Quick calculation check please

[name123]

Could someone just check these calculation results for me please. I know this is really basic, and I'm assuming they aren't right but I couldn't see where they were wrong, so any help would be appreciated. (I know I've been given loads on this forum already, and I am grateful).

v    = 29,979,245.8
gam  = 1.005

t        0             1              2
x        0             0              0
t'       0             1.005          2.010
x'       0             -30,130,275.7  -60,260,551.4

t        0             1              2
x        0             29,979,245.8   59,958,491.6
t'       0             0.9950         1.990
x'       0             0              0

 

[jedishrfu]

I edited your post and used code tags which give you a mono spaced font.

You may want to clean it up a little more though.

 

[name123]

I edited your post and used code tags which give you a mono spaced font.

You may want to clean it up a little more though.

Thank you :)

 

[DrGreg]

...I'm assuming they aren't right...

Why are you assuming that?

Note that you have rounded $\gamma$ to 4 significant figures, so don't expect to get more than 4-sig-fig precision from your calculations.

 

[name123]

Could someone just check these calculation results for me please. I know this is really basic, and I'm assuming they aren't right but I couldn't see where they were wrong, so any help would be appreciated. (I know I've been given loads on this forum already, and I am grateful).

v    = 29,979,245.8
gam  = 1.005

t        0             1              2
x        0             0              0
t'       0             1.005          2.010
x'       0             -30,130,275.7  -60,260,551.4

t        0             1              2
x        0             29,979,245.8   59,958,491.6
t'       0             0.9950         1.990
x'       0             0              0

Why are you assuming that?

Note that you have rounded $\gamma$ to 4 significant figures, so don't expect to get more than 4-sig-fig precision from your calculations.

The problem with it being correct seems to me to be that I could imagine a conveyor belt going at 0.1c relative to the floor. And I could imagine observers along the floor where conveyor belt would have moved to after each second (from the floors perspective). The second table seems to be showing me that for any given observer on the conveyor belt they could consider t'=0 x'=0 to be at the point of passing any observer on the floor, and they will all agree that their clocks are only doing roughly 0.9950 ticks for each tick the floor observer's clocks, and this can be checked each time they subsequently pass an observer (so in table 2 it shows that after it sets t'=0 x'=0 as it passes a floor observer, when it passes the next that next ones clock has gone on a second while its has only progressed 0.995, and when it passes the next, it has also gone on a second while its has only progressed 0.995 since the last time. But it also seems to be saying (in the first table), that if they kept their eye on the clock they were passing when they decided t'=0 x'=0 they would be thinking that for each tick it made their clock had made 1.005 ticks, even though everyone on the conveyor belt can agree that their clocks aren't going faster but slower compared with the ones they were passing, and that can be checked each time they pass a floor observer.

I think I can see it now from the conveyor belt's perspective the t' isn't for the one that set t'=0 x'=0 it is for the one on the conveyor belt at the x' coordinate given (the one that set t'=0 x'=0 remains at x' =0 from its perspective), but am still not clear on the different ratio.

 

[Ibix]

First a tip to make arithmetic easier: measure distance in light seconds (ls) and time in seconds. Then c=1 ls/s and v=0.1 ls/s.

Is it the lack of symmetry that's bothering you? By that, I mean were you expecting the second table to look a bit more like the first?

What's going on here is that you've set up your experiment in a way designed to show you what's going on in the floor frame. In the first table you've got one clock on the floor at x=0 and three clocks on the belt at x'=0, -0.1005, -0.2010. In the second table you've got one clock on the belt at x'=0 and three clocks on the floor at x=0, 0.1000, 0.2000. Basically, you have pre-positioned clocks on the belt so that they pass clocks on the floor at a convenient time for clocks on the floor in both tables. This means that your experimental setup is privileging the floor frame; that's why it kind of looks like there's something special about the floor frame.

If you use (x',t')=(0,0), (0,1), (0,2) and (x',t')=(0,0), (0.1,1), (0.2,2) you'll have privileged the belt frame. If you then calculate the (x,t) coordinates you'll reproduce your tables upside down, except for a sign.

Also, draw a space-time diagram. They make visualisation a lot easier.

Hope that's helpful.

 

[name123]

First a tip to make arithmetic easier: measure distance in light seconds (ls) and time in seconds. Then c=1 ls/s and v=0.1 ls/s.

Is it the lack of symmetry that's bothering you? By that, I mean were you expecting the second table to look a bit more like the first?

What's going on here is that you've set up your experiment in a way designed to show you what's going on in the floor frame. In the first table you've got one clock on the floor at x=0 and three clocks on the belt at x'=0, -0.1005, -0.2010. In the second table you've got one clock on the belt at x'=0 and three clocks on the floor at x=0, 0.1000, 0.2000. Basically, you have pre-positioned clocks on the belt so that they pass clocks on the floor at a convenient time for clocks on the floor in both tables. This means that your experimental setup is privileging the floor frame; that's why it kind of looks like there's something special about the floor frame.

If you use (x',t')=(0,0), (0,1), (0,2) and (x',t')=(0,0), (0.1,1), (0.2,2) you'll have privileged the belt frame. If you then calculate the (x,t) coordinates you'll reproduce your tables upside down, except for a sign.

