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Could SR not be built from only one postulate?

  1. May 18, 2014 #21

    bcrowell

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    This is a nice way of stating what's unsatisfactory about Einstein's 1905 axiomatization. It assumes the state of the art in 1905, which was that there were two main theories of physics: Newton's laws and Maxwell's equations. If you want an axiomatization that reads more like the modern view of how relativity works -- as a theory of the geometry of spacetime -- then you probably want something more like Ignatowsky's axiomatization.
     
    Last edited: May 18, 2014
  2. May 19, 2014 #22
    strangerep, thanks very much for your detailed explanation, and the references! Unfortunately, I don't know much algebra so there's some of what you said that was beyond me, not to mention the papers you referenced (I could follow them pretty much through the introduction but nothing more :\ ). I had heard a bit about de Sitter and anti-de Sitter space, but didn't know what it was about. You made that a bit clearer for me, so thanks for that as well!

    Matterwave, that's a good point- there's no guarantee that the current physics is correct either, so it would probably be better for *every* postulate of SR to be stated (like in Ben's book- again, Ben thanks for that, it looks like a very refreshing take on SR :D).

    Thanks for the interesting responses everyone, you've given me a lot to think about.
     
  3. May 19, 2014 #23

    Fredrik

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    I like Matterwave's reply (post #19) the best. You could also say that it implies that the velocity of a massive particle influenced by a constant force must satisfy the formula ##v=(F/m)t+v_0## in every inertial coordinate system. This implies that c is not the same in every inertial coordinate system.

    I have a lot more to say about this subject, but unfortunately I don't have time right now.
     
  4. May 19, 2014 #24

    atyy

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    As Matterwave, DaleSpam and others have pointed out - you need to specify what the "laws of physics" are. If the laws of physics include Maxwell's equations then 2 is contained in 1. If Maxwell's equations are not in the "laws of physics" then 2 is not contained in 1.

    The reason that Maxwell's equations are stated explicitly in 2 in most books is that for many years between Newton and Maxwell, Maxwell's equations were not in the "laws of physics". For example, if Newton's universal gravitation but not Maxwell's equations are in the "laws of physics", then 1 alone would produce Galilean relativity.

    Whether you want to count the axioms as 1 or 2 is a matter of taste, depending on what you include in the "laws of physics".
     
  5. May 19, 2014 #25

    xox

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    The above is an excellent way of stating the answer.
     
  6. May 19, 2014 #26
    Yeah, I liked that. But then there's the problem of subtleties that Ben mentioned in his book, isn't there?
    I think I prefer strangerep's way of looking at it best; as far as I can tell, it doesn't rely on the rest of physics, and is very precise. However, it seems like a much more mathematical way of looking at the problem and I feel there should be more physics (like in Ben's approach).
     
  7. May 19, 2014 #27

    atyy

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    Yes. The most modern way is to simply postulate that Poincare symmetry is a symmetry of the laws of physics (whatever those may be).

    However, it is worth remembering the old ways, especially because the Principle of Relativity ("axiom 1" in the old way) goes all the way back to Galileo, and is still very useful. It says one can drink coffee in an aeroplane that is moving very fast, just as well as on the ground.

    Also, in Einstein's formulation of general relativity, the Principle of Relativity can be said to fail as a global principle, but hold as a local principle. This "hold as a local principle" is the Principle of Equivalence, which again goes back to Galileo: bodies of different masses (as long as their mass is small relative to the earth's) will fall and reach the ground at the same time.

    So GR can be seen as reconcilation of 2 important "principles" of Galileo, and the fact that both Newtonian gravitation and Maxwell's equations are "laws of physics" in some regime.

    Edit: I'm not sure I agree with Ben's criticism of Einstein postulates in 2.4.1. of http://www.lightandmatter.com/sr/. It is true that special relativity can be handle accelerated frames. However, one can think of the first postulate as stating the existence of global inertial frames. Stating the postulate in this way does not depend on not being able to handle accelerated frames, but merely states the existence of a special class of frames which we call "inertial". I feel that Ben's criticism based on accelerated frames is not valid criticism of Einstein's SR postulates, but is valid criticism of Einstein's (initial) postulates for GR.

    I do agree with Ben's criticism that "the speed of light is the same in all inertial frames" is a slightly less general postulate than is possible, in the sense that if the photon were found to have a mass, then the speed of light would not be the same in all inertial frames. However, special relativity (Poincare symmetry of the laws of physics) could still hold, even if the photon were found to have a mass.

