Thread: Light clock in a gravitational field

If a light clock, made out of two parallel mirrors is placed perpendicular to the lines of gravitational field, the photon bouncing between the two mirrors will describe a series of arcs of spirals that progressively converge towards the center of the planet. This means that eventually, the photon will "fall off" the mirrors and the clock will stop working. I am trying to fins a software package that will draw the progression of inspiralling photon trajectories, anyone can help? The photon trajectory looks like the trajectory of an oscillating, ever shortening windshield wiper (an animation would be even better but I am happy with a static representation as well. The math is pretty simple, the trajectory, in Schwarzschild coordinates is going in one direction and going in the opposite direction, the arcs of spiral are linked together.

Due to the equivalence principle, the angle of the light, at any given moment, will correspond with the rate at which a physical object would be falling.

If this is Earth, and after one second, an object would be falling at about 10 meters per second, then the beam of light bouncing between two mirrors will be moving at an angle of Sin(10/300,000) after one second. (Using 300,000 meters per second as the speed of light, so really Sin(10/c) ) So using my calculator it looks like that 0.0000005818 degrees. Assuming I read the e(-7) notation right.

Originally Posted by kojax

Due to the equivalence principle, the angle of the light, at any given moment, will correspond with the rate at which a physical object would be falling.

Yes, I am aware of this. What I need is the plotting software.

If this is Earth, and after one second, an object would be falling at about 10 meters per second, then the beam of light bouncing between two mirrors will be moving at an angle of Sin(10/300,000) after one second.

You surely mean Sin(10/300,000,000) since c=300,000,000 m/s

By my calculator it will be 5.76E-10. More precisely using 9.8m/s we have an angle of 5.71E-10

Oh dear. Yeah three zeros would.... tend to make quite a difference.


I was suggesting it because it's a lot easier to program. Even without the trig function, you could just use a two dimensional vector, with one set to 10 m/s, and the other set to sqrt(c^2- 100)

I know a little bit of Python, and that doesn't seem like that would be too hard to set up. If you want to do it properly, I guess that's another story.

Originally Posted by kojax

Oh dear. Yeah three zeros would.... tend to make quite a difference.


I was suggesting it because it's a lot easier to program. Even without the trig function, you could just use a two dimensional vector, with one set to 10 m/s, and the other set to sqrt(c^2- 100)

I know a little bit of Python, and that doesn't seem like that would be too hard to set up. If you want to do it properly, I guess that's another story.

Yes, I know but if I use the actual values, they will result into a bunch of horizontal lines plotted on top of each other. This is why I need to exaggerate the drawing , in order to illustrate the pedagogical point.

Originally Posted by xyzt

Originally Posted by kojax

Oh dear. Yeah three zeros would.... tend to make quite a difference.


I was suggesting it because it's a lot easier to program. Even without the trig function, you could just use a two dimensional vector, with one set to 10 m/s, and the other set to sqrt(c^2- 100)

I know a little bit of Python, and that doesn't seem like that would be too hard to set up. If you want to do it properly, I guess that's another story.

Yes, I know but if I use the actual values, they will result into a bunch of horizontal lines plotted on top of each other. This is why I need to exaggerate the drawing , in order to illustrate the pedagogical point.

Yeah I can see what you mean. If they're like only 1 meter apart, then the light's going to bounce between them 300 million times in one second (I got the zero's right this time.) (150 million round trip bounces.)

Maybe you could put the mirrors really really far apart, and then kind of "suspend disbelief" about the Earth not really being big enough for its gravity to act uniformly on a distance that large?

Like if they're 50 million meters apart, then it only takes 3 round trip bounces to account for the first second. And that's only at first. The progression downward will grow.

the angle will be too small to be noticeable.if the drawing must fit the calculation.

Originally Posted by kojax

Originally Posted by xyzt

Originally Posted by kojax

Oh dear. Yeah three zeros would.... tend to make quite a difference.


I was suggesting it because it's a lot easier to program. Even without the trig function, you could just use a two dimensional vector, with one set to 10 m/s, and the other set to sqrt(c^2- 100)

I know a little bit of Python, and that doesn't seem like that would be too hard to set up. If you want to do it properly, I guess that's another story.

