Time dilation vs the Equivalence principle

[Prometeus]

Some time ago there was a similar thread
https://www.physicsforums.com/threa...me-dilation-and-equivalence-principle.929838/

but what I want to discuss is similar but not the same and I would like to specify my question in such way, that it hopefully won't go sideways as in cited previous thread.

So we have two labs.

First lab is on surface of Earth, time dilation is equivalent to gravitational potential of surface of Earth, gravity is 1 g.

Second lab is on spaceship which is let's say 1 light year away and there is no star or planet around. The starship accelerates with proper acceleration 1 g and its speed relative to Earth is quite low, let's say 0,00001 % of speed of light. So time dilation relative to Earth caused by speed is close to zero. Due to relativity theory, acceleration causes no time dilation, so there all together there is no time dilation on the spaceship.

Now both labs have equipment (accelerator?) which can produce muons. Muons are created in both labs and they decay in time t1 in first lab (on Earth) and time t2 in second lab (in spaceship). Based on my understanding of relativity, gravitational time dilation on Earth is very slightly slowing down the decay of muons on Earth and therefore time t1 is bigger than time t2 on spaceship, where is no time dilation and muons decay faster than on Earth.

Just to make the example precise, muons don't jump up and down in the ship, so I will just ignore front and rear potential time difference in the spaceship.

So finally, based on Equivalence principle there should be no measurable differences between proper acceleration and gravity, but in the described example the scientist which knows how much time it takes for muon to decay on Earth can distinguish if the lab is on Earth or inside a spaceship.

What is wrong in my statement that you can use muons to measure difference between acceleration and gravity?

 

[stevendaryl]

First lab is on surface of Earth, time dilation is equivalent to gravitational potential of surface of Earth, gravity is 1 g.

Second lab is on spaceship which is let's say 1 light year away and there is no star or planet around. The starship accelerates with proper acceleration 1 g and its speed relative to Earth is quite low, let's say 0,00001 % of speed of light. So time dilation relative to Earth caused by speed is close to zero. Due to relativity theory, acceleration causes no time dilation, so there all together there is no time dilation on the spaceship.

When you say that there is or is not time dilation, that is a coordinate-dependent statement. You can turn it into a related coordinate-independent statement by operationalizing it. The way you might operationalize it on Earth is this:

• Take two identical clocks on the surface of the Earth and synchronize them.
• Leave one clock on the ground and carry the second clock to the top of a mountain, a height of $L$ above the ground.
• Let the two clocks run for 10 years (or however long).
• Now, bring the clock down from the mountain, and compare the elapsed times on the clocks.
• The clock that was on the mountain will shows $10 (1+\frac{gL}{c^2})$ years has passed, compared with $10$ for the clock that was on the ground the whole time. (where $g$ is the acceleration due to gravity and $c$ is the speed of light). The mountain clock will show a little more elapsed time.

Now, you can do the same experiment on an accelerating spaceship: synchronize two clocks at the rear of the spaceship. Carry one clock to the front of the spaceship. Wait 10 years. Then bring the front clock back to the rear. The clock that had been in the front will show $10 (1+\frac{gL}{c^2})$ years have passed, where $L$ is the length of the rocket ship, compared with $10$ for the clock in the rear.

So operationally, there is time dilation in both cases.

 

[p1l0t]

What is the difference between gravity and acceleration? I thought gravity was just acceleration caused by the curvature of spacetime or something.

 

[Prometeus]

When you say that there is or is not time dilation, that is a coordinate-dependent statement. You can turn it into a related coordinate-independent statement by operationalizing it. The way you might operationalize it on Earth is this:

• Take two identical clocks on the surface of the Earth and synchronize them.
• Leave one clock on the ground and carry the second clock to the top of a mountain, a height of $L$ above the ground.
• Let the two clocks run for 10 years (or however long).
• Now, bring the clock down from the mountain, and compare the elapsed times on the clocks.
• The clock that was on the mountain will shows $10 (1+\frac{gL}{c^2})$ years has passed, compared with $10$ for the clock that was on the ground the whole time. (where $g$ is the acceleration due to gravity and $c$ is the speed of light). The mountain clock will show a little more elapsed time.

