Equivalence of Clocks in Gravitational Fields: A Thought Experiment

[K.J.Healey]

Do you mean force or potential?

I meant potential and I meant 1/R, which still isn't linear.

 

[paw]

I've followed this thread with interest. It seems like there is a lot of disagreement as to whether there is a 'gravitational potential' in an accelerating 'elevator' type of experiment. I just found this article at Wiki that seems to claim there is.

Strong Eqivalence Principle

Einstein combined the equivalence principle with special relativity to predict that clocks run at different rates in a gravitational potential, and light rays bend in a gravitational field, even before he developed the concept of curved spacetime. It is important to note that any accelerated frame of reference has a gravitational potential associated with it. Therefore clocks displaced in the direction of acceleration with respect to an accelerating rocket will be found to be going faster or slower by the observer in the accelerating rocket in accord with gravitational time dilation. The same applies to other gravitaitional effects such as gravitational red shifting and the bending of light.

So the original equivalence principle, as described by Einstein, concluded that free fall and inertial motion were physically equivalent. This form of the equivalence principle can be stated as follows. An observer in a windowless room cannot distinguish between being on the surface of the Earth, and being in a spaceship in deep space accelerating at 1g. This is not strictly true, because massive bodies give rise to tidal effects (caused by variations in the strength and direction of the gravitational field) which are absent from an accelerating spaceship in deep space.

While I wouldn't ever cite Wiki as a definitive source it seems like this quote clearly claims an accelerated frame DOES have a gravitational potential associated with it. Does this seem correct?

 

[pervect]

The full theory of GR doesn't really offer any exact defintion of "potential". The Newtonian concept of potential is used is some linearized approximate versions of GR. For some idea of what's going on I'd suggest reading Steve Carlip's usenet posts on the topic. (Why Steve Carlip? He's a recognized authority on gravity (see http://www.physics.ucdavis.edu/Text/Carlip.html#Honors) a good writer, and he's written enough about the topic in public venues like usenet, that you can often find things he's said about a particular topic.

So, this advice takes us to

http://groups.google.com/group/sci....d099b?lnk=st&q=&rnum=2&hl=en#30482700341d099b

which says in part:

1. In the weak field approximation, the time-time component of
the metric (in a ``nearly rectilinear coordinate system'') depends
linearly on the Newtonian gravitational potential, and you can
read off the potential energy from that.

This actually works in this case, if you take "zero energy" as being someone at the origin of the coordinate system. You have a metric coefficeint of (1+gh)^2 and get a "potential" of gh. But to make this meaningful, you'd have to describe how you got it, just talking about "gravitational potential" in the full theory as if it were defined by the theory is a mistake - it is not a part of the theory.

In Newtonian theory, and also in electromagnetism, the notion of potential works so well that one can reduce the problem of solving a problem of the Newtonian gravitational field or the electromagnetic field to a simple scalar partial differential equation.

This is not the case in GR - one can represent GR by a family of differential equations, there is no known way to reduce it to a scalar equation in terms of a singe scalar potential from which the "field quantities" can be derived.

 

[yogi]

Well, c=1, G=1, and the sun has a mass of 3 km. How would you derive that from cosmological properties?

When you disregard the units you throw away valuable information - this is one of the areas where modern physics has handicapped itself.

 

[yogi]

Different folks have weighed in on the question of whether two spaced apart clocks in an accelerating free space rocket will read different values when combined. While the rocket experiment is difficult to perform, I do not see why a spinning disk could not be used with 3 clocks- the first (R/0) at the center, the second at radial distance R/2 and the third at R. Run the disk for an extended period and compare the clocks at R/2 and R to the center clock from time to time while the disk is spinning. My guess is that the R/2 and R clock will exhibit only SR time losses during the experiment, i.e., there will be no additional time difference(s) due to the fact that the R/2 and R clocks experience different gravitational potentials.

Now stop the rotation and compare the accumulated readings on the three clocks to each other. What would you find?

Why isn't this a do-able experiment?

 

[Hurkyl]

That is a subjective opinion
...
It is great as an interum, but I think it will ultimately be viewed as a constructive consequence of a something more fundamental.

I was not stating an opinion. Whether or not GR is correct is entirely irrelevant to the question of whether GR is complete.