Yes, I know but if I use the actual values, they will result into a bunch of horizontal lines plotted on top of each other. This is why I need to exaggerate the drawing , in order to illustrate the pedagogical point.

Yeah I can see what you mean. If they're like only 1 meter apart, then the light's going to bounce between them 300 million times in one second (I got the zero's right this time.) (150 million round trip bounces.)

Maybe you could put the mirrors really really far apart, and then kind of "suspend disbelief" about the Earth not really being big enough for its gravity to act uniformly on a distance that large?

Like if they're 50 million meters apart, then it only takes 3 round trip bounces to account for the first second. And that's only at first. The progression downward will grow.

Yes, the progression downward grows fast, as in , so the angle increases rather fast.

that equation ought to be the height attained by an object. in contrast think the downward propagation should be v=2gh^1/2(squared).

Originally Posted by merumario

that equation ought to be the height attained by an object. in contrast think the downward propagation should be v=2gh^1/2(squared).

Nonsense.

Originally Posted by kojax

Originally Posted by xyzt

Originally Posted by kojax

Oh dear. Yeah three zeros would.... tend to make quite a difference.


I was suggesting it because it's a lot easier to program. Even without the trig function, you could just use a two dimensional vector, with one set to 10 m/s, and the other set to sqrt(c^2- 100)

I know a little bit of Python, and that doesn't seem like that would be too hard to set up. If you want to do it properly, I guess that's another story.

Yes, I know but if I use the actual values, they will result into a bunch of horizontal lines plotted on top of each other. This is why I need to exaggerate the drawing , in order to illustrate the pedagogical point.

Yeah I can see what you mean. If they're like only 1 meter apart, then the light's going to bounce between them 300 million times in one second (I got the zero's right this time.) (150 million round trip bounces.)

Maybe you could put the mirrors really really far apart, and then kind of "suspend disbelief" about the Earth not really being big enough for its gravity to act uniformly on a distance that large?

Like if they're 50 million meters apart, then it only takes 3 round trip bounces to account for the first second. And that's only at first. The progression downward will grow.

So, if you have a light clock with mirrors that are high and apart from each other, the photon will "fall off" the mirror after . If this means only , so the light clock will stop functioning after only 1 year!

Originally Posted by xyzt

Originally Posted by kojax

Originally Posted by xyzt

Originally Posted by kojax

Oh dear. Yeah three zeros would.... tend to make quite a difference.


I was suggesting it because it's a lot easier to program. Even without the trig function, you could just use a two dimensional vector, with one set to 10 m/s, and the other set to sqrt(c^2- 100)

I know a little bit of Python, and that doesn't seem like that would be too hard to set up. If you want to do it properly, I guess that's another story.

Yes, I know but if I use the actual values, they will result into a bunch of horizontal lines plotted on top of each other. This is why I need to exaggerate the drawing , in order to illustrate the pedagogical point.

Yeah I can see what you mean. If they're like only 1 meter apart, then the light's going to bounce between them 300 million times in one second (I got the zero's right this time.) (150 million round trip bounces.)

Maybe you could put the mirrors really really far apart, and then kind of "suspend disbelief" about the Earth not really being big enough for its gravity to act uniformly on a distance that large?

Like if they're 50 million meters apart, then it only takes 3 round trip bounces to account for the first second. And that's only at first. The progression downward will grow.

So, if you have a light clock with mirrors that are high and apart from each other, the photon will "fall off" the mirror after . If this means only , so the light clock will stop functioning after only 1 year!

I'm confused. If the mirror is of any reasonable height, the light will fall off of it a lot sooner than a year.

Only H and g matter to the time. Putting the mirrors further apart just makes it look better visually, but the amount of time required to fall off will be the same regardless of the distance apart. It bounces between the mirrors fewer times, but each bounce takes longer to complete.

Also, since g is an acceleration, and not a velocity, the time "t" required to travel a height "h" from rest will be


For example, if the height were 45 meters, and g =10, then it would take 3 seconds to fall that far. Your average speed would be 15 meters per second. (0 as your starting velocity, and 30 meters/second is your final velocity after 3 seconds, so your average speed is 1/2 of 30).