Also, draw a space-time diagram. They make visualisation a lot easier.

Hope that's helpful.

I think I get it now. If an observer on the floor was to set its clock to 0 as the observer that is passing it on the conveyor belt sets its clock to 0, and the next observer on the floor also sets its clock to zero, then when the observer on the conveyor belt that had set its clock to 0 reaches it, they can compare and see that while its has progressed a second the one on the conveyor belts has only progressed 0.9950.
But if it were done the other way around and the observers on the floor thought they were stationary and it was the observers on the floor that were moving, and the conveyor belt observers could set their clocks to zero as their clocks showed a certain time, but the observers on the floor might object and say that the clocks on the conveyor belt weren't showing the same time at the same time and if they waited until that to set the clocks to zero they'd be out of synch. Instead the observers on the floor might suggest, that the conveyor belt observers set their clocks to zero as they pass the next floor observer, and then from the conveyor belt perspective each person on the conveyor belt has set its clock to 0 when a one of the traveling floor observers was passing it, and the next was a fixed distance away, and for each of them by the time the next reaches it the passing observer's clocks have ticked 1 while theirs have only ticked 0.995. And that is for all the observers on the conveyor belt from the conveyor belts perspective and they all agree that the distance the next moving floor observer was the same for each of them and that the floor observers were moving at a fixed velocity. From that I would have thought they could have concluded that it takes a fixed amount of time for the floor observers moving at a fixed velocity to go between one conveyor belt observer and another a fixed distance away, and that this could be checked on each passing. But the distance isn't fixed if you use the speed of light as being invariant to measure things. When doing so whether the conveyor belt observers are both equal distance apart or not depending on where you measure it from.

 

[Ibix]

Sorry for the slow reply - I typed up a response on my phone yesterday, but the browser crashed just before I sent it and I haven't had a chance to respond since.

I think I get it now. If an observer on the floor was to set its clock to 0 as the observer that is passing it on the conveyor belt sets its clock to 0, and the next observer on the floor also sets its clock to zero, then when the observer on the conveyor belt that had set its clock to 0 reaches it, they can compare and see that while its has progressed a second the one on the conveyor belts has only progressed 0.9950.

It's very important when talking about "at the same time" to specify who is defining "the same time", since observers in different frames won't necessarily agree. I think you mean "at the same time in the floor frame" here, in which case, I agree.

But if it were done the other way around and the observers on the floor thought they were stationary and it was the observers on the floor that were moving, and the conveyor belt observers could set their clocks to zero as their clocks showed a certain time, but the observers on the floor might object and say that the clocks on the conveyor belt weren't showing the same time at the same time and if they waited until that to set the clocks to zero they'd be out of synch. Instead the observers on the floor might suggest, that the conveyor belt observers set their clocks to zero as they pass the next floor observer, and then from the conveyor belt perspective each person on the conveyor belt has set its clock to 0 when a one of the traveling floor observers was passing it, and the next was a fixed distance away, and for each of them by the time the next reaches it the passing observer's clocks have ticked 1 while theirs have only ticked 0.995. And that is for all the observers on the conveyor belt from the conveyor belts perspective and they all agree that the distance the next moving floor observer was the same for each of them and that the floor observers were moving at a fixed velocity.

I think you've got this, although your description is rather confusing. What I think you are realising is that the floor-based observers see the belt-based clocks ticking slowly, so they are happy that the belt-based clocks only show 0.995s passing in the 1s it takes them to move from one floor-based observer to the next. Meanwhile, the belt-based observers see the distance between adjacent floor-based observers length contracted to 0.0995ls, so are happy that only 0.995s elapses between successive floor-based observers passing them.

But the distance isn't fixed if you use the speed of light as being invariant to measure things.

This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one.

 

[name123]

This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one.

So if I call the observers on the floor Team A, and the observers on the conveyor belt that is going at 0.1c Team B, and let's still imagine that in the Team A reference frame the observers are spaced out 0.1c distance from each other in their frame of reference, and they measure it in feet for example so roughly 98,357,105.643 ft apart, and let's say they it measure out in rulers from their rest frame. Let's also imagine that they have some rulers from Team B's rest frame, and they measure the distance out with however many of them it takes. Team A can agree that the distance of a certain amount of Team B rulers is equal to a certain amount of their rulers, and then it would be a fixed equivalent distance whichever rulers you used wouldn't it, or are you suggesting that there would be a disagreement about the amount of rulers (of either type) laid out on the floor between the Team A observers?

 

[Ibix]

Seriously, use a sensible unit for the problem. Using feet to measure experiments on the scale of light seconds is asking for transcription errors and, as Dr Greg noted, pointless unless you retain 11 significant figures on everything.

You have a conceptual problem here. Rulers are not in one frame or the other, they are in both. That means that "rulers from Team B's rest frame" doesn't make sense. Frames are just choices of coordinates. Imagine that we're discussing a trip. We both have maps on our phones, but yours has flipped into landscape mode. Roads that are going up my screen are going across yours. In other words, my phone is using a reference frame in which north is up, but yours is using one where west is up. It doesn't make sense to talk about "bringing a road from your map to mine". The same is true of relativistic frames.

Team A could lay out rulers on the floor (which would be at rest in Team A's frame), and would measure 0.1ls between observers. Team B could lay out rulers on the belt (which would be at rest in Team B's frame), and would measure 0.0995ls.

