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Relative velocity

  1. Sep 5, 2014 #1
    A simple experiment can be build to demonstrate a possible contradiction between thinking theory and reality.

    Use two hand electrical light torches set upon a table and back to back. Switch them on simultaneously, and calculate the relative velocity of their opposite leaving flashes.

    ←c← O= =O →c→

    In this case, the relative velocity between two leaving flashes is twice the speed of light.
    A matter of discussion
    Well, now put yourself in a teacher position and try to explain to your students, how this could be possible?
    I leave this answer to you!
  2. jcsd
  3. Sep 5, 2014 #2


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    Last edited: Sep 5, 2014
  4. Sep 5, 2014 #3

    Doc Al

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    No, the separation rate of the two flashes (according to you) is 2c. But that's not the velocity of anything.

    Further: There's no problem at all with having two things separate at a rate greater than c (as observed by a third frame), but their relative velocity will always be less than c.
  5. Sep 5, 2014 #4


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    As DaleSpam and Doc Al have stated, you misunderstand the technical meaning of "relative velocities". Google "relativistic velocity addition"
  6. Sep 5, 2014 #5


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    Furthermore, we cannot use light as the thing that we measure or define the velocity of anything else relative to. We can use any speed short of c but not c. It would be a lot easier to explain and understand if you start with two objects traveling in opposite directions at a speed of something less than c but for which their simple addition is c or more such as 50% c which results in a velocity of one object relative to the other of 80% c. Or if you want something closer to c you could have the two objects move at 75% c in opposite directions with a resultant relative speed of 96% c between them. If you follow the progression, you can see that no matter how close to c the two objects are moving in opposite directions, their relative velocity never reaches c.
  7. Sep 5, 2014 #6


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    Here are some spacetime diagrams to illustrate the first example that I gave. In all the diagrams, the thick blue lines represents the table or other object from which the other two objects in black and red start off moving at 50% c in opposite directions:


    First, we see how the blue object measures the speed of the red object. To do this he needs to know the distance away that the red object is at some particular time which is marked off in seconds by the blue dots. I'm showing one example where he sends a radar signal at his time of 4 seconds (count the blue dots from the bottom up starting with zero) which hits the red object and reflects back to him at his time of 12 seconds. The difference between these two times is 8 seconds and since the radar signal made a round trip, he divides that by two to get 4 light-seconds as the distance. Now the question is what time does he assume that measurement applies to? The answer according to Special Relativity is that it is the average or midpoint of the two times which is 8 seconds. Since the red object started out at his time of 0 seconds at a distance of 0 light-seconds, he establishes that the object has moved 4 light-seconds in 8 seconds for a speed of 50% c.

    The blue object can do the same thing for the black object to establish its speed at 50% c in the opposite direction.

    Now let's see how the black object would measure the speed of the red object relative to himself. He does exactly the same thing that the blue object did as shown in this diagram:


    He sends a radar signal at his time of 1 second which reflects back to him at his time of 9 seconds (count the black dots from the bottom up). Since the difference is 8 seconds, he concludes that the red object was 4 light-seconds away and he establishes the time to be the average of 1 and 9 which is 5 seconds and since the red object was coincident with him at his time zero, he establishes that the red object has moved 4 light-seconds in 5 seconds for a speed of 80% c.

    If we want, although it is not necessary, we can transform this diagram to the rest frame of the black object which more directly shows the established speed:


    The red object can also measure the speed of the black object relative to himself and will establish the same speed of 80% c in the opposite direction.

    We could do the same thing with my second example but that will result in either a very large diagram or one where everything is scrunched together so I'll leave that exercise up to you.

    Attached Files:

  8. Sep 5, 2014 #7
    I actually started a thread on this exact same topic not too long ago. You can review it for some supplementary discussion..

  9. Sep 5, 2014 #8
    I've no doubt you are right, but this is typically an example where the wording used by physicists triggers misunderstandings. Why is it so difficult to explain that there are two formulas which both hold in SR for combining / composing speeds?

    One formula deals with the transformation of the speed of a single object as represented in a reference frame R into the speed of the same object if represented in another frame R' (ONE OBJECT / TWO FRAMES). It is precisely analogous to the Lorentz transformation which transforms the coordinates (space and time) of the object as represented in R into its coordinates in R'. Hence it should be called a “speed transformation formula” in order to highlight this analogy.

