Light clock moving to demonstrate time dilation

[TcheQ]

Bcrowell, the text is very dense, full of technicalities and most interesting. But with regard to this particular point..., I don't think it manages to derive the LTs without postulating that the speed of light is the same for both frames.

The author says “For convenience, let's adopt time and space units in which c=1”. Assuming the formulas are derived and right and useful, it is actually very convenient to equate c with 1 (the so called geometric or natural units). But if one uses that trick in a “derivation”, one should warn what it means, since it’s a major step. What it means is that, in frame A, a rod is said to be 1 light-second long if light takes 1 second to traverse its length (or, rather, 2 s to complete the round-trip). And if that same rod is now moving relative to A, that is to say, it’s at rest with frame B, is it still 1 light-second long? It wouldn’t if light didn’t take 1 s to traverse its length. But we assume it does, so c = 1 also in frame B. Conclusion: the very fact of using that convention (c=1) equates to assuming, from the outset of the derivation, that both frames measure the same speed for light.

39:00 onwards, he deals with how you arrive at the conclusion c=c', using c=1 for simplicity

 

[Saw]

39:00 onwards, he deals with how you arrive at the conclusion c=c', using c=1 for simplicity

TcheQ, thanks for pointing at the relevant part of the video.

I think the lecturer himself confirms the point, although what he says does help to explain it better.

First, he considers that a light beam has, in an unprimed frame, a speed = c, whatever it is. Without special choice of units, that means that:

x (distance traversed by the beam in a given time = L = the length of a rod where that distance is marked) = ct (speed of light x that time lapse).

If we then choose to measure distance in light seconds, the expression becomes, by definition (a light-second = distance traversed by light in a second):

x = t

So far, so good. The choice of units is innocuous. No major step. Just a convenient approach.

Second, he wonders about the coordinates of that very same light beam in a primed frame, moving relative to the other. Without choice of units and without any special assumption, the equation will be:

x' = c't'

Now it’s time to make your choice of units. Can you simply say, like we did before…

x’ = t’?

Well, classical mechanics would say you can’t. Its reasoning would be:

(a) If the rod whose length we are using as reference is the same one as before, then we have x’ = x.
(b) In the primed frame the beam does not travel at c, but at (c-v) in the go trip (since the target is escaping), (c+v) in the return trip (since it is heading towards its target) and at the average between the two in the round trip.
(c) With (b)’s assumption, no matter which reference you take (light speed at the go trip, at the return trip or at the average of the round trip), if you make the calculation, you’ll conclude that the time that the beam takes to traverse the length of the rod (x=x’=L=L’) is never x light-second.

Hence with the classical assumptions you *cannot* say that x’=t’!

Of course you can change the assumptions. You can postulate that c = c’ and so x’=t’, even if as a consequence of that x’ may be different from x and t’ different from t. If you take that step (a major step, by the way), then all the rest follows. But without that postulate, you go nowhere. The lecturer somehow recognizes it when he states that, in a later part of the derivation, x^2-t^2=0 does not necessarily imply x’^2-t’^2=0, but he could have made it clearer.

To sum up: c = c’ is not a conclusion you arrive at but a postulate you derive consequences from.

Unlike what oversimplified explanations suggest, you cannot change a physical theory (you cannot shift from classical to special relativity) just with algebra or geometry, unless the original theory contained some algebraic or geometric mistake. And I wouldn’t advise anyone to tell Newton that in his face. If you wish to change, you can (and I think you must) but you need a new physical assumption for that purpose, not just math or drawings.

 

[bcrowell]

Bcrowell, the text is very dense, full of technicalities and most interesting. But with regard to this particular point..., I don't think it manages to derive the LTs without postulating that the speed of light is the same for both frames.

