A well defined thought experiment

[Mikael17]

Bob lives on top of a skyscraper which is 1000 meters high.
There are exactly 10 meters between 2 (glass) walls in his penthouse apartment.
Between these walls, - a photon moves back and forth (is reflected) for exactly 100 billion years.
Bob can measure the elapsed time to be 3153600000000000000 s
Based on the speed c, - Bob can also calculate the distance the photon has traveled = t x c = 946080000000000000000000000 meters.
Bob can both calculate and see (and count) that the photon travels 94608000000000000000000000 times back and forth between the 2 walls.
Because Bob knows the distance between the two walls and he has counted how many times the photon traveled back and forth, - and because Bob also knows the speed c , - it is easy for Bob now to double check and be 100% sure that the photon has traveled 94608000000000000000000000 meters.

Alice lives at the bottom of the skyscraper.
The entire skyscraper is built in glass.
From her apartment, Alice can look up into Bob's apartment and see that the 2 walls (between which the photon moves) in Bob's Penthouse are the same walls that continue 1000 meters vertically down and that they are the same walls that are in her apartment on 1st floor.

Alice can from here 1st floor apartments see and hear that the photons move between these 2 same walls as in Bob's apartment.
Because each time the photon turns around she gets a "beep" and a a light signal, - sent from Bob's apartment.
Alice also measures, just like Bob, the distance between these glass-walls to be exactly 10 meters (in her apartment) .

After exactly 100 billion years have passed (as the experiment took), Alice can see on her watch that it has lost 3153.6 seconds relative to Bob's watch.
Just like Bob, Alice has counted, - have seen / heard, - how many times the photon has moved back and forth, and completely agrees with Bob that it happened 946080000000000000000000000 times.

The problem is now that when Alice wants to calculate how far the photon has traveled, the distance will be : t x c - (3153.6 s * c) = less distance - compared Bobs calculation result, - of coursee because of the 3153,6 s time dilation.

The calculation shows that the distance the photon was traveling is much shorter than Bob has measured and calculated.

946080000000 meters has somehow been lost, - relative to what she herself has seen the photon travel with her own eyes.

What is the explanation for this?

Why don't Bob and Alice agree?

 

[PeterDonis]

The problem is now that when Alice wants to calculate how far the photon has traveled, the distance will be : t x c - (3153.6 s * c) = less distance - compared Bobs calculation result, - of coursee because of the 3153,6 s time dilation.

This is wrong. Alice cannot assume that Bob's photon is traveling at $c$, because she is not at the same height as Bob. In curved spacetime, the apparent speed of light at a remote location from an observer is not always $c$ for that observer. Curved spacetime means we have to use General Relativity, not Special Relativity, and the rule that the speed of light is always $c$ is only valid in Special Relativity (i.e., in flat spacetime). There is a corresponding rule in GR, but it is not as simple and does not support the reasoning you are using.

 

[Ibix]

To rephrase Peter's comment, Alice is using a coordinate system where her measured second does not match up to a measured second further up the building. Thus her naive calculation doesn't correctly measure the local speed of light.

More generally, "light always moves at $3\times 10^8\mathrm{ms^{-1}}$" is not a general statement. It's only true in inertial coordinates in flat spacetime, and you have a coordinate system that isn't inertial and isn't in flat spacetime.

 

[Mikael17]

This is wrong. Alice cannot assume that Bob's photon is traveling at $c$, because she is not at the same height as Bob. In curved spacetime, the apparent speed of light at a remote location from an observer is not always $c$ for that observer.

Alice will continue to ask which of my parameters are comparable different for me than they are for Bob.

Is it d or m?

There are no other possibilities.

It has nothing to do with the speed of light, because even if it was just a mouse running from one wall to the other, billion of times, the problem / challenge to really understand this will be the same.

Alice wants to know what changes deeper in a gravity field relative to higher up in the gravity field. .

After all, Alice has the right to compare all relativistic effects just as she has the right to recognize / measure that the passage of time is a variable.?

Let's keep this in GR (we can imagine that the Earth is at absolute rest, i.e. no motion and no rotation) and hence no SR effect

 

[Ibix]

It has nothing to do with the speed of light,

You multiplied by $c$! Of course whether that's the right thing to multiply by or not matters.

