What's wrong with Michelson's explanation of ether dragging?

In summary: However, the deeper truth that Lorentz believed in was that the ether must be connected to the physical world in a fundamental way. Although this explanation seems rather simplistic, perhaps there is some deeper truth to it (such as, the gravitational fields and the light propagation medium/ether are "connected" at the most fundamental level).
  • #1
joey_m
22
0
The null result of the Michelson-Morley experiment led Lorentz to believe that the ether can never be detected because it distorts things in the direction that they move with respect to it. But from what I've read, Michelson himself explained the result by saying that the ether must somehow be "dragged along" by the movements of the earth. Although this explanation seems rather simplistic, perhaps there is some deeper truth to it (such as, the gravitational fields and the light propagation medium/ether are "connected" at the most fundamental level).

Isn't this explanation at least as sensible as one that says that space and time are relative? (I'm still not sure what the heck that's supposed to mean!)
 
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  • #2
Nothing was wrong with the Ether dragging explanation. It followed the normal route of physical enquiry. It was proposed, experimentally tested and found to be incorect.

Matheinste.
 
  • #3
joey_m said:
The null result of the Michelson-Morley experiment led Lorentz to believe that the ether can never be detected because it distorts things in the direction that they move with respect to it. But from what I've read, Michelson himself explained the result by saying that the ether must somehow be "dragged along" by the movements of the earth.
See the problem with stellar aberration discussed here.
joey_m said:
Isn't this explanation at least as sensible as one that says that space and time are relative? (I'm still not sure what the heck that's supposed to mean!)
Basically it means that the different inertial coordinate systems related by the Lorentz transform disagree about distances and times between events, but the laws of physics are found to work exactly the same in all these coordinate systems, so there's no reason to label one system's measurements as more correct than any other's. Did you read my last comment on the Problem with thought experiment thread? If so, did you have an answer to this question?
If you think there is some "real truth" about whose clocks are in-sync and whose clocks are out-of-sync, how would you propose to determine this experimentally?
 
  • #4
I'll read up on the phenomenon of stellar aberration. I don't think it'll change my basic point that, assuming that a light-propagation medium exists (which even Einstein assumes), there must be a fundamental relationship between it and the "gravity-propagation medium" (for want of a better term).

(Jesse, I know it means those series of words, but I just can't wrap my mind around those series of words. That is, I can't "feel it", if you know what I mean!)
 
  • #5
joey_m said:
I'll read up on the phenomenon of stellar aberration. I don't think it'll change my basic point that, assuming that a light-propagation medium exists (which even Einstein assumes), there must be a fundamental relationship between it and the "gravity-propagation medium" (for want of a better term).
I don't think Einstein assumed there was a medium for the propogation of either! What quote of his are you thinking of?
 
  • #6
joey_m said:
The null result of the Michelson-Morley experiment led Lorentz to believe that the ether can never be detected because it distorts things in the direction that they move with respect to it. But from what I've read, Michelson himself explained the result by saying that the ether must somehow be "dragged along" by the movements of the earth. Although this explanation seems rather simplistic, perhaps there is some deeper truth to it (such as, the gravitational fields and the light propagation medium/ether are "connected" at the most fundamental level).

Isn't this explanation at least as sensible as one that says that space and time are relative? (I'm still not sure what the heck that's supposed to mean!)

The problem is that it was falsified experimentally. See here.
 
  • #7
And not only did the Michelson-Morley experiment invalidate the existing ether theories of that time but relativity was confirmed, not just in general but with very specific predictions like the precession of the perihelion of Mercury, and since then by all sorts of astronomical phenomena and by the relativistic effects experienced by satellites orbiting the Earth.
 
  • #8
joey_m said:
I'll read up on the phenomenon of stellar aberration. I don't think it'll change my basic point that, assuming that a light-propagation medium exists (which even Einstein assumes), there must be a fundamental relationship between it and the "gravity-propagation medium" (for want of a better term).

(Jesse, I know it means those series of words, but I just can't wrap my mind around those series of words. That is, I can't "feel it", if you know what I mean!)

It is a difficult thing to wrap your mind around without practice. There also must be a good intuitive reason for practicing which seems to be what you are searching for. Just being asked to believe the Lorentz transform explains it is not good enough for many of us. So let's take a closer look.