The author says “For convenience, let's adopt time and space units in which c=1”. Assuming the formulas are derived and right and useful, it is actually very convenient to equate c with 1 (the so called geometric or natural units). But if one uses that trick in a “derivation”, one should warn what it means, since it’s a major step. What it means is that, in frame A, a rod is said to be 1 light-second long if light takes 1 second to traverse its length (or, rather, 2 s to complete the round-trip). And if that same rod is now moving relative to A, that is to say, it’s at rest with frame B, is it still 1 light-second long? It wouldn’t if light didn’t take 1 s to traverse its length. But we assume it does, so c = 1 also in frame B. Conclusion: the very fact of using that convention (c=1) equates to assuming, from the outset of the derivation, that both frames measure the same speed for light.

No, that's incorrect. The existence of a constant c has already been established at that point. Setting c=1 after that is simply a choice of units.

If you still have any doubts about the possibility of deriving the Lorentz transformation without assuming a constant speed of light, please take a look at the two references I gave in #26. Amazon will probably let you see the relevant parts with their "look inside" feature. Wolfgang Rindler is an extremely well known relativist. I really don't think he's deluding himself when he says you can derive the Lorentz transformations without assuming constant c.

Critical thinking is great, but when I cite multiple published references to demonstrate a particular point, the burden then falls on you to explain why so many published sources are saying the same thing. I think at a minimum you either need to (a) read all the references and explain why *all* of them are wrong, or (b) demonstrate that there is some published controversy on this point among researchers in the field.

 

[yuiop]

39:00 onwards, he deals with how you arrive at the conclusion c=c', using c=1 for simplicity

I watched this lecture and all he does is show that the Lorentz transformations have the property that the speed of light is constant in all inertial frames while the Newtonian (Galliean) transformations do not have that property and on that basis rejects the Newtonian transformation. At time 43:00 he states that he is looking for a transformation that preserves the constancy of the speed of light and then picks the Lorentz transformation (without any derviation or postulates shown) because it has the desired property. It is not surprising that using a transformation that was formulated with the initial assumption that the speed of light is constant in all frames should predict that the speed of light is constant in all frames.

From the point of view of this derivation, constancy of c is something that is derived from the Lorentz transformation, and in fact once you've discovered this universal velocity c, it takes a little more work to convince yourself that light must also travel at that velocity.

As above, if you start with a transformation that was formulated with the initial assumption that the speed of light is constant in all frames then it is not surprising that the transformation should predict that the speed of light is constant in all frames. Circular reasoning.

Did you read what I said in my post? I like "Light and matter" site and will read the link carefully. But, in logical terms, the task seems impossible: deriving a formula where c is a constant without assuming that c is constant...? How do you do it: you start by writing c and c' and in some step you forget yourself and start writing c everywhere?

Essentially I am agreeing with Saw that most derivations that do not have the constanty of the speed of light explicity stated as an assumption or postulate, have it impicitly assumed at the outset or have a conditional that it is a required outcome. However, it might not be "impossible" to have such a derivation but it would need a large and unreasonable set of alternative postulates.

For example let us take this set of postulates or initial assumptions:

1) Objects length contract with relative motion, in accord with the Lorentz transformations.
2) Clocks with motion relative to the observer are measured to run slower than clocks at rest with the observer, in accord with the Lorentz transformations.
3) Simultaneity is relative, in accord with the Einstein simultaneity equations.
4) Velocities add according to the Relativistic velocity addition equations.

Starting with all those assumptions we could probably claim that the constanty of the speed of light in all inertial reference frames is an inevitable logical conclusion if we accept that the above 4 postulates are self evident and reasonable assumptions supported by the available experimental evidence. However at the time Lorentz, Einstein, Fitzgerald and others were formulating the Lorentz transformations, none of the above 4 postulates were self evident and reasonable initial assumptions. In fact they were conclusions that were hard to swallow at the time because there was no direct experimetal evidence they were true and it was certainly way outside of everyday experience. Now if someone had doubts about the validity of the constancy of the speed of light, it would be difficult to argue they must accept it as inevitable conclusion, because everyone accepts that length contraction is self evident, obvious and inalienable fact.

On the other hand if someone had doubts about time dilation or length contraction and it was pointed out that if they accept that the speed of light is measured to be the same and constant in all inertial reference frames (and the laws of physics are the same in all IRFs) they would probably have to concede they are inevitable logical outcomes. After all there was already experimetal evidence that the speed of light is constant and independent of the motion of the source and Maxwell's equations also strongly suggested it.

