Problem with thought experiment

[joey_m]

Please be patient, as you are dealing with a relativity newbie here!

I've read Einstein's book, "Relativity: The Special and the General Theory", and I can't seem to figure out chapter nine, "The Relativity of Simultaneity".

To refresh your memory, there is a train moving at a constant velocity, and it is marked with two markers that are to be hit by lightning at the same moment. In the mid-point of these markers is an observer. There is also an observer on the railway platform at the same position when all of this is happening.

In the chapter, Einstein says that the observer on the train will see the lightning flash that he is approaching before the other flash. This makes sense, but it is an argument based on classical mechanics. I thought that regardless of the relative motion between an observer and a light source, the speed of light is supposed to be the same, meaning that the moving observer will see the light flashes coming to him at the same speed, from both sides, causing the observer on the platform to think that the moving guy's clock had to slow down and speed up, at the same time.

In other words, think of a light source that emits a beam of light at the moment that two objects, in the same position and of different velocities, are moving away from it. According to a neutral observer, the faster object's time slows down more, and they both witness the light coming towards them at the same speed. Well, what if the situation is exactly the same, except that one of the objects is going towards the light source at the moment of emission?

Because the speed of light is reference frame invariant, won't the object that is moving towards the light source still see the light beam coming towards it at the same constant speed? And won't the neutral observer only be able to explain this by way of saying that time speeds up for this object (coming to the light) while it slows down for the other object?

I thought special relativity only supports the slowing down of time. But if, according to a rest frame, objects are moving towards a light beam, it seems that its time would necessarily need to get quicker (according to the rest frame) in order for it to make sense that the moving object sees the light coming at the constant speed of light.

What am I missing here?

(No complicated math please! )

 

[JesseM]

Because the speed of light is reference frame invariant, won't the object that is moving towards the light source still see the light beam coming towards it at the same constant speed?

Yes, it will. The meaning of the "relativity of simultaneity" is that different frames can actually disagree about whether two events happened at the same time (simultaneously) or at different times. In the frame of the observer on the tracks, both lightning strikes happen simultaneously, so since the observer on the train is moving towards one and away from the other, and the light from both strikes is moving at the same speed in this frame, naturally in the frame of the tracks the light from one strike must reach the observer on the train before the light from the other strike. The only way to reconcile this with the notion that the light from both strikes also moves at the same speed in the frame of the observer on the train is to postulate that in his frame, the two strikes actually happen non-simultaneously, one after the other. That way, even though the light from each strike has the same distance to reach his eyes, and travels at the same speed in both cases, he can agree that the light will from each strike reaches his eyes at a different moment.

 

[Doc Al]

In the chapter, Einstein says that the observer on the train will see the lightning flash that he is approaching before the other flash. This makes sense, but it is an argument based on classical mechanics.

Why do you say that?

I thought that regardless of the relative motion between an observer and a light source, the speed of light is supposed to be the same, meaning that the moving observer will see the light flashes coming to him at the same speed, from both sides,

It's true: All observers see light as traveling at the same invariant speed with respect to themselves.

Let's stick to the point of that chapter: Showing that simultaneity is relative.

Let's view things from the platform. It is given that the lightning strikes occur simultaneously according to platform observers. We also know, per relativity, that the light flashes travel at speed "c" as seen by the platform observers. And since the observer on the train is moving towards one flash and away from the other, we can deduce that he will see the light from those flashes at different times. This logic is perfectly clear, right? And everybody (both train and platform observers) will agree.

Now let's see what else we can deduce by looking at things from the train viewpoint. Since we've already established that the train observer sees the light from the flashes at different times, and since we know that he's in the middle of the train so that each light flash must have traveled the same distance, and since we know that according to him the light travels at the usual speed "c", the train observer is forced to conclude that according to him the lightning flashes could not have happened simultaneously.

Make sense?

(Edit: JesseM beat me again. Pay attention to what he tells you.)

 

[joey_m]

Why do you say that?

If you think about it in terms of baseballs instead of light beams, then you can understand why I say that. If the observers have their eyes closed and pitching machines are hurling baseballs in their directions, then it is easy to see that the moving guy will feel two different thumps while the stationary one will feel them at the same time. This is plain to see no matter what kind of scientist you are: classical or relativist. The problem is that the concept of light confuses things because that is the very thing that we use to see anything at all.

My point, in other words, is that this entire chapter, on its face, leads to no new physical insights. It seems that Einstein is just playing with words when he correlates the obvious concept of simultaneity/event sequencing with the nebulous one of "time" (he even puts that word in quotes in the book!).

Let's stick to the point of that chapter: Showing that simultaneity is relative.

Again, since the point of the chapter seems to be nothing new, I am trying to deepen my understanding of relativity by way of figuring out how this thought experiment works according to the logic that Einstein presents in the following few chapters (esp. chapter 12: "The Behaviour of Measuring-Rods and Clocks in Motion").

So, my issue has nothing to do with "simultaneity" and it has everything to do with how the stationary observer makes sense of the report from the moving observer that both of the light beams hit him at the same speed. According to the stationary guy, the rear beam hit the moving guy at c because the moving guy's clock must have slowed down. But according to this same logic, if the forward beam were to hit the moving guy at c, then the moving guy's clock would have had to seem to speed up.

It seems to me that special relativity only talks about light beams and observers moving in the same direction and never coming towards one another. So, this is my only real issue: if, relative to a stationary observer, a light beam and an observer are coming towards one another, and the moving observer reports that the light beam hit him at c, then isn't the stationary guy's only option to assume that the moving guy's clock was ticking faster?

 

[HallsofIvy]

If you think about it in terms of baseballs instead of light beams, then you can understand why I say that. If the observers have their eyes closed and pitching machines are hurling baseballs in their directions, then it is easy to see that the moving guy will feel two different thumps while the stationary one will feel them at the same time. This is plain to see no matter what kind of scientist you are: classical or relativist. The problem is that the concept of light confuses things because that is the very thing that we use to see anything at all.

This is not a matter of "simultaneity". One "observer" feels two "thumps" as the baseballs hit him, the other feels them hit at once. But the baseballs that hit one person are different from the baseballs that hit the other. Baseballs A and B striking striking person I has nothing to do with baseballs C and D striking person II. Every observer, whatever their speed would see the two balls striking the stationary person as simultaneous and then see the two balls striking the moving person as not simultaneous. Can you give an example of one person seeing baseballs A and B striking person I as simultaneous while another sees baseballs A and B striking person 1 as not simultaneous?

My point, in other words, is that this entire chapter, on its face, leads to no new physical insights.

Then you did not understand the chapter. Go back and read it again.

 

[JesseM]

So, my issue has nothing to do with "simultaneity" and it has everything to do with how the stationary observer makes sense of the report from the moving observer that both of the light beams hit him at the same speed. According to the stationary guy, the rear beam hit the moving guy at c because the moving guy's clock must have slowed down.

The fact that all observers measure the same light beam to move at c cannot be explained purely in terms of time dilation, if that's what you're suggesting. To measure the speed of anything, you need to measure its position at one time and its position at another time, with the time of each measurement defined in terms of a local reading on a synchronized clock in the same local region as the measurement; then "speed" is just (change in position)/(change in time). So, time dilation, length contraction, and the relativity of simultaneity all come into play. From the stationary observer's perspective, the ruler which the moving observer used to measure the distance was shrunk by a factor of $$\sqrt{1 - v^2/c^2}$$, the time between ticks on the moving clocks is expanded by $$1 / \sqrt{1 - v^2/c^2}$$, and the two clocks are out-of-sync by $$vx/c^2$$ (where x is the distance between the clocks in their own rest frame, where they are synchronized). On another thread I posted an example of how these factors come together to ensure both observers measure the same light beam to move at c:

Say there's a ruler that's 50 light-seconds long in its own rest frame, moving at 0.6c in my frame. In this case the relativistic gamma-factor (which determines the amount of length contraction and time dilation) is 1.25, so in my frame its length is 50/1.25 = 40 light seconds long. At the front and back of the ruler are clocks which are synchronized in the ruler's rest frame; because of the relativity of simultaneity, this means that in my frame they are out-of-sync, with the front clock's time being behind the back clock's time by vx/c^2 = (0.6c)(50 light-seconds)/c^2 = 30 seconds.

Now, when the back end of the moving ruler is lined up with the 0-light-seconds mark of my own ruler (with my own ruler at rest relative to me), I set up a light flash at that position. Let's say at this moment the clock at the back of the moving ruler reads a time of 0 seconds, and since the clock at the front is always behind it by 30 seconds in my frame, then in my frame the clock at the front must read -30 seconds at that moment. 100 seconds later in my frame, the back end will have moved (100 seconds)*(0.6c) = 60 light-seconds along my ruler, and since the ruler is 40 light-seconds long in my frame, this means the front end will be lined up with the 100-light-seconds mark on my ruler. Since 100 seconds have passed, if the light beam is moving at c in my frame it must have moved 100 light-seconds in that time, so it will also be at the 100-light-seconds mark on my ruler, just having caught up with the front end of the moving ruler.

Since 100 seconds passed in my frame, this means 100/1.25 = 80 seconds have passed on the clocks at the front and back of the moving ruler. Since the clock at the back read 0 seconds when the flash was set off, it now reads 80 seconds; and since the clock at the front read -30 seconds, it now reads 50 seconds. And remember, the ruler was 50 light-seconds long in its own rest frame! So in its frame, where the clock at the front is synchronized with the clock at the back, the light flash was set off at the back when the clock there read 0 seconds, and the light beam passed the clock at the front when its time read 50 seconds, so since the ruler is 50-light-seconds long, the beam must have been moving at 50 light-seconds/50 seconds = c as well! So you can see that everything works out--if I measure distances and times with rulers and clocks at rest in my frame, I conclude the light beam moved at 1 c, and if a moving observer measures distance and times with rulers and clocks at rest in his frame, he also concludes the same light beam moved at 1 c.

