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Could SR not be built from only one postulate?

  1. Jun 2, 2014 #61
    Yes, however one cannot demonstrate that the devices called “accelerometer”and “gyroscope” actually measure “accelerations” and “changes in orientation” without invoking somehow the laws of physics. Such devices can provide a more accurate assessment of our state of motion than our senses, once it has been demonstrated that they are fit to purpose, but beforehand the need remains for an independent definition of “inertial” (see below).

    I agree with many of your statements which I find better than mine, in particular the need to refer to a consensus between observers. However I think a more logic presentation can be proposed if the postulates leading to SR are set at a deeper level, as follows.

    This is not derived from your assumptions, it is a postulate, the first postulate from which everything will flow: my sensation reveals an objective property, a qualification upon which all observers will agree. Then, assuming I do not sense any acceleration or rotation in my body, my state of motion can be defined as “inertial”. Hence a 4-coordinate system attached to my body and reflecting my time flow provides an inertial frame of reference. I think this definition is immune from contradictions whilst removing the indirect reference to laws of physics (via "accelerometers").

    A second postulate is required in order to derive the Lorentz transformation (assuming no further constraint will reduce the generality of the development alongside Rindler's approach), whereby the difference between representing oneself at rest or in constant motion is non-objective, conventional.

    Thanks to the first postulate, the equivalence relationship “to be in constant relative motion” structures the family of all possible frames of reference in such a way that the associated transformation will map an inertial frame onto an inertial frame. Thanks to the arbitrariness set by the second postulate, one may conclude that:

    In conclusion two independent postulates are required for developing SR, which are of a more general nature that those often proposed.
     
  2. Jun 2, 2014 #62

    Fredrik

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    There's no significant difference between specifying that human senses should be used and specifying that an accelerometer (defined by instructions on how to build one) should be used. All you have done is to use a different device to detect acceleration.
     
  3. Jun 2, 2014 #63

    PAllen

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    That's fine so far. But to conclude SR is true rather than Galilean relativity, you must do some experiments that distinguish them. Your distinguishing experiment need not have anything to do with light, which you may consider a virtue of this approach (e.g. it could be time dilation).
     
  4. Jun 2, 2014 #64
    I'm not sure how to interpret your statement. In the following I will assume you mean that a reference to actual experimental results is necessary before one can state that the postulates we are dealing with lead to the Lorentz transformation of SR as opposed to the Galilean transformation of the Newtonian mechanics.

    If this is the case I disagree. I already explained why in a previous input but I welcome comments.

    The Lorentz- or the galilean- transformation are the only possible outcomes but they are exclusive. So the choice between both must be sorted out before reaching any conclusion. On the one hand, the Lorentz transformation necessarily deals with a 4-dimensions space-time coordinate system since it entails a dependency between space coordinates (typically x) and the time coordinate. This dependency is constitutive of SR: it cannot be eliminated without abandoning the theory.

    On the other hand, the Lorentz transformation gets squeezed down to the galilean transformation by negating this dependency between space and time so that only a 3-dimensions space coordinate system is required, complemented with a separate, independent, invariant 1-dimension time coordinate system. This reduction from 4 to 3+1 dimensions corresponds to a loss of generality. This loss can be triggered by various hypotheses or constraints acting on top of the postulates, for example by imposing that the time coordinate is left invariant.

    The Lorentz transformation appears to be the most general solution for the required transformation, the only one which deals with a 4-dimensions integrated coordinate system. Since both possible outcomes are exclusive, a conclusion in favor of the galilean transformation could only be reached following the conscious acceptance of a reduction of generality, e.g. by adding that the time coordinate remains invariant or equivalently by adding the possibility for a signal propagating at an infinite speed, or by adding that simultaneity at a distance is a reality. The addition of any of the above constraints on top of the postulates will trigger the derivation of the galilean transformation. Otherwise, the Lorentz transformation will be arrived at.
     
  5. Jun 2, 2014 #65

    PAllen

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    That just becomes another assumption. In addition to POR, isotropy, homogeneity you add: of the two remaining possibilities, pick the one you like. Any 'natural philosopher' of circa 1800, brought up to speed on the math, would say you obviously want to add a postulate of simultaneity to rule out the nonsensical alternative (what we call SR). In fact, Newton had such postulate: time flows equably and consistently for all observers. Unlike for symmetries, I don't see any convincing argument for one conclusion over another except experiment.
     
