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Could SR not be built from only one postulate?

  1. May 24, 2014 #41

    Meir Achuz

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    That is excluded by the word "constant in the postulate:
    "any two observers moving at constant speed and direction with respect to one another will obtain the same results for all experiments"
    If gravity were included the phrase "at the same location" would have to be included, but I think that is implied in the postulate, which gives only relative velocity in the difference beteen the two observers.
     
  2. May 24, 2014 #42

    Fredrik

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    All theories are wrong. Some are just less wrong than others. So if a statement that's part of a "wrong" theory of physics can't be a law of physics, there are no laws of physics.
     
  3. May 24, 2014 #43
    Thanks for your answer. I have the strong feeling that your rejection of my first postulate is due to a misunderstanding since it deals with physical objects, not with observers.

    Although I agree with the statements quoted above, I'm trying to eliminate any direct reference to “observers” performing measurements or experiments. Yes, only an “observer” who feels unaccelerated can imagine being attached to an inertial frame of reference and the transformation between inertial frames of reference will map zero-accelerated observers to other zero-accelerated observers. But assuming one of these non-accelerated observers observes a non-zero-accelerated object, the said transformation between inertial frames of reference will map this non-zero-accelerated object onto a non-zero-accelerated object: if an object is accelerated when represented in one inertial frame of reference, it must have an accelerated motion in any other inertial frame of reference, irrespective of the presence of any “observer”. This is what I tried to express in my first postulate which deals with objects and not with observers. Therefore there is no need to involve complex maths.


    Yes, homogeneity and isotropy of space, as well as homogeneity of time merely reflect the absence of good reasons to inject asymmetries in our representations of space and time. Any alternative would require a justification counteracting the empirical evidence. The two postulates I formulated are very general in their nature: since we do not sense any difference between velocity and rest, it would be irrational to assume a priori an absolute difference between uniform motion and rest. Conversely, our sense of acceleration suggests the opposite a priori assumption about the difference between inertial and non-inertial state of motion.

    It is noticeable that the second postulate proposed by Einstein in 1905 about the invariance of the speed of light is of a less general nature, so that the set of postulates and assumptions from which he derived the Lorentz transformation lacks homogeneity. However the main effect of his second postulate is to inject a dependency between space and time quantities, a parameter (c) homogeneous to a velocity which is left invariant through changes of the inertial frame of reference: this imposes correlated changes in the transformation between space (x coordinate) and time physical quantities. Therefore the most general transformation compatible with all constraints cannot be squeezed down to a mere transformation of space coordinates. De facto it deals with space-time and this rules out the galilean transformation.

    No doubt, injecting Maxwell's equations as a constraint has the same effect: the transformation will induce correlated changes in space (x coordinate) and time physical quantities and this also rules out the galilean transformation. However, injecting Maxwell's equations leads to the same pattern as Einstein's second postulate insofar the set of conditions which leads to the Lorentz transformation lacks homogeneity: Maxwell's equations relate to a specific range of phenomena whereas the postulate on relativity of motion and the homogeneity / isotropy symmetries encompass all phenomena.

    On the other hand, one may decide to inject laws of the Newtonian mechanics as a constraint, which are incompatible with the perspective of a parameter homogeneous to a velocity remaining invariant. This would also lack homogeneity but more importantly it rules out the perspective of an invariant correlation between changes in space (x coordinate) and changes in time. The most general transformation between inertial frames of reference falls down to a mere transformation of space coordinates, not of space-time. This will lead to the galilean transformation.


    As a conclusion, I think the above shows that:

    i) the two postulates I proposed, complemented with space and time symmetries established empirically lead to either the Lorentz transformation or the galilean transformation, which are exclusive;

    ii) the Lorentz transformation which embraces space-time is more general than the galilean transformation insofar the former reduces to the latter proviso the addition of one constraint, e.g. imposing that time is not affected by the transformation between inertial frames of reference, or imposing that c is infinite, or imposing that simultaneity at a distance makes sense, etc... in which case the Lorentz transformation is ruled out, leaving the galilean transformation as the only possible outcome.

    Overall, I believe that the Lorentz transformation is the most general solution that can be derived from the two postulates I proposed, maximising the impact of empirical symmetries without injecting any additional constraint. This should not come as a surprise since the universality of time is somehow a hidden postulate of the Newtonian mechanics.
     
  4. May 24, 2014 #44

    strangerep

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    Then you're not doing physics. A frame of reference is an abstract construction of a particular observer.