Now, you can do the same experiment on an accelerating spaceship: synchronize two clocks at the rear of the spaceship. Carry one clock to the front of the spaceship. Wait 10 years. Then bring the front clock back to the rear. The clock that had been in the front will show $10 (1+\frac{gL}{c^2})$ years have passed, where $L$ is the length of the rocket ship, compared with $10$ for the clock in the rear.

So operationally, there is time dilation in both cases.

This is exactly the sideway discussion I have tried to avoid with very specific example. Please could you answer the question: Would be the time of decay of muons different?

 

[stevendaryl]

What is the difference between gravity and acceleration? I thought gravity was just acceleration caused by the curvature of spacetime or something.

That's a little backwards, in my opinion. The equivalences are:

• Freefall in a gravitational field is locally equivalent to floating in gravity-free space.
• Staying in place in a gravitational field of strength $g$ is locally equivalent to accelerating upward at rate $g$ in gravity-free space.

So it's not the gravity that is equivalent to acceleration---it's opposing gravity (thwarting the natural tendency to fall) that is equivalent to acceleration.

 

[PeroK]

What is the difference between gravity and acceleration? I thought gravity was just acceleration caused by the curvature of spacetime or something.

The precise statement of the "or something" is quite important.

 

[p1l0t]

That's a little backwards, in my opinion. The equivalences are:

• Freefall in a gravitational field is locally equivalent to floating in gravity-free space.
• Staying in place in a gravitational field of strength $g$ is locally equivalent to accelerating upward at rate $g$ in gravity-free space.

So it's not the gravity that is equivalent to acceleration---it's opposing gravity (thwarting the natural tendency to fall) that is equivalent to acceleration.

It's all relative I suppose.

 

[stevendaryl]

This is exactly the sideway discussion I have tried to avoid with very specific example. Please could you answer the question would be the time of decay of muons different?

But your discussion started out assuming that there was time dilation in one case but not in the other. That's not true. Operationally, there is time dilation in both cases, and they affect muons in exactly the same way (locally).

Why I put the qualifier "locally" in is because you can obviously tell the difference between a gravitational field and acceleration, because gravity various as $1/r^2$, but the g-forces on board a rocket do not. However, if you don't move too high above the surface of the Earth, the acceleration of gravity is approximately constant, and so is the g-forces aboard a rocket. So to a certain approximation, they're indistinguishable.

 

[stevendaryl]

Now both labs have equipment (accelerator?) which can produce muons. Muons are created in both labs and they decay in time t1 in first lab (on Earth) and time t2 in second lab (in spaceship). Based on my understanding of relativity, gravitational time dilation on Earth is very slightly slowing down the decay of muons on Earth and therefore time t1 is bigger than time t2 on spaceship, where is no time dilation and muons decay faster than on Earth.

Near the Earth, it's not correct to say that gravity causes time dilation in any absolute sense. What's relevant is comparing a clock at one height to a clock at another height. Gravitational time dilation says that the higher clock will show more elapsed time (when you bring the clocks back together to compare).

 

[Prometeus]

But your discussion started out assuming that there was time dilation in one case but not in the other. That's not true. Operationally, there is time dilation in both cases, and they affect muons in exactly the same way (locally).

Why I put the qualifier "locally" in is because you can obviously tell the difference between a gravitational field and acceleration, because gravity various as $1/r^2$, but the g-forces on board a rocket do not. However, if you don't move too high above the surface of the Earth, the acceleration of gravity is approximately constant, and so is the g-forces aboard a rocket. So to a certain approximation, they're indistinguishable.

Could you please elaborate a little more on the time dilation of muons. Why they should be locally time dilated in the same way? It contradicts experiments with muons:
https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation#Clock_hypothesis

 

[PeroK]

In another sense, gravity is not like acceleration. If you have two labs in space and one accelerates past the other and they compare clocks, then they will measure a symmetric, mutual time dilation that depends only on the varying relative speed between them.

