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    Velocity: Stuff That Just Doesn't Add Up
    By Johannes Koelman | January 31st 2012 09:25 PM | 34 comments | Print | E-mail | Track Comments
    About Johannes

    I am a Dutchman, currently living in India. Following a PhD in theoretical physics (spin-polarized quantum systems*) I entered a Global Fortune

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    We are all familiar with velocities. Velocities tell us how positions change with time. Velocities can not be assigned to individual objects, as they describe a relation between pairs of objects. We know this since Galileo Galilei. Yet, in common day language this profound fact is mostly ignored.

    Make a statement like "The ball has a distance of 30 meters", and sure enough, people will frown and reply with a question for clarification: "Distance to what?". Make the equally silly statements "The ball has a velocity of 20 meters per second" and the same folks will nod to indicate they perfectly understand what you are saying.



    Galilean Relativity

    Relativity is commonly associated with Einstein, yet the first person to formulate a principle of relativity was Galileo Galilei. Core of Galilean relativity is the mere fact that velocities are relative. In the following, I will make this explicit by writing vXY for the velocity of object X as seen from the perspective of object Y.

    Imagine a railroad track. Velocities along this track are measured using real (positive or negative) numbers that denote the velocity expressed in terms of the speed of light. A train moves along the track. Travis is sitting still in the train. A housefly flies forward in the train and passes Travis. Travis (T) observes the fly (F) to have a velocity vFT. The train with Travis and the fly passes a railway station. Stacey (S), who is standing still at the station, sees Travis passing by at a velocity vTS. She doesn't notice the fly, but the fly does notice Stacey, and sees her passing by at a velocity vSF

    Galilean relativity tells us how all these velocities relate to each other. They simply add up to zero:

    vFT + vTS + vSF = 0

    Notice the cyclic structure of the subscripts in this equation. This cyclicity is essential: swap the subscripts in one of the velocity terms, and the equation becomes invalid. Swap all three pairs of subscripts, and the equation is again cyclic, and again valid:

    vFS + vST + vTF = 0

     In fact, the above equation is not just valid for Stacey, Travis and the Fly, but for any three objects moving along a line.

    This velocity summation equation even holds true when applied to only two objects. Such a special case results when identifying one object to another. Let's say we ignore the fly and substitute the subscript T (Travis) for F (Fly). It follows that:

    vTS + vST + vTT = 0

    However, the velocity of an object seen from the object's own perspective is identical to zero:  vTT = 0. Therefore:

    vTS + vST = 0

    or

    vST = -vTS

    In other words: if Stacey sees Travis in the train moving with a certain speed in one direction, Travis - when looking out of the train - sees Tracey moving with the same speed in the opposite direction. We are all familiar with this velocity reciprocity.

    Let's go back to the first equation, and put in some numbers for the relative velocities. To make thing interesting, we assume a more than capable ultra-high-speed train and a very fast fly. Let's say the train moves relative to the station at 2/3 of the speed of light (vTS = 2/3) and the fly propels itself through the train at 1/2 the speed of light (vFT = 1/2). From above equation it follows that the fly sees Tracey moving backward at 7/6 times the speed of light (vSF = -7/6). And hence, Tracey sees the fly moving forward at 7/6 times the speed of light (vFS = 7/6). 



    Lorentzian Relativity

    Already as a teenager, Einstein was concerned about outcomes predicted by Galilean relativity. A fly (or anything else) overtaking a beam of light didn't make sense to him. Yet this is exactly what would happen in the above example if Tracey would shine a flashlight in the direction of movement of the train. How would the fly perceive the beam of light? The laws of electromagnetism which describe the propagation of light could not answer this question. Electromagnetism is not compatible with Galilean relativity. 

    Anyone else would have tried to modify the equations of electromagnetism to render them compatible with Galilean relativity. Not so Einstein. His deep physical intuition and thorough understanding of electromagnetism told him that something else had to give. He became convinced that the concept of relativity of position and motion is in itself correct, but that Galilean relativity is rotten. So he modified the venerable 273 years old Galilean relativity to render it compatible with the laws of electromagnetism published by James Maxwell 40 years earlier.  