If you want to also consider what happens if, after reaching the front end of the moving ruler at 100 seconds in my frame, the light then bounces back towards the back in the opposite direction towards the back end, then at 125 seconds in my frame the light will be at a position of 75 light-seconds on my ruler, and the back end of the moving ruler will be at that position as well. Since 125 seconds have passed in my frame, 125/1.25 = 100 seconds will have passed on the clock at the back of the moving ruler. Now remember that on the clock at the front read 50 seconds when the light reached it, and the ruler is 50 light-seconds long in its own rest frame, so an observer on the moving ruler will have measured the light to take an additional 50 seconds to travel the 50 light-seconds from front end to back end.

 

[Doc Al]

If you think about it in terms of baseballs instead of light beams, then you can understand why I say that. If the observers have their eyes closed and pitching machines are hurling baseballs in their directions, then it is easy to see that the moving guy will feel two different thumps while the stationary one will feel them at the same time. This is plain to see no matter what kind of scientist you are: classical or relativist.

The point is not that the moving guy would feel two lumps--or see the flashes arrive separately--that's the easy part. The point is what will he deduce about when the balls were thrown--or when the lightning flashes struck--from that fact? If we were using classical baseballs instead of beams of light, we would not be able to easily deduce anything unusual about simultaneity because baseballs do not have the same speed in all frames. Light behaves very nonclassically!

My point, in other words, is that this entire chapter, on its face, leads to no new physical insights.

Better reread the chapter--it looks like you missed the point and the meaning of the relativity of simultaneity.

 

[joey_m]

Jesse,

You've brought us to the brink of my point with your illustration, and now let us consider its implications. Let us begin with the assumption that "you" (the one who is observing the 50 light-second (ls) flying ruler) are totally ignorant about relativistic effects and you are trying to understand how the universe works.

In the first scenario (when the light beam and the 50 ls ruler are going in the same direction), you observe that light beam takes 100 seconds to travel the apparent 40 ls length of the ruler (because of space contraction). According to your measurements, the guy on the flying ruler should have witnessed the light to be moving at 0.4*c (40 ls/100 s). But later, the flying guy tells you that he witnessed the light to be moving at c. Since you totally believe the flying guy (he's never lied to you before!), your only choice is to "disbelieve" what you know about time. In moving frames, it seems, each second will have to tick slower in order for 0.4c to be "readjusted" to c.

In the second scenario (when the light and the flying ruler are going in opposite directions), you observe that light only takes 25 seconds to travel the apparent 40 ls length. This time, you calculate that the guy should have registered the speed of light to be 1.6*c (40 ls/25 s). In this case, the flying guy also claims that he witnessed the light to be moving at c. The problem here is obvious. In order for 1.6c to be "readjusted" back to c, each second will have to tick relatively faster!

So, in your experiment, since the effect on the clock (whether it speeds up or slows down) of a relatively moving observer depends completely upon the question of whether a light beam is going in the same or in the opposite direction, then there is absolutely no way to logically deduce the "true" behaviour of a moving clock when light beams are approaching from both directions (which is the scenario in the original thought experiment that I had a problem with).

Bottom line: given these considerations, the entire theoretical basis of special relativity doesn't make any sense to me!



Everyone else,

I come from an extensive philosophical background, and I have had many extraordinarily deep meditations on the "meaning" of time. So, when I see a physicist come along and construct what seems to be a highly superficial definition of this grand concept (based on the relative ordering of witnessed events, in this case), I am none too impressed. In other words, I know precisely what Einstein was trying to do in chapter nine, but I've seen far too many sophisticated philosophical arguments concerning temporality for Einstein's to have any effect on me whatsoever. I just don't think that reading that section for an eleventh time will be any great revelation for me!

 

[jcsd]

joey_m, the light does not cover 40ls in your frame because in both cases the ruler is moving in your fra,e. To compare rates of clocks you must comapre how far the light travels in your frame with how far the light travels in 'the guy's' frame rather than comapring the error between a Gallilean transformation and a Lorentz transformation.

Your scheme for comparing clocks cannot be self-consistant, as it is perfectly possible to have light beams going in both directions at the same time.

When learnign special relativity it's very easy to get yourself in a tangle.

 

[JesseM]

Jesse,

You've brought us to the brink of my point with your illustration, and now let us consider its implications. Let us begin with the assumption that "you" (the one who is observing the 50 light-second (ls) flying ruler) are totally ignorant about relativistic effects and you are trying to understand how the universe works.

In the first scenario (when the light beam and the 50 ls ruler are going in the same direction), you observe that light beam takes 100 seconds to travel the apparent 40 ls length of the ruler (because of space contraction). According to your measurements, the guy on the flying ruler should have witnessed the light to be moving at 0.4*c (40 ls/100 s). But later, the flying guy tells you that he witnessed the light to be moving at c. Since you totally believe the flying guy (he's never lied to you before!), your only choice is to "disbelieve" what you know about time. In moving frames, it seems, each second will have to tick slower in order for 0.4c to be "readjusted" to c.

In the second scenario (when the light and the flying ruler are going in opposite directions), you observe that light only takes 25 seconds to travel the apparent 40 ls length. This time, you calculate that the guy should have registered the speed of light to be 1.6*c (40 ls/25 s). In this case, the flying guy also claims that he witnessed the light to be moving at c. The problem here is obvious. In order for 1.6c to be "readjusted" back to c, each second will have to tick relatively faster!

But you're ignoring the relativity of simultaneity. In your frame, the flying guy's clocks at the front and back of the ruler, which he is using to measure the time of the light passing each end of the ruler (and which are synchronized in his own frame), are out-of-sync by 30 seconds. If the ruler is moving to the right, then in your frame the clock on the left end of the ruler is ahead of the clock on the right end by 30 seconds. So when the light is moving from left to right (same direction as the ruler), although you see it taking 100 seconds for it to get from left to right, and although each of the flying clocks only advances forward by 80 seconds in this time (because they are slowed down by a factor of 0.8), this does not mean that the reading of the right clock as the light passes it is 80 seconds ahead of the reading on the left clock as the light passes it; instead, because you see the clock on the right end as 30 seconds behind, its reading when the light passes it is only 80 - 30 = 50 seconds greater than the reading of the clock on the left end passed it.

On the other hand, when the light goes in the opposite direction from right to left, it only takes 25 seconds in your frame to go from one end to the other, and since the clocks are slowed down by a factor of 0.8 they will only advance forward 20 seconds in this time. But, the clock on the left is 30 seconds ahead of the clock on the right, so the time on the right clock as the light passes it will be 20 + 30 = 50 seconds greater than the time on the left clock when the light passed it.

So in both cases you see both clocks slowed down by the same factor of 0.8, but because of the way the two clocks are out-of-sync in your frame, in both cases you also see that the time on the second clock the light passes will be 50 seconds ahead of the time on the first clock the light passes.

What's the problem with this?

 

[joey_m]

What's the problem with this?

Okay, let's stick to your method of comparing clocks for the moment.

In the first case, 100 of your seconds is equal to 50 of the flying guy's seconds.

In the second case, 25 of your seconds is equal to 50 of the flying guy's seconds.

On the one hand, his clock is ticking two times slower (100 s/50 s), and on the other, it is ticking twice as fast (25 s/50 s)!

In other words, I am not talking about the fact that they will both measure light to be the same speed, because this is a postulate of the theory of relativity. I am only saying that in order for these real world measurements to make sense to you (the stationary observer), you will necessarily have to think two contradictory things about the relative "speed" of the flying guy's clocks!

To me, this seems to be a big problem!

 

[JesseM]

Okay, let's stick to your method of comparing clocks for the moment.

In the first case, 100 of your seconds is equal to 50 of the flying guy's seconds.

Why do you say that? I see both clocks tick 80 seconds forward in the 100 seconds it takes in my frame for the light to go from left to right. But since the left clock started at 0 seconds and the right clock started at -30 seconds, by the time the light reaches the right end, the left clock reads 0 + 80 = 80 seconds and the right clock reads -30 + 80 = 50 seconds.

In the second case, 25 of your seconds is equal to 50 of the flying guy's seconds.

No, I see each of the two clocks ticks forward by 20 seconds in the 25 seconds it takes for the light to go back from right to left in my frame. But since at the moment the light changed directions the left clock read 80 seconds while the right clock read 50 seconds, the left clock reads 80 + 20 = 100 seconds and the right clock reads 50 + 20 = 70 seconds when the light reaches the left end again.

In other words, I am not talking about the fact that they will both measure light to be the same speed, because this is a postulate of the theory of relativity. I am only saying that in order for these real world measurements to make sense to you (the stationary observer), you will necessarily have to think two contradictory things about the relative "speed" of the flying guy's clocks!

Not true, you can see above the numbers work out fine--in the first case I see both clocks tick forward by 80 seconds from their starting time (though they have different starting times because of the relativity of simultaneity), in the second case I see both clocks tick forward by 20 seconds from the time they showed when the light changed direction (which again is different for both clocks). And since I saw the light take 100 seconds to go from left to right and 25 seconds to go from right to left, and 0.8*100 = 80 and 0.8*25 = 20, in both cases I am seeing both clocks slowed down by the same factor of 0.8. Do you disagree with my math here?