Can nothing be done to increase the time the light bounces off the mirrors?

Originally Posted by kojax

Originally Posted by xyzt

Originally Posted by kojax

Originally Posted by xyzt

Originally Posted by kojax

Oh dear. Yeah three zeros would.... tend to make quite a difference.


I was suggesting it because it's a lot easier to program. Even without the trig function, you could just use a two dimensional vector, with one set to 10 m/s, and the other set to sqrt(c^2- 100)

I know a little bit of Python, and that doesn't seem like that would be too hard to set up. If you want to do it properly, I guess that's another story.

Yes, I know but if I use the actual values, they will result into a bunch of horizontal lines plotted on top of each other. This is why I need to exaggerate the drawing , in order to illustrate the pedagogical point.

Yeah I can see what you mean. If they're like only 1 meter apart, then the light's going to bounce between them 300 million times in one second (I got the zero's right this time.) (150 million round trip bounces.)

Maybe you could put the mirrors really really far apart, and then kind of "suspend disbelief" about the Earth not really being big enough for its gravity to act uniformly on a distance that large?

Like if they're 50 million meters apart, then it only takes 3 round trip bounces to account for the first second. And that's only at first. The progression downward will grow.

So, if you have a light clock with mirrors that are high and apart from each other, the photon will "fall off" the mirror after . If this means only , so the light clock will stop functioning after only 1 year!

I'm confused. If the mirror is of any reasonable height, the light will fall off of it a lot sooner than a year.

You are right:

1. While light advances horizontally by , the vertical amount is at the first bounce
2. At the second bounce, light advances horizontally by , the vertical amount is
3. At the third bounce, light advances horizontally by , the vertical amount is
N. At the N-th bounce, light advances horizontally by , the vertical amount is

The total vertical path traveled by light after N bounces is

We can find the number of bounces N from the equation:




Then, the total time is . So the light clock, placed "horizonatally" in the gravitational field degrades extremely fast, to the point of becoming useless.

But something is not quite right with the above application of the equivalence principle since I know for a fact that the photon orbit has , in fact, a very small curvature, so it should take a very long time for the photon to "fall off" the mirrors. I think that is the wrong answer.

Originally Posted by xyzt

\

Then, the total time is . So the light clock, placed "horizonatally" in the gravitational field degrades extremely fast, to the point of becoming useless.

If you tried to build one in real life, a light clock would be useless for other reasons also. After one second traveling at 300 million meters per second, your beam of light has traveled 300 million meters. Probably it's pretty dim by now. Even if it had been the focused beam of a laser.

Also if your mirrors are real mirrors, that aren't infinity flat, you'll get some problems from that after millions of reflections.

Originally Posted by xyzt

But something is not quite right with the above application of the equivalence principle since I know for a fact that the photon orbit has , in fact, a very small curvature, so it should take a very long time for the photon to "fall off" the mirrors. I think that is the wrong answer.

You have to remember how fast C is. It is very very very fast, and light's forward velocity is always C.

If you progress downward at 0.001% of C, then you're still progressing downward at 3000 meters per second. A slight angle isn't necessarily a slow speed of travel.

Originally Posted by kojax

Originally Posted by xyzt

\

Then, the total time is . So the light clock, placed "horizonatally" in the gravitational field degrades extremely fast, to the point of becoming useless.

If you tried to build one in real life, a light clock would be useless for other reasons also. After one second traveling at 300 million meters per second, your beam of light has traveled 300 million meters. Probably it's pretty dim by now. Even if it had been the focused beam of a laser.

Also if your mirrors are real mirrors, that aren't infinity flat, you'll get some problems from that after millions of reflections.

Very true.

Originally Posted by xyzt

But something is not quite right with the above application of the equivalence principle since I know for a fact that the photon orbit has , in fact, a very small curvature, so it should take a very long time for the photon to "fall off" the mirrors. I think that is the wrong answer.

You have to remember how fast C is. It is very very very fast, and light's forward velocity is always C.

If you progress downward at 0.001% of C, then you're still progressing downward at 3000 meters per second. A slight angle isn't necessarily a slow speed of travel.

