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    Was Einstein Wrong?
    By Barry Barrett | May 24th 2012 04:35 PM | 49 comments | Print | E-mail | Track Comments
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    I have a B.S. in Mathematics and a minor in Philosophy....

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    Was Einstein wrong about 

    No, the equation correctly relates an objects rest mass to its energy content. However, in 1952, Herbert Ives published Derivation of the Mass-Energy Relation in the Journal of the Optical Society of America. It purports to show that Einstein's derivation was invalid because Einstein committed the error of begging the question mathematically (its in the Appendix if you're curious). According to Ives, Einstein implicitly assumed that E=mc^2 in his derivation of E=mc^2. I believe Ives' mistake results from the false assumption that Einstein put in the relativistic equation for the kinetic energy of a particle, when he merely notes that his expression resembles it. Recall that Einstein argues that the energy in the last step is purely kinetic because it is constructed from terms that are only due to the motion of the box. He never puts in the formula for relativistic kinetic energy explicitly, as Ives does. Instead, he makes an a priori argument that the term is kinetic, expands it, and then compares it to the classical formula for kinetic energy. The classical formula is appropriate because the box is moving slowly. A criticism of Ives by John Stachel and Robert Torretto expertly confirms this analysis, "the reader should note how carefully Einstein avoids any assumption about how the relativistic energy depends on velocity, mass, or any other parameters..."(Stachel. Einstein from B to Z. page 218)


    I should add that, even if the derivation is incorrect, even if all derivations are incorrect, the equation has been verified countless times in countless controlled experiments (for example, it is verified constantly at particle accelerators).    

    Again, if properly interpreted -as giving the relationship between an objects rest mass and its energy content -the equation is correct, but according to esteemed physicist, Tony Rothman, although Einstein's 1905 derivation of is correct, Einstein was mistaken about why it was correct. But, before I address that, let me explain why I'm interested in Rothman's opinion. In my last blog, I presented Einstein's first derivation of E=mc^2 and wrote that Einstein oppositely oriented the light rays in his thought experiment to "simplify the mathematics". It does simplify the mathematics, but if the same argument is applied to one light ray, then the mathematics does not give the same result because it lacks a cancellation of terms and, therefore, the orientation of the light rays is essential. First, I'll show this by doing the calculation with one light ray and then I'll try to explain physically why the orientation of the rays is essential to deriving the relativistic effect. Finally, I will ask the reader to comment on my argument because, according to Rothman, Einstein was wrong and the derivation isn't relativistic.   


    I've searched my local library for a more detailed evaluation of Einstein's paper, but the best reference I've found, the only reference in fact, is Abraham Pais' great scientific biography of Einstein, Subtle is the Lord. It has a quick encapsulation of the paper that mentions the opposite orientation of the light rays but doesn't explain it or the physical meaning of the cancellation.  According to Pais, Einstein had doubts about the paper.

    But Einstein was not quite sure. In the fall of 1905 he wrote to Habicht, "The line of thought is amusing and fascinating, but I cannnot know whether the dear Lord doesn't laugh about this and has played a trick on me..." 

    Unlike the original thought experiment, begin by assuming that a single light ray leaves the right side of the box.

    Then 



    and

    because 



    and recall that  

     

    so




    And applying the argument from the last blog gives the following expression for the difference in the kinetic energy of the box, due to the observed motion of the box, after the emission of a light ray from it.



    If the light ray is going in the opposite direction then, -v/c is replaced by v/c and this is why oppositely orienting the light rays produced only the kinetic energy term of the form 1/2mv^2 (where m=L/c^2) through cancellation. Einstein pointed his light rays in opposite directions to cancel out this term. The derivation depends crucially on this cancellation.

    I've shown what pointing the light rays in opposite directions accomplishes mathematically, but what is the physical interpretation of this cancellation? Here is my initial answer (see * and ** below for my final answer).