 

[joey_m]

Okay, let's start over here, as I am once again interested in arguing this question.

To set the stage, there is a ruler (#1) that we see as stationary and another one that we see as flying at 0.6c (#2).

Ruler #1's rest length is 100 ls and #2's is 50 ls.

Now we see a light beam shooting at the same time that ruler #2 starts moving.

Let us question concerning the fact of what each observer measures concerning his own ruler.

When the light beam is going in the same direction as ruler #2, these two statements are true (are they not?):

1) Observer #1 measures that the light beam travels the 100 ls of his ruler's length in exactly 100 seconds, yielding c.
2) Observer #2 measures that the light beam travels the 50 ls of his ruler's length in exactly 50 seconds, yielding c.

However, the only way that observer #1 can possibly make sense of the fact that observer #2 did not witness the beam to be going at 0.4c (40ls/100s) is if observer #2's clock in fact ticks relatively slower, thereby causing the beam (from #2's perspective) to appear to move faster within the 100 seconds of #1's time. This means that observer #1 must "adjust" #2's 100s down to 40s, so that 40ls/40s = c. This is the essence of time dilation.

Now, when the light beam reverses course, are not these statements true:

1) Observer #1 measures that the light beam travels 25 ls of his ruler's length in exactly 25 seconds, yielding c.
2) Observer #2 measures that the light beam travels the 50 ls of his ruler's length in exactly 50 seconds, yielding c.

In this case, observer #1 would assume that #2's light beam is going at 1.6c (40ls/25s). Now, the only way that it can make sense for observer #1 that #2's beam moves at c is if (conversely to the above scenario), #2's clock ticks relatively faster, thereby causing the beam (from #2's perspective) to appear to move slower within the 25 seconds of #1's time. This means that observer #1 must "adjust" #2's 25s up to 40s, so that 40ls/40s = c. This is the opposite of time dilation (which is not allowed in special relativity).


***

The problem is that Einstein only came up with his theory when considering what would happen if he tried to catch up with a beam of light. The reason why he would never be able to catch it, according to a neutral witness, is that Einstein's clock ticks relatively slower, allowing the beam to seem to be going faster (from Einstein's perspective) than what the witness would have calculated it to be, according to classical Newtonian mechanics.

However, for some strange reason, Einstein never asked what would happen if he charged in the opposite direction to a beam of light. How, according to the same neutral witness, could it be explained that the light is moving at the same velocity relative to Einstein as in the above case? The only possible way for the witness to make sense of things is if Einstein's clock ticks faster.

I smell something fishy here...

 

[JesseM]

Okay, let's start over here, as I am once again interested in arguing this question.

To set the stage, there is a ruler (#1) that we see as stationary and another one that we see as flying at 0.6c (#2).

Ruler #1's rest length is 100 ls and #2's is 50 ls.

What frames are these lengths in? If you're using my example above, #2 is 50 ls in its own rest frame, which means that because of length contraction it's shrunk down to sqrt(1 - 0.6^2)*50 = 0.8*50 = 40 ls in our frame.

Now we see a light beam shooting at the same time that ruler #2 starts moving.

Yes, and in my example, at the moment the light is shot out, the back ends of each ruler are lined up with the position the light departs from. In our frame the back end of ruler #1 will remain there, while the back end of ruler #2 will continue on at 0.6c.

Let us question concerning the fact of what each observer measures concerning his own ruler.

When the light beam is going in the same direction as ruler #2, these two statements are true (are they not?):

1) Observer #1 measures that the light beam travels the 100 ls of his ruler's length in exactly 100 seconds, yielding c.
2) Observer #2 measures that the light beam travels the 50 ls of his ruler's length in exactly 50 seconds, yielding c.

However, the only way that observer #1 can possibly make sense of the fact that observer #2 did not witness the beam to be going at 0.4c (40ls/100s) is if observer #2's clock in fact ticks relatively slower, thereby causing the beam (from #2's perspective) to appear to move faster within the 100 seconds of #1's time. This means that observer #1 must "adjust" #2's 100s down to 40s, so that 40ls/40s = c. This is the essence of time dilation.

Wrong. Time dilation plays a part, but as I keep telling you, you also must remember the relativity of simultaneity, which says that different frames have a different definition of what it means for two clocks to be synchronized. This means that if there are clocks mounted at the front and back end of ruler #1, and also clocks mounted at the front and back of ruler #2, and each pair of clocks is synchronized in the frame of the ruler they're attached to, then in "our" frame (the frame of ruler #1), the clocks at the front and back of ruler #1 will be synchronized, but the clocks at the front and back end of ruler #2 will be out-of-sync by vx/c^2, where x is the distance between the clocks in the ruler's rest frame (50 ls) and v is the speed of the ruler in our frame (0.6c). So, the clocks at the front and back end will be out-of-sync by a constant amount of 50*0.6 = 30 seconds in our frame, with the clock on the back end ahead of the clock on the front end by this amount at any given moment in our frame. So, if we look at what each of the two clocks on ruler #2 reads at different moments in our frame, it would look like:

t=0 seconds in our frame
clock at back of ruler #2 reads 0 seconds
clock at front of ruler #2 reads -30 seconds

t=10 seconds in our frame
clock at back of ruler #2 reads 8 seconds
clock at front of ruler #2 reads -22 seconds

t=20 seconds in our frame
clock at back of ruler #2 reads 16 seconds
clock at front of ruler #2 reads -14 seconds

t=30 seconds in our frame
clock at back of ruler #2 reads 24 seconds
clock at front of ruler #2 reads -6 seconds

t=40 seconds in our frame
clock at back of ruler #2 reads 32 seconds
clock at front of ruler #2 reads 2 seconds

t=50 seconds in our frame
clock at back of ruler #2 reads 40 seconds
clock at front of ruler #2 reads 10 seconds

t=60 seconds in our frame
clock at back of ruler #2 reads 48 seconds
clock at front of ruler #2 reads 18 seconds

t=70 seconds in our frame
clock at back of ruler #2 reads 56 seconds
clock at front of ruler #2 reads 26 seconds

t=80 seconds in our frame
clock at back of ruler #2 reads 64 seconds
clock at front of ruler #2 reads 34 seconds

t=90 seconds in our frame
clock at back of ruler #2 reads 72 seconds
clock at front of ruler #2 reads 42 seconds

t=100 seconds in our frame
clock at back of ruler #2 reads 80 seconds
clock at front of ruler #2 reads 50 seconds

So you can see that for any given 10 second interval in our frame, each of the two clocks on ruler #2 advances forward by 8 seconds; that's because the time dilation factor here is sqrt(1 - 0.6^2) = 0.8. But there is also the relativity of simultaneity which makes it so if the two clocks on ruler #2 are synchronized in their own rest frame, they are out-of-sync by 30 seconds in our frame; indeed, you can see from these numbers that the reading on the back clock is always 30 seconds ahead of the reading on the front clock at any given moment in our frame.

Now, in relativity it is conventional to assign coordinates to events using local measurements on synchronized clocks, so in the frame of ruler #2, the time coordinate assigned to the moment when the light beam was emitted next to the back end of the ruler is the same as the reading on the clock at the back end at that moment, and likewise the time coordinate assigned to the moment when the light beam catches up to the front of the ruler is the same as the reading on the clock at the front end of the ruler at that moment. Assigning time-coordinates to events using only local measurements in this way ensures you don't have to worry about delays between the time the event happens and the time it is measured.

So, if the light is next to the back end of ruler #2 at t=0 seconds in our frame, and it catches up with the front end of ruler #2 at t=100 seconds in our frame, then the relevant readings on ruler #2's clocks are:

t=0 seconds in our frame
clock at back of ruler #2 reads 0 seconds
clock at front of ruler #2 reads -30 seconds

t=100 seconds in our frame
clock at back of ruler #2 reads 80 seconds
clock at front of ruler #2 reads 50 seconds

This means that in ruler #2's coordinates, the light passing the front end happens 50 seconds after the light passes the back end. And someone at rest in this frame will use a ruler at rest in the frame to measure the distance between events, so since ruler #2 is at rest in this frame and measures 50 ls long in its own rest frame, the spatial distance between these events is 50 ls. Since speed = (distance covered)/(change in time) in any given frame, this means the speed of the light beam in the frame of ruler #2 is 50 ls/50 s = c.

Now let's say that there is a mirror on the end of ruler #2, so as soon as the light reaches the front end, it is reflected in the opposite direction. In our frame, it will catch up with the back end 25 seconds later, at t=125 seconds. We can see what the clocks at either ends of ruler #2 read at 5-second intervals from t=100 to t=125 in our frame:

t=100 seconds in our frame
clock at back of ruler #2 reads 80 seconds
clock at front of ruler #2 reads 50 seconds

t=105 seconds in our frame
clock at back of ruler #2 reads 84 seconds
clock at front of ruler #2 reads 54 seconds

t=110 seconds in our frame
clock at back of ruler #2 reads 88 seconds
clock at front of ruler #2 reads 58 seconds

t=115 seconds in our frame
clock at back of ruler #2 reads 92 seconds
clock at front of ruler #2 reads 62 seconds

t=120 seconds in our frame
clock at back of ruler #2 reads 96 seconds
clock at front of ruler #2 reads 66 seconds

t=125 seconds in our frame
clock at back of ruler #2 reads 100 seconds
clock at front of ruler #2 reads 70 seconds

Just as before, we see that each clock is slowed down by a factor of 0.8 in our frame--for every 5 seconds of time in our frame, each clock advances forward by 0.8*5 = 4 seconds. And as before, the clock at the back end is always 30 seconds ahead of the clock at the front end in our frame.

