Langevin Twins Paradox Proof

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  • #1
Yehuda
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Paradox was verified by Hafele-Keating in 1971. They placed atomic clocks in commercial airplanes and then compared to a reference clock. Recently some people/groups are considering their data as questionable & unreliable. Does somebody know later experiment that also verified the paradox in a similar way?
 

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  • #2
Meir Achuz
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It is not a paradox, unless you don't understand it.
 
  • #3
russ_watters
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It isn't a paradox, its a well-proven consequence of Relativity.

Currently, there are 24 GPS satellites circling the globe (plus, iirc, 4 spares) at various orbital inclinations. Each contains a clock(actually, 3 clocks, iirc) superior to the ones used in the 1971 experiment and since each is going ~40x faster and flying 10x higher than the planes in the 1971 experiment, the Relativistic effects are far more pronounced. The GPS system verifies the predictions of Relativity to an extremely high precision.
 
  • #4
yogi
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Almost all authors agree there is no paradox - but there is much disagreement as to why.
 
  • #5
ahrkron
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yogi said:
Almost all authors agree there is no paradox - but there is much disagreement as to why.

There are different ways to explain the effect, but there is no disagreement on the fact that it comes from relativity.
 
  • #6
yogi
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Some of the explanations are mutually inconsistent - what is agreed upon, in general, is the validity of time dilation. What is not explained is the physical mechanism that produced an actual difference in the Hafele-Keating clocks when they were compared, or any other experiment where one clock has lost time relative to another.
 
  • #7
pervect
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All of the explanations arrive at the same numerical results. Any "inconsistencies" are not the result of different predictions, but of different philosophies.

I believe that yogi is attributing properties to clocks that they simply don't have, and that his search for a "physical" explanation of their behavior is based on the faulty assumption that two clocks following a different path "should" both read the same time when they next meet. There is no reason to suppose that clocks have or should have this property, though in older Newtonian theory clocks did have this property.
 
  • #8
yogi
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Pervect - not at all - I don't know how you could interpret my post(s) as postulating that two clocks which follow different space-time paths will read the same when later compared.

Sometime back on another thread I raised the question re the mechanism that produced actual time differences - you posted a short explanation which much intrigued me - then the thread went off in a different direction. I would appreciate your reiteration of that idea - and perhaps some additional embellishment.
 
  • #9
Hurkyl
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So what do you mean by "What is not explained is the physical mechanism that produced an actual difference"? More precisely, what do you mean by physical mechanism, and why do you think one is needed?


Suppose I had marked two spots on the floor, and I had drawn two different paths that led from the first spot to the second, one a straight line, and the other a semicircle.

I find two people who happen to have devices that measure the length of a path, and set them on each path. They roll their device along their path. I claim that the devices will give different readings: the device rolled along the semicircle will measure a greater distance than the device rolled along the straight line.

Would you object that I have not given a physical mechanism that produced an actual difference in the measurements of the devices? Would you take this as a flaw of Euclidean geometry, and any physics based upon it?


The situation with clocks is similar. Wait, that's misleading... it's virtually identical. The paths in my experiment are directly analogous to trajectories through space-time. The length measuring devices are directly analogous to clocks.
 
  • #10
yogi
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Hurkyl - that is a description - I find no fault with the description, but it is not an explanation - how does a clock that is put in uniform motion wrt to another clock to which it has been synchronized know at what rate to run?
 
  • #11
derz
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yogi said:
Hurkyl - that is a description - I find no fault with the description, but it is not an explanation - how does a clock that is put in uniform motion wrt to another clock to which it has been synchronized know at what rate to run?

"Laws of nature are the same to every inertial observer".

This is the basic postulate of SR. The value of the interval (or the speed of light, c) has to remain constant to every observer, no matter what their relative speed is. Otherwise laws of nature would be chaotic.