I have to disagree with this, the angle made by the light with the horizontal is very, very small, so the photon deflection at each traversal of the clock width is also very small, . Nothing to do with the speed of light.

Time clocks measure by comb frequencies. Minute time differences.
Also the idea of an optical cavity is that the light is trapped. So a deviation will make it go back into it's original path due to the mechanical detuning of mirrors. The Photon is never lost unless by absorption or scattering on an impurity.

Originally Posted by xyzt

Originally Posted by xyzt

But something is not quite right with the above application of the equivalence principle since I know for a fact that the photon orbit has , in fact, a very small curvature, so it should take a very long time for the photon to "fall off" the mirrors. I think that is the wrong answer.

You have to remember how fast C is. It is very very very fast, and light's forward velocity is always C.

If you progress downward at 0.001% of C, then you're still progressing downward at 3000 meters per second. A slight angle isn't necessarily a slow speed of travel.

I have to disagree with this, the angle made by the light with the horizontal is very, very small, so the photon deflection at each traversal of the clock width is also very small, . Nothing to do with the speed of light.

The deflection isn't the problem. It's the angle. Even with no deflection at all, if the angle of the light stops being perfectly horizontal, then the beam will be moving down the mirror.

If an object is moving at C in a direction 1 degree off of horizontal , then:

1) - It is moving at 0.9998 * C in the horizontal direction

and

2) - It is moving at 0.0175 * C in the downward direction.

That's how the vectors split up. If you were going down a hill that was 1 degree, those numbers would be the "rise" and "run" of the hill.

0.0175 * C is 5,246,368 meters per second.

Originally Posted by kojax

Originally Posted by xyzt

Originally Posted by xyzt

But something is not quite right with the above application of the equivalence principle since I know for a fact that the photon orbit has , in fact, a very small curvature, so it should take a very long time for the photon to "fall off" the mirrors. I think that is the wrong answer.

You have to remember how fast C is. It is very very very fast, and light's forward velocity is always C.

If you progress downward at 0.001% of C, then you're still progressing downward at 3000 meters per second. A slight angle isn't necessarily a slow speed of travel.

I have to disagree with this, the angle made by the light with the horizontal is very, very small, so the photon deflection at each traversal of the clock width is also very small, . Nothing to do with the speed of light.

The deflection isn't the problem. It's the angle. Even with no deflection at all, if the angle of the light stops being perfectly horizontal, then the beam will be moving down the mirror.

If an object is moving at C in a direction 1 degree off of horizontal , then:

1) - It is moving at 0.9998 * C in the horizontal direction

and

2) - It is moving at 0.0175 * C in the downward direction.

That's how the vectors split up. If you were going down a hill that was 1 degree, those numbers would be the "rise" and "run" of the hill.

0.0175 * C is 5,246,368 meters per second.

Except! the angle is much, much smaller than 0.0175.
In fact, the angle is given by the formula found here.
For the Earth, this boils down to about . This means that the component of the light speed along the mirror is about 0.9m/s.
For each photon trip, the beam descends roughly meters.
It will take bounces for the photons to "slip off" the vertical mirrors.
Since each bounce takes seconds, the total time will be seconds, consistent with the 0.9m/s speed.


By comparison, the prior formula , so the light clock becomes useless in about 1 second. The derivation using photon trajectories is consistent now with the derivation using the equivalence principle, I am happy now.

Originally Posted by Kerling

Time clocks measure by comb frequencies. Minute time differences.
Also the idea of an optical cavity is that the light is trapped. So a deviation will make it go back into it's original path due to the mechanical detuning of mirrors. The Photon is never lost unless by absorption or scattering on an impurity.

Interesting point. Nevertheless, we have just found out (see discussion with kojax) that , if placed "horizontally" in a gravitational field, the light clock stops working in about 1 second (because the photons follow a downwards path when bouncing between the mirrors)! This "feature" renders them virtually useless.

Originally Posted by xyzt

Originally Posted by Kerling

Time clocks measure by comb frequencies. Minute time differences.
Also the idea of an optical cavity is that the light is trapped. So a deviation will make it go back into it's original path due to the mechanical detuning of mirrors. The Photon is never lost unless by absorption or scattering on an impurity.

