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It depends on exactly what one means by [itex]\Delta t[/itex] and [itex]\Delta t^{\prime}[/itex].
Grimble said:So would I be right in deducing that (2) is, in fact, the correct formula??
Grimble said:Should we not take Einstein's own usage as the convention in this case?
He referred to the stationary system as K and the moving system as K'; and his time t was that of the stationary system and t' that of the moving system as transformed by the Lorentz equations.
Rasalhague said:In section 11 of the book you linked to, Einstein uses t' to denote a time which results from applying the Lorentz transformation to a time labelled t. But in section 12, he uses t to denote a time which results from applying the Lorentz transformation to a time labelled t'. So if there is a convention here, it's not defined relative to the Lorentz transformation, which fortunately does have a conventional form.
http://www.bartleby.com/173/11.html
http://www.bartleby.com/173/12.html
Einstein doesn't use the term "moving system" in either of these sections, as far as I can see. (I haven't read the whole book, so maybe it's used elsewhere.) But I don't know how the terms "moving system" and "stationary system" could be used to determine a convention for which frame to call K', since these terms are just as arbitrary, given that each system is moving relative to the other, and an observer at rest in either system will observe the same effect (slowness) in a clock at rest in the other system.
and K' is the moving system in Figure 2.A co-ordinate system K then corresponds to the embankment, and a co-ordinate system K' to the train.
Grimble said:But in chapter 11 he writes:
"A co-ordinate system K then corresponds to the embankment, and a co-ordinate system K' to the train."
and K' is the moving system in Figure 2.
Grimble said:Unfortunately it is confusing until we realize that what he is saying with relation to the Lorentz transformations is that the time in an inertial frame of reference (IFoR) is proper time and we apply the Lorentz transformation to convert it to co-ordinate time, the time perceived in another inertial frame of reference moving at a constant velocity with respect to the first.
If we then compare the resultant co-ordinate time with with the proper time in the 2nd IFoR we, not surprisingly, find that the relationship is the Lorentz Factor.
I believe we can change labels like system A and system B to either IFoR but, because they are fundamental to the Lorentz transformation equations, we need to have a solid convention for what they are ferring to.
Grimble said:...if the number of units can in one view increase and in the other decrease, in one sense the moving clock reads more time has passed and in the other that less time has passed, for do we not reckon time by the number of units of time passing rather than by the size of them?
So in what way does the clock slow?
Grimble said:Another interesting consideration is that however one measures it the total duration of whatever we are measuring is the same, moving or not.
The number of seconds multiplied by the length of one second gives the same total whether it is proper time or co-ordinate time. The difference is, that the unit of measurement changes: take the muon experiment referred to earlier where we have 2.2 microseconds proper time and approximately 65 microseconds co-ordinate time and the conversion is made applying the Lorentz factor which was 29.4
So in which way is it slowing?
Grimble said:And surely, whichever way we calculate it, the co-ordinate seconds have to be smaller than proper seconds, in the same way that co-ordinate metres are smaller than proper metres.
For how else can the speed of light in the moving system - measured in co-ordinate units from the stationary system - still be c?
If the transformed lengths are contracted and the times are dilated how can [tex]c = \frac{d}{t} = \frac {d^'}{t^'}[/tex]
Grimble.
I'm sorry, I haven't explained it well but in the following quote you say that Freund et al. take "dilated time" (co-ordinate units?) to mean an expanded total (of reduced units).Rasalhague said:I don't understand your first point here. We can conceive of the calculation as making the total bigger, or equivalently as making the units smaller, but not both, because that would be like multiplying by gamma squared, wouldn't it? Or if you made the total of one clock bigger, and the units of the clock you're comparing it to smaller, then that would be like making them both bigger, or both smaller, i.e. multiplying by gamma and dividing by gamma, i.e. no change. But there is a change, the calculation converts 2.2 into 65.
Rasalhague said:My point is just that Freund, like Adams, Lerner, Petkov and Schröder, takes "dilated time" to mean an expanded total (of reduced units). So for all of these authors, dilation seems to refer to the quantity of units, the total, rather than--as I thought you originally suggested--the size of individual units. If these authors had taken dilation to refer to the size of units, then surely they'd have used the label "dilated time" for the interval made up of a reduced quantity of these dilated units, wouldn't they?