 

[PeterDonis]

Alice will continue to ask which of my parameters are comparable different for me than they are for Bob.

Is it d or m?

It isn't d. I don't know what m is supposed to be.

There are no other possibilities.

Yes there are. Alice's speed of light $c$ for Bob's light beam is different from Bob's.

It has nothing to do with the speed of light

Yes, it does. c explicitly appears in the formula you are saying that Alice uses to calculate the distance.

Alice wants to know what changes deeper in a gravity field relative to higher up in the gravity field. .

The rate of time flow, which affects lots of things, including what speed Alice will measure Bob's light to travel.

After all, Alice has the right to compare all relativistic effects just as she has the right to recognize / measure that the passage of time is a variable.?

I don't know what you mean by this or why you think it matters.

Let's keep this in GR (we can imagine that the Earth is at absolute rest, i.e. no motion and no rotation) and hence no SR effect

Sure, that's fine, but do you actually know GR? Do you know how to make the relevant calculations? Do you know how to modify the formula you give in the OP to take GR effects into account?

If, as I strongly suspect, the answers to these questions are "no", then I fail to see what your apparent confidence in your (wrong) claims is based on. If you can't do the calculation right, how can you possibly think you have the right answer?

 

[Mikael17]

To rephrase Peter's comment, Alice is using a coordinate system where her measured second does not match up to a measured second further up the building. Thus her naive calculation doesn't correctly measure the local speed of light.

Are you saying that both the photon and the mouse on the floor, (both of which move back and forth from one wall to another, - billions of times) - - both move in a curved path " ?

 

[Mikael17]

Yes, it does. c explicitly appears in the formula you are saying that Alice uses to calculate the distance.

I understand why you think the motion of light not is perfect.
We can repeat the experiment by replaing the photon with any kind of object moving from wall to wall, in Bobs apartment, - at his total horizontal floor, - the time dilation will be the same, - and the challenge to understand why Alice and Bob not agree, is not vanished. right ?

 

[PeterDonis]

Are you saying that both the photon and the mouse on the floor, (both of which move back and forth from one wall to another, - billions of times) - - both move in a curved path " ?

In terms of spacetime, the mouse on the floor will definitely have to move in a curved path, yes. The mouse has nonzero proper acceleration; it feels weight. That means its path through spacetime is curved.

You can approximate the spatial path of the mouse as being a straight line.

For the light, it gets more complicated, because a freely moving light beam will travel in a straight path in spacetime, but in space its path will bend downward due to gravity. So if you want the light to go back and forth billions of times, it can't be freely moving; it would need to be inside a waveguide or a fiber optic cable or something that kept its spatial path at the same height. And in that case the light's path in spacetime, like the mouse's, would be curved.

 

[PeterDonis]

I understand why you think the motion of light not is perfect.

I don't know what you mean by this. You really, really need to work on clarity of communication.

We can repeat the experiment by replaing the photon with any kind of object moving from wall to wall, in Bobs apartment, - at his total horizontal floor, - the time dilation will be the same, - and the challenge to understand why Alice and Bob not agree, is not vanished. right ?

The time dilation is the same no matter what kind of object you use, yes.

An object other than light does not have a speed that is naively expected to be invariant in the first place, so it should be even easier to see that, for example, your mouse traveling across Bob's floor will have a different speed as measured by Alice than it does as measured by Bob, since Alice is further down in the gravity well and her clocks run slower than Bob's. If there's any challenge at all in this scenario, it's understanding how Alice can measure Bob's light beam to be traveling at a different speed than Bob does, because of her time dilation. She does, but it takes some GR analysis to see why.

 

[Ibix]

Are you saying that both the photon and the mouse on the floor, (both of which move back and forth from one wall to another, - billions of times) - - both move in a curved path " ?

The mouse does. Light will be on a free fall path, so a straight line with sharp corners at the reflection events. But the more important fact is the curvature of spacetime and how Alice is dealing with it, which leads to the elapsed time between what she is calling "now" and "one second later" not being one second everywhere.

 

[PeterDonis]

Light will be on a free fall path

Not if it's going to be reflected back and forth for 100 billion years. As I said in post #9, freely falling light will bend downwards. Long before 100 billion years have passed it will have hit the floor.

