I need someone who knows the math

Discussion in 'Physics & Math' started by BigBangIsGod, Mar 4, 2016.

  1. paddoboy Valued Senior Member

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    I'm sure you have been informed many times in the past.
    The reasons have been explained to you. If you deny SR, you will obviously deny the fact that there is no paradox.
    People can't continue to spoon feed you my friend.
     
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  3. The God Valued Senior Member

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    First of all it was much before Dingle, it probably started with Paul Langevin and even Einstein acknowledged it. Dingle pushed it to no success.

    Lets cut everything. I am asking you a very simple question, A and B move in two different direction with respect to Earth at the same speed. On return they find that they have aged slowly by the same amount with respect to Earth, suggesting that there is no dilation betwen them. Now the crux is both had relative speed between them, so there should have been time dilation between them too. Where is the catch ?

    Please note that I am not harping who has slowed and who has not, that is the classic twin paradox...even acknowledged by Einstein himslef and he proposed turn around/acceleration etc as solution.
     
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  5. paddoboy Valued Senior Member

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    Herbert Dingle??? Being ignorant of this person I did some checking......
    http://www.mathpages.com/home/kmath024/kmath024.htm
    Beginning in the late 1950's the retired British philosopher of science Herbert Dingle (1890-1978) wrote several letters to the science magazine "Nature" and other publications, claiming that special relativity did not predict asymmetric ages for the re-united twins in the famous twins paradox. Eventually he was convinced by several other correspondents that he was wrong, i.e., he became convinced that special relativity did, in fact, predict unequal ages for the re-united twins. After accepting this, Dingle began another letter-writing campaign, now claiming that special relativity is logically inconsistent. After printing many of Dingle’s letters, along with refutations from various scientists (including Max Born), these publications stopped accepting his letters, so Dingle felt compelled to write a book, “Science At The Crossroads”, in which he recounted his struggle against what he regarded as the deluded and corrupt scientific establishment. In 1972, just as his book was about to be published, he engaged in one more debate, this time with Professor Ray Lyttleton, in the “letters to the editor” section of The Times. In his final letter, Dingle presented his “irrefutable proof” as follows:

    Suppose clocks A and B move along the same straight line at uniform speeds differing by 161,000 miles a second. At the instant at which B passes A both read noon. Then, according to special relativity, at the instants when B reads 1 and 2 o'clock, A reads 2 and 4 o'clock respectively… Einstein himself made just this calculation, and concluded that since B recorded a smaller interval than A between the same events, it was working more slowly. But if he had similarly calculated the reading of B for the readings 1 and 2 o'clock of A he would have got 2 and 4 o'clock respectively, and must have reached the opposite conclusion: he did not do this, so missed the contradiction. I invite Ray to fault these calculations, or convince your [readers] that each of two clocks can work faster than the other. I do hope he will not disappoint them.


    In a nutshell, Dingle considers two systems of inertial coordinates x,t and x',t' with a relative velocity of v, and then notes that the partial derivative of t' with respect to t at constant x is equal to the partial derivative of t with respect to t' at constant x’. He declares this to be logically inconsistent. Needless to say, Dingle’s “reasoning” is incorrect. It consists of the claim that those two partial derivatives must be the algebraic reciprocals of each other, which of course is false.


    To elaborate on this point, the Lorentz transformation is x' = (x−vt)g, t' = (t−vx/c2)g, and its algebraic inverse is x = (x'+vt')g, t = (t'+vx'/c2)g, where g = 1/(1−v2/c2)1/2. These equations imply t' = gt at x = 0, and t = gt' at x' = 0. Dingle alleged that these two facts are mutually contradictory, because the first implies t'/t = g and the second implies t/t' = g. However, these ratios apply to two different conditions, namely, x = 0 and x' = 0 respectively. Hence, contrary to Dingle's assertion, there is no contradiction, nor are these relations merely "appearances". They are the actual ratios of the inertial time coordinates along two different directions in space-time. (For a more on Dingle’s misunderstandings, see the note onHerbert Dingle and the Twins.)