The teams could then swap rulers, rewind the belt, and repeat if they wish; they'll get the same result.

 

[name123]

Seriously, use a sensible unit for the problem. Using feet to measure experiments on the scale of light seconds is asking for transcription errors and, as Dr Greg noted, pointless unless you retain 11 significant figures on everything.

You have a conceptual problem here. Rulers are not in one frame or the other, they are in both. That means that "rulers from Team B's rest frame" doesn't make sense. Frames are just choices of coordinates. Imagine that we're discussing a trip. We both have maps on our phones, but yours has flipped into landscape mode. Roads that are going up my screen are going across yours. In other words, my phone is using a reference frame in which north is up, but yours is using one where west is up. It doesn't make sense to talk about "bringing a road from your map to mine". The same is true of relativistic frames.

Team A could lay out rulers on the floor (which would be at rest in Team A's frame), and would measure 0.1ls between observers. Team B could lay out rulers on the belt (which would be at rest in Team B's frame), and would measure 0.0995ls.

The teams could then swap rulers, rewind the belt, and repeat if they wish; they'll get the same result.

In this example though there is no assumption that the speed of light was invariant whatever the rest frame. So it doesn't matter that the light travels the rulers faster in one rest frame than the other. The reason I chose feet instead of meters is that meters are measured in light with the principle that the speed of light is invariant whatever the rest frame. Remember, I had earlier wrote:

"But the distance isn't fixed if you use the speed of light as being invariant to measure things"

And you replied:

"This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one."

So I've given an example where distance is measured by rulers (in feet), and you haven't explained why the distance would be considered different if the speed of light wasn't considered invariant in each rest frame.

 

[PAllen]

In this example though there is no assumption that the speed of light was invariant whatever the rest frame. So it doesn't matter that the light travels the rulers faster in one rest frame than the other. The reason I chose feet instead of meters is that meters are measured in light with the principle that the speed of light is invariant whatever the rest frame. Remember, I had earlier wrote:

"But the distance isn't fixed if you use the speed of light as being invariant to measure things"

And you replied:

"This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one."

So I've given an example where distance is measured by rulers (in feet), and you haven't explained why the distance would be considered different if the speed of light wasn't considered invariant in each rest frame.

Because that's the way the universe works? All physical phenomena transform between frames using the Lorentz transform, irrespective of whether they are based in any way on light. You should be aware that from the indistinguishability of inertial frames, isotropy, and homogeneity, you can derive that either the Galilean transform or the Lorentz transform (with some finite invariant speed), are the only possibilities. Then, all physics will be consistent with one or the other, based on experiment. In our universe, it is the Lorentz transform.

 

[Ibix]

So I've given an example where distance is measured by rulers (in feet), and you haven't explained why the distance would be considered different if the speed of light wasn't considered invariant in each rest frame.

Take a 1m ruler. Mark it "1m" on one side and "39.37in" on the other. Do you really think one side will length contract and not the other? If you want to, mount a mirror and clocks on each end and start light pulses bouncing between them, so you're measuring the distance in terms of light speed. How will distance measured one way change without the other changing?

The meter isn't defined in terms of light speed because of relativity. It's defined that way because we're really good at measuring time, so it makes sense to define distance units in terms of a speed and a time.

I may have misled you with my point about the frame invariance of "[t]he speed of light [being] one possible starting point". For the record, the speed of light is invariant in all inertial frames. This was one of Einstein's postulates, and if it is not true in any given scenario then we are not discussing relativity.

 

[name123]

Because that's the way the universe works? All physical phenomena transform between frames using the Lorentz transform, irrespective of whether they are based in any way on light. You should be aware that from the indistinguishability of inertial frames, isotropy, and homogeneity, you can derive that either the Galilean transform or the Lorentz transform (with some finite invariant speed), are the only possibilities. Then, all physics will be consistent with one or the other, based on experiment. In our universe, it is the Lorentz transform.

I never said it wasn't. All the calculations used the Lorentz transformation.

 

[name123]

Take a 1m ruler. Mark it "1m" on one side and "39.37in" on the other. Do you really think one side will length contract and not the other? If you want to, mount a mirror and clocks on each end and start light pulses bouncing between them, so you're measuring the distance in terms of light speed. How will distance measured one way change without the other changing?

The meter isn't defined in terms of light speed because of relativity. It's defined that way because we're really good at measuring time, so it makes sense to define distance units in terms of a speed and a time.

I may have misled you with my point about the frame invariance of "[t]he speed of light [being] one possible starting point". For the record, the speed of light is invariant in all inertial frames. This was one of Einstein's postulates, and if it is not true in any given scenario then we are not discussing relativity.

I never said that the ruler would shrink on one side and not the other. The point was that what is a meter is determined by light. So with the supposition that the speed of light isn't invariant, a meter ruler would be a different length according to Team A members than Team B members, and while the scenario did show how it could handle them being different lengths (they might not be the same number of feet), I thought it might get confusing to use meters as a measurement since it would sound as though it implied a ruler of the same length (same name) but actually it would be a different length ruler (if the speed of light wasn't considered to be invariant).

