The Light Speed Postulate and its Interpretation in Derivations of the Lorentz Transformation

Discussion in 'Pseudoscience' started by tsmid, Apr 24, 2016.

  1. Neddy Bate Valued Senior Member

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    1,727
    We are only interested in one event located at x2, and it is the event represented by (x2, t2). You might as well start representing events with both spacial and temporal coordinates, since you are not making any progress trying to do everything your own way.

    And we are also only interested in one event located at x1, and that is the event represented by (x1, t1). Of course, by definition, (x1', t1') must be the same event as (x1, t1), where one representation is recorded by system 1, using its own coordinate system and clocks, and the other representation is recorded by system 2, using its own coordinate system and clocks.

    And because those two systems are in relative motion to each other, it turns out that x1' is not numerically equal to x1 or x2, and it also turns out that t1' is not numerically equal to t1 or t2. What the light postulate really requires is that t1'=x1'*c.

    If you were correct in your wild assumptions, the LT would have been x1=x1' and t1=t1' and x2=x2' and t2=t2' but they most certainly are not!

    The class 1 event (x1, t1) occurs in system 2 at (x1', t1') where x1' is not co-located with x2, and therefore t1' is not equal to t2. This does not violate the light postulate provided x1'/t1'=c.
     
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  3. tsmid Registered Senior Member

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    That doesn't answer my question: assuming you have a continuous row of detectors in frame 1 that react to the emission of the light signal, you have a corresponding continuous progression of these event locations x1' in frame 2. I was asking you to give the functional form of this progression in terms of the system time t2.
    I just want one equation

    x1'(t2)=??
     
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  5. Neddy Bate Valued Senior Member

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    Your notation does not make sense to me, because x1' is a function of x1 and t1. It is not related to t2 at all. And the moment when the light reaches x1' in system 2 is t1' which is not equal to t2.

    The closest thing I can think of to what you might be looking for is x1'=c*t1' where t1' is the time in system 2, just as x1' is the location in system 2.

    You seem to be trying to generalise the time in system 2 to simply be t2 at all times. But according to what we've been discussing up until very recently, t2 is only one particular moment in the time of system 2, namely the moment when the light reaches one particular location x2.

    I thought you agreed much earlier in the thread that x1 and x2 could be considered to be the endpoints of two rods of identical proper length, each one at rest in each system. Now you seem to be trying to generalise x1 to be the entire x axis of system 1, which makes x1' the entire x axis of system 2. Why don't you concentrate on the two events at the ends of the rods instead of needlessly complicating things with an infinite number of events?
     
    Last edited: Aug 26, 2016
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  7. Neddy Bate Valued Senior Member

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    Sorry, I meant to write t1'=x1'/c.
     
  8. tsmid Registered Senior Member

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    I am sure you are not suggesting that the light speed postulate only applies to events related to detectors at the endpoints of the rod but not potentially at any other points. That's why one should take x1 and x2, as well as t1 and t2 as variables, after all they are variables as they don't have any specific numerical values. The clocks in frame 1 do not show just one specific time, but, as clocks should do, they progress in time, and with it progresses the location of the light signal i.e. x1(t1)=c*t1 as indicated by the detector responses at these locations. As you indicated yourself earlier, the sequence of events associated with this will also be a sequence of events in frame 2. But in order to indicate that this sequence is related to the frame 1 detector events we use the location variable x1' rather than x2. And if we want to work out the speed which which this location progresses in frame 2, we must record these locations as a function of the system time t2 i.e. x1'(t2). The system time in frame 2 is not t1' (whatever that may be). The frame 2 observer has no knowledge about any other clocks other than his own i.e. the time t2. He can only record the locations x1' in terms of this time.
     
  9. Neddy Bate Valued Senior Member

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    1,727
    That is correct, but when the variable x1 essentially refers to the proper length of the rod, then the equation t1=x1/c represents only to one specific moment in time in frame 1, namely the moment when the light reaches the endpoint of the rod. That is one event which can then be transformed from system 1 coordinates (x1, t1) to system 2 coordinates (x1', t1'). Note that here x1' represents one specific location on the x axis of system 2, and here t1' represents one specific moment in time in system 2.

    If instead you want to change that and make it so that ALL of the points that the light passes on the x axis are all called x1 progressively, and that the equation t1=x1/c represents all moments in time in frame 1, then you have an infinite number of events. Now it becomes difficult to talk about any ONE of those events, because if I try to say that I would like to transform from system 1 coordinates (x1, t1) to system 2 coordinates (x1', t1') now we no longer know WHICH EVENT AM I TALKING ABOUT?

