B SR equation seems to depend on orientation of the 'light clock'

[bwana]

TL;DR Summary

In the derivation of the SR equation using a 'light clock', the final equation seems to depend on the direction of the photon path relative to the train

One of my first exposures to special relativity was looking at a 'light clock' where a photon is emitted and captured by a detector whilst traveling on a train. The passenger sees the photon go straight up.

The distance traveled by this photon in a given time, t, is c t

An observer however sees the photon traveling diagonally as the train moves.

The observer then connects the the two frames of reference with a triangle.

And this allows the derivation of the equation.
However, on a different train, the passenger sets up the light clock differently.

And the observer sees things somewhat differently.

And so he gets a different equation.Yet a third passenger sets things up differently again.

And now the observer sees something yet different and is totally thrown for a loop

It's SOOO CONFUSING!

Please make it stop!

We are still dealing with inertial frames so which is the 'correct' model? Obviously, the first depiction is always portrayed so there must be an explanation supporting its validity over the others.

 

[Office_Shredder]

I think the main point here is that an observer cannot measure the time it takes for light to travel from one location to another. They can only measure how long it takes light to travel to a location, and then back again. When the light makes a round trip, the two ratios become the same.

 

[PeterDonis]

One of my first exposures to special relativity was looking at a 'light clock' where a photon is emitted and captured by a detector whilst traveling on a train.

As @Office_Shredder has remarked, it's not a light clock unless the emitter and the detector are at the same spatial location. None of your examples have that property.

Where are you getting these examples from?

 

[Dale]

Summary:: In the derivation of the SR equation using a 'light clock', the final equation seems to depend on the direction of the photon path relative to the train

Obviously, the first depiction is always portrayed so there must be an explanation supporting its validity over the others.

As others mentioned, none of those is a proper light clock without the return back to the same place in the frame where it is measuring time.

That said, the perpendicular light clock avoids the complication of length contraction. For any other orientation you must also include length contraction. If you do so then you will find that all orientations are consistent. The perpendicular orientation is used for simplicity.

 

[Janus]

Light clock example with arms both perpendicular and parallel to the relative motion.

As already noted the observer stationary with respect to the animation frame, measures a length contraction of the parallel arm.
Once this is accounted for, the math give the same time dilation( for a round trip) no matter what angle the light is aimed for the light clock.

 

[Ibix]

Light clock example with arms both perpendicular and parallel to the relative motion.

Aside: I find looking at that animation fascinating. Watching the light pulse in the moving horizontal light clock, the pulse looks to return much faster than its outward leg. I know its speed should always be the same and the time difference is from the movement of the mirrors. I trust Janus to do that right, and I know the clock numbers are there for me to check if I want. But it looks fast. I think I intuitively do a Galilean transform into the frame of the mirrors (wildly inappropriate for a system moving at 0.87c).

I wonder if people sometimes struggle with frames because we use them all the time, but can struggle to step back from our intuitive use of them. I've certainly caught myself looking out of a train window and thinking something like "the trees are going past really fast, we'll be at the station soon" which is perfectly understandable to anybody and involves a frame change mid-sentence.

 

[Sagittarius A-Star]

I've certainly caught myself looking out of a train window and thinking something like "the trees are going past really fast, we'll be at the station soon" which is perfectly understandable to anybody and involves a frame change mid-sentence.

You should have thought:

... the "station" will arrive here soon.

 

[bwana]

Thank you all for your replies. Yes, the details make all the difference. It is the observer who makes the calculation. He can measure the time it takes for the train to go from a to b and he can also measure the time it takes for light to go from a to b because he has a good stopwatch. He can record these numbers at his location. The observer gets the time measurement from the passenger who has his own stopwatch, after he gets off the train. The calculation is performed after the experiment is over.

I do not understand the statement that the ‘emitter and detector have to be at the same location’. Of course once I accept that statement then the animation of Janus is a clear demonstration that the calculation is covariant with the orientation of the light clock and yields the same equation. (The math is messier but gives the result). I question the physical intuition that the emitter and detector have to be ‘in the same place’. Isn’t the point of relativity that there is no ‘absolute placement’ in the universe?

 

[pervect]

I do not understand the statement that the ‘emitter and detector have to be at the same location’. Of course once I accept that statement then the animation of Janus is a clear demonstration that the calculation is covariant with the orientation of the light clock and yields the same equation. (The math is messier but gives the result). I question the physical intuition that the emitter and detector have to be ‘in the same place’. Isn’t the point of relativity that there is no ‘absolute placement’ in the universe?

If you know what a world-line is, saying that the emitter and the detector have to be at "the same location" is really saying that they have to be on the same worldline, the worldline of some specific observer. It's also generally assumed in SR that this is the worldline of an inertial observer.

Hope that helps, not sure it will.

 

[Ibix]

I question the physical intuition that the emitter and detector have to be ‘in the same place’. Isn’t the point of relativity that there is no ‘absolute placement’ in the universe?