 

[Ibix]

Not if it's going to be reflected back and forth for 100 billion years. As I said in post #9, freely falling light will bend downwards. Long before 100 billion years have passed it will have hit the floor.

Sure - it'll hit the floor in the same half a second or so a dropped ball will take. That's why I said "sharp corners at the reflection events".

 

[PeterDonis]

That's why I said "sharp corners at the reflection events".

Ah, got it. I'm not even sure that will work by itself, but I don't think the point is important for this discussion. The exact path taken by the light and how it is realized is not the issue.

 

[PeroK]

I understand why you think the motion of light not is perfect.
We can repeat the experiment by replaing the photon with any kind of object moving from wall to wall, in Bobs apartment, - at his total horizontal floor, - the time dilation will be the same, - and the challenge to understand why Alice and Bob not agree, is not vanished. right ?

If Bob and Alice are facing each other, then Alice might say the mouse is running to the left, while Bob says it's running to the right. They might not "agree" about the direction the mouse is running. So what?

 

[Mikael17]

For the light, it gets more complicated, because a freely moving light beam will travel in a straight path in spacetime, but in space its path will bend downward due to gravity. So if you want the light to go back and forth billions of times, it can't be freely moving; it would need to be inside a waveguide or a fiber optic cable or something that kept its spatial path at the same height. And in that case the light's path in spacetime, like the mouse's, would be curved.

What then if the light is reflected at the bottom and at the top of the skyscraper (vertical motion)

 

[PeterDonis]

What then if the light is reflected at the bottom and at the top of the skyscraper (vertical motion)

If you are asking whether the light's spacetime path would be curved in this case, no, it wouldn't have to be for purely vertical motion.

But if you are asking what the light's speed would be in this case, its speed would still not be $c$ (and in the vertical case neither Alice nor Bob would measure its speed to be $c$ over the entire vertical extent of travel).

 

[PeroK]

What then if the light is reflected at the bottom and at the top of the skyscraper (vertical motion)

You can simplify your whole experiment. Alice, at the top of the tower, measures the round trip for a light signal to the bottom of the tower and back. Bob, at the bottom of the tower measures the round trip from the bottom to the top and back. They measure different times for the light signal round trip, over a common distance. But, in GR, it's the locally measured speed of light that is invariant. Not a coordinate speed. This is discussed here.

https://www.physicsforums.com/threads/what-is-coordinate-speed.997660/

 

[PeterDonis]

They measure different times for the light signal round trip, over a common distance.

Note, btw, that they can verify the common distance by extending a ruler (or a tape measure or some similar distance measuring device) from top to bottom and verifying that they both agree on the distance it reads.

 

[Ibix]

Ah, got it. I'm not even sure that will work by itself, but I don't think the point is important for this discussion. The exact path taken by the light and how it is realized is not the issue.

Up to ray optics it's possible - just arrange a symmetric (approximately) parabolic path. Beam spread will eventually do for it, but the billions of bounces is pointless anyway. One would have been enough, to quote Einstein out of context.

 

[Vanadium 50]

946080000000000000000000000

Could you possibly have made this any more difficult to follow?

 

[berkeman]

Could you possibly have made this any more difficult to follow?

Hey, that's called "Unscientific Notation", which is a standard. Somewhere. I'm not able to find it on my HP calculator at the moment...

 

[Dale]

Alice also measures, just like Bob, the distance between these glass-walls to be exactly 10 meters (in her apartment) .

Can you please post your work deriving this? I don’t think it is a correct claim.

What is the explanation for this?

Most likely an incorrect calculation.

There are no other possibilities.

Sure there are.

It has nothing to do with the speed of light, because even if it was just a mouse running from one wall to the other, billion of times, the problem / challenge to really understand this will be the same.

Can you please post your work deriving this claim also? I don’t think it is correct either. Especially since we are talking about light, it seems likely that the speed of light would be important, and the claim that this holds for all speeds is suspicious.

After all, Alice has the right to compare all relativistic effects just as she has the right to recognize / measure that the passage of time is a variable.?

Certainly, but both she and you should be prepared to justify your claims with the relevant calculations.

Let's keep this in GR

Indeed, please justify your claims using the mathematics of GR.

 

[PeterDonis]

Can you please post your work deriving this? I don’t think it is a correct claim.