But you really missed my point entirely -- asking the question "why?" is entirely pointless, unless you have already assumed some collection of 'things' in terms of which you will accept explanations. If you do not wish to use some modern physical theory for that purpose, then you need to specify your alternative theory. (You cannot expect people to read your mind. ) Of course, if this isn't already crackpot territory, it's dangerously close. (Unless, of course, you're doing actual research)

 

[Hurkyl]

When you disregard the units you throw away valuable information - this is one of the areas where modern physics has handicapped itself.

No information is lost; the process is reversible. e.g. one can easily convert a velocity of 1/3 in natural units to a velocity of 10^8 m/s in mks units.

Incidentally, why do you feel so strongly that, say, the meter should not be considered a derived unit expressed in terms of the second? Do you have any reason other than tradition?

 

[meopemuk]

Different folks have weighed in on the question of whether two spaced apart clocks in an accelerating free space rocket will read different values when combined. While the rocket experiment is difficult to perform, I do not see why a spinning disk could not be used with 3 clocks- the first (R/0) at the center, the second at radial distance R/2 and the third at R. Run the disk for an extended period and compare the clocks at R/2 and R to the center clock from time to time while the disk is spinning. My guess is that the R/2 and R clock will exhibit only SR time losses during the experiment, i.e., there will be no additional time difference(s) due to the fact that the R/2 and R clocks experience different gravitational potentials.

Now stop the rotation and compare the accumulated readings on the three clocks to each other. What would you find?

Why isn't this a do-able experiment?

I think it is doable in principle, but not in practice. The effect is very small, and you need very precise atomic clocks to measure it. You cannot put an atomic clock on a spinning disk, it would just stop working.

As far as I know, experiments with spinning disks used the Mossbauer effect to measure the influence of rotation on photon frequencies. However, it is impossible to make a clock based on gamma ray frequencies. These frequencies are too high to count oscillations.

Maybe I am missing some new experimental developments, but in my opinion, we are very far from experiments with clocks on a spinning disk.

Eugene.

 

[pervect]

When you disregard the units you throw away valuable information - this is one of the areas where modern physics has handicapped itself.

No information is lost by the use of geometric units. Units are a lot like "types" in computer programming. They don't add anything fundamental, but they do help prevent careless errors.

 

[yogi]

Yes to Hurkyl and pervect - you can always recover the units at the end - but in the process of developing relationships, carrying the units along provides insight -it frequently leads to paths not obvious otherwise. If it doesn't help you, that is your M.O. I have found it extremely useful for my areas of interest.

Hurkyl - Asking "why" is my only interest in these forums - and if you read many other posters, you will see a similar curosity.

What makes you think i should have a worked out theory to question the completeness of GR? Einstein continually questioned his own works throughout his life - something can be recogonized as missing or in dispute without having an alternative - Hawking made the same criticism, too wit: "We have two theories of gravity, but neither can explain its strength, nor do we know why the electric charge has the value it has" I guess we should chastise Stephen as a borderline crackpot (to use your words).

 

[yogi]

I think it is doable in principle, but not in practice. The effect is very small, and you need very precise atomic clocks to measure it. You cannot put an atomic clock on a spinning disk, it would just stop working.

As far as I know, experiments with spinning disks used the Mossbauer effect to measure the influence of rotation on photon frequencies. However, it is impossible to make a clock based on gamma ray frequencies. These frequencies are too high to count oscillations.

Maybe I am missing some new experimental developments, but in my opinion, we are very far from experiments with clocks on a spinning disk.

Eugene.

Why would a mild G field be disruptive of atomic clocks - they work in the Earth's field. For example, the experiment could be limited to one or two G's and carried on for many months which should yield data at least as good as Hafle and Keating which involved changing altitudes and non uniform accelertions.

 

[meopemuk]

Why would a mild G field be disruptive of atomic clocks - they work in the Earth's field. For example, the experiment could be limited to one or two G's and carried on for many months which should yield data at least as good as Hafle and Keating which involved changing altitudes and non uniform accelertions.

Yes, you have a good point. Atomic clocks are now more compact and stable than in the times of Hafele and Keating. So, it should be possible to put them in a centrifuge and spin for months. Are there experimentalists willing to do that?

My best guess would be that this experiment will show a real permanent time difference between the moving clock and the clock at rest. This kind of difference was already observed for the lifetime of muons in a cyclotron ring. However, I don't think this would be a proof that the same time difference can be found in an uniformly accelerated rocket. In the spinning disk experiment there is an inherent asymmetry between two clocks. The same kind of asymmetry that is used in explaining the "twin paradox". This asymmetry is absent in the uniformly accelerated rocket.