They have to be in the same place as "place" is defined in your frame. They can't be in different places because you would need two clocks, one at the emitter and one at the detector, and how do you know the clocks are zeroed correctly? The answer turns out to be that you cannot know without having those clocks exchange light pulses (or some equivalent process), thereby making your synchronisation process a light clock.

 

[Dale]

I question the physical intuition that the emitter and detector have to be ‘in the same place’.

If a clock does not stay at in the same place according to some frame then it is not at rest in that frame and therefore it is not measuring time according to that frame.

Isn’t the point of relativity that there is no ‘absolute placement’ in the universe?

Certainly. But if I want to measure things from my perspective then I need to use devices at rest with respect to me.

 

[Janus]

Aside: I find looking at that animation fascinating. Watching the light pulse in the moving horizontal light clock, the pulse looks to return much faster than its outward leg. I know its speed should always be the same and the time difference is from the movement of the mirrors. I trust Janus to do that right, and I know the clock numbers are there for me to check if I want. But it looks fast.

The perceived quickness is likely due to the fact to the combination of the shortness of the parallel arm, and the fact the back ground is uniform. As a result, you tend to judge the left moving pulse's speed relative to the right moving light clock, rather than relative to the animation frame.

 

[Sagittarius A-Star]

I do not understand the statement that the ‘emitter and detector have to be at the same location’.

All "tick"-events of the light clock must be at the same location with reference to the rest frame of the light clock. Generally, only then the time dilation factor $1/\gamma$ for the tick-rate is valid.

Therefore, you must mount a "tick"-counter at only one of the two mirrors of the light clock.

(The math is messier but gives the result)

In the rest frame of the longitudinal light clock with rest length $L' = L_0$, the "tick" period is:
$\Delta t' = \frac{2L_0}{c}$.

For the moving longitudinal light clock, the "tick" period is (with Lorentz contraction $L=L_0/\gamma$):
$\Delta t = \frac{L}{c-v} + \frac{L}{c+v} = L (\frac{c+v}{c^2-v^2} + \frac{c-v}{c^2-v^2}) = \frac{2L}{c} \frac{1}{1 -v^2/c^2} = \frac{2L}{c} \gamma^2 = \frac{2L_0}{c} \gamma = \Delta t' * \gamma$.

 

[Sagittarius A-Star]

I question the physical intuition that the emitter and detector have to be ‘in the same place’. Isn’t the point of relativity that there is no ‘absolute placement’ in the universe?

You can derive the time dilation factor for $\Delta t$ from the inverse Lorentz transformation for time:

$\Delta t = \gamma (\Delta t' + \frac{v}{c^2}*\Delta x')$

The reference frame for ‘in the same place’ (for consecutive clock-"tick"-events) is obviously the rest frame of the clock: $\Delta x' = 0$. Then it follows:

$\Delta t = \gamma \Delta t'$.

 

[Ibix]

As a result, you tend to judge the left moving pulse's speed relative to the right moving light clock, rather than relative to the animation frame.

Exactly - I've switched to the Galilean reference frame of the clock (which, of course, is only possible because your "light" isn't actually traveling at c and nor is your clock doing 0.87c). I just find it fascinating how easy that is to do, compared to the struggles we see with the concept when it's treated formally.

 

[robphy]

Aside: I find looking at that animation fascinating. Watching the light pulse in the moving horizontal light clock, the pulse looks to return much faster than its outward leg. I know its speed should always be the same and the time difference is from the movement of the mirrors. I trust Janus to do that right, and I know the clock numbers are there for me to check if I want. But it looks fast. I think I intuitively do a Galilean transform into the frame of the mirrors (wildly inappropriate for a system moving at 0.87c).

I wonder if people sometimes struggle with frames because we use them all the time, but can struggle to step back from our intuitive use of them. I've certainly caught myself looking out of a train window and thinking something like "the trees are going past really fast, we'll be at the station soon" which is perfectly understandable to anybody and involves a frame change mid-sentence.

Watch the Frame of Reference video (Ivey & Hume, 1960) from time stamp 10:00 ...

Not quite the same... but the Push Me Pull You setup displays some interesting motions, in the presence of possible visual distractions.
[ www.youtube.com/watch?v=amfw2nABke4 ]

 

[bwana]

So many answers. some seem circular, some shed light.
Ibix said
"They can't be in different places because you would need two clocks, one at the emitter and one at the detector, and how do you know the clocks are zeroed correctly?"
My rebuttal →the passenger is holding a stopwatch-it clicks when the light leaves the bulb and clicks again when the light hits the detector. There is only one clock on the train. Likewise the observer is using one clock. Yes there are two clocks in the example but one is on the train and another is with the observer-these can never be synchronized nor do they need to be. Each clock just measures elapsed time in its frame.

Dale said
"If a clock does not stay at in the same place according to some frame then it is not at rest in that frame and therefore it is not measuring time according to that frame."
→the stopwatch of the passenger is staying still relative to his frame-it is in his hand. Likewise the stopwatch of the observer. Sagittarius A-Star shed some light with
"All "tick"-events of the light clock must be at the same location with reference to the rest frame of the light clock. Generally, only then the time dilation factor 1 / γ for the tick-rate is valid. Therefore, you must mount a "tick"-counter at only one of the two mirrors of the light clock."