Yes, the 'clock' by which the muon's lifetime is measured is the muon's lifetime and that is occurring in the muon's rest frame(2.2 microseconds - proper time); the 'lab' time here (approx. 65 microseconds) is that time 'observed' from the "lab's" rest frame; and so it is in co-ordinate time; i.e. it is in co-ordinate microseconds not proper seconds.We're talking about the interval of time between two events on the muon's worldline, its creation and its annihilation. This interval is the muon's lifetime. The muon lasts 2.2 microseconds in its own rest frame and 65 microseconds in the rest frame of the lab. The time between events that happen in the same place is the proper time between them (a timelike spacetime interval). Since the muon's birth and death happen in the same place in the muon's rest frame, the proper time between them is 2.2 microseconds. In the lab's rest frame, the coordinate time of this interval is 2.2 * 29.4 = approx. 65 microseconds. If we think of the muon as a clock, we can say informally that the muon is running slow in comparison to a clock at rest in the lab's rest frame in the sense that the muon only counts to 2.2 microseconds while the lab clock counts to 65. From the perspective of an observer at rest with respect to the lab, a 2.2-microsecond muon is slow to disappear; it lasts for 65 seconds.
Grimble said:I'm sorry, I haven't explained it well but in the following quote you say that Freund et al. take "dilated time" (co-ordinate units?) to mean an expanded total (of reduced units).
Grimble said:And what I am saying is that if the co-ordinate time is an expanded number of reduced units, and if the 'expansion' and 'reduction' are both according to the Lorentz factor, then the only conclusion is that the total duration remains constant. It is precisely the number of units and the size of the units that change.
Grimble said:Yes, the 'clock' by which the muon's lifetime is measured is the muon's lifetime and that is occurring in the muon's rest frame(2.2 microseconds - proper time); the 'lab' time here (approx. 65 microseconds) is that time 'observed' from the "lab's" rest frame; and so it is in co-ordinate time; i.e. it is in co-ordinate microseconds not proper seconds.
But wherever and however it is measured, the duration of the muon's lifetime is and can only be 2.2 microseconds proper time, that is surely a physical constant. We are only discussing how that one fixed interval is measured in different circumstances.
So 2.2 proper microseconds are equal in duration to approx. 65 co-ordinate microseconds?
Grimble said:You have been a tremendous help so far, my friend,(if I may call you that?) and I see much clearer now and appreciate your patience, and following your suggestion I am putting all that I have learned into a new diagram that I hope will pull all these different threads together. I should be able to post it later today - it is certainly helping me to see things more clearly and I hope I am not too presumptious if I ask you to view it?
Thanks, once again for your help and guidance.
Grimble
But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity v is
of a metre.
**Let us now consider a seconds-clock which is permanently situated at the origin (x' = 0) of K'. t' = 0 and t' = 1 are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks:
t = 0
and
As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but
seconds, i.e. a somewhat larger time.
collectedsoul said:Is there a way to derive the time dilation formula without using light clocks? Specifically, without using Euclidean geometry, by simply using mechanics, maybe?
collectedsoul said:Hi,
I've just recently learned about relativity by watching some videos and reading parts of some books and my understanding of it is very basic and quite shaky. It just seems so bizarre to me! So please bear with any errors on my part and correct me.
I think I understood what you said about the light clock derivation using Galilean relativity and simply keeping the speed of light constant and therefore stretching time to account for the increased distance. And this is where I had a doubt in the first place - isn't length being contracted in the moving frame wrt the stationary frame?
If not the vertical length traveled by the light, then the horizontal length (is my reasoning in this regard valid)? If not at all, then in what cases are length contracted?
About your discussion of the coordinate units in comparison to proper units, are you referring in the first case to the relative measurements of a moving object wrt a stationary one, and in the second case to measurements within each frame itself? Then that makes sense to me.
I didn't follow you on the last bit though (not sure what an unprimed system is).
Coming back to the light clock derivation from your second link http://www.answers.com/topic/lorentz-factor#Derivation, I quote the premise:
Imagine two observers: the first, observer A, traveling at a constant speed v with respect to a second inertial reference frame in which observer B is stationary. A points a laser “upward” (perpendicular to the direction of travel). From B's perspective, the light is traveling at an angle.
When I asked the original qs., I was thinking of a case where A points the laser in the horizontal direction rather than vertical. Or simply the case where a stationary light source emits beams measured by two observers at a same distance x, one moving with velocity v and the other standing still. I was trying to find out the effect of light having the same measured speed by both observers using mechanics equations and getting really confused in the process. It'd be a big help if you could explain what is happening in this situation, and whether or not its possible to derive time dilation from this.
