# Relativity without the aether: pseudoscience?

JesseM said:
The points in his article would apply to any theory that posits a special reference frame (for example, a theory that says there is only one frame where the speed of light is 'really' c in all directions) and yet does not make any predictions about the results of actual experiments which are different from those of SR (so all observers will measure the speed of light to be c in all directions, even if this is explained as a faulty measurement because their clocks are not ticking the 'correct' time and their rulers are not reading the 'correct' length).
Aether said:
I'm not sure what you mean by the statement in parenthesis.
Well, think of it this way. Suppose we live in a universe governed by purely Newtonian laws, where light always moves at speed c with respect to the rest frame of the aether, and in other frames it is actually possible to measure your velocity relative to the aether by seeing how fast light moves in one direction vs. the other using ordinary rulers and clocks, just like in our actual universe we could measure our velocity relative to the atmosphere by measuring how fast sound waves move in one direction vs. the other. In this hypothetical universe we have two observers, A who is at rest with respect to the aether, and B who is moving at velocity v with respect to the aether. We give both of them a set of rulers and clocks which they use to define their own coordinate systems, but as a joke, observer B is given special gag rulers that are shorter than normal by a factor of $$\sqrt{1 - v^2/c^2}$$, and gag clocks whose ticks are longer than normal by a factor of $$1/\sqrt{1 - v^2/c^2}$$. What's more, we tell observer B that he is the one at rest with respect to the aether, so that he can synchronize his clocks using the assumption that light travels at the same speed in both directions relative to himself. The result will be that the coordinate systems of observer A and observer B will be related by the Lorentz transformation equation, no? And that both will measure light to move at c in all directions relative to themselves, using their own rulers and clocks? But isn't it true that in this universe, observer A's frame is the only one where light "really" moves at c in both directions, while B's measurement was faulty because his clocks are not ticking the "correct" time and their rulers are not reading the "correct" length?

Along the same lines, a believer in the Aether could believe that the real situation is pretty close to this, except that instead of having to give any observer rulers and clocks which we know to work incorrectly, it's just a property of the laws of nature that rulers moving at v relative to the aether will naturally shrink by $$\sqrt{1 - v^2/c^2}$$ and clocks moving at v relative to the aether will naturally have their ticks extended by $$1/\sqrt{1 - v^2/c^2}$$. If this was true there might be no empirical way to decide who was really at rest with respect to the aether and thus whose rulers and clocks were really measuring correctly, but one might believe there was some objective truth about this nonetheless (just like in some interpretations of QM, there is an objective truth about the simultaneous position and momentum of every particle even if there is no way to measure this empirically).
JesseM said:
I don't know exactly what you mean by "LET", does it fit both these criteria?
Aether said:
I am using "LET" as a label the ether transformation equations from M&S-I (see my post #92 in the "consistency of the speed of light" thread for details); this may not be exactly what anyone else, particularly H.A. Lorentz, means by LET.
OK, but as Hurkyl says, the choice of coordinate systems is just a convention, simply choosing a different coordinate system does not give you a different theory of physics. I had assumed that LET involved some hypothesis about there being a particular frame which is actually the rest frame of the aether, and that rulers moving relative to this frame shrink and clocks slow down, even if it cannot be determined experimentally which frame this is. Was I misunderstanding?
JesseM said:
That quote doesn't say that aether theories don't exhibit local Lorentz-invariance, it just says that it is an unexplained "happenstance" if they do. In aether theories you need a multitude of separate coincidences to explain why every new phenomena happens to exhibit lorentz-invariance, and you have no reason to predict that new phenomena will exhibit it, whereas SR makes a clear prediction that all phenomena must exhibit local lorentz-invariance, and gives a single conceptual explanation for why they all do.
Aether said:
Is this how you would explain the essential differences between the SR and LET transformation equations (from post #92 referenced above)? It seems like a very simple choice of synchronization convention to me when I compare those two sets of equations.
Again, simply choosing a different coordinate system does not give you a different theory; without some physical assumption about there being a particular frame that is "objectively" special in some way (because it is the rest frame of the aether, perhaps), this is just the theory of SR described in terms of a different choice of coordinates.

Perspicacious
Aether said:
Here's the first two lines from a paper from Kostelecky & Mewes for example: http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:hep-ph/0111026 [Broken] "Lorentz violation is a promising candidate signal for Planck-scale physics. For instance, it could arise in string theory and is a basic feature of noncommutative field theories...". So, when I say "yet" I simply mean that I am aware of many physicists who expect to find violations eventually.
A violation of Lorentz invariance wouldn't prove that an aether exists. There are trivial models of spacetime where an absolute frame of reference exists without a material aether fluid.

Aether said:
Simultaneity is relative in the Lorentz transformation, but absolute simultaneity is maintained with a variable speed of light in LET transformation. Both are equally valid.
It's not that easy. Clocks can be resynchronized in Galilean relativity so that events which are simultaneous in one frame are not simultaneous in another frame. Does that mean that instantaneousness is an option in a Newtonian universe?

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No. That is naive and thoroughly disproven. You are confusing the concept of simultaneity with the concept of inertial reference frames. You cannot calibrate clocks that way.

Chronos said:
No. That is naive and thoroughly disproven. You are confusing the concept of simultaneity with the concept of inertial reference frames. You cannot calibrate clocks that way.
Are you responding to the quote "Clocks can be resynchronized in Galilean relativity so that events which are simultaneous in one frame are not simultaneous in another frame"? You are certainly free to calibrate clocks any way you like, even in a Newtonian universe. For example, I could give a bunch of observers their own sets of rulers and clocks, and tell each one "if your velocity is v relative to the rest frame of the atmosphere, then any two of your clocks lying along your direction of motion and a distance d apart should be set so that the front clock is behind the back clock by vd/(s^2 - v^2), where s is the speed of sound." Then voila, each observer will have a different definition of simultaneity, and each observer will measure sound to travel at the same coordinate speed in both directions, regardless of his velocity relative to the atmosphere (so you could also get this definition of synchronization by telling each observer to synchronize their clocks using the assumption that sound waves travel at the same speed in all directions in their own frame).

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Perspicacious said:
Why aren't you holding out the option of rejecting both? Or do you enjoy wasting everyone's time? I reject both extremes.
I offered that as my preferred option in post #11. If everyone here agrees with that, then that would be an interesting result.

Perspicacious said:
The axiom of a luminiferous aether fluid is a religious belief without scientific consequences.
Who said anything about a luminiferous aether fluid? We're talking about clock synchronization conventions where aether=absolute syncrhony, and SR=Einstein synchrony.

