Equivalence of Clocks in Gravitational Fields: A Thought Experiment

[yogi]

In a G field, clocks at a lower potential (closer to the mass producing the field) are known to run slower. When the two clocks are brought togther, the upper clock should be found to have accumulated more time than the lower clock.

A rocket accelerating at "a" is equivalent to a G field during the acceleration phase i.e., a nose clock in the rocket will observe the frequency of signals transmitted from a lower tail clock to be reduced in frequency, and the lower tail clock will see signals transmitted by the upper clock arriving at higher frequency. The two experients are equivalent during the acceleration phase. When the rocket stops accelerating, the two clocks are then in the same frame and their readings can be compared

For complete equivalence - the nose clock and the tail clock should show a different reading after the experiment just as do G field clocks emersed for an extended period at different gravitational potentials. Can anyone point to a reference that deals with an experiment designed to measure accumulated time differences between rigidly separated clocks undergoing identical uniform accelerations

 

[meopemuk]

In a G field, clocks at a lower potential (closer to the mass producing the field) are known to run slower. When the two clocks are brought togther, the upper clock should be found to have accumulated more time than the lower clock.

A rocket accelerating at "a" is equivalent to a G field during the acceleration phase i.e., a nose clock in the rocket will observe the frequency of signals transmitted from a lower tail clock to be reduced in frequency, and the lower tail clock will see signals transmitted by the upper clock arriving at higher frequency. The two experients are equivalent during the acceleration phase. When the rocket stops accelerating, the two clocks are then in the same frame and their readings can be compared

For complete equivalence - the nose clock and the tail clock should show a different reading after the experiment just as do G field clocks emersed for an extended period at different gravitational potentials. Can anyone point to a reference that deals with an experiment designed to measure accumulated time differences between rigidly separated clocks undergoing identical uniform accelerations

I don't think such an experiment has ever been performed. It would encounter substantial technical difficulties. A very long rocket and very high accelerations would be needed to see any visible effect.

You can find an (incomplete) list of references to experimental measurements of the gravitational time dilation and redshift on Earth and in space in
http://www.arxiv.org/physics/0612019

Eugene.

 

[Garth]

I don't think such an experiment has ever been performed. It would encounter substantial technical difficulties. A very long rocket and very high accelerations would be needed to see any visible effect.
Eugene.

In fact an experiment was performed in 1976 and called Gravity Probe A, although the two clocks were situated with one in the rocket and the other on the ground.

Garth

 

[yogi]

Thanks Eugene. I am not surprised considering the degree of difficulty. I wonder if any meaningful data could be obtained using a large rotating disc with one clock mounted at the perhipery and a second clock at a radial position of 1/2 the perhipery - when the disc is rotated at a constant angular rate, the two clocks are subjected to different acceleration potentials as well as different velocities - after subtracting out the SR velocity effects, it would seem there should be a difference in the total amount of accumulated time between the two clocks. But if I remember correctly, these type of centrifuge experiments do not reveal time dilation other than that attributable to the velocity as per SR. Don't know whether that conclusion was arrived at by taking data while the disc was in motion or by comparing accumulated times logged by the clocks after the disc is brought to rest.

 

[yogi]

Garth - that is interesting - it may be the closest thing that has been done along this line...but I may be not be correctly reading the result. As I understand the experiment, the measurements were made while the rocket was accelerating, specifically "The clock rate was measured for most of the duration of the flight and compared to theoretical predictions." This validates the equivalence principle during the dynamic phase. But is there a residual time difference between two separated clocks undergoing the same acceleration. In other words, we make a comparison between the times based upon gh/c^2 during flight - but this does not show that the two clocks are running at different rates - only that there will be an observational difference - not a permanent age difference as is the case with clocks in different gravitational potentials for extended periods. If the lower clock were actually running at a slower rate than the upper clock, the effect would be doubled since each pulse is delayed by gh/c^2 plus it would also have an added delay determined by the lower emission frequency of the lower clock.