The problem with stellar aberration has already been pointed out. Even if we assumed that Special Relativity (SR) and Lorentz Ether Theory (LET) agreed in all predictions there are problems. On closer inspection it is no different than defining a certain object in the Universe as not moving and the speed of everything else must be measured relative to that one object. Only in LET this object/s we are supposed to use for this cannot be detected much less measured in any way so we can't even know what is moving relative to it. On Earth we use ground speed to do this but ground speed is a very poor choice when doing orbital mechanics of the solar system or our galaxy. We could still use ground speed and correctly say the Earth is not moving and say everything orbits around the Earth using some VERY complex math, but why? It would, like LET (mostly), make the same predictions. LET runs into similar difficulties when we talk about many phenomena. LET is even worse because it uses something we cannot detect, measure, or determine any aspect of how it works to define what is at rest. At least we can see and measure ground speed. What use is something we can't measure to define what is at rest when another observer can say its motion is different (because there is no way to measure it) and this other observer is just as correct as the first, even if they both claim the ether as their rest frame? This alone means that it doesn't even make a good interpretation as claimed, even if stellar aberration didn't falsify it.

Suppose that at some level there is something to the ether idea. Suppose you could re-define some kind of ether that successfully predicts stellar aberration, gravity consistent with General Relativity, etc. What value would that be? If all it did was tell us what we already know it would be of no value whatsoever. It would still leave us with no way to test if it really was the right explanation. It would just be an ontological construct designed to soothe the philosophical prejudices of a few. If you think such a model could make new testable predictions that couldn't be predicted by standard means then I for one will hear you out. Before you can even dream of doing something like that it is imperative that you fully understand why the ether as conceived is not a viable construct to explain relativistic effects. So the main reason for objecting to postulating a classical ether connection to gravity is not because it is fundamentally impossible, though it probably is, but history has shown it to be at best superfluous and at worse flat out wrong. If you want to object then deliver the goods. If you wish to pursue this model then more power to you, just don't claim what you haven't got.
 
  • #9
JesseM said:
I don't think Einstein assumed there was a medium for the propogation of either! What quote of his are you thinking of?
If you will read his Leiden address of 1920 and/or his essay "On the Ether" from 1924, you will see that he was quite explicit about the need for an ether to support the propagation of EM through "empty" space. He also claimed that gravitation and inertial effects arise from matter's interaction with the local ether in which it is embedded, and not from some Machian "spooky action-at-a-distance". These ideas are given little notice these days, and in fact are generally dismissed out-of-hand, but that was his line of reasoning for many years. He felt that this was a viable route to incorporate electromagnetism into GR, though he never managed to accomplish it in his lifetime. Here's the final paragraph of the Leiden address.

Einstein said:
Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.
 
  • #10
turbo-1 said:
If you will read his Leiden address of 1920 and/or his essay "On the Ether" from 1924, you will see that he was quite explicit about the need for an ether to support the propagation of EM through "empty" space. He also claimed that gravitation and inertial effects arise from matter's interaction with the local ether in which it is embedded, and not from some Machian "spooky action-at-a-distance". These ideas are given little notice these days, and in fact are generally dismissed out-of-hand, but that was his line of reasoning for many years. He felt that this was a viable route to incorporate electromagnetism into GR, though he never managed to accomplish it in his lifetime. Here's the final paragraph of the Leiden address.
That quote doesn't say what you're saying it all--it only says that space has its own properties, and in that sense it's like an ether. But he doesn't say it's like an ether in the crucial sense of having its own distinct local rest frame.
 
  • #11
JesseM said:
That quote doesn't say what you're saying it all--it only says that space has its own properties, and in that sense it's like an ether. But he doesn't say it's like an ether in the crucial sense of having its own distinct local rest frame.
He was far more specific in his essay "On the Ether", which is Chapter One of "The Philosophy of Vacuum" by Saunders and Brown. The translation is copyrighted material, but here is an authorized preview. Unfortunately, two critical pages are missing, but you'll get the gist of it. Some of the missing material deals with his rejection of the Machian notion of action-at-a-distance. He kept pursuing this line of reasoning for many years, and I wish more of his works from this period were available in English translations.

http://books.google.com/books?id=ZU...ig=82St7DUG9LXyCKuuKiS8BU4smkA&hl=en#PPA17,M1
 
  • #12
turbo-1 said:
He was far more specific in his essay "On the Ether", which is Chapter One of "The Philosophy of Vacuum" by Saunders and Brown. The translation is copyrighted material, but here is an authorized preview.
Specific about what? Did he claim that there was an ether with a single definite local rest frame? If you've read this book, can you provide a relevant quote?
 