If you claim that the Lorentz transformation can be derived without any assumtion of the constancy of the speed of light you should make it clear what initial assumptions you are making. However, it would be unreasonable and circular to derive the Lorentz tranformations from the postulate "the Lorentz transformations are a correct description of time, space and motion in nature".

Basically I think you need to show that the Lorentz transformations can be derived without any assumption of the constancy of the speed of light. One way of doing this is it to pick random transformations, until you find one that has the property that the speed of light is constant for inertial observers. Once you have done that it would be unreasonable to claim that the speed of light MUST be constant because your randomly picked transformation predicts it. For example I could pick random transformations until I find one that satifies the condition that the speed of light is observer dependent and then claim that "proves" the speed of light is not constant, which is of course utter rubbish, because I have effectively made it an initial condition that the speed of light is not constant.

 

[Saw]

No, that's incorrect. The existence of a constant c has already been established at that point. Setting c=1 after that is simply a choice of units.

I think we may have a little misunderstanding here. What I am saying, really, is not that the choice of units itself is the postulate. The postulate is assuming the speed of light is the same in all frames, with whatever units you play. If you choose units so that c=1 in one frame, then the postulate is assuming that c=1 in all frames. You may want to look at post #32 where I try to express the idea more at length than what you just quoted. Do you agree to that?

If you still have any doubts about the possibility of deriving the Lorentz transformation without assuming a constant speed of light, please take a look at the two references I gave in #26. Amazon will probably let you see the relevant parts with their "look inside" feature. Wolfgang Rindler is an extremely well known relativist. I really don't think he's deluding himself when he says you can derive the Lorentz transformations without assuming constant c.

Where does he say so? In page 45 he states that:

Newton’s axiom t=t’ would lead to (…) the GT. Instead, we now appeal to Einstein’s Law of propagation of light. According to it, x=ct and x’=ct’ are valid simultaneously, being descriptions of the same light signal in S and S

And then he goes on to derive the LTs.

That’s just what I was saying: to derive the LTs you have to leave aside the classical assumption (t=t’) and rely on a different assumption (c=c’, at the expense of admitting that t≠t’ and x≠x’).

I am not aware that there is any “published controversy” on this issue. In fact, I don’t think it’s controversial. From what I have read in the forum, it is usually commented that the habitual thought experiments are not proofs of SR, but ways to quantitatively derive its equations, assuming of course that you buy its postulates (which are true or not as ruled by experiment).

Edit: I agree with kev's post.

 

[TcheQ]

Hence with the classical assumptions you *cannot* say that x’=t’!

There is a shorthand here that isn't stated, that might not be obvious. c=1m/s i.e. it is not dimensionless

x=m
t=s
c=m/s

PS Relativity is classical mechanics

I watched this lecture and all he does is show that the Lorentz transformations have the property that the speed of light is constant in all inertial frames while the Newtonian (Galliean) transformations do not have that property and on that basis rejects the Newtonian transformation. At time 43:00 he states that he is looking for a transformation that preserves the constancy of the speed of light and then picks the Lorentz transformation (without any derviation or postulates shown) because it has the desired property. It is not surprising that using a transformation that was formulated with the initial assumption that the speed of light is constant in all frames should predict that the speed of light is constant in all frames.

But that's exactly what Einstein did, due to those early experiments showing lightspeed was constant regardless of object velocity. It was a response to the experimental observation of the constant speed of light that the concepts were combined - the theory didn't come before the hypothesis.

I am sure you can search Elsevier (or even google? :S) for a paper that is referenced that shows how these are derived where a constant speed of light is NOT assumed (I bet at least one person did it in the last 105 years)


oh and x²-c²t²=x'²-c'²t'² is what requires solving to show c=c' (i really don't feel like going through the basic math). If i can I will try ans find where this particular property is derived in another lecture.