And as before, the time-coordinate of the light being reflected from the front end of ruler #2 in ruler #2's frame should just be the local time on the clock at the front end when that happens; likewise, the time-coordinate of the light reaching the back end of ruler #2 in ruler #2's frame should be the local time on the clock at the back end when that happens. So the two relevant readings are:

t=100 seconds in our frame
clock at back of ruler #2 reads 80 seconds
clock at front of ruler #2 reads 50 seconds

t=125 seconds in our frame
clock at back of ruler #2 reads 100 seconds
clock at front of ruler #2 reads 70 seconds

So we see that even though in our frame it took 100 seconds for the light to go from the back end to the front end of ruler #2 and then only 25 seconds to go from the front end to the back end, in ruler #2's own frame it took 50 seconds for the light to go from the back end to the front end, and another 50 seconds for the light to go from the front end to the back end. And in ruler #2's frame the distance between the two ends is always 50 ls, so the light moved at c both ways.

Do you follow now? If not, please be specific on what step you disagree with, don't just keep ignoring my point about the relativity of simultaneity as you have done on this thread so far.

 

[yogi]

Joey: "In the chapter, Einstein says that the observer on the train will see the lightning flash that he is approaching before the other flash. This makes sense, but it is an argument based on classical mechanics. I thought that regardless of the relative motion between an observer and a light source, the speed of light is supposed to be the same, meaning that the moving observer will see the light flashes coming to him at the same speed, from both sides, causing the observer on the platform to think that the moving guy's clock had to slow down and speed up, at the same time"

This is a situation where the observer at the center of the the moving train will see one flash before the other - that is perfectly correct. It is also correct that both light flashes will pass by at the same velocity c according to SR. In the first case, you can think of the velocity of light as being measured by a coordinate system attached to the location of the point where the flash occurred - the velocity of light is not c as between the source and the moving receiver. Same for a coordinate system attached to the other flash point (where the lightning strikes). But a local coordinate system attached to the moving observer will measure the speed of light in each direction as equal to c - he uses his own local rulers and clocks that are fixed to his own moving frame -

A good example of this is sometimes cited in the literature when attempting to explain the correction needed to allow for the Earth's rotation when signals are sent from a GPS satellite - its commonly referred to as the "one way sagnac correction" It occurs anytime the point of reception is moving wrt to the light source - the measured velocity will not be c when measurments are made from the source to the moving object.

 

[joey_m]

So we see that even though in our frame it took 100 seconds for the light to go from the back end to the front end of ruler #2 and then only 25 seconds to go from the front end to the back end, in ruler #2's own frame it took 50 seconds for the light to go from the back end to the front end, and another 50 seconds for the light to go from the front end to the back end. And in ruler #2's frame the distance between the two ends is always 50 ls, so the light moved at c both ways.

It boggles the mind how you fail to fully appreciate the issue that I have with this result. You go through such great pains to elaborate on the method used to attain this result without examining what the result fundamentally means as concerning the theory of special relativity.

I have no argument with your method (including the relativity of simultaneity). Let's just focus on what the results of the experiment mean.

Einstein's major revelation is that moving clocks tick slower. That is it. Not even the notion of space contraction was original, because Lorentz already had that figured out (hence the term, "Lorentz contraction"). What I'm trying to explain is that this revelation is logically absurd.

Let me dissect your paragraph to make things abundantly clear:

in our frame it took 100 seconds for the light to go from the back end to the front end of ruler #2

in ruler #2's own frame it took 50 seconds for the light to go from the back end to the front end

This simply means that, according to us, an event that takes up 100 seconds of our time only takes up 50 seconds of #2's time. To us, #2's clock ticks slower. (Conclusion A)

in our frame it took ... only 25 seconds to go from the front end to the back end

in ruler #2's own frame it took ... 50 seconds for the light to go from the front end to the back end

Contrarily, this means that, from our perspective, an event that takes up 25 seconds of our time takes up 50 seconds of #2's time. To us, #2's clock ticks faster. (Conclusion B: contradicts conclusion A!)

Now, are you once again going to do a lecture on the method used to obtain these two conclusions and ignore my concerns about the logical contradiction that these conclusions represent?

And then are you going to again accuse me of failing to acknowledge the "profound importance" of the method that I have entirely no argument with?

The point is, I don't disagree with you at all. You just seem to be utterly unwilling (or unable?) to logically examine the results that you have undertaken so much effort to arrive at.

 

[JesseM]

in our frame it took ... only 25 seconds to go from the front end to the back end

in ruler #2's own frame it took ... 50 seconds for the light to go from the front end to the back end

Contrarily, this means that, from our perspective, an event that takes up 25 seconds of our time takes up 50 seconds of #2's time. To us, #2's clock ticks faster.

No they don't! Did you completely ignore everything I just wrote about the relativity of simultaneity? Just look at the numbers for the phase of the trip where the light is going in the opposite direction as ruler #2:

t=100 seconds in our frame
clock at back of ruler #2 reads 80 seconds
clock at front of ruler #2 reads 50 seconds

t=105 seconds in our frame
clock at back of ruler #2 reads 84 seconds
clock at front of ruler #2 reads 54 seconds

t=110 seconds in our frame
clock at back of ruler #2 reads 88 seconds
clock at front of ruler #2 reads 58 seconds

t=115 seconds in our frame
clock at back of ruler #2 reads 92 seconds
clock at front of ruler #2 reads 62 seconds

t=120 seconds in our frame
clock at back of ruler #2 reads 96 seconds
clock at front of ruler #2 reads 66 seconds

t=125 seconds in our frame
clock at back of ruler #2 reads 100 seconds
clock at front of ruler #2 reads 70 seconds

So, for example, from t=100 to t=105, the clock at the front of ruler #2 goes from reading 80 seconds to reading 84 seconds. If a clock advances forward by 4 seconds in 5 seconds of our time, would you say it is running slower or faster? Please answer this question specifically. And if you agree it's running slower, look at the rest of the numbers here--do you agree that in every time interval, each of the two clocks is running slow in our frame? If so, how can you say that "we see #2's clocks ticking faster"? Again, please don't ignore this question if you wish to respond.

Now, are you once again going to do a lecture on the method used to obtain these two conclusions and ignore my concerns about the logical contradiction that these conclusions represent?

There's no logical contradiction, because you're wrong that we ever see any of the #2 clocks running faster. Rather, what we see is that they are measuring time using a method that to us seems wrong, because they measure the start of the journey with one clock and the end of the journey with another, and these two clocks are not in sync with each other. To us this method doesn't match with the time elapsed on their clocks at all, in fact we only see each clock move forward by 20 seconds in the 25 seconds it takes for the light to go from one end to the other in our frame. But in their own frame, the situation is reversed, with their clocks being in sync in this frame, and our own clocks being out-of-sync; in relativity there is no "real truth" about whether two events at different locations (like two clocks showing the same time) happened at the same time or at different times, each frame will have a different opinion on this and all frames are equally valid, a feature known as the relativity of simultaneity. So to us, the light leaves the right clock when it reads 50 seconds, and by the time the light reaches the left end, the right clock has only advanced forward by 20 seconds (less than the 25 seconds in our frame), so the event of the light reaching the left end is simultaneous with the event of the right clock reading 70 seconds in our frame. In ruler #2's frame, these events are not simultaneous, instead the event of the light reaching the left end is simultaneous, where the clock reads 50 seconds, is simultaneous with the event of the right clock reading 50 seconds as well, since the two clocks are synchronized in ruler #2's frame. And the situation is totally symmetrical--the clocks of ruler #2 are out-of-sync in our frame, but in ruler #2's frame it is our clocks on ruler #1 (which we are using to make local measurements of the time that the light hits each end of the ruler) than are out-of-sync. To help see the symmetry, please check out the diagrams I posted on this thread, showing a situation where you have two rulers A and B with clocks mounted at regular intervals on them moving past each other at high speed, and how in the frame of ruler A it is B's clocks that are running slow and out-of-sync, but in the frame of ruler B it is A's clocks that are slow and out-of-sync.

And then are you going to again accuse me of failing to acknowledge the "profound importance" of the method that I have entirely no argument with?

The point is, I don't disagree with you at all. You just seem to be utterly unwilling (or unable?) to logically examine the results that you have undertaken so much effort to arrive at.

What is there to logically examine? If a clock ticks only 4 seconds forward in 5 seconds in my frame, doesn't that mean it's running slow in my frame, by definition? So why do you keep insisting the clocks are running fast when the light is going in the other direction, when the numbers clearly show that both clocks are running slow?

 

[robert Ihnot]

Joey: "In the chapter, Einstein says that the observer on the train will see the lightning flash that he is approaching before the other flash. This makes sense, but it is an argument based on classical mechanics.

In the Classic argument the beam proceeding from the tracks outside the front of the train will arrive first. However, classically, if both beams are fired from within the train, then the front beam will travel against the train at c-v, while the beam in the rear will proceed at c+v. This leads to the equation regarding Distance/velocity, where we have,

$$\frac{M+vt}{(c+v)t} =\frac{M-vt}{(c-v)t}$$, which leads to a time of t=M/c, where M is half the length of the train.

Thus the Classic result tells us the beams meet in the middle of the train, exactly like we would have had expected before Einstein.