    Last edited: Jun 2, 2014
  6. Jun 5, 2014 #66
    Indeed it was this a priori assumption which prevented physicists looking for a transformation of 4-coordinate events. Once Einstein understood that his postulate on the invariance of the speed of light was incompatible with this a priori assumption, he could set more general 4-dimensions equations in view of producing a genuine transformation of space-time events, … and later on he understood that a further generalization could encompass the non-uniform gravity field... This is the way science is progressing.

    Imposing the invariance of the time coordinate also acts as an asymmetry since it restricts the transformation to 3 out of 4 coordinates of any event. There is no fundamental difference.
     
  7. Jun 5, 2014 #67

    PAllen

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    How is "no time invariance" fundamentally more natural "time invariance"? Note, time is distinguishable in SR (the geometry is pseudo-Riemannian, not Riemannian). Or you can say, manifolds are preferable to fiber bundles (?).

    Also, note that the value of c only chooses units not physics. Thus, there exactly two physically distinguishable choices to make (there is or is not an invariant speed), not infinite versus one. You can choose "time invariance" or "c invariance". You are not making headway convincing me that anything other than experiment chooses SR over Galilean relativity.
     
  8. Jun 5, 2014 #68

    strangerep

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    :smile:
     
  9. Jun 5, 2014 #69

    WannabeNewton

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    :cry:
     
  10. Jun 10, 2014 #70

    Yes, I agree that in order to derive the Lorentz transformation a finite constant (c) with the dimension of a speed must somehow be injected according to which “instantaneous actions at a distance” get excluded. Obviously my previous inputs overlooked it. Thanks for this lesson.

    Still I wish to challenge the rationale for invoking either a “law of physics” (such as Maxwell's equations or the “invariance of the speed of light”) or some experimental results (such as the Michelson and Morley experiment) as a valid foundation for the formal derivation of the Lorentz transformation. In my view, statements about the world, how it is, how it works, what happens there,... are just inappropriate. We should not accept any a priori statements in this range.

    SR provides a formal framework into which a model / description / simulation of the world and associated phenomena will get hosted. The purpose of that framework is to specify how our formal description of the same phenomena should be evolved when we change the perspective from which this description is proposed. The SR framework should be physically neutral, it should provide an empty structure, in the same way as in GR the actual curvature of space-time relies upon the effective presence of energy or mass.

    The two postulates I have proposed for SR are not about the world, they are about us: we sense accelerations and rotations whereas we do not sense speed or rest. The symmetries we have discussed are not about the world itself, they deal with our a priori concepts of space and time: we only grasp differences in position, in orientation, in time, not their absolute values ... But the addition of c as an external constraint, somehow linked to a belief in the existence of a “law of nature”, in order to complement this set of postulates and symmetry rules does not fit well. It has no bearing to the meta-rules which the SR formal framework must comply with.

    Actually we all know why c is necessary, why “no signal can travel at an infinite speed”, why we must impose this constraint on our formal framework. It is not a postulate about the world and neither an external constraint derived from experiments. It reflects the causal structure we impose to any abstract construction deserving to be labelled as a “physics theory”. We can't accept that our theories claim “explaining” phenomena through “instantaneous actions at a distance”. Causes and effects must be ordered in time otherwise they can't be distinguished from each other. As long as our physics theories abide to some concept of causality, they must fit within a formal framework imposing a maximum limit for the speed of any signal invoked in a causal explanation.

    My conclusion is that we impose the existence of c as a consequence of our own internal mental structure, it is not imposed to us by external experiments and neither by some miraculous knowledge about the world. Comments are, of course, welcome.
     
  11. Jun 10, 2014 #71

    strangerep

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    It is not imposed as an "external constraint, linked to a belief [...]". Only its value is determined by experiment.

    The Galilean case is simply an approximation of what happens in the Poincare case as ##|v|/c## becomes small. (Most people just say "as ##c## becomes large", but it's better to have a dimensionless quantity when taking limits.)

    In this sense there are not 2 separate cases, but only one -- and it comes with a universal invariant speed ##c##, whose value must be determined by experiment.