Perspicacious said:
As I've already illustrated with The Santa-Reindeer Postulate, LET is SR with an added, meaningless assumption with no observable consequences.
LET picks up one assuption, and drops one assumtion for a net zero of assumtions.

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JesseM said:
OK, but as Hurkyl says, the choice of coordinate systems is just a convention, simply choosing a different coordinate system does not give you a different theory of physics. I had assumed that LET involved some hypothesis about there being a particular frame which is actually the rest frame of the aether, and that rulers moving relative to this frame shrink and clocks slow down, even if it cannot be determined experimentally which frame this is. Was I misunderstanding?
I think that describing the difference between SR and LET as a simple choice of coordinate systems is probably accurate.

JesseM said:
Again, simply choosing a different coordinate system does not give you a different theory; without some physical assumption about there being a particular frame that is "objectively" special in some way (because it is the rest frame of the aether, perhaps), this is just the theory of SR described in terms of a different choice of coordinates.
SR & LET are empirically equivalent theories, so I suppose that means that they aren't different theories at all but rather different coordinate systems. Nevertheless, teaching one coordinate system to the exclusion of the other leads to widespread and firmly held superstitious beliefs about physical reality: for example, that the speed of light is actually a constant, and that simultaneity is actually relative.

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What is the geometrical difference between SR and LET?
Once you get to Minowski geometry, there isn't one, but some physical concepts have different definitions.

E.G. in 2-D space, a line of simultaneity in SR is defined by constructing a line perpendicular to the observer's velocity vector. But, in LET, it's defined by being parallel to the postulated "absolute" line.

Of course, in LET, you can still talk about SR-simultaneity, since it's just a geometric definition. And in SR, if you pick out a special line, you could talk about LET-simultaneity, as if the special line is the "absolute" line.

I am of the understanding, though, that LET doesn't really start with Minowski geometry -- it starts with something more Newton like, but postulates that the aether conspires so that things look like a Minowski geometry.

And that is the reason why SR is preferred over LET's: SR's definitions of things like "simultaneous" are geometric, and can be determined by experiment. LET's at least have this additional assumption about what is "absolutely simultaneous" that cannot be defined purely from geometry, and of which no experiment is known that can determine if two things really are absolutely simultaneous.

Or, do you operate in a coordinate independent geometry where there is no SR & LET per se?
That's the point of Minowski geometry.

Just like Euclidean geometry, you can do it analytically (with coordinates), but it is also possible to do Minowski geometry synthetically.

For example, you can still study triangles, and do trigonometry in Minowski geometry. (Of course, there are a lot of cases depending on which sides are space-like and which are time-like. Light-like lines segments are annoying because they have length zero)

For example, in a right-triangle with one space-like leg of length x, and one time-like leg of duration y, you can compute s² = y² - x². If s² is positive, then the hypotenuse has duration |s| and is timelike. Otherwise, it has length |s| and is spacelike.

In a 1+1 space-time plane, the Minowski analog of circles are the hyperbolas with null vectors as asymptotes. Using this, you can introduce trigonometry, but it uses hyperbolic functions (like sinh) instead of the circular functions (like sin). Lorentz boosts are the analog of rotations, since they "rotate" along these hyperbolas.

Of course, it's hard to draw pictures, because your paper is Euclidean -- unfortunately, it seems that trying to draw Minowski geometry on Euclidean paper does single out a special class of (Minowski-)orthogonal coordinate charts -- the ones whose coordinate axes are drawn as Euclidean-orthogonal lines. So, there are essentially different ways of drawing the Minowski plane on Euclidean paper -- if their "special class" of orthogonal charts are different, then you'll get differently-proportioned Euclidean pictures, despite the fact they're both the same Minowski shape.

And while you can physically do the most natural Euclidean transformations to a sheet of paper (translation, reflection, and rotation), you cannot do all of the most natural Minowski transformations in these pictures: (translation, reflection, and Lorentz boost) -- translation is fine, the results of most Minowski reflections would be different than Euclidean reflections, and you simply can't do a Lorentz boost to a sheet of paper.

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Hurkyl said:
That's the point of Minowski geometry.
Great! Then that's where I want to go, and it underlines the point of this thread. There is a point in their training where physicists must shed SR per se, and it is a mistake to equate SR with physical reality?

Hurkyl said:
LET's at least have this additional assumption about what is "absolutely simultaneous" that cannot be defined purely from geometry, and of which no experiment is known that can determine if two things really are absolutely simultaneous.
It isn't an "additional" assumption, it is an alternate assumption taking the place of the constancy of the speed of light. Which assumption would an impartial observer, not from our culture, choose? Is it not a coin toss (today)?

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I should warn you that I'm a mathematician, not a physicist!

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Oh, here's another nifty thing:

You may recall that there is a sort of "map" between Euclidean geometry and the complex numbers.

For example:

The length of the line segment from p to q is given by $|q - p|$, where $|z|^2 := z \bar{z}$.

Multiplication by $e^{i \theta}$ is the same thing as rotating (counterclockwise) about the origin by the angle θ.

Addition by z is the same thing as translation by the vector from the origin to z.

Well, there is a number system called the "hyperbolic numbers" that plays the same role for Minowski geometry. It has the hyperbolic unit h, which satisfies h² = 1. (As opposed to i² = -1 of the complex numbers)

The hyperbolic numbers aren't as nice as the complexes: for example, if x is a real number, then it doesn't have a hyperbolic square root if x<0, but if x>0, it has four hyperbolic square roots: √x, -√x, h√x, -h√x. In particular, 1 has four square roots, 1, -1, h, and -h.

You can't divide by all hyperbolic numbers either: for example, 1+h is a "null" hyperbolic number, and you cannot divide by it.

But these numbers do serve the same role as the complexes do. For example, the metric ds² along the line segment from p to q is given by ds²=|p-q|², where $|x + hy|^2 = (x + hy) (x + hy)^* = (x + hy) (x - hy) = x^2 - y^2)$. (I use * to denote hyperbolic conjugation because I can't remember how to make an overbar go over the whole factor)

Multiplication by $e^{h\beta}$ is a Lorentz boost fixing the origin. (the Minowski analog of rotating about the origin)

Addition by z is the same thing as translation by the vector from the origin to z.