But, from the standpoint of Relativity it does not matter whether the beams were fired from within the train or from along the tracks, because to all observers the beam always travels at c, and thus outside or inside the train they travel side by side at the same rate.

 

[joey_m]

Okay, now let's start getting into the nitty-gritty details...

In reference to this:

how can you say that "we see #2's clocks ticking faster"?

... I never said anything about what "we see". That's your deal. I'm only talking about the objective disagreement between the number of ticks that each observer counts while the light beam is transversing from one point to another.

To me, the importance of this whole "relativity of simultaneity" (ROS for short) thing seems to be a complete non-issue. I don't understand what difference it makes that two clocks show different "times of day" when they are both ticking at the same rate. When it comes to making a determination concerning a temporal duration, the question of whether the rates of ticking of two clocks are equal is the important factor.

Let me try to illustrate what I mean thusly:

Say we are at a track meet, and we are about to watch the 100 meter dash. There is a clock at the start-line and there is another one at the finish-line. When the gun goes off, the clock at the start-line reads 5:00:00 pm and the one at the finish-line reads 4:59:51 pm. That is, the clock at the finish-line is precisely nine seconds behind the one at the start-line. However, they are both perfectly "in sync", when it comes to the rate at which they are ticking!

Now, say that it takes both clocks precisely 10 ticks for the fastest man to traverse the 100 meters that separates start from finish. I guess my major malfunction is that I fail to understand the significance of the one-second difference between what the finish-line clock reads at the end of the race (5:00:01 pm) and what the start-line clock reads at the beginning (5:00:00 pm). If the officials of the track meet were to come along and say that the duration of the race is actually supposed to be determined by this perfectly arbitrary one-second differential, then everyone observing the event would think that they were mad. For every reason, everyone knows that duration is determined simply by counting the number of ticks on a clock!

So, with your thought experiment, the following results are obtained:

In the forward direction:

We see 100 seconds tick off both of our clocks.
We see 80 seconds tick off both of the flying clocks.

In the reverse direction:

We see 25 seconds tick off both of our clocks.
We see 20 seconds tick off both of the flying clocks.

However, the guy that is moving along with the flying clocks sees the following:

In the forward direction:

He sees 50 seconds tick off both of his clocks.

In the reverse direction:

He sees 50 seconds tick off both of his clocks.

So, in the forward direction, according to what "we see", 80 seconds have elapsed on both of the flying clocks. (The 50 second difference that you came up with between the front and back clocks is wholly meaningless.) However, we are at the same time supposed to believe the flying guy when he tells us that only 50 seconds have elapsed according to both of his clocks. If we don't think of the flying guy as a liar, our only option is to disbelieve our own eyes. In this case, what we thought were 80 ticks must really only have been 50 ticks. Time, in fact, has apparently gotten even slower (from 100 seconds to 50 seconds) than what we initially (and incorrectly) observed (from 100 seconds to 80 seconds).

In the reverse direction, we see that only 20 seconds have elapsed on both of the flying clocks. (Again, the difference between the front and back clocks is perfectly meaningless.) And we are yet again supposed to believe the flying guy when he says that 50 ticks have occurred on both of his clocks. So, what we thought were 20 ticks must have really been 50 ticks. In this case, we must disbelieve our own eyes that 20 seconds have ticked off of the flying clocks. Our only option is to believe that 25 seconds of our time was equal to 50 seconds of the flying guy's time. Time, in other words, has apparently gotten faster.

My point is that you don't seem to grasp the fact that Einstein's theory of special relativity has nothing to do with what "we see" about moving clocks and it has everything to do with the way in which relatively moving observers are supposed to make sense of the notion that they both always measure light to move at the same velocity.

If a naive, independent observer sees someone trying to catch up with a beam of light at velocity v (as in Einstein's original thought experiment), he will originally think that the velocities are subtractive, and that the moving person will measure the speed of light to be a relatively smaller value (c-v). However, when the moving guy comes back and tells the observer that he measured the speed of the light beam to be c, then the only conclusion that the observer can make is that the moving guy's clock must have slowed down (in order to give the light beam the chance to appear to move faster, from the moving guy's perspective).

If the situation is reversed, with the moving guy moving towards a beam of light at velocity v, then the naive observer will originally think that the velocities are additive, and that the moving guy will measure a relatively larger speed of light (c+v). In this case, the only way for the observer to make sense of the fact that the moving guy measures light to move at c is if the moving guy's clock had sped up (in order to give the light beam the chance to appear to move slower, from the moving guy's perspective).

Now, I think that I've done more than an adequate job in terms of describing why it is that your ROS "obsession" is misplaced. ROS has precisely nothing to do with the concept of temporal duration, in terms of the number of physical ticks that are seen to have elapsed on one or more clocks. Please do not once again accuse me of ignoring "everything" that you are saying about this non-issue. It is an unnecessary complication that sheds absolutely no light on anything.

So, I ask again that you please not ignore me when it comes to the discrepancies between these two observations:

1) In the forward direction, we count 80 seconds to have physically ticked off of both of the flying clocks while the flying guy counts 50 ticks off of both of the same clocks.

2) In the reverse direction, we count 20 seconds to have physically ticked off of both of the flying clocks while the flying guy counts 50 ticks off of both of the same clocks.

Both of these statements are internally logically contradictory if we are supposed to believe ourselves as well as the flying guy (that is, how can 80 ticks equal 50 ticks, and how can 20 ticks equal 50 ticks?). Furthermore, if we disbelieve ourselves, these statements taken together are absurd, because the question of whether a moving clock slows down or speeds up depends entirely on which way it is moving in relation to the beams of light in its surroundings.

And if these problems are indeed irresolvable, then the only thing left to do is to abandon Einstein's "improvements" of Galileo's otherwise perfectly sensible concept of relativity.

 

[JesseM]

... I never said anything about what "we see". That's your deal. I'm only talking about the objective disagreement between the number of ticks that each observer counts while the light beam is transversing from one point to another.

That's what I meant by "see", how fast each clock was ticking in each observer's own frame. Look at the numbers I gave. Do you agree that in our frame, between 0 and 100 seconds of our time, the clock at the back end of ruler #2 ticks 80 seconds forward (from 0 to 80 seconds), and the clock at the front end of ruler #2 also ticks 80 seconds forward (from -30 to 50 seconds)? Do you also agree that in our frame, between 100 and 125 seconds, the clock at the back end of ruler #2 ticks 20 seconds forward (from 80 to 100 seconds) and the clock at the front end of ruler #2 also ticks 20 seconds forward (from 50 to 70 seconds)? Do you agree that in each case, the clocks on ruler #2 were only ticking forward 4 seconds for every 5 seconds of our own time?

To me, the importance of this whole "relativity of simultaneity" (ROS for short) thing seems to be a complete non-issue. I don't understand what difference it makes that two clocks show different "times of day" when they are both ticking at the same rate. When it comes to making a determination concerning a temporal duration, the question of whether the rates of ticking of two clocks are equal is the important factor.

Let me try to illustrate what I mean thusly:

Say we are at a track meet, and we are about to watch the 100 meter dash. There is a clock at the start-line and there is another one at the finish-line. When the gun goes off, the clock at the start-line reads 5:00:00 pm and the one at the finish-line reads 4:59:51 pm. That is, the clock at the finish-line is precisely nine seconds behind the one at the start-line. However, they are both perfectly "in sync", when it comes to the rate at which they are ticking!

Of course I agree with that last statement, and the clocks at the front and back of ruler #2 are indeed ticking at the same rate in my example. It was you who claimed that the clocks would have to be ticking at a different rate when the light was traveling in one direction than they were when the light was traveling in the other, but you can see from the numbers in my example that this isn't true, in both directions, each clock is ticking forward 4 seconds for every 5 seconds of my time (slowed down by a factor of 0.8). But because the observer traveling on ruler #2 believes the two clocks are not out-of-sync, he believes the light took 50 seconds to go from the back end to the front end (because the clock at the back end read 0 seconds when the light passed it, and the clock at the front end read 50 seconds when the light reached it), and also believes the light took 50 seconds to return from the front end to the back end (because the clock at the front end read 50 seconds when the light began the trip back, and the clock at the back end read 100 seconds when the light reached it again). Do you disagree that this shows that the observer on the ruler will believe the light traveled at the same speed in both directions, in spite of the fact that both his clocks were ticking at the same rate both before and after the light turned around, and despite the fact that in my frame it took the light 100 seconds to go from back to front but only 25 seconds to return from front to back?

Now, say that it takes both clocks precisely 10 ticks for the fastest man to traverse the 100 meters that separates start from finish. I guess my major malfunction is that I fail to understand the significance of the one-second difference between what the finish-line clock reads at the end of the race (5:00:01 pm) and what the start-line clock reads at the beginning (5:00:00 pm). If the officials of the track meet were to come along and say that the duration of the race is actually supposed to be determined by this perfectly arbitrary one-second differential, then everyone observing the event would think that they were mad. For every reason, everyone knows that duration is determined simply by counting the number of ticks on a clock!

But that's because in your example, everyone would agree that the clock at the starting line and the clock at the finish line were out-of-sync. In relativity, different frames disagree on whether two clocks are "synchronized" or not, and all the laws of physics work exactly the same in these different frames, so there is no experiment you can do to decide whose clocks are "really" synchronized.