    When Einstein was done, a new principle of relativity emerged. A beautiful principle of relativity that is compatible with particle motion as well as with electromagnetism. The mathematical equations for this form of relativity were already written down by Hendrik Lorentz, without him realizing what they really meant.  

    Lorentzian (aka 'special') relativity modifies the cyclic velocity sum equation by adding another cyclic term to it, a term that is the product of the relative velocities:

    vFT + vTS + vSF + vFT . vTS . vSF = 0

       The result is a modified velocity composition equation that no longer guarantees the three velocities to add to zero, but that remains compatible with velocity reciprocity. This reciprocity can be derived fromthe above equation by again ignoring the fly (F) and substituting the subscript T (Travis) for F (Fly):

    vTT + vTS + vST + vTT . vTS . vST = 0

    Using vTT = 0 the product term drops out of the equation, and the velocity reciprocity relation follows:  vTS + vST = 0.

    More importantly, the Lorentzian velocity composition relation is fully compatible with electromagnetism. This can be seen by eliminating the fly (F), and replacing it with the light from a flashlight (F) that Travis shines in the forward direction of the train. Travis sees the flashlight photons traveling forward at the speed of light (vFT = 1) and, according to the laws of electromagnetism, Stacey sees the same flashlight photons also moving forward at the speed of light (vFS = 1). Using reciprocity (vSF = -vFS) it follows that vSF = -1. Substituting the values for vFT and vSF in the above Lorentzian velocity composition equation it follows that 

    1  + vTS -1  -  vTS  = 0

    This equation is valid for any relative velocity vTS between Travis and Stacey. In other words: no matter how fast the train moves, both Travis in the train and Stacey outside the train see the photons move at the same velocity. Exactly as expected from the equations of electromagnetism.

    Now let's investigate again the situation of the fly moving through the train. How does the Lorentzian velocity formula change the outcomes when the fly moves at half the speed of light relative to the train, and the train moves at 2/3 of the speed of light relative to Stacey? Substituting vTS = 2/3 and vFT = 1/2, it follows that vSF = -7/8. (Check for yourself.) Again using reciprocity, it follows that vFS = 7/8. In other words: based on the modified equation Stacey sees the fly moving at less than the speed of light. A result compatible with the fact that nothing, no train, no fly, no particle, no neutrino (!?) can overtake an observer with a speed larger than the speed of light.



    Lost Elegance?

    The Lorentzian velocity equation has an additional term compared to the corresponding Galilean equation. This triple velocity product term is negligible compared to the other terms when all relative velocities are small compared to the speed of light. Yet, it still is an extra term. Somehow it seems that Lorentzian/special relativity is less elegant than Galilean relativity.

    This is no more than an appearance. An appearance that disappears when one looks deeper into the concepts playing a role in both relativity theories. Galilean relativity places a lot of emphasis on velocities, not so Lorentzian relativity. Einstein's theory that leads to Lorentzian relativity is based on a spacetime description in which velocities are represented as slopes. Would the time and space dimensions behave similarly, velocities relative to an observer would be the tangent of angles with the time-axis relevant to that observer. However, time behaves in a way subtly different from the way spatial dimensions behave, and as a result velocity is measured by the hyperbolic tangent of an angle with the time-axis. This angle is referred to as rapidity. And the point is: these angles (rapidities) do add up to zero. That is to say: a Lorentzian rapidity composition formula would take the shape of three rapidities adding to zero. A Galilean rapidity composition formula, on the other hand, would be ugly. In terms of rapidities, Lorentzian relativity is the most elegant form of relativity.

    Moreover, in contrast to velocities, rapidities can be determined as the sum of subsequent accelerations experienced by an object. Arguably, rapidities in Lorentzian relativity are the closest thing to the velocities that appear in Galilean relativity.  Small rapidities are indistinguishable from velocities, but the rapidity of light is infinite. 