    At the time I was attempting to calculate  from the quantum formula  where h is Plank's constant and f is the frequency of the light wave and E is the energy of a photon. Instead of using the formula for energy shift, I applied the relativistic Doppler shift formula to f. This gives the shift in energy, E, of due to the shift in frequency, f.  That is very simple to do because the equation for relativistic frequency shift is



    Because this has the same form as the energy shift formula seen in my last blog (Einstein derived that formula from this one in On the Electrodynamics of Moving Bodies) taking the energy, L, equal to hf, produces the same result as before. Therefore, again, for the one light ray case, 

     


    it appears that oppositely orienting the light rays serves to cancel out the first order shift due to the non relativistic Doppler effect and leave the higher order shift due to relativity. This is confirmed by noting that the purely relativistic transverse Doppler effect is given by 





    Unlike the nonrelativistic case, relativity includes a shift that is not due to longitudinal motion. This transverse Doppler Effect is purely relativistic, invoking it implies that v is large enough that v^/c^2 is no longer negligible. It is the observed frequency shift of a light ray as the observer passes the axis of the light ray at a 90 degree angle with velocity, v. We could redo the original thought experiment again, this time considering the mass loss due to the transverse effect and this gives the original result. This is a sketchy argument, meant only to be suggestive of a possible rigorous argument to be developed later, but- at the very least- relativity introduces a term that doesn't cancel out (how could a transverse effect "cancel out"?) and therefore relativity predicts mass loss via radiation.





    But, then I read this in an article on physicsworld.com, titled Did Einstein discover E = mc^2? 
    While Einstein's celebrated 1905 paper, "On the electrodynamics of moving bodies", clearly laid down the foundations of relativity by abandoning the ether and making the speed of light invariant, his derivation of E = mc^2 did not depend on those assumptions. You can get the right answer with classical physics, says Rothman, all in an ether theory without c being either constant or the limiting speed. "Although Einstein begins relativistically, he approximates away all the relativistic bits, and you are left with what is basically a classical calculation."
    So do I have it exactly reversed? Instead of deriving a relativistic result Did Einstein actually "approximate away all the relativistic bits"?!  

    I haven't found a more detailed mathematical argument by Rothman.

     What do you think?  

    * In my last blog, I listed the assumptions implicit in the derivation. I should have included conservation of momentum. Based on the comment by Anonymous below, making the two light rays oppose each other implies, from conservation of momentum, that the total momentum of the light in the box frame of reference is 0. If only one light ray leaves, then the momentum of light is positive and conservation of momentum implies that the box has gained momentum ("recoiled"). That would, at the very least, complicate the derivation and would be reason enough to oppositely orient the light rays. But, on the other hand, the amount of recoil from light would be negligible. However, assuming that the recoil is negligible would have been equivalent to assuming that the "mass" or energy content of light is small and that is what Einstein was attempting to prove quantitatively. 

     ** In quantum mechanics E=pc=hf implies that p=(h/c)f and in units where h/c=1, p=f. This suggests that the results about Doppler shift can be restated in terms of momentum and recoil. 
    Because non relativistic Doppler shift is given by


    and p=f implies


     and for the left moving light ray

    and, because 

    In the Box Frame of reference, the momentum is cancelled and the box experiences no recoil, but in the frame of reference moving with velocity, v, the total momentum shift is  

     


    the light itself has a net momentum in one direction. 

    But recall that, in the original derivation, (1+v/c) and (1-v/c) were added, not subtracted (energy is a scalar), and that resulted in cancellation. Therefore, the cancellation of the v/c terms is a trivial consequence of the condition that the total momentum of light is cancelled so the box isn't recoiled (if this isn't obvious to you, take the v in the total momentum shift formula equal to 0 or think about what happens as v gets smaller). The box must not recoil because it was essential to Einstein's original argument that the kinetic energy of the box is only due to the relative motion of Moving Frame with respect to the Box Frame. 

    The net momentum shift calculated above leads to another, manifestly nonrelativistic, derivation of E=mc^2 based on the the fact that p=E/c

    ***I'm still unsure of how the original derivation can really be said to "approximate out all of the relativistic bits" when the final step depends on expansion of the explicitly relativistic gamma factor. Although the same argument probably goes through by merely considering c to be very large, the introduction of the gamma factor itself would not have been introduced without relativity.  But, until I've read more about it, it is probably reasonable to assume that I'm over-analyzing his statement. He did say "basically" nonrelativistic. 
      
    ****Reading the earlier paper, On the Electrodynamics of Moving Bodies, I came across the relativistic formula for the kinetic energy of a particle. It includes mc^2 and it is probable that Einstein intuited that mc^2 must be the energy content of the body from that and simply designed his thought experiment such that this result came out.

    *****Today, 5/25/2012, Stanford uploaded a lecture by Leonard Susskind that begins with a derivation of the relativistic doopler effect. 