Alternatively, send an email to susskind :P i think it's at the start of lecture 3

 

[JesseM]

I think we may have a little misunderstanding here. What I am saying, really, is not that the choice of units itself is the postulate. The postulate is assuming the speed of light is the same in all frames, with whatever units you play. If you choose units so that c=1 in one frame, then the postulate is assuming that c=1 in all frames.

The issue isn't setting the constant c=1 in itself, it's that he's assuming that this constant c is the actual measured speed of light in every frame. At about 40:30 he says that in these units a light ray should obey the equation x² - t² = 0 in the original unprimed frame. Then he says he wants a new coordinate transformation (different from the Galilei transformation x'=x-vt and t'=t) which "had the property that if x² - t² = 0, then we would find that x'² - t'² = 0", so that if you're the unprimed observer and he's the primed observer then we'll find "both of us agreeing that light rays move with unit velocity". He points at 41:35 that this wouldn't work with the Newtonian coordinate transformation; and yet, nothing would stop you from using units where c=1 in a Newtonian universe (where c is just the constant which equals 299792458 meters/second in units of meters and seconds), it's just that light wouldn't actually move at c in every frame under the Newtonian coordinate transformation. Along the same lines, if we define the speed of sound in the rest frame of the air as s=340.29 meters/second, we can use units where s=1, that doesn't in itself imply that the measured speed of sound waves in frames other than the rest frame of the air will be equal to 1 in these units. The part where he asserts the second postulate is not in his choice of units, but rather in his assumption that if x² - t² = 0 in the unprimed frame then x'² - t'² in the primed frame.

 

[yuiop]

Let's keep it really simple. The emission is a multiple photon event. Each observer sees a different photon, the one that has the correct angle to intercept the mirror.

This is not correct. The light clock is done with single photon and all observers observe that same photon. The light clock works just as well with a highly directional focused laser beem as it does with a regular light bulb. You can imagine a person on a train tossing a ball directly up and down. To a person outside the train the ball is moving along a zig zag path. It is clearly the same ball that appears to take different paths depending on the relative motion of the observer and it is the same with photons.

 

[JesseM]

I am sure you can search Elsevier (or even google? :S) for a paper that is referenced that shows how these are derived where a constant speed of light is NOT assumed (I bet at least one person did it in the last 105 years)

If you don't assume a constant speed of light, then what you get is a more general coordinate transformation which includes a constant that can either be set to infinity (in which case you get the Galilei transformation) or to a finite value (in which case, if you set this finite value to c, you get the Lorentz transformation). You can't uniquely derive the Lorentz transformation without explicitly assuming a constant finite speed that's the same in all frames. See this paper:

http://arxiv.org/pdf/physics/0302045

 

[yuiop]

Alternatively, send an email to susskind :P i think it's at the start of lecture 3

I am not claiming anything Susskind said is wrong and nor is Susskind claiming that the Lorentz trasformations can be derived without the second postulate. As JesseM quite correctly points out, Susskind introduces the second postulate when he states the transformation he is looking for must have the property that x2 - t2 = x'2 - t'2.

 

[Saw]

The issue isn't setting the constant c=1 in itself, it's that he's assuming that this constant c is the actual measured speed of light in every frame. At about 40:30 he says that in these units a light ray should obey the equation x² - t² = 0 in the original unprimed frame. Then he says he wants a new coordinate transformation (different from the Galilei transformation x'=x-vt and t'=t) which "had the property that if x² - t² = 0, then we would find that x'² - t'² = 0", so that if you're the unprimed observer and he's the primed observer then we'll find "both of us agreeing that light rays move with unit velocity". He points at 41:35 that this wouldn't work with the Newtonian coordinate transformation; and yet, nothing would stop you from using units where c=1 in a Newtonian universe (where c is just the constant which equals 299792458 meters/second in units of meters and seconds), it's just that light wouldn't actually move at c in every frame under the Newtonian coordinate transformation. Along the same lines, if we define the speed of sound in the rest frame of the air as s=340.29 meters/second, we can use units where s=1, that doesn't in itself imply that the measured speed of sound waves in frames other than the rest frame of the air will be equal to 1 in these units. The part where he asserts the second postulate is not in his choice of units, but rather in his assumption that if x² - t² = 0 in the unprimed frame then x'² - t'² in the primed frame.