    In essence, Dingle simply noted the reciprocity of the Lorentz transformation and its inverse, which of course, far from being overlooked by Einstein, was the whole basis of the special theory of relativity. Indeed Lorentz summarized Einstein’s theory in 1909 by saying


    Attention must be drawn now to a remarkable reciprocity that has been pointed out by Einstein... The behavior of measuring rods and clocks in translational motion, when viewed superficially, give rise to a remarkable paradox which on closer examination, however, vanishes.


    Thus Dingle’s claim that Einstein never noticed this reciprocity is simply bizarre, as was Dingle’s "proof" of the logical inconsistency of special relativity. His reasoning can just as well be applied to “prove” that Euclidean geometry is logically inconsistent. Consider two Cartesian coordinate systems x,y and x',y' with a common origin, but rotated by an angle ofq with respect to each other. Given the x,y coordinates of any point, we can compute the x',y' coordinates by means of the equations


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    We can also solve these equations for x and y in terms of x' and y' to give the inverse transformation


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    This is just elementary linear algebra. Now, if we hold y constant and vary x, how much does x' vary? In other words, what is the partial derivative of x' with respect to x? This is denoted by ¶x'/¶x, and clearly we have ¶x'/¶x = cos(q). Now we ask a different question, namely, if we hold y' constant and vary x', how much does x vary? This is equivalent to asking for the partial derivative of x' with respect to x, and of course we have ¶x/¶x' = cos(q). Dingle's confusion is due to the fact that (like some befuddled freshman calculus students) he imagines ¶x'/¶x and ¶x/¶x' are algebraic reciprocals of each other, which would imply that 1/[¶x'/¶x] = ¶x/¶x', and therefore cos(q) = 1/cos(q), which is impossible for anyq other than 0. Does this prove that Euclidean geometry (and linear algebra) is logically inconsistent? Of course not, because Dingle's argument is obviously specious; the partial derivatives ¶x'/¶x and ¶x/¶x' are not the algebraic reciprocals of each other.


    As noted previously, the application of Dingle's argument to the Lorentz transformation is exactly the same. Two inertial coordinate systems x,t and x',t' with a mutual relative velocity v are related according to the equations


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    where g = 1/(1-v2)1/2. The inverse transformation is


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    In this case we have the two partial derivatives ¶t'/¶t = g and ¶t/¶t' = g. Dingle erroneously assumed that 1/[¶t'/¶t] = ¶t/¶t', and so he arrived at 1/g = g, which is impossible for any v other than 0. Again the fallacy is the erroneous assumption that partial derivatives can be algebraically inverted. Of course, we can invert total derivatives, so let's see what happens if we take the absolute differentials (for any constant v) of the time transformation equations. We have


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    Dividing the left hand equation by dt and the right hand equation by dt' gives


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    Considering the first of these equations, notice that for an object at rest in the unprimed reference frame we have dx/dt = 0 (by definition), and so we have dt'/dt = g. However, for an object at rest with respect to the primed coordinates we have dx/dt = v, which gives


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    This shows how we arrive at either one of the partials, depending on which direction in spacetime we are considering. Likewise for the second transformation equation we can consider the cases when dx'/dt' = 0 or -v, giving the results g and 1/g respectively. Also, since the total derivatives are reciprocals of each other, we can multiply them together to give unity, i.e.,


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    Notice that dx/dt is the velocity of one worldline with respect to some arbitrary reference frame, and dx'/dt' is the velocity of a different worldline with respect to that same reference frame. Let us denote these velocities by u and w respectively. Recalling that g2 = 1/(1-v2), the above equation becomes


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    Solving for u gives the familiar formula


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    which is the relativistic speed composition formula. Needless to say, there's nothing inconsistent or self-contradictory here. Dingle was simply making an elementary error.


    This is all quite elementary, and not terribly interesting. The more interesting question is what happened to Dingle. For most of his life he wrote approvingly about what he called “relativity”, beginning with his 1922 essay “Relativity for All”. He even wrote a monograph on special relativity (1940) in which he dutifully noted the relativity of simultaneity, and so on. But in his later years (beginning in the mid to late 1950s), after embarking on a passionate anti-relativity crusade, he exhibited a complete inability to grasp the most fundamental aspects of special relativity, culminating in the grotesquely crackpotish "Science at the Crossroads" in 1972. The obvious question is, how could he write and lecture about a theory for almost forty years, and then suddenly be unable to grasp it? Granted, Dingle was 82 when “Crossroads” was published, but can this really account for his confusion?
     