If, however, one lab is in a gravitational field and the other isn't, they will measure an asymmetric time dilation, with the clock in the gravitational field running slower.

 

[Prometeus]

In another sense, gravity is not like acceleration. If you have two labs in space and one accelerates past the other and they compare clocks, then they will measure a symmetric, mutual time dilation that depends only on the varying relative speed between them.

If, however, one lab is in a gravitational field and the other isn't, they will measure an asymmetric time dilation, with the clock in the gravitational field running slower.

Thats what I am trying to say. But can the time dilation be used to distinguish between gravity and acceleration in a way I have described? And if yes, is this a "violation" of Einstein Equivalence principle?

 

[stevendaryl]

Could you please elaborate a little more on the time dilation of muons. Why they should be locally time dilated in the same way? It contradicts experiments with muons:
https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation#Clock_hypothesis

Well, the clock hypothesis is about clocks in gravity-free space, so it doesn't apply to clocks on the surface of the Earth.

On board a rocket, you can analyze the operational comparison of the clocks using Special Relativity:

• Initially, the rocket is at rest in some frame (the "launch frame"), and the two clocks are synchronized.
• The rocket launches and starts accelerating.
• According to relativity, the length of the rocket starts shrinking (Assuming Born-rigid motion: If in the rocket's own reference frame, the rocket has a constant length, then in the launch frame, the rocket will be length-contracted. The faster the rocket goes, the more length-contracted it will be.)
• Because of length contraction, the front of the rocket doesn't travel quite as fast as the rear of the rocket, according to the "launch" frame. This is just elementary logic: If the front and rear start off a certain distance apart, the length of the rocket, and later they are a different distance apart, the contracted length, then that means that the front and the rear have traveled at different speeds. In particular, if the length contracts, it means that, according to the launch frame, the rear of the rocket has traveled faster than the front, by a small amount.
• Therefore, a clock in the rear of the rocket will experience more time dilation than a clock in the front.
• So when the clocks are brought back together, the clock in the front will show more elapsed time.

This is all perfectly consistent with the clock hypothesis.

 

[stevendaryl]

Thats what I am trying to say. But can the time dilation be used to distinguish between gravity and acceleration in a way I have described? And if yes, is this a "violation" of Einstein Equivalence principle?

Absolutely not. Einstein DERIVED gravitational time dilation by first showing that clocks on an accelerating rocket (or actually, an elevator in his thought experiment) would experience time dilation, and then using the equivalence principle to predict that the same effect would be observed by clocks in a gravitational field.

So gravitational time dilation can't possibly contradict the equivalence principle since it was derived using the equivalence principle.

 

[PeroK]

Thats what I am trying to say. But can the time dilation be used to distinguish between gravity and acceleration in a way I have described? And if yes, is this a "violation" of Einstein Equivalence principle?

It's not because the equivalence principle deals with local experiments in an accelerating reference frame. The emphasis is local. If an external non-accelerating observer is involved they can easily tell whether you are accelerating or in a gravitational field.

 

[Dale]

Now both labs have equipment (accelerator?) which can produce muons. Muons are created in both labs and they decay in time t1 in first lab (on Earth) and time t2 in second lab (in spaceship). Based on my understanding of relativity, gravitational time dilation on Earth is very slightly slowing down the decay of muons on Earth and therefore time t1 is bigger than time t2 on spaceship, where is no time dilation and muons decay faster than on Earth.

No, t1=t2 in this scenario.

 

[stevendaryl]

No, t1=t2 in this scenario.

Of course, that's the simple answer---regardless of what gravity or acceleration does to clocks, if it affects muons and clocks the same way, then you'll measure the same decay times.

 

[Dale]

if it affects muons and clocks the same way, then you'll measure the same decay times.

Yes, precisely. The observer at infinity will detect a difference, but the local clocks will not.

The view of the remote observer is not covered by the equivalence principle since the Earth's gravitational field is not uniform that far away. I.e. tidal effects are not negligible.

 

[Prometeus]

Well, the clock hypothesis is about clocks in gravity-free space, so it doesn't apply to clocks on the surface of the Earth.