 

[meopemuk]

Thanks Eugene. I am not surprised considering the degree of difficulty. I wonder if any meaningful data could be obtained using a large rotating disc with one clock mounted at the perhipery and a second clock at a radial position of 1/2 the perhipery - when the disc is rotated at a constant angular rate, the two clocks are subjected to different acceleration potentials as well as different velocities - after subtracting out the SR velocity effects, it would seem there should be a difference in the total amount of accumulated time between the two clocks. But if I remember correctly, these type of centrifuge experiments do not reveal time dilation other than that attributable to the velocity as per SR. Don't know whether that conclusion was arrived at by taking data while the disc was in motion or by comparing accumulated times logged by the clocks after the disc is brought to rest.

The question about the action of acceleration on the rate of clocks always puzzled me. The ultimate experiment of this kind was performed at CERN in 1970's. They accelerated a pulse of muons in a cyclotron ring and measured the increase of the muons' lifetime. They found that in full agreement with the velocity effect of special relativity the lifetime increased 27x. However, surprisingly, no effect of acceleration on the lifetime was found. This was in spite of really huge accelerations of the order of 10^18 g. Apparently
acceleration had no effect on the clock's rate. I read in many places that this doesn't contradict the principle of equivalence, but I just can't understand why?

Eugene.

 

[meopemuk]

If the lower clock were actually running at a slower rate than the upper clock, the effect would be doubled since each pulse is delayed by gh/c^2 plus it would also have an added delay determined by the lower emission frequency of the lower clock.

Could you please explain a bit more why you think there should be a double effect?

Eugene.

 

[yogi]

Eugene - the double effect could only happen if there was an actual alteration of the emission frequency at the source - and one would have to conclude that acceleration somehow affects time. But this doesn't occur, at least there does not appear to be any experimental evidence that the lower clock would be affected because it was subjected to a different acceleration potential (That is confirmed by the Gravity A experiment cited by Garth). Moreover, there does not seem to be any physical reason that would convey to the rocket clocks that they are in different gravitational potentials. So Equivalence in the sense of a closed elevator seems to be limited only to the duration of acceleration - as in SR each clock runs at the same rate in its own frame and the time difference is an observational one. What bothers me is the G field - an uncompensated GPS clock will gain 38 usec per day because of its altitude. This ongoing accumulation of additional time by the clock that is furtherest removed from the gravitational source would be measured as a real age difference between the two clocks when they are brought together for comparison at some later date.

I guess I am questioning whether the mechanism that leads to time dilation in a G field is the same as that involved in other accelerations.

 

[Voltage]

You'll never obtain experimental evidence for this, yogi, because it's based upon a misconception. The principle of equivalence does not confer absolute equivalence. In the accelerating rocket, your two clocks experience the same acceleration. In the rocket standing on the surface of the earth, they do not. They can only experience the same acceleration if they're in what's called a uniform gravitational field, and in the real world, gravitational fields are not uniform.

 

[meopemuk]

What bothers me is the G field - an uncompensated GPS clock will gain 38 usec per day because of its altitude. This ongoing accumulation of additional time by the clock that is furtherest removed from the gravitational source would be measured as a real age difference between the two clocks when they are brought together for comparison at some later date.

I think that gravitational time dilation is an absolute effect: all observers would agree that clocks in space tick faster than identical clocks on the Earth surface. If you bring the GPS satellite back to Earth, you'll see a real effect of extra aging of its clock.

I know one experiment in which two sets of atomic clocks were used: one clock at the ground level and another clock high in the mountains. After some time the mountain clock was transported back to the valley and readings of both clocks were compared side-by-side. The gravitational time dilation was confirmed with the accuracy of 10-20%.

L. Briatore and S. Leschiutta, "Evidence for the Earth gravitational shift by direct atomic-time-scale comparison", Nuovo Cimento B, 37 (1977), 219

Eugene.