    Also rapidities are directly observable as the change in 'tone' of light due to Doppler shift. So when you are stopped by a police officer who accuses you of speeding, ask him how he established your velocity. If his answer contains the words "radar gun", you politely point out that radar Doppler measurements yield rapidities and not velocities. And you finish the conversation with an empathic "Surely you must have learned at the police academy that rapidities are always larger than velocities". 

    By the way: next time you meet folks who believe in tachyonic neutrinos: ask them what rapidities these neutrinos attain...  

    Comments

    By the way: next time you meet folks who believe in tachyonic neutrinos: ask them what rapidities these neutrinos attain...  
    Well, yes that'll fix those damned heretics, won't it?

    Let's try an alternative to rapidity. Since the metric in SR is s^2=x^2+y^2+z^2-(ct)^2 it has always made sense to label the time axis as imaginary time, i.t.
     
    So using the correct, logical units, lets define another measure of speed. In keeping with the spirit of Johannes' question, let's call it bogosity. So bogosity, b, is related to things we can measure by b = -i. dx/dt
     
    So, next time you meet folks who believe in brachyonic motor cars: ask them what bogosity these cars attain? :)

    Okay, so you take the velocity and put it through a nice non-linear function like this: 




     
    http://mathworld.wolfram.com/InverseHyperbolicTangent.html 

    which happens to blow up at x  >= 1 and you then ask "believers" what happens for v > c.
     
    Doh?

    Of course hardly anybody has actually admitted to "believing" in tachyonic neutrinos, but some people are willing to consider the implications if the CERN/OPERA results are confirmed. These silly old crackpots seem to deserve strawmen arguments behind their backs... 
     
    The answer, by the way, is of course, that it goes complex (imaginary?). Something has to go complex at v > c: we've already seen it may mean the (rest) mass or it may mean the transformed dimensions. We knew that from the moment we found the Lorentz factor blew up and then went imaginary. So why the fuss about yet another function that blows up at c?

    And, on another tack, perhaps someone who can remember the maths could say whether the above logarithmic definition relates simply to the Lorentz factor? Otherwise I'll have to work it out myself and that would suck a great deal.

     
    ''Bogusity' is the right term. You can define any math nonsense here, fact is only rapidities add up. The author doesn't state it, but I would argue that relativistic rapiditiy is the true generalization of Newtonian velocity. So rapidities should have been called velocities. All the rest is bogusity.

    ''Bogusity' is the right term.
    No, the correct term is bogosity, with an o, though it does derive from "bogus" with a u.  Google it if you don't believe me.
    fact is only rapidities add up
    What you call "relativistic rapidity" is an additive quantity in Minkovski spacetime.  Newtonian velocity is additive in Galilean spacetime.
    I would argue that relativistic rapiditiy is the true generalization of Newtonian velocity
    Oh, that is so wrong. It so happens we live in a spacetime that is locally Minkovski but that does not make rapidity as discussed here by Johannes, a generalization of Newtonian velocity. It's one alternative of many. There doubtless is such a generalization, but it's not the case discussed here which is just as much a special case as the Newtonian one.  Even with simple spacetime signatures like [1,1,1,1] and [-1,1,1,1] there are another 14 for the cheerleaders to root for.

    Though you may want to discount some as equivalent. At the very minimum you have 5 - according to how many -1s you have. There again you could expand it to a 4 x 4 tensor. And make it complex.
    Halliday

    Derek:

    You said:

    ... Even with simple spacetime signatures like [1,1,1,1] and [-1,1,1,1] there are another 14 for the cheerleaders to root for.

    Though you may want to discount some as equivalent. At the very minimum you have 5 - according to how many -1s you have. ...

    Well, since there is no physically distinguishable difference between a metric and its negative, this reduces the number of unique (non-degenerate, so the metric is non-singular) choices to three (3):  Euclidean [1,1,1,1] (=[-1,-1,-1,-1]), Minkowskian [-1,1,1,1] (=[-1,-1,-1,1]=[1,-1,-1,-1], and other permutations), and Symplectic [-1,-1,1,1] (=[1,1,-1,-1], and other permutations).