    ******Sixty Symbols on E=mc^2

    Comments

    vongehr
    You can get the right answer with classical physics, says Rothman, all in an ether theory without c being either constant or the limiting speed. ... What do you think?
    You get the whole of special relativity from an ether. That one may even drop the one or the other assumption (here of c being a limiting speed) in order to get little parts of the theory is perfectly fine. What is problematic is: Why has special relativity, which is little more than the SO(1,d) symmetry afflicting certain measurements, become such a dogmatic religion that people find these facts so revolutionary and problematic? (Certainly neither empirical verification of GR, whose parts that may destroy an ether theory cannot be verified except you like to jump into a black hole so at least you can find out for yourself, nor the quantum foundation, which people do not accept anyway.)
    mathematical_investigations
    Why has special relativity, which is little more than the SO(1,d) symmetry afflicting certain measurements, become such a dogmatic religion that people find these facts so revolutionary and problematic?
    I am not familiar with ether theories, but it doesn't surprise me that they can also capture some aspects of relativity. What I don't understand is how, in this specific case, Einstein can be said to be "approximating out all the relativistic bits." I think it could be the step where gamma is approximated by 1+v^2/c^2 which of course doesn't impose a limit on v the way gamma does. But how would you get a 1/2(v^2/c^2) term out of the classical theory?
    vongehr
    in this specific case, Einstein can be said to be "approximating out all the relativistic bits."
    I won't look into this specific detail, but if you appreciate a useless smartass comment: If he derives something that is valid "classically", it will not depend on the "truly relativistic" bits ("" indicating Rothman's definitions of these terms, whatever they are), so in that sense, he is almost forced to effectively remove the "truly relativistic" bits in order to arrive at the result.
    mathematical_investigations
    Useless smartass comments are always welcome. :)
    The subtle answer is that the kinetic term is due to innertia as relativistic back-reaction: http://arxiv.org/abs/physics/9802031. Here what matters is the acceleration or change in momentum, which is different in the two cases.

    Its easy to loose sight of momentum in the dazzle of high NRG physics. Its entropy that tracks momentum, but that involves the order parameter and renormalization, which are now esoteric concerns.

    mathematical_investigations
    That is interesting and, if it is correct, would be a much deeper answer than what I had in mind.
    Bonny Bonobo alias Brat
    Barry, you also might find this recent article by Robert Matthews interesting or at least relevant to this blog, as its called 'Why E may not equal MC squared'. Maybe you could also explain how the findings reported there in any way justify the title of the article?
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    mathematical_investigations
    Thank you.  It could be evidence of that, but it gets somewhat philosophical because you can say that it is incorrect or you can introduce somewhat ad hoc "fictituous forces" and such to fix it up.
    Bonny Bonobo alias Brat
    Actually Barry,  was hoping that you could explain why or how they were saying in this article or rather in its title, that this could somehow be evidence of E not equalling MC squared. Couldn't see it myself :(
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    mathematical_investigations
    Thanks, I should relate my blogs to current events. He didn't explain anything, but there are these two problems presented, the problem of dark energy and the problem of find more high energy particles than expected.  Dark Energy, if I understand it correctly, is a consequence of experimental evidence (the accelerating expansion of the universe) and classical General Relativity. Classical General Relativity cannot explain the evidence without postulating the existence of a new form of energy that is inherent in space. However, some physicists believe that there really isn't Dark Energy and Einstein's equations are wrong. Once the equations are corrected to account for QM, the accelerated expansion will be explained without postulating Dark Energy. An analogous situation has occurred with these Cosmic Ray findings. The evidence doesn't match up. The energy of these rays is too strong so maybe E=mc^2 is wrong. It could also be that E=mc^2 is correct and that these particles are something new and different from the particles they expected.  It seems to me that there is always this choice when confronted with new evidence- keep the theory and introduce new elements into it (dark energy or weird new particles or make a new theory without the new elements). This would be an interesting topic to blog about. It relates to many issues in the philosophy of science that are fascinating. 
    Without their details, all I can really say is they must be using E=mc^2 (actually the are using the generalization of it) in their calculations of the particle energy, and because they aren't getting the result they expect, there is always a chance that E=mc^2 isn't quite correct. (but my guess is that the particles are different in nature or they are simply mistaken)