  • #13
He said in part "The fact that the general theory of relativity has no preferred space-time coordinates which stand in a determinate relation to the metric is more a characteristic of the mathematical form of the theory than of its physical content."

and

"The metric tensor which determines both the gravitational and inertial phenomena on the one hand and the tensor of the electromagnetic field on the other, still appear as fundamentally different expressions of the state of the ether: but their logical independence is probably more to be attributed to the imperfection of our theoretical edifice than to a complex structure of reality itself."
 
  • #14
turbo-1 said:
He said in part "The fact that the general theory of relativity has no preferred space-time coordinates which stand in a determinate relation to the metric is more a characteristic of the mathematical form of the theory than of its physical content."
Looking at the context on p. 17-18, I don't think the first quote is talking about the issue of a single preferred frame at all, when he says "no preferred space-time coordinates" I think he's comparing it to special relativity which has a family of preferred coordinate systems, namely the inertial frames (see the preceding discussion on p. 17 where he talks about the specialness of inertial coordinate systems in SR); so in that quote I think he's talking talking about the fact that general relativity is stated in a "generally covariant" form which works the same in all coordinate systems, but as I talked about in post #8 on this thread, it was indeed realized after Einstein formulated this principle that it was really more a feature of the mathematical formulation, and not a physical feature, because it's possible to rewrite any theory (including Newtonian mechanics) in a generally covariant tensor form which will work in arbitrary coordinate systems.
turbo-1 said:
and

"The metric tensor which determines both the gravitational and inertial phenomena on the one hand and the tensor of the electromagnetic field on the other, still appear as fundamentally different expressions of the state of the ether: but their logical independence is probably more to be attributed to the imperfection of our theoretical edifice than to a complex structure of reality itself."
This second quote from p. 18 doesn't talk about the issue of a preferred frame at all, I think he's talking about the issue of whether gravity and electromagnetism can be unified or whether they're two separate forces (note that the preceding sentence is 'Furthermore, in my opinion, we have not as yet succeeded in going beyond a superficial integration of the electromagnetic forces into the general scheme of relativity', so it's clear he's making a speculative comment about the idea that in the future we might find a new theory which unifies them, perhaps by treating electromagnetism in terms of curved spacetime as in the Kaluza-Klein theory). He does state this idea in terms of "ether", but throughout the paper it's clear that he's using the word "ether" just to refer to the notion of space having intrinsic properties and fields associated with it, without implying the additional notion that there is a single preferred frame as in pre-relativistic notions of the ether. I note that on p. 17 he says:
In this way the Maxwell-Lorentz theory finally influenced our understanding of the theoretical foundations of physics to such an extent that it led to the founding of the special theory of relativity. It was realized that the electromagnetic equations do not in truth determine a particular state of motion, but that, in accordance with these equations--just as in classical mechanics--there is an infinite manifold of coordinate systems, moving uniformly with respect to each other, and all on a par, so long as one applies suitable transformation formulae for the space coordinates and the time. It is well known that this realization brought about a deep modification of kinematics and dynamics as a result. The ether of electrodynamics now no longer had any special or particular state of motion. It had the effect, like the ether of classical mechanics, of giving preference not to a particular state of motion, but only to a particular state of acceleration.
So, I think you've misinterpreted the two quotes you posted if you thought Einstein was suggesting an ether with a single preferred rest frame, and I think he makes it clear in the article that he isn't defining "ether" in this way.
 
  • #15
Let me just say that I am not advocating a literal phenomenon of "ether dragging" whereby mass and the ether simply interact with one another. I am thinking more along the lines of the notion that the null result of the MM experiment simply tells us that the ether is perfectly at rest with respect to the surface of the earth. Instead of going full steam ahead into "relativity mode", can't we consider the possibility that the ether is an extraordinarily dense, yet highly fluid medium? Lorentz assumed it to be more like a static, solid block. Can't we have more imagination than that?