    Rubbish. If someone puts an axe through your skull, the causal consequence (your death) is not dependent on your mental structure (i.e., it doesn't depend on whether you're awake, asleep, or in a coma).
     
  12. Jun 10, 2014 #72

    PAllen

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    I don't quite agree with this. Infinity is not a value, so the Galilean case is more properly viewed as the alternative where there is no invariant speed. As for different values of c, IMO this is just units. There is no way to physically distinguish between different values of c without fixing many other things. Thus, for me, the two choices are between a pseudo-riemannian metric of (1,-1,-1,-1) on a 4-manifold vs. a fiber bundle with invariant t and invariant distance (in the Euclidean 3-manifold). These two (no more) physically distinguishable options fall out of the derivation.
     
  13. Jun 10, 2014 #73

    Fredrik

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    I think of what we're doing simply as finding all groups of permutations of ##\mathbb R^4## that take straight lines to straight lines, so to me it makes the most sense to acknowledge that for each such group, there's a set of lines that aren't just taken to straight lines, but are invariant under transformations that preserve the origin. In the case of Galilean transformations, these are the lines that are drawn horizontally in a spacetime diagram. It makes sense to think of them as representing motion with infinite speed.

    That's the physical way of looking at it. (Nothing wrong with that of course :smile:). The mathematical way is that the groups with different positive values of ##c^2## are all isomorphic. The ones with negative values of ##c^2## have to be ruled out by other methods. In the 1+1-dimensional case, it's sufficient to assume that 0 is an interior point of the set of velocities associated with the elements of the group, i.e. that there's an open interval containing 0 such that for each v in that interval, there's a transformation with velocity v.
     
  14. Jun 10, 2014 #74

    strangerep

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    In the derivation, the constant emerges most naturally as having dimension of inverse speed squared. So one could just as easily think of it as taking a limit as an inverse speed constant approaches 0, (which is certainly a number). :smile:

    Physical experiments can only give results within error bounds applicable for the apparatus. E.g., an apparatus with insufficient accuracy to probe relativistic scales could only say that the value of "c" is greater than <whatever> (expressed in terms of local length and time standards, of course).

    (Strictly speaking, there are certainly cases in mathematics where a sequence of elements of an abstract space ##S## converges, yet the limit is not in ##S##. But in our case, ##S## is a 1-parameter space of groups of linear transformations of the solution manifold of ##d^2x/dt^2=0## (with ##c## being the "parameter" in ##S##). I don't think such subtleties affect this case unless one invokes some more subtle topological issues. But I could be wrong about that.)
     
    Last edited: Jun 10, 2014
  15. Jun 11, 2014 #75

    PAllen

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    But once you go from looking just at transforms to mathematical structure, there are just two possibilities. There either is or isn't a pseudo-riemanian manifold, and if there is, it can always be given the metric with determinant -1. Here we do have the case that the limit is not a Minkowski space, while all the other elements are.

    As for 1/speed squared, that is 1, in natural units. Then , the limit is 1.
     
  16. Jun 11, 2014 #76

    strangerep

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    Indeed.

    This reminds me of an observation that Gerry Kaiser made a long time ago. If one performs the Inonu--Wigner contraction while remaining in a Poincare irrep, one gets instead the centrally-extended Galilei group, including canonical (position--momentum) commutation relations like those of QM. This is very different from what we get from a "naive" contraction to ordinary Galilei. Yet, (afaik), the centrally-extended Galilei group doesn't show up if we start from the POR alone, but only if we go to Poincare and then contract within an irrep. [IIUC]

    Evidently, one gets different answers depending on whether the group's homogeneous space is considered more (or less) important physically than its irreps. (QFT would suggest that the irreps are more important.)

    I recall that there can be subtleties in the distinction between representation--contraction vs abstract group contraction, but I'm no expert on that.

    Yes, but that just shows that one cannot usefully perform the Inonu--Wigner group contraction (i.e., Poincare->Galilei) in that way.
     