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  7. rpenner Fully Wired Valued Senior Member

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    No, it's a complex question because it assumes facts not in evidence when you said what it "suggests".

    There are two. One is relative time dilation is a property of relative velocity and the second is both A and B are non-inertial in that they each change their relative velocities in order to return to the world-line of the Earth. So since neither A or B remains in the same state of inertial motion, there is no universally applicable time dilation factor that describes them. That's why I worked out the OPQR space-time parallelogram example for Earth coordinates, A outbound coordinates and A inbound coordinates. It requires an inhomogeneous Lorentz transform so that the end of the outbound journey has the same representation as the start of the inbound journey. As a result, the simple concept of relative time dilation between two inertial trajectories is inadequate to describe how bent world-lines transform. Here's a nice concrete example with V=0.8 c, T=15 s.

    \( \begin{array}{c|ll|ll|ll} \textrm{Event} & x_{Earth} & t_{Earth} & x_{outbound} & t_{outbound} & x_{inbound} & t_{inbound} \\ O & 0 & 0 & 0 & 0 & - 40 c \cdot \textrm{s} & - 32 \, \textrm{s} \\ P & 12c \cdot \textrm{s} & 15 \, \textrm{s} & 0 & 9 \, \textrm{s} & 0 & 9 \, \textrm{s} \\ Q & -12c \cdot \textrm{s} & 15 \, \textrm{s} &- 40 c \cdot \textrm{s} & 41 \, \textrm{s} & - 40 c \cdot \textrm{s} & -23 \, \textrm{s} \\ R & 0 & 30 \, \textrm{s} & - 40 c \cdot \textrm{s} & 50 \, \textrm{s} & 0 & 18 \, \textrm{s} \end{array}\)

    For Earth, first O happens, then P and Q happen simultaneously, then R happens.
    For the inertial system where outbound A (and inbound B) are at rest, the events happen in the order O, P, Q, R.
    For the inertial system where inbound A (and outbound B) are at rest, the events happen in the order O, Q, P, R.

    When A switches the standard of rest from outbound to inbound at event P, relative to A's changing standard of rest, the event Q changes from an event in the future to an event in the past. That's captured in the math of the inhomogeneous Lorentz transform and not merely in the toy concept of relative time dilation.

    Now
    \(( t_R - t_O)^2 - c^{-2}(x_R-x_O)^2 = (30 \, \textrm{s})^2, \\ ( t_P - t_O)^2 - c^{-2}(x_P-x_O)^2 = ( t_Q - t_O)^2 - c^{-2}(x_Q-x_O)^2 = (t_R - t_P)^2 - c^{-2}(x_R-x_P)^2 = (t_R - t_Q)^2 - c^{-2}(x_R-x_Q)^2 = (9 \, \textrm{s})^2\)
    no matter which inertial coordinates one uses to compute. So if you had just worked out the example long-hand you never would have run into any appearance of contradiction. The fault lies solely in bad intuition and bad preconceptions, not in the physical theory.
     
    Last edited: Mar 9, 2016
  8. The God Valued Senior Member

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    You have some interesting quotes, which may be useful and hence I have taken them and put in a single frame.

    With respect to your analysis of 0.8c and T = 15 seconds, simple toy application of SR, will get you a factor of 0.6, and thus elapsed time on earth being 15 X 2 = 30 Seconds and on the craft it would simply be = 2 X 9 = 18 Seconds...ignoring the turn around, no dispute.

    This is also not under dispute that a craft shot in any direction from Earth, will come back with the same proper elapsed time.

    Now we go back to your Pt#2 as in your above quote, which is intuitively very complex to appreciate, possibly because of mixing up of coordinate time with proper time, inability to see that measurement/observation is c dependent, but can you make a comment that A and B will age same. For example instead of taking Earth Twin as the age reference, what if A is taken as the age reference.
     
    Last edited: Mar 9, 2016
  9. Ophiolite Valued Senior Member

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    Dementia?
     
  10. danshawen Valued Senior Member

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    Guys, please...