For this to be valid, time and distance can't be the same to every observer, but are relative concepts that depend on the relative speed of the observers.

btw. hi I'm new here. Sorry if I don't write perfect english, I'm finnish :smile:
 
  • #12
Hurkyl
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how does a clock that is put in uniform motion wrt to another clock to which it has been synchronized know at what rate to run?

The same way that a measuring tapes "know" to give different readings when they measure different paths: the measuring tapes measure length, so when they measure paths with different lengths, they give different readings.

In the same way, clocks measure proper time. So, if used to measure two paths along which the proper time is different, the clocks will measure different durations.
 
  • #13
Hurkyl
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Hurkyl - that is a description - I find no fault with the description, but it is not an explanation

If you're going to make these sorts of objections, you had better say precisely what you mean by "description" and "explanation".
 
  • #14
DrGreg
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yogi said:
how does a clock that is put in uniform motion wrt to another clock to which it has been synchronized know at what rate to run?

I have two rulers (measuring rods) and lay one in a north-to-south direction and the other in a northeast-to-southwest direction. How does the second ruler "know" to measure distance traveled northeastwards instead of distance traveled northwards ?

(The question is, of course, rhetorical. This is another way of saying what Hurkyl just said.)
 
  • #15
yogi
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Drgreg: That is the big difference - when one observes lengths in a relatively moving reference frame, the contraction is an illusion, an apparency - the lengths do no really change - as Dr Resnick puts it: The contraction is real only in the sense that the measurements are real, There is no permanent physical change as Lorentz proposed to save the ether. Or as Eddinton once said: The contraction of lengths is true, but its not really true (whatever that means).

As for temporal changes - there is a permanent record - relatively moving clocks log different times, and this time differential can be ascertained when the two clocks are brought to rest in the same frame as per Hafele-Keating. Time dilation is not an fleeting phenomena - this is the significance of Einsteins explanation of the physical effects described in part IV of his 1905 paper - and that is the element that distinquishes SR from all the assertions that it was Poincare or Lorentz or somebody else who should be accorded recognition for the concept - Einstein stuck his neck out and predicted actual time dilation. No one else had the courage to go that far.

My question remains - perhaps it is a consequence of spatial motion relative to Minkowski Spacetime - the problem with this interpretation is that it smacks of a preferred global reference frame.
 
  • #16
russ_watters
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yogi said:
The contraction is real only in the sense that the measurements are real, There is no permanent physical change as Lorentz proposed to save the ether. [emphasis added]
There is a difference between a clock and a tape measure that no one has picked-up on: a clock is a recording device (actually, two separate pieces: a measuring device and a recording device) and a tape measure isn't. Once a clock ticks off a second, that second is gone, never to be seen again. The only thing left is the record of that second, which for a clock is the time output by the display. So the measurements taken by a clock are exactly as permanent as those taken by a person with a clipboard standing over a tape measure. Better yet, one of those newfangled laser tape measures could keep a "permanent" record of length contraction.
 
  • #17
Hurkyl
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Given two different trajectories in space-time, the proper time along those paths is generally unequal. Thus, the time measured by clocks traversing each trajectory will generally be different.


P.S. when length measurements of the same "object" are conducted in two different frames, what is actually measured is different for each frame. Thus, just like the clocks, different paths have different proper lengths, and thus the results of the measurement are different.
 
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  • #18
yogi
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Hurkyl - in the usual case, we consider one observer to be in the rest frame, so there is no proper distance to be attributed to the at rest clock. Of course in the general case, (as per your post 9 above) three adjacent clocks A, B and C could be sychronized, and A could travel to a distant point D measured along a straight line connecting C and D and B could follow a semicircle in the CD rest frame and arrive at D. So each clock would have a different trajectory - and a different proper path length - but does this reveal anything about why the two moving clocks do not run at the same rate.

What is interesting about the development of the conclusions in part IV of the 1905 paper is the shift from observational symmetry with regard to length contraction, to the conclusion that temporal differences are real - and consequently they are "one way" since at the end of the experiment A cannot read more than B and at the moment and in the same frame B cannot read more than A. This all comes with very little comment.
 