    Sorry, but I'm not going to switch from trying to get you to understand 3 perfectly well-defined events, to an infinite number of poorly defined events. I'll pass.
     
  10. rpenner Fully Wired Valued Senior Member

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    Well, to the extent (x1,t1) is the generic name for an event on the x1 = c × t1 half-line where t1 > 0, we may Lorentz transform the line via the algebraic methods of substitution.

    If our transform is x' = (x + v t)/√(1 − v²/c²), t' = (t + v x/c²)/√(1 − v²/c²) where v is some constant such that –c < v < c, then we have

    x1 = c × t1 [given]
    x1' = (x1 + v t1)/√(1 − v²/c²) , t1' = (t1 + v x1/c²)/√(1 − v²/c²) [via substiution x→x1, x'→x1', t→t1, t'→t1' ]
    x1' = (c × t1 + v t1)/√(1 − v²/c²), t1' = (t1 + v c× t1 /c²)/√(1 − v²/c²) [via substiution x1 → c × t1 ]
    x1' = c × t1 (1 + v/c)/√(1 − v²/c²), t1' = t1 ( 1 + v/c) /√(1 − v²/c²) [via distributive property and definition of division ]
    t1 = t1' √(1 − v²/c²) / ( 1 + v/c) [via distributive property and definition of division ]
    x1' = c × { t1' √(1 − v²/c²) / ( 1 + v/c) } (1 + v/c)/√(1 − v²/c²) [via substiution t1 → t1' √(1 − v²/c²) / ( 1 + v/c) ]
    x1' = c × t1' [via definition of division ]

    Likewise
    t1 > 0 [given]
    0 < c [given]
    –c < v < c [given]
    0 < c + v [ via addition ]
    0 < 1 + v/c [via division by positive number]

    0 < c² [via squaring]
    0 ≤ v² < c² [via squaring –c < v < c ]
    0 < c² – v² [ via addition ]
    0 < 1 – v²/c² [via division by positive number]
    0 < √(1 − v²/c²) [via property of square roots]
    0 < (1 + v/c)/√(1 − v²/c²) [via division by positive number]
    0 < t1 ( 1 + v/c) /√(1 − v²/c²) [via multiplication by positive number]
    t1' > 0 [ via substiution t1' → t1 ( 1 + v/c) /√(1 − v²/c²) ]

    So for all admissible values of v, the half-line consistent with a description of a pulse of light propagating with velocity +c in one frame, x1 = c × t1 and t1 > 0, is described in the other frame as: x1' = c × t1' and t1' > 0.

    More generally, if the coordinates of the origin event in frame 1 are (x0, t0) then the pulse of light pulse of light propagating with velocity +c in one frame, x1 = x0 + c × (t1 – t0) and t1 > t0, is described in the other frame as: x1' = x0' + c × (t1' – t0') and t1' > t0', x0' = (x0 + v t0)/√(1 − v²/c²), t0' = (t0 + v x0/c²)/√(1 − v²/c²). Algebra, the Lorentz transform and the consistency of the speed of light in vacuum are all consistent.

    If tsmid disagrees with this, he is wrong.
    If tsmid claims algebra cannot do this, he is wrong.
    If tsmid claims this is not how to use the Lorentz transform, he is wrong.
    If tsmid claims the above is not a description of propagation with velocity +c, he is wrong.
    If tsmid claims not to understand this, he is trolling for these are the definitions of the terms he uses in his claims and to claim not to understand the language one's own position is couched in is trolling.
    If tsmid claims trolling is acceptable, he is wrong.
     
    Last edited: Aug 27, 2016
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  11. Neddy Bate Valued Senior Member

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    Of course I agree, but then there is the problem that tsmid also has been claiming that every time recorded in system 2 must be called t2. He does not even seem to accept that the variable t1' might have some usefulness in distinguishing it from t2, as two different times in system 2. So, he will probably want to replace your t1' with t2 and write x1' = c × t2 again. *Sigh*
     
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  12. tsmid Registered Senior Member

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    368
    First of all, there is no need for a separate transformation variable t1'. If you know the transformed location x1', the corresponding time in frame 2 must be x1'/c (assuming, like Einstein did, that the light speed postulate applies to these secondary photon events as well (secondary, because the detection of the class 1 events in frame 2 requires the detection of photon detection events, not directly of photons)).