    If your rest frame is Earth and you LEAVE it in a relativistic projectile, you will of course accelerate, but that is not important to time dilation effects. What IS important is, so long as you remain in relative motion with respect to the rest inertial reference frame on Earth from which you started, time will proceed at a different (slower) rate from that experienced on Earth. This will be evident when you return, whether you use a twin or just a pair of clocks which were synchronized when you left.

    The God is right. There is no 'paradox' or inconsistency, mathematical or otherwise, associated with this. It is what it is. Reality does not yield to inconsistent math or math that is in error because it is based on static geometry and its inherent single observer point of view. And you are not hallucinating. Time (coupled with the propagation of energy AND rotation) is literally the ONLY dimension needed to CONSISTENTLY describe a universe of energy transfer events.

    If your geometry is based on physical relationships wth respect to bits of bound energy, good luck with that. The light travel time between them is not static or the same to all observers. It never was. Neither does time proceed at the same rate in different inertial reference frames. This is why only REST ENERGY, not relative energy, is invariant. Without that particular invariant, the speed of light would have no real meaning. That's why +/- c is the real invariant. It always requires TWO observers. Like the twin 'paradox', you cannot understand this with math written to satisfy the observations of a single static observer.
     
    Last edited: Mar 9, 2016
  11. rpenner Fully Wired Valued Senior Member

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    You are still missing the point. There is no simple concept of differential aging relating to traveling crafts. The simple concept of relative time dilation between two inertial trajectories is inadequate to describe how bent world-lines transform.

    OR has a proper time of 30 s, OP, OQ, PR and QR have proper times of 9 s.
    Thus in Earth Coordinates, at t=0 A & B have age 0, at t=15 s, A&B have age 9s, and t=30s, A&B have age 18s. This creates the illusion that the remote age of A&B has some meaning which perverts everything Special Relativity teaches about proper time and relativity of simultaneity.

    In A's outbound coordinates, at t=0 A&B have age 0, at t=9 A has age 9s but B has age 81/41 s because B is (9/41) along OQ which will age B 9 s.
    Then A switches to A's inbound coordinate so at t=9, A has age 9s but B now has age 651/41 s so that at t=18 s A & B have age 18s.

    Did A's turn around cause B to age 570/41 ≈ 13.9 s in an instant? No, because that exact value is a function of both B's remote bent path through space-time and A's change in standard of rest. If' A's turnaround is modeled more realistically, then A's description of B's accelerated aging can be described continuously. But it is still just a figment of relativity of simultaneity. Just as the figment that relative aging is going on at the same rate in Earth coordinates is figment of relativity of simultaneity.

    Therefore the concept of "what age is a traveller, remote from me, at this same time" is inextricably bound up in the definition of (and therefore relativity of) simultaneity which means it has no absolute physical meaning and is strongly dependent on the choice of coordinates. Therefore the ratio of age of B to age of A has no intrinsically meaningful answer when A and B are no longer at the same location. Therefore, there is no simple concept of differential aging relating to traveling crafts in special relativity.
     
  12. paddoboy Valued Senior Member

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    ??? What have you been doing for the last few days?
    The god is very rarely right and in this instant your claim is also wrong.
    Our friend has been the one claiming there is a paradox, when there is not a paradox.
    Your own take on the situation appears complicated and confused and wrong as is most of your take on physics it seems.
    Worth noting also that this latest sidetrack is just another example of off topic debate introduced by the god, when he has been shown to be wrong on the original subject of the OP...Obviously again to avoid recognition of his gross errors on cosmology in general.
    Aligning yourself with what you presume to be the underdog Dan, is not always the wise thing to do. Particularly in this case.
     
  13. danshawen Valued Senior Member

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    The only "coordinates" that are meaningful, are in relation to other inertial reference frames, or the propagation of unbound energy that was sourced in a different rest frame.

    Those coordinates are best expressed as light travel time between reference frames, including any relative Doppler shifts. Coordinates of any other kind are nonsensical. Inertialess space does not support geometry, coordinates or origins, other than those relative to rest energies of other particles comprised of bound energy. And while the quantum field supports an origin for time, there is no uniformity that can be expected in terms of the relative rates at which time proceeds. Time itself is a layer deeper than the relative propagation of bound or unbound energy, not completely described or explained in detail either by relativity nor QM.