It seems to me that using a meter as a measurement of distance relies on the assumption that clocks don't slow with motion. Whereas using a ruler only relies on the assumption that the distance is a single distance as opposed to multiple distances and which one it is for an observer depending upon the relative motion of their rest frame compared to the rest frame containing the distance. So given that your response was your reply to:

"So I've given an example where distance is measured by rulers (in feet), and you haven't explained why the distance would be considered different if the speed of light wasn't considered invariant in each rest frame."

You replied, but I didn't notice your answer. Because remember I had written:

"But the distance isn't fixed if you use the speed of light as being invariant to measure things"

So it was clear that I understood that if the speed of light was considered as invariant (as with relativity) that the distance wasn't considered fixed. And yet you replied:

"This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one."

So if you just want to say "well in relativity the speed of light is considered to be invariant", then that is fine, but it wouldn't be right to suggest what I had stated wasn't quite right, and state that "distances are different when measured in different frames whatever method you use to measure them" ** if ** one could consider the speed of light to not be invariant and measure distances with rulers, and the distances remain the same. And it is your lack of an answer to why the distances wouldn't be considered to be the same under such circumstances. that led me to write "you replied, but I didn't notice your answer".

I was thinking that if the team A observers were separated by a distance of 0.1c (in their rest frame) marked out in 1m rulers from Team A's rest frame and 1m rulers from Team B's rest frame, so that if there was any difference in length it would be reflected in the amount of rulers of that type compared to the amount of rulers of the other type, and if the all observers (both team A and team B) were to set their clocks to zero when they next passed an observer (so x = x'=0, t = t' = 0) then they could all agree that when they next passed an observer, the floor observers body clocks were showing t = 1 while the conveyor belt observers body clocks were showing t = 0.9950. What I'm not clear on is why if that is so, it couldn't be concluded that if light wasn't considered to be invariant the distance between the team A members could be considered to be fixed (a certain amount of rulers of whatever type) and that Team B's clocks could be considered to be going slower than Team A's. So is it so, and is it ok to make that conclusion?

 

[PeterDonis]

with the supposition that the speed of light isn't invariant, a meter ruler would be a different length according to Team A members than Team B members

You have to be careful in specifying exactly what is meant by "the speed of light isn't invariant". What appears as "the speed of light" in the table of SI units is just a defined number, 299,792,458; but if you go out and measure some light and how it is moving, you will be using a ruler that is some number of atoms long (and to be perfectly precise, you would have to specify exactly what kind of atoms), and measuring with a clock whose "ticks", if you are using the SI definition of the second, are determined by the frequency of a particular energy level transition in a particular kind of atom. So if Team A and Team B want to compare their measurements, they have to compare how many atoms long their rulers are (and make sure both rulers are made of the same kind of atoms), and exactly what kind of atom and which energy level transition they are using to define their clock ticks.

What "the speed of light isn't invariant" would mean is that, after both teams have verified that their rulers are exactly the same number of the same kind of atoms long, and that their clocks are using exactly the same energy level transition in the same kind of atom for their tick rate, they get different results for the ratio of (ruler lengths traveled by light) / (clock ticks). The reason SI units now define the speed of light as a particular number is that, whenever such experiments have actually been done, Teams A, B, C, D, ... W, X, Y, Z have all gotten the same number, 299,792,458, for the ratio of (ruler lengths traveled by light) / (clock ticks).

All of the SR stuff about length contraction, time dilation, etc., applies to rulers and clocks defined in this way: i.e., if we have a ruler that is $L$ atoms long, moving relative to us, then a ruler at rest relative to us will have the same length as the moving ruler (i.e., the ends will line up at a particular instant of our time) if it is only $L / \gamma$ atoms long. And if we have a clock moving relative to us, it will tick $T$ ticks (defined by the appropriate atomic energy level transition) in the same time as an identical clock at rest relative to us will tick $T \gamma$ ticks (defined by the same energy level transition).

It seems to me that using a meter as a measurement of distance relies on the assumption that clocks don't slow with motion

No, it only relies on the assumption that the clock and the ruler are at rest relative to each other and to the experimenter. That is the condition that ensures that you get the ratio 299,792,458 for the ratio of (ruler lengths traveled by light) / (clock ticks) (where the ruler is a particular number of atoms long and the clock ticks are determined by a particular atomic energy level transition). The fact that identical rulers and clocks in motion relative to the experimenter (and to his rulers and clocks) appear shorter and tick slower is not an assumption; it's an experimental fact.

 

[name123]

You have to be careful in specifying exactly what is meant by "the speed of light isn't invariant". What appears as "the speed of light" in the table of SI units is just a defined number, 299,792,458; but if you go out and measure some light and how it is moving, you will be using a ruler that is some number of atoms long (and to be perfectly precise, you would have to specify exactly what kind of atoms), and measuring with a clock whose "ticks", if you are using the SI definition of the second, are determined by the frequency of a particular energy level transition in a particular kind of atom. So if Team A and Team B want to compare their measurements, they have to compare how many atoms long their rulers are (and make sure both rulers are made of the same kind of atoms), and exactly what kind of atom and which energy level transition they are using to define their clock ticks.