(1+h is "null" because |1+h|²=(1+h)(1-h)=1-1=0)

And we have the hyperbolic Euler's identity, as well as the circular one:
$$e^{i\theta} = \cos \theta + i \sin \theta$$
$$e^{h\beta} = \cosh \beta + h \sinh \beta$$

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Perspicacious
In a Newtonian universe

JesseM said:
I could give a bunch of observers their own sets of rulers and clocks, and tell each one "if your velocity is v relative to the rest frame of the atmosphere, then any two of your clocks lying along your direction of motion and a distance d apart should be set so that the front clock is behind the back clock by vd/(s^2 - v^2), where s is the speed of sound." Then voila, each observer will have a different definition of simultaneity, and each observer will measure sound to travel at the same coordinate speed in both directions, regardless of his velocity relative to the atmosphere.
Or, more simply, in a Newtonian universe, clocks can be reset by a fixed amount in every moving frame such that

x' = x-vt
t' = (t - vx/s^2)/(1-v^2/s^2)

where s' = (s^2-v^2)/s is the speed of sound in the frame that moves at speed v wrt the atmosphere.

Aether said:
SR & LET are empirically equivalent theories, so I suppose that means that they aren't different theories at all but rather different coordinate systems.
Not necessarily--like I said, you could have a theory that made no predictions different from relativity, but which assumed there actually was an unobservable physical entity called "the ether" which had its own rest frame. If LET makes no such assumption, why is "ether" in the name at all?
Aether said:
Nevertheless, teaching one coordinate system to the exclusion of the other leads to widespread and firmly held superstitious beliefs about physical reality: for example, that the speed of light is actually a constant, and that simultaneity is actually relative.
The Lorentz transform is the most physical one, since the laws of physics will have the same form in every frame, and each observer can construct a physical version of this coordinate system in isolation (using rulers and clocks inside a windowless box), without having to know his velocity relative to some "special" observer. Neither of these would be true of other coordinate transforms.

By the way, would you say the same sort of thing about the Galilei transform in Newtonian physics? Is it "superstitious" to say that simultaneity is absolute in Newtonian physics, since you could describe Newtonian physics using a coordinate transform where simultaneity is relative? If you really take this argument to its logical conclusion, you'd have to say it's superstitious to say anything at all about absolute vs. relative simultaneity, regardless of what the laws of physics are like, since you always can use coordinate systems where either one is true, even if they are ungainly or unphysical.

pmb_phy
Perspicacious said:
Special relativity is physics according to the definition of physics whereas adherence to the aether is a religious belief that doesn't generate any physics.
The OP did not speak of ether theory. The OP spoke only of Lorentz ether theory up until this point (and I have yet to read past this). However it was Lorentz himself who believed in an ether until the day that he died. I'm not sure what his thoughts were on the matter after that day.

It should be known that Einstein never proved there was no ether. What he proved was that if there is an either than it plays no role in the laws of physics. You can call infnite space in which there is no matter in the normal sense as a region of space filled with virtual particles. This "sea" of virtual particles are what a few physicists today refer to as the ether.

I believe what I have said above is accurate but I'm not certain. I think that this - http://www.aip.org/history/einstein/essay-einstein-relativity.htm
speaks to it.

Pete

pmb_phy said:
It should be known that Einstein never proved there was no ether. What he proved was that if there is an either than it plays no role in the laws of physics. You can call infnite space in which there is no matter in the normal sense as a region of space filled with virtual particles. This "sea" of virtual particles are what a few physicists today refer to as the ether.
Unlike the ether, though, the sea of virtual particles is not thought to have its own natural rest frame, so it doesn't violate Lorentz symmetry even in an unobserved way.

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Hurkyl said:
Oh, here's another nifty thing:

You may recall that there is a sort of "map" between Euclidean geometry and the complex numbers.

For example:

The length of the line segment from p to q is given by $|q - p|$, where $|z|^2 := z \bar{z}$.

Multiplication by $e^{i \theta}$ is the same thing as rotating (counterclockwise) about the origin by the angle θ.

Addition by z is the same thing as translation by the vector from the origin to z.

Well, there is a number system called the "hyperbolic numbers" that plays the same role for Minowski geometry. It has the hyperbolic unit h, which satisfies h² = 1. (As opposed to i² = -1 of the complex numbers)

The hyperbolic numbers aren't as nice as the complexes: for example, if x is a real number, then it doesn't have a hyperbolic square root if x<0, but if x>0, it has four hyperbolic square roots: √x, -√x, h√x, -h√x. In particular, 1 has four square roots, 1, -1, h, and -h.

You can't divide by all hyperbolic numbers either: for example, 1+h is a "null" hyperbolic number, and you cannot divide by it.

But these numbers do serve the same role as the complexes do. For example, the metric ds² along the line segment from p to q is given by ds²=|p-q|², where $|x + hy|^2 = (x + hy) (x + hy)^* = (x + hy) (x - hy) = x^2 - y^2)$. (I use * to denote hyperbolic conjugation because I can't remember how to make an overbar go over the whole factor)

Multiplication by $e^{h\beta}$ is a Lorentz boost fixing the origin. (the Minowski analog of rotating about the origin)

Addition by z is the same thing as translation by the vector from the origin to z.

(1+h is "null" because |1+h|²=(1+h)(1-h)=1-1=0)

And we have the hyperbolic Euler's identity, as well as the circular one:
$$e^{i\theta} = \cos \theta + i \sin \theta$$
$$e^{h\beta} = \cosh \beta + h \sinh \beta$$
What's a good book to learn this from, Hurkyl?

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JesseM said:
If LET makes no such assumption, why is "ether" in the name at all?
I suppose that it is because absolute simultaneity implies some sort of instantaneous connection between events.

JesseM said:
By the way, would you say the same sort of thing about the Galilei transform in Newtonian physics? Is it "superstitious" to say that simultaneity is absolute in Newtonian physics, since you could describe Newtonian physics using a coordinate transform where simultaneity is relative? If you really take this argument to its logical conclusion, you'd have to say it's superstitious to say anything at all about absolute vs. relative simultaneity, regardless of what the laws of physics are like, since you always can use coordinate systems where either one is true, even if they are ungainly or unphysical.
It is not superstitious to choose a coordinate system. But when a popular majority of the inhabitants of Salem winds up saying things like "experiments prove that the speed of light is a constant", and then proceeds to burn people at the stake for simply pointing out that "it's just a coordinate system, stupid", then you're into the realm of superstition. Oh, that's exactly what did happen to both Galileo and Newton wasn't it?

