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

[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.

 

[Dale]

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.

I like it. It seems well done to me. Thanks for sharing!

 

[PeroK]

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

I'm using Firefox on a laptop and when the clock is moving, all four light pulses hit the mirrors simultaneously.

 

[Dale]

I'm using Firefox on a laptop and when the clock is moving, all four light pulses hit the mirrors simultaneously.

Interesting. I just tried it on a laptop running Chrome and it was good, rear mirror hits first, side mirrors second, and front mirror last, with all four returning to the center at the same time. Are you simulating the lab frame or the rocket frame? I was simulating lab frame.

 

[PeroK]

Interesting. I just tried it on a laptop running Chrome and it was good, rear mirror hits first, side mirrors second, and front mirror last, with all four returning to the center at the same time. Are you simulating the lab frame or the rocket frame? I was simulating lab frame.

Tried it on MS Edge. No luck. I just selected rocket. In any case, the clock is moving with no loss of synchonisation.

 

[Dale]

Try the lab frame instead of the rocket frame.

 

[PeroK]

Try the lab frame instead of the rocket frame.

Okay, but in the rocket frame the clock shouldn't be moving.

 

[joekahr]

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

OK. I am here to learn. What did I get wrong about the physics?

Perhaps you object to not showing the light clock in its own frame when the Center Clock option is set. I address this in the FAQ under "Why does the Light Clock stay contracted and its time dilated when 'Center Clock' are selected?" Is that an adequate answer?

 

[robphy]

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.

Neat.

If you make a "circle" of clocks, the contracted arrangement in the lab frame will look like an ellipse of mirrors.
If you locate the spatial components of the reflections you get a different ellipse (with the foci locating the emission event and the reception event at the source).

The full spacetime diagram will look like my avatar.
(The animated version of my avatar appears when you click on my avatar and bring up my PF profile.)

Here are some old videos, recently uploaded to YouTube:
https://www.youtube.com/channel/UCkqNyIlz3XF6IhyYbnmoJoA

 

[Dale]

Okay, but in the rocket frame the clock shouldn't be moving.

I agree. The mirrors are not moving in the rocket frame regardless of v.

 

[joekahr]

Okay, but in the rocket frame the clock shouldn't be moving.

You will see that the light clock is not moving relative to the Rocket Frame x'y' grid, whether the 'Center Clock' is set or not.

 

[joekahr]

No - I did read the instructions before commenting.

Sorry about that. I never tested on Android. Please try on a computer, iPhone or iPad.

 

[joekahr]

The OP's question was about the orientation of the mirrors of a light clock. I noticed that my app (https://joekahr.github.io/lightclock/) had a bug that made changing the orientation difficult. It's fixed now.
I also made the Center Clock default settings more intuitive: clock is moving in the Lab frame, clock is centered in the Rocket frame.

 

[Grasshopper]

That's so cool. Interesting that the light pulses always seem to intersect at the center. That surely indicates some sort of symmetry. Maybe that length contraction applies evenly across the direction of travel.

 

[Ibix]

Interesting that the light pulses always seem to intersect at the center.

They must. You could build an arbitrarily small bomb trigger that would go off if illuminated by all four pulses at once and not if only three or fewer pulses illuminate it. If all four pulses reach the center simultaneously in one frame they must in all, because whether the bomb goes off or not can't be frame dependent.

 

[ergospherical]

They must. You could build an arbitrarily small bomb trigger that would go off if illuminated by all four pulses at once and not if only three or fewer pulses illuminate it. If all four pulses reach the center simultaneously in one frame they must in all, because whether the bomb goes off or not can't be frame dependent.

Physicists are so dramatic — there's always got to be explosion of some sort
What's wrong with a little electronic counter; I mean, it'd probably be cheaper...

 

[robphy]

That's so cool. Interesting that the light pulses always seem to intersect at the center. That surely indicates some sort of symmetry. Maybe that length contraction applies evenly across the direction of travel.

Check out the videos of the "circular light clocks" linked from my earlier reply
https://www.physicsforums.com/threa...ation-of-the-light-clock.1003748/post-6506712