I agree with all that. I suppose you didn´t mean it as contradicting what I had said in the passage you quoted from me...

 

[yuiop]

I am sure you can search Elsevier (or even google? :S) for a paper that is referenced that shows how these are derived where a constant speed of light is NOT assumed (I bet at least one person did it in the last 105 years)

Maxwell's equations predict that the speed of light is c and independent of the velocity of the source, according to an observer at rest with the light medium (aether). A lot of sources claim that they also predict that the speed of light is the same measured value in all inertial frames, but I am not clear on that. In Susskind's lecture, he states Maxwell assumed that the speed of light is c relative to the aether and that maxwell also assumed that the speed of light would not be c is an observer was moving relative to the aether. On a side note, Susskind also states that various corrections were tried to "rescue the aether" but none were successful. Unfortunately, I have to differ from Susskind on this historical point, because the corrections made by Lorentz in his Lorentz Ether Theory DO rescue the ether and have predictions identical to those of SR. (The downside is that Lorentz's corrections also mean it impossible to detect the ether by any physical measurement).

Anyway, it would be interesting if anyone could clearly state whether or not Maxwell's equations predict the constancy of the speed of light as measured in any inertial reference frame, without making that an initial condition. My guess is that the answer is no or Maxwell's equations would be the Special Theory of Relativity.

 

[Mentz114]

The important thing about the LT is that the proper interval is invariant under it. This connects to reality if we identify proper time as the time on local clocks.

This means x^2 - c^2t^2 = x&#039;^2 - c^2t&#039;^2[/itex]. If the same constant c did not appear on both sides, the equation would be untrue ( the primes mean after LT ).<br /> <br /> From which it appears that if the LT is have the desired properties, c must be the same in all inertial frames.

 

[JesseM]

I agree with all that. I suppose you didn´t mean it as contradicting what I had said in the passage you quoted from me...

Yeah, I realize my comment may have made it sound like I was disagreeing with something you said, but I didn't mean to imply a disagreement, I was just trying to clarify the point you were bringing up with some more details.

 

[TcheQ]

Anyway, it would be interesting if anyone could clearly state whether or not Maxwell's equations predict the constancy of the speed of light as measured in any inertial reference frame, without making that an initial condition. My guess is that the answer is no or Maxwell's equations would be the Special Theory of Relativity.

http://www.phys.unsw.edu.au/einsteinlight/jw/module3_Maxwell.htm

Maxwell's equations predict the speed of light using ε0 and μ0 (vacuum permittivity and permeability)
All experiments indicate that permittivity and permeability of a vacuum is unchanged, no matter how fast you are traveling - if they did change, it would be measurable as they in turn determine the magnetic and electric fields of particles.

As Mentz alludes to, ct is a fundamental assumption of fourth-dimensional geometry. if ct does not equal c't', then the property of all physics laws behaving the same in inertial frames would be untrue (and you can start an entirely new branch of physics on that assumption i believe ;p)

 

[JesseM]

http://www.phys.unsw.edu.au/einsteinlight/jw/module3_Maxwell.htm

Maxwell's equations predict the speed of light using ε0 and μ0 (vacuum permittivity and permeability)
All experiments indicate that permittivity and permeability of a vacuum is unchanged, no matter how fast you are traveling - if they did change, it would be measurable as they in turn determine the magnetic and electric fields of particles.

Yes, but in Maxwell's day it was assumed that Maxwell's equations would only be exactly correct in the rest frame of a hypothesized substance filling space called the luminiferous aether--the idea was that light was a vibration in the this substance analogous to sound waves in air. They wouldn't have expected Maxwell's equations to still apply exactly in a frame that was in motion relative to the aether.

 

[TcheQ]

Yes, but in Maxwell's day it was assumed that Maxwell's equations would only be exactly correct in the rest frame of a hypothesized substance filling space called the luminiferous aether--the idea was that light was a vibration in the this substance analogous to sound waves in air. They wouldn't have expected Maxwell's equations to still apply exactly in a frame that was in motion relative to the aether.