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What's a good book to learn this from, Hurkyl?
I don't know -- I only remember seeing it in an article mathematics journal probably over a decade ago. I worked out the application to Minowski geometry myself a couple years back. (I connected the two through the importance of the hyperbola. Of course, this connection was probably mentioned in the article as well)

You can develop a great deal of it through analogy -- pull out a text on complex numbers, and then work out how things must be modified to work with h²=1 instead of i²=-1. For example, the Euler identity is proven as:

$$\begin{equation*} \begin{split} e^{h\beta} &= 1 + (h\beta) + \frac{1}{2!}(h\beta)^2 + \frac{1}{3!}(h\beta)^3 + \cdots \\ &= 1 + h\beta + \frac{1}{2!}h^2\beta^2 + \frac{1}{3!}h^3\beta^3 + \cdots \\ &= (1 + \frac{1}{2!}\beta^2 + \frac{1}{4!}\beta^4 + \cdots) + h(\beta + \frac{1}{3!}\beta^3 + \frac{1}{5!}\beta^5 + \cdots) \\&= \cosh \beta + h \sinh \beta \end{split} \end{equation*}$$

Which is exactly the method used in proving $e^z = \cos z + i \sin z$.

Aether said:
I suppose that it is because absolute simultaneity implies some sort of instantaneous connection between events.
I doubt that's what most people who discuss the "Lorentz Ether Theory" mean. Also, if you do believe there is some sort of real instantaneous connection between events, then wouldn't that mean there is a single relativistic reference frame whose definition of simultaneity is "really" the correct one? If it was just a matter of coordinate systems, then you'd be free to pick any relativistic reference frame and then make it so all the frames in the LET coordinate transformation used that frame's definition of simultaneity.
Aether said:
It is not superstitious to choose a coordinate system. But when a popular majority of the inhabitants of Salem winds up saying things like "experiments prove that the speed of light is a constant", and then proceeds to burn people at the stake for simply pointing out that "it's just a coordinate system, stupid", then you're into the realm of superstition. Oh, that's exactly what did happen to both Galileo and Newton wasn't it?
I don't think any physicist would disagree that you are free to pick a coordinate system where the speed of light is not constant, but they might argue that such coordinate systems are unphysical (they could not be constructed by observers in windowless boxes using rulers and clocks, for example). After all, you could also pick a weird coordinate system where clock speed varies by location and thus a given particle like a muon would have a different half-life depending on its position in space, but it would seem a bit pedantic to disagree with the statement "experiments show that all muons have the same half life" on this basis.

Perspicacious
pmb_phy said:
I believe what I have said above is accurate but I'm not certain.
Pete, your comments are totally irrelevant. The title of this thread is "Relativity without the aether: pseudoscience?" The first sentence on page one says, "Special relativity (SR) and Lorentz ether theory (LET) are empirically equivalent systems for interpreting local Lorentz symmetry."

I countered the accusation by assuming a minimal axiom set for SR and adding The Santa-Reindeer Postulate. Russ Watters responded similarly by adding the invisible Purple Elephant conjecture.

The starter of this thread (Aether) doesn't see the absurdity of adding an invisible, empty postulate to SR that has no logical or observable consequences. My position is that if an axiom doesn't generate any quantifiable predictions, then it's worthless and needs to be thrown out.

As a mathematician, I understand games. I can accept definitions, the meaning of words and the logical consequences of adding to SR the silly Santa-Reindeer Postulate or the invisible Purple Elephant conjecture. Where's the logic? How can adding an unobservable Santa or a non-interacting purple elephant to SR turn a pseudoscientific theory into real science? Aether didn't answer this question. You can give it a try if you like.

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Hurkyl said:
I don't know -- I only remember seeing it in an article mathematics journal probably over a decade ago. I worked out the application to Minowski geometry myself a couple years back. (I connected the two through the importance of the hyperbola. Of course, this connection was probably mentioned in the article as well)

You can develop a great deal of it through analogy -- pull out a text on complex numbers, and then work out how things must be modified to work with h²=1 instead of i²=-1. For example, the Euler identity is proven as:

$$\begin{equation*} \begin{split} e^{h\beta} &= 1 + (h\beta) + \frac{1}{2!}(h\beta)^2 + \frac{1}{3!}(h\beta)^3 + \cdots \\ &= 1 + h\beta + \frac{1}{2!}h^2\beta^2 + \frac{1}{3!}h^3\beta^3 + \cdots \\ &= (1 + \frac{1}{2!}\beta^2 + \frac{1}{4!}\beta^4 + \cdots) + h(\beta + \frac{1}{3!}\beta^3 + \frac{1}{5!}\beta^5 + \cdots) \\&= \cosh \beta + h \sinh \beta \end{split} \end{equation*}$$

Which is exactly the method used in proving $e^z = \cos z + i \sin z$.
Actually, I just started reading my first book on complex analysis yesterday, so I'll print this out and use it as a bookmark for awhile until I understand it better. Thanks!

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JesseM said:
I doubt that's what most people who discuss the "Lorentz Ether Theory" mean. Also, if you do believe there is some sort of real instantaneous connection between events, then wouldn't that mean there is a single relativistic reference frame whose definition of simultaneity is "really" the correct one? If it was just a matter of coordinate systems, then you'd be free to pick any relativistic reference frame and then make it so all the frames in the LET coordinate transformation used that frame's definition of simultaneity.
I am using LET as a label for the ether transformation equations that I posted from Mansouri-Sexl. If we ever find a way to detect a locally preferred frame, then LET takes charge. Failing that, then SR and LET are at least empirically equivalent. That is the state of affairs today, and for the puposes of this discussion I haven't made any predictions for future observations.

JesseM said:
I don't think any physicist would disagree that you are free to pick a coordinate system where the speed of light is not constant, but they might argue that such coordinate systems are unphysical (they could not be constructed by observers in windowless boxes using rulers and clocks, for example). After all, you could also pick a weird coordinate system where clock speed varies by location and thus a given particle like a muon would have a different half-life depending on its position in space, but it would seem a bit pedantic to disagree with the statement "experiments show that all muons have the same half life" on this basis.
Why can't such a coordinate system be constructed by an observer in a windowless box? I presume that any coordinate system constructed by an observer in a windowless box is undefined outside the box, and inside the box the lack of windows isn't relevant.

Aether said:
I am using LET as a label for the ether transformation equations that I posted from Mansouri-Sexl. If we ever find a way to detect a locally preferred frame, then LET takes charge.
Why does LET take charge then? This seems like a double standard, since your position is that despite the fact that the laws of nature look much simpler if we use the Lorentz transformation, that isn't a reason to favor it over the LET transformation; so if we discovered some new laws that looked simpler if we used the LET transform, to be consistent you should say that we should have no reason to favor the LET transform over the Lorentz transform in this case. Also, if LET is just a set of transformation equations (why do you call them 'ether' transformation equations if you don't assume a physical substance called 'ether', BTW?) then we'd have no obligation to make the physically preferred frame match the one whose coordinate time ticks the fastest in the LET transform.
Aether said:
Why can't such a coordinate system be constructed by an observer in a windowless box? I presume that any coordinate system constructed by an observer in a windowless box is undefined outside the box, and inside the box the lack of windows isn't relevant.
Unless I am misunderstanding something, the LET coordinate systems can't be constructed by a bunch of observers in a windowless box because they require each observer to know his velocity relative to a particular preferred coordinate system in order to synchronize his clocks correctly. My physical interpretation of the LET transformation equations is that each observer defines coordinates in terms of a network of rulers and clocks just like in SR, except that instead of each observer synchronizing his clocks using the assumption that light travels at the same speed in all directions in his frame, there is only a single observer who synchronizes his clocks this way, and all other observers adjust their clocks so that their definition of simultaneity matches this special frame. This is not possible unless each observer knows his velocity relative to this special frame.