Imagine the surface of a body of water in which vortexes are embedded inside of vortexes. Inside of each particular vortex, water energy waves will always propagate at the same rate with respect to a frame of reference that is swirling along with it.

We can now correlate our water vortexes with local, inwardly directed compressions of the light propagation medium. Likewise, the entire body of water can be equated with the totality of the universal ether.
 
  • #16
Joey_m:
Instead of going full steam ahead into "relativity mode", can't we consider the possibility that the ether is an extraordinarily dense, yet highly fluid medium? Lorentz assumed it to be more like a static, solid block. Can't we have more imagination than that?

General relativity, relativistic electrodynamics and quantum theory work without an aether. So why introduce this complicated thing when it isn't needed, and probably won't work in any case.
 
  • #17
joey_m said:
Instead of going full steam ahead into "relativity mode", can't we consider the possibility that the ether is an extraordinarily dense, yet highly fluid medium? Lorentz assumed it to be more like a static, solid block. Can't we have more imagination than that?

No, see here. Why do you keep reposting the same ideas and you refuse to read?
 
  • #18
Mentz114 said:
Joey_m:


General relativity, relativistic electrodynamics and quantum theory work without an aether. So why introduce this complicated thing when it isn't needed, and probably won't work in any case.

I think it is needed because when theories don't quite fit observation then people start artificially modifying them to an absurd degree. Relativity and quantum theory are both very successful models up to a point. I think replacement is inevitable.

Incidentally, the Michelson Morley experiment failed because the aether moves towards the centre of the earth. The Earth is not traveling through it so no modification to the speed of light beams would be expected.

Nick
 
  • #19
Nickelodeon said:
I think it is needed because when theories don't quite fit observation then people start artificially modifying them to an absurd degree. Relativity and quantum theory are both very successful models up to a point. I think replacement is inevitable.

Incidentally, the Michelson Morley experiment failed because the aether moves towards the centre of the earth. The Earth is not traveling through it so no modification to the speed of light beams would be expected.

Nick

That second to last sentence of yours gives away the model you speak of as if it has authority. Personally I like analyzing aether theories and seeing exactly when where and how they break down empirically. The version you are speaking so authoritatively about fails any basic comprehension of thermodynamics, even at the level of aerodynamics.

Isn't the model you are implying here "artificially modifying" a theory over a hundred years old? What do you mean by replacement? Even if some version of an aether theory actually worked it does not imply a replacement for the standard formalism.

Hint: Steer clear of your personal theories unless you think your prepared to defend it in the Independent Research thread. Learn everything you can about the standard formalism you can. I can almost guarantee it's not what you think it is from reading descriptions.

Good luck...
 
  • #20
Mentz114 said:
Joey_m:


General relativity, relativistic electrodynamics and quantum theory work without an aether. So why introduce this complicated thing when it isn't needed, and probably won't work in any case.

my_wan said:
That second to last sentence of yours gives away the model you speak of as if it has authority. Personally I like analyzing aether theories and seeing exactly when where and how they break down empirically. ...

Hi,

There are two versions of aether being discussed in this thread.

1) Old aether

This version of aether is a fluid medium that llight waves travel in, rather like waves in an ocean or sound in air. The problem with this version was that the motion of the Earth would be expected to leave a detectable wake as the Earth moved through it and that it would cause the Earth to slow down in its orbit as the aether dragged on it. The null result of the MM experiment caused people to conjecture that the aether was dragged along with the Earth rather like the atmosphere. This conjecture was disproved by measurements of stellar aberration.

2) Lorentz aether.

Lorentz proposed a more sophisticated version of the aether. His aether had the properties that anything moving relative to the Lorentz aether is length contracted and time dilated according to the Lorentz transformations. His aether is completely frictionless and so is not dragged along and does not leave a wake. (It passes through normal matter with little or no interaction rather like the WIMP particles that are proposed as an explanation of dark matter). Mathematically, the results of assuming a Lorentz aether are (for most trivial calculations) identical to those of Einstein. Einstein correctly claimed that since the mathematical results of not assuming an aether are no different to assuming an aether, then there is no need of an aether. It is also correct to state that it is not possible to detect the aether by any experimental means and therefore it is not possible to determine an absolute reference frame.

