Light clock moving to demonstrate time dilation

[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 [tex]x^2 - c^2t^2 = x'^2 - c^2t'^2[/itex]. If the same constant c did not appear on both sides, the equation would be untrue ( the primes mean after LT ).

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 [tex]x^2 - c^2t^2 = x'^2 - c^2t'^2[/itex]. If the same constant c did not appear on both sides, the equation would be untrue ( the primes mean after LT ).

From which it appears that if the LT is have the desired properties, c must be the same in all inertial frames.

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 $$\Delta t'$$ 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 $$\Delta x'^2 - c^2 \Delta t'^2$$

 

[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.
.
.
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?
.
.
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.

 

[JesseM]

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.

Interesting, I didn't realize there was a third causality-violating case. Do you have the equations for the causality-violating coordinate transformation handy?

If you assume causality and nonsimultaneity, then the only possible case is the one that gives SR.

Fair enough, although assuming nonsimultaneity at the start seems much less physically-motivated than assuming that the frame-invariant speed is finite (and equal to the speed of light).

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

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.

Yeah, I didn't think to distinguish those two. I suppose I should modify my statement above to say something like "You can't uniquely derive the Lorentz transformation without explicitly assuming that the frame-invariant constant speed (whose existence follows from the first postulate) has a finite value." Although, where does the assumption of causality fit in? Does the causality-violating case still have a frame-invariant constant speed of its own (and if so can it be either finite or infinite?), or does the conclusion of such a frame-invariant speed only follow from the first postulate plus the assumption of causality?

 

[yuiop]

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.

If you assume causality and nonsimultaneity, then the only possible case is the one that gives SR.

In SR if two events are causally connected, then one event is within the light cone of the other. If you do not have an initial clear idea of the speed of light (or what a light cone looks like) in different reference frames, then defining exactly what you mean by causality as a postulate would seem to be a difficult task.

As for nonsimultaneity, there must be an infinite number of ways that non simultaneity could be defined for different reference frames. To be useful postulate it would have to be defined mathematically as L*sqrt(1-v^2/c^2)*v/c^2 where we have to make initial assumptions about how length transforms and presence of the c^2 terms in the formula is a bit of a nuisance in a derivation that claims to make no assumptions about the speed of light.

It would seem that you are starting with these 3 postulates:

1) The laws of physics are the same in all reference frames.
2) If 2 events separated by a distance (L) are simultaneous in one inertial reference frame then the non-simultaneity of the 2 events in another inertail reference frame with relative velocity (v) is L*sqrt(1-v^2/c'^2)*v/c'^2.
3) If one event is caused by another event in one inertial reference frame, then the order of that sequence of events is the same in any other inertial reference frame.

Not as neat and tidy as Einstein's two postulates and the non-simultaneity postulate seems especially messy with a lot of built in hidden assumptions.

 

[JesseM]

As for nonsimultaneity, there must be an infinite number of ways that non simultaneity could be defined for different reference frames.

But most of those ways would violate the first postulate, no? This paper which I linked to in post #39 says the first postulate alone is sufficient to derive a general coordinate transformation that can reduce either to the Galilei transformation or the Lorentz transformation depending on the value assigned to a certain constant. Since simultaneity isn't relative in the Galilei transformation, that suggests that just assuming simultaneity is relative--without any specific assumption about how it works--should be sufficient to derive the equations of the Lorentz transformation.

Although looking at that paper, it doesn't seem they make any assumptions about causality, so I'm not sure how this fits with what bcrowell was saying about that being an essential assumption when deriving the Lorentz transformation...

 

[yuiop]

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?

Briefly, the answer is yes.