 

[yogi]

You'll never obtain experimental evidence for this, yogi, because it's based upon a misconception. The principle of equivalence does not confer absolute equivalence. In the accelerating rocket, your two clocks experience the same acceleration. In the rocket standing on the surface of the earth, they do not. They can only experience the same acceleration if they're in what's called a uniform gravitational field, and in the real world, gravitational fields are not uniform.

That was my thought also - in the G field (rocket at rest on the Earth's surface) there is not only a difference in potential, but there is a difference in force acting upon the upper and lower clocks. In the free space accelerating rocket, there is only a difference in potential. So does this lead to a proposition that says: Time dilation for the rocket sitting on the Earth is real and permanent, whereas time dilation for the free space accelerating rocket is apparent only? If this is true, then, as you say, "there is no absolute equivalence." The observer in the sealed elevator can thus determine which kind of field he is subjected too by using two clocks - one on the floor - one the ceiling - if the operator monitors them from a midpoint and they read the same after a sufficient period, the elevator cannot be in a G field.

Of course, this same result can be arrived at from the divergence of the G field - and while tidal and divergence effects distinguish uniform fields from mass created attractions, the force differences are usually regarded as indicative rather than causal. Here we seem to be dealing with a change in principle - in a G field the clock rate (and hence time) appears to be substantively affected.

 

[CaptainQuasar]

I think what he's saying is that because a real gravity field is a gradient, two clocks at different points in the gradient isn't the same thing as two clocks accelerating at the exact same rate. I bet you that if you varied the acceleration slightly between the two clocks - in fact, if you varied it in proportion to the ratio of the force of gravity on the two clocks in the gravity field - you would get the same amount of time dilation.

You'd need a telescoping rocket. Or better yet, one that was like a Chinese yo-yo. But then you'd have to account for the angular acceleration, too. Hmm... two Chinese yo-yo's, wound in opposite directions, attached end-to-end! But someone would put their eye out.

 

[Voltage]

I wouldn't say "apparent", yogi, perhaps "relative" is a better word. If you and I passed each other in the dark depths of space I'd say your time was dilated and you'd say mine was, and we might get into an argument about who's time was really dilated. But as meopemuk says above, if I was on a planet and you were up in space, we'd both agree that my time was dilated. We'd say it was absolute rather than relative. As to whether it's correct to say that time dilation is real and permanent in a gravity situation, but is not in the accelerating elevator situation, is debateable. Personally I wouldn't describe this as a change in principle. I'd say the principle of equivalence still applies, and whilst it isn't a total exact equivalence, the time dilation still occurs and is measurable in both situations, even though one situation lacks some agreed baseline.

PS: I believe the observer in the sealed elevator can perform a Pound-Rebka experiment to determine his situation, but please check this independently.

 

[meopemuk]

Time dilation for the rocket sitting on the Earth is real and permanent, whereas time dilation for the free space accelerating rocket is apparent only?

I think this is a very good point. I think an observer in a sealed elevator cabin should be able to decide whether the cabin is accelerating or standing still in a gravitational field. To do that he would need to place identical clocks at the ceiling and on the floor of the cabin. Then wait for a while. Then bring these clocks together and compare their readings. If their readings are the same, then the cabin was accelerating. If the ceiling clock shows later time, then he was in the gravity field. If the floor clock shows later time, then he needs to wake up.

Eugene.

 

[meopemuk]

PS: I believe the observer in the sealed elevator can perform a Pound-Rebka experiment to determine his situation, but please check this independently.

In my understanding, Pound-Rebka experiment in an accelerated elevator cabin will show the same result as in the gravity field. This is usually shown by Doppler-effect-like arguments. I think yogi is right that Briatore-Leschiutta type experiment may do the trick and distinguish between acceleration and gravity.

Eugene.

 

[Ich]

Acceleration and gravity are exactly equivalent (in a differential way). They are the same. Different acceleration at different points changes the result only quantitatively. The clocks in the elevator would read different times, just as clocks on Earth would.