Eugene.

 

[Ich]

My guess is that the R/2 and R clock will exhibit only SR time losses during the experiment, i.e., there will be no additional time difference(s) due to the fact that the R/2 and R clocks experience different gravitational potentials.

That is true, acceleration does not make clocks run faster or slower.
In flat spacetime, you have two possibilities: you analyze the problem in an inertial frame, and you get the standard SR time dilations. Or you switch to accelerated frames where the objects are at rest, and you recover the very same time dilation, but this time in terms of gravitational potential, as there is no more motion. It is just a different point of view, not a different physical effect.

I even didn't have to rephrase my answer.

 

[yogi]

That is true, acceleration does not make clocks run faster or slower.
In flat spacetime, you have two possibilities: you analyze the problem in an inertial frame, and you get the standard SR time dilations. Or you switch to accelerated frames where the objects are at rest, and you recover the very same time dilation, but this time in terms of gravitational potential, as there is no more motion. It is just a different point of view, not a different physical effect.

I even didn't have to rephrase my answer.

So, can I conclude from your post there is no permanent age difference to be measured after an extended trial except those that correspond to velocity differences. And if so, there would be no residual affect to indicate that R is in a deeper gravitational well than R/2. And if that is so, can we not conclude that all free space rocket accelerations with spaced apart clocks will never reveal a permanent age difference?

It seems to me there is a critical flaw in complete equivalence - its not just a matter of degree to be rationalized away by corrections that depend from the shape of the field (tidal effects and divergence) - In GR we are dealing with geometric phenomena that affect time and space - in the free space acceleration, there is no perceived affect upon local space and time. This is one of the reasons I have questioned Einstein's 1918 rationale of the twin paradox - the substitution of a G field for what is in reality a free space turn around acceleration is questionable. The free space turn around acceleration "a" should only produce apparent time dilations (ah/c^2) at a distant location - and then only during the turn around period. This cannot result in permanent time dilation and age difference - but that is what is required under the 1918 explanation - real age differences apparently do not occur except in G fields.

 

[pervect]

That is true, acceleration does not make clocks run faster or slower.
In flat spacetime, you have two possibilities: you analyze the problem in an inertial frame, and you get the standard SR time dilations. Or you switch to accelerated frames where the objects are at rest, and you recover the very same time dilation, but this time in terms of gravitational potential, as there is no more motion. It is just a different point of view, not a different physical effect.

So, can I conclude from your post there is no permanent age difference to be measured after an extended trial except those that correspond to velocity differences.

No, re-read Ich's post.

In this particular post, Ich hasn't said anything about an extended trial.

What Ich has said is that the explanation offered by different observers will be different. So if we take the example of the extended trial that you are interested in, where we have two clocks in an elevator, one clock is raised to a higher height, left there, and then returned, everyone will agree (regardless of their frame) that the clock that was raised to a higher height shows a longer elapsed time.

One way of looking at it: the clock at the top of the elevator accelerates at a lower proper acceleration for a longer period of proper time than the clock at the bottom of the elevator, which accelerates at a higher proper acceleration for a shorter period of proper time. Ignoring the accelerations needed to separate and re-unite the clocks, the total velocity change of the top and bottom of the rocket must be the same (because the rocket is Born rigid), thus proper acceleration * proper time (lower) = proper acceleration * proper time (upper), but only the product is identical, the proper times and proper accelerations of the upper and lower clocks are different.What will be different is the explanation offered as to why this happens. In an inertial frame, the explanation will not involve any considerations gravitational potential. You will find that the proper time is different simply due to velocity - acceleration does NOT enter into the Lorentz transforms.

The point of Ich's post is just the above - that the explanations are different.

Why are the explanations different? Because they are phrased in terms that depend on an "observer", rather than being phrased in an observer-independent manner. It's not an issue of the mathematics, it's the baggage that people carry over from taking coordinate too seriously.

 

[yogi]

Pervect - thanks for your post. But as usual, I still have questions.


Lets confine the discussion to a free space elevator accelerating upward - no gravitational mass anywhere to be found. You say "(because the rocket is Born rigid), thus proper acceleration * proper time (lower) = proper acceleration * proper time (upper), but only the product is identical, the proper times and proper accelerations of the upper and lower clocks are different."