    Of course, if we allow for degenerate cases (so the metric is singular, as it is in Galilean relativity), then we have a few more possibilities, based upon how many zeros we choose.

    David

    Okay, I knew I was out of my depth, but thanks for the correction.

    But even three cases start to illustrate that one case is NOT a generalization of another, even if it is a better approximation to experimental reality.
    Three cases? You are misled by your futile math exercises.There is only one case: you might not like it, but nature can't care less, it tells us straight in our face it's locally Minkowskian.

    For goodness sake calm down and stop blustering.
     
    [added later]
     
    Okay. Now for the last time.

    Relativity, as you call it, is not a generalization of Newtonian physics. Neither is Minkovski space a generalization of Galileo. That is what you claimed and that is what I rebutted. The fact that relativity/Minkovski applies to the real world doesn't make it a generalization of its predecessor.  A particular spacetime cannot, by definition, be a generalization of another. 

    This actually a common misconception, largely because Galilean and Minkovski spacetime converge at low velocities or as c goes to infinity. (Newtonian space-and-time, however is incorrigible because the metric excludes time altogether.) Now if you were to claim that relativity provides a mathematical framework for generalizing Pythagorus to 4 (or more) dimensions - with as many spacetime signatures as you can count (pace David!) - I would agree completely, but saying that nature is in your face with just one case, confirms that you are not talking about generalization at all. Perhaps you meant some other word?

    Ok, I will spell it out for you. A special case is obtained as a limiting behavior of a generalization. Galilean relativity is obtained from special relativity in the limit c goes to infinity. Newtonian space and Newtonian time (not spacetime, and not Galilean!) are obtained from Minkowskian spacetime in the limit c goes to infinity. Additive velocities are obtained from additive rapidities in the limit c goes to infinity.

    Yes I know that's what you meant. That's why I said "This actually a common misconception, largely because Galilean and Minkovski spacetime converge at low velocities or as c goes to infinity. (Newtonian space-and-time, however is incorrigible because the metric excludes time altogether.)" 
     
    The fact that two models converge at a physically impossible limit for both of them doesn't make one a special case of the other.

    If you still can't see it then so be it.
     
    "You still don't see it!" said the blind to the deaf... As always, the best comedy dialogues are read in comment sections on the Internet.
    One last time before I give up: you are misleading yourself as you are discussing the limit c -> infinity, while working isith units in which c=1. The Minkowski metric reads [c2, -1, -1, -1]. Got that? Now take the limit c -> infinity, and observe your math arguments (what convergence?) breaking down. The limit c -> infinity is only meanigful in the context of spacetime splitting into a separate space and time.

    Doesn't matter how you spin it, you're repeating the same mistake
    spacetime splitting
    and adding a few more.
    Still, their is something wrong with Einsteins model.

    According to Einstein’s relativity theory, is the speed of light for every observer the same in
    all reference frames.
    However, there seem to be tiny differences in the lightspeed if we observe the outliers of
    satellite to satellite distance measurements.
    At the same time tiny structural irregularities in Planetary radar-pulse reflection
    Measurements can be observed, see: I.I. Shapiro in 1964, radar reflection measurements between the Earth and Venus and Mercury.
    Both observations support the idea of some adding up of lightspeed velocities and the existence of ellipsoidal lightspeed extinction volumes around massive objects like the earth.
    As a consequence I would propose new lightspeed experiments between the earth and dual
    satellites or dual balloons and even in the laboratory to support these lightspeed extinction
    ideas.
    See:
    Experiments to Determine the Mass Related Lightspeed Extinction Volume Around the Earth and Around Spinning Objects in the Lab.
    http://vixra.org/pdf/1102.0056v1.pdf

    The Stand-Up Physicist
    This article was about special relativity. Shapiro's work were tests of general relativity. One can get into a long discussion about whether general relativity is a generalization of special relativity. None-the-less, Shapiro's work is one part of the experimental work that shows a dynamic metric similar to the Schwarzschild metric of general relativity must be used to describe gravity.
    Thank you Dough,