 

[Voltage]

In my understanding, Pound-Rebka experiment in an accelerated elevator cabin will show the same result as in the gravity field. This is usually shown by Doppler-effect-like arguments. I think yogi is right that Briatore-Leschiutta type experiment may do the trick and distinguish between acceleration and gravity.

Noted, Eugene. Your comment also noted Ich. How does one get an expert to chip in on this?

 

[Ich]

This is introductory level SR, an area which I would claim to have mastered meanwhile.
Of course you don't have to believe me, it is enough to draw a spacetime diagram and see that events of equal proper time are no longer simultaneous in a comoving frame.
I might add that I don't know the "Briatore-Leschiutta experiment". I'm talking about my comment "The clocks in the elevator would read different times, just as clocks on Earth would."

 

[CaptainQuasar]

Acceleration and gravity are exactly equivalent (in a differential way). They are the same. Different acceleration at different points changes the result only quantitatively. The clocks in the elevator would read different times, just as clocks on Earth would.

The effect of a given spacetime curvature due to gravity - a given value of the Riemann tensor - and its equivalent rate of acceleration are the same. But gravity is a gradient - it shows up as a field and varies across space - and acceleration does not, at least not in an elevator.

What do you mean "only quantitatively"? Quantitative things are all we're talking about - things which can be measured by numbers. Apart from what you're going to name your quarks, quantitative is all there is in physics.

 

[CaptainQuasar]

This is introductory level SR, an area which I would claim to have mastered meanwhile.

Special relativity is a constrained case of general relativity that does not involve acceleration. There's no changing between inertial reference frames in special relativity. This is GR.

(I am not an expert either, BTW, but I know the difference between SR and GR.)

 

[Ich]

But gravity is a gradient - it shows up as a field and varies across space - and acceleration does not, at least not in an elevator.

Well, it does. See Bell's spaceship paradox.
I'm not a native speaker; what I meant to say with "in a differential way" is "in the limit of small regions of spacetime". Simply the equivalence principle.

What do you mean "only quantitatively"?

I mean that small deviations from uniform acceleration will produce only small errors when you omit these deviations in your calculation. There is no reason to expect a fundamentally different result like "no time dilation at all".

 

[Ich]

There's no changing between inertial reference frames in special relativity. This is GR.

No, that is still SR. You might use the mathematical formalism of GR with advantage, but you don't have to. The shifting of simultaneitiy which produces the effect in question is not more than basic SR.

 

[CaptainQuasar]

I mean that small deviations from uniform acceleration will produce only small errors when you omit these deviations in your calculation. There is no reason to expect a fundamentally different result like "no time dilation at all".

But there isn't any time dilation between subjects in the same inertial frame. Clocks in the accelerating elevator would be in the same inertial frame at all times. Time dilation occurs between different inertial frames, which is why you would need to accelerate them at different rates - to put them in different inertial frames and produce the same effect as the other clocks being at points in the gravity field where the curvature of spacetime is different.

Bravo for coming to debate physics in a 2nd language, BTW.

 

[CaptainQuasar]

No, that is still SR. You might use the mathematical formalism of GR with advantage, but you don't have to. The shifting of simultaneitiy which produces the effect in question is not more than basic SR.

So the time dilation between the clocks in the gravity well - you're saying that's special relativity too? I have never seen any discussion of gravity in a text on special relativity - certainly not in an introductory one, considering that explaining it involves tensor calculus - but perhaps I am sheltered.

Also, you mentioned drawing spacetime diagrams above. I am familiar with Minkowski diagrams for relating events and observers in different inertial frames, but I don't know of a way to represent acceleration in them.

 

[Ich]

Clocks in the accelerating elevator would be in the same inertial frame at all times.

An accelerating frame is of course not inertial. You could describe it by ever changing inertial frames ("comoving" frames), which is where things get tricky. For example, if you want to keep distances in all comoving frames constant (which is necessary to define the comoving frames properly), you will find that you have to apply different accelerations to each point of the system.
But this is going too far, I would still recommend that you read about "Bell's spaceship" if you're interested.