    That phrase doesn't make sense. I guess you mean "a parameter (c) with dimensions of speed".

    You also seem to have used the word "homogeneous" in a distorted way elsewhere in your post. Have you actually studied the Rindler textbook reference I gave? He explains it reasonable detail. (I sense that you are uncomfortable with math, but without math it's all just hand-waving.)
     
  5. May 26, 2014 #45
    I can't see any way to adapt my “first postulate” and the subsequent definition of an “inertial frame of reference” in order to replace “physical objects” with “observers”. However nothing prevents attaching an hypothetical “observer” to each inertial frame of reference as I defined it, but what he/she will actually “observe” will be “distorted” by the Doppler effect over the signals transporting the information about remote events. It depends on what you wish to represent... But you're right, I'm not a physicist.

    Yes, my command of English is rather limited. Does my statement make sense once properly worded?

    I meant that some assumptions whereby Maxwell's equations are valid, or whereby Newton's first law is valid, are less general than other assumptions like the principle of relativity of motion or the isotropy and homogeneity of space. May be non- “homogeneous” was inappropriate to qualify the association of different categories ... But my main reservation in respect to such assumptions is that they appear to bring circularity: they invoke some specific “laws of physics” in order to set / define the mathematical framework into which “laws of physics” should be stated.

    No, I'm quite convinced that the Rindler textbook will fly far above my head, for both maths and physics. But I found your presentation interesting, because it seems to indicate that the above “circular” references can be overcome, the only one which remains according to your post #32 being the definition of the “Einsteinian inertial frame” which still refers to Newton's first law, so that I suggested a different approach for that specific definition through a “first postulate” leading to a new definition of an inertial frame of reference. That's all, I'm afraid.
     
  6. May 26, 2014 #46

    strangerep

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    Yes, in the most general treatment, one need not appeal to other laws of physics to restrict the possibilities. A remarkable amount can be derived just from the concept of preservation of "zero acceleration".

    BTW, the Rindler definition of "Einsteinian inertial frame" I quoted previously mentions "universal time". This is sufficient for ordinary special relativity, but in fact this is relaxed even further in the more general fractional--linear approach. Instead, one works with a weaker assumption: that spacetime (in the absence of gravitation) seems the same to inertial observers anywhere and anywhen.

    Well, I would encourage you to at least look at some of his early chapters before adopting such a defeatist attitude.
     
  7. May 31, 2014 #47
    As long as the definition of an "Einsteinian inertial frame" refers to “inertial observers”, one still needs to provide an acceptable definition for the word “inertial”. The issue at stake is whether this can be done without invoking “laws of physics” such as "Newton's first law" or circular definitions like the "absence of forces".
     
  8. May 31, 2014 #48

    WannabeNewton

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    I'm still at a complete loss as to why we need to talk about the "laws of physics" or any of the other verbiage.

    An inertial frame is simply one which has both zero 4-rotation and 4-acceleration; every excellent textbook on relativity I have ever seen defines it in this way, in terms of kinematical concepts. What's the issue?
     
  9. May 31, 2014 #49

    atyy

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    Couldn't it be possible that spacetime is Lorentzian and flat, but the laws of physics do not have Poincare symmetry?
     
  10. May 31, 2014 #50

    PAllen

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    Because the topic is to derive SR, not assume it. The concept of 4-rotation and 4-acceleration presumes it. A physical definition of inertial frame needs some further physical assumption or experimental finding to select between Galilean spacetime and Minkowski spacetime.
     
  11. May 31, 2014 #51

    WannabeNewton

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    @atyy and PAllen, the distinction between Galilean space-time and Minkowski space-time is the latter presumes a space-like foliation with separate Euclidean temporal and spatial metrics whereas the latter contains no such foliation and presumes the Minkowski metric. From here one can demand that equations (or "the laws of physics") be Lorentz (or Poincare) covariant and get SR. Why are inertial frames required for any of this? Equations will be Lorentz covariant as long as they are written in terms of spinor or tensor representations of the Lorentz group which is a completely frame-independent condition. After the dust settles we can simply define an inertial frame in terms of zero rotation and acceleration. I honestly don't see any need to talk about inertial frames before the dust settles.
     
  12. May 31, 2014 #52

    PAllen

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    The discussion was do you need anything other than a physical principle of relativity+homgeneity+isotropy, to get SR. To apply POR, you need a non-circular physical definition of inertial frames. Then you need something physical (experiment or law) to select SR vs. GR (Galilean relativity, not General relativity).
     