    The second formula combines the speeds of two different objects if both are represented in the same reference frame R (TWO OBJECTS, ONE FRAME): a simple addition as is the case in newtonian physics (the “relative speed between both objects in R”), it being noted that since each of the incoming speeds is necessarily lower than c, the result of their addition is necessarily lower than 2c.

    A meaningful exercise would be to discuss how the “relative speed between both objects in R” evolves for various choices of R.
  10. Sep 5, 2014 #9


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    I think this exercise seems meaningful to you because you are limiting the transformations to be in line with the motion between the objects, aren't you? What if you considered transformations at other angles? Would you still see meaning in the exercise?
  11. Sep 5, 2014 #10


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    I do agree that the terminology is confusing. The terminology developed historically, and could certainly be improved with the benefit of hindsight. However, that never seems to work. We are still trying (unsuccessfully) to get rid of relativistic mass. In the end, students need to learn the bad terminology.
  12. Sep 6, 2014 #11
    I feel it is important to show that various quantities which were considered as “physical quantities” in the newtonian paradigm (I mean quantities characterising intrinsic properties of physical objects or of the relations between those) must at best be considered as features of the representation scheme in the SR paradigm (length, mass, speed, period,...). Their statute is pretty much the same as the statute of coordinates: they provide different but equivalent descriptions of the (simulated) world depending on the selected representation scheme (coordinate system for position and orientation, frame of reference for velocities,...). Features such as “time dilation” and “length contraction” stem from / relate to ... the degree of freedom for choosing a frame of reference for velocities, they do not relate to any intrinsic physical process taking place in the (simulated) world. In this respect, I highly praise the way you separate, in your diagrams and explanations, the layer of “observations” or “measurements” on top of the pure representation layer. I believe all this is easier to explain if the transformations of the reference frame are chosen in line with the motion between the objects, nothing more.

    …. and since we are looking at improper wordings which have disastrous consequences in terms of misunderstandings, I would mention as well that there are no such things as “relativistic effects”: there are phenomena which can be properly explained and predicted in the SR paradigm, whereas they can't be properly explained or predicted in the newtonian paradigm. In the same way, there is no “relativistic Doppler effect”: the classical formula holds in all cases, provided the emitted and the received periods are expressed in the same representation scheme.
  13. Sep 6, 2014 #12


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    And how are such phenomenon NOT "relativistic effects"? I'm not following your reasoning here. You seem to be saying that they are relativistic effects but you don't like calling them relativistic effects.
  14. Sep 6, 2014 #13
    SR explains the non-decaying of muons until they reach the ground, whereas the newtonian theory fails to provide a correct explanation. The discovery of the SR theory did not change anything to the behaviour of muons. Phenomena are not “relativistic” of their own and a theory has no physical effects: it does not effect the world. It has consequences insofar it can or cannot explain or predict specific phenomena. Because it derives the full consequences of the compulsory need for a physical speed limit for any physical object, SR enables a more consistent modelling of the world than the newtonian paradigm, essentially by providing a space-and-time framework which structurally prevents any possibility of involving instantaneous actions at a distance. On that basis, the “time dilation” feature assigns a different value to the mean lifetime of a muon depending on the reference frame chosen for analysing its behaviour. Nothing changes in the world and nothing changes in the behaviour of muons due to your decision to select one or the other frame of reference for performing this analysis. But representing both the distance covered by the muon over its lifetime and the thickness of the atmosphere in one and the same reference frame has remarkable consequences in respect to the predictions that can be derived in the SR paradigm as opposed to the newtonian paradigm. SR has no physical “effects”; its “consequences” relate to improving our ability to explain or predict phenomena.
  15. Sep 6, 2014 #14
  16. Sep 6, 2014 #15


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    I think I will keep on using the term "relativistic effects".
  17. Sep 6, 2014 #16


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    OK, I see what you mean now, but your attempt at changing standard terminology to suit your liking is doing what in the military is called "pissing up a rope".
  18. Sep 6, 2014 #17


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    Thanks for the praise, and high praise at that!!