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JesseM said:
Unless I am misunderstanding something, the LET coordinate systems can't be constructed by a bunch of observers in a windowless box because they require each observer to know his velocity relative to a particular preferred coordinate system in order to synchronize his clocks correctly. My physical interpretation of the LET transformation equations is that each observer defines coordinates in terms of a network of rulers and clocks just like in SR, except that instead of each observer synchronizing his clocks using the assumption that light travels at the same speed in all directions in his frame, there is only a single observer who synchronizes his clocks this way, and all other observers adjust their clocks so that their definition of simultaneity matches this special frame. This is not possible unless each observer knows his velocity relative to this special frame.
If you could detect a locally preferred frame from within a windowless box, then everyone could synchronize to that without reference to the walls of the box, and that would be great; everyone inside the box would be synchronized with everyone outside the box. However, the observers in the box can at least all agree on using the rest frame of the box itself as a common reference, and they can all ping the walls of the box with their radars, and they can all synchronize their clocks to maintain absolute simultaneity with each other.

Aether said:
If you could detect a locally preferred frame from within a windowless box, then everyone could synchronize to that without reference to the walls of the box, and that would be great; everyone inside the box would be synchronized with everyone outside the box. However, the observers in the box can all agree on using the rest frame of the box as a common reference, and they can all ping the walls of the box with their radars, and they can all synchronize their clocks to maintain absolute simultaneity with each other.
When I talk about "windowless boxes" I mean that each observer has his own windowless box, not that you have a bunch of observers within the same windowless box. In SR, each observer can construct a network of rulers and clocks in his own windowless box without any knowledge of things outside his box, and if these boxes are moving alongside each other arbitrarily close by in space, then the Lorentz transform will map between the readings on each observer's clock/ruler system as they pass by each other. With the LET transform, there is no way each observer in his own box can physically construct the different coordinate systems unless they have windows and can communicate, so that they can agree on which of them will have the preferred coordinate system and the rest can synchronize their clocks by seeing how fast they're moving relative to this preferred observer.

By the way, I added a little to the beginning of my previous post after you responded to it...

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pmb_phy
Perspicacious said:
Pete, your comments are totally irrelevant. The title of this thread is "Relativity without the aether: pseudoscience?"
Wow! Its like you didn't even read my post!

Its overly obvious from my response is that the answer to "Relativity without the aether: pseudoscience?" is no (if the question posed is even a valid question to ask in the first place). russ and yourself have assumed a definition of "ether" (i.e. ether - that which supports the propagation of light) whose existence has never been detected either directly or indirectly and you both start off with this assumption. I chimmed in. So now you're surelyt asking who these "people" are right? I do recall that the name of one of these chaps is Albert Einstein. Albert Einstein - An address delivered on May 5th, 1920, in the University of Leyden
http://www.mountainman.com.au/aether_0.html

I may have the wrong idea between Eistein's 1920's address but that is just one person who looks at the the term "ether" as being different from that used by Maxwell and the ancient's. The ancient's used the term "ether" to refer to the element which permeated all of, otherwise empty, space.

Pete

Gold Member
JesseM said:
When I talk about "windowless boxes" I mean that each observer has his own windowless box, not that you have a bunch of observers within the same windowless box. In SR, each observer can construct a network of rulers and clocks in his own windowless box without any knowledge of things outside his box, and if these boxes are moving alongside each other arbitrarily close by in space, then the Lorentz transform will map between the readings on each observer's clock/ruler system as they pass by each other. With the LET transform, there is no way each observer in his own box can physically construct the different coordinate systems unless they have windows and can communicate, so that they can agree on which of them will have the preferred coordinate system and the rest can synchronize their clocks by seeing how fast they're moving relative to this preferred observer.

By the way, I added a little to the beginning of my previous post after you responded to it...
That could turn out be a practical advantage to SR in the absence of a locally preferred frame, but I'm not convinced of that yet. Can you show a simple example using the transformation equations that I provided? There is already an example relating to momentum being discussed in the "Einstein's clock synchronization convention" thread. Until it is proven otherwise, I will assume that LET is empirically equivalent to SR because Mansouri-Sexl say that they are; but if it can be shown not to be exactly so then that would make a real difference to how I look at this.

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Perspicacious
Pete,

I'm not aware of anyone, including Einstein, who successfully defined the meaning of "aether" so that it would result in definite, quantifiable predictions. Aether (the poster) doesn't even want to extend the current range of SR's quantifiable predictions. Aether wants a new theory that is empirically equivalent to SR. He is arguing for the addition of an undetectable absolute frame of reference. I don't see how his revision is any different than adding to SR the Santa-Reindeer Postulate or the invisible Purple Elephant conjecture.

pmb_phy
Perspicacious said:
I'm not aware of anyone, including Einstein, who successfully defined the meaning of "aether" so that it would result in definite, quantifiable predictions.
The terms "aether," "ether" and "superluminal ether" are all syonyms for the same thing. I used the term because Einstein was using it. I thgought it'd be the least confusing way to get this straight. I guess not. This is a fact which has not escaped aether. Right aether.

Sorry folks but I must leave the board for the rest of the day. If there is something here beyone semantics then please let me know.