    Secondly, as I said before, t1' is not an independent variable, but a function of other variables that in general would be completely inaccessible for the frame 2 observer (and indeed completely irrelevant for his determination of the speed). What you are doing when using t1' is effectively re-tuning the time standards of frame 2 behind the frame-2 observers back just to suit your purpose (namely to apply the classical principle of vectorial velocity addition (albeit in some modified form) to light signals despite the fact that the speed of light is invariant).
    What you are doing here is reminiscent of the ancient philosopher Zeno, who also used some similar re-scaling of length and time units to 'prove' that Achilles can never overtake a tortoise, an arrow never reach its target, and such like.

    Tell me where you see the usefulness of replacing the equation x2=c*t2 with x2*q=c*t2*q where q is some arbitrary numerical factor. As t2 and x2 are continuous variables, all time and length values are already covered anyway by the first equation
     
  13. Neddy Bate Valued Senior Member

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    1,727
    Trying to be as clear as I can be, what I have been claiming is that x1' is not numerically equal to x2 for any one single event as the photon travels down the two system's x axes in your chosen scenario. (There is an exception for the light-emission-event at x1'=x2=0 of course).

    As such, x1'/c would not generally be equal to x2/c. We all agree that the variable t2 is equal to x2/c, so we have the equation t2=x2/c. But you refuse to agree that there is any need for a variable which would be equal to x1'/c, such as t1'=x1'/c. And so it appears that you might be stonewalling to prevent any progress.
     
    Last edited: Aug 28, 2016
  14. Neddy Bate Valued Senior Member

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    1,727
    As rpenner showed, there is no problem with x1=c*t1 transforming as x1'=c*t1' for all such photon events, while x2=c*t2 remains true. But it sure becomes more difficult to explain what the difference is between x2=c*t2 and x1'=c*t1' when system 2 claims that all of its clocks always display the same time. One might become confused as to whether that time is to be called t1' or t2. That is why I would rather have discussed just a few well-defined events, rather then a whole continuum of them. However, the answer is that t1' and t2 are both equally valid times in system 2, but one simply occurs earlier/later than the other.
     
    Last edited: Aug 28, 2016
  15. Neddy Bate Valued Senior Member

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    1,727
    Just a friendly reminder that in tsmid's opening post, at his equation (7), tsmid restricted this exercise to x1=x2:


    Not only did he restrict this exercise to x1=x2, but fully to x1'=x1=x2 as well. And note the time variable -- no need for t1, t2, t1', or t2'. Just t. Classic.
     
    Last edited: Aug 28, 2016
  16. Confused2 Registered Senior Member

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    501
    I have to admit I found relativity extremely difficult and I still do. Never trolling or willfully obtuse - just having difficulty with the concept.
    The Lorentz Transform is an unwieldy beast as demonstrated by the fact that it took 100 posts of ? before rpenner finally demonstrated how to use it properly.
    As we are already in pseudoscience is it worth trying something similar but simpler?
    I think so.

    We have a flat railway wagon. On the wagon is a laser that fires a pulse vertically between two sheets of paper separated by a vertical distance of AB metres. When the laser passes through the paper it makes a flash of light and a little puff of smoke. First puff of smoke is at A and the second puff of smoke is at B. The time between the puffs of smoke is t_ab= AB/c

    We now send the wagon through a station at velocity v. Lets make A (the first puff) at coordinates (0,0) on the platform and the wagon - this is Tsmids starting point.

    Because the wagon is moving past the platform while the pulse travels the second puff of smoke is seen at point C from the platform where C is (clearly) not vertically above A. By using two dimensions hopefully this keeps Tsmid's end points away from each other.

    AB is the vertical distance in the wagon frame.
    BC is the distance the wagon advances in the station frame
    AC is the distance the pulse travels in the station frame.
    Staying with high school notation...
    Pythagoras tells us (AC)² = (AB)²+(BC)² (1)
    Since we know the speed of light is the same on the wagon and on the platform...
    t_ab = AB/c
    or
    AB = ct_ab
    t_ac = AC/c
    or
    AC = ct_ac
    since t_ac is the elapsed time in the platform frame the distance the wagon advances (BC) will be velocity*time so
    BC = vt_ac
    substituting into (1)
    (ct_ac)² = (ct_ab)²+(vt_ac)²
    (ct_ab)² = (ct_ac)²-(vt_ac)²
    (ct_ab)² = t_ac²(c²-v²)
    (t_ab)² = t_ac²(1-v²/c²)
    t_ab = t_ac√(1-v²/c²)

    Clearly the elapsed time between the first puff of smoke and the second puff in the wagon frame (t_ab) is not the same as the elapsed time between the same events (t_ac) in the platform frame.
     