One of the most difficult thought experiments for newcomers to understand in the twin's paradox as is evidenced by the hundreds of threads on the subject in this forum. If LET was the accepted way of thinking of relativity the twin's paradox would not be an issue as it very easy to understand why one twin ages differently to the other when analysed in LET terms. The LET ether background gives a very clear mental reference that is easy to analyse even if it is imaginary. It is a bit like the Greenwich time meridian. Time does not really start and stop on a line parallel with Greenwich and it could equally have been drawn anywhere on the globe, but once we drawn a line (even an arbitrary and imaginary one) it gives us a sensible and useful reference to work from. I think it is a pity that LET is not used as an educational visualisation tool and once it is understood, it is also easy to demonstrate that the LET ether is an an arbitray and imaginary (but useful) reference tool.

 

[Saw]

Briefly, the answer is yes.

One of the most difficult thought experiments for newcomers to understand in the twin's paradox as is evidenced by the hundreds of threads on the subject in this forum. If LET was the accepted way of thinking of relativity the twin's paradox would not be an issue as it very easy to understand why one twin ages differently to the other when analysed in LET terms. The LET ether background gives a very clear mental reference that is easy to analyse even if it is imaginary. It is a bit like the Greenwich time meridian. Time does not really start and stop on a line parallel with Greenwich and it could equally have been drawn anywhere on the globe, but once we drawn a line (even an arbitrary and imaginary one) it gives us a sensible and useful reference to work from. I think it is a pity that LET is not used as an educational visualisation tool and once it is understood, it is also easy to demonstrate that the LET ether is an an arbitray and imaginary (but useful) reference tool.

I fully agree. Personally I use LET as a visualisation tool to educate myself and it never fails me... to understand SR! But here I have raised a side-issue that deserves its own treatment, while the question raised by the original poster has taken an interesting turning with bcrowell latest comments and others' answers. I suggest leaving LET's pedagogical value for another thread and focusing here on the issue raised by the original poster in his post #17: can you derive LTs (or other SR formulas) without assuming as an axiom that c (in S) = c' (in S')?

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.

Yeah, I didn't think to distinguish those two. I suppose I should modify my statement above to say something like "You can't uniquely derive the Lorentz transformation without explicitly assuming that the frame-invariant constant speed (whose existence follows from the first postulate) has a finite value."

If I summarise well, what bcrowell is arguing is that a derivation where you start by introducing a constant c in both sides of the transformation, gives off a result with three possible solutions: (a) c is infinite (which leads to the GT), (b) c is finite and causality is respected (which leads to the LT and SR) and (c) causality is not respected (with a finite c as well?).

I am sure this is true. Unfortunately, like kev, I cannot find the relevant text in Rindler’s book. Bcrowell, maybe some short copy and paste, if you can manage to do it, would not violate copyrights and would enlighten the discussion.

Anyhow, if the question is presented in these terms, I do not think we have any real disagreement at all. There is absolutely no logical difference between these two approaches:

1. I derive introducing a constant c, just to test what happens. Of the three possible solutions that I thus come out with, there is one that is absurd and I discard it (c). Of the other two, I prefer (b) because it agrees with experiment at relativistic speeds, while (a) is only true at “slow” speeds.
2. I assume that (b) is true. I derive introducing it as an axiom in the derivation and logically find it. Yes, I do obtain together with it (a) and (c), but the latter is absurd and the former clashes with modern experiments.

Unless, what you are saying, bcrowell, is that (a = infinite c = the GT) is to be forcefully discarded, even in the absence of experimental evidence, out of purely logical grounds: because an infinite speed is intrinsically absurd. If so, well, yes, that may certainly be a point, it would deserve serious consideration, although it should be made explicit and further discussed.

Finally, I’d like to note that all this vaguely reminds me of the derivation of final velocity after collisions, based on combining conservation of momentum and energy. Mathematically, you have multiple possible solutions. You choose bteween them based on empirical observation of what happens after the collision. For example, if the objects stick together, you take coefficient of restitution = 0. And there is a possible solution you discard from scratch because it's absurd and violates Pauli exclusion principle (one body goes through the other...).

 

[yuiop]

But most of those ways would violate the first postulate, no?

I have done some research and concluded you are right about this.