On board a rocket, you can analyze the operational comparison of the clocks using Special Relativity:

• Initially, the rocket is at rest in some frame (the "launch frame"), and the two clocks are synchronized.
• The rocket launches and starts accelerating.
• According to relativity, the length of the rocket starts shrinking (Assuming Born-rigid motion: If in the rocket's own reference frame, the rocket has a constant length, then in the launch frame, the rocket will be length-contracted. The faster the rocket goes, the more length-contracted it will be.)
• Because of length contraction, the front of the rocket doesn't travel quite as fast as the rear of the rocket, according to the "launch" frame. This is just elementary logic: If the front and rear start off a certain distance apart, the length of the rocket, and later they are a different distance apart, the contracted length, then that means that the front and the rear have traveled at different speeds. In particular, if the length contracts, it means that, according to the launch frame, the rear of the rocket has traveled faster than the front, by a small amount.
• Therefore, a clock in the rear of the rocket will experience more time dilation than a clock in the front.
• So when the clocks are brought back together, the clock in the front will show more elapsed time.

This is all perfectly consistent with the clock hypothesis.

I would like to give emphasis on the experiments described in the link and not on the clock hypothesis which is limited to Special relativity. What I am trying to say, that experiments show no effect of acceleration on time dilation (from whatever theoretical source). It is not about front and rear of rocket which is irrelevant in this specific case.

 

[Prometeus]

No, t1=t2 in this scenario.

I appreciate your specific answer, but I think experiments show that t1 is different to t2. If t1 would be the same as t2 it would mean that acceleration is having an effect on time dilation which was not confirmed by experiments.

 

[stevendaryl]

I would like to give emphasis on the experiments described in the link and not on the clock hypothesis which is limited to Special relativity. What I am trying to say, that experiments show no effect of acceleration on time dilation (from whatever theoretical source). It is not about front and rear of rocket which is irrelevant in this specific case.

Then what you're talking about has nothing to do with gravitational time dilation or the equivalence principle. Gravitational time dilation is about comparing clocks at different locations within a gravitational field. Muon decay works like a clock, it will be observed to happen slower the lower the muon is in a gravitational field.

The equivalence principle says the same thing about accelerating rockets: muons at the rear of a rocket decay slower than muons at the front of the rocket.

 

[stevendaryl]

I appreciate your specific answer, but I think experiments show that t1 is different to t2. If t1 would be the same as t2 it would mean that acceleration is having an effect on time dilation which was not confirmed by experiments.

You're asking questions, but you aren't accepting the answers. You are misunderstanding time dilation and muon experiments.

What relativity predicts is that muons decay slower the lower in a gravitational field than they do higher in a gravitational field, and it predicts that the same is true of muons in the front/rear of rockets. That's what the equivalence principle says.

No offense, but you're asking questions that you don't have the background to understand. People have tried to give you the information, but you have rejected it. So that makes it frustrating.

 

[stevendaryl]

You're asking questions, but you aren't accepting the answers. You are misunderstanding time dilation and muon experiments.

What the experiments show is that muons decay slower the lower in a gravitational field than they do higher in a gravitational field, and the same is true of muons in the front/rear of rockets. That's what the equivalence principle says.

I want to make a correction. I don't think that any experiments have shown the effect of gravitational time dilation on muon decay. What has been shown is just regular velocity-dependent time dilation, and the gravity of the Earth has a negligible effect. So I think that the basis for your original question is not well-founded.

 

[Dale]

but I think experiments show that t1 is different to t2.

Which specific experiment? It would probably be a famous experiment since it would show a violation of SR or GR or both.

If t1 would be the same as t2 it would mean that acceleration is having an effect on time dilation which was not confirmed by experiments.

No. It would mean that clocks are time dilated by gravity in the same manner as muons.

 

[Prometeus]

I want to make a correction. I don't think that any experiments have shown the effect of gravitational time dilation on muon decay. What has been shown is just regular velocity-dependent time dilation, and the gravity of the Earth has a negligible effect. So I think that the basis for your original question is not well-founded.