Pete

Aether said:
That could turn out be a practical advantage to SR in the absence of a locally preferred frame, but I'm not convinced of that yet. Can you show a simple example using the transformation equations that I provided?
A simple example of what?
Aether said:
There is already an example relating to momentum being discussed in the "Einstein's clock synchronization convention" thread. Until it is proven otherwise, I will assume that LET is empirically equivalent to SR because Mansouri-Sexl say that they are; but if it can be shown not to be exactly so then that would make a real difference to how I look at this.
I think you're confused here--if LET is defined solely in terms of a different coordinate system, without any different assumptions about the laws of physics, then obviously the theory is not predicting any new empirical consequences, because it's assuming the same laws of physics! Changing your coordinate system doesn't change the laws of physics, it only changes the equations used to express the laws of physics in that coordinate system. You can pick any crazy coordinate system you want! (I still doubt that Mansouri and Sexl share your view that the LET refers to nothing more than a coordinate transformation with no new physical assumptions, but that's another issue) But if you want to physically construct a measuring device such that the reading on a clock and the marking on the ruler next to a particular event correspond to the coordinates of the event in that coordinate system, then it should be equally obvious that this measuring device will be different if you choose a different coordinate system. The set of measuring devices that correspond to each coordinate system in the LET transformation are different that the set of measuring devices that correspond to each coordinate system in the Lorentz transformation, and one characteristic that's different is that there's no way (according to the current known laws of physics) for a bunch of observers to construct the coordinate system of their rest frame without knowing their velocity relative to a single preferred frame, whereas in the Lorentz transform each observer's rest frame can be constructed with no knowledge of the outside world (in a windowless box). This is not an "empirical difference" in the sense of the laws of physics being different, it's just a difference in what would be needed to construct a set of measuring devices corresponding to each coordinate system allowed by the transformation. It is a good reason to see the LET transformation as more unphysical than the Lorentz transformation, though.

Speaking of the physical basis for using one coordinate system vs. another, did you have any comment on the part I added to my earlier post after you responded to it?
JesseM said:
Aether said:
I am using LET as a label for the ether transformation equations that I posted from Mansouri-Sexl. If we ever find a way to detect a locally preferred frame, then LET takes charge.
Why does LET take charge then? This seems like a double standard, since your position is that despite the fact that the laws of nature look much simpler if we use the Lorentz transformation, that isn't a reason to favor it over the LET transformation; so if we discovered some new laws that looked simpler if we used the LET transform, to be consistent you should say that we should have no reason to favor the LET transform over the Lorentz transform in this case. Also, if LET is just a set of transformation equations (why do you call them 'ether' transformation equations if you don't assume a physical substance called 'ether', BTW?) then we'd have no obligation to make the physically preferred frame match the one whose coordinate time ticks the fastest in the LET transform.

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Gold Member
JesseM said:
A simple example of what?
Of two observers in separate windowless boxes who are able to do something meaningful using the Lorentz transform that they can't do equally well with the ether transform. If you show me something that they can do with the Lorentz transform, then I will try to match it with the ether transform; and if I can't do it, then that will give me a practical reason to think that an impartial observer would prefer the Lorentz transform over the ether transform. (However, the next paragraph may make that unnecessary for now.)

JesseM said:
I think you're confused here--if LET is defined solely in terms of a different coordinate system, without any different assumptions about the laws of physics, then obviously the theory is not predicting any new empirical consequences, because it's assuming the same laws of physics! Changing your coordinate system doesn't change the laws of physics, it only changes the equations used to express the laws of physics in that coordinate system. You can pick any crazy coordinate system you want! But if you want to physically construct a measuring device such that the reading on a clock and the marking on the ruler next to a particular event correspond to the coordinates of the event in that coordinate system, then it should be equally obvious that this measuring device will be different if you choose a different coordinate system. The set of measuring devices that correspond to each coordinate system in the LET transformation are different that the set of measuring devices that correspond to each coordinate system in the Lorentz transformation, and one characteristic that's different is that there's no way (according to the current known laws of physics) for a bunch of observers to construct the coordinate system of their rest frame without knowing their velocity relative to a single preferred frame, whereas in the Lorentz transform each observer's rest frame can be constructed with no knowledge of the outside world (in a windowless box). This is not an "empirical difference" in the sense of the laws of physics being different, it's just a difference in what would be needed to construct a set of measuring devices corresponding to each coordinate system allowed by the transformation. It is a good reason to see the LET transformation as more unphysical than the Lorentz transformation, though.
That sounds like a potentially plausible reason to prefer the Lorentz transformation over the ether transformation for all practical purpose unless/until a locally preferred frame is detected. However, it does lead most people to a false belief that the one-way speed of light has been measured by endless numbers of experiments, and in general I think that it tends to warp one's perception of local Lorentz invariance.

JesseM said:
Speaking of the physical basis for using one coordinate system vs. another, did you have any comment on the part I added to my earlier post after you responded to it?

Why does LET take charge then? This seems like a double standard, since your position is that despite the fact that the laws of nature look much simpler if we use the Lorentz transformation, that isn't a reason to favor it over the LET transformation; so if we discovered some new laws that looked simpler if we used the LET transform, to be consistent you should say that we should have no reason to favor the LET transform over the Lorentz transform in this case. Also, if LET is just a set of transformation equations (why do you call them 'ether' transformation equations if you don't assume a physical substance called 'ether', BTW?) then we'd have no obligation to make the physically preferred frame match the one whose coordinate time ticks the fastest in the LET transform.
I am still forming my position and how to describe it, but I think that it is this: that the laws of nature are described by the Minkowski metric, but that the Lorentz transformation isn't any different (mathematically at least) from the ether transform. Mansouri-Sexl say that they are also empirically equivalent, not just mathematically equivalent. If we discovered some new laws that enable us to define a locally preferred frame, then I would presume that would mean that the Minkowski metric itself would be asymmetrical in some way and the natural choice of transformation would then resolve into something that maintains absolute simultaneity. I call them ether transformation equations because that is what Mansouri-Sexl call them.

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Homework Helper
Gold Member
Aether said:
Special relativity (SR) SR and Lorentz ether theory (LET) are empirically equivalent systems for interpreting local Lorentz symmetry.

While I don't have the time to contribute to this lively thread, let me just point to
"A Comparison between Lorentz's Ether Theory and Special Relativity in the Light of the Experiments of Trouton and Noble", Michel Janssen's dissertation, which is available at http://www.tc.umn.edu/~janss011/ . Here is the first paragraph in his Introduction
In this dissertation, I want to compare the ether theory of the great Dutch physicist Hendrik Antoon Lorentz (1853–1928) to Einstein’s special theory of relativity. To the end of his life, Lorentz maintained, first, that his theory is empirically equivalent to special relativity, and, second, that, in the final analysis, it is a matter of taste whether one prefers the standard relativistic interpretation of the formalism of the theory or his own ether theoretic interpretation (see, e.g., Nersessian 1984, pp. 113–119). I will argue that Lorentz’s first claim, when understood properly, should be accepted, but that the second should be rejected.

I haven't read the dissertation yet. (I don't have too much free time right now.) However, this and his other papers on the History of Relativity look interesting.