This paper which I linked to in post #39 says the first postulate alone is sufficient to derive a general coordinate transformation that can reduce either to the Galilei transformation or the Lorentz transformation depending on the value assigned to a certain constant.

It turns out that there are 3 transformation systems that satisfy the first postulate. (Ben was right about this*). Namely the Galilean Transformation (GT), the Lorentz Transformation (SR) and the causality violation transformation (CV). The K parameter in the paper you mentioned has the values K=0 (GT), K=1/c^2 (SR) and k=-1/c^2 (CV). Dimensional considerations do not allow the K constant to take on any other values, so there is no spectrum of solutions as I initially assumed.

Since simultaneity isn't relative in the Galilei transformation, that suggests that just assuming simultaneity is relative--without any specific assumption about how it works--should be sufficient to derive the equations of the Lorentz transformation.

From the above considerations, non simultaneity (i.e t ≠ t’) excludes GT but still allows the SR and CV transformations as Ben said.

(I eventually managed to find the text Ben was referring to. See http://books.google.com/books?id=fU...e=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false page 57 section 2.11)

Although looking at that paper, it doesn't seem they make any assumptions about causality, so I'm not sure how this fits with what bcrowell was saying about that being an essential assumption when deriving the Lorentz transformation...

The CV transformation that has K = -c^2 is dismissed in the paper and Rindler calls it un-physical. It is interesting to analyse it a bit further and see why it called causal violating. The CV transformations are:

$$t' = \frac{t + vx/c^2}{\sqrt(1 + v^2/c^2)}$$

$$x' = \frac{x - vt}{\sqrt(1 + v^2/c^2)}$$

so the gamma factor is [itex] \sqrt\left(1 + v^2/c^2\right)}and time speeds up and lengths expand with relative velocity and lengths and time remain real for velocities greater than c.

From the CV transforms, it can be worked out that the invariant interval in CV is:

$$dS = \sqrt{\left(c^2T^2 + X^2 \right)}$$

This implies that the relationship $C^2T^2 -X^2 = C^2T^2 ' - X^2 '$ that is valid in SR (for real values of X and T), is also valid in CV coords if we treat capital C as the speed of light with a value of i*c where small c is a unit conversion factor from distance to time.

The velocity addition equation in CV coordinates is:

$$w = \frac{u + v}{1-uv/c^2}$$

From the above it can be seen that the invariant speed in CV coords is i*c where i is the imaginary number. Clearly if we put a condition that our desired transformation system should have an invariant finite velocity then the CV transformation qualifies and we need to specify that the invariant velocity is both finite and real to exclude GT and CV transformations.

As a consequence of the CV addition formula, two real finite velocities that are both less than c, can have a total velocity that is greater than c and there is no upper bound to real velocities in CV coords. This is where the causal violation comes in. While the temporal ordering of events remains invariant in CV coords for velocities less than c, causality is violated because the CV transformation allows velocities greater than c.

So in summary, the first postulate (P1) of SR (the Relativity Principle) admits 3 three possible transformations, SR, GT and CV.

Any one of this group of further postulates (conditions) exclude the GT transformations:

P2) There exists a invariant velocity that is finite.
P3) Non simultaneity (t ≠ t’).
P4) Length transformation (x ≠ x').

Any one of this goup of additional postulates (conditions) excludes the CV transformations:

P6) There exists an invariant velocity that is real.
P7) Causality invariance.
P8) An upper bound to real velocitites. i.e. the addition of two real velocities both less than c, results in a velocity that is also less than c.

P8 is difficult to define for CV coordinates, because if c is a unit conversion factor from length to time and CV coordinats do not admit an invariant velocity that is real, then it would seem to be impossible to define a real conversion factor that is consistent in different reference frames.

With the postulates expressed as above, even SR requieres 3 postulates, but postulates P2 and P6 are effectively combined as one postulate, i.e. there exists an invariant velocity that is both real and finite.