    Roughly speaking, the Principle of Relativity says global inertial frames exist. However, we know that Newton's law of gravitation also obeys the Principle of Relativity - Galilean relativity. So to specify special relativity, we must add either (a) Maxwell's equations are a law of physics, or (b) speed of light if the same in all inertial frames, or (3) Poincare symmetry is the symmetry of the laws of physics.
     
    Last edited: May 19, 2014
  8. May 19, 2014 #28

    Meir Achuz

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    I just noticed this thread. The best single postulate for relativity was almost that of Galileo in 1632:
    "any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments".
    If he hadn't been so careful to include the word "mechanical", his quote would include special relativity.
    Removing that word leaves a better single postulate than the two of Einstein.
    Since measuring the speed of light is an experimental measurement, no additional postulate is necessary.
     
    Last edited: May 19, 2014
  9. May 19, 2014 #29

    PAllen

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    No, you are making the assumption that c is frame invariant. Newton would have proposed that, like bullets, light speed would be frame variant in precisely the right way for the Galilean transform to be valid. You cannot arrive at the Lorentz transform without some additional fact. You could add the second postulate as an experimental fact (circa 1900, not much earlier) rather than a postulate, but either way it has to be added (or something equivalent, e.g. Maxwell's equations). Let me add, since the speed of bullets is arbitrary, you would have to go from measurements seeming to come out the same for light to proposing that it is (or follows from) a law that it is constant.
     
    Last edited: May 19, 2014
  10. May 19, 2014 #30

    robphy

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  11. May 19, 2014 #31
    No, not along my line of thinking but interesting nevertheless- at least the parts that I could understand.

    Nice 'signature', by the way :D.
     
  12. May 19, 2014 #32

    strangerep

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    Well, the physical content (using only a very small set of intuitive assumptions), and its mathematical development proceed along the following lines:

    Following Rindler [1],

    Postulate 1 (Principle of Relativity):
    The laws of physics are identical in all inertial frames,
    or, equivalently,
    the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame.

    This requires a definition of inertial frames. Still following Rindler, an Einsteinian inertial frame is a reference frame in which spatial relations, as determined by rigid scales at rest in the frame, are Euclidean and in which there exists a universal time in terms of which free particles remain at rest or continue to move with constant speed along straight lines (i.e., in terms of which free particles obey Newton's first law).

    The boundary between the physics and the maths lies in this: an observer can reasonably possess local length scales (a very short rigid rod), and a local clock (measuring short time intervals). That much is physical. One then imagines that the rod could be successively laid end-over-end indefinitely to create a spatial coordinate grid. Similarly, one imagines that the clock could be duplicated endlessly, with the duplicates moved to spatially remote locations.

    The (abstract) space of dynamical parameters needed to describe such an arrangement is assumed to correspond to (possibly a subspace of) ##R^4##, i.e., 3 space and 1 time. Similarly, velocities are assumed to correspond to (possibly a subspace of) ##R^3##. Thus we imagine a 7-dimensional velocity extended phase space of parameters. (We need not extend any further to higher dimensional phase spaces involving acceleration, jerk, etc, since the requirement of inertial motion restricts acceleration to be zero.)

    These imaginings are made more precise using group theory. E.g., the basic physical spatial displacement defined by the rod is expressed in terms of a transformation of these dynamical parameters. Demanding that such transformations preserve inertial motion (i.e., that zero acceleration is mapped to zero acceleration), and that they form a Lie group, one can derive quite strong restrictions for the possible form that such transformations may take.

    It's similar for temporal displacements, and velocity boosts (preserving the origin). One takes the basic physical operation of a small temporal displacement expressed via a local clock, or a velocity difference (holding the spatiotemporal origin invariant), expresses these as transformations, and imposes the same group theoretic requirements when composing multiple such transformations.

    Summarizing, the physical content consists of the concept of inertial motion of an observer, and the availability of means for measuring very local spatial displacements, very local temporal delay, and relative velocities of other such observers who are momentarily at the 1st observer's origin. One also assumes spatial isotropy: that there is no preferred spatial direction.

