Light Clock Traveling in Z Direction

[sqljunkey]

Hi,

I'm studying GR these days. But then I got to thinking about something about SR and got kinda stuck.

I know that if a light clock is moving in the x direction(from left to right), the light beam has to traverse more space and then you will see this person's clock running slower in the train.

What if this person was moving away from you in the Z direction. If I ignored the perspective effect in my Cartesian coordinate system and just left the orthogonal view, how can I see that the light has to travel more distance and move slower given that the distance between the ceiling of the clock and the floor of the clock remains the same.

And then even if it has the perspective effect, as the train is moving away, I would see the distance of the floor and the ceiling of the clock shrink. Within this logic since the light source has less distance to cover between the two walls of the clock it would look as though it's going faster?

 

[Dale]

the perspective effect

What do you mean by the perspective effect?

 

[sqljunkey]

Just this,

"The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object's dimensions along the line of sight are shorter than its dimensions across the line of sight.", https://en.wikipedia.org/wiki/Perspective_(graphical) .

 

[PeroK]

Hi,

I'm studying GR these days. But then I got to thinking about something about SR and got kinda stuck.

I know that if a light clock is moving in the x direction(from left to right), the light beam has to traverse more space and then you will see this person's clock running slower in the train.

What if this person was moving away from you in the Z direction. If I ignored the perspective effect in my Cartesian coordinate system and just left the orthogonal view, how can I see that the light has to travel more distance and move slower given that the distance between the ceiling of the clock and the floor of the clock remains the same.

Have you tried to calculate the working of a light clock in this scenario? In terms of actually doing the maths and physics?

 

[Dale]

Just this,

"The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object's dimensions along the line of sight are shorter than its dimensions across the line of sight.", https://en.wikipedia.org/wiki/Perspective_(graphical) .

That has nothing to do with the light clock. I don’t understand the question with that unrelated concept used multiple times.

 

[Sorcerer]

Are you asking what you see as opposed to what actually happens?

 

[Mister T]

I know that if a light clock is moving in the x direction(from left to right), the light beam has to traverse more space and then you will see this person's clock running slower in the train.

The usual language used to describe this is you will observe the person's clock running slower. Figuring out what you see depends on the additional complication due to the delay in light travel time. The usual analysis of the light clock does not include this.

Lest you think this a quibble, what you will actually see depends on the direction of motion. If the clock moves towards you then you see it running faster, but if it's moving away from you then you see it running slower. See the analysis of the relativistic Doppler effect.

What if this person was moving away from you in the Z direction.

Do you mean the light clock is moving in a direction parallel to the clock's light beam? Then it's a more complicated analysis, but you get the same result as you do with the conventional light clock.

how can I see that the light has to travel more distance and move slower given that the distance between the ceiling of the clock and the floor of the clock remains the same.

But the light doesn't move slower. And the floor to ceiling distance is Lorentz contracted.

When the light beam is moving away from you it has to move some distance to reach the mirror that's facing you. That distance is further than it would be if the clock were at rest. After reflection it moves towards you and has to move some distance to reach the mirror that faces away from you. That distance is shorter than it would be if the clock were at rest. But these two effects, while opposite, are not equal. One is larger, and it turns that overall, in a to-and-fro cycle that constitutes one tick of the clock, the beam moves further than it would if the clock were at rest. You need to do the calculation, or see it done, to be convinced. I know of no way to explain it qualitatively. Or rather, the explanation of the calculation would be very very hard to follow if not accompanied by the the calculation itself.

 

[sqljunkey]

Hi I think I figured it out.

Even though it's traveling away from the person watching it's still moving in the Z direction so the photon has to travel in that direction. I got confused a bit since you wouldn't see the z axis. Thanks all!

 

[Sorcerer]

The usual language used to describe this is you will observe the person's clock running slower. Figuring out what you see depends on the additional complication due to the delay in light travel time. The usual analysis of the light clock does not include this.

Lest you think this a quibble, what you will actually see depends on the direction of motion. If the clock moves towards you then you see it running faster, but if it's moving away from you then you see it running slower. See the analysis of the relativistic Doppler effect.
Do you mean the light clock is moving in a direction parallel to the clock's light beam? Then it's a more complicated analysis, but you get the same result as you do with the conventional light clock.
But the light doesn't move slower. And the floor to ceiling distance is Lorentz contracted.

When the light beam is moving away from you it has to move some distance to reach the mirror that's facing you. That distance is further than it would be if the clock were at rest. After reflection it moves towards you and has to move some distance to reach the mirror that faces away from you. That distance is shorter than it would be if the clock were at rest. But these two effects, while opposite, are not equal. One is larger, and it turns that overall, in a to-and-fro cycle that constitutes one tick of the clock, the beam moves further than it would if the clock were at rest. You need to do the calculation, or see it done, to be convinced. I know of no way to explain it qualitatively. Or rather, the explanation of the calculation would be very very hard to follow if not accompanied by the the calculation itself.

Is that necessarily always true? Could it be, depending on the specific speed value, that a clock moving toward you will appear to be ticking faster than your own clock but still slower than it should be based on the Doppler effect alone? Or could it still appear to be moving slower than your clock but faster than it should be based on time dilation calculations?

That is, could it be that in some instances the Doppler effect as the clock approaches you is weaker than the time dilation effect?

 

[Mister T]

Is that necessarily always true?

Yes.

Could it be, depending on the specific speed value, that a clock moving toward you will appear to be ticking faster than your own clock but still slower than it should be based on the Doppler effect alone?

Huhh? A clock moving towards you will always appear to be ticking faster by a factor of $\sqrt{\frac{1+\beta}{1-\beta}}$. I don't know what you mean by "should", but if you mean what it's actually doing, it will be ticking slower by a factor of $\frac{1}{\sqrt{1-\beta^2}}$ compared to the rate at which it would tick if $\beta$ were zero.

That is, could it be that in some instances the Doppler effect as the clock approaches you is weaker than the time dilation effect?

The (relativistic) Doppler effect includes the time dilation effect. Look at the derivation. The derivation on Wikipedia is easy to find, easy to follow, and it's correctly done.