Well my example did not use mirrors. But reflecting on your words 'tick events', I would say one tick event occurs upon emission of the photon and another tick event occurs upon detection. In my example, the tick events are occurring in the stopwatch which is in the observer's hand- it is the same location in the passenger's frame for both events-emission and detection.

But did you intend to say
"All "tick"-events of the light clock must be AT THE LOCATION WHERE THE EMISSION AND DETECTION EVENTS ACTUALLY HAPPEN"

Stated another way, Are you saying that the stopwatch has to be physically located AT THE PLACE OF EMISSION AND THE PLACE OF DETECTION as well? What ever for?? If I have two photodetectors on the same electrical circuit(with one detector at the emitter and another at the detector) then I can achieve your requirement-tho I cannot see the physical intuition that would make one stipulate that. I feel like this restriction was slipped in silently and without explanation - the way a magician would use misdirection to perform a trick.

Of course, anyone could say that all events are actually events in SPACETIME-so if you want to measure time and time alone(like we do when we calculate the lorentz-fitzgerald factor), you have to hold the spatial coordinates constant-in other words, emission and detection events have to occur at the same location EVEN WITHIN AN INERTIAL FRAME. So when you craft an experiment you must respect this constraint.

But step back for a moment, why did we ever imagine that SPACETIME was a thing? the stipulation of SPACETIME seems like a circular argument. We first require that events occur in SPACETIME, and then we derive an equation(SR) FROM THAT ASSUMPTION. I cannot then see how SR gives us independent evidence that SPACETIME must be the fundamental fabric we inhabit. I know that SR works. But then this is just an inductive proof.

As you all can see, I am not using equations at this point- I am talking about physics- the actual physical requirements that guide our intuition. Please just tell me if my preceding three paragraphs are touching on the essential argument or if they are so way off that they are not even wrong.

 

[Office_Shredder]

So many answers. some seem circular, some shed light.
Ibix said
"They can't be in different places because you would need two clocks, one at the emitter and one at the detector, and how do you know the clocks are zeroed correctly?"
My rebuttal →the passenger is holding a stopwatch-it clicks when the light leaves the bulb and clicks again when the light hits the detector. There is only one clock on the train.

Think about how you would actually do this. If you are in a totally dark room with completely black walls that absorb all light, and you turn on a laser pointer and point it at a wall, how do you know when it hits the wall? How do you even know if a wall exists? You need light to reflect back to you in order to know you should click the stopwatch off.

 

[PeterDonis]

Are you saying that the stopwatch has to be physically located AT THE PLACE OF EMISSION AND THE PLACE OF DETECTION as well?

If you want to call it a "light clock", yes, that's exactly what's required: the light pulse gets emitted, bounces off a mirror, and comes back to the point of emission, and a clock at that point measures how long it took to make the round trip.

What ever for??

Because it's the only way to eliminate all issues regarding synchronization of clocks that are spatially separated, and therefore to make the experiment completely free of arbitrary choices of convention. See below.

f I have two photodetectors on the same electrical circuit(with one detector at the emitter and another at the detector) then I can achieve your requirement

No, you can't. You have to synchronize clocks at the two spatially separated photodetectors, and there is no way to do that without adopting some kind of arbitrary convention. (The usual term for such a convention is "simultaneity convention".)

 

[PeterDonis]

the passenger is holding a stopwatch-it clicks when the light leaves the bulb and clicks again when the light hits the detector.

If the bulb and the detector are spatially separated, the passenger and his stopwatch can only be at one of the two locations, not both. So how can the passenger's stopwatch possibly click "when" both events happen?

 

[Dale]

the stopwatch of the passenger is staying still relative to his frame-it is in his hand. Likewise the stopwatch of the observer.

Please use the quote feature rather than copy-pasting quotes. It makes the conversation more readable and enables automatic forum messages.

The stopwatch cannot measure when the light reaches the distant receiver.

The stopwatch cannot tell you the time of anything that doesn’t take place at the location of the stopwatch without assuming a simultaneity convention.

I feel like this restriction was slipped in silently and without explanation - the way a magician would use misdirection to perform a trick.

That isn’t surprising. The light clock is a helpful pedagogical tool for teaching SR. It is not and was not intended to be a derivation of SR. Once you have a solid derivation of SR then you can do two things:

1) you can gather experimental evidence to support or refute it

2) you can study the math and derivations to get a better understanding

Both of those come after a good derivation from first principles. The light clock is part of 2). So a proper derivation of the whole of SR is already assumed and in the discussion of the light clock we freely bring in concepts from the derivation as needed.

But step back for a moment, why did we ever imagine that SPACETIME was a thing?

That happened when Minkowski noticed that the Lorentz transform preserved a quantity that looks very similar to the Euclidean formula for distance, but included time also.

the stipulation of SPACETIME seems like a circular argument. We first require that events occur in SPACETIME, and then we derive an equation(SR) FROM THAT ASSUMPTION.