    Bonny Bonobo alias Brat
    An analogous situation has occurred with these Cosmic Ray findings. The evidence doesn't match up. The energy of these rays is too strong so maybe E=mc^2 is wrong. It could also be that E=mc^2 is correct and that these particles are something new and different from the particles they expected.
    Barry, you say that the reason E = MC2 might be wrong was used as the title of this article was because the evidence doesn't match up and because the energy for these cosmic rays is too strong, but I personally can't see that this is what the authors are claiming. To me, all they seem to be saying is that they can't find the source of these incredibly strong cosmic rays, excerpts from the article say the following :-
    Forget the large Hadron collider (LHC); as the Austrian physicist Victor Hess discovered in 1912, our universe harbours natural accelerators reaching energies far beyond anything humans could wield.Yet the whereabouts and nature of these awesome "machines" remains a mystery.
    Analysis of the paths of cosmic rays suggested they came from within our own galaxy, perhaps from exploding stars.Then in February 1962, just weeks before the anniversary, nature fired a shot across the bow of science. Instruments set up in the New Mexico desert detected a single subatomic particle smashing into the Earth with an energy 10 billion times higher than any man-made accelerator could then achieve (and 10 million times what the LHC can reach, even now).
    Suddenly, the theories claiming to explain cosmic rays were found to be wanting. Now an explanation was needed for how nature could accelerate particles to even greater energies than anyone thought possible. 
    Astrophysicists realised these incredibly violent events - equal to our sun releasing its lifetime energy output in a few seconds - were probably due to the collapse of gigantic stars into black holes, or the collision between the remnants of such stars.
    Calculations suggested the resulting gamma-ray bursts (GRBs) could also give cosmic rays their energy. Theorists even proposed a way of confirming the link, by looking for ghostly particles called neutrinos streaming out of GRBs, as these should be produced along with the cosmic rays.
    But last month, right on schedule, nature yet again rained on the astrophysicists' parade. Just in time for the centenary of Hess's discovery, the science journal Nature reported new results blowing a hole in the GRB theory. Researchers at the IceCube neutrino observatory in Antarctica looked for bursts of neutrinos coinciding with the 300 GRBs spotted by satellites between May 2008 and April 2010 - and failed to find a single one.
    Naturally they're trying to put a brave face on things. Maybe their theories just need tweaking to weaken the link between cosmic rays and neutrinos. Or maybe the GRB theory is just plain wrong - in which case, it's time to look for an even more violent cosmic event.
    Short of the Big Bang, the most powerful objects in the universe are the huge black holes lurking at the centre of most galaxies, including our own.Their titanic gravity tears apart whole stars, forming a searingly hot disc of swirling debris. Could these be the origin of cosmic rays?
    I don't see how any of this implies that E = MC2 is wrong, how can the energy be 'too strong'?
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    mathematical_investigations
    "you say that the reason E = MC2 might be wrong was used as the title of this article was because the evidence doesn't match up and because the energy for these cosmic rays is too strong, but I personally can't see that this is what the authors are claiming. To me, all they seem to be saying is that they can't find the source of these incredibly strong cosmic rays


    Based on

    our universe harbours natural accelerators reaching energies far beyond anything humans could wield.Yet the whereabouts and nature of these awesome "machines" remains a mystery.
    and

    Suddenly, the theories claiming to explain cosmic rays were found to be wanting. Now an explanation was needed for how nature could accelerate particles to even greater energies than anyone thought possible.

    I believe that what has happened is, because the measured energies aren't coming out right, the response can either be 1.e=mc^2 is wrong or 2.e=mc^2 is correct and there is some new source of energy. 


    The two claims are opposite sides of the same coin. 

    I would include a third option 3.experimental error.

    I read "Now an explanation was needed for how nature could accelerate particles to even greater energies than anyone thought possible" as 1. "e=mc^2 is wrong" 

    And 

    "maybe the GRB theory is just plain wrong - in which case, it's time to look for an even more violent cosmic event." as 2."e=mc^s is correct and and there are some new sorts of violent events out there."




    That is my interpretation. I could be wrong.
    Bonny Bonobo alias Brat
    Researchers at the IceCube neutrino observatory in Antarctica looked for bursts of neutrinos coinciding with the 300 GRBs spotted by satellites between May 2008 and April 2010 - and failed to find a single one. 
    Thanks Barry, and I would like to include a fourth option, that E = MC2 is correct but the researchers at the Icecube observatory just failed to find any neutrino evidence to support the GRB theory. Surely this doesn't necessarily mean that the neutrinos accompanying cosmic rays from GRBs don't exist, it might just be because neutrinos are notoriously difficult to trap and measure or maybe these neutrinos from GRBs are oscillating and/or behaving in some manner that we don't yet understand?