Good point. Experiments with muons showed only velocity dependent time dilation and absence of time dilation in acceleration. Time dilation effect in gravity of the Earth would be negligible and it was only my conjecture that there should be also time dilation effect.

 

[Janus]

Some time ago there was a similar thread
https://www.physicsforums.com/threa...me-dilation-and-equivalence-principle.929838/

but what I want to discuss is similar but not the same and I would like to specify my question in such way, that it hopefully won't go sideways as in cited previous thread.

So we have two labs.

First lab is on surface of Earth, time dilation is equivalent to gravitational potential of surface of Earth, gravity is 1 g.

Second lab is on spaceship which is let's say 1 light year away and there is no star or planet around. The starship accelerates with proper acceleration 1 g and its speed relative to Earth is quite low, let's say 0,00001 % of speed of light. So time dilation relative to Earth caused by speed is close to zero. Due to relativity theory, acceleration causes no time dilation, so there all together there is no time dilation on the spaceship.

Now both labs have equipment (accelerator?) which can produce muons. Muons are created in both labs and they decay in time t1 in first lab (on Earth) and time t2 in second lab (in spaceship). Based on my understanding of relativity, gravitational time dilation on Earth is very slightly slowing down the decay of muons on Earth and therefore time t1 is bigger than time t2 on spaceship, where is no time dilation and muons decay faster than on Earth.

Just to make the example precise, muons don't jump up and down in the ship, so I will just ignore front and rear potential time difference in the spaceship.

So finally, based on Equivalence principle there should be no measurable differences between proper acceleration and gravity, but in the described example the scientist which knows how much time it takes for muon to decay on Earth can distinguish if the lab is on Earth or inside a spaceship.

What is wrong in my statement that you can use muons to measure difference between acceleration and gravity?

The big problem is where you state that you will ignore the front and rear potential time difference in the spaceship. You can't make a comparison to gravitational time dilation at the Earth's surface without taking into a account this potential time difference, Because gravitational time dilation is a measurement of clocks at different potentials. To make any comparison, you would have to do the following type of experiment:

At the Earth, you have two identical clocks. One sitting on the surface and one 10 meters above it. On the ship, you have two identical clocks, one at the rear and one 10 meters closer to the nose. You then compare how much the clocks on Earth differ after the lower clock has ticked off some interval to how much the clocks on the ship differ after the rear clock ticks off the same interval.

This difference will not be the same, it will turn out that the clocks on the ship will differ from each other just a tad more than those on the Earth. This is because the potential between the clocks is the equivalent of that for a gravity field that maintains a constant 1g over the distance separating the clocks, while on the Earth, gravity falls off slightly between the two heights of the clocks. Thus you get a larger potential difference in the ship than you do for the Earth.

This does not violate the Equivalence principle however. For example, what if I have another two clock, again 10 meters apart, but this time the lower clock is twice the radius of the Earth away from a planet with 4 times the Earth's mass. That lower clock would be at 1g of gravity just like the Earth surface clock is. However, since 10 meters is a smaller fraction of 2 Earth radii than 1 Earth radius, gravity will fall off less between these two clocks, and the potential between them will be larger than that for the Earth clocks, and thus the difference in tick rate between this pair will be larger than for the Earth pair. In both cases the differential tick rate is caused by gravitational time dilation, and even though the lower clock of each pair is experiencing the same force of gravity, the differential between of the clocks of each pair is different.

You will also notice that our new pair of clocks more closely matches the clocks in the spaceship than the Earth clocks do. If we increase their distance from the planet while simultaneously increasing the mass of the planet in order to maintain 1g at the lower clock, the behavior of these clocks become closer and closer to that of those in the ship. The behavior of these two set of clocks begin to converge. At some point they will become all but indistinguishable. This is the main point of the equivalence principle: that over a small enough region gravity and acceleration are equivalent.