Thanks for your patience!
collectedsoul said:isn't length being contracted in the moving frame wrt the stationary frame? If not the vertical length traveled by the light, then the horizontal length (is my reasoning in this regard valid)? If not at all, then in what cases are length contracted?
collectedsoul said:When I asked the original qs., I was thinking of a case where A points the laser in the horizontal direction rather than vertical. Or simply the case where a stationary light source emits beams measured by two observers at a same distance x, one moving with velocity v and the other standing still. I was trying to find out the effect of light having the same measured speed by both observers using mechanics equations and getting really confused in the process. It'd be a big help if you could explain what is happening in this situation, and whether or not its possible to derive time dilation from this.
Rasalhague said:If A emits light in the direction of movement, then length contraction does come into play. In fact this is a typical scenario used by textbooks to derive the length contraction formula.
http://www.pa.msu.edu/courses/2000spring/PHY232/lectures/relativity/contraction.html
Where did you get that from?and t = 2L(c2 - v2)-1/2
Ich said:Where did you get that from?
It doesn't. It calculates the time it takes for light moving at c to reach an object moving at v, that's where those denominators come from.But as this uses light traveling at three different speeds c, c-v and c+v
Rasalhague said:Another link with animations:
http://webphysics.davidson.edu/physlet_resources/special_relativity/illustration4.html
No, the text says moving clocks tick slower as measured in the frame we label as "stationary":Grimble said:Again purely Galilean transformations. With the addition of shortening the moving clock, which ticks slower although the text says it is the stationary clock that ticks slower.
Therefore it takes more clicks as measured by the stationary clock to measure a time interval of a moving clock. Observed from stationary frames, moving clocks run slower. This is called time dilation.
The geometry of space at a given instant in any inertial frame is Euclidean, even if the geometry of spacetime is not. The spatial distance between points that are a distance dx apart along the x-axis of some inertial frame, dy along the y-axis and dz along the z-axis would just be [tex]\sqrt{dx^2 + dy^2 + dz^2}[/tex] in that frame for example.Grimble said:The problem is that these derivations/demonstrations/proofs or whatever are only using Euclidean geometry so how can they have anything to do with SR? Merely adding a length contraction or time dilation proves nothing.
JesseM said:The geometry of space at a given instant in any inertial frame is Euclidean, even if the geometry of spacetime is not. The spatial distance between points that are a distance dx apart along the x-axis of some inertial frame, dy along the y-axis and dz along the z-axis would just be [tex]\sqrt{dx^2 + dy^2 + dz^2}[/tex] in that frame for example.
What do you mean it's not? The Euclidean formula for distance still works fine when dealing with objects that are moving at relativistic velocities in your frame. For example, if you have an object which in its own frame looks like a square 10 light-seconds on each side, and in your frame it's moving at 0.6c so the side moving parallel to the direction of motion is 8 light-seconds long in your frame while the side perpendicular to the direction of motion is 10 light-seconds long in your frame, then the distance between opposite corners of the square is just going to be given by the ordinary Euclidean formula [tex]\sqrt{8^2 + 10^2}[/tex]. Relativity only plays a role in making one side shorter in your frame than it is in the object's own rest frame, but other than that nothing about the object's shape in your frame is any different than it would be if you were just dealing with an 8 by 10 rectangle in classical Newtonian physics.Grimble said:Yes of course, and I certainly agree with that, but one inertial frame observed from another is not, it is relativistic.
In relativity as in Galilean physics, velocity is defined as distance/time, so the time T for an object traveling at speed v to travel a given distance D must be given by T = D/v. And as I said, distances are given by the ordinary Euclidean formula, so if an object travels a horizontal distance Dh between points and a vertical distance Dv, it is just as true in an inertial SR frame as it is in Galilean physics that the total distance the object traveled must be [tex]\sqrt{{D_h}^2 + {D_v}^2}[/tex].Grimble said:Where in all these references is there one allusion to proper time, co-ordinate time or Lorentz transformations?
All I am seeing, as I say, are Galilean transformations, where the addition of an extra movement results in a longer path, and a longer time.
I don't understand, why do you think the 1st postulate implies that the time should be constant? The 1st postulate implies that if you run the same experiment in different frames each frame will see the same result, so if you construct a light clock at rest in frame A and measures the time in frame A, it should give the same answer that you'd get if you constructed an identical light clock at rest in frame B and measured the time in frame B. But the 1st postulate does not say that if you construct a light clock at rest in frame A but in motion in frame B, and measure the time in frame B, that the result would be the same as if you construct a clock at rest in frame B and measure the time in frame B. In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical.Grimble said:IF they hadn't complied with Einstein's 2nd postulate they could have complied with the 1st and kept the time the same but increased the velocity. THAT is the problem with Galilean transformations that Einstain was addressing - HOW to comply with the 1st and keep the time constant whilst AT THE SAME TIME complying with the 2nd and keep the speed of light constant. With these references they only do one at the expense of the other.