    I agree that in line scenarios are easier to explain, especially if we want to make spacetime diagrams but we don't want to jump to the conclusion that only in line transformations are preferred.

    I'm also concerned that since Time Dilation and Length Contraction are purely coordinate effects, you seem to have come to the conclusion that changing times and lengths associated with objects as a result of their accelerations are not physical. Is that the case?

    Another concern: the classical Doppler formula does not hold in all cases. There is a "relativistic Doppler effect" which is different from the classical Doppler effect. The classical formula incorporates two speeds whereas the relativistic formula has only one and you can't get it by just setting one of the speeds in the classical formula to zero. And like all "observations" or "measurements", it doesn't matter which representation layer you use to depict a scenario involving Doppler, they all work equally well.

    Please explain your position on Doppler.
  19. Sep 7, 2014 #18
    Thanks for challenging my views.

    I need to be careful with the wording and English is a foreign language for me... First I'll address my understanding of the background of your question. I understand that we both agree that Time Dilation and Length Contraction relate to our formal representation of the physical time and space, so according to SR they “effect” (I would better say “constrain”) delta-time and delta-space quantities in a variable way depending on the selected reference frame. An analogy with space “coordinates” can be found with the representation of a rectangular stick in a 2D coordinate system, for which the values of the delta-x and delta-y projections of the stick onto the x and y coordinate axes will depend on the relative orientation of the axes system in respect to the stick. The delta-x and delta-y values are attributes to the representation of the stick for a given choice of the coordinate system (its “image”, if you prefer), they are not directly attributes to the stick itself. They are not "physical" as per your wording. The only direct attributes to the stick are its “proper” length and width, independent of any representation scheme. There are however a few peculiar cases where the values of the attributes to the image of the stick match the values of the “proper” attributes to the stick, I mean when the axes are selected to be parallel to the rectangle lines. But please note that the matching of the values does not imply the identity of the concepts behind them: an attribute to the image in a given coordinate system shall not be confused with an attribute to the object itself.

    My statement is that SR constrains the changing values of delta-time and delta-space associated with (or which are attributes to) the formal representation (the image), in any given representation scheme, of the physical objects. According to SR, the only viable representations of the physical world which are compatible with the existence of a speed limit must constrain the attributes to the representations (images) of the physical objects according to the Time Dilation and the Length contraction formulas. Valid predictions can only be derived in the framework of a valid representation scheme. SR does not tell anything else and this is enough to reach predictions which are verified by experiments.

    Let's now come to your wording. You are “concerned” about my suspected “conclusion” whereby “changing times and lengths associated with objects as a result of their accelerations are not physical”. First I hope that the two paragraphs above have convinced you that my conclusion is not in line with your wording, for two reasons: i) “changing times and lengths” are not “associated with objects”, but are attributes to their representations, just like the delta-x and delta-y are attributes to the representation of the stick in the example above; ii) SR only deals with those indirect attributes and does not constrain the direct ("physical") attributes to objects, just like my choice for the orientation of the coordinate system does not "effect" the “proper” ("physical") dimensions of the stick. Second, I will in turn challenge your wording insofar I believe it can be interpreted as if you consider that SR constrains the physical attributes of objects. How is that possible unless you think (contrary to my expectations) that my choice of a representation scheme “effects” the physical world? I'm quite convinced that your wording goes beyond what you actually think. Is that the case?
  20. Sep 7, 2014 #19


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    Your English (spelling, grammar, etc.) betrays the fact that it is foreign to you. However, I believe you have made one common mistake that even people for whom English is their native tongue often make which is mixing "effect" and "affect". Most of the time, "effect" is a noun and "affect" is a verb. It would make more sense to me if your three uses of the word "effect" were replaced with "affect".

    You have just restated the position which we agree on that Time Dilation and Length Contraction are coordinate effects. In other words, given the coordinates in one frame of a scenario, we can use the Lorentz Transformation process to establish the coordinates in another frame moving with respect to the original frame.

    And this is your conclusion which we don't agree on. One of the most important aspects of Special Relativity is that the Lorentz Transformation process applies not just to coordinates but also to the formulas and equations of physics. They must remain unchanged after undergoing the Lorentz Transformation. For example, under Newtonian physics, the force equation is F=ma but this does not transform unchanged using the Lorentz Transformation. SR revolutionized all of physics by requiring all laws to be changed so that they remain intact through the Lorentz Transformation process. Maxwell's equations already fit the requirement.