 

[PeterDonis]

absence of time dilation in acceleration

The muon experiments show that acceleration itself has no effect on clock rates. But all of the muons in those experiments (the ones where the muons were trapped in a storage ring) were at the same "height" relative to the acceleration, so these experiments say nothing about the rates of clocks at different "heights" in an accelerating rocket. Other experiments, such as the Pound-Rebka experiment, clearly show that clocks at different heights on Earth run at different rates, even though they are at rest relative to each other; and by the equivalence principle, the same will be true of clocks at different heights in a rocket accelerating at 1 g in free space. This difference in clock rates is not due to a difference in accelerations; it's due to a difference in height.

 

[Prometeus]

The muon experiments show that acceleration itself has no effect on clock rates. But all of the muons in those experiments (the ones where the muons were trapped in a storage ring) were at the same "height" relative to the acceleration, so these experiments say nothing about the rates of clocks at different "heights" in an accelerating rocket. Other experiments, such as the Pound-Rebka experiment, clearly show that clocks at different heights on Earth run at different rates, even though they are at rest relative to each other; and by the equivalence principle, the same will be true of clocks at different heights in a rocket accelerating at 1 g in free space. This difference in clock rates is not due to a difference in accelerations; it's due to a difference in height.

Yes, I am writing specifically about zero height difference and pure acceleration effect and everybody seems like repeating the only mantra they know - the difference in height in the rocket. Seems like common obsession of all who had studied physics. It surely has some hidden Freudian meaning :)

 

[stevendaryl]

Yes, I am writing specifically about zero height difference and pure acceleration effect and everybody seems like repeating the only mantra they know - the difference in height in the rocket. Seems like common obsession of all who had studied physics. It surely has some hidden Freudian meaning :)

Well, you were asking about gravity's effect on time dilation. It has no measurable effect unless you compare time rates at different heights. So the reason people kept bringing that up was because you seemed to think that gravity was involved somehow.

It's very annoying--you keep asking about gravity's effect on time dilation, and people keep telling you, and you keep saying: I don't mean that. Well, there is no other effect.

 

[sweet springs]

Say muon lifetime is 2.2 micro second, both the Earth lab and the 1g accelerating rocket observe the same value using their clocks.
Time dilation works when we compare two positions in gravitational field, e.g. surface vs center of the Earth, upward vs downward position of the accelerating rocket.
For an example you should provide more information, height or depth of the Earth observed in the rocket system and also its speed, for time comparison of the two.

 

[Dale]

Good point. Experiments with muons showed only velocity dependent time dilation and absence of time dilation in acceleration. Time dilation effect in gravity of the Earth would be negligible and it was only my conjecture that there should be also time dilation effect.

Nevertheless, we can analyze what SR/GR would predict. It predicts that they would be equal.

Yes, I am writing specifically about zero height difference and pure acceleration effect and everybody seems like repeating the only mantra they know - the difference in height in the rocket. Seems like common obsession of all who had studied physics. It surely has some hidden Freudian meaning :)

It means that experts naturally assume that if you are asking about gravitational time dilation then you would want to discuss a scenario where there would be measurable gravitational time dilation. Your proposed experiment is not sensitive to the acceleration/gravity.

 

[Janus]

Yes, I am writing specifically about zero height difference and pure acceleration effect and everybody seems like repeating the only mantra they know - the difference in height in the rocket. Seems like common obsession of all who had studied physics. It surely has some hidden Freudian meaning :)

Zero height difference in the rocket is the equivalent of zero height distance on the Earth. Zero height difference in a gravitational field produces zero time dilation. Neither gravity or acceleration produce time dilation in of themselves. Time dilation in both cases is only caused by a difference in potential. Either different altitudes in the Earth's gravity or different positions in the accelerating frame of the rocket.
When someone says that the time dilation at the Earth's surface has some value, they mean they mean compared to a point that is at a maximum gravitational potential relative to the Earth. (By convention we set this maximum as being zero, with all potentials closer to the Earth having negative values. Thus the specific gravitational potential at the Earth surface is ~ -62511759 j/kg

 

[PeterDonis]

Im writing specifically about zero height difference and pure acceleration effect

And we're telling you, repeatedly, that there is no such thing. So your question is answered. And there is no point in continuing to go around in circles about it.

Thread closed.