    Remember when I first told you, time dilation is different EVERYWHERE? That's exactly what I meant.

    And "curved world lines" mean exactly nothing. There is no quadratic/complex relationships between space and time. There is a relationship of time, linear propagation, and rotational propagation of energy, but it is not anything like what Minkowsi described mathematically. Nor is his math consistent with what is actually observed, because he apparently believed only one observer was sufficient.

    And before you ask, I've no clue as to why a first derivative with respect to time should work so spectacularly well (the speed of light is a VELOCITY) to make an invariant, and yet the SECOND derivative (acceleration) with respect to time has basically no comparable effect whatsoever in terms of time dilation or a prediction of a fundamental physical dynamic. Perhaps it is because in order to calculate acceleration, you must have an unambiguous definition of both velocity AND of time itself, and the latter is something we yet lack. But there is a tantalizing clue that it might be because space really is simply another means of describing a dimension that is in all cases proportional to time. That being the case, the second derivative would always yield a constant, would it not? I'm not talking about math. I'm talking about the physical reality seems to be that acceleration isn't possible at all on the quantum level because fundamentally time itself is already an acceleration.

    I told you your calculus would likewise avail you nothing, because it is fundamentally based on a static geometry AND also absolute time. But did anyone listen? Noooo…..
     
    Last edited: Mar 9, 2016
  14. Bruinthor Registered Member

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    Something to consider for those who think the SR deniers are on to something. Special Relativity is the an intrinsic part of Quantum Electrodynamics https://en.wikipedia.org/wiki/Quantum_electrodynamics
    QED is in turn is arguably the most thoroughly tested and verified theory ever created by humans. If there is something seriously wrong with SR how can it possibly contribute to so many accurate predictions.
     
  15. rpenner Fully Wired Valued Senior Member

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    This marks you as a physics outsider, and in the context of this discussion, a waste of resources. Cartesian coordinate systems have been vital in physics' role in describing phenomena since Newton. The point of Special Relativity is while the choices of physical units, standard of rest, spatial orientation and position of the origin make no difference to the physics being described, they make all the difference to the form in which the description takes. That's why invariants like \(c^2 (t_2 - t_1) ( t_4 - t_3) - (\vec{x}_2 - \vec{x}_1) \cdot ( \vec{x}_4 - \vec{x}_3) \) matter – they represent concrete physical quantities that all inertial coordinate systems, and thus all valid physical descriptions agree upon.
    There is no discussion of energy, bound or otherwise, in the discussion. Only the geometry of Lorentzian space-time, which is a more fundamental concept.

    A "reference frame" is not a physical thing, but an imaginary choice of labeling space-time with coordinates so that every inertial trajectory is described by the equation of a straight line and thus is equivalent to a particular choice of physical units, standard of rest, spatial orientation and position of the origin that describes all of the relevant region of space-time (in the approximation that Special Relativity is an acceptable model of space-time). As a reference frame is not a physical ( or even localized) object, there is no concept of "travel time between" them and certainly none in play in this discussion. Again, this marks you not only as a physics outsider, but as a sterile source of misinformation who can't even be bothered to pay attention to the discussion.

    For the above given reasons, this is nonsense and not part of the discussion. We are discussing descriptions of behavior, not descriptions of physical behavior by beings limited to light-based observations, so Doppler shifts nowhere enter the discussion.

    One can, of course, calculate the proper time of arrival (according to A) of 18 (or more) regularly-timed (according to B) flashes of light but such discussion would be off-topic to correcting "The God"'s fixation of ratios of elapsed proper time of A to elapsed proper time of B as such a measure implies a universal standard of simultaneity when none exists in Special Relativity.

    Baseless in entirety without empirical or other authority. For the above given reasons, this is nonsense and not part of the discussion. We are discussing descriptions of behavior in the context of the model of special relativity. Thus your pet peeves are off-topic and unwarranted.

    This marks you as a physics outsider, for if you have been paying attention you would know that: no one agrees with you, remembers when you first asserted such a baseless claim, or bears the duty of supporting your claims. You need to support your own claims and cite your own references or be dismissed as a waste of time.