  17. Confused2 Registered Senior Member

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    501
    Deleted by me - need time to think.
     
  18. tsmid Registered Senior Member

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    As far as I recall, you agreed earlier that x1'=x2 if the two system are stationary with regard to each other (v=0). Now assume the event happens again at the same point x1 in frame 1, but now frame 1 is moving with velocity v in the positive x2-direction of frame 2. This logically means you must now have x1'>x2 in frame 2. But instead you would now even need x1'<x2 in order to be able to apply the light speed postulate to the class 1 events in frame 2. This is only possible if you actually physically change the length units in frame 2 (and with it the time units in order to keep c constant) for the class 1 events but not for the class 2 events, and that is what the use of x' and t' in the Lorentz transformation implies. This has nothing to do with physics. The length and time units of both frames are set by definition, and you can't just change either of them as you please for certain events in order to get the results you want (in this case be able to apply the classical principle of a velocity dependent transformation to light signals).
     
    Last edited: Aug 29, 2016
  19. Confused2 Registered Senior Member

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    501
    This is the conceptual difficulty I referred to earlier. Reality doesn't have a conceptual problem with Special Relativity and does it naturally. See (for example) the Muon experiment where there is not a 1 to correspondence between time in the fast moving (relative to Earth) muon frame and time in the Earth frame.
    http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html
    The relationship is as predicted in my post #113
     
  20. origin In a democracy you deserve the leaders you elect. Valued Senior Member

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    10,578
    It does not 'imply' that, it flat out categorically states that.
    That my good fellow IS physics.
    And both observers in their own frame will agree that a meter is a meter and a second is a second.
    The observers will not agree that the other frames meter stick is a meter or that the other frames clock records the time the same as theirs. You are right that you cannot change them as you please - however SR and LT shows how they are changed in a specific way.
     
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  21. expletives deleted Registered Senior Member

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    @ Confused2:

    I find that these difficulties (and seeming paradoxes also) effectively disappear once one understands where to draw the line between: the 'calculating usefulness' of SR maths construct; and the 'consistent understanding' of the SR perspective within the overarching GR construct. One must be aware always, when 'using' SR, that it involves an inescapable relativity between two (or more, in complex scenarios under study) observational and/or interacting 'frames' having their own inherent parameters and properties irrespective of what either 'frame' measures from their respective 'point of view' of the effective outcome of such measurements in either SR 'frame'.

    The way your difficulties (and the seeming paradoxes) are avoided, is by realizing that SR is limited by its own NON-absoluteness limitations (re such things as motion, energy etc) in any particular observation/interaction under study.

    Hence why GR is required to be overlaid on SR scenarios like Einstein's "Twin Paradox" gedanken. It's the absolute physical effects of acceleration (irrespective of each Twins' respective SR perspective and measurement of each other's relative motions) that rescues the physical sense from NON-physical SR-only derived non-sense. It takes both Twins' points of view, combined and overlain with GR point of view, to make SR anything more than just a convenient (but severely logically and physically constrained) algorithm for comparing different 'frame views' for the sake of mere relative calculation rather than absolute understanding (which latter is where GR came in after SR; due to Einstein's own recognition of the continuing requirement of logical and physical consistency and understanding which SR itself could never by itself provide).


    That is because "reality" doesn't have to distinguish between human maths abstractions (such as 'relative only' SR model) and natural physical phenomena reflecting the actual absolute respective parameters and properties of whatever entities and quantities are involved in the actual interaction under study (such as GR's 'absolute acceleration' concepts and effects; as distinct from SR's relative motions only). Unlike humans, the "reality" isn't interested in impressing anyone with obviously limited and arbitrarily constrained partial "relativity only" perspective "situations" which ignores the overarching "reality" situation as it is irrespective of such partial perspectives and measurement constructs.


    This is again where SR's own constraints prove fatal, both to interpreting as well as understanding the "reality"; which nature absolutely represents and deals only with "real things". Unlike those humans who insist on treating and interpreting SR-only maths abstractions and modeling results as somehow being "real" instead of the abstractions that they are and will remain until GR considerations and effects are overlain on same (eg, as in the Twins' acceleration history overlaid on relative SR motion 'measurements' from either twin).