Aether said:
Of two observers in separate windowless boxes who are able to do something meaningful using the Lorentz transform that they can't do equally well with the ether transform. If you show me something that they can do with the Lorentz transform, then I will try to match it with the ether transform; and if I can't do it, then that will give me a practical reason to think that an impartial observer would prefer the Lorentz transform over the ether transform. (However, the next paragraph may make that unnecessary for now.)
OK, but I already did that. The different observers in windowless boxes can create ruler/clock systems such that, when an outside observer looks at the coordinates that two different observers assign to the same event using these systems, he will see that they are related by the Lorentz transform. They do this just by creating a system of rulers with clocks attached to each ruler marking, and synchronizing the clocks using light signals, under the assumption that light moves at c in their own box's rest frame; then to assign coordinates to a given event, they just look at the reading on the ruler and clock that were at the same position as the event at the moment it happened (the Lorentz transformation is usually derived based on the assumption that each observer assigns coordinates to events using exactly this physical setup). In contrast, there's no way for observers in windowless boxes to build measuring devices such that the coordinates different observers assign to the same event are related by the LET transform, although they can do this if they can communicate and agree on which observer has the preferred frame, and then the rest can measure their velocity relative to this preferred observer and use this information to synchronize their clocks.
Aether said:
That sounds like a potentially plausible reason to prefer the Lorentz transformation over the ether transformation for all practical purpose unless/until a locally preferred frame is detected. However, it does lead most people to a false belief that the one-way speed of light has been measured by endless numbers of experiments, and in general I think that it tends to warp one's perception of local Lorentz invariance.
I disagree that this is "false" just because it depends on using a particular coordinate system--you could equally well say it's a false belief that people can measure the speed of anything, like, say, cars. Likewise the same reasoning would force you to say it's "false" that scientists have measured that muons all have the same decay rate regardless of your position in space, since you could in principle pick a weird coordinate system where clocks ticked at different coordinate speeds depending on position. I think it's understood implicitly in statements like this that you're talking in terms of the most physically natural coordinate system, otherwise you'd always have to qualify every single statement about physics that refers to position, time, velocity, acceleration, etc.
Aether said:
I am still forming my position and how to describe it, but I think that it is this: that the laws of nature are described by the Minkowski metric, but that the Lorentz transformation isn't any different (mathematically at least) from the ether transform.
Of course it's different--they're different coordinate systems. The Lorentz transformation transforms time coordinates like $$t' = \gamma (t - vx/c^2)$$ while the LET transformation transforms them like $$t' = \gamma t$$. What else do you think it means to say that two coordinate transformations are mathematically different?
Aether said:
Mansouri-Sexl say that they are also empirically equivalent, not just mathematically equivalent.
Again, I suspect that Mansouri-Sexl don't mean the "Lorentz Ether Theory" to refer just to a coordinate transformation, but to a hypothesis that there is an unobserved physical entity called "ether" out there which has a particular rest frame, and which has the property that clocks slow down and rulers shrink the faster they move relative to this rest frame. To say that this theory was "empirically equivalent" to SR would just be to note that there's no experiment we can do that would tell us our velocity relative to this ether rest frame, even if there is some unknowable objective answer to this question.
Aether said:
If we discovered some new laws that enable us to define a locally preferred frame, then I would presume that would mean that the Minkowski metric itself would be asymmetrical in some way and the natural choice of transformation would then resolve into something that maintains absolute simultaneity.
Like I said, no matter what the laws of physics are you could still use either set of coordinate systems. If you want to say that the LET coordinate systems would be more of a "natural choice" in this case I agree, but then you should agree that in terms of the laws of physics as we know them now, the Lorentz transform is more of a "natural choice", since the known laws of physics will obey the same equations in every reference frame using the Lorentz transform but not using the LET transform, and because observers in windowless boxes could create physical versions of the coordinate systems of the Lorentz transform but not of the LET transform.

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Gold Member
JesseM said:
OK, but I already did that. The different observers in windowless boxes can create ruler/clock systems such that, when an outside observer looks at the coordinates that two different observers assign to the same event using these systems, he will see that they are related by the Lorentz transform.
I don't recall hearing you say anything about an outside observer before.

JesseM said:
They do this just by creating a system of rulers with clocks attached to each ruler marking, and synchronizing the clocks using light signals, under the assumption that light moves at c in their own box's rest frame; then to assign coordinates to a given event, they just look at the reading on the ruler and clock that were at the same position as the event at the moment it happened (the Lorentz transformation is usually derived based on the assumption that each observer assigns coordinates to events using exactly this physical setup). In contrast, there's no way for observers in windowless boxes to build measuring devices such that the coordinates different observers assign to the same event are related by the LET transform, although they can do this if they can communicate and agree on which observer has the preferred frame, and then the rest can measure their velocity relative to this preferred observer and use this information to synchronize their clocks.
The outside observer is the one who defines the preferred frame. Now that you have introduced him, and said that he is the one who can actually relate the two coordinate systems from the two windowless boxes, then I suspect that the LET transform might perform equally well. This outside observer merely needs to syncrhonize the clocks in the windowless boxes, and then do an LET transform; this is no different than working the Lorentz transform which has the exact same syncrhonization function embedded within it.

JesseM said:
I disagree that this is "false" just because it depends on using a particular coordinate system--you could equally well say it's a false belief that people can measure the speed of anything, like, say, cars.
The speed of anything can be measured to have the same value regardless of direction or clock synchronization using a round-trip radar pulse for example. The round-trip speed of light can also be measured to have the same value regardless of direction or clock synchronization.

JesseM said:
Likewise the same reasoning would force you to say it's "false" that scientists have measured that muons all have the same decay rate regardless of your position in space, since you could in principle pick a weird coordinate system where clocks ticked at different coordinate speeds depending on position. I think it's understood implicitly in statements like this that you're talking in terms of the most physically natural coordinate system, otherwise you'd always have to qualify every single statement about physics that refers to position, time, velocity, acceleration, etc.
First we need to establish that SR coordinates really are more physcially natural than LET in the absence of a locally preferred frame, and then we need to establish that SR+LET is not even more physically natural than that. Even if we can eventually come to an agreement that SR is more natural in the absence of a locally preferred frame, we have already agreed that LET is more natural when there is a locally preferred frame. What about coordinate free geometry, and being able to at least see the problem from both the SR and the LET perspectives? I don't think it matters which coordinate system that you use so long as you are aware of its inherent limitations, or at least that it has some.