Lastly, one question. The paper does not seem to admit the possibility of a transformation system where the speed of light is (c +/- v). Such a balistic velocity for light is consistent with MMX null result. What theoretically excludes the ballistic speed of light idea? Obviously it is excluded by the Maxwell equations and the abberation of light, but the paper makes no mention of those factors.



* andcoughIcoughwascoughwrongcoughagain.

 

[bcrowell]

Lastly, one question. The paper does not seem to admit the possibility of a transformation system where the speed of light is (c +/- v). Such a balistic velocity for light is consistent with MMX null result. What theoretically excludes the ballistic speed of light idea? Obviously it is excluded by the Maxwell equations and the abberation of light, but the paper makes no mention of those factors.

These derivations don't have anything to do with light. They simply show that there is some invariant velocity c. If we discovered tomorrow that the photon had a nonvanishing rest mass, these derivations would be unaffected.

When you refer to a ballistic velocity $c\pm v$, are you talking about the idea that the velocity of a beam of light could depend on the source's velocity, i.e., have the source's velocity added on to it? I don't think that relates directly to the issue that these derivations are addressing, which is how time and space look to different observers. But anyway there are only three cases (SR, Galilean, and causality-violating), and the linear form of the expression $c\pm v$ is not consistent with the SR case, since velocities don't add linearly in SR. It's not consistent with the Galilean case if you equate $c_L$, the speed of light, to $c_{CE}$, the maximum speed of cause and effect. These derivations prove that there's an invariant $c_{CE}$, and in the Galilean case $c_{CE}=\infty$, so you can't equate it to $c_L$, which is finite.

 

[bcrowell]

Although looking at that paper, it doesn't seem they make any assumptions about causality, so I'm not sure how this fits with what bcrowell was saying about that being an essential assumption when deriving the Lorentz transformation...

I didn't say it was essential, just that it was one way to rule out one of the three cases. There are lots of different axiomatic systems you can set up from which the Lorentz transformations can be derived.

Pal says that the causality-violating case is inconsistent, because the thing that plays the role of $\gamma$ (he calls it A) can be negative. Rindler says (in the edition I have, which is Essential Relativity, 1977): "Evidently, $\gamma$ must be positive, because x and x' increase together at t=0." (p. 32); and: "The corresponding group has many unphysical properties if x and t have the significance of the present context," because $\gamma$ can be negative, there is an infinite velocity discontinuity, and causality can be violated (p. 52).

So Pal seems to be saying that the causality-violating case is not even mathematically self-consistent, while Rindler says that it's just highly unphysical. I believe what's going on here is that Pal wants v to be a continuous function of $\gamma$ that goes to 0 when $\gamma=1$. He considers those to be self-consistency requirements. Rindler seems to consider continuity to be a physical requirement, not a self-consistency requirement.

If you take Rindler's attitude about this, then it is essential to introduce some other physical criterion (a postulate) that rules out this case. One such postulate is causality.

 

[yuiop]

When you refer to a ballistic velocity $c\pm v$, are you talking about the idea that the velocity of a beam of light could depend on the source's velocity, i.e., have the source's velocity added on to it?

Yes, I am saying if there was a hypothetical system where the the speed of light is (c+v) where v is the velocity of the source, then a MM type experiment would give the required null result, without requiring length contraction. To an observer co-moving with the MM apparatus the speed of light from the source mounted on the apparatus would be c. I know that real experiments rule out this possibility but I was wondering if it could be ruled out on purely theoretical considerations such as violation of causality or the relativity principle?

 

[bcrowell]

Yes, I am saying if there was a hypothetical system where the the speed of light is (c+v) where v is the velocity of the source, then a MM type experiment would give the required null result, without requiring length contraction. To an observer co-moving with the MM apparatus the speed of light from the source mounted on the apparatus would be c. I know that real experiments rule out this possibility but I was wondering if it could be ruled out on purely theoretical considerations such as violation of causality or the relativity principle?

Well, if you take a set of physical postulates that suffice to derive the Lorentz transformations, then they suffice to rule it out, for the reasons I gave in #68.