    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    mathematical_investigations
    That could be the case. This is why I somewhat avoid new experimental results. 
    Note that E=mc^2 is meant to be applied to a system at rest.

    The full equation is:
    E^2 - (pc)^2 = (mc^2)^2
    or alternatively, if discussing the energy of a moving system
    E = gamma m c^2

    So be careful that in your thought experiments the systems have no net momentum. The system of two oposing light rays has no net momentum. The system of one light ray does.

    mathematical_investigations
    That could very well be the deep reason why the derivation works with two light rays and not one, but, because this is a derivation of E=mc^2, I began by pretending I don't know E=mc^2 or its generalization. 

    In fact, I think that is the correct explanation in retrospect, but it wouldn't have been available to Einstein at the time. I believe that came from Minkowski (even though it is implicit in SR).  

    Actually, I meant to mention blue greens proof, using the more general formula, in the comments section of my last blog that a system of two photons moving in  opposite directions gains mass.
    fundamentally
    Barry,
    Why don't you  explain the meanings of your symbols?
    If you think, think twice
    mathematical_investigations
    I've assumed that the reader has read my last blog entry (but, if you haven't, skip it and read Einstein's paper, "Does the Inertia of a Body..." Being a retired physicist, you'll find it an easy read). If you read that, I believe the meanings of the symbols should be clear. But I admit the part about the doopler effect is sketchy. It is really meant to be a heuristic argument only. It really boils down to "hey look the relativistic effect has a 1/2v^2/c^2 term! and that looks like the term that doesn't get cancelled out so maybe that's the relativistic term" 
    Right question,

    Is c the initial letter of "celeritas"? [symbolic]

    The phenomenon of light has no meaning at all without interaction between [2 or more] baryonic masses, which is time dependant. And mass is relative to an observer [other baryonic mass]. Without observer, "c" or "m" is completely meaningless.

    When the speed of light is absolute, an expanding or contracting universe is not very likely.

    Light cannot be observed by other light. A temperature of light is meaningless without measurement [interaction of particles].

    mathematical_investigations
    Can you provide evidence for these claims? Are you the Oracle of Delphi? 
    fundamentally
    I think Hannes is partly right. Without observer and an observed item the whole concept of relativity loses its meaning.

    The observer can be a particle.
    If you think, think twice
    I think Hannes is partly right. Without observer and an observed item the whole concept of relativity loses its meaning.
    The observer can be a particle.
    Or even a turtle?

    I have never understood why relativity is supposed to depend on an observer. Certainly one can take a nice long journey into quantum anti reality and emerge with a religious zeal for observer-dependent worlds, but last time I looked, which was a very long time ago, the observer played the same role in relativity (SR) as in classical physics, i.e. none at all except to receive information passively.

    In that sense, any statement about Life, the Universe and Everything is meaningless without someone to hear it.

    However, the grand assumption of relativity is that physical laws can be cast in covarient form. Alice's observation of Bob's relativistic mass is fair enough - though it's a tricky beast to measure using inertia because the mass keeps changing and this soaks up a lot of the applied force instead of it all creating acceleration like Newton would have expected. So her observation is better thought of as that of his proper mass, which is invariant, but observed from the perspective of her frame of reference. The Lorentz transforms showing a gamma times increase in this and decrease in that are not about physical processes but about natural perspective applied in what turns out to be 4D Minkovskian spacetime.
     

     
    fundamentally
    My previous reaction was incomplete. The relativity that is described by Lorentz transformations requires three ingredients: The observer, the observed item and a maximum speed of information transfer.
    If you think, think twice
    Halliday
    Hans:

    It is entirely incorrect to state or even suggest that "The relativity that is described by Lorentz transformations requires ... a maximum speed of information transfer."

    "The relativity that is described by Lorentz transformations requires" no such thing!

    "c", the speed of light (the speed of electromagnetic waves in vacuum) is not "a maximum speed of information transfer."  However, it is an invariant speed.  The one and only invariant speed.

    David

    mathematical_investigations
    How could you transfer information faster? There are some supposed exceptions to special relativity, and the reply to them is usually "but you can't transmit information that way" 
    David's not talking about FTL transmission of information. He is saying that relativity does not depend on transmission of information at all. 
     

    mathematical_investigations
    I understand that relativity, as it is traditionally explained, doesn't depend of information transfer, but he seemed to go further than that and say that relativity doesn't put a restriction on information transfer at all when he said "The relativity that is described by Lorentz transformations requires no such thing!"
    Yes, this is a very common ambiguity.