Interesting point. Nevertheless, we have just found out (see discussion with kojax) that , if placed "horizontally" in a gravitational field, the light clock stops working in about 1 second (because the photons follow a downwards path when bouncing between the mirrors)! This "feature" renders them virtually useless.

Well, then let me give you a alternate possibility.
Let's say we construct this clock, and put it near the gravitational field, one, powerfull enough to change the path of the mirror.
Then we will look at this clock, in such a way that it will at best deform, and not break. Say that considering the vacuum of space, and the rather appearant black hole. These mirror needn't any additional vacuum measuring devices. As, well, they work in earthly atmosphere too, so in space they should be great.
Then,
Why on earth, wouldn't the gravitational field affect the geometry of the mirror's?
In saying so, nothing would happen, provided the change is not too large over the linear span of the circular parabolic mirrors?

Originally Posted by Kerling

Originally Posted by xyzt

Originally Posted by Kerling

Time clocks measure by comb frequencies. Minute time differences.
Also the idea of an optical cavity is that the light is trapped. So a deviation will make it go back into it's original path due to the mechanical detuning of mirrors. The Photon is never lost unless by absorption or scattering on an impurity.

Interesting point. Nevertheless, we have just found out (see discussion with kojax) that , if placed "horizontally" in a gravitational field, the light clock stops working in about 1 second (because the photons follow a downwards path when bouncing between the mirrors)! This "feature" renders them virtually useless.

Well, then let me give you a alternate possibility.
Let's say we construct this clock, and put it near the gravitational field, one, powerfull enough to change the path of the mirror.
Then we will look at this clock, in such a way that it will at best deform, and not break. Say that considering the vacuum of space, and the rather appearant black hole. These mirror needn't any additional vacuum measuring devices. As, well, they work in earthly atmosphere too, so in space they should be great.
Then,
Why on earth, wouldn't the gravitational field affect the geometry of the mirror's?
In saying so, nothing would happen, provided the change is not too large over the linear span of the circular parabolic mirrors?

I am sorry, I cannot comprehend your post, must be a language problem. Are you saying that the clock is getting also distorted in such a way that it compensates the spiralling orbit of the photons? This is certainly incorrect.

spirals? Wat spirals?

Originally Posted by Kerling

spirals? Wat spirals?

OK,

Photon trajectories in a gravitational fiel are of 3 types:

1. Radial (along the r Schwarzschild coordinate)
2. Spherical (unstable), easily deteriorating into the third type:
3. Spiral (going to the infinity or going towards the singularity, depending on a special parameter, called impact parameter)

You can learn all this from any of the several very good books on GR.

ow of course. I'm just really wondering how you concluded that it would break. Or slip. let me put it like this; have you guys ever hear of decoherence time? (Of a cavity)?This is the time for a single photon to show a spontaneous change in fase. this is wel below the speed of th cavity.secondly. the cavity is quantized, a regukar deflection would change the allowed modes due to GR, since the energy if th photon doesn't change it would destructively decohere long before it falls off. hence we need to broaden the state by saying it allows more states.also cavities are designed to capture slightly rogue beams of light back into their original alignment. since it is rather small i don't think the gravity would b of any relevance whatsoever

Originally Posted by Kerling

ow of course. I'm just really wondering how you concluded that it would break. Or slip. let me put it like this; have you guys ever hear of decoherence time? (Of a cavity)?This is the time for a single photon to show a spontaneous change in fase. this is wel below the speed of th cavity.secondly. the cavity is quantized, a regukar deflection would change the allowed modes due to GR, since the energy if th photon doesn't change it would destructively decohere long before it falls off. hence we need to broaden the state by saying it allows more states.also cavities are designed to capture slightly rogue beams of light back into their original alignment. since it is rather small i don't think the gravity would b of any relevance whatsoever

No, resonating cavities is NOT what we are talking about, they definitely do not degrade.
What we are talking about is a simple light clock, two mirrors distance apart and "tall". In our exercise . As you can see, from the calculations, such a light clock, when placed horizontally in the gravitational field is rendered useless in about one second, unlike the resonating cavities.