  • #19
Hurkyl
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but does this reveal anything about why the two moving clocks do not run at the same rate.

Yes it does: the paths they took are different. The paths have different proper times. Clocks measure proper time. Therefore the clocks give different measurements.



Let's try this from the top.


Space-time is four-dimensional. We have a metric on space-time: the measure of a straight line segment is given by, in any inertial frame: (using the +--- convention, and units where c = 1)

d(P, Q)² = (Δt² - Δx² - Δy² - Δz²)

And through the methods of calculus, we can turn this into an infinitessimal metric which can be used to measure curves through space-time.

A bit of work shows that d(P, Q)² is independent of the inertial reference frame used to compute it, thus the metric is a "pure" geometric object -- we can speak of its value without referring to any sort of coordinate system.


When d(P, Q)² > 0, we say that the line segment PQ is time-like, and that |d(P, Q)| is the proper time along the line segment. When d(P, Q)² < 0, we say that PQ is space-like, and that |d(P, Q)| is the proper length of the line segment.

This terminology also applies to curves through space-time where the metric does not change sign.


Now, if we have a point-like object traveling through space-time, its trajectory is a time-like curve, called its world-line. If it's traveling inertially, it's actually a time-like line! We can pick two points on the worldline, and ask about the proper time experienced by the object as it traveled from one point to the other.

A clock, Special Relativistically speaking, is a device that measures the proper time along the worldline it traverses.

So, if I have two (time-like separated) points in space-time, and two clocks follow two different worldline (segments) joining the two points, then they've measured the metric along two entirely different curves, and it should be clear that the circumstances would have to be very special for the readings to be the same.

(Note that in this discussion of traveling points and clocks, I've not invoked a single mention of any sort of coordinates!)

We can draw space-like curves in space-time, and also ask about the lengths of these things.


Measuring the length of an actual object is another question all together. Suppose we have a one-dimensional object traveling through space-time. It traces out a two-dimensional shape through space-time, called its worldsheet.

Now, it doesn't make sense to ask about the length of a two-dimensional shape! However, I can ask about the proper length of a cross-section of a worldsheet. Clearly, differently angled cross-sections will have different lengths.

Now, I'll step back to an inertial reference frame: when we ask about the length of an actual (one-dimensional) object, what we really mean is that we want to compute the proper length of a cross-section of the object's worldsheet. We use our coordinates to specify exactly which cross-section: usually, we want something like the intersection of the worldsheet with the hyperplane of a given, constant time coordinate. (A hyperplane of simultaneity)

Different inertial reference frames have different hyperplanes of simultaneity, and thus will use different cross-sections when measuring the length of a one-dimensional object.


Inertial reference frames are also good for selecting cross-sections of worldlines as well (a.k.a. a point on the worldline). Given a worldline and two values of coordinate time (or hyperplanes of simultaneity, if you prefer), we can speak of the segment of the worldline between those coordinate times. Since we can compute the difference of the chosen coordinate times, and the proper time along the segment, we can form the ratio Δτ / Δt. The infinitessimal version is interpreted as the "rate change of proper time with respect to coordinate time", or time dilation.

Note that time dilation is always with respect to coordinate time! It doesn't make sense to ask about the rate of one clock with respect to another clock directly! You have to do it indirectly by comparing both clocks to a coordinate time!

It should be clear that we should have no expectation of clocks running at the same rate with respect to coordinate time: if two clocks are traveling along different worldlines, and we select intervals between two coordinate times, the clocks are used to measure the proper time along two entirely different worldline segments, and entirely different worldline segments will generally have entirely different proper times along them.



This is all pure geometry: the only physics involved is the hypotheses that this geometry describes the universe, and that there is a device capable of measuring proper time. This is why I say that when you ask for an "explanation" of why two clocks run at different rates (with respect to a chosen coordinate system), that it's tantamount to asking for an explanation of why two line-segments in Euclidean geometry can have different lengths.


edit: fixed mistake involving the metric
 
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  • #20
robphy
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typo: d(P, Q) = (Δt² - Δx² - Δy² - Δz²) [no square root]... or else use d² to distinguish the classes of 4-vectors.
 