That is historically false. Einstein developed SR first and Minkowski came up with spacetime a few years later.

However, you should be aware that circular reasoning is not problematic. The equations need to be self consistent and lead to experimental predictions that are consistent with the evidence. Circularity helps the self-consistency and doesn’t interfere with experimental consistency. So this particular point is not circular, but it wouldn’t matter if it were.

Please just tell me if my preceding three paragraphs are touching on the essential argument or if they are so way off that they are not even wrong

What essential argument are you looking for? Are you trying to understand a derivation of SR, or the concept of spacetime, or just the light clock example?

 

[robphy]

But step back for a moment, why did we ever imagine that SPACETIME was a thing? the stipulation of SPACETIME seems like a circular argument. We first require that events occur in SPACETIME, and then we derive an equation(SR) FROM THAT ASSUMPTION. I cannot then see how SR gives us independent evidence that SPACETIME must be the fundamental fabric we inhabit. I know that SR works. But then this is just an inductive proof.

"Spacetime" is a fancy word for the "position-vs-time graph",
which is used in pre-Einsteinian physics. So, Spacetime is not exclusive to Special Relativity.

The main difference between the Galilean diagram and the Minkowski diagram is the
specific non-Euclidean metric geometry that is used.

 

[Sagittarius A-Star]

"Spacetime" is a fancy word for the "position-vs-time graph",
which is used in pre-Einsteinian physics. So, Spacetime is not exclusive to Special Relativity.

I don't think so. In pre-Einsteinian physics, nomally "Space-Time"-diagram is used and in Einsteinian physics "Spacetime"-diagram.

A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity.

Source:
https://en.wikipedia.org/wiki/Spacetime_diagram

Before Einstein/Minkowski, there was no reason to combine the words "space" and "time" to "spacetime".

 

[Ibix]

My rebuttal →the passenger is holding a stopwatch-it clicks when the light leaves the bulb and clicks again when the light hits the detector.

How is the passenger going to click the stopwatch "when the light hits the detector" unless they are at the detector? They can't know it's reached the detector until some signal has reached the passenger. That means the passenger isn't measuring the time for the light to travel from bulb to detector - they're measuring the time for light to travel from bulb to detector to the passenger.

There are only two ways out of that. One way is to design the experiment so the bulb and detector are in the same place. The other is to have two synchronised clocks, one at the bulb and one at the detector. But how do you synchronise those clocks? There isn't a way to do it without making assumptions about the one-way speed of light, which means that your experiment's results depend on your assumption.

 

[Sagittarius A-Star]

Summary:: In the derivation of the SR equation using a 'light clock', the final equation seems to depend on the direction of the photon path relative to the train

Usually, a 'light clock' is defined with mirrors and with a counter only at one mirror:

The constancy can be exploited to construct a special kind of clock in thought, a so-called light clock. Its operating principle is very simple: Two mirrors are placed at a constant distance from each other. A light pulse runs up and down between them. Each arrival of the pulse at the upper mirror corresponds to a “tick” of the clock. A detector receives the pulse when it reaches the upper mirror and passes this information on to a counter, which counts how many times the pulse has already arrived

Source:
https://www.einstein-online.info/en/spotlight/light-clocks-time-dilation/

Well my example did not use mirrors.

Then you contradict yourself, because in the O.P., your called it a 'light clock'.

I feel like this restriction was slipped in silently and without explanation - the way a magician would use misdirection to perform a trick.

You did introduce this restriction, by calling it a 'light clock'.

Of course, anyone could say that all events are actually events in SPACETIME-so if you want to measure time and time alone(like we do when we calculate the lorentz-fitzgerald factor), you have to hold the spatial coordinates constant-in other words, emission and detection events have to occur at the same location EVEN WITHIN AN INERTIAL FRAME. So when you craft an experiment you must respect this constraint.

Yes (I assume, that you mean the 4D-Minkowski spacetime).

I cannot then see how SR gives us independent evidence that SPACETIME must be the fundamental fabric we inhabit.

Evidence came for example from the Michelson-Morley experiment, not from theory alone.

 

[Ibix]

If I have two photodetectors on the same electrical circuit(with one detector at the emitter and another at the detector) then I can achieve your requirement-tho I cannot see the physical intuition that would make one stipulate that.

Note that there is a finite and frame dependent signal transmission time in the wires. So introducing your two photodetectors is a way of sneaking in a return trip, just using electrical signals instead of light. That's much harder to analyse without first understanding how a light clock works.

 

[Ibix]

But step back for a moment, why did we ever imagine that SPACETIME was a thing? the stipulation of SPACETIME seems like a circular argument.

Historically, Einstein derived the Lorentz transforms from an assumption that the speed of light is invariant in all inertial frames and the law of physics are the same in all inertial frames. The concept of spacetime was introduced three years later by Minkowski, who observed that the Lorentz transforms could be seen as coordinate transforms on a 4d manifold.