    I said bent world-lines as OP is clearly not co-linear with PR. Please try to pay attention.
    This marks you as a mathematics outsider, because the field of geometry marks you as completely wrong and have you been under the burden to be aware of the wrongness of your claim since Minkowski's 1908 lecture: Raum und Zeit.

    Your baseless claim naturally requires your demonstration of correctness, because others will not do it for you. But it is certainly off-topic for this discussion.

    It is more approximately correct than Newtonian or Galilean descriptions of space and time, but not as approximately correct as General Relativity. To the extent that you are claiming otherwise, your baseless claim naturally requires your demonstration of correctness, because others will not do it for you.
    You have not supplied any reason why, if one understands how physical descriptions change covariantly so that the same physical laws are held to be true by any one reference frame, knowledge of the description of a phenomenon in one reference frame augmented by knowledge of how physical descriptions change does not allow one an adequate summary of any information possibly provided by an alternate description of the same physical phenomena. In short, you have claimed that there is a potential for different observers to disagree about what happened. Did the clock strike 11 times or 12? Did the radioactive decay happen or not happen? The current discussion is predicated on the limited claim that to good approximation, when gravity is neglected, Special Relativity is just such knowledge of how the same physics is described differently by difference choices of inertial coordinate systems without the differences in descriptions amounting to a physical difference. To the extent that you are claiming otherwise, your baseless claim naturally requires your demonstration of correctness, because others will not do it for you and is certainly off-topic for this discussion.

    To the extent that you conflate "observers" with choices of inertial coordinate systems, that fault lies with you and marks you as a physics and math outsider who has not been paying attention to the discussion.
     
    Last edited: Mar 9, 2016
  16. paddoboy Valued Senior Member

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    While we have the malady of delusions of grandeur, and tall poppy syndrome, we will always have cranks trying their best to trump the greatest scientist we have ever seen. [well at least up there in the top two or three

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  17. paddoboy Valued Senior Member

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    There are not many posts by this bloke that are not peppered with references to bound and unbound energies, whether relevant to the discussion at hand or not.
    Bingo also to that. I truly have trouble deciphering Dan's posts as I inferred at post 109:
     
  18. The God Valued Senior Member

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    But could you really decipher Rpenner posts ? Or will you ever learn to right sensibly on the topic the way he does..
     
  19. The God Valued Senior Member

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    Thanks rpenner....

    Since if I view from B's point of view (instead of A's as done by you), things will swap thus leading to confusion on the ageing....because true ageing cannot be coordinate dependent.

    So, to eliminate this, can we conclude.

    1. Time Dilation is more to do with coordinate time which may or may not not be linked with proper time.
    2. The ageing comparison between two can be done if both are at the same spatial point.
    3. Coordinate time dilation may mislead if it is considered as proper time differential.
     
  20. rpenner Fully Wired Valued Senior Member

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    True differential assignment of aging would be very nice if it were coordinate independent, but such a hypothesis almost immediately leads one to conclude a universal notion of simultaneity and thus absolute time, in direct contradiction with the math of special relativity. Conflicting math hypotheses leads to inconsistency and invokes the principle of explosion.

    But we can look events as describe from at B's two standards of rest: O to Q outbound and Q to R inbound:

    \( \begin{array}{c|ll|ll|ll} \textrm{Event} & x_{Earth} & t_{Earth} & x_{outbound} & t_{outbound} & x_{inbound} & t_{inbound} \\ O & 0 & 0 & 0 & 0 & 40 c \cdot \textrm{s} & - 32 \, \textrm{s} \\ P & 12c \cdot \textrm{s} & 15 \, \textrm{s} & 40 c \cdot \textrm{s} & 41 \, \textrm{s} & 40 c \cdot \textrm{s} & -23 \, \textrm{s} \\ Q & -12c \cdot \textrm{s} & 15 \, \textrm{s} &0 & 9 \, \textrm{s} &0 & 9 \, \textrm{s} \\ R & 0 & 30 \, \textrm{s} & 40 c \cdot \textrm{s} & 50 \, \textrm{s} & 0 & 18 \, \textrm{s} \end{array}\)