  • #21
Hurkyl
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You're right. Except for when I wrote down the formula, I was thinking of d(P, Q)², and not d(P, Q).
 
  • #22
yogi
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Hurlyl - very good tutorial - I will comment tomorrow when not so tired.
 
  • #23
yogi
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Hurkyl - its all well and correct to conclude that clocks following different world lines will measure different proper times - and that all is geometric, and that this geometry follows from the metric - - but there is still a gap as to how the reality of different clock rates arises from a theory that is based upon observations in another frame - in particular, measurements of spatial lengths in another frame (improper lengths) and the notion that events that are simultaneous in one frame are not necessarily simultaneous in a relatively moving frame. From these foundations the transforms are derived - still there is no physics per se. I keep going back (and everyone is probably tired of hearing it) to The 1905 paper - it would seem that the observed slowing of clocks is reciprocally observed (in which case there is no reality of one clock slowing wrt to another) until Einstein proposes to synchronizes two clocks in one frame and then move one until it reaches the other - in which case there results an actual time difference when the two clocks are compared. Einstein calls it a "pecular consequence"
 
  • #24
dgoodpasture2005
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the clocks may read different times, but the same amount time has passed. It's just the clocks at fault, don't be fooled... it's gravity and gravity forces acting on the clocks mechanisms.
 
  • #25
russ_watters
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dgoodpasture2005, time dilation is observed in clocks that have no mechanical parts whatsoever. It is also the same in all clocks capable of measuring it - if it were some sort of "clock effect", different clocks in the same frame would not necessarily agree.
 
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  • #26
Hurkyl
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yogi: I'm not really sure what you're trying to say in #23. Are you asking "How does one derive Minowski space-time from the hypotheses of Special Relativity?"

(Minowski space-time is what I was describing in #19. I figure you already know this, but I want to be sure)

It's somewhat easier to work the other direction: to show that Minowski space-time satisfies the constraints laid out by Special Relativity.
 
  • #27
yogi
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Hurkyl - I have struggled with the evolution of Einstein's development of the physical reality to be attributed to a time differential between the "at rest" clock and the moving clock that has been previously sychcronized in the same frame - let me ask a couple of questions to establish a base.

If two non accelerating spaceships pass each other at relative velocity v - would you say (as is commonly asserted) that each would observe the other clock to be running slow? For example, each spaceship could set up two clocks separated by a known proper length in their own frame and measure the time passage on a single clock in the other spacecraft as it traversed the distance. Will they both yield the same value as to the slowing of time in the other frame?

If yes - then there is no absolute (real) time dilation in this fact situation

Now if the answer is yes - what if the clocks in the two spaceships had been previously synchronized thousands of years ago, and one of them had done all the traveling (big circle at constant velocity to Altair and back) while one remained at rest - but the historic information as to their previous togetherness had long since been lost - we then have two spaceships bearing clocks that should not be running at the same rate according to Einstein's assertions in part IV of the 1905 paper. With no historic information - how does one know the difference between the two circumstances when encountering a passing spaceship?
 
  • #28
Hurkyl
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I'm going to work through some exercises that will help with this problem. I really working on things geometrically, but I'm probably biased because I didn't feel I understood what was going on until I really understood these space-time diagrams.

If my directions aren't clear, I can try drawing these diagrams and attaching the image.

Exercise 1: synchronization of clocks

Here is a method of geometrically synchronizing clocks. This is theoretically important because it demonstrates that you can do such a thing with absolutely no notion of a reference frame. (And we will use it to prove a theorem!)


First off, we have to figure out how to draw space-time. Take out a sheet of paper. Let's decide that time runs vertically and space runs horizontally, and that light-like lines are at 45° angles.

Now, let's say we have a clock traveling inertially. Represent this by drawing a line on your paper. Since anything travels in a time-like direction, this means the line will be closer to vertical than horizontal.