    Is he talking about (not) "requiring it as an assumption before relativity can be derived", i.e. as an axiom, or (not) "requiring it of all systems to which relativity is applied" i.e. as a theorem?  Or even as (not) a general rule?
     
    Ho hum. It won't be the first time arguments have blown up over taking the wrong sense of the word :/

     
    mathematical_investigations
    Exactly.   
    Halliday
    Barry:

    Your question, above ("How could you transfer information faster?"), was quite far from the "mark", as Derek pointed out.  He even did a good job of showing some different interpretations of what I was saying.

    Incidentally, there are multiple correct interpretations in Derek's list.  Additionally, the question of "How could you transfer information faster?" is actually not even truly applicable.  There is nothing in the Special Theory of Relativity (nor even General Relativity) that requires or enforces any such limitation.  The only issue of whether one even can "transfer information faster" is actually outside of relativity theory, even though relativity theory is a cause for why the question should even arise:  It really pertains to questions about what is meant by "causality", and whether such even exists.  (General Relativity adds to this issue with its potential to have closed time-like paths.  Thus it can cause issues with "causality" even without "transfer[ing] information faster".)

    Do you want details?

    David

    mathematical_investigations
    Do you want details
    Yes. Could you explain
    There is nothing in the Special Theory of Relativity (nor even General Relativity) that requires or enforces any such limitation.
    Halliday
    Barry:

    You ask:

    Could you explain
    There is nothing in the Special Theory of Relativity (nor even General Relativity) that requires or enforces any such limitation.

    To be clear, the "limitation" referred to is that "The relativity that is described by Lorentz transformations requires ... a maximum speed of information transfer."

    You have already stated that "I understand that relativity, as it is traditionally explained, doesn't depend of information transfer".  Of course, it's not only as "it is traditionally explained".  Relativity simply "doesn't depend" upon "information transfer", in general.  (On the other hand, the approach to what is supposedly "Einstein's relativity" that is being advocated by Frank Wappler, supposedly taking his ques from J. L. Synge, certainly appears to take an "information" centric approach, and seems to try to impose an information transfer limitation.  However, that is most certainly not Einstein's theory.)

    Einstein's original derivation of his relativity theory used only two postulates (like axioms):

    1. The Principle of Relativity:  "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."
    2. "that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body."

    (Note:  Since the first postulate includes "electrodynamics and optics", and since the original derivation of the Lorentz Transformations was obtained simply from investigating the symmetries of electrodynamics [in the form of Maxwell's equations], one could simply use the first postulate alone.  However, the first postulate also includes "the equations of mechanics" which, in the form of Newtonian mechanics, have a significantly different symmetry.  So, the second postulate seems to serve as the tie breaker.)

    As you can see, "information transfer" isn't even mentioned.  Certainly, there is no postulate/axiom that holds any speed as any kind of maximum.

    The same is true of the approach I am using in my series.  In fact, in both Einstein's paper (though only in an implied form) and my series, the nature of the space being used is that of a vector space.  This means that there is simply no way to restrict the speeds, that correspond with different spacetime directions/intervals (Δx, Δy, Δz, Δt), to only those that are less than any given value.

    In fact, if such were to be attempted, then one would not be able to have space-like as well as time-like intervals, though both are used quite extensively in relativity.

    So, there is neither a maximum speed postulated (given as an axiom), nor is there any limitation derived or used as a part of Einstein's Special Theory of Relativity.  (Since the tangent spaces of General Relativity are precisely the same vector space as in Special Relativity, this conclusion holds equally there.)

    OK.  Now, we all know we have heard that the speed of light is the "universal speed limit".  So, what gives?

    Well, it is most certainly true that the speed, v, seen within the Lorentz Transformations, can only fall within the range |v| < c in order for the transformations to be valid.  Is this not a "speed limit"?  Is this not "required" or "imposed" by the validity of the Lorentz Transform?

    While this is indeed true, and is a "limitation", it is, strictly speaking, only a limitation on a parameter used in the transformations.  The transformations can just as well be formulated in terms of hyperbolic sines and cosines.

    Oh, but wait a minute.  Doesn't that speed, v, have an actual, physical significance?  Isn't that why the transformations are actually parameterized the way they are?

    Absolutely!  It is the difference in the speed of one reference frame relative to another (for a restricted class of what one considers as a "reference frame").