Please understand the fact that although in my frame ruler #2's clocks are out-of-sync by 30 seconds, in ruler #2's frame they are perfectly in sync, and if I in turn had two clocks that were in sync in my frame and 50 light-seconds apart on my ruler, then the observer on ruler #2 would judge my clocks to be 30 seconds out-of-sync. If you think there is some "real truth" about whose clocks are in-sync and whose clocks are out-of-sync, how would you propose to determine this experimentally? (please give me a specific answer to this question if you reply to this post) Again, every possible statement about the laws of physics that's true in one observer's coordinate system is true in the other's as well. Any experiment I do in my frame, with the results measured in terms of my own rulers and clocks, will give exactly the same results as an identical experiment performed in the frame of ruler #2, with the results measured in terms of rulers and clocks in that frame. No experiment can break the symmetry and prove that one observer's definition of clock synchronization is "correct" while the other's is "incorrect" in some objective sense.

So, with your thought experiment, the following results are obtained:

In the forward direction:

We see 100 seconds tick off both of our clocks.
We see 80 seconds tick off both of the flying clocks.

In the reverse direction:

We see 25 seconds tick off both of our clocks.
We see 20 seconds tick off both of the flying clocks.

However, the guy that is moving along with the flying clocks sees the following:

In the forward direction:

He sees 50 seconds tick off both of his clocks.

In the reverse direction:

He sees 50 seconds tick off both of his clocks.

So, in the forward direction, according to what "we see", 80 seconds have elapsed on both of the flying clocks. (The 50 second difference that you came up with between the front and back clocks is wholly meaningless.) However, we are at the same time supposed to believe the flying guy when he tells us that only 50 seconds have elapsed according to both of his clocks. If we don't think of the flying guy as a liar, our only option is to disbelieve our own eyes.

We don't think he's a liar or disbelieve our own eyes--we just think he is using an "incorrect" method to measure time, since he's defining the time elapsed in terms of readings on two different clocks which are out-of-sync according to the definition of simultaneity used in our frame. But according to the definition of simultaneity used in his frame, his clocks are in-sync, while it is we who are using an "incorrect" method to measure time! After all, the reason we think that the light took 100 seconds to get from back to front is that at the moment the light left the back end of the ruler, the back end was lined up with the 0 l.s. mark on our own ruler (ruler #1), and the clock sitting on that mark read 0 seconds; then at the moment the light reached the front end of the ruler, the front end was lined up with the 100 l.s. mark on our own ruler #1, and the clock sitting on that mark read 100 seconds. These two clocks are synchronized according to our definition of simultaneity, but according to the definition of simultaneity used by the guy sitting on ruler #2, these two clocks were actually 60 seconds out-of-sync. In the frame of the ruler #2 guy, at the moment the light left the back end, the clock on the 0 l.s. mark of our ruler #1 read 0 seconds, while the clock on the 100 l.s. mark of ruler #1 already read 60 seconds. And in the frame of ruler #2 guy, the clocks on ruler #1 are only ticking at 0.8 the correct rate, so 50 seconds later when the light reaches the front end of ruler #2, the clocks on ruler #1 have only advanced forward by 0.8*50 = 40 seconds, so the clock at 0 l.s. now reads 0 + 40 = 40 seconds while the clock at 100 l.s. now reads 60 + 40 = 100 seconds.

Now say we have another clock at the 75 l.s. mark of our ruler #1, which is also synchronized with the other two according to our definition of simultaneity. In the frame of the ruler #2 guy, this clock is 15 seconds behind the clock at the 100 l.s. mark, and like the other two clocks it is slowed down by a factor of 0.8 in his frame. So, in the frame of the ruler #2 guy we have:

1. At the moment the light leaves the back end of ruler #2 (when it is lined up with the 0 l.s. mark on ruler #1), the clock at 0 l.s. on ruler #1 reads 0 s, the clock at 75 l.s. on ruler #1 reads 45 s, and the clock at 100 l.s. on ruler #1 reads 60 s.

2. 50 seconds later in the ruler #2 frame, at the moment the light reaches the front end of ruler #2 (when it is lined up with the 100 l.s. mark on ruler #1), the clock at 0 l.s. on ruler #1 reads 0 + 40 = 40 s, the clock at 75 l.s. on ruler #1 reads 45 + 40 = 85 s, and the clock at 100 l.s. on ruler #1 reads 60 + 40 = 100 s.

3. 50 seconds later in the ruler #2 frame, at the moment the light returns to the back end of ruler #2 (when it is lined up with the 75 l.s. mark on ruler #1), the clock at 0 l.s. on ruler #1 reads 40 + 40 = 80 s, the clock at 75 l.s. on ruler #1 reads 85 + 40 = 125 s, and the clock at 100 l.s. on ruler #1 reads 100 + 40 = 140 s.

So in the frame of the ruler #2 guy, it is us who are measuring time incorrectly by comparing readings on clocks which are out-of-sync according to his definition of simultaneity--he can see why we say the time for the light to go from back to front was 100 s while the time to go from front to back was only 25 s in terms of the readings on our out-of-sync clocks, but he would say that each of our clocks "really" ticked forward by 40 seconds on each phase of the trip.

So you see, the situation really is quite symmetrical, and there's no way to break the symmetry and decide who's "really" right! Each observer thinks that the other is measuring time-intervals "incorrectly" by using a pair of out-of-sync clocks, and each observer also thinks the other guy's clocks are slowed down by a factor of 0.8 relative to his own.

To make the symmetry as clear as possible, I drew up some illustrations of two rulers with clocks on them moving side-by-side in this thread, showing how the situation looks in each ruler's own rest frame. Please take a look at the illustrations, and make sure you understand what they are showing!

So, I ask again that you please not ignore me when it comes to the discrepancies between these two observations:

1) In the forward direction, we count 80 seconds to have physically ticked off of both of the flying clocks while the flying guy counts 50 ticks off of both of the same clocks.

2) In the reverse direction, we count 20 seconds to have physically ticked off of both of the flying clocks while the flying guy counts 50 ticks off of both of the same clocks.

Both of these statements are internally logically contradictory if we are supposed to believe ourselves as well as the flying guy (that is, how can 80 ticks equal 50 ticks, and how can 20 ticks equal 50 ticks?).

We don't "believe" the flying guy's measurement of the time-interval, because in our frame he is measuring the interval "incorrectly", comparing readings on two clocks which are out-of-sync according to our definition of simultaneity. But he says exactly the same thing about our measurements of the time interval, using his own definition of simultaneity! Since all the laws of physics work exactly the same in both frames, there is no possible way to decide which one of us is "really" correct, even if you believe there must be an objective truth about whose definition of simultaneity is "really" the correct one. That's the relativity of simultaneity!

 

[Mentz114]

And if these problems are indeed irresolvable, then the only thing left to do is to abandon Einstein's "improvements" of Galileo's otherwise perfectly sensible concept of relativity.

Arrogant. Do you really think you've discovered a contradiction in SR ? Gallilean relativity is wrong. It breaks down for the EM field. If Gallilean velocity addition is correct I could catch up to a beam of light and hold it in my hands.

You've been given very good answers to your 'paradoxes' but you'll never be convinced by rational argument because you're convinced you're right.

You should be banned from this forum.

 

[joey_m]

If you think there is some "real truth" about whose clocks are in-sync and whose clocks are out-of-sync, how would you propose to determine this experimentally?

There are two types of "in-sync":

1) The type which deals with absolute position
2) The type which deals with relative rates of change

In the first case, two points can only be "in-sync" if they are the same point. In other words, all we are talking about is the law of identity (aka the law of non-contradiction.) In this case, it is perfectly senseless to talk about different points. Applying this logic to your thought experiment, it is perfectly senseless to talk about different clocks. Different clocks can never be "in-sync" (of the first type) precisely because they are not the same clock.

In the second case, we can start talking about different points (or clocks). If two points are moving at the same rate, then they are indeed "in-sync" (of the second type). In your experiment, if two clocks are ticking at the same rate, then they are likewise "in-sync", in this way. The positions of the hands on a clock have nothing to do with this version of synchronicity.

As far as your question is concerned, then, the only way it makes sense is if we distinguish between the two types of "in-sync" that I have outlined. Translated into the first type, we arrive at the following question:

If you think there is some "real truth" about whose clocks are the same clock and whose clocks are different clocks, how would you propose to determine this experimentally?

And translated into the second type:

If you think there is some "real truth" about whose clocks are ticking at the same rate and whose clocks are ticking at different rates, how would you propose to determine this experimentally?

The first question is absurd and unanswerable, as far as I'm concerned. But the second question might very well be worth considering. The problem is that you aren't asking the second question, you are asking the first one. I can't answer it because it is fundamentally meaningless.

As far as your method of time measurement is concerned, I would love to know of someone other than you who advocates subtracting the "hand positions" of different clocks in order to make a determination of the temporal duration of an event.

The bottom line is that temporal duration can only mean the difference in hand positions of the same clock. We can then say that a measurement of temporal duration is accurate if this difference agrees with the difference that would have been obtained by a standard clock. If this is the case, then the clocks are "in-sync", in the meaningful sense of that term.

 

[JesseM]

There are two types of "in-sync":

1) The type which deals with absolute position
2) The type which deals with relative rates of change

In the first case, two points can only be "in-sync" if they are the same point. In other words, all we are talking about is the law of identity (aka the law of non-contradiction.) In this case, it is perfectly senseless to talk about different points. Applying this logic to your thought experiment, it is perfectly senseless to talk about different clocks. Different clocks can never be "in-sync" (of the first type) precisely because they are not the same clock.