    As a result, all objects will undergo some sort of physical deformation during acceleration which can approximate the same effects that Time Dilation and Length Contraction yield but it's not exactly the same and Special Relativity is inadequate to predict exactly what will happen.

    As I explained earlier, the choice of a representation scheme does not affect the physical world but there is more to it than that. Let me use a couple of scenarios to explain what I mean.

    If we take a clock and express its Proper Time according to a frame in which it is at rest and then transform to a second frame in which it is moving, we will see that it is Time Dilated in this new frame. Nothing physical has happened as a result of this process.

    But let's take that same clock that is at rest in the first frame and accelerate it so that it acquires the same speed that it had when we transformed it to the second frame. Now something physical has happened to it and we have to ask ourselves if it is now Time Dilated in exactly the same way as in the first scenario? In one sense we can answer "yes" because otherwise it doesn't fit the definition of a clock but in a more practical sense we have to answer "we don't know" because depending on how we accelerated it, we may have stressed it to a degree that it is now deformed and ticks at a different rate or we could have even destroyed it.

    Take for example an ideal light clock that we accelerate along the axis of the propagating light. The two mirrors have to be supported by some structure. We could accelerate one end of the structure such that we initially compress the structure or we could accelerate the other end resulting in an initial stretching of the structure. Then when we end the acceleration, the opposite stresses would apply. In either case, SR can't tell us if the structure ends up with all the stresses removed or whether the clock has been compromised.

    For the second scenario, consider a meter stick that has been calibrated against the old official meter stick standard. We can verify that its length is one meter using the new official meter standard using a cesium atomic clock and measuring how long it takes for light to propagate from one end of the meter stick to the other (where we have attached a mirror) and back to the starting point.

    Now we accelerate the meter stick along its length and again verify its length. Will it be the same? Depending on how we accelerate it, probably not or definitely not. Again, SR is inadequate to answer that question precisely but we do know that it will have undergone a physical deformation.

    Hopefully, this clears up any confusion in my wording.

    Now what about my concern over your understanding of Doppler?
  21. Sep 8, 2014 #20
    I think I now understand what you had in mind in #17 with the expression: “changing times and lengths associated with objects as a result of their accelerations” but this is clearly a non-issue. If a physical acceleration or any other physical process has altered the proper length of the stick or the proper period of the clock, the same length contraction and time dilation factors as beforehand will apply to their representation as “in motion” in a given IRF, but they will of course apply to the new proper (“physical”) characteristics of the objects. SR does not describe physical processes, it only provides a consistent framework for representing their effects. So I invite you to reconsider your statement whereby “... SR is inadequate to predict exactly what will happen ...” since asking whether the proper length is affected by a physical acceleration is clearly out of scope of SR.

    The newtonian mechanics had the same requirements, except that it invoked the galilean transformation. For me, part of the “revolution” brought by SR was to generalise the concept of “invariance through a change of the coordinate system” into the concept of “invariance through a change of the representation scheme”, the latter also encompassing a change of the inertial frame of reference for velocities. The consequence is indeed that “laws of physics” must now be understood as “laws of newtonian mechanics” and “laws of SR mechanics” respectively, since their expression is necessarily theory-laden, due to the difference in the invariance criterion (combined with the existence in SR of a max speed limit). In both cases it is clear that equations and formulas describing the effects of “physical” processes must be invariant through the relevant transformations. So I don't see any matter for a serious disagreement there.

    My statement in #18: “SR does not tell anything else and this is enough to reach predictions which are verified by experiments.” … (to which you negatively reacted) purposely insisted on the fact that SR does not make any statement about the “physical” behaviour of (e.g.) clocks: it does not state that they physically “slow down” when being “in motion”, it only states that their coordinate-like period when they are represented “in motion in respect to an IRF" must comply with the Time Dilation formula. Here I see a potential for disagreement since most (if not all?) text books on SR at best leave this issue open and very often display an ambiguous wording, if not worse. Please comment.

    I'll come back to you about the Doppler effect asap. Thanks.
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