A different way of structuring the learning is to start with the concept of spacetime and insist that it has certain properties. Then you derive the consequences of that and show that they are consistent with experiment. That is a somewhat cleaner approach than Einstein starting in the middle and working up to experiment and Minkowski picking up where Einstein started and working down to spacetime.

Neither approach is circular, but you may be getting a mix of the two - which probably means a mix of different starting assumptions. The more modern approach is the one of assuming spacetime is a thing and deriving the consequences of that - just stating what symmetries you require and going from there has been an incredibly successful approach over the last century or so.

Some sources prefer to teach the historical approach, but I think it's a mistake now. It's a bit like a detective story - most of the story is the detective stumbling around finding bits and pieces of clues in random order, but at the end he gets everybody together and lays it out in a neat structure to explain how the crime was committed. It can be fun to read the stumbling around, but if you just want to know who did it then skip to the end and read the neatly packaged version. Then, if you go back and look at the historical version, you can see who spotted what and why it was important.

 

[Sagittarius A-Star]

Stated another way, Are you saying that the stopwatch has to be physically located AT THE PLACE OF EMISSION AND THE PLACE OF DETECTION as well? What ever for??

Maybe, the following explanation helps:

Caution, fake light clock!

Occasionally one sees animations with light clocks ticking twice as fast as the ones shown here. For them the counter counts when the pulse reaches the upper mirror and also when it reaches the lower mirror. Such light clocks lead the simple functional principle ad absurdum, because how does the counter know when the pulse arrives at the lower mirror? This information would first have to be laboriously transferred from the lower mirror to the counter. But this transmission cannot be done faster than the speed of light. In particular, the information would not reach the counter before the light pulse itself already arrives again at the upper mirror.

Source:
https://www.einstein-online.info/en/spotlight/light-clocks-time-dilation/

 

[robphy]

"Spacetime" is a fancy word for the "position-vs-time graph",
which is used in pre-Einsteinian physics. So, Spacetime is not exclusive to Special Relativity.

The main difference between the Galilean diagram and the Minkowski diagram is the
specific non-Euclidean metric geometry that is used.

I don't think so. In pre-Einsteinian physics, nomally "Space-Time"-diagram is used and in Einsteinian physics "Spacetime"-diagram.Source:
https://en.wikipedia.org/wiki/Spacetime_diagram

Before Einstein/Minkowski, there was no reason to combine the words "space" and "time" to "spacetime".

Um... My "which statement" refers to the "position-vs-time graph" (not "spacetime" or "space-time").

Spoiler

"Spacetime" is a fancy word for

the "position-vs-time graph",
which is used in pre-Einsteinian physics. So, Spacetime is not exclusive to Special Relativity.

The context of the OP's question is not about the term SPACETIME or SPACE-TIME
but about the idea of SPACETIME or SPACE-TIME

But step back for a moment, why did we ever imagine that SPACETIME was a thing? the stipulation of SPACETIME seems like a circular argument. We first require that events occur in SPACETIME, and then we derive an equation(SR) FROM THAT ASSUMPTION. I cannot then see how SR gives us independent evidence that SPACETIME must be the fundamental fabric we inhabit. I know that SR works. But then this is just an inductive proof.

 

[Sagittarius A-Star]

The context of the OP's question is not about the term SPACETIME or SPACE-TIME
but about the idea of SPACETIME or SPACE-TIME

But according to Sean Carroll:

This is Idea #6, "Spacetime." Which, naturally, is about the major idea underlying special relativity -- that both space and time are different parts of one unified, four-dimensional spacetime.

Source:


So I think it is clear, that the OP meant the 4D-Minkowski spacetime.

 

[robphy]

@Sagittarius A-Star ,

A position-vs-time graph is a non-euclidean two-dimensional space.

A space-vs-time graph is a non-euclidean four-dimensional space,

Interesting...
search for "space-time" in this 1903 textbook (Technical mechanics Part 2 By Edward Rose Maurer, 1903)
https://www.google.com/books/edition/Technical_Mechanics/Al8vAQAAMAAJ?hl=en&gbpv=0
while the term is "space-time" hyphenated, the idea of a four-dimensional space of events is there.
Also,
search for "space time" in this 1897 textbook (The Polarizing Photo-chronograph, Albert Cushing Crehore, George Owen Squier, 1897)
https://www.google.com/books/edition/The_Polarizing_Photo_chronograph/1-SgAAAAMAAJ?hl=en&gbpv=0

Special Relativity is really about the Minkowski-metric structure on this abstract space of events.
Einstein didn't interpret it that way in 1905, but slowly began to appreciate it after Minkowski's formulation (and Minkowski's death).

 

[vanhees71]

As you all can see, I am not using equations at this point- I am talking about physics- the actual physical requirements that guide our intuition. Please just tell me if my preceding three paragraphs are touching on the essential argument or if they are so way off that they are not even wrong.

Talking about physics without the adequate language leads naturally to confusion. The only adequate language for physics is math (mostly differential and group theory if it comes to the foundations ;-)).

