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#1
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The Measurement of Contraction
The theory of relativity is presented as one that cannot be understood by most people. This aura of complexity and difficulty is maintained by using ambiguity and vagueness in describing and defining the theory and defending it with even more ambiguity and vagueness against logic. But what is it really? Stripped of all esoteric language it simply means that physical objects (we may call them frames) that are capable of motion (cars, trains, planes, the earth, planets, stars) will shrink along the axis parallel to the motion and have their clocks slow down. The effects are in proportion to the velocity of the object and are called length contraction and time dilation. They can be calculated using the transformation formulas that are the substance of the Lorentz contraction hypothesis. But as the tools of measurement (rulers, clocks) are also adjusted relative to the speed, any change in metres and clock rates will not be noticed by anyone measuring length or time in his environment (the local frame). Regardless of the velocity of the local frame, things measured within the local frame by local frame occupants always have the same size and the local frame time always flows at the same rate. However, the supporters of the relativity theories (both SR and LET) have introduced the concepts of observed frame and observer frame. If an observed frame and an observer frame travel at different speeds, it is claimed that any contraction and dilation affecting the observed object can be measured by an observer in the observer frame. A procedure of measurement which will lead to 'the visibility of contraction' is described in a textbook (Resnick, Introduction to Special Relativity, 1968) as: "An observer is an INFINITE set of recording clocks distributed throughout space, AT REST and synchronized with respect to one another. The space time co-ordinates of an event (x,y,z,t) are recorded by the clock at the location (x,y,z) of the event at the time (t) it occurs. Measurements thus recorded throughout space time (we might call them local measurements) are then available to be PICKED UP and ANALYSED by an EXPERIMENTER who collects the measurements made in this way. Each inertial frame is IMAGINED to have such a set of recording clocks, or such an observer. The relations between the space- time co-ordinates of a physical event measured by one OBSERVER (S) and the space-time co-ordinates of the same physical event measured by another OBSERVER (S') are the equations of transformation". Resnick uses the terms INFINITE, AT REST, PICKED UP, ANALYSED, EXPERIMENTER, OBSERVER (S) and OBSERVER (S'). While the phraseology may not prevent calculating contraction according to the Lorentz formulas with a pencil on a piece of paper, it can surely never be done in the field. It is a smokescreen to support the illusion that contraction is a physical reality. But let us ask questions about such a measurement. Event 1 is the recording of location A of one end of a rod in space at time t by observers S and S'. Event 2 is the recording of location B of the other end of the rod also at time t by the same observers. The experimenter picks up and analyses the four space time co-ordinates made by the two observers. Will the answer be the contracted length of the rod? How will the movement of the two observers relative to the rod disclose the contracted length of the rod and not a distorted view of it? How are optical, communicative and angular aspects included in the measurement? What single fixed co-ordinate system is common to both observers or is it only relative to observer S or observer S'? How will the experimenter establish that the times t of the two observers are simultaneous given that SR imposes a whole set of ifs and buts on simultaneity? How will any difference in the speed the observed object, S and S' be taken into account? It all can be done on paper making assumptions but not as a real test. The more ardent supporters of the contraction hypothesis assert that proof is not required. Their argument is that to 'make predictions' is sufficient or they will do the calculations manually and if these are rejected or ignored by anyone, they accuse him of being incapable of the maths and he should study the subject to understand the theory. With understanding they mean accepting it without reservation. It is the approach of all monotheistic religions which ostracise all non- believers, excommunicate them, vilify them, and ultimately stone or burn them. In this NG it is automatically practised against any member who dares to reject SR or even question it. Peter Riedt |
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"Peri of Pera" wrote in message