    I am not talking about Shapiro's work in general, but his special research on the RADAR REFLECTION astronomy in 1964. which is different from so the Shapiro solar time delay effect.
    See the pdf link I gave:

    Figure 3, Arrows are pointing to the tiny irregularities or radar residuals.
    Figure from I.I.Shapiro, in “Radar Astronomy” p. 171. by Evans and Hagfors, 1968.
    About the Time delay residuals in figure 3, I.I.Shapiro wrote:
    “Preceding inferior conjunction, the residuals are negative, whereas following they become
    positive.
    This behaviour is readily explained by Venus being ahead of its orbit relative to earth, since in
    that case, it would be closer to earth than predicted before conjunction and further away (from
    earth) afterwards in agreement with figure 3-4.
    Quantitatively too, the amount seems to be in accord with the earlier determinations.
    Remarkably although the residuals shown are enormous relative to errors associated with
    some of the more accurate measurements.”
    My conclusion: Shapiro did NOT account for the possibility that he measured the mutual
    influences of the both LASOF lightspeed ellipsoids of the Earth and Venus, as we do in
    figure 4.
    In figure 4, calculations are made which tell us that the major axes of the LASOF ellipsoids
    for the Earth and Venus are estimated to be respectively 70 and 54 million kilometres.
    Future measurements however will be able to give these numbers a more accurate foundation,
    because only then we are perhaps able to calculate more intensely focussed on this subject.

    Johannes Koelman wrote (January 31st 2012 09:25 PM):
    > [...] writing vXY for the velocity of object X as seen from the perspective of object Y.
    > [...] vFT + vTS + vSF + vFT . vTS . vSF = 0
    > [...] Stacey sees the same flashlight photons also moving forward at the speed of light (vFS = 1). Using reciprocity (vSF = -vFS) it follows that vSF = -1.

    Is it correct and intended to conclude that for every pair, H and K, where "K sees vHK = -1", there may be found some "J" such that "J sees vKJ = 1" and "H sees vJH = 1" ?

    Or is it not perhaps a rather silly statement to assign a real value to vSF as "velocity of Stacey as seen from (the perspective of) certain flashlight photons", for instance ?

    Johannes Koelman
    "Is it correct and intended to conclude that for every pair, H and K, where "K sees vHK = -1", there may be found some "J" such that "J sees vKJ = 1" and "H sees vJH = 1" ?"

    Not sure if I interpret you correctly, but I don't think you are drawing the right conclusion. A correct statement would be that a Lorentzian velocity triplet vAB, vBC, vCA, for which one velocity is +1 and another is -1, the third velocity can have any value.
    Johannes Koelman wrote (02/01/12 | 23:23 PM):
    > A correct statement would be that a Lorentzian velocity triplet vAB, vBC, vCA,
    > for which one velocity is +1 and another is -1, the third velocity can have any value.

    Then let's take the specific case with values
    vCA = +1, vBC = -1, and some particular value vAB < 0.

    Moreover, consider a triplet of numbers labelled "vXB, vBC, vCX" of values
    vXB = +1, vBC = -1 (from above), and some particular value vCX < 0.

    Does this triplet of numbers "vXB, vBC, vCX" correspond to (values of) a "Lorentzian velocity triplet" in this case,
    i.e. compatible and consistent with the above triplet "vAB, vBC, vCA" being (values of) a "Lorentzian velocity triplet" ?

    (If so ... the notion of "Lorentzian velocity" didn't quite seem to match the notion of
    "velocity" familiar for instance from ball games --
    where pitcher and catcher are required to determine their distance between each other,
    and to determine "simultaneity" between the pitcher releasing the ball and the catcher
    starting his watch and/or between the catcher receiving the ball and the pitcher stopping his watch, in
    order to obtain their duration of "having kept the ball in play",
    and finally to divide their distance by this duration to evaluate the (average) velocity of the ball they threw to each other.)

    p.s.
    Sorry --
    "<i>italicised text</i>" is rendered as: "italicised text",
    text"<sub>subscript</sub> is rendered as: "textsubscript" ...