Although it would be nice to see the OP work this out, and I'll leave the OP to work out (or not) the rest of the open questions, for this one item I'm willing to chip in. Since the distance in question is tangential, I think it is correct (or at least correct enough for this discussion) to say that Alice and Bob would both measure it the same. There are complexities involved with radial distances in Schwarzschild spacetime, but not so much with tangential ones.

 

[Dale]

Since the distance in question is tangential, I think it is correct (or at least correct enough for this discussion) to say that Alice and Bob would both measure it the same.

I don’t think so. If the spatial geometry were flat then definitely two lines in the radial (vertical) direction definitely don’t keep the same distance. Perhaps the spatial curvature compensates for it, but I doubt it. Or perhaps the time dilation somehow compensates, but again I doubt it.

I don’t think the walls can be both vertical and equidistant. In my opinion that claim requires some justification at least.

 

[PeterDonis]

If the spatial geometry were flat then definitely two lines in the radial (vertical) direction definitely don’t keep the same distance.

Ah, I see: I had missed the fact the OP says that Alice measures the distance between the walls in her apartment, not Bob's. Yes, if the walls are perfectly vertical (i.e., radial), the two distances won't be the same. I think that is equally true whether the spatial geometry is Euclidean or the Flamm paraboloid-like shape of a constant time slice in Schwarzschild spacetime (although the exact numerical value of the difference might not be the same in the two cases).

 

[Dale]

So if you want the light to go back and forth billions of times, it can't be freely moving; it would need to be inside a waveguide or a fiber optic cable or something that kept its spatial path at the same height.

I think that there is an angle that you could (in principle) place the mirrors on each side so that it bounces back and forth at the same height. It wouldn’t be vertical. And I suspect it would be different in the two apartments.

 

[PeterDonis]

I think that there is an angle that you could (in principle) place the mirrors on each side so that it bounces back and forth at the same height.

Yes, @Ibix said much the same thing in post #20.

I suspect it would be different in the two apartments.

I suspect you're right.

 

[Mikael17]

If you are asking whether the light's spacetime path would be curved in this case, no, it wouldn't have to be for purely vertical motion.

But if you are asking what the light's speed would be in this case, its speed would still not be $c$ (and in the vertical case neither Alice nor Bob would measure its speed to be $c$ over the entire vertical extent of travel).

I would think that both Alice and Bob measure the height of buildings the same, i.e. 1000 meters.
It is therefore ruled out that d (distance) the photon travels up and down is different for Alice and Tom, - right ?
The speed of the photon is not affected by both Bob and Alice watching it move (up and down) for 100 billion years.
Bob and Alice must (off course) also agree on how many times the photon has traveled up and down. (?)

When we assume that Bob and Alice receive a light signal every time the photon hits the bottom and top of the skyscraper, they both just have to count these signals and multiply by distances of 1000 meters (which is the height of the skyscraper) - then they both have to be 100% sure of how far the photon is have traveled.
Both bob and Alice are therefore both calculated that the photon has traveled (94608000000000000000000 * 1000 meters).
Bob and Alice agree that the photon has traveled 9.4608e26 meters.

But when Bob and Alice now calculate the distance that the photon has traveled, based on the time shown by their clocks, - they will not agree.
Alice has again measured the travel time to be 3153.6 seconds less than Bob and thus also calculates the travel distance to be 3153600 meters less than Bob.

How can Alice and Bob's calculations lead to Alice and Bob first agreeing and then disagreeing with each other about how far the photon has traveled?
--------------------------------------------------------------------------------------------------------------------
Should the mystery lie in the fact that Bob and Alice do not perceive the speed of light the same (?)

In that case, let's say that Bob and Alice have also measured how many times the elevator has gone up and down.

After all, Bob and Alice can only agree that the elevator has run up and down 8.64e15 times
When Bob and Alice calculate 8.64e15 * 1000 meters, they both agree and get the same result (8.64e18 meters).

Let's say the elevator speed is 500m/s
Alice claims that when she calculates the distance the elevator has traveled based on the time she has measured, the result is :(3153.6 seconds * 500m/s) = 1576800 meters less than the travel distance that Bob has calculated based on the time Bob has measured..

It is then a paradox that Bob and Alice again first agree and then disagree (?)