 

[Sagittarius A-Star]

Talking about physics without the adequate language leads naturally to confusion. The only adequate language for physics is math (mostly differential and group theory if it comes to the foundations ;-)).

Yes. Feynman: Mathematicians versus Physicists

 

[pervect]

Bwana, the original poster, initially expressed some confusion about how different observers regarded the measures of time and space he's used to using as being different depending on the observer, for instance, length contraction.

The good news is that there are formulations of special relativity that focus on the things that are the same for all observers (often called invariants), and not the rules about how different observers see things differently.

The bad news is that one of the simplest of these observer invariant concepts, called proper time, is being resisted by the very same poster, in spite of numerous attempts to explain it to him :(.

If the OP is familiar with space-time diagrams, the invariant length of a particular curve representing a worldline on said diagram can be associated with a number, which can be thought of as an abstract measure of the length of this curve / worldline that is the same for everyone. For the case when the worldline in question is the worldline of some observer, this invariant length corresponds to the difference in clock readings of an ideal clock that traverses this worldline (as the observer ages).

However, it is necessary to appreciate the difference between proper time (which is associated with this invariant quantity), and time differences that rely on a notion of simultaneity, which are n ot associated with this invariant quantity, in order to get anywhere. That's the point we're at now in this discussion.

 

[Grasshopper]

Could we maybe go into detail about why measurements at a distance require sychronizarion conventions/calibration, and why that presents a problem? Thanks.

 

[PeterDonis]

Could we maybe go into detail about why measurements at a distance require sychronizarion conventions

Say you are in New York and I am in Los Angeles. You send me a light signal. We want to figure out how long it took the signal to get from you to me. You measure the time by your clock when you emit the signal, and I measure the time by my clock when I receive the signal. In order for us to say that the difference between those times is the time it took the signal to travel, we have to assume that when your clock and my clock read the same time, those events are simultaneous--they happen at the same time. That is assuming a synchronization convention. (Note that we are both at rest relative to each other so there is no issue of relative motion affecting the rates of our clocks; both clocks run at the same rate. The issue is solely how we define which events at the two clocks are simultaneous.)

Of course the convention I just described seems natural and obvious (assuming that we have previously undergone a procedure like Einstein clock synchronization to initialize both clocks in a way that matches the convention). But that doesn't mean it isn't a convention. Also, it is actually only natural and obvious if both clocks are at rest in a global inertial frame. And that never actually happens in our actual universe, because spacetime in our actual universe is not flat, and because objects in our actual universe, like the Earth, do things like rotate on their axis. In a rotating frame, like the actual "rest frame" of the actual rotating Earth, Einstein clock synchronization does not work globally. So any actual clock synchronization between an actual clock in New York and an actual clock in Los Angeles cannot be Einstein clock synchronization and cannot satisfy all the properties of clocks synchronized that way at rest in a global inertial frame. We can use a local inertial frame as an approximation, but it's just an approximation and is only useful in limited circumstances.

For an example of a different synchronization convention, consider GPS. The GPS system, roughly speaking, uses two different reference frames. For the underlying clock synchronization and base calculations, it uses an Earth Centered Inertial (ECI) frame, which is an inertial frame whose spatial origin is the center of the Earth but which is not rotating with the Earth, and which defines clock synchronization using this non-rotating frame. (It also adjusts the clock rate in this frame to match the actual clock rate on the Earth's geoid--roughly speaking, at "mean sea level" on an imaginary Earth that is covered entirely by ocean; such a mean sea level is an equipotential surface, i.e., it has the same clock rate everywhere on it.) But to actually display results to the user, it uses an Earth Centered Earth Fixed (ECEF) frame, which of course must be rotating with the Earth, but whose clock synchronization and clock rate is the same as the ECI frame. In other words, roughly speaking, it has the clock synchronization and time coordinate of the ECI frame, but spatial coordinates which are fixed to the rotating Earth. So both frames have the same clock synchronization, and it is not a "natural and obvious" one for observers on the rotating Earth (but it is for imaginary observers not rotating with the Earth but traveling in the same orbit about the Sun).

why that presents a problem

It doesn't present a problem in many circumstances. GPS works fine with the conventions I described above.[

But it does present a problem if you're trying to understand the fundamentals of relativity with a light clock, because understanding the fundamentals means understanding the invariants--the things that don't depend on any choice of convention. To do that, you have to eliminate anything from your scenario that does depend on a choice of convention. That's why the light clock is defined with the emission and detection of the light pulse at the same place, and one "tick" of the clock corresponds to one round trip from emission, bouncing off the mirror, back to detection at the same place.

 

[pervect]

Could we maybe go into detail about why measurements at a distance require sychronizarion conventions/calibration, and why that presents a problem? Thanks.

One might measure the length of an object by finding the position of the right end of the object, and subtracting the position of the left end.

If the object isn't moving, synchronization doesn't matter, but if the object is moving, the position of the left end of the object is a function of time, and so is the position of the right end of the object. Thus, any error in sycnchronization leads to an incorrect length measurement.