What "the speed of light isn't invariant" would mean is that, after both teams have verified that their rulers are exactly the same number of the same kind of atoms long, and that their clocks are using exactly the same energy level transition in the same kind of atom for their tick rate, they get different results for the ratio of (ruler lengths traveled by light) / (clock ticks). The reason SI units now define the speed of light as a particular number is that, whenever such experiments have actually been done, Teams A, B, C, D, ... W, X, Y, Z have all gotten the same number, 299,792,458, for the ratio of (ruler lengths traveled by light) / (clock ticks).

All of the SR stuff about length contraction, time dilation, etc., applies to rulers and clocks defined in this way: i.e., if we have a ruler that is $L$ atoms long, moving relative to us, then a ruler at rest relative to us will have the same length as the moving ruler (i.e., the ends will line up at a particular instant of our time) if it is only $L / \gamma$ atoms long. And if we have a clock moving relative to us, it will tick $T$ ticks (defined by the appropriate atomic energy level transition) in the same time as an identical clock at rest relative to us will tick $T \gamma$ ticks (defined by the same energy level transition).

I don't see that we need to go into the technical details of making rulers of near equal length. You could even use light to make a really long ruler in your rest frame.

I had said:
"It seems to me that using a meter as a measurement of distance relies on the assumption that clocks don't slow with motion"
and you replied:

No, it only relies on the assumption that the clock and the ruler are at rest relative to each other and to the experimenter. That is the condition that ensures that you get the ratio 299,792,458 for the ratio of (ruler lengths traveled by light) / (clock ticks) (where the ruler is a particular number of atoms long and the clock ticks are determined by a particular atomic energy level transition). The fact that identical rulers and clocks in motion relative to the experimenter (and to his rulers and clocks) appear shorter and tick slower is not an assumption; it's an experimental fact.

It wasn't the shorter rulers and ticks were an assumption, but I was assuming that if light wasn't invariant, and clocks slowed with motion, then a 1m ruler in one rest frame could be a different length from a 1m ruler in another rest frame. And could be measured to be so side by side. The Team B people on the conveyor belt could use their Team B 1m rulers to measure out how far apart the Team A members were, and the Team A members used their Team A 1m rulers to measure how far they are apart, and they could stop the conveyor belt and check. Whether it would be found that the rulers were different lengths when in the same rest frame or whether they were the same length in the same rest frame, and that the team B members were spaced out further than the team A members wouldn't change the thought experiment fact that while in motion the Team B 1m rulers on the conveyor belt measured the distance between the Team A members differently from the Team A 1m rulers. Has there been an experiment to tell btw?

The team A members have their clocks and rulers at rest relative to each other (and they are the experimenters), and the team B members have their clocks and rulers at rest relative to each other, but they both come up with different answers when it comes to how many meters it is. I'm not suggesting that the distance they are giving is relative, it is just that it seems to me you'd need to get into their rest frame and do an experiment with light and a clock to find the distance (from there to there) that was being talked about, or use the transformation to compare it to a 1m distance in another rest frame. So when I meant a meter as a measurement of distance, I meant from a "there to there" perspective, and if the universe had been different, and the clocks had slowed in a different way, it could have been really impractical to assume that light would always go the same distance each tick of the clock.

 

[PeterDonis]

You could even use light to make a really long ruler in your rest frame.

But that would be measuring distance by light travel time, which makes the speed of light invariant by definition; the question "what would it be like if the speed of light was not invariant" would be meaningless. In order to even ask that question, you need to have a way of measuring distance (like "number of atoms long") that isn't dependent on light travel time.

I was assuming that if light wasn't invariant, and clocks slowed with motion, then a 1m ruler in one rest frame could be a different length from a 1m ruler in another rest frame.

But what defines a "1m ruler"? Is it "a certain number of atoms long"? If so, what does the speed of light have to do with it? Each Team makes sure their rulers are the same number of atoms long. Then you bring both rulers to rest and compare them.

The Team B people on the conveyor belt could use their Team B 1m rulers to measure out how far apart the Team A members were, and the Team A members used their Team A 1m rulers to measure how far they are apart, and they could stop the conveyor belt and check.

This assumes that the process of stopping the conveyor belt does not affect the physical structure of the rulers. Experimentally, this appears to be a good assumption, at least for reasonable accelerations; but it's worth pointing it out since we're trying to be explicit about assumptions.

Also, experimentally, we find that whenever rulers that are the same number of atoms long are brought to rest and compared, they have the same length (their ends match up), regardless of their past history of relative motion (subject to the above assumption being valid). But, again, that by itself doesn't prove anything about whether the speed of light is invariant; it just proves that "number of atoms long" is a good criterion for how to define the "length" of a ruler. Then you can use these rulers, along with clocks whose tick rates are set by atomic energy level transitions, to measure the speed of light. So whether or not the speed of light is invariant is independent of whether or not rulers of identical construction (same number of atoms) have identical lengths when they are at rest relative to each other.

 

[name123]

I had said:

"You could even use light to make a really long ruler in your rest frame."

To which you replied:

But that would be measuring distance by light travel time, which makes the speed of light invariant by definition; the question "what would it be like if the speed of light was not invariant" would be meaningless. In order to even ask that question, you need to have a way of measuring distance (like "number of atoms long") that isn't dependent on light travel time.

