if there's no response and you just
keep repeating a statement from somewhere then things get pretty useless
after a while. I did the work, the math, the physics, showed you the images
from my simulations.

Now it's up to you.



Regards, Hans.

Hans,

I appreciate your passion. Have a little patience, RMS will go away. Soon.
It has hung around for nearly 30 years, it has evolved into the Standard Model (SM) which has, under Alan Kostelecky evolved into the Standard Model Extension. Test theories that look for "Lorentz symmetry violations" and find ...nothing. But they keep a lot of test theorists employed. No harm done, an interesting exercise.
RMS is very rarely taught in school (maybe because it is exceptionally ugly?, maybe because it assumes that light speed is anisotropic-something refuted by experiment?). Who knows? It is good to discuss such things but we also must tell the truth: there are quite a few OWLS experiments and they all concluded that the light speed is...isotropic. This is the theme of this thread, so we need to set the things right.

 

[clj4]

Equivalent? Only in the preferred frame.

Note that 3.6 makes both length and speed anisotropic in any other
reference frame besides the preferred frame. Round wheels would be
only really round in the preferred frame.

We at Earth would see al our wheels changing shape in a 24 hours cycle
corresponding with the rotation of the earth.


Regards, Hans

Hi Hans

Can you show this mathematically. I am just curious how one approaches this issue mathematically.

 

[Aether]

No experiment has ever been able to distinguish between these two points of view, and if one ever does it could only favor the ether view.

What makes you think that? In light of the experimental proof that shows exactly the opposite...see our little discussion

When going this far back, please give the post# that you are quoting from.

I no longer think that exactly, and I haven't said exactly that recently. The inherently coordinate-system dependent nature of speed measurements simply can't be overcome by any classical experiment, not even in principle. The eventual detection of any violation of local Lorentz invariance would allow a coordinate-system that maintains absolute simultaneity (e.g., the ether view) to be realized within any laboratory in a consistent way (e.g., a locally preferred frame would be measurable from within a windowless laboratory); such a coordinate system can be realized right now, but the choice of the locally preferred frame would be arbitrary since there isn't any known experiment to detect it. SR could still continue to be used for many purposes even if Lorentz symmetry is violated.

I agree that there is ample experimental evidence to constrain the possible violation of local Lorentz invariance to a small value (not counting gravity!) within the error-bars of the experiments. What you are claiming is that one-way speed of light measurements aren't coordinate-system dependent, and that is what I disagree with. Why aren't you proposing that the "postulates" of SR are actually proven facts, and therefore SR needs to be updated to reflect that? What about the quote from Albert Einstein in my signature below, what do you think that he meant by this?

 

[clj4]

What you are claiming is that one-way speed of light measurements aren't coordinate-system dependent, and that is what I disagree with. Why aren't you proposing that the "postulates" of SR are actually proven facts, and therefore SR needs to be updated to reflect that? What about the quote from Albert Einstein in my signature below, what do you think that he meant by this?

Sorry, I should have known better than to give you an opportunity for a diversion. We are still on the Gagnon experiment.

Let's not play games , I don't claim anything else than the fact that there are OWLS experiments that show light speed to be clearly isotropic. We are still on the Gagnon experiment, you have to provide a valid answer to it. We are waiting...

 

[Aether]

Suppose that LET turns out only to be useful as a waypoint (for some) on the philosophical journey to SR. Even so, what is the justification for tolerating false (however well intentioned) claims that the constancy of the speed of light is an empirically determined fact? Why not just state up front that partiality to SR is simply a matter of coordinate choice, and not an empirical necessity?

You know (by now) that the claims of one way light speed constancy are far from being false.

There are a handfull of published papers (listed here in sextion 3.2: http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html) that lend (meager) support to such claims, but still any claim that the constancy of the one-way speed of light is an empirically determined fact is false.

You also know that there are several experiments proving it.

I know that there are several old experiments claiming to prove it, I also know that Mansouri-Sexl and Zhang say that this isn't possible even in principle. Please cite a modern experiment to test for local Lorentz violations (or claiming to measure the one-way speed of light) that doesn't either cite Mansouri-Sexl directly, or at least indirectly; and then show where they claim (as you do) that Mansouri-Sexl is wrong. Or, cite one modern (or not so modern) reference that claims flatly (as you do) that Mansouri-Sexl is wrong.

 

[clj4]

There are a handfull of published papers (listed here in sextion 3.2: http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html) that lend (meager) support to such claims, but still any claim that the constancy of the one-way speed of light is an empirically determined fact is false.

I know that there are several old experiments claiming to prove it, I also know that Mansouri-Sexl and Zhang say that this isn't possible even in principle. Please cite a modern experiment to test for local Lorentz violations (or claiming to measure the one-way speed of light) that doesn't either cite Mansouri-Sexl directly, or at least indirectly; and then show where they claim (as you do) that Mansouri-Sexl is wrong. Or, cite one modern (or not so modern) reference that claims flatly (as you do) that Mansouri-Sexl is wrong.

We've been through this, let's not go around in circles. We agreed that you have an experiment to work on, you tried (and failed) to refute it, please stop regurgitating the same quotes from MS and/or Zhang.

You need to continue from here:

https://www.physicsforums.com/showpost.php?p=930823&postcount=107

 

[vladb]

Hi:

Einstein's second postulate states that the speed of light is constant as viewed from any frame of reference. <...>

Actually Einstein didn't postulate that. Just today I've read an article (which I'm going to attach. See part III there) about misconceptions about special relativity.
As far as I understood, Einstein postulated that the speed of light doesn't depend on the velocity of the source of light _only_ (in an inertial frame). And only from this he derived that the speed of light is also constant in all inertial frames of reference.

 

[Aether]

Lorentz transformations amount to an arbitrary choice of Einstein's clock synchronization convention. Lorentz symmetry is equally well represented by transformations in which absolute simultaneity is maintained.

Correct.

Good, I'm glad that we agree on this. Then our discussion on the coordinate-system dependent nature of one-way speed measurements is concluded in my favor?

The problem starts when the experiments come back and show that the Robertson-Mansoury-Sexl HYPOTHESIS of light speed anisotropy is wrong.

This is a different discussion entirely (about violations of Lorentz symmetry as opposed to coordinate-system dependency), but we can continue-on with this now as long as we're both on the same page with respect to the inherently coordinate-system dependent nature of one-way speed of light measurements.

We've been through this, let's not go around in circles. We agreed that you have an experiment to work on, you tried (and failed) to refute it, please stop regurgitating the same quotes from MS and/or Zhang.

You need to continue from here:

https://www.physicsforums.com/showpos...&postcount=107

OK.

Actually Einstein didn't postulate that. Just today I've read an article (which I'm going to attach. See part III there) about misconceptions about special relativity.
As far as I understood, Einstein postulated that the speed of light doesn't depend on the velocity of the source of light _only_ (in an inertial frame). And only from this he derived that the speed of light is also constant in all inertial frames of reference.

Thanks for the the article vladb. It looks like it's hot off the press. Here's Einstein's 1905 paper on special relativity: http://home.tiscali.nl/physis/HistoricPaper/Historic Papers.html.

 

[clj4]

Good, I'm glad that we agree on this. Then our discussion on the coordinate-system dependent nature of one-way speed measurements is concluded in my favor?

Not. we've been thru this before, please stop the diversions.

The only thing we agree on is that you need to finish the refutation of the Gagnon paper, pick up from here, please:

https://www.physicsforums.com/showpost.php?p=930823&postcount=107