    Frank:As an experiment, I'm trying to see whether I can get italics (emphasis) and subscripting when not logged in.Ok italics (using the <em> tag):  This is supposed to be in italics.Now for a subscript:  Textsubscript.Unfortunately, while the <em> tag works for italics, the <p> tag (for paragraph breaks), and the <sub> tag (for subscripts) don't work.  This is contrary to what the "Allowed HTML tags" seem to indicate.Even, <br> (for line break) doesn't work.David

    Halliday

    Frank:

    I just finished doing an experiment, in which I tried to see whether I could get italics (emphasis) and subscripting when not logged in.

    Ok italics (using the <em> tag):  This is supposed to be in italics.

    Now for a subscript:  Textsubscript.

    Unfortunately, while the <em> tag did work for italics, the <p> tag (for paragraph breaks), and the <sub> tag (for subscripts) didn't work (in the non-logged-in experiment).  This is contrary to what the "Allowed HTML tags" seemed to indicate (both then, and now).

    Even, <br> (for line break) didn't work.

    However, as you can see, even using the "Plain Editor", when logged in, everything works just fine.

    I'm very sorry for the trouble.  This is worse than an inconvenience, in my opinion.

    David

    Halliday

    Frank:

    I'm not quite sure what you are trying to say with

    Then let's take the specific case with values
    vCA = +1, vBC = -1, and some particular value vAB < 0.

    Moreover, consider a triplet of numbers labelled "vXB, vBC, vCX" of values
    vXB = +1, vBC = -1 (from above), and some particular value vCX < 0.

    Does this triplet of numbers "vXB, vBC, vCX" correspond to (values of) a "Lorentzian velocity triplet" in this case,
    i.e. compatible and consistent with the above triplet "vAB, vBC, vCA" being (values of) a "Lorentzian velocity triplet" ?

    OK.  In both cases, you have completely allowable "Lorentzian velocity triplet[s]".  The first is a "Lorentzian velocity triplet" for absolutely any velocity -1 < vAB < 1.  (You chose -1 < vAB < 0.  OK.)  The second is a "Lorentzian velocity triplet" for absolutely any velocity -1 < vCX < 1.  (You chose -1 < vCX < 0.  OK.)

    I suppose you want to know whether your choices are consistent.  Is that correct?

    OK.  Going from the first "Lorentzian velocity triplet" to the second involves introducing a fourth "player"/object X.

    In the first case, we have A and B moving relative to one another at sub-light speed, with A moving in the negative direction as seen by B.

    In the second case, we ignore A, and introduce a new "player"/object X.  In this case, X and C are moving relative to one another at sub-light speed, with C moving in the negative direction as seen by X.

    So, the question appears to boil down to whether there is a consistent velocity choice for X relative to A that can make this happen.  Well, I can already tell that X would be having to move at the speed of light, in the positive direction, as seen by A, in order to go from vCA =1 to any finite vCX.  So, I can see that you have an ill-posed problem here, since the relative velocity of two objects moving at the speed of light, in the same direction is ill-defined.

    In fact, relative velocity between two objects moving in the same direction at the speed of light is not even truly meaningful, since there is no non-zero proper time (the time one sees on one's own wristwatch, for instance) for finite (as seen by any time-like observer, like us) light-like (null) spacetime paths.

    However, this most certainly has nothing whatever to do with your concern expressed as:

    If so ... the notion of "Lorentzian velocity" didn't quite seem to match the notion of
    "velocity" familiar for instance from ball games --
    where pitcher and catcher are required to determine their distance between each other,
    and to determine "simultaneity" between the pitcher releasing the ball and the catcher
    starting his watch and/or between the catcher receiving the ball and the pitcher stopping his watch, in
    order to obtain their duration of "having kept the ball in play",
    and finally to divide their distance by this duration to evaluate the (average) velocity of the ball they threw to each other.