And how would that detract from current day observations of electromagnetism in relativistic inertial fields?

 

[JesseM]

And how would that detract from current day observations of electromagnetism in relativistic inertial fields?

Not sure what observations you're referring to, or what you mean by "relativistic inertial fields"...but certainly there are plenty of modern observations that make the old classical aether model untenable, if that's what you meant. I was just talking about how Maxwell's equations were interpreted before relativity, which is what kev was asking about.

 

[TcheQ]

Not sure what observations you're referring to, or what you mean by "relativistic inertial fields"...but certainly there are plenty of modern observations that make the old classical aether model untenable, if that's what you meant. I was just talking about how Maxwell's equations were interpreted before relativity, which is what kev was asking about.

It was a rhetorical question. Maxwell's equations are immutable in relativity, and the theory is supported by evidence. It does not matter how they were derived, they predict c.

 

[yuiop]

The important thing about the LT is that the proper interval is invariant under it. This connects to reality if we identify proper time as the time on local clocks.

This means x^2 - c^2t^2 = x&#039;^2 - c^2t&#039;^2[/itex]. If the same constant c did not appear on both sides, the equation would be untrue ( the primes mean after LT ).<br /> <br /> From which it appears that if the LT is have the desired properties, c must be the same in all inertial frames.

<br /> <br /> This is an interesting point. However I doubt the assumption of the invariance of proper time can be inserted into the generalised transformation equations mentioned in the #39 by Jesse and the Lorentz transformations and the constancy of the speed of light pops out. I imagine that invariance of proper time intervals is implicit in both the Galilean and Lorentz transformations, but I would I have to check that out a bit more.

 

[JesseM]

It was a rhetorical question. Maxwell's equations are immutable in relativity, and the theory is supported by evidence. It does not matter how they were derived, they predict c.

The equations alone don't predict anything without an interpretation of how the equations are supposed to relate to physical experiments...for example, even today you wouldn't say that Maxwell's equations predict light moves at c in a non-inertial frame would you? The modern interpretation is that they work in any inertial frame (but not non-inertial ones), the old interpretation was that they worked in the rest frame of the aether (but not other inertial frames). Nothing inherently illogical about the old interpretation, it just didn't turn out to be supported by the experimental evidence.

 

[yuiop]

It was a rhetorical question. Maxwell's equations are immutable in relativity, and the theory is supported by evidence. It does not matter how they were derived, they predict c.

I think you need to be a bit more precise than that. Maxwell's equations predict that light waves move at c relative to a medium in much the same way as sound has a characteristic speed relative to the medium it is propogating in. This implies that both sound and light propogate at a velocity that is independent of the source. Sound waves change in frequency when the source is moving but the speed relative to the medium remains unchanged. In Maxwell's time it was probably assumed the speed of light would vary when the observer is moving relative to the medium in much the same way as the speed of sound changes when the observer is moving relative to the medium. I think this is pretty much what Susskind was getting at in his lecture when he discussed Maxwell's equations and the aether.

 

[JesseM]

This is an interesting point. However I doubt the assumption of the invariance of proper time can be inserted into the generalised transformation equations mentioned in the #39 by Jesse and the Lorentz transformations and the constancy of the speed of light pops out. I imagine that invariance of proper time intervals is implicit in both the Galilean and Lorentz transformations, but I would I have to check that out a bit more.

In a universe with Galilei-invariant laws of physics, the proper time between any two events on a clock's worldline would just be equal to the coordinate time between those events, in any inertial frame (since all inertial frames agree on the time between a pair of events according to the Galilei transform, and a clock's rate of ticking in a given inertial frame always keeps pace with coordinate time regardless of the clock's motion). So if you just define "proper time intervals" as \Delta t and \Delta t&#039; then proper time intervals would be invariant under the Galilei transform (trivially so since t = t' in the transformation equations), but the quantity \Delta x^2 - c^2 \Delta t^2 would of course not be equal to \Delta x&#039;^2 - c^2 \Delta t&#039;^2

 

[Saw]

certainly there are plenty of modern observations that make the old classical aether model untenable

And just for clarification (I count on your agreement on this) what has been made untenable is, as you said, the "classical aether model", that is to say, a model where the speed of light is constant only in the aether frame and variable in any other frame, but not the aether itself, which is neither an illogical idea nor has been disproved, it's just unprovable (since it is immeasurable, as kev said) and unnecessary (since you can do anything in physics on the basis of the geometric description of Minkowski spacetime without the need to go into endless, complex and little remunerating discussions about whether that hypothetical aether has these or those properties).