A key point here is that synhronization is special relativity is known to be observer dependent. See any discussion on "the relativity of Simultaneity" and/or "Einstein's train".

 

[Lynch101]

I feel like this restriction was slipped in silently and without explanation - the way a magician would use misdirection to perform a trick.

I used to feel like this when I was first trying to learn about relativity, but it was usually just the case that I had either forgotten something I had learned previously, or there was something that I wasn't aware of.

If at all possible, try to park your intuition about how nature works and try instead to focus on what relativity says. A cognitive trick that helped me was telling myself that I don't necessarily need to agree with what relativity says, I just need to develop an understanding of what it says. Eventually, the basics will click and you'll see that it does work out.

EDIT: Questions will still arise when your intuitive ideas about nature are challenged, but I found that letting go of the idea that my intuition must be right, was a big help.

 

[Lynch101]

Could we maybe go into detail about why measurements at a distance require sychronizarion conventions/calibration, and why that presents a problem? Thanks.

I'm going to try and give an example, but if other members point out flaws in it, go with those other posters. I'm only at a stage where I have an understanding of relativity, without necessarily being able to work through more complex problems. So, this attempted explanation is as much to test my ability to explain basic ideas in relativity.

Interstellar Olympics
A very basic example would be to consider the 100m race in the Olympics or perhaps a longer distance race where the start and finish lines are not visible to each other - we might actually imagine an interstellar race with start and finish lines light years apart. While it would be easy to tell who won the race - the first person past the post - timing how long each runner takes to complete the race is a different matter.

While the person at the start line knows exactly when to start their clock, how does the person at the finish line know when the race has started, so they can start their clock? Similarly, how would the person at the start line know when the winner has crossed the finish line, so that they can record their time?

In order for both to start their clocks at the same time, they would need to send a signal to each other. For our interstellar race, a pulse of light would be the best method. We might suggest that the person at the start line send a pulse to the finish line when the race starts, but there would be a delay from the time it is sent to the time it arrives at the finish line and this would skew the race times. This delay would need to be accounted for.

Back to Square One
We might suggest measuring the distance form start to finish and just dividing by the speed of light, but how do we measure the speed of light? We might think that we can just send a pulse of light from start to finish and measure the time it takes but that just returns us to our original problem of starting the clocks at the same time to measure the race times of the racers.

What we might do instead is send out a light pulse from the start, have it reflected back from the finish line to the start line, and record the time it took to do the roundtrip. Then distance divided by time will give us our speed. So, knowing the distance and the speed of light, we can now send the light pulse from the start to the finish line and adjust the clock at the finish line accordingly.

Convention
There is one slight issue with this, however. How do we know that the length of time it took the light pulse to go from start line to finish, was the same as the time from finish to start? We don't. This is why we have to take it as a convention i.e. take it as a matter of definition. If we could be sure, then there would be no clock synchronisation issue in the first place.

 

[stevendaryl]

Getting back to the original post,

Let me restate the results that are purely kinematic.

We have two observers, one at rest in frame $F$, and one at rest in frame $F'$. The second observer is moving at velocity $v$ in the $x$-direction relative to the first observer. This observer has a "light clock", which is just a pair of mirrors facing each other connected by a rod, with a light pulse bouncing back and forth between the mirrors. The kinematic results are these:

1. If the rod is oriented in the x-direction, and the length of the rod is $L_{parallel}$, then the time for a round-trip for the light pulse is $\dfrac{\gamma^2 L_{parallel}}{c}$, where $\gamma$ is defined by $\gamma = \dfrac{1}{1-\frac{v^2}{c^2}}$ (lengths and times measured in frame $F$).
2. If the rod is oriented in the y-direction, and the length of the rod is $L_{perpendicular}$, then the time for a round-trip for the light pulse is $\dfrac{\gamma L_{perpendicular}}{c}$ (lengths and time measured in frame $F$).

In order for the light clock to give a consistent time interval, regardless of its orientation, it has to be that, as measured in frame $F$, the length of the rod connecting the mirrors must change when the orientation of the rod changes. Specifically,

$L_{parallel} = \frac{1}{\gamma} L_{perpendicular}$

So without additional assumptions, the lightclock thought experiment doesn't by itself imply time dilation, but rather, it implies length contraction: A moving rod is contracted in the direction of its motion compared to its length when oriented perpendicular to its motion.

To get the usual time dilation factor, you need an extra assumption:

$L_{perpendicular} = L$, where $L$ is the length of the light clock as measured in frame $F'$

In other words, you need to assume that there is NO length contraction in the direction perpendicular to the direction of motion. But this assumption is the only assumption that can satisfy the principle of relativity (Consider two light clocks, one at rest in frame $F$ and one at rest in frame $F'$, both oriented perpendicular to the direction of motion. They can't both be contracted relative to each other.)

 

[Grasshopper]

Getting back to the original post,

Let me restate the results that are purely kinematic.