    Then one asks for the most general transformation of the parameters in the (abstract) velocity phase space which preserve the condition of zero acceleration. That encapsulates all the ways in which 2 inertial observers could be labelling the 7-dimensional abstract phase space in "different" (though physically equivalent) ways. The parameters of the transformation between 2 such observers may then be identified as the physically meaningful relative observables (spatial displacement, temporal delay, relative velocity, spatial orientation, etc). One naturally adopts an additional principle of "physical regularity": that finite values of these relative observables must be mapped by the transformations to finite values. That restricts the allowable transformations a bit further.

    Then it's just a matter of grinding through the math of Lie group theory applied to such transformations to find the most general possibility, as I described earlier. The method is firmly grounded in realistic physics, which can be expressed quite concisely. The detailed math is extensive, of course, taking many pages if one performs all calculations explicitly. But overall it's a good thing: from a small set of physical concepts based on intuitive local operations, one derives an extensive theory. The possibility of an invariant limiting speed constant, and an invariant length constant, emerge as derived consequences. Only their values need be determined experimentally.


    References:

    [1] W. Rindler, Introduction to Special Relativity,
    Oxford University Press, 1991 (2nd Ed.), ISBN 0-19-85395-2-5.
     
  13. May 20, 2014 #33

    Fredrik

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    I will briefly describe my view on the postulates and "derivations" of the Lorentz transformation here.

    The postulates are nowhere near as significant as most texts will make you think. It should be emphasized that they're not even part of the theory. SR is defined by (purely mathematical) definitions of terms like "Minkowski spacetime" and "proper time", and a few correspondence rules that tell us how to interpret the mathematics as predictions about results of experiments.

    The "derivations" of the Lorentz transformation that start with the postulates are certainly interesting and fun, but they shouldn't be viewed as proofs. They should be viewed as ways to guess how to define a new theory, or rather, a new framework in which to define theories. Once we have defined the mathematics of the theory properly, we can prove theorems that resemble the postulates.

    The proper way to turn the "derivations" into actual derivations (i.e. proofs) is to first interpret the postulates as mathematical statements. Then you can take those statements as the starting point of a proof. The question is, what are we really proving? There's no obvious answer to the question of what mathematical statement best corresponds to the principle of relativity. So we still won't be able to say that we have (rigorously) derived the Lorentz transformation from the postulates. We have derived it from one mathematical interpretation of the postulates.

    Because of this, I prefer to do those "derivations" in a way that's not completely rigorous, and to use language that indicates what parts of the argument are really just clever guesses. See e.g. this post. (Start reading at the line that starts with "The explicit". The "numbered statements" that I'm referring to in that post are the postulates).

    I also think that some of the theorems (with rigorous proofs) that take a mathematical interpretation of the principle of relativity as a starting point are very interesting. The ones I've looked at can be interpreted as saying that SR and Newtonian mechanics are the only possible theories of physics in which ##\mathbb R^4## is the underlying set of spacetime (the mathematical structure that represents real-world space and time), and inertial coordinate systems can be defined on that set.
     
  14. May 23, 2014 #34
    It would seem that the physical world can be split into two classes of objects: those which have an inertial state of motion and those which have a non-inertial state of motion. Neglecting gravitational effects, the Lorentz transformation of SR could be essentially derived from two postulates:

    1) the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”;

    2) all inertial frames of reference are physically equivalent, so that the laws of physics (i.e. the internal rules belonging to a physics theory) are invariant through a change of the inertial frame of reference.

    Is this correct?
     
  15. May 23, 2014 #35

    PAllen

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    For (2), the result you get depends on what laws of physics you include. If you use Newtonian machanics, you get the Galilean transform from these assumptions. If you use Maxwell's equations, you get the Lorentz transform. Note, part of the genesis of SR was working out what dynamical laws of mechanics replace Newtonian mechanics. If you include as the laws of physics Newtonian mechanics + Maxwell, you have a self contradictory system.
     
  16. May 23, 2014 #36

    Meir Achuz

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    ...or Newtonian mechanics is wrong, and not a law of physics. That is why it is better to say something like 'all experimental results'.
     
  17. May 23, 2014 #37
    I'm trying to elaborate on Rindler's approach (I assume it is valid and recognized) as presented by strangerep in #32 according to which it seems possible to derive the Lorentz transformation of SR, referring neither to the invariance of the speed of light nor to Maxwell's equations, from constraints applying to the transformation between inertial frames of reference, in particular the need for transformations to “preserve inertial motion (i.e., that zero acceleration is mapped to zero acceleration), and that they form a Lie group”. I find this approach interesting and the purpose of my contribution is to challenge my own understanding of Rindler's approach. Actually I assume it entails as well that finite accelerations map onto finite (non-zero) accelerations, hence the first postulate I suggest:

    First postulate: the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”.