Now draw a line parallel to the first line. This could be another clock that is "stationary" with respect to the first one.


Now, we synchronize clock synchronization!

Pick a point A on the first line. Draw the two light-like lines passing through A. These will both intersect the second line, let's label those points B and C.

(Physically, it means we send a light flash from point B that is received and reflected at point A, and received at point C... or swap B and C if you labelled them the other way)

Let D be the midpoint of B and C.

We synchronize the clocks by specifying that the times at points A and D are equal. We can call the line AD a line of simuntaneity.

(Take a moment to convince yourself this gives the "right" result when the clocks are "stationary")

Theorem: The angle bisector of angle <ADB is light-like. Similarly for <ADC.

A tricky theorem. It has lots of useful corollaries.

Corollary: If two clocks are synchronized by the above method, and their worldlines are at a (Euclidean) angle of x° from vertical, then the line of simultaneity are at an angle of (90-x)° from vertical.
Corollary: You only need one clock to determine the lines of simultaneity.
Corollary: Multiple clocks on parallel worldlines may be simultaneously synchronized by the above method.

These set the foundation of the notion of an inertial reference frame. Also, it gives a quicker geometric way of synchronizing two clocks traveling parallel worldlines: you can simply draw a line of simultaneity wherever you need to do so.


Exercise 2: time dilation

Now that we know how to draw lines of simultaneity, we can work through one of the classical experiments. We're going to have to use the metric this time.

We have two spaceships, each with a clock on its front and rear. The clocks on each ship are synchronized by the above method.


Let's assume, for simplicity, that one of the spaceships is traveling vertically. Draw two vertical lines denoting that ships clocks.

Draw a single slanted line denoting one of the clocks on the other ship. This intersects the two vertical lines in points A and B.

Let the coordinates of point A be (x, t), and of point B be (x', t'). (We don't need their exact values)

Now, since the lines of simultaneity for the vertical clocks are horizontal, we can compute a time difference of |t - t'|.

Using the metric, I can compute the proper time along the line segment AB, which is &radic;( (t-t')² - (x - x')² ), which is clearly less than |t - t'|. Thus, we've geometrically determined time dilation.

Notice, in particular, that this experiment requires three clocks. We can reduce the number to two, by using the (here, horizontal) lines of simultaneity instead of the other vertical clock, allowing us to define the rate of the other clock with respect to the coordinate time defined by the lines of simultaneity.

In fact, we don't need the first clock at all -- we can simply specify some parallel space-like lines as being the lines of simultaneity of some reference frame, and then observe the rate of a particular clock with respect to this coordinate time.

(Again, emphasizing that time dilation is a clock vs coordinate time, not one clock vs another clock)


The answer to your question is yes: in this circumstances, in the reference frame defined by the clocks of one ship, the clocks of the other ship will be observed to be dilated, and vice versa.



one of them had done all the traveling (big circle at constant velocity to Altair and back) while one remained at rest

That's impossible: I think you meant constant speed. Constant velocity would mean it was traveling in a straight line through space-time. Then, we have the initial and final points lying on both (straight) worldlines, and thus they must be the same line.

Anyways, one of these lines will be straight, and one will be a curve (some sort of helix-like thing in 3D space-time). There's a theorem that the longest time between any two time-like separated points is a straight line. (Analogous to the shortest distance between two space-like separated points is a straight line)


... how does one know the difference between the two circumstances when encountering a passing spaceship?

I'm not sure exactly what you mean...


Anyways, let's observe what acceleration does to lines of simultaneity, since we can do that now! (I'm not sure this relates to your question or not, but it's still a good exercise!)


Let's draw a worldline of a clock that accelerates. Start off by drawing a vertical line from the bottom of your paper. (I'm assuming forward in time runs upward). Then, gently arc clockwise a short distance, then draw it the rest of the way as straight (but now slanted). Put a little point where it started and finished arcing, and we'll call the time at those points 1 and 2. Mark two other points on the world line corresponding to times 0 and 3.