I'm not sure why that would make the speed of light invariant if the rulers could be different lengths. For example, if team A considered there to be 0.1c Team A 1m rulers between the A team members, but the B team considered there to be more Team B 1m rulers, and when they stop the conveyor belt they find they are right. A 1m ruler would the distance light traveled in 1 second in the rest frame divided by 299 792 458, and just considering the distance between the team A members, the ruler length seems to me to be different in Team B's rest frame than in Team A's rest frame.

 

[PeterDonis]

I'm not sure why that would make the speed of light invariant if the rulers could be different lengths.

I don't understand. You're proposing defining the "length" of the ruler by how long it takes light to go from one end to the other. (There's no other way to make a "ruler" from light.) That means the "speed of light" is "one ruler length per ruler length" by definition; it can't change.

 

[name123]

I don't understand. You're proposing defining the "length" of the ruler by how long it takes light to go from one end to the other. (There's no other way to make a "ruler" from light.) That means the "speed of light" is "one ruler length per ruler length" by definition; it can't change.

The speed of light in that rest frame, you seem to be assuming an invariance in the speed of light between rest frames.

Consider the scenario where the Team B members agree on the spots where they pass the Team A members. You could imagine that the difference is that the Team B rulers are smaller than the Team A rulers so that the speed of light wasn't invariant, as in one rest frame it doesn't go so far in a second, which is why the ruler is smaller and they disagree on what the correct distance between the A team members is. I'm not saying that is the case, I'm just pointing out that unless you assume an invariance in the distance light travels in every rest frame, that a particular ruler is based on the distance the light traveled in a certain clock time in a certain frame of reference doesn't imply that it is the same length (covers the same distance) as other such rulers in other rest frames even if identical clocks were used to measure the time light took to travel the length of the ruler (it could be a description of a universe in a fictional book)

 

[PeterDonis]

The speed of light in that rest frame, you seem to be assuming an invariance in the speed of light between rest frames.

No, I'm not. I'm just reasoning from what using "light" to define the "length" of a ruler means. It doesn't matter what frame you do the measurement in.

Let me try describing it a bit differently. I have a ruler and I want to know how "long" it is, in terms of light. What do I do? Light cannot be at rest, so there's no way to grab a light beam, bring it to rest, and lay it alongside my ruler to see how many "light units" long the ruler is. The only thing I can do is to send the light from one end of the ruler to the other and time how long it takes. So the "length" of the ruler is just the time it takes light to go from one end to the other.

Now I ask what the "speed" of the light is. The speed is defined as "the distance from one end of the ruler to the other" divided by "the time it takes light to go from one end to the other". But we have just defined the first thing to be equal to the second thing; so dividing one by the other can only give the answer 1. In other words, we have (speed of light) = (distance from one end to the other) / (time it takes light to go from one end to the other) = (time it takes light to go from one end to the other) / (time it takes light to go from one end to the other) = 1. There's no other possible answer.

Now, where in the above did I specify any frame? The same reasoning goes through regardless of the frame, i.e., regardless of the state of motion of the ruler. (The only thing you need to assume, as I said before, is that the clock that measures the light travel time is at rest relative to the ruler. But they can both be moving at any speed you like relative to anything else.) Even if two teams, Team A and Team B, run the same experiment in different frames, they must both come up with the same answer for the speed of light, namely 1. That is, they must if light is used to define what "length" means. So the only way to even ask the question "what if the speed of light were not invariant?" meaningfully is to define "length" some other way than using light.

 

[name123]

No, I'm not. I'm just reasoning from what using "light" to define the "length" of a ruler means. It doesn't matter what frame you do the measurement in.

Let me try describing it a bit differently. I have a ruler and I want to know how "long" it is, in terms of light. What do I do? Light cannot be at rest, so there's no way to grab a light beam, bring it to rest, and lay it alongside my ruler to see how many "light units" long the ruler is. The only thing I can do is to send the light from one end of the ruler to the other and time how long it takes. So the "length" of the ruler is just the time it takes light to go from one end to the other.

Now I ask what the "speed" of the light is. The speed is defined as "the distance from one end of the ruler to the other" divided by "the time it takes light to go from one end to the other". But we have just defined the first thing to be equal to the second thing; so dividing one by the other can only give the answer 1. In other words, we have (speed of light) = (distance from one end to the other) / (time it takes light to go from one end to the other) = (time it takes light to go from one end to the other) / (time it takes light to go from one end to the other) = 1. There's no other possible answer.

Now, where in the above did I specify any frame? The same reasoning goes through regardless of the frame, i.e., regardless of the state of motion of the ruler. (The only thing you need to assume, as I said before, is that the clock that measures the light travel time is at rest relative to the ruler. But they can both be moving at any speed you like relative to anything else.) Even if two teams, Team A and Team B, run the same experiment in different frames, they must both come up with the same answer for the speed of light, namely 1. That is, they must if light is used to define what "length" means. So the only way to even ask the question "what if the speed of light were not invariant?" meaningfully is to define "length" some other way than using light.

But light isn't being used to define length, length is distance. Light would just be being used to make a ruler of a certain distance. The distance the ruler covered would depend on the distance light traveled in the rest frame within the clock-time of that rest frame, and without the assumption that the distance covered in each rest frame would be the same, the light ruler lengths can be different in different rest frames. So the distance between two points (positions of the A-Team observers for example) could be equivalent to the distance covered by x 1m rulers from one rest frame (the A-Team's for example) and equivalent to the distance covered by y 1m rulers from another rest frame (the B-Team's for example), where x != y.