JesseM said:
Of course it's different--they're different coordinate systems. The Lorentz transformation transforms time coordinates like $$t' = \gamma (t - vx/c^2)$$ while the LET transformation transforms them like $$t' = \gamma t$$. What else do you think it means to say that two coordinate transformations are mathematically different?
They are not really any different, but you have obscured that from presenting the equations in a different form than Mansouri-Sexl do and by neglecting to mention that Einstein synchronized clocks are readjusted to LET synchronization before using the LET transform. The $$-vx/c^2[/itex] term from the Lorentz transform is merely pulled out of the equation and then plugged directly into the clock at x to synchronize it. JesseM said: Again, I suspect that Mansouri-Sexl don't mean the "Lorentz Ether Theory" to refer just to a coordinate transformation, but to a hypothesis that there is an unobserved physical entity called "ether" out there which has a particular rest frame, and which has the property that clocks slow down and rulers shrink the faster they move relative to this rest frame. To say that this theory was "empirically equivalent" to SR would just be to note that there's no experiment we can do that would tell us our velocity relative to this ether rest frame, even if there is some unknowable objective answer to this question. Do you have a copy of their papers? I can provide you with one if you don't. They don't call their ether theory "LET", I'm just using it as a convenient label. JesseM said: Like I said, no matter what the laws of physics are you could still use either set of coordinate systems. If you want to say that the LET coordinate systems would be more of a "natural choice" in this case I agree, but then you should agree that in terms of the laws of physics as we know them now, the Lorentz transform is more of a "natural choice", since the known laws of physics will obey the same equations in every reference frame using the Lorentz transform but not using the LET transform, and because observers in windowless boxes could create physical versions of the coordinate systems of the Lorentz transform but not of the LET transform. We might be able to come to an agreement along these lines at some point, but now that you have thrown an "outside observer" into the example I'm not yet ready to say that the Lorentz transform is a more natural choice in the absence of a locally preferred frame. Last edited: Gold Member pmb_phy said: The terms "aether," "ether" and "superluminal ether" are all syonyms for the same thing. I used the term because Einstein was using it. I thgought it'd be the least confusing way to get this straight. I guess not. This is a fact which has not escaped aether. Right aether. It seems to me that these terms all imply that the Minkowski metric is, or at least has the potential to be, asymmetric somehow. I suppose that's what GR is all about (e.g., that the metric is, or at least has the potential to be, asymmetric...asym-metric?), and that Einstein is saying that the metric is the aether (however, all of this may ultimately depend on what your definition of "is" is ). Last edited: Science Advisor Aether said: I don't recall hearing you say anything about an outside observer before. The outside observer isn't necessary for their coordinates to be related by a Lorentz transform, but since their boxes are all windowless, none of them can see what each other's coordinate readings are--that's the only reason I introduced him. Alternately, you could have each of them build their measuring devices in windowless boxes, then once the devices were up and running the boxes could be opened so they could all see the devices that everyone else built. Aether said: The outside observer is the one who defines the preferred frame. No, certainly not--the outside observer's velocity is irrelevant, all he's doing is noting "hmm, observer A assigned that event coordinates x=3.5 meters, t=9 seconds while observer B assigned the same event coordinates x'=7 meters, t'=2 seconds", something like that. Aether said: Now that you have introduced him, and said that he is the one who can actually relate the two coordinate systems from the two windowless boxes, then I suspect that the LET transform might perform equally well. This outside observer merely needs to syncrhonize the clocks in the windowless boxes, and then do an LET transform See, your outside observer needs to actually interact with the different measuring devices, to take a part in how they are constructed, for them to work right--my outside observer is only noting the readings on measuring devices which have already been completely constructed by the observers inside their windowless boxes. My outside observer can just be a ghost noting facts about the world without being able to have any effect on it (and even if there is no outside observer, it's still true that the coordinates of different observers are related by the Lorentz transform even if none of them are aware of this fact), yours is taking an active role in building the different measuring devices, making use of information he obtains by seeing the velocity of different devices relative to one another. JesseM said: I disagree that this is "false" just because it depends on using a particular coordinate system--you could equally well say it's a false belief that people can measure the speed of anything, like, say, cars. Aether said: The speed of anything can be measured to have the same value regardless of direction or clock synchronization using a round-trip radar pulse for example. The round-trip speed of light can also be measured to have the same value regardless of direction or clock synchronization. You can't measure the speed of anything in a way that doesn't depend on your choice of coordinate system. For example, suppose a spaceship is flying at 0.8c in the preferred frame of the ether transform, and you want to know how fast it's flying in the ether-transform frame of an observer moving at 0.6c in the same direction relative to the preferred frame. In the preferred frame, two points along the ship's path are x=0,t=0 and x=0.8,t=1, so using the ether transform equations you provided in post #92, we find that these events correspond to x'=0,t'=0 and x'=0.25,t'=0.8 (actually I'm not sure about this last one--you wrote the equation [tex]t_1=(1-v^2/c_0^2)^{1/2}T_1$$ for the time transformation in the ether equation, but then for the Lorentz transform you wrote $$t_1=(1-v^2/c_0^2)^{1/2}T_1-vx_1/c_0^2$$ when it's actually supposed to be $$t_1=(T_1-vx_1/c_0^2)/(1-v^2/c_0^2)^{1/2}$$, so should we also divide by $$(1-v^2/c_0^2)^{1/2}$$ rather than multiply by it in the ether transform equation?). This means that the velocity of the ship in this second observer's rest frame would be 0.3125c according to the ether transformation. On the other hand, if you use the Lorentz transform then the coordinates of these same two events are x'=0,t'=0 and x'=0.25,t'=0.65, so the velocity of the ship in this observer's rest frame is 0.3846c according to the Lorentz transformation. So you can see that even for objects moving slower than light, your notion of the speed of slower-than-light objects like spaceships or cars depends on your choice of coordinate system.
JesseM said:
Of course it's different--they're different coordinate systems. The Lorentz transformation transforms time coordinates like $$t'=\gamma(t - vx/c^2)$$ while the LET transformation transforms them like $$t'=\gamma t$$. What else do you think it means to say that two coordinate transformations are mathematically different?
Aether said:
They are not really any different, but you have obscured that from presenting the equations in a different form than Mansouri-Sexl do
OK, show me the form that they present the time transformation in then.
Aether said:
and by neglecting to mention that Einstein synchronized clocks are readjusted to LET synchronization before using the LET transform. The [tex]-vx/c^2[/itex] term from the Lorentz transform is merely pulled out of the equation and then plugged directly into the clock at x to synchronize it.
I'm still not understanding how this means there is "no mathematical difference" between them. If you synchronize the clocks in a different way, then isn't that a mathematically different coordinate system? Won't the actual time-coordinates I assign to particular events be changed as a result? What type of change in the coordinate systems would qualify as "mathatically different" in your use of the term?
Aether said:
Do you have a copy of their papers? I can provide you with one if you don't.
I don't have a copy, so if you could provide one that would be helpful, thanks.

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