I say this, because -if this issue (which is sometimes very controversial; could I propose a FAQ for it?)- were clarified, one could think of a more interesting discussion, for another thread, like: are LET and SR empirically indistinguishable, do they make the same predictions and do they share the same formulas, in ALL respects?

 

[TcheQ]

I think you need to be a bit more precise than that. Maxwell's equations predict that light waves move at c relative to a medium in much the same way as sound has a characteristic speed relative to the medium it is propogating in. This implies that both sound and light propogate at a velocity that is independent of the source. Sound waves change in frequency when the source is moving but the speed relative to the medium remains unchanged. In Maxwell's time it was probably assumed the speed of light would vary when the observer is moving relative to the medium in much the same way as the speed of sound changes when the observer is moving relative to the medium. I think this is pretty much what Susskind was getting at in his lecture when he discussed Maxwell's equations and the aether.

They were initially intended to be used in aether calcs, but that doesn't mean we then discard them when the concept of the aether was dissolved due to experimentation. It would be like discarding the microwave background because it wasn't intended to be detected.

x²-c²t²=0 is just an elaborate coordinate transfer. The established laws of physics are those that are intended to satisfy the condition that they will remain the same under any conditions, and this includes Maxwell's equations.

This argument can be countered if you can show experimental evidence that Maxwell's equations do not hold under relativistic conditions.

 

[JesseM]

And just for clarification (I count on your agreement on this) what has been made untenable is, as you said, the "classical aether model", that is to say, a model where the speed of light is constant only in the aether frame and variable in any other frame, but not the aether itself, which is neither an illogical idea nor has been disproved, it's just unprovable (since it is immeasurable, as kev said) and unnecessary (since you can do anything in physics on the basis of the geometric description of Minkowski spacetime without the need to go into endless, complex and little remunerating discussions about whether that hypothetical aether has these or those properties).

I say this, because -if this issue (which is sometimes very controversial; could I propose a FAQ for it?)- were clarified, one could think of a more interesting discussion, for another thread, like: are LET and SR empirically indistinguishable, do they make the same predictions and do they share the same formulas, in ALL respects?

I would say that a Lorentz Ether Theory could be indistinguishable from SR, in which case the LET is more of a philosophical interpretation as opposed to a physical theory (a bit like the different interpretations of quantum mechanics). But on the other hand you could also come up with (probably fairly contrived) LET theories where all the laws of physics tested so far are Lorentz-symmetric (or any deviation from Lorentz-symmetry is too small to have been detected by experiments done so far), but there might be some new laws of physics found in the future that were not Lorentz-symmetric and which would allow you to define a preferred frame. It would seem a strange coincidence that so many seemingly unrelated previous laws had been apparently Lorentz-symmetric, though. And even in a LET indistinguishable from SR, the fact that all types of clocks and rulers are affected in the same way by movement relative to the aether (regardless of whether they are based on the electromagnetic force or some other force like gravity or the strong nuclear force) has an oddly coincidental and contrived feel, which is one of the aesthetic/philosophical reasons why most physicists reject this sort of interpretation...there's an extended discussion of this problem here:

http://groups.google.com/group/sci.physics.relativity/msg/a6f110865893d962?pli=1

 

[bcrowell]

I think we may have a little misunderstanding here. What I am saying, really, is not that the choice of units itself is the postulate. The postulate is assuming the speed of light is the same in all frames, with whatever units you play.

No, that is not used as a postulate in these derivations.