However, even though Einstien correctly points out that there is no (mathematical) need of an aether, for some people an aether might be helpful for intuitively viewing relativity (even if just to prove to themselves that it is undetectable). Often, it is suggested (as hinted at by Metz114 and my_wan) that assuming a Lorentz aether would produce incorrect results in some situation, yet I have never seen a clear demonstration of where classic relativity and assuming a Lorentz aether produce different results, mathematically. Can anyone here demonstrate (or point to a link) where they are not mathematically equivalent?
 
  • #21
my_wan said:
Personally I like analyzing aether theories and seeing exactly when where and how they break down empirically.
Good luck...

Using the word 'aether' is probably asking for trouble. I, perhaps, should have used 'dark energy' as a modern replacement or perhaps something akin to 'Hubble flow', or maybe 'quintessence'.
 
  • #22
Lorentz's problem is that he made a fatal flaw in interpreting the MM results: he had a "directional bias"...

He only asked why it was that a light beam shot in the same direction of Earth's motion around the sun yields the same speed as a beam shot in the perpendicular direction. He did not ask why a beam shot in the opposite direction of Earth's motion is also the same speed as a beam shot perpendicularly. In the first case, his explanation was that objects shrink along the axis that is in the direction of movement relative to the ether. Using this logic, his explanation in the second case would have been that objects grow along this same axis. Had he only considered the second case, the Lorentz transformation would have been the exact opposite of what it currently is, and Einstein's theory of special relativity would have said that relatively moving bodies see each other as larger (space dilation), and that they see each other's clocks ticking faster (time contraction). Einstein only bought Lorentz's explanation because (according to Michio Kaku) Einstein used to wonder about what it would be like to try to catch up with a beam of light rather than rushing headlong towards a beam of light. There was a fundamental "directional bias" in both of their arguments.

If you understand the issues that I am raising, then there is no choice but to go back to the "drawing board", by way of reinterpreting the meaning of the null result of the MM experiment. After the experiment, Michelson thought that the ether is carried along with the movement of the Earth on a global scale, rather than in the way that those supposed ether-drag disproving experiments tried to show. But if you make the assumption, like me, that the Earth's "ether context" is identical with its "gravitational context", then you have an explanation that both makes logical sense (as opposed to Lorentz/Einstein) and conforms to observable reality.
 
  • #23
joey_m said:
Lorentz's problem is that he made a fatal flaw in interpreting the MM results: he had a "directional bias"...

He only asked why it was that a light beam shot in the same direction of Earth's motion around the sun yields the same speed as a beam shot in the perpendicular direction. He did not ask why a beam shot in the opposite direction of Earth's motion is also the same speed as a beam shot perpendicularly. In the first case, his explanation was that objects shrink along the axis that is in the direction of movement relative to the ether. Using this logic, his explanation in the second case would have been that objects grow along this same axis. Had he only considered the second case, the Lorentz transformation would have been the exact opposite of what it currently is, and Einstein's theory of special relativity would have said that relatively moving bodies see each other as larger (space dilation), and that they see each other's clocks ticking faster (time contraction). Einstein only bought Lorentz's explanation because (according to Michio Kaku) Einstein used to wonder about what it would be like to try to catch up with a beam of light rather than rushing headlong towards a beam of light. There was a fundamental "directional bias" in both of their arguments.

If you understand the issues that I am raising, then there is no choice but to go back to the "drawing board", by way of reinterpreting the meaning of the null result of the MM experiment. After the experiment, Michelson thought that the ether is carried along with the movement of the Earth on a global scale, rather than in the way that those supposed ether-drag disproving experiments tried to show. But if you make the assumption, like me, that the Earth's "ether context" is identical with its "gravitational context", then you have an explanation that both makes logical sense (as opposed to Lorentz/Einstein) and conforms to observable reality.