    So, isn't this a "speed limitation"?  It is quite correct that the Lorentz transformations will never transform the speed of something that is moving in a time-like trajectory (less than the speed of light) to the speed of something that is moving in a space-like trajectory (greater than the speed of light), or even to a light-like trajectory (moving right at the speed of light) except in some limiting sense (as v approaches c).

    So, the speed of any time-like trajectory is limited to being less than the speed of light.  This can take the form of a theorem, based upon the postulates uses by Einstein.

    However, what about space-like trajectories?  These have a rather similar "limitation" imposed upon them:  The Lorentz transformations will never transform the speed of something that is moving in a space-like trajectory (greater than the speed of light) to the speed of something that is moving in a time-like trajectory (less than the speed of light), or even to a light-like trajectory (moving right at the speed of light) except in some limiting sense (as v approaches c).

    Furthermore, the Lorentz transformations can never transform a light-like trajectory into anything other than a light-like trajectory.

    (Note:  The classifications of time-like, light-like, and space-like can be formulated in a completely invariant form.  So, one doesn't even have to refer to transformations at all, since invariants are unchanged by transformations.  Hence the name.)

    So, the speed of light does, indeed, form a "limit", but not a prohibition against the existence of anything faster, but a limit from both below and above:  Nothing time-like (slower) can go faster, and nothing space-like (faster) can go slower.

    There is absolutely nothing there that claims that "information transfer" can only occur via light-like trajectories, or anything except space-like trajectories.

    It is only when one asks questions about the meaning of "causality" in the light of the possibilities of transferring information via various trajectories that one finds any possible issues with "information transfer" along space-like trajectories (various "paradoxes" and such).

    So, does that just about cover it?

    David

    mathematical_investigations
    Thanks. There is nothing in special relativity that disallows information to be conveyed using faster than light particles like Tachyons so it cannot be derived from that assumption. That is indisputable. 

    But, it is trivial that you could derive the equations of SR from the assumption that there is "a universal maximum" because universal means invariant. I don't know what other meaning it could have. So, such an assumption would actually lead to the correct equations, but the "maximum" part adds a new assumption that isn't contained in SR.
    Halliday
    Barry:

    You are correct that if one takes "universal" to "mean invariant", that the invariance then provides the necessary characteristic to derive SR.  (Of course, you are further correct that "the 'maximum' part adds a new assumption that isn't contained in SR.")

    However, one could also take "a universal maximum" to mean "A universe wide maximum" (to use Hans' term), or to mean that the maximum is to apply universally to all "kinds of things", without exception (such as not allowing tachyons to be an exception), in which case invariance is not necessarily implied.

    David

    mathematical_investigations
    That interpretation didn't occur to me. In his defense, he could have meant information travel within the domain of known physics. He didn't really explain himself in any detail. 
    blue-green

    I have a hunch for a tact that avoids any talk of c. Instead, of talking about causality via light-cones, focus instead on how simultaneity cannot be absolute and how different observers cannot agree on "before" and "after". Perhaps that in itself would lead to a post-Newtonian inner-product with a hyperbolic signature.

    fundamentally
    A universe wide maximum is invariant when it is taken as a unit.
    If you think, think twice
    Halliday
    Hans:

    Even if there were "A universe wide maximum", there is no reason, whatsoever, that such need be formulated/imposed/whatever in such a way that it is invariant.

    David

    mathematical_investigations
    but last time I looked, which was a very long time ago, the observer played the same role in relativity (SR) as in classical physics
    When solving problems in SR you should always be clear about frames of reference. "Acceleration" should be replaced by "acceleration in this frame of reference" because its meaning isn't well defined otherwise. In classical physics, the acceleration of an object is the same in any two inertial systems. In Newtonian physics you are very rarely talking about what another observer measures because dynamically both systems are the equivalent. Even the classical doopler shift found by taking c equal to infinity in the relativistic formula gives no frequency shift . ,
    What has that got to do with anything? 
    mathematical_investigations
    I interpreted
    Without observer and an observed item the whole concept of relativity loses its meaning.
    as merely stating that, in relativity problems, you must speak about observers and be explicit about which frame you are considering (because I didn't really do that in this blog). So your reply that they play the same role in classical and relativistic physics made little sense to me, but now I believe you were trying to make some broader philosophical point that doesn't interest me.
    Fair enough, but note that Hans has already added THREE unnecessary entities to the system:
     The observer, the observed item and a maximum speed of information transfer.
    You have started an unstoppable avalanche. :) 


    Johannes Koelman
    "I haven't found a more detailed mathematical argument by Rothman."
    You might want to have a peek at p170-175 of the entertaining booklet "Instant Physics" by Tony Rothman.