    First, this example consists of only three "players"/objects:  The pitcher, the catcher, and the ball.

    One can introduce a couple of additional light-like players, for the purpose of "determin[ing] their distance between each other" (the pitcher and the catcher), and "determin[ing] 'simultaneity' between the pitcher releasing the ball and the catcher starting his watch and/or between the catcher receiving the ball and the pitcher stopping his watch".  However, never at any point do we have any need, or even desire to determine the relative velocities between any of these light-like players.

    No, for pitcher and catcher having zero relative velocity to each another (the usual "ball game"), there are no new issues, whatsoever, introduced by special relativity over Galilean relativity.

    The only case where new issues come into play are cases where the pitcher and catcher are moving relative to one another.  However, even then, the only truly new issue is the question of "simultaneity".  (There is also the issue of the velocities, and distances the pitcher will measure vs. the catcher.  [The differences in times is more directly related to the issue of "simultaneity" already mentioned.  Actually, even the differences in distances is highly related to the issue of "simultaneity".]  However, even Galilean relativity recognizes that the measurement of velocities and distances can be troublesome for observers moving relative to one another.  The difference is in the form of these issues.)

    David

    David Halliday wrote (02/02/12 | 13:07 PM):
    > So, I can see that you have an ill-posed problem here

    Fair enough. Perhaps I can summarize the point of my example thus:
    There's more involved in the (familiar, i.e. relativistic) notion of velocity than merely the
    "cyclic velocity sum equation of Lorentzian (aka 'special') relativity".

    Specificly:
    > In fact, relative velocity between two objects moving in the same direction at the speed of light is not even truly meaningful, since since there is no non-zero proper time [...]

    Surely the case "vCA = +1, vBC = -1, and some particular value vAB < 0"
    by itself can be considered meaningful. It seems more questionable whether in this case A (as well as B) can be considered
    "moving at the speed of light" "as seen by C"".

    I'd argue that each participant's history of observations (of signals having been exchanged with others) is an essential operational requirement for evaluating any geometric relations of such participants among each other.

    Therefore, for instance, if A (together with a "system" of other suitable participants -- or to stay in the baseball analogy: A as member of a particular "pitcher-catcher-matchup") succeeded in evaluating the velocity of C as "vCA = +1" then there is no implication that C would in turn succeed by the same method in evaluating the velocity of A.
    Still setting "vAC = -1" is not wrong; but that's based on first A having ("actually") evaluated "vCA = +1".

    > The pitcher, the catcher, and the ball.
    > One can introduce a couple of additional light-like players [...]
    > However, never at any point do we have any need, or even desire to determine the relative velocities between any of these light-like players.

    But yes, we do:
    since in my proposed example, for short: "vAB < 0, vBC = -1, vCX < 0",
    both vAB and vCX were supposedly particular, definite values;
    yet A and B are also (meant to be) "lightlike" to C and X, and vice versa.

    > for pitcher and catcher having zero relative velocity to each another (the usual "ball game")

    (It seems actually quite a problem, to be tackled as part of the GTR, how a sufficient number of participants should agree on whether they are "at rest to each other", or not. However, for the present discussion, we should take this issue as solved and consider only this sort of "systems" or "pitcher-catcher-matchups".)

    > there are no new issues, whatsoever, introduced by special relativity over Galilean relativity.

    So how exactly did Galileo prescribe (or at least envision) that pitcher and catcher should obtain their duration(s) of "having kept the ball in play"? ...

    I don't think this is historically accurate -- the Lorentz transformation was known as the symmetry for Maxwell's equations. It wasn't Einstein who decided to change Galilean Relativity -- what he did was to apply Lorentz to everything -- and understand what that meant.

    Johannes Koelman
    The Lorentz transformations were known, but it was Einstein who was bold enough to interpreted them correctly. That is what I meant when I wrote "The mathematical equations for this form of relativity were already written down by Hendrik Lorentz, without him realizing what they really meant."
    I cannot recall having ever been so far out of my depth in trying to follow a discussion. And so I rely upon a rather crude instinctual pivot towards the pre-eminent voice of civility, humanity and self-discipline. Tone wins, I tell myself. I'm wondering whether this hapless strategy would really avail in a situation in which I encountered, say, another life form of manifest dominance. Dunno.