In the second case, we can start talking about different points (or clocks). If two points are moving at the same rate, then they are indeed "in-sync" (of the second type). In your experiment, if two clocks are ticking at the same rate, then they are likewise "in-sync", in this way. The positions of the hands on a clock have nothing to do with this version of synchronicity.

Neither of these examples covers the ordinary definition of what it means for two clocks to be "synchronized", namely, showing the same reading at the same time. When people planning a complicated operation in a movie (say, a bank heist) say they need to "synchronize watches" beforehand, do you think they are just saying they want their clocks to tick at the same rate, and don't care if, for example, one guy's clock reads 3:10 at the same moment the other guy's clock reads 3:22? You gave your own example of unsynchronized clocks in a previous post:

Say we are at a track meet, and we are about to watch the 100 meter dash. There is a clock at the start-line and there is another one at the finish-line. When the gun goes off, the clock at the start-line reads 5:00:00 pm and the one at the finish-line reads 4:59:51 pm. That is, the clock at the finish-line is precisely nine seconds behind the one at the start-line.

Surely if one clock is "nine seconds behind" the other, that means they are not "synchronized" in the ordinary English language sense of the word, even if they are "in sync" by some weird definition which is concerned solely with the rate of ticking. Can you come up with a definition to cover this ordinary English meaning, a definition under which the clock at the start-line and the finish-line of your example would not be "synchronized" or "in sync" or whatever your preferred term is?

As far as your method of time measurement is concerned, I would love to know of someone other than you who advocates subtracting the "hand positions" of different clocks in order to make a determination of the temporal duration of an event.

Well, Einstein for one, and every other author of any relativity textbook I've seen. Read Section 1 of his 1905 paper, where he says that to define the time of an event in a given coordinate system, we should use "synchronous stationary clocks located at different places", where he defines what it means for clocks at different places to be "synchronous" using light signals. (Basically, he says if you send a light signal from clock A to clock B, then bounce it back from B to A, then look at the difference in readings between A and B when the light passed each one on the first half of the trip, and compare with the difference in readings between B and A when the light passed each one on the return trip, the two clocks are defined to be 'synchronous' if these two differences are equal to one another. An equivalent method would be to set off a flash at the midpoint of the two clocks, and make sure they read the same time when the light reaches each one.)

Likewise, here's an excerpt from the textbook Spacetime Physics by John Wheeler and Edwin Taylor, where they write on pages 37-38:

The fundamental concept in physics is event. An event is specified not only by a place but also by a time of happening. Some examples of events are emission of a particle or a flash of light (from, say, an explosion), reflection or absorption of a particle or light flash, a collision.

How can we determine the place and time at which an event occurs in a given free-float frame? Think of constructing a frame by assembling meter sticks into a cubical latticework similar to the jungle gym system seen on playgrounds (Figure 2-6). At every intersection of this latticework fix a clock. These clocks are identical. They can be constructed in any manner, but their readings are in meters of light-travel time (Section 1-4).

How are the clocks to be set? We want them all to read the "same time" as one another for observers in this frame. When one clock reads midnight (00.00 hours = 0 meters), all clocks in this frame should read midnight (zero). That is, we want the clocks to be synchronized in this frame.

How are the several clocks to be synchronized? As follows: Pick one clock in the lattice as the standard and call it the reference clock. Start this reference clock with its pointer set initially at zero time. At this instant let it send out a flash of light that spreads out as a spherical wave in all directions. Call the flash emission the reference event and the spreading spherical wave the reference flash.

When the reference flash gets to a slave clock 5 meters away, we want that clock to read 5 meters of light-travel time. Why? Because it takes light 5 meters of light-travel time to travel the 5 meters of distance from reference clock to slave clock. So an assistant sets the slave clock to 5 meters of time long before the experiment begins, holds it at 5 meters, and releases it only when the reference flash arrives. (The assistant has zero reaction time or the slave clock is set ahead an additional time equal to the reaction time). When assistants at all slave clock in the lattice follow this prearranged procedure (each setting his slave clock to a time in meters equal to his own distance from the reference clock and starting it when the reference light flash arrives), the lattice clocks are said to be synchronized.

...

Use the latticework of synchronized clocks to determine location and time at which any given event occurs. The space position of the event is taken to be the location of the clock nearest the event. The location of this clock is measured along three lattice directions from the reference clock: northward, eastward, and upward. The time of the event is taken to be the time recorded on the same lattice clock nearest the event. The spacetime location of an event then consists of four numbers, three numbers that specify the space position of the clock nearest the event and one number that specifies the time the event occurs as recorded by that clock.

So, do you have another idea of what physical apparatus we could use to assign time coordinates to events (such as the event of a runner crossing the starting line and the event of a runner crossing the finish line), and to calculate time intervals between events (the time it took the runner to get from start to finish line), other than using the readings of synchronized clocks which were sitting still (in your frame) right next to the events as they happened (i.e. using a clock at rest at the starting line and another, synchronized clock at rest at the finish line)? If so, what is your alternate idea about how to determine these numbers experimentally?

Also, if you agree that using local readings on synchronized clocks is a decent idea (perhaps you did not recognize that my use of the term 'in sync' was meant to be synonymous with 'synchronized'), then do you have an alternative idea for how each observer should synchronize their own clocks that's different from the light-signal method (which will result in observers in different frames disagreeing about simultaneity)? In Newtonian physics this would of course be a simple matter, we could just bring two clocks to a common location, make sure they're synchronized when right next to each other, then move them apart again. But in relativity this method is not trustworthy, since we know that moving clocks around seems to change their rate of ticking--if you bring two clocks together and synchronize them, then move one away and bring it back, it will now be behind the clock that stayed at rest in your frame.

 

[joey_m]

Neither of these examples covers the ordinary definition of what it means for two clocks to be "synchronized", namely, showing the same reading at the same time.

You say that you are not talking about my first definition of synchronicity (absolute position) but I still say that you are, and I'll tell you why (I fully admit that I bungled the previous explanation)...

A clock is simply an embodiment of a theoretical construct. In this case, the theoretical construct is the dimension of temporality. Of course, clocks exist in many different places, but each clock in the world is an attempt to embody the very same temporal context in which the world exists, in the form of a "world time-line".

In terms of the comparison of the time-lines that two given clocks trace out, there is an important distinction to make between:

1) An agreement in the absolute locations of any of the given points.
2) An agreement in the units of measure, so that the rates of ticking are equal.

Your version of synchronicity is only giving weight to the first case. You are saying, "As long as both clocks agree on the absolute location of both zero-points, then they are synchronous".

Well, my point is simply this... if the hands of the clocks tick at different rates (say one ticks every second, and the other ticks every two seconds), then the first case of synchronocity is all for naught. Granted, the hands on both of the clocks were synchronous for one instant, but for every instant thereafter, they were asynchronous.

Of course you've heard the saying that even a broken clock is correct twice a day. This just means that particular instants of agreement of the absolute locations of the hands on a clock are of no consequence if the hands themselves are not moving at the same continuous rate, in relation to Earth's rate of rotation.

A more abstract way of putting this is that the first case of synchronicity is purely derivative of the second case. Or, it is only because of synchronicity #2 that synchronicity #1 has any real-world meaning.

If we take synchronicity #1 as being of primary importance (as do you and Einstein), then we are attaching a priori significance to a particular, static material arrangement above and beyond the manner in which material objects interact and flow through space. But I say that, if there are to be such things as fundamental natural laws, then the particular moment at which we begin observing nature is of no consequence. The laws of nature, in other words, must be "time-line origin neutral". (I don't want to hear anything about the "Big Bang" theory here. We can talk about that on another thread.)

But in relativity this method is not trustworthy, since we know that moving clocks around seems to change their rate of ticking--if you bring two clocks together and synchronize them, then move one away and bring it back, it will now be behind the clock that stayed at rest in your frame.

This point is nothing other than an abstraction of the "twin paradox", which is very much related to my opening post. I have other ways of stating this conundrum...

Imagine a large, featureless, massive body, that is at rest in the middle of empty space. Now imagine two small satellites that are in motion with respect to each other. Each one can say that the other is orbiting this body around its circumference. Each one of these satellites is carrying a clock. Now, just as they are passing one another, the clocks happen to be perfectly in-sync. When they again pass each other, are the clocks, according to relativity, still not supposed to be perfectly in-sync? And if not, how can we possibly make sense of this fact?

Or, using your point, how about if you moved both clocks away in the same manner, but in opposite directions? They should both be in-sync when you bring them together. But, according to the relativist, since they were both moving relative to one another, they should both show earlier times than each other!

 

[JesseM]

You say that you are not talking about my first definition of synchronicity (absolute position) but I still say that you are, and I'll tell you why (I fully admit that I bungled the previous explanation)...

A clock is simply an embodiment of a theoretical construct. In this case, the theoretical construct is the dimension of temporality. Of course, clocks exist in many different places, but each clock in the world is an attempt to embody the very same temporal context in which the world exists, in the form of a "world time-line".

In terms of the comparison of the time-lines that two given clocks trace out, there is an important distinction to make between:

1) An agreement in the absolute locations of any of the given points.
2) An agreement in the units of measure, so that the rates of ticking are equal.

Your version of synchronicity is only giving weight to the first case. You are saying, "As long as both clocks agree on the absolute location of both zero-points, then they are synchronous".