    The advantage I see is that this postulate caters for a definition of an “inertial frame of reference” which avoids the circularities of usual definitions: on the one hand they refer to Newton's first law which refers to the absence of forces which refers to inertial motion (first circularity), and on the other hand they refer to a law of physics (Newton's first law) in order to define the framework into which laws of physics should be formulated (second circularity).

    However your comment shows that my second postulate is inappropriate. So let me replace it with something less ambiguous, as follows:

    Second postulate: the difference between the rest state and an inertial state of motion is arbitrary, so that all inertial frames of reference (as defined above) are physically equivalent. Therefore the laws of physics (i.e. the internal rules belonging to a physics theory) which hold in a given inertial frame of reference should also be valid in any other inertial frame of reference. This in turn implies that their mathematical formulation shall remain invariant through the transformation between inertial frames of reference.

    Hopefully the above is only a rewording of the constraints presented in #32. Assuming this can lead to the Lorentz transformation (as per my limited understanding of Rindler's approach), I doubt it can equally lead to the galilean transformation unless an additional constraint is imposed whereby the time coordinate remains invariant through the transformation.
     
  18. May 23, 2014 #38

    PAllen

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    No, the reasoning in #32 only establishes the either Galilean transform or Lorentz transform with some invariant speed TBD are the only possibilities. It is then either choice of what you consider to be laws (if you approach this axiomatically) or what experimental data you have that selects between Galilean and Lorentz, and also determines what the invariant speed is.

    [edit: I think you fail to grasp the last sentence of #32:

    "The possibility of an invariant limiting speed constant, and an invariant length constant, emerge as derived consequences. Only their values need be determined experimentally."

    This says, based on the POR and the additional plausible assumptions and much math, you get some invariant speed as a possibility. The other possibility is no invariant speed = Galilean relativity.]
     
    Last edited: May 23, 2014
  19. May 23, 2014 #39

    strangerep

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    That doesn't really make sense.

    For present purposes, one only needs the physical input that an observer can distinguish between his own accelerated/unaccelerated state -- hence that an equation like $$\frac{d^2 x}{dt^2} ~=~ 0$$is applicable to all unaccelerated observers. One then seeks the maximal set of symmetries of that equation.

    If one admits non-zero acceleration, then one must begin with a much larger abstract phase space. E.g., for constant acceleration, the phase must be enlarged to 10 dimensions, and the group of possible transformations is much larger than fractional-linear -- since one is now preserving an equation of motion like $$\frac{d^3 x}{dt^3} ~=~ 0 ~.$$The maximal symmetry group of this equation has been (partially) investigated, but I'm not sure if anything physically-useful has come out of that.

    For non-constant accelerations, the phase space becomes ever larger, maybe even infinite-dimensional for truly arbitrary acceleration.

    To understand this setting properly, one needs to know about continuous symmetries of differential equations. There's a significant body of theory about this -- I found the textbook of Stephani [1] quite useful. (The case of inertial motion symmetries is actually an exercise in one of his early chapters: the generators derived in that exercise correspond to fractional-linear transformations when integrated).

    Note that I only quoted Rindler insofar as his 1st postulate and definition of inertial frame. Although he does perform a group-theoretic derivation of the Lorentz transformation, he adds an extra postulate about homogeneity, which restricts fractional-linear transformations down to linear transformations. The more general case is discussed in the other references I gave.

    Actually, all these possibilities are contained in the most general (fractional-linear) approach. In the limit where spatial distances are small compared to the universal length constant, one recovers the Poincare transformations, i.e., standard special relativity. Then, in the limit where relative speed is much smaller than ##c##, one recovers Galilean transformations.

    ---------------
    References:

    [1] H. Stephani,
    Differential Equations -- Their solution using symmetries,
    Cambridge University Press, 1989C, ISBN 0-521-36689-5.
     
    Last edited: May 23, 2014
  20. May 23, 2014 #40

    atyy

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    I prefer to state specifically that Maxwell's equations are a "law of physics", or to assume that Poincare symmetry is a symmetry of the "laws of physics". If one says all experimental results (to date), that includes GR with accelerated expansion of the universe, in which case SR does not hold.
     
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