    Einstein's initial insight was that someone moving in a very fast train that could achieve the speed of light would see a beam of light as a static, pulsating electromagnetic field; and that was plainly absurd. It is a testament to Einstein's glorious era that, to this day, people perform their relativity thought experiments with trains and not with, say, airplanes or (bog assist us) spaceships.

    The fact is that our mammal brains cannot avoid referring to a "static" frame of reference for velocities, such as the "static" Earth. Of course then you hear things like "Earth moves through space at a speed of...", which moves the frame of reference to the Sun, and sometimes from our Sun to our galaxy.

    By the way, you have spam. I suggest you use rel=nofollow on comment links, if at all possible -- or at least suggest it to science20.com.

    OK, I get it. Relative to my magnet out of my old hardrive that I have charged to a static potential, I am moving at the speed of light right?

    http://www.mathpages.com/home/kmath216/kmath216.htm "Probabilities and Velocities"!!!!!!!!!!!!!!!!!!!!!!!! but of course,no one knows what really is "probability"....

    let a particle be in a infinite potential hole.the integral between boundaries from wave function squared =1,"the chicken is in the pot".speed over c=probability like>zitterbewegung speed = c or else...

    hk physicist wrote about "mikado universe" long ago.let me:the string theorists(would be theological master puppeteers) have attain a record with 26 dimensions.why don't we see them,the rest till 26?because of Compactification....but the entropic gravity does the opposite-only a "holographic screen" ,a 2D object is needed.waw!opposing "Compactification" is "Emerging".E-merrrrrrrrrrrrrrrrrrr-gennnt!(mel brooks style)."mikado universe" is just too big to beat verlinde,let's try with 1 D ,unidimensional universe!!!!!!!!!!!!let's say we scoop up 3 or 4 sets on the real line,each set of numbers wisely sculptured,gathered,and represent with 1 set 1 dimension in our physical space.rotation in 3D,4D?some morphism!gravity?something mathematically "beautiful",i presume

    hk physicist wrote about "mikado universe" long ago.let me:the string theorists(would be theological master puppeteers) have attain a record with 26 dimensions.why don't we see them,the rest till 26?because of Compactification....but the entropic gravity does the opposite-only a "holographic screen" ,a 2D object is needed.waw!opposing "Compactification" is "Emerging".E-merrrrrrrrrrrrrrrrrrr-gennnt!(mel brooks style)."mikado universe" is just too big to beat verlinde,let's try with 1 D ,unidimensional universe!!!!!!!!!!!!let's say we scoop up 3 or 4 sets on the real line,each set of numbers wisely sculptured,gathered,and represent with 1 set 1 dimension in our physical space.rotation in 3D,4D?some morphism!gravity?something mathematically "beautiful",i presume

    Prompted by Alex's observation that "Our mammal brains cannot avoid referring to a "static frame of reference", I should like to make the following remarks:

    Einstein insisted that a concept, such as that of simultaneity, is meaningless for a physicist unless it can be defined properly; but the idea of a stationary frame of reference is important in the Theory of Relativity because it is implied by Einstein’s "operational” definition of simultaneity. (ie. no stationary - no simultaneous). This is because, according to the theory, clocks in "moving" frames are not all synchronized.

    It seems however that the argument so far is flawed because it lacks a definition for the term "stationary". Perhaps a reader of this blog could furnish a good-enough definition, with extra brownie points for an operational one?

    An operational definition of stationarity is "absence of a signal on an accelerometer". Note that this allows you to also operationalize (relative) rapidity as the integral over proper time of the accelerometer signal.

    Thanks for your interest, Time Traveller, but according to your definition, an object cannot move unless a force is acting on it.

    Only a true Time Traveller could have proposed this venerable idea, which has been out of use for aeons, with such assurance.