The MM experiment considers the two way speed of light so it covers both cases. The beam going forward along the parallel arm is reflected at a mirror and returned to the origin of the aparatus, so for part of the experiment the beam IS traveling in the opposite direction to the motion of the Earth. Lorentz's explanation does not require the arm to "grow" in any direction. During the actual experiment the apparatus was floated on a bed of mecury and rotated in a variety of directions, giving a null result whichever way it was orientated with respect to the Earth's supposed motion. Lorentz formulated his transformation rules for length and time to be in agreement with any direction of motion and to give a consistent measured speed of light in any direction. Quite simply, the Lorentz transformations are mathematically correct and in agreement with actual experimental observations. How they are interpreted physically is another matter.
 
  • #24
joey_m said:
He only asked why it was that a light beam shot in the same direction of Earth's motion around the sun yields the same speed as a beam shot in the perpendicular direction. He did not ask why a beam shot in the opposite direction of Earth's motion is also the same speed as a beam shot perpendicularly. In the first case, his explanation was that objects shrink along the axis that is in the direction of movement relative to the ether. Using this logic, his explanation in the second case would have been that objects grow along this same axis.
As kev said, the MM experiment was based on the round-trip time for a light beam to go from one end of the arm of an interferometer to the other end and back--Lorentz was already assuming the light went in both directions. If you actually do the math, you see that assuming the arm always shrinks in proportion to its speed relative to the aether will indeed lead to the prediction that the time for the light to make the round trip is the same regardless of the arm's motion relative to the aether.
 
  • #25
joey_m said:
He only asked why it was that a light beam shot in the same direction of Earth's motion around the sun yields the same speed as a beam shot in the perpendicular direction. He did not ask why a beam shot in the opposite direction of Earth's motion is also the same speed as a beam shot perpendicularly. In the first case, his explanation was that objects shrink along the axis that is in the direction of movement relative to the ether. Using this logic, his explanation in the second case would have been that objects grow along this same axis.

There is no experimental confirmation for any growth in the direction opposite to movement. Quite the opposite :rofl:
 
  • #26
Nickelodeon said:
Using the word 'aether' is probably asking for trouble. I, perhaps, should have used 'dark energy' as a modern replacement or perhaps something akin to 'Hubble flow', or maybe 'quintessence'.

It doesn't matter what terms you use, actually it makes it worse. You said, "Incidentally, the Michelson Morley experiment failed because the aether moves towards the centre of the earth". Strike "aether" and replace the term with "dark energy" and you only compound your problem. "The Michelson Morley experiment failed because dark energy moves towards the centre of the earth" makes no sense whatsoever. It's so distorted that I wouldn't even know where to begin explaining the problem. The 'Hubble flow' is even worse still. Using 'quintessence' is no different than saying something is made of something. However, I do understand the generalizations you have made and why. Your authoritative tone is off the deep end.
 
  • #27
I think we have allowed this thread enough breathing time, and it is appropriate to point out the https://www.physicsforums.com/showthread.php?t=5374", especially for those of you who are new here. Pay particular attention to speculative, unverified personal theories and where exactly you can try these. PF is not the place to simply throw out an idea without any valid premise, so if you only wish to do that, you've found the wrong place.

Zz.
 
Last edited by a moderator:

1. What is Michelson's explanation of ether dragging?

Michelson's explanation of ether dragging is based on the idea that the ether, a hypothetical substance that was believed to fill all space and serve as the medium for the propagation of light, is dragged by the Earth as it moves through space.

2. Why is there controversy surrounding Michelson's explanation?

There is controversy surrounding Michelson's explanation because it conflicts with the principles of special relativity, which state that the speed of light is constant in all inertial frames of reference and does not depend on the movement of the source or the observer.

3. How did Michelson's experiment contribute to the understanding of ether dragging?

Michelson's famous experiment, known as the Michelson-Morley experiment, was designed to detect the motion of the Earth through the ether. The results of the experiment showed no evidence of ether dragging, leading to the development of the theory of special relativity.

4. What evidence supports the rejection of Michelson's explanation?

Aside from the results of the Michelson-Morley experiment, there is also evidence from other experiments such as the Kennedy-Thorndike experiment and the Fizeau experiment that contradict Michelson's explanation of ether dragging.

5. Is there any modern support for Michelson's explanation of ether dragging?

No, there is no modern support for Michelson's explanation of ether dragging. The principles of special relativity have been extensively tested and confirmed, and no evidence has been found to support the idea of ether dragging.

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