If you choose units so that c=1 in one frame, then the postulate is assuming that c=1 in all frames. You may want to look at post #32 where I try to express the idea more at length than what you just quoted. Do you agree to that?

No, you're mistaken for the reasons I explained in #33.

Where does he say so? In page 45 he states that [...]



And then he goes on to derive the LTs.

Rindler presents two different derivations in that book. First he does a derivation that uses constancy of the speed of light as a postulate. Then he does one that doesn't use that as a postulate. The second one is the one on the page number I gave in #26 (p. 51 in the edition I have). The one you're quoting from is the first one.

That’s just what I was saying: to derive the LTs you have to leave aside the classical assumption (t=t’) and rely on a different assumption (c=c’, at the expense of admitting that t≠t’ and x≠x’).

No, that's incorrect. None of the derivations I referred you to make an assumption of c=c'.

 

[bcrowell]

As above, if you start with a transformation that was formulated with the initial assumption that the speed of light is constant in all frames then it is not surprising that the transformation should predict that the speed of light is constant in all frames. Circular reasoning.

None of the derivations I referred to make any such assumption.

Essentially I am agreeing with Saw that most derivations that do not have the constanty of the speed of light explicity stated as an assumption or postulate, have it impicitly assumed at the outset or have a conditional that it is a required outcome. However, it might not be "impossible" to have such a derivation but it would need a large and unreasonable set of alternative postulates.

No, it doesn't require a large and unreasonable set of postulates. None of the three derivations I referred to require a large and unreasonable set of postulates. Have you read them, or are you just imagining what you think they might say?

If you claim that the Lorentz transformation can be derived without any assumtion of the constancy of the speed of light you should make it clear what initial assumptions you are making.

All three of the derivations I cited make this clear. Have you read them?

 

[bcrowell]

If you don't assume a constant speed of light, then what you get is a more general coordinate transformation which includes a constant that can either be set to infinity (in which case you get the Galilei transformation) or to a finite value (in which case, if you set this finite value to c, you get the Lorentz transformation).

This is correct. The Morin and Rindler derivations that I referenced in #26 discuss this very clearly. There are three possible cases: (a) Galilean, (b) SR, or (c) a case that violates causality.

You can't uniquely derive the Lorentz transformation without explicitly assuming a constant finite speed that's the same in all frames.

If you assume causality and nonsimultaneity, then the only possible case is the one that gives SR. In any case, I think you're confounding two issues: (1) whether c is frame-invariant, and (2) whether c is finite. Morin and Rindler prove #1. You can have frame-invariance and finiteness (SR), and you can also have frame-invariance and infiniteness (Galilean). The two are almost logically independent, except that of course if c is infinite then it's not a number than you can measure, so it can't be frame-dependent.

 

[yuiop]

None of the derivations I referred to make any such assumption.
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No, it doesn't require a large and unreasonable set of postulates. None of the three derivations I referred to require a large and unreasonable set of postulates. Have you read them, or are you just imagining what you think they might say?
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All three of the derivations I cited make this clear. Have you read them?

Of the references in #26 I the relevant parts are not available in the Google snapshot or Amazon peek inside. The Rindler book appears to be out of print and the later paperback version does not seem to include a derivation without assuming c'=c. I was able to see Susskind's lecture which starts with the Lorentz transformations, which have been derived assuming the constancy of c. You yourself later stated in your Light and matter website that "From the point of view of this derivation, constancy of c is something that is derived from the Lorentz transformation". Therefore you have to demonstrate how the Lorentz transformations can be derived without assuming the constancy of c in the first place. I am sure it can be done, but it I have a hunch it would take more than the two simple postulates of SR.

... , except that of course if c is infinite then it's not a number than you can measure, so it can't be frame-dependent.

Curiously, if you plot the path of a particle with infinite velocity on a SR spacetime diagram (a horizontal line) and transform to another reference frame the path has finite (but superluminal) velocity and can go forwards or backwards in time. Counter-intuitively, infinite velocity is not infinite in all inertial reference frames in SR. Of course in a Galilean system where simultaneity is the same in all reference frames, infinite velocity would be the same in all reference frames.