... The Measurement of Contraction The theory of relativity is presented as one that cannot be understood by most people. Not 'cannot', just 'is difficult' This aura of complexity and difficulty is maintained by using ambiguity and vagueness in describing and defining the theory No .. the theory is very clear and precise .. however, some popularisation of it are vague and ambiguous, because they use colloquial terms that the layman is more familiar with, and those terms are vague and ambiguous. and defending it with even more ambiguity and vagueness against logic. There is nothing illogical about SR But what is it really? Stripped of all esoteric language it simply means that physical objects (we may call them frames) that are capable of motion (cars, trains, planes, the earth, planets, stars) will shrink along the axis parallel to the motion and have their clocks slow down. No .. nothing happens to the object .. it can't do. Someone moving quickly past you does not change you. But something happens to how they are measured by things that move relative to them The effects are in proportion to the velocity of the object No .. they are not in propoertion at all. and are called length contraction and time dilation. You missed relativity or simultaneity .. so many people do, yet it is probably the most important effect They can be calculated using the transformation formulas that are the substance of the Lorentz contraction hypothesis. Why not just say "Lorentx transforms" But as the tools of measurement (rulers, clocks) are also adjusted relative to the speed, Now you are talking LET, not SR. LET is the one that claim physical shrinking and an absolute frame of reference and aboslute velocities etc (its a bit like what KenSteo talks about, only LET is self-consistent) any change in metres and clock rates will not be noticed by anyone measuring length or time in his environment (the local frame). Regardless of the velocity of the local frame, things measured within the local frame by local frame occupants always have the same size and the local frame time always flows at the same rate. Again, the above is not SR However, the supporters of the relativity theories (both SR and LET) have introduced the concepts of observed frame and observer frame. Well, derr .. there isn't much physics able to be done if there is no observer !! If an observed frame and an observer frame travel at different speeds, it is claimed that any contraction and dilation affecting the observed object can be measured by an observer in the observer frame. There is no contraction and dilation in the observed object in SR .. there is only a measurement of it from the observer frame. A procedure of measurement which will lead to 'the visibility of contraction' is described in a textbook (Resnick, Introduction to Special Relativity, 1968) as: "An observer is an INFINITE set of recording clocks distributed throughout space, AT REST and synchronized with respect to one another. The space time co-ordinates of an event (x,y,z,t) are recorded by the clock at the location (x,y,z) of the event at the time (t) it occurs. Measurements thus recorded throughout space time (we might call them local measurements) are then available to be PICKED UP and ANALYSED by an EXPERIMENTER who collects the measurements made in this way. Each inertial frame is IMAGINED to have such a set of recording clocks, or such an observer. The relations between the space- time co-ordinates of a physical event measured by one OBSERVER (S) and the space-time co-ordinates of the same physical event measured by another OBSERVER (S') are the equations of transformation". Resnick uses the terms INFINITE, AT REST, PICKED UP, ANALYSED, EXPERIMENTER, OBSERVER (S) and OBSERVER (S'). So? While the phraseology may not prevent calculating contraction according to the Lorentz formulas with a pencil on a piece of paper, it can surely never be done in the field. We measure and calculate the positions, speeds, and lengths of moving objects all the time .. what are you alking about? It is a smokescreen to support the illusion that contraction is a physical reality. It isn't in SR. In LET it is a physical reality that we cannot measure (which is rather pointless) But let us ask questions about such a measurement. Event 1 is the recording of location A of one end of a rod in space at time t by observers S and S'. Event 2 is the recording of location B of the other end of the rod also at time t by the same observers. Yes .. that is what length is. The thing is, time t for S and time t for S' are not simultaneous The experimenter picks up and analyses the four space time co-ordinates made by the two observers. Will the answer be the contracted length of the rod? The rod isn't contracted (in SR) .. but he will see possible different measured lengths by the two observers, depending on the relative velocities of them compare to the rod. In LET the situation is the same, excpe that the rod is said to possibly be contracted due to its absolute motion in the ether, but what the observers observe is the same in either case. How will the movement of the two observers relative to the rod disclose the contracted length of the rod and not a distorted view of it? There is no contracted length in SR, and it won't disclose the contracted lenth in LET (you cannot determine what the contracted length is in LET) How are optical, communicative and angular aspects included in the measurement? When we talk about what we the length of the rod is for the two observers to be here, we make adjustments for those things .. ie it is what we measure after tkaing into account the possible differences in transit time for light, aberation etc etc.. What single fixed co-ordinate system is common to both observers or is it only relative to observer S or observer S'? It is relative. In LET there is an unknown absolute ether system of coordinates, but that doesn't really make any difference. How will the experimenter establish that the times t of the two observers are simultaneous They aren't given that SR imposes a whole set of ifs and buts on simultaneity? How will any difference in the speed the observed object, S and S' be taken into account? In what way 'taken into account' .. the observers measure what they measure. If you want to do some calculation using Lorentx transforms back to the frame of the rod itself, then you get that the rod is the length of the rod ... but what I assume you are actually interested in is the length of the rod in the frames S and S' .. ite how much space the rod takes up at a given time in those two frames respectively It all can be done on paper making assumptions but not as a real test. No .. it can be done as a real test. The more ardent supporters of the contraction hypothesis assert that proof is not required. We have experiments that indicate length contraction. It is more difficult to measure as we need to get something up to very high speeds and measure its length .. and that is tricky to do. Their argument is that to 'make predictions' is sufficient or they will do the calculations manually and if these are rejected or ignored by anyone, they accuse him of being incapable of the maths and he should study the subject to understand the theory. Noone argues that. With understanding they mean accepting it without reservation. It is the approach of all monotheistic religions which ostracise all non- believers, excommunicate them, vilify them, and ultimately stone or burn them. In this NG it is automatically practised against any member who dares to reject SR or even question it. Nonsense |
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On Feb 7, 8:04*pm, Peri of Pera wrote:
The Measurement of Contraction The theory of relativity is presented as one that cannot be understood by most people. Unfortunately that is how it is often presented. This aura of complexity and difficulty is maintained by using ambiguity and vagueness in describing and defining the theory Quite true. I often wondered why they would present the theory in such a vague and confusing way, when it can be simply presented w/o vagueness and confusions. Rare are the SR books which are well presented. I have come to the conclusion that SR authors are either clueless about relativity and thus remain vague in its descriptions or that those authors are just really bad pedagogists. I am not saying that SR is wrong and I tend to say the opposite. But I do say that SR is badly represented or described. If relativity authors really wanted to, they can introduce SR in simple terms well accessible to highschool students. But then SR would loose all of its "mystery" and relativist would no longer be viewed as "brilliant". |
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"Peri of Pera" wrote in message ... The Measurement of Contraction The theory of relativity is presented as one that cannot be understood by most people. This aura of complexity and difficulty is maintained by using ambiguity and vagueness in describing and defining the theory and defending it with even more ambiguity and vagueness against logic. But what is it really? Stripped of all esoteric language it simply means that physical objects (we may call them frames) that are capable of motion (cars, trains, planes, the earth, planets, stars) will shrink along the axis parallel to the motion and have their clocks slow down. The effects are in proportion to the velocity of the object and are called length contraction and time dilation. They can be calculated using the transformation formulas that are the substance of the Lorentz contraction hypothesis. Note: the transformation formulas do much more than that: they also account for time dilation and define relativity of simultaneity. But as the tools of measurement (rulers, clocks) are also adjusted relative to the speed, any change in metres and clock rates will not be noticed by anyone measuring length or time in his environment (the local frame). Regardless of the velocity of the local frame, things measured within the local frame by local frame occupants always have the same size and the local frame time always flows at the same rate. That is roughly the way Lorentz formulated the theory, and also close to Einstein's earliest formulation. Presented like that, it's indeed rather easy to understand. However, the supporters of the relativity theories (both SR and LET) have introduced the concepts of observed frame and observer frame. "Observer frame" or "Local frame", what does it matter?! As long as one knows what is meant. It becomes less good when next "observer frame" is replaced by "observer": due to such jargon, all too many people fall in the trap of confusing inertial coordinate system procedures with quantum-mechanical "observer" effects. If an observed frame and an observer frame travel at different speeds, it is claimed that any contraction and dilation affecting the observed object can be measured by an observer in the observer frame. A procedure of measurement which will lead to 'the visibility of contraction' is described in a textbook (Resnick, Introduction to Special Relativity, 1968) as: "An observer is an INFINITE set of recording clocks distributed throughout space, AT REST and synchronized with respect to one another. The space time co-ordinates of an event (x,y,z,t) are recorded by the clock at the location (x,y,z) of the event at the time (t) it occurs. Measurements thus recorded throughout space time (we might call them local measurements) are then available to be PICKED UP and ANALYSED by an EXPERIMENTER who collects the measurements made in this way. Each inertial frame is IMAGINED to have such a set of recording clocks, or such an observer. The relations between the space- time co-ordinates of a physical event measured by one OBSERVER (S) and the space-time co-ordinates of the same physical event measured by another OBSERVER (S') are the equations of transformation". Resnick uses the terms INFINITE, AT REST, PICKED UP, ANALYSED, EXPERIMENTER, OBSERVER (S) and OBSERVER (S'). While the phraseology may not prevent calculating contraction according to the Lorentz formulas with a pencil on a piece of paper, it can surely never be done in the field. It is a smokescreen to support the illusion that contraction is a physical reality. What do you mean with "physical reality"? But let us ask questions about such a measurement. Event 1 is the recording of location A of one end of a rod in space at time t by observers S and S'. Event 2 is the recording of location B of the other end of the rod also at time t by the same observers. The experimenter picks up and analyses the four space time co-ordinates made by the two observers. Will the answer be the contracted length of the rod? In principle, yes. In practice it's hard to test length contraction directly because of the difficulty of creating high-speed extended objects and as the effect is not cumulative. It's different however with time dilation which is regularly measured. How will the movement of the two observers relative to the rod disclose the contracted length of the rod and not a distorted view of it? How are optical, communicative and angular aspects included in the measurement? What single fixed co-ordinate system is common to both observers or is it only relative to observer S or observer S'? ??? "observer S" *is* a co-ordinate system. How will the experimenter establish that the times t of the two observers are simultaneous given that SR imposes a whole set of ifs and buts on simultaneity? It is *defined* as simultaneous. If the PoR is correct then more cannot be done, if that is what you mean. How will any difference in the speed the observed object, S and S' be taken into account? ??? S and S' *are* the co-ordinate systems. It all can be done on paper making assumptions but not as a real test. The more ardent supporters of the contraction hypothesis assert that proof is not required. Their argument is that to 'make predictions' is sufficient or they will do the calculations manually and if these are rejected or ignored by anyone, they accuse him of being incapable of the maths and he should study the subject to understand the theory. With understanding they mean accepting it without reservation. Certainly not. :-)) Instead, when taking into account every imagined possibility, no other reasonable option remains. It is the approach of all monotheistic religions which ostracise all non- believers, excommunicate them, vilify them, and ultimately stone or burn them. In this NG it is automatically practised against any member who dares to reject SR or even question it. No problem with questioning it - that's science! However, be prepared to receive answers. :-) Harald |
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"Peri of Pera" wrote in message ... The Measurement of Contraction [snip] But let us ask questions about such a measurement. Event 1 is the recording of location A of one end of a rod in space at time t by observers S and S'. This is wrong already. First you decide in whose coordinate system the rod is at rest. Let's assume that this is S. So for S' the rod is moving, so S' must make sure he measures the events simultaneously - otherwise he is just daft. Event1 is something that happens on one end of a rod at time t1' according to observer S'. According to observer S who is at rest w.r.t. the rod, this happens at some time t1. According to S' the spatial coordinate of this event is x1'. According to S the spatial coordinate of this event is x1 Event 2 is the recording of location B of the other end of the rod also at time t by the same observers. Wrong again Event2 is something that happens on the other end of the rod at the same time t2' = t1' according to the same observer S'. According to observer S who is at rest w.r.t. the rod, this happens at some time t2. According to S' the spatial coordinate of this event is x2'. According to S the spatial coordinate of this event is x2 If S' is to measure the lenght of the rod, he must make sure that t1' = t2' , so he can *DEFINE* the "coordinate length" as L' = | x1' - x2' | For S the times t1 and t2 don't matter since he is at rest w.r.t. the rod, so he *DEFINES* the "proper length" as L = | x1 - x2 | The experimenter picks up and analyses the four space time co-ordinates made by the two observers. I have counted 8 space time co-ordinates, two of which must be identical: t1' = t2'. Will the answer be the contracted length of the rod? How will the movement of the two observers relative to the rod disclose the contracted length of the rod and not a distorted view of it? If you do it correctly, you will get L' = L / gamma which says that the coordinate lenght is contracted w.r.t. to the proper length. How are optical, communicative and angular aspects included in the measurement? For both observers the times and the distance of events are recorded by sending light signals to the events. This has been explained to you before. See if you can find how it is done. In the one dimensional case there are no angles involved. What single fixed co-ordinate system is common to both observers none. or is it only relative to observer S or observer S'? How will the experimenter establish that the times t of the two observers are simultaneous given that SR imposes a whole set of ifs and buts on simultaneity? See above. How will any difference in the speed the observed object, S and S' be taken into account? See above. It all can be done on paper making assumptions but not as a real test. Indeed, this particular test never has been done. We also never dropped you from a high tower to test whether you will indeed fall. The