I'm sorry, but I have a hard time understanding where the solution to this dilemma is?

 

[Mikael17]

You can simplify your whole experiment. Alice, at the top of the tower, measures the round trip for a light signal to the bottom of the tower and back. Bob, at the bottom of the tower measures the round trip from the bottom to the top and back. They measure different times for the light signal round trip, over a common distance. But, in GR, it's the locally measured speed of light that is invariant. Not a coordinate speed. This is discussed here.

https://www.physicsforums.com/threads/what-is-coordinate-speed.997660/

Bob and Alice also calculated the distance the elevator was travelling up and down, first they agreed,on calculation and right after they disagree, read my post above, so it cannot be limited to be a speed of light problem

 

[Ibix]

They don't agree on the elapsed time. They do agree on the distance travelled by the lift or light pulse. Therefore they don't agree about the speed.

I don't understand why this is difficult.

 

[PeterDonis]

I would think that both Alice and Bob measure the height of buildings the same, i.e. 1000 meters.

Yes.

It is therefore ruled out that d (distance) the photon travels up and down is different for Alice and Tom, - right ?

For the case where the photon travels purely vertically, yes.

The speed of the photon is not affected by both Bob and Alice watching it move (up and down) for 100 billion years.

Not by that, but it is different for Alice and Bob because their relative clock rates are different. That means the speed of the photon is different for each of them.

Bob and Alice must (off course) also agree on how many times the photon has traveled up and down. (?)

Yes.

When we assume that Bob and Alice receive a light signal every time the photon hits the bottom and top of the skyscraper, they both just have to count these signals and multiply by distances of 1000 meters (which is the height of the skyscraper) - then they both have to be 100% sure of how far the photon is have traveled.

Yes, they agree on the distance the photon traveled. But they do not agree on the time it took to travel; that time is different for each of their clocks.

when Bob and Alice now calculate the distance that the photon has traveled, based on the time shown by their clocks, - they will not agree.

Yes, they will, because unlike you, they will not be using the same speed of light. This point has already been made in previous posts.

Should the mystery lie in the fact that Bob and Alice do not perceive the speed of light the same (?)

Yes, as has already been said in previous posts, as well as above.

In that case, let's say that Bob and Alice have also measured how many times the elevator has gone up and down.

After all, Bob and Alice can only agree that the elevator has run up and down 8.64e15 times
When Bob and Alice calculate 8.64e15 * 1000 meters, they both agree and get the same result (8.64e18 meters).

Yes, but they will disagree on the time it took the elevator to do that, so they will disagree on the elevator's speed.

Let's say the elevator speed is 500m/s

We can't, because the elevator speed is not the same for Alice and Bob.

Alice claims that when she calculates the distance the elevator has traveled based on the time she has measured, the result is :(3153.6 seconds * 500m/s) = 1576800 meters less than the travel distance that Bob has calculated based on the time Bob has measured..

No, she doesn't, because, once again, unlike you, she understands that the speed of the elevator is different for her than it is for Bob.

It is then a paradox that Bob and Alice again first agree and then disagree (?)

No, because they don't. They always agree on the number of transits and the distance, and they always disagree on the elapsed time.

I'm sorry, but I have a hard time understanding where the solution to this dilemma is?

That's because you have ignored the statements already made in previous posts (which I have repeated above) about the speeds being different. You need to stop doing that. You can't possibly get the right answer if you ignore it when it's put right in front of your face.

 

[PeterDonis]

Bob and Alice also calculated the distance the elevator was travelling up and down, first they agreed,on calculation and right after they disagree, read my post above, so it cannot be limited to be a speed of light problem

Wrong. See post #32.

 

[Mikael17]

They don't agree on the elapsed time. They do agree on the distance travelled by the lift or light pulse. Therefore they don't agree about the speed.

I don't understand why this is difficult.

I can easily understand that. But the question is which parameter is what changes in a gravitational field if you compare Bob's and Alice's reality. We know that Alice's clock ticks slowing . We know d is the same, - The only possibility is therefore purely mathematical that m (the ruler) must also be a relative variable?

 

[Ibix]

But the question is which parameter is what changes in a gravitational field if you compare Bob's and Alice's reality.

The elapsed times. You just agreed that!