I'm not sure what you mean by "space and time not being absolute"--certainly it's true that the distance and time between a pair of events will vary depending on what frame you use, but there is nothing in SR that implies we can't use the Euclidean formula for the distance between events in a single frame, or that we can't use the ordinary kinematical formula that velocity = distance/time in each frame. The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity, along with the uniquely relativistic notion that if the light is bouncing between mirrors at c in the light clock's own rest frame, it must also be bouncing between them at c in the frame where the light clock is in motion.Grimble said:Einstein had to conceive of space and time not being absolute in order to comply with both postulates - thereby giving rise to transformations, and co-ordinate time.
Ich said:Don't you see that you are looking at very basic derivations of SR effects? There are no Galilean or Lorentzian transformations, they only rely on very general properties that you can derive from the postulates.
At the end of the process, one would find out how the transformations have to look like.
Your assertion "To use c+v and c-v it can only be describing a Galilean transformation" is plain nonsense. Just try to follow what they are saying, and try to understand.
JesseM said:What do you mean it's not? The Euclidean formula for distance still works fine when dealing with objects that are moving at relativistic velocities in your frame. For example, if you have an object which in its own frame looks like a square 10 light-seconds on each side, and in your frame it's moving at 0.6c so the side moving parallel to the direction of motion is 8 light-seconds long in your frame while the side perpendicular to the direction of motion is 10 light-seconds long in your frame, then the distance between opposite corners of the square is just going to be given by the ordinary Euclidean formula [tex]\sqrt{8^2 + 10^2}[/tex]. Relativity only plays a role in making one side shorter in your frame than it is in the object's own rest frame, but other than that nothing about the object's shape in your frame is any different than it would be if you were just dealing with an 8 by 10 rectangle in classical Newtonian physics.
I don't understand, why do you think the 1st postulate implies that the time should be constant? The 1st postulate implies that if you run the same experiment in different frames each frame will see the same result, so if you construct a light clock at rest in frame A and measures the time in frame A, it should give the same answer that you'd get if you constructed an identical light clock at rest in frame B and measured the time in frame B. But the 1st postulate does not say that if you construct a light clock at rest in frame A but in motion in frame B, and measure the time in frame B, that the result would be the same as if you construct a clock at rest in frame B and measure the time in frame B. In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical.
I'm not sure what you mean by "space and time not being absolute"--certainly it's true that the distance and time between a pair of events will vary depending on what frame you use, but there is nothing in SR that implies we can't use the Euclidean formula for the distance between events in a single frame, or that we can't use the ordinary kinematical formula that velocity = distance/time in each frame. The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity, along with the uniquely relativistic notion that if the light is bouncing between mirrors at c in the light clock's own rest frame, it must also be bouncing between them at c in the frame where the light clock is in motion.
I am referring to Einstein's statement in http://www.bartleby.com/173/11.html" where he saysspace and time not being absolute
THE RESULTS of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section VII) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows:
1. The time-interval (time) between two events is independent of the condition of motion of the body of reference.
2. The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.
I still don't see it. The problem Einstein refers to arises if you say that w=c-v is the speed of light as measured by the carriage. It isn't.if you read this you should see why I am saying what I am saying.
Ich said:I still don't see it. The problem Einstein refers to arises if you say that w=c-v is the speed of light as measured by the carriage. It isn't.
That does not mean that the equation w=c-v is evil and mustn't be used for whatever purposes. It's just that w is not the speed of light as measured by the carriage. It is the difference of the velocity of light and the velocity of the carriage as measured in the embankment system. That has nothing to do with any transformation laws, as it is a description in only one reference frame.
However to an observer who sees the clock pass at velocity v, the light takes more time to traverse the length of the clock when the pulse is traveling in the same direction as the clock, and it takes less time for the return trip.
- the moving clockHowever to an observer who sees the clock pass at velocity v
Light takes longer - moves slowerthe light takes more time to traverse the length of the clock when the pulse is traveling in the same direction as the clock
Light takes less time - moves fasterand it takes less time for the return trip.
They didn't write what you read. There is no such thing as "observing the clock's frame of reference". You may observe the clock, recording time and date of such measurements as given by your frame of reference.Now I may be wrong, but I read that as saying: "to an observer of the moving clock, when observing the CLOCK'S FRAME OF REFERENCE, that the pulse of light moves slower one way and faster the other..."
(emphasis mine)Light takes less time - moves faster
- whilst traveling the same distance.