more ardent supporters of the contraction hypothesis assert that proof is not required. I don't think that you need a proof for your falling down from a high tower either. Their argument is that to 'make predictions' is sufficient or they will do the calculations manually and if these are rejected or ignored by anyone, They should also be consistent with all other experiments which *have* been conducted (and which, by the way, are generally ignored by most people on this forum) they accuse him of being incapable of the maths and he should study the subject to understand the theory. With understanding they mean accepting it without reservation. It is the approach of all monotheistic religions which ostracise all non- believers, excommunicate them, vilify them, and ultimately stone or burn them. In this NG it is automatically practised against any member who dares to reject SR or even question it. No, rest assured, on this NG these people are not stoned or burned. We just have a good laugh over them, and we use them for practice for when we have to explain to our kids. Thanks for that. It has worked superbly! Dirk Vdm |
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Peri of Pera (Peter Riedt) wrote:
The Measurement of Contraction The theory of relativity is presented as one that cannot be understood by most people. That's called "projecting". This aura of complexity and difficulty is maintained by using ambiguity and vagueness in describing and defining the theory and defending it with even more ambiguity and vagueness against logic. But what is it really? Stripped of all esoteric language it simply means that physical objects (we may call them frames) that are capable of motion (cars, trains, planes, the earth, planets, stars) will shrink along the axis parallel to the motion and have their clocks slow down. The effects are in proportion to the velocity of the object and are called length contraction and time dilation. The "shrink" and dilation are not proportional to the velocity. Peter, you've seen the formula for Lorentz "gamma". How you could possibly think it's proportional to velocity? SR is precise. Peter Riedt just doesn't know what the words mean, not even words he uses. -- --Bryan |
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On Feb 8, 12:04*pm, Peri of Pera wrote:
The Measurement of Contraction [...] But let us ask questions about such a measurement. Event 1 is the recording of location A of one end of a rod in space at time t by observers S and S'. Event 2 is the recording of *location B of the other end of the rod also at time t by the same observers. The experimenter picks up and analyses the four space time co-ordinates made by the two observers. Will the answer be the contracted length of the rod? How will the movement of the two observers relative to the rod disclose the contracted length of the rod and not a distorted view of it? How are optical, communicative and angular aspects included in the measurement? What single fixed co-ordinate system is common to both observers or is it only relative to observer S or observer S'? How will the experimenter establish that the times t of the two observers are simultaneous given that SR imposes a whole set of ifs and buts on simultaneity? How will any difference in the speed the observed object, S and S' be taken into account? It all can be done on paper making assumptions but not as a real test. The more ardent supporters of the contraction hypothesis assert that proof is not required. Their argument is that to 'make predictions' is sufficient or they will do the calculations manually and if these are rejected or ignored by anyone, they accuse him of being incapable of the maths and he should study the subject to understand the theory. With understanding they mean accepting it without reservation. It is the approach of all monotheistic religions which ostracise all non- believers, excommunicate them, vilify them, and ultimately stone or burn them. In this NG it is automatically practised against any member who dares to reject SR or even question it. Peter Riedt One cannot really be surprised if SR appears to people to be not well or clearly explained. It is in the peculiar situation of something that is largely correct, and looks like it could be completely correct, but is actually partly wrong. In a situation like this, the most brilliant people can explain it in all ways ad nauseum, and point to the right final results but, somehow, the mind is always left partially unsatisfied. Why? I think that it is like the case of a room with obstacles in it. No matter how people show you how you can move about the room, you are always at least slightly irritated and dissatisfied with the obstacles that never get out of the way. They can tell you that they are not obstacles, and you can easily move around them, but you are not, and never can be persuaded. A perfectly true theory will be experienced as something of clarity, with everything manifestly in order, and in its place. A theory hiding a subtle falsehood, however small, however, will never give such an experience, and people will therefore be forever questioning it and feeling perplexed, even when they cannot put their finger on what it is about the theory that seems unsatisfactory. Such is the case with SR Alen |