I don't really understand any of this, you're speaking in very abstract terms and I can't follow what you're trying to say at all. What does "agreement in the absolute location of any of the given points" mean physically? Can you give me a specific, concrete example? I originally took you to mean "location" in space, i.e. the two clocks could only be synchronized" according to your first definition if they were right next to each other (which wouldn't allow us to make sense of the idea that clocks on either end of a racetrack can be either synchronized or out-of-sync), but maybe you're talking about "location" in time or spacetime or something? And likewise what does "given points" mean? Points in space, points in time, events like readings on particular clocks, what? What does it mean for clocks to "agree" on these points? What are the "zero-points"--the events of different clocks reading zero? As you can see I'm pretty lost! A concrete illustration involving specific numbers would be helpful.

Well, my point is simply this... if the hands of the clocks tick at different rates (say one ticks every second, and the other ticks every two seconds), then the first case of synchronocity is all for naught.

What clocks are you talking about? Each observer is assigning coordinates to events using clocks which are all ticking at the same rate as one another. It is only when an observer compares his own clocks to those of a different observer in motion relative to himself that he sees his clocks ticking at a different rate than the other observer's clocks, but no one ever said the clocks of one observer were supposed to be "synchronized" with those of a different observer!

Imagine a large, featureless, massive body, that is at rest in the middle of empty space. Now imagine two small satellites that are in motion with respect to each other. Each one can say that the other is orbiting this body around its circumference. Each one of these satellites is carrying a clock. Now, just as they are passing one another, the clocks happen to be perfectly in-sync. When they again pass each other, are the clocks, according to relativity, still not supposed to be perfectly in-sync? And if not, how can we possibly make sense of this fact?

Why do you think there's a problem? In the inertial rest frame where the body is at rest, both clocks have exactly the same speed at all times, so of course they tick at the same rate (slowed down by the same amount relative to clocks at rest in this frame).

Or, using your point, how about if you moved both clocks away in the same manner, but in opposite directions? They should both be in-sync when you bring them together. But, according to the relativist, since they were both moving relative to one another, they should both show earlier times than each other!

Your "according to the relativist" is wrong, because if you move the clocks apart and then bring them together, they cannot possibly have been moving inertially (constant speed and direction) the whole time. The prediction that moving clocks run slow as predicted by the time dilation equation is only meant to work in inertial frames, like the frame of the observer who remains at the midpoint of the two clocks and measures their velocities as symmetrical. And of course, if in his frame their speeds are identical at all moments, the amount they're slowed down will be the same at all moments too, so it's no surprise that they show the same elapsed time.

 

[joey_m]

The problem here is that we have two distinctly different ideas of the meaning and significance of "synchronicity". I am convinced that this is all the result of the misapplication of the term "simultaneity" as well as a bizarre use of the word "event". To me, an event cannot be a mathematical point, and two different points can never be simultaneous. I also don't see the possible physical significance of whether two different clocks are ever showing the same hand positions. It might be an interesting logical issue to discuss, but it is of no fundamental importance. I only care about whether they are ticking at the same rate.

Because of these problems that I have, I don't think there is much hope of us finding a common language to use in order to discuss this particular issue much further.

Also, about the orbiting satellite problem, both satellites are in motion relative to each other. Observers on each one could say that the other is orbiting around the body, while he is hovering above it. The fact that the body around which they are orbiting is motionless has nothing to do with this. If their clocks are showing the same time when they cross paths once, and they are always moving relative to each other, then according to relativity, they should both show times earlier than each other when they cross paths again.

If you say that they are not in rectilinear motion relative to each other, and therefore time dilation does not apply, then all you have to do is read Chapter 23 of Relativity to see why that explanation does not apply: http://www.bartleby.com/173/23.html

 

[yuiop]

Why do you think there's a problem? In the inertial rest frame where the body is at rest, both clocks have exactly the same speed at all times, so of course they tick at the same rate (slowed down by the same amount relative to clocks at rest in this frame).

I assume by "the inertial rest frame where the body is at rest" you mean the central massive body that both satellites are orbiting around. How do you define that the central massive body is the body really at rest while the satellites are not at rest? For example, let's say the satellites are orbiting Earth. One of the satellites is in a geostationary orbit above Earth so it remains stationary above a specific point on the Earth while the other satellite is at exactly the same height but orbiting in the opposite direction. Can the geostationary satellite now claim to be the one "really" at rest because it as at rest with respect to the central massive body?

Your "according to the relativist" is wrong, because if you move the clocks apart and then bring them together, they cannot possibly have been moving inertially (constant speed and direction) the whole time. The prediction that moving clocks run slow as predicted by the time dilation equation is only meant to work in inertial frames, like the frame of the observer who remains at the midpoint of the two clocks and measures their velocities as symmetrical.

In general relativity, a free falling body feels no force and is considered to be in an inertial reference frame. An orbiting satellite also feels no force and is also in an inertial reference frame, so the argument that the satellites are not in inertial reference frames is not strictly correct. However, if we have a ring of satellites all orbiting in the same direction it will be impossible to syncronise them by the classical Einstein method.

And of course, if in his frame their speeds are identical at all moments, the amount they're slowed down will be the same at all moments too, so it's no surprise that they show the same elapsed time.

Of course I agree with this last statement and I do not disagree with your overall position. Just being a sort of Devil's advocate. Joey, being the "anti relativist" will ultimately lose his argument, but he has to be shot down correctly ;)

 

[JesseM]

The problem here is that we have two distinctly different ideas of the meaning and significance of "synchronicity". I am convinced that this is all the result of the misapplication of the term "simultaneity" as well as a bizarre use of the word "event". To me, an event cannot be a mathematical point, and two different points can never be simultaneous. I also don't see the possible physical significance of whether two different clocks are ever showing the same hand positions. It might be an interesting logical issue to discuss, but it is of no fundamental importance. I only care about whether they are ticking at the same rate.

Well, have you ever studied the mathematics of any area of physics, including Newtonian mechanics? The only way to describe the laws governing the motion of objects in physics is to write down equations for their motion which are based on some coordinate system that tells you the position of objects as a function of some global time-coordinate. How else would you suggest anyone could write down physical laws mathematically? If you had a separate time-coordinate for every position in space, it seems to me you'd have no way to describe the motion of an object through different points in space using equations. Likewise, if you didn't idealize by assuming that that each object has a precise position at every time-coordinate--that, for example, the event of the object's center being at position x=6 meters, y=3.7 meters, z=8.2 meters happens at precisely t=15.6 seconds--then I also don't see how you can describe motion mathematically using equations. And that's all that "event" means, some physical fact which can be pinpointed as happening at a precise set of position and time coordinates (even if you drop the notion of objects having sharp edges or centers and try to describe physics in terms of continuous fields of density or force or whatever, you still have to be able to talk about the precise value of this field at any given precise position and time, and the field taking that value at that position and time is an 'event' too). I don't see how you can reject these kinds of ideas without rejecting the entire idea of describing laws of physics mathematically using equations, not just relativity but also Newtonian physics.

Anyway, once you do have a coordinate system with a global time-coordinate, then all the "simultaneity" means is that two events have the same time-coordinate. We don't need to worry about whether about whether these events "really" happened at the same time or not, but the notion of "happening at the same time-coordinate" is perfectly well-defined as long as you have some physical recipe for assigning coordinates, like the relativistic notion of having a lattice of inertial rulers and clocks at rest relative to one another, with the clocks' times set using light-signals in the way I (and Wheeler) described earlier.

If this stuff seems foreign to you, I really recommend A) learning some basic calculus if you don't know it already, particularly the notion of derivatives and integrals, and then B) learning basic Newtonian mechanics, either taking a course or working through a textbook. You'll soon see that all this is inherent to writing down equations to describe physics, I can't imagine any alternative way. For example, take the Newtonian equation for the acceleration a(t) of an object as a function of time in the presence of the gravitational field of another object with mass M, where r(t) represents the precise distance between the centers of the objects, and G is the "gravitational constant". Then the equation will be a(t) = GM/r(t)^2. How could this be rewritten without making use of the notion that we have a global time coordinate which can describe the distance between two objects at different spatial locations as a function of the time coordinate?

Also, about the orbiting satellite problem, both satellites are in motion relative to each other. Observers on each one could say that the other is orbiting around the body, while he is hovering above it.

To deal with gravity you have to deal with curved spacetime which is outside the domain of special relativity, so let's simplify this and ignore gravity by imagining two observers being swung in circles in opposite directions by ropes attached to some central point. In this case, yes, each observer could adopt a coordinate system where they are at rest (although I don't know how you could even describe the idea of motion vs. rest without a coordinate system that tells you an object's position coordinate as a function of the time coordinate, and whether it's the same or different at different time coordinates). However, each observer will realize they are not at rest in an inertial coordinate system because they will feel G-forces--the "centrifugal force"--while an observer in an inertial coordinate system always feels completely weightless.

The fact that the body around which they are orbiting is motionless has nothing to do with this. If their clocks are showing the same time when they cross paths once, and they are always moving relative to each other, then according to relativity, they should both show times earlier than each other when they cross paths again.

Nope, because again, relativity doesn't say the same time dilation equation still works in non-inertial coordinate systems.

If you say that they are not in rectilinear motion relative to each other, and therefore time dilation does not apply, then all you have to do is read Chapter 23 of Relativity to see why that explanation does not apply: http://www.bartleby.com/173/23.html

Einstein never says here that the standard equations of special relativity still work in the frame K' of the rotating observer; the whole point of the passage is to show how laws work differently in this frame, how the rotating observer seems to experience a "gravitational field" which affects how objects move in his frame, and also influences the rate that clocks ticking due to "gravitational time dilation", an effect which is different than the purely velocity-based time dilation of inertial frames in SR.