    Baring apparently minor language quibbles I would tend to agree with your conclusions.
    I would expand on those three takeaways like this:
    1. Time dilation as a simple numerical factor is relative to two inertial standards of rest. Coordinate time of an inertial system coordinates is an appropriate stand-in for one or both inertial standard of rests. (The figmentary time axis of a particular system of coordinates has equation of motion x=0 so it has coordinate velocity 0.) And so the relative time dilation factor for one from the other is \(\sqrt{1 - \left( \frac{\vec{v}}{c} \right)^2}\) where v is the relative velocity of one standard of rest in terms of the other.
    2. Comparing total elapsed proper time (aging) between two objects/clocks which were originally together, and are now together again involves no ambiguity related to relativity of simultaneity and thus represents a fundamental physical quantity susceptible to identical measurement, independent of one's choice of a standard of rest. For a generic trajectory through space time, total elapsed proper time may be calculated as \(\Delta \tau = \int \sqrt{1 - \left( \frac{1}{c} \frac{d\vec{x}}{dt} \right)^2 } \, dt\) and for a piece-wise inertial trajectory this simplifies as a sum of \(\sqrt{ ( t_1 - t_0)^2 - \frac{1}{c^2} ( \vec{x}_1 - \vec{x}_0 )^2 } \) for each inertial segment of the path.
      1. Continuity of elapsed proper time along a non-inertial object/clocks's
    3. Since the given of objects/clocks which were originally together, at some point not together and now together again presumes at least one of the object/clocks were traveling a non-inertial path (at least in SR), it must be true that for at least one of the object/clocks there is no single state of inertial motion which is universally that of the object/clock. Thus it is not true that both object/clocks can be used a standard of rest to compute both its and the other's total elapsed proper time (aging). In the case where neither is inertial, then neither can be used. Nonetheless, total elapsed proper time (aging) can be correctly calculated given knowledge of the trajectory as described in any inertial coordinate system, including one where just one of the object/clocks instantaneously achieves or a hypothetical mean inertial system that connects where and when the clocks separated to the where and when of their reunion.
     
    Last edited: Mar 10, 2016
  21. paddoboy Valued Senior Member

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    I certainly understood the principals which apparently escaped you: Nice to see you finally admitting error though.
    Keep it up!

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  22. danshawen Valued Senior Member

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    Observers on the ends of different radii hands of an analog clock literally cannot even agree on how long it takes for an observer on another hand to complete one revolution, as a time dilation effect. The twin paradox works as well for circular paths as it does for linear ones, and that's an incontrovertible fact.

    The differing decay of pi mesons in cosmic rays is an example of relativity's (time dilation) effect on radioactive decay. In that case, the half life is very much dependent on frame of reference, yes.

    I have provided ample reasons to reject the covariant view you expound. A failure to provide a coordinate origin that is consistent with different observers, that is consistent with length contraction / rotation that describes an ACTUAL ORIGIN CONSISTENT WITH DIFFERENT OBSERVERS for any such contraction or rotation other than by mathematical convention makes any claims of covariance inconstent both mathematically and physically. Don't explain to me that you have established an origin for a consistent coordinate system in an inertialess medium for which this is both a physical and mathematical impossibility. Don't tell me you understand the first thing about relativity theory and then go on to expound your mathematics based on a geometry of absolute space, taking place in an absolute time where all clocks are synchronized because motion is suspended and time is rendered static. This is fantasy, besides not being the universe in which we all reside.

    The consistent view is that energy transfer events in the frames in which they were created only have meaning in terms of their relation to other energy transfer events in other or the same reference frame, not an arbitrary coordinate system based on any particular choice of reference frame. There is no preferred reference frame, nor a preferred rate at which time progresses relative to any other reference frame.
     
    Last edited: Mar 10, 2016
  23. The God Valued Senior Member

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    3,546
    No errors baby, I just highlighted a point, which clearly helped in getting few conclusions not known to many, error was in yours and origin's approach. It is amply demonstrated and proved time again that you guys take the mainstream line without even understanding how it is manifested...many a times even I have seen that you guys interpretation of mainstream itself is faulty on basics..

    Nonetheless I am thankful to rpenner, who has gotten few things concluded by lucid maths and clearly articulated points. Recall how you guys were calling and still insisting time dilation as time travel, thus pushing popscience...
     
    danshawen likes this.

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