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"Alen" wrote in message ... [snip] One cannot really be surprised if SR appears to less gifted people to be not well or clearly explained. It is in the peculiar situation of something that is largely correct, and looks like it could be completely correct, but is in the eyes of less gifted people actually partly wrong. In a situation like this, the most brilliant people can explain it in all ways ad nauseum, and point to the right final results but, somehow, the less gifted mind is always left partially unsatisfied. Why? I think that it is like the case of a room with obstacles in it. No matter how people show you how you can move about the room, you are always at least slightly irritated and dissatisfied with the obstacles that never get out of the way. They can tell you that they are not obstacles, and you can easily move around them, but you are not, and never can be persuaded. A perfectly true theory will be experienced as something of clarity, with everything manifestly in order, and in its place. A theory hiding a subtle falsehood, however small, however, will never give such an experience, and people will therefore be forever questioning it and feeling perplexed, even when they cannot put their finger on what it is about the theory that seems unsatisfactory. Such is the case with your and most crackpot's understanding of SR Severy less gifted Alen Dirk Vdm |
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"jeckyl" :
"Peri of Pera" : The Measurement of Contraction The theory of relativity is presented as one that cannot be understood by most people. Not 'cannot', just 'is difficult' Restricting oneself to the usual algebraic axiomatics and the establishment of Lorentz transforms, I had the opposite problem: it seemed too simple, yet it evaded our intuition... This aura of complexity and difficulty is maintained by using ambiguity and vagueness in describing and defining the theory No .. the theory is very clear and precise .. however, some popularisation of it are vague and ambiguous, because they use colloquial terms that the layman is more familiar with, and those terms are vague and ambiguous. and defending it with even more ambiguity and vagueness against logic. There is nothing illogical about SR .... so I once decided to have a go at a truly intuitive approach of its axiomatics. In my case (and I take it, in many people's that would mean:geometry and graphic rendering. But what is it really? Stripped of all esoteric language it simply means that physical objects (we may call them frames) that are capable of motion (cars, trains, planes, the earth, planets, stars) will shrink along the axis parallel to the motion and have their clocks slow down. No .. nothing happens to the object .. it can't do. Someone moving quickly past you does not change you. But something happens to how they are measured by things that move relative to them The effects are in proportion to the velocity of the object No .. they are not in propoertion at all. and are called length contraction and time dilation. You missed relativity or simultaneity .. so many people do, yet it is probably the most important effect Agreed. A result of an "isotropy of = equality of light clocks" axiom (*), easily taken for granted in a rest frame, but a real hypothesis when realising that there is no such thing as an all round rest frame... (*) actually my axiom 4 in the page below, under "MySRT". See also axiom 5 and comments following: " Axiom 4, put to comparison between different inertial systems, yields differential simultaneity. Axioms 4 and 5 yield length contraction and time dilation. Another result is the constancy of light speed, as a common feature of all light clocks, their isotropy meaning same light time for same distance, but being aware of what 'distance' means in different systems. Actually, understanding the properties of light clocks, and their role as basic gauge tools for measuring length and time in any system, with any ruler and clock equipment, is my cornerstone for an intuitive and geometrical understanding of SRT. The Michelson-Morley experiment may with hindsight be re-interpreted as an attempt to prove light clocks not being true (isotropic) clocks ! " SRT made easy, intuitive, geometrically and logically satisfactory: http://home.scarlet.be/~pin12499/paratwin.htm SRT in my QBasic pages with some graphic examples: http://home.scarlet.be/~pin12499/qbRelaty.html guido |
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: The Measurement of Contraction The theory of relativity is presented as one that cannot be understood by most people. Unfortunately that is how it is often presented. This aura of complexity and difficulty is maintained by using ambiguity and vagueness in describing and defining the theory Quite true. I often wondered why they would present the theory in such a vague and confusing way, when it can be simply presented w/o vagueness and confusions. Rare are the SR books which are well presented. I have come to the conclusion that SR authors are either clueless about relativity and thus remain vague in its descriptions or that those authors are just really bad pedagogists. I am not saying that SR is wrong and I tend to say the opposite. But I do say that SR is badly represented or described. If relativity authors really wanted to, they can introduce SR in simple terms well accessible to highschool students. But then SR would loose all of its "mystery" and relativist would no longer be viewed as "brilliant". guido: Let's see if you don't find my approach perhaps less vague, clueless, confusing (if a bit messy, yes:-), mysterious : SRT made easy, intuitive, geometrically and logically satisfactory: http://home.scarlet.be/~pin12499/paratwin.htm SRT in my QBasic pages with some graphic examples: http://home.scarlet.be/~pin12499/qbRelaty.html guido |
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