(So, it went straight from 0 to 1, arced from 1 to 2, then straight from 2 to 3)

Draw the lines of simultaneity at points 0 and 1: they should be horizontal. These lines correspond to coordinate times 0 and 1.

Do the same at points 2 and 3: the lines should be slightly counterclockwise of horizontal. These correspond to coordinate times 2 and 3.

Notice that way off to the left of the diagram, you have the same point being marked as both coordinate time 0 and coordinate time 2! Yuck! And off to the right, you can see a larger gap than "normal" between coordinate times 1 and 2! This is the yuckiness that is acceleration.

If you want, try to fill in lines of simultaneity corresponding to some times between 1 and 2, and see if you can "watch" the lines rotating as you progress along the worldline.

Really, one ought only to work with coordinates in an "infintiessimal" neighborhood of an accelerating particle, to avoid this weirdness. The only reason we get nice, global coordinates in the inertial case is because our space is so nice. When we progress to General Relativity, we don't get any niceness, so in all cases, lines of simultaneity only define coordinates in an "infinitessimal" neighborhood.
 
  • #29
yogi
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As before - i want to take a little time to digest your epistle before responding

Thanks

Yogi
 
  • #30
Hurkyl
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I need to make an addendum: I wouldn't feel right leaving this statemet unqualified:


That's impossible: I think you meant constant speed.

"Constant speed", of course, must mean relative to some coordinate system, because a constant speed trajectory in one frame is generally not constant speed in another frame. (This is true of Galilean relativity as well)
 
  • #31
Gir
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russ_watters said:
There is a difference between a clock and a tape measure that no one has picked-up on: a clock is a recording device (actually, two separate pieces: a measuring device and a recording device) and a tape measure isn't. Once a clock ticks off a second, that second is gone, never to be seen again. The only thing left is the record of that second, which for a clock is the time output by the display. So the measurements taken by a clock are exactly as permanent as those taken by a person with a clipboard standing over a tape measure. Better yet, one of those newfangled laser tape measures could keep a "permanent" record of length contraction.


no, clocks are a method of measure, they are not time itself. they make the passage of time but if you sping the hands backwords time doesn't reverse itself. clocks are exactly like a tape measure in that they both measure something. besides which, this is all totally besides the point.
 
  • #32
russ_watters
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Gir said:
no, clocks are a method of measure, they are not time itself.
Um, I didn't say they 'are time'. I did say they are a measuring device. Did you read incorrectly? :confused:
sping the hands backwords time doesn't reverse itself. clocks are exactly like a tape measure in that they both measure something.
Again, that's all I was saying: that they are exactly like tape measures. So what's the problem? :confused:
 
  • #33
CeeAnne
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Well, I feel it's good to question these things ... question everything. Okay, we have the observation ... we have the mathematical relationships describing it ... so, now let's do search for the mechanism causing it. Time is merely a measurement relative to a frame and something happens to that measurement ... relativity doesn't cause things ... there is a mechanism.
 
  • #34
derz
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CeeAnne said:
Well, I feel it's good to question these things ... question everything. Okay, we have the observation ... we have the mathematical relationships describing it ... so, now let's do search for the mechanism causing it. Time is merely a measurement relative to a frame and something happens to that measurement ... relativity doesn't cause things ... there is a mechanism.

The mechanism is the postulate "laws of nature are equal to every inertial observer". If this wouldn't be true the laws of nature would be totally chaotic. From this postulate follows that the value of c has to be constant to every observer, since c defines the value of the interval. If the interval would be different to different observers, the maximum signal speed would also seem like changing, depending on the speed of the observer.

If c wouldn't be constant there could be photons with different velocities, same frequencies but different energies etc. It just wouldn't make sense.

And because c is constant to every inertial observer, time and length can't be absolute.
 
  • #35
yogi
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Hurkyl - your graphical analysis is appreciated - but I am still left without an answer to the question which concerns me. I will send you a private message since this thread is about to close
 

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