 

[PeterDonis]

Light would just be being used to make a ruler of a certain distance.

You cannot use light to make a ruler. You cannot bring a beam of light to rest and lay it alongside something else to measure it. It cannot be done. That is what I have been telling you. The only way to use light to define "distance" is the way I described. There is no other way.

The distance the ruler covered

A ruler doesn't cover distance. A ruler sits at rest alongside whatever is being measured. If the ruler is moving, you are not using it to measure distance.

 

[name123]

I had said:

Light would just be being used to make a ruler of a certain distance.

You cannot use light to make a ruler. You cannot bring a beam of light to rest and lay it alongside something else to measure it. It cannot be done. That is what I have been telling you. The only way to use light to define "distance" is the way I described. There is no other way.

I meant by measuring how long it takes for a beam of light to reach a detector, and then marking how long it would therefore have gone in one second, assuming the speed of light to be constant in a rest frame, and making rulers of equivalent distance and re-measuring them to get the accuracy. But normally I'd assume they wouldn't make a ruler but just measure the time for the light to get from the emitter to the detector and describe it in meters where light travels 299 792 458m/s by definition. So a meter is by definition the distance light travels in the frame of reference the measurement is being taken in one second of that rest rest frame's clock time divided by 299 792 458. Why was it that number by the way?

A ruler doesn't cover distance. A ruler sits at rest alongside whatever is being measured. If the ruler is moving, you are not using it to measure distance.

By covering I meant spanning. A ruler spans a distance, it is a measure of that distance by virtue of it spanning it. So both the A team and the B team agree that the observers on their team are equally spaced out.

 

[maline]

This discussion about "what if the speed of light wasn't frame-invariant" is rather confusing. Let's get straight what happens in reality. Your question (or one of them) was

So if I call the observers on the floor Team A, and the observers on the conveyor belt that is going at 0.1c Team B, and let's still imagine that in the Team A reference frame the observers are spaced out 0.1c distance from each other in their frame of reference, and they measure it in feet for example so roughly 98,357,105.643 ft apart, and let's say they it measure out in rulers from their rest frame. Let's also imagine that they have some rulers from Team B's rest frame, and they measure the distance out with however many of them it takes. Team A can agree that the distance of a certain amount of Team B rulers is equal to a certain amount of their rulers, and then it would be a fixed equivalent distance whichever rulers you used wouldn't it, or are you suggesting that there would be a disagreement about the amount of rulers (of either type) laid out on the floor between the Team A observers?

In other words I think you're asking "how can two frames disagree on a distance when they can just compare whose rulers are longer?"
The answer, as you might have guessed, is that they can't. First of all, you need to see that there's no difference between various types of rulers or units. Your two teams can pass the rulers back and forth and agree that they are all the same length -when they are at rest. The problem is that each team "thinks" the other team's rulers are currently moving and therefore contracted, while their own rulers are at rest and therefore proper (longer) length.
Now, what happens if we place the rulers side by side? Well, they can't just sit side by side. At least one of them is moving. So the most you can say is that one end of each ruler is at x=x'=t=t'=0. Now we want to know, at that moment, which ruler has its other end further away? Ah, there lies the rub! The two teams can't agree when "that moment" is. At some point in time, the other ends of the rulers pass each other. One team will say that t=0 is a time before the other ends pass, and the others will say t'=0 is after the other ends passed, so each can keep claiming that their ruler is longer.
So yes, in your words, there will not be a disagreement about the amount of rulers laid out on the floor between the Team A observers, because if they're on the floor they're at rest in frame A. But there will be disagreement about the number of ruler on the belt that are between team A members, because each end of each ruler passes each team member at some moment, and the question is at a given moment how many already passed each person, but you get different answers depending on which events you consider simultaneous.

 

[maline]

So if they stop the conveyor belt are the B team Observers spaced out wider than the A team Observers?

After the belt stops, the distances are all the same.Team A will say "you guys just stopped, and the back end stopped first, and you all got stretched back to proper size", but if team B wants to keep using their old frame, they can say "no, we just started moving, and (what you guys call) our front end started first, so now we are contracted just like you".

 

[name123]

This discussion about "what if the speed of light wasn't frame-invariant" is rather confusing. Let's get straight what happens in reality. Your question (or one of them) was

In other words I think you're asking "how can two frames disagree on a distance when they can just compare whose rulers are longer?"
The answer, as you might have guessed, is that they can't. First of all, you need to see that there's no difference between various types of rulers or units. Your two teams can pass the rulers back and forth and agree that they are all the same length -when they are at rest. The problem is that each team "thinks" the other team's rulers are currently moving and therefore contracted, while their own rulers are at rest and therefore proper (longer) length.
Now, what happens if we place the rulers side by side? Well, they can't just sit side by side. At least one of them is moving. So the most you can say is that one end of each ruler is at x=x'=t=t'=0. Now we want to know, at that moment, which ruler has its other end further away? Ah, there lies the rub! The two teams can't agree when "that moment" is. At some point in time, the other ends of the rulers pass each other. One team will say that t=0 is a time before the other ends pass, and the others will say t'=0 is after the other ends passed, so each can keep claiming that their ruler is longer.