    Last edited: Jun 11, 2014
  17. Jun 11, 2014 #77
    I will confess that I skipped to the end, I have another lecture this evening. If I missed this point being introduced, I apologize. All of this hinges on one other assumption as yet unclaimed - that of invariant physical space. If in fact any of the dimensional constants change - as they do, I have been led to believe - then the two postulates cannot be combined into one. The proposal suffers from one other deficit, in that they are co-dependent. SR was ever meant to measure conditions in local groups, and in fact could never state with any authority that these conditions held at every point. Those ideas are left over from Newtonian Era thinking - which admittedly still serves for all kinds of ballistics problems.

    During my skim, I remember seeing a statement to the effect that all infinities are equal. In fact, they are not. It matters a great deal HOW you got there, and how fast. Physics hates infinities, and except as a concept, there is no proof they exist (a geometry proof notwithstanding). You can calculate with them, and you can avoid calculating with them. If you have performed a division by zero, umm, look for an error somewhere. I remember as a lark once in high school physics I ran some calculations with infinities, but labeled them. I got some interesting results. Some of those equations showed up in college texts later. No notes, sorry. The same went for imaginary numbers, which I was told I could safely ingnore, since it 'didn't make sense to carry it through'. Twistor theory.

    The bottom line, I think, is to carefully examine your base assumptions, or at least state ALL of them.
     
  18. Jun 12, 2014 #78

    strangerep

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    That's unwise -- it's too easy to make yourself look silly. :yuck:

    If you had studied earlier posts in this thread you might have seen some where I tried to make a distinction between an abstract space of certain parameters (chosen by reference to an inertial observer's experience) used to express the free equations of motion, and finding the maximal symmetry group thereof. The more concrete notions of relative physical quantities emerge as parameters in those symmetry transformations.

    So assumptions about the enveloping parameter space were not "unclaimed" -- but merely implicit in the use of the theory of dynamical symmetry groups.

    Really? I bet you don't have experimental evidence of variability of the local speed of light in vacuum.

    Again, if you had studied the whole thread you might have seen that issues surrounding spacetime homogeneity were not ignored. Without spacetime homogeneity, one gets a more general theory (known as "projective" or "de Sitter" relativity). Adding the assumption of homogeneity reduces this to the usual Poincare relativity.

    Really? I don't remember that. If you see silly statements like that you should reference them in a quote box. Anyone who has done some university--level pure math knows that there is more than one kind of infinity.

    These are just sweeping statements, conveying little meaning. How one "got to infinity"?? What are you talking about? (And if you answer, make sure it remains on-topic for this thread.)

    No idea what you're talking about here. One certainly can't "ignore" imaginary numbers in modern theoretical physics.

    Is this just a random phrase or did you have a point about twistor theory?

    Actually, the "bottom line" is that you should study nontrivial threads properly before relieving yourself in them by ill-informed brain--f*rts. :grumpy:
     
    Last edited: Jun 12, 2014
  19. Jun 12, 2014 #79

    Fredrik

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    While it's true that there's more than one infinity in math, I don't think it's relevant here. A line in a spacetime diagram is either horizontal or not.
     
  20. Jun 12, 2014 #80
    I agree. However, I also agreed to be neutral and productive. In that spirit, that is the only thing I'm going to say about the matter of the rest of the of the response.

    Sorry - as you say - I do look silly by not reading all of the intervening text. Did, at some point, someone come to the conclusion that SR could not be built from one postulate? Because there seem to be a great number of constraints.

    My original statement;

    -->If in fact any of the dimensional constants change - as they do, I have been led to believe -

    And my response; no, I have not performed any experiments that show the variability of the speed of light under any conditions. In a vacuum is deliberately left out. In fact, I will go out on a limb and say that it has shown itself to be remarkably steady in the brief period that we have been measuring it. Some other conditions may support a longer (unobserved) flat value. But, I would not have made the interjection without at least some knowledge of the opinions of others who had performed experiments and come to the conclusion that some of the constants were 'drifting'. I have no knowledge of their equipment nor experimental method. I do, however, have a thought experiment whereby the speed of light could change and you could not measure it.

    --->During my skim, I remember seeing a statement to the effect that all infinities are equal. In fact, they are not.

    I have a little bit of trouble with that for at least two reasons. The first, and primary one is the process of arriving at one's answer is often non-commutative. Another is citing university math in support of a statement and then not connecting the two. The third is sniping and then admonishing.
     
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