    Which means that even by SR's own perspective, the motion of both Muon and Earth is "relative" only; with neither being the "proper reference frame" for whatever we deduce as to the motions of either. As far as 'purely relative SR view' is concerned, it may just as well be the Earth, instead of the Muon, that is the one suddenly accelerated to near lightspeed! But we know from GR energy and momentum and other "absolute accelerations and effects" considerations that it is the Muon whose energy and motion (and therefore internal time rate for decay processes etc) that has been affected by its sudden creation/acceleration in the atmosphere, and not the Earth's such like parameters.

    That is my observation regarding the difficulties and understandings of SR and its limitations and confusions when not considering the wider GR perspective treating the absolute whole of the interaction under study, rather than treating it 'relatively' from the partial perspective of one side or the other.

    The above is only my opinions based on my own objective understandings got from wide researching of the relevant theories and their implications in the contexts of utility for modeling as well as reasonable and consistent understanding of the "reality" you mentioned". I trust they are helpful rather than exacerbate your "difficulty" re SR, Confused2. Best.
     
    Last edited: Aug 29, 2016
  22. Confused2 Registered Senior Member

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    With all due respect may I suggest you aren't seeing what you are looking at. The muon has an internal self-destruct clock which is never further than (say) 10^-30 metres from 'it' (the muon). Eric the Earthling is using two clocks which are effectively x apart (say 10km) to time arrival at clock A and arrival at clock B.
    Assuming we know (and have proved to our satisfaction) that we have a frame invariant spacetime interval s so that
    s²=x²-c²t²
    In the muon frame elapsed time =t we have
    s²=-c²t'² //ignoring 10^-30
    In the Earth frame t' and x' (where x' ~10km) we have
    s²=x'²-c²t'²
    since s² is the same in both frames
    c²t²=x'²-c²t'²
    we know x' = vt' from Newton and the definition of velocity
    so
    -c²t²=(vt')²-c²t'²
    c²t²=c²t'²-(vt')²
    t= t'√(1-v²/c²)
    so t=t_muon is less than t'=t_Earthclock and the factor has been calculated.
    The Hyperphysics analysis of muons assumes you (and Tsmid) are already familiar with the concepts involved.

    Fair enough, now demonstrate one of your alternative theories which give a good prediction of the muon lifetime.

    Huge thanks to rpenner and many others for enabling (and possibly even allowing) me to make this post.
     
  23. expletives deleted Registered Senior Member

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    @ Confused2:

    That's exactly what I was pointing out when I said that the respective entities and processes have their own inherent properties and parameters which they bring to any interaction/study involving two (or more) entities. Hence why mere abstract relativity-only measurements and conclusions cannot provide the actual situational reality unless all entities' properties and parameters are taken into account together and not arbitrarily separated within an algorithmic analysis which is fundamentally unsound unless all states and effects are already known as per the inherent properties and parameters (as I explained when the Twin case needed the real acceleration information before making any sense in reality physics terms).

    Realize that all of that is an abstract theoretical exercise which is not informed by information applying to the Muon. Only a comparison (just like in the Twins' comparison) of the actual entities and parameters involved, based on actual acceleration etc information, can ever resolve the SR relativity impasse which would have one treat the Earth equally with the Muon as to respective acceleration etc in fact as opposed to mere SR relativity view switching from real to abstract analysis.

    Just because some algorithm (think epicycles) give a useful prediction, it doesn't make the epicycle analytical construct a real phenomenological representation. In any case, challenging some self-inconsistent 'explanation' (in this instance re Muon lifetime in various motional states) does not mandate an alternative be offered; only that the claim for that 'explanation' be supported against the challenge based in real observational and logical situation as outlined in that challenge. Falsifying of an existing theory/claim is not conditional upon an alternative being ready at time of falsifying of said present claim/theory (in this instance,the challenge is not so much in the nature of falsification, as it is one of asking you to support your Muon lifetime etc theory/claims given what I just outlined in my questioning of same).

    Obsequiousness to persons, no matter their status or rank or power etc, is not a good attitude to bring when discussing issues on their own present scientifically and logically tenable merits as presented in present discussion. The only obsequiousness acceptable is to scientific methodology and principles of objectivity and relevance to real phenomenological analysis and understanding. I understand the concept of showing respect; but personal obsequiousness is taking things too far and may be seen as sucking up etc by those not so charitably disposed as myself. I hope my honest view in that regard has not offended anyone personally. Best.
     
    Last edited: Aug 30, 2016

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