I had already stated that the distance isn't fixed if the speed of light was invariant. The discussion was about it it wasn't considered invariant.

Though thank you because you pointed out something I hadn't noticed which was that it cannot be shown not be invariant because it isn't that there is a disagreement about the distance, there is a disagreement about the time of the event.

 

[maline]

The discussion was about it it wasn't considered invariant.

Why are you worrying about that? It is!
And the question I quoted from you is a valid question in relativity, which I answered.

it isn't that there is a disagreement about the distance, there is a disagreement about the time of the event.

No, there are disagreements about both distance & time, but they depend on each other.

 

[PeterDonis]

I meant by measuring how long it takes for a beam of light to reach a detector,

Ok, in that case all of my previous comments still apply. You measure the travel time of a beam of light using a clock that is at rest with reference to the source and the detector. Then you define the "distance" between the source and the detector as this travel time. (You state it in somewhat different words, but that's the meaning.) With this definition, it is impossible for the speed of light to be anything but 1. And this will be true regardless of the state of motion of the clock, source, and detector relative to anything else. It will be the same for multiple labs, each with its own clock, source, and detector, all in motion relative to each other; all of them will obtain "1" for the speed of light.

Why was it that number by the way?

Because, when the change of definition was made, that was the most accurate value for the speed of light measured according to the old definition. So to minimize the practical effect of the change, they made sure it would not change the actual number used for practical calculations. (It would just prevent that number from ever changing again.)

 

[name123]

I had written:
"it isn't that there is a disagreement about the distance, there is a disagreement about the time of the event."
and you replied:

No, there are disagreements about both distance & time, but they depend on each other.

Wouldn't both team A observers and team B observers agree on the distance in meters between the team A observers?

 

[name123]

I had said:

"I meant by measuring how long it takes for a beam of light to reach a detector,"

to which you replied:

Ok, in that case all of my previous comments still apply. You measure the travel time of a beam of light using a clock that is at rest with reference to the source and the detector. Then you define the "distance" between the source and the detector as this travel time. (You state it in somewhat different words, but that's the meaning.) With this definition, it is impossible for the speed of light to be anything but 1. And this will be true regardless of the state of motion of the clock, source, and detector relative to anything else. It will be the same for multiple labs, each with its own clock, source, and detector, all in motion relative to each other; all of them will obtain "1" for the speed of light.

The distance traveled by light in one rest frame isn't assumed to be the distance traveled by light in another rest frame. If it was assumed that the distance traveled by light wasn't the same in each rest frame, a 1m ruler in one frame of reference spans a different distance from a 1m ruler in a different frame of reference. I'm assuming this could be a fictional universe we are talking about and we are just talking about what is implied by what has been said.

Also I asked about why the number 299 792 458 was used as the number of meters per second that light travels, and you replied:

Because, when the change of definition was made, that was the most accurate value for the speed of light measured according to the old definition. So to minimize the practical effect of the change, they made sure it would not change the actual number used for practical calculations. (It would just prevent that number from ever changing again.)

Thanks I hadn't realized there was an old definition for metre :)

 

[PAllen]

The distance traveled by light in one rest frame isn't assumed to be the distance traveled by light in another rest frame. If it was assumed that the distance traveled by light wasn't the same in each rest frame, a 1m ruler in one frame of reference spans a different distance from a 1m ruler in a different frame of reference. I'm assuming this could be a fictional universe we are talking about and we are just talking about what is implied by what has been said.

So, to clarify: you are allowing for the possibility that after two observers in relative motion construct meter sticks using the same procedure involving light; then one changes speed to match the other, they compare their rulers and they are different?

 

[name123]

So, to clarify: you are allowing for the possibility that after two observers in relative motion construct meter sticks using the same procedure involving light; then one changes speed to match the other, they compare their rulers and they are different?

Yes since as mentioned in the conversation the descriptions could be considered to be descriptions in a fictional universe, and the discussion is what would be implied by such a description. It could also be imagined that they only have different lengths while in the rest frames and if you brought them back into the same rest frame there wouldn't be a difference.

 

[PeterDonis]

It could also be imagined that they only have different lengths while in the rest frames and if you brought them back into the same rest frame there wouldn't be a difference.

No, this is the part that doesn't work. If you bring them to rest relative to each other and there is no difference, then there is no meaning to the statement that "they have different lengths while in different rest frames" apart from the ordinary length contraction effects of relative motion. In order to define "length" in a way that allows comparison between different rest frames, you have to define it in some way that can be constructed in different rest frames. Light travel time is one such way. Another way is the one I mentioned before, to define a "meter", for example, by the number of atoms of a particular type that equate to a meter of length if you line them up side by side.

But if, for example, Team A and Team B each have rulers that are $10^{10}$ atoms long, say, and both rulers line up with each other when they are at rest relative to each other, then there is no meaning to the question of whether their rest lengths can "change" if they are moving relative to each other. The "length" of each ruler is defined by how many atoms long it is; there is no meaning to the statement "yes, Team B's ruler is the same number of atoms long as Team A's, but it has a different length because it's in a different rest frame". And if you substitute any other direct observable for "number of atoms long", and define "length" to mean that observable, the same argument goes through.