    Last edited: May 31, 2014
  13. May 31, 2014 #53

    atyy

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    Yes, for defining SR, that's the modern way. But the old way using the Principle of Relativity and the speed of light still works.

    Incidentally, were you actually commenting on Sugdub's question whether an inertial frame can be determined without reference to the laws of physics, assuming SR is true? In theory, yes. In practice, no, since one has to build some instruments to measure acceleration, rotation etc. And in calibrating them, the laws of physics will be used.
     
  14. May 31, 2014 #54
    In which way can a mathematical concept such as a coordinate system be physically zero-accelerated? May be you assume its origin remains collocated with a zero-accelerated physical body? Then how can one characterize a zero-accelerated body unless a postulate states that its accelerated or non-accelerated state of motion is an objective property of this body?

    Apart from setting a postulate as suggested above, one necessarily comes back to invoking physical laws, leading to circular statements as per the Wikipedia dedicated article:

    Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid. However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law.... According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, ...
     
  15. Jun 1, 2014 #55

    WannabeNewton

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    Who said anything about a coordinate system having zero acceleration? All I said was the frame has zero acceleration. All this means is the object of interest at rest in the frame has zero acceleration.

    There is no need for such a postulate. No such postulate exists in SR. It is simply a consequence of the definition in both Newtonian and relativistic mechanics.

    http://articles.adsabs.harvard.edu//full/1967QJRAS...8..252D/0000252.000.html

    Also just because inertial frames are defined in a certain way in Newtonian mechanics doesn't mean we need to follow the same tired route in relativity. As atyy mentioned there is a much more coherent and fundamental way to approach SR, as opposed to the antiquated approach taken by Einstein and some of his contemporaries.
     
  16. Jun 1, 2014 #56

    WannabeNewton

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    Ah I see; I probably should have read the entire discourse.

    I don't disagree there.
     
  17. Jun 1, 2014 #57

    strangerep

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    (Not sure whether I should stay involved with this, but... maybe one more post...)

    First, let's replace the phrase:

    "The laws of physics are identical in all inertial frames."

    by the equivalent:

    "The outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame."

    Then we can seek a physical definition of "inertial frame"...

    The task is not to express an observer's local experiences without referring to "laws of physics" (or, equivalently, the "outcome of any physical experiment [...]"). Rather, the task is to relate one observer's experiences to those of others. That's why it's called "relativity". :wink:

    Of course each observer already possesses some physical concepts such as (local) position, time, and devices for measuring such things locally, and hence also a concept of differential ratios thereof (velocity, acceleration, etc). An observer equipped with suitable accelerometers, gyroscopes, etc, can tell whether heshe is accelerating or not. If heshe detects no acceleration, then heshe is an inertial observer. (In this sense, non-acceleration is indeed a property that an observer can ascribe to hisherself.)

    The "inertial reference frame" imagined by an inertial observer is simply an intuitively natural extrapolation of locally performable operations, e.g., moving 1 step to the right, waiting until 1 minute has elapsed according to hisher clock, etc. To be an "inertial motion", such operations must be non-accelerative once completed, meaning that (e.g.,) after the spatial translation of moving 1 step to the right heshe still detects no acceleration.

    Then we assume that (the mathematical expressions of) these operations form a Lie group, since that seems to be the case for strictly local operations, at least as far as they can reach.

    Such local experience can be extended a bit further by the "radar" method, if the observer has a light emitting source (e.g., a torch) and a device for receiving light and noting its direction (e.g., a pair of eyes that implement binocular vision). Such parallax methods allow an observer to relate remote events to hisher imagined reference frame.

    (I'll skip the additional complications/ambiguities that arise beyond the useful range of the radar method or more sophisticated parallax techniques.)
     
    Last edited: Jun 1, 2014
  18. Jun 1, 2014 #58

    strangerep

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    Tsk, tsk. :wink:
     
  19. Jun 1, 2014 #59

    PAllen

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    Well, if you are talking about drawing conclusions from the POR + experiments, before using the Radar method you first have to establish the constancy of light speed (no need to worry about one way / two way if we are assuming isotropy). Having done such an experiment, you already find SR selected rather than Galilean relativity.
     
  20. Jun 1, 2014 #60

    strangerep

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    Actually, I was trying to describe how one might reach the concept of an inertial frame, beginning at a physically plausible starting point. Probably, I should have ditched the radar stuff in my previous post, since it confuses the logic -- as you pointed out.

    [Edit: ... and thank you for pointing it out, btw. :biggrin: ]
     
    Last edited: Jun 1, 2014
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