...to put them in different inertial frames and produce the same effect as the other clocks being at points in the gravity field where the curvature of spacetime is different.

Different curvature is not necessary. Even in GR, time dilation is not a local property of spacetime (but curvature is). It is defined only as a relation between two points, its magnitude (in small fields) is proportional to the difference in gravitational potential, not to its first or second derivative (gravitational acceleration or tidal acceleration respectively, where tidal acceleration corresponds to curvature). A "difference in potential" is also present in flat spacetime when you change to accelerating frames.

Bravo for coming to debate physics in a 2nd language, BTW.

Thanks, I do my best to improve my skill in physics as well as in English. Still a long way to go.

 

[Ich]

So the time dilation between the clocks in the gravity well - you're saying that's special relativity too?

Nope, but the time dilation in acceleratin frames is. It's just a bit hard to accelerate in every direction simultaneously without exploding, that's where tensor calculus comes in.

Also, you mentioned drawing spacetime diagrams above. I am familiar with Minkowski diagrams for relating events and observers in different inertial frames, but I don't know of a way to represent acceleration in them.

You draw two starting points, say 1 space unit apart. Then you draw to identical curved lines, which resemble qalitatively the world line of an accelerated body (the are hyperbolae in fact, but the exact shape doesn't matter). After time t0 in the initial rest frame, you stop accelerating and continue with a straight line. Note that the stop events occur at the same proper time for both observers.
Now compare the stop events in the frame where both observers are at rest after acceleration. You will find different times, which means that when you bring both clocks slowly together, they will read different times.
You will also find that the distance between the observers did increase in their new frame. Both effects are "real".

 

[CaptainQuasar]

Well, I concede! The references I'm finding concur that the effect of acceleration is a gradient or differential effect across the entire body. Thank you for pointing out Bell's spaceship paradox, it was interesting to read about.

But if I can make excuses for myself to save face - this doesn't appear to be introductory stuff! What I read of Bell's paradox said that many learned physicists, even at places like CERN, do not accept his solution to the problem. (Though the majority agree with him.)

 

[HallsofIvy]

In a G field, clocks at a lower potential (closer to the mass producing the field) are known to run slower. When the two clocks are brought togther, the upper clock should be found to have accumulated more time than the lower clock.

A rocket accelerating at "a" is equivalent to a G field during the acceleration phase i.e., a nose clock in the rocket will observe the frequency of signals transmitted from a lower tail clock to be reduced in frequency, and the lower tail clock will see signals transmitted by the upper clock arriving at higher frequency. The two experients are equivalent during the acceleration phase. When the rocket stops accelerating, the two clocks are then in the same frame and their readings can be compared

For complete equivalence - the nose clock and the tail clock should show a different reading after the experiment just as do G field clocks emersed for an extended period at different gravitational potentials. Can anyone point to a reference that deals with an experiment designed to measure accumulated time differences between rigidly separated clocks undergoing identical uniform accelerations

This not correct. Yes, "a rocket accelerating at "a" is equivalent to a G field during the acceleration phase" but it is a uniform field, not one in which being "closer to the mass producing the field" increases the potential. The two clocks in the rocket will have the same "potential" and so the same time rate. Of course, that is not true for one clock in the rocket and the otheer remaining on the ground.

 

[Ich]

The two clocks in the rocket will have the same "potential" and so the same time rate.

They have the same acceleration, hence their potential (the integral over acceleration) is different.

@CaptainQuasar: I concede, too. This is not really introductory stuff, though it was mentioned in my standard textbook at university. To draw a slanted line of simultaneity in the diagram I described surely is, but after studying a subject some things might seem obvious which are not in fact.

 

[Voltage]

Thanks for that, Halls of Ivy. Can you clarify this point: If I was in the rocket conducting a Pound-Rebka experiment, would I be able to tell whether the rocket was sitting on the ground rather than accelerating through space?