We have two observers, one at rest in frame $F$, and one at rest in frame $F'$. The second observer is moving at velocity $v$ in the $x$-direction relative to the first observer. This observer has a "light clock", which is just a pair of mirrors facing each other connected by a rod, with a light pulse bouncing back and forth between the mirrors. The kinematic results are these:

1. If the rod is oriented in the x-direction, and the length of the rod is $L_{parallel}$, then the time for a round-trip for the light pulse is $\dfrac{\gamma^2 L_{parallel}}{c}$, where $\gamma$ is defined by $\gamma = \dfrac{1}{1-\frac{v^2}{c^2}}$ (lengths and times measured in frame $F$).
2. If the rod is oriented in the y-direction, and the length of the rod is $L_{perpendicular}$, then the time for a round-trip for the light pulse is $\dfrac{\gamma L_{perpendicular}}{c}$ (lengths and time measured in frame $F$).

In order for the light clock to give a consistent time interval, regardless of its orientation, it has to be that, as measured in frame $F$, the length of the rod connecting the mirrors must change when the orientation of the rod changes. Specifically,

$L_{parallel} = \frac{1}{\gamma} L_{perpendicular}$

So without additional assumptions, the lightclock thought experiment doesn't by itself imply time dilation, but rather, it implies length contraction: A moving rod is contracted in the direction of its motion compared to its length when oriented perpendicular to its motion.

To get the usual time dilation factor, you need an extra assumption:

$L_{perpendicular} = L$, where $L$ is the length of the light clock as measured in frame $F'$

In other words, you need to assume that there is NO length contraction in the direction perpendicular to the direction of motion. But this assumption is the only assumption that can satisfy the principle of relativity (Consider two light clocks, one at rest in frame $F$ and one at rest in frame $F'$, both oriented perpendicular to the direction of motion. They can't both be contracted relative to each other.)

Wait, does this imply that if the relative motion between two frames is exactly perpendicular to the line of two events that happen simultaneously in one of the frames, that the events will also be simultaneous in the moving frame as well?

Edit: In particular, if the origins coincide, and the two simultaneous events are on the y-axis. If the “at rest” sees both light flashes simultaneously at the origin, and there is no vertical length contraction, it seems the moving frame should also see the events simultaneously, since the light is approaching vertically.

Unless the angle is changed in the moving frame.

 

[Ibix]

Wait, does this imply that if the relative motion between two frames is exactly perpendicular to the line of two events that happen simultaneously in one of the frames, that the events will also be simultaneous in the moving frame as well?

Yes. The time Lorentz transform only depends on distance parallel to the relative velocity of the frames. It can't depend linearly on perpendicular distance (e.g. $t'=\gamma\left(t-\frac{v}{c^2}x-\frac{v}{c^2}y\right)$) since there's no non-arbitrary way to pick your positive $y$ and $z$ directions (positive $x$ comes from the velocity direction). And they can't depend on higher powers of $y$ and $z$ because there's no non-arbitrary way to pick an origin for $y$ and $z$. Any non-arbitrary way of doing either picks out a special direction or position in spacetime, which we've assumed to be isotropic and homogeneous.

 

[vanhees71]

For two events $x_A$ and $x_B$ the time difference measured by an observer moving with four-velocity $u$ ($u \cdot u=1$), is given by $c \Delta t(u)=u \cdot (x_A-x_B)$.

Now if $t_A=t_B$ in one reference frame then an observer moving with velocity $\vec{v} \perp \vec{x}_A-\vec{x}_B$, his four-velocity is $u=\gamma (1,\vec{\beta})$ and $\vec{\beta} \cdot (\vec{x}_A-\vec{x}_B)=0$ from which $c \Delta t_u=u \cdot (x_A-x_B)=\gamma c (t_A-t_B)=0$, i.e., for such an observer the events are also simultaneous.

 

[joekahr]

I wrote a light clock app (https://joekahr.github.io/lightclock/) that allows the user to change the orientation of the mirrors. Please try it. Comments are welcome.

 

[PeroK]

I wrote a light clock app (https://joekahr.github.io/lightclock/) that allows the user to change the orientation of the mirrors. Please try it. Comments are welcome.

What is this supposed to show?

 

[Ibix]

I wrote a light clock app (https://joekahr.github.io/lightclock/) that allows the user to change the orientation of the mirrors. Please try it. Comments are welcome.

User interface doesn't work in Firefox on Android. You can start the animation but no further interaction works.

 

[joekahr]

Bring up the control panel by double clicking. Does that work?

 

[PeroK]

Bring up the control panel by double clicking. Does that work?

Yes, but your physics is wrong! Everythings stays simultaneous in a frame where the clock is moving.

 

[Dale]

Yes, but your physics is wrong! Everythings stays simultaneous in a frame where the clock is moving.

The physics looked good to me. The things that were supposed to be simultaneous were and the things that were not supposed to be simultaneous were not.

I ran it on an iPhone

 

[Ibix]

Bring up the control panel by double clicking. Does that work?

No - I did read the instructions before commenting.