    By considering two equal masses, one of which sends out a photon that gets absorbed by the other mass, Rothman derives E=mc2 by using momentum and energy conservation and the dispersion equation for the photon (E=pc).

    A lot is wrong with E=mc2, not so much with the equation itself, but rather with the way the equation is interpreted. This famous equation has little to do with relativity. As was commented above, the real relativistic equation reads E2 - p2 = m2 (insert appropriate factors of c if you insist on using non-rational units).
    mathematical_investigations
    Thank you! I will read both articles. 

    The derivation by Rothman sounds similar to the one I linked above.  That derivation, because of the approximations, only really assumes that light moves much faster than the box and therefore the Doopler shift is given by the classical formula.  SR plays a minimal role in that calculation because c is basically a constant for small v in classical physics (c'=v+c  for small v implies c'~c).
    But I'm finding it difficult to use this information to understand precisely how Einstein's original derivation "approximates out the relativistic bits" because the c^2 terms come about differently in both cases. Those articles might help. 
    Einstein needed Newton's binomial theorem [Taylor series] to come to
    m[before] - m[after] = L / c^2.

    According to Einstein
    E'0 – E0 = K0 + C [energy difference before emitting light]

    and [afterwards]
    E1'– E1 = K1 + C

    Both equations have [+ c], so you can ignore c all together.

    What remains in these equations is the actual meaning of 'c'.
    It is not being emitted or measured in reality in E'0 – E0 = K0 + C.
    It is only a part of an equation.

    See also Isaac Asimov in his book The Intelligent Man's Guide to Science; Vol 2. Biological Science (1960).
    I don't know Rothmann's book unfortunately. I visited a good explanation in detail of E=mc^2 (in Dutch however, see http://www.einsteingenootschap.nl/uitleg%20E=mc2.htm ). The equations can be followed easily.

    David Bodanis biography of Einstein's E=mc^2 is also interesting, I suppose.

    I wrote +C on purpose with a big letter C above in E1'– E1 = K1 + C and E1'– E1 = K1 + C

    But is has nothing to do with the speed of light (small teller c) being a constant. It is just an arbitrary additive Constant. Einstein wrote it with capital C in his paper.

    There is no place for C in E^2 - p^2c^2 = m^2c^4.
    m^2c^4 equals the rest energy, when p=0.

    Maybe that is the reason people talking about a small letter c being a constant? ;-)

    mathematical_investigations
    It is not being emitted or measured in reality in E'0 – E0 = K0 + C.It is only a part of an equation.
    Energy comes with a constant and the energy of the box does as well. Because H_0 and E_0 are the energy of a box at the same moment in time with respect to different frames, there energy difference is kinetic, but it would be imprecise to say that it is exactly kinetic. You could imagine a pulley in the box with a weight and gravity pulling the weight down to power an L.E.D.  by magnetic induction. Gravitational (and all conservative forces) come with a constant because of how they are defined. Without the cancellation of C the argument doesn't work at all and the "kinetic energy" could come out as anything depending on what sort of mechanism is in the box.
    Barry, I found some information about Einstein being incorrect.

    About the opposing light rayes, I think Einstein had in his mind the situation where electrons are moving in a straight wire, producing electromagnetic radiation, Maxwell's speciality. Electrons move at the speed of light.

    The electrons move only at the x-axis, compared to the molecules in the wire. If you think away the electron all you have is one object and two frames where one of those can be arbitrary chosen as a frame of reference [relativity].

    From the view of the moving frame L* = L (1-(v/c)cosφ)/(1-v2/c2)1/2

    When cosφ is 90° than in L = L 1-v/c cosφ, the equation gives a nice L* = L (1-0).

    So it becomes L* = L / (1-v2/c2)1/2.

    Because an electron moves at the speed of light then v2/c2 approaches to one. At a lower speed the limit approached by the kinetic energy 1/2 mv^2 < E'< mv^2 .

    See A revision of the Fundamental Laws of Matter and Energy (1908) by Gilbert Newton Lewis.
    http://en.wikisource.org/wiki/A_revision_of_the_Fundamental_Laws_of_Matt...

    So I don't know if Einstein made a real failure here.

    See http://vixra.org/pdf/1203.0014v1.pdf for his 'failure'.

    mathematical_investigations
    The paper on vixra is garbage. The 1908 paper isn't broad enough in scope.