That is very good -

But what I would like to know is: Why is it true that the accelerations will be different as observed by the holder of the upper clock and the holder of the lower clock? In a uniform acceleration field, where is the physics that tells the upper clock it is the upper clock? Granted, we can send signals between the clocks during the acceleration phase - and get the observed ah/c^2 time difference... but this cannot affect the intrinsic rate of the two clocks. And if the intrinsic clock rate is not altered, how then can there be a net time difference when the clocks are later compared?

If you are saying, this is as it must be to satisfy the equivalence principle, I concur. But my original intent was to see if there were any experimental confirmation of age differences in uniform acceleration fields - which I believe has been answered in the negative.

 

[meopemuk]

But my original intent was to see if there were any experimental confirmation of age differences in uniform acceleration fields - which I believe has been answered in the negative.

Hi yogi,

I know only one experiment which (in a loose language) can be described as "measuring the influence of uniform acceleration on the rate of clocks"

C. E. Roos, J. Marraffino, S. Reucroft, J. Waters, M. S. Webster, E. G. H.
Williams, A. Manz, R. Settles, G. Wolf, "\Sigma^{\pm} lifetimes and longitudinal acceleration", Nature, 286, (1980), 244.

This experiment is somewhat similar to the lifetime dilation of muons experiencing "transverse" acceleration in a cyclotron, which I mentioned earlier. However, in this case, the acceleration is "longitudinal" (produced by breaking charged particles in hydrogen or photoemulsions). Of course, this is far from uniform acceleration of clocks, and not directly relevant to our discussion here, but it is the closest experimental thing that comes to mind. Just as in the muon experiment, these authors found that acceleration (up to 5 \cdot 10^{15}g) had no effect on the lifetime.

Eugene.

 

[pervect]

The reason why is basically geometrical. Look at a spacetime diagram, for instance the one on the Bell spaceship paradox,

http://en.wikipedia.org/wiki/Image:Bell_observers_experiment2.png

or my crude sketch (attached). Better yet would be to draw one for yourself.

The left red curve in my sketch is the front of the rocket. The rightmost red curve is the back of the rocket. Time runs up the page.

It should be reasonably obvious from the sketch that CD is longer than AB from the diagram. AB is the proper time measured by a clock on the front of the spaceship, and CD is the proper time measured by a clock on the rear.

This demonstrates the desired result, that the clocks at the front and back of the rocketship don't run at the same rate.

How were AC and BD, the lines of simultaneity constructed? From the Lorentz transform, and the velocity of the spaceship at points B and D. The line of simultaneity drawn is the one appropriate given the velocity of the rocketship at events B and D. (Because the rocketship is Born rigid, these velocities must be the same, i.e. the rocket is not getting longer or shorter, so the tail is not moving relative to the head in the "rocket frame").

So it is the relativity if simultaneity (the angle between AC and BD) that causes CD to be longer than AB - as viewed by the inertial observer, at least. (This space-time diagram was drawn from the standpoint of an inertial observer, so it will "explain" things from the viewpoint of an inertial observer.)

 

[yogi]

Thanks Eugene

I think pervect is saying that Born rigidity requires the trailing clock to have a greater acceleration or order to keep the distance beteen the floor and the elevator the same.

I will check out the experiment you cited. Read one of your papers recently - would be interested in chatting more about your conclusion as to the instaneous propagation of the G field.

 

[yogi]

Thanks pervect - our posts overlapped - I was reading some of your material on another thread whill responding to Eugene.

 

[meopemuk]

I will check out the experiment you cited. Read one of your papers recently - would be interested in chatting more about your conclusion as to the instaneous propagation of the G field.

You are welcome to join discussion at
https://www.physicsforums.com/showthread.php?t=175965

 

[Hurkyl]

Hurkyl - Asking "why" is my only interest in these forums - and if you read many other posters, you will see a similar curosity.

What makes you think i should have a worked out theory to question the completeness of GR? Einstein continually questioned his own works throughout his life - something can be recogonized as missing or in dispute without having an alternative - Hawking made the same criticism, too wit: "We have two theories of gravity, but neither can explain its strength, nor do we know why the electric charge has the value it has" I guess we should chastise Stephen as a borderline crackpot (to use your words).

I'm not objecting to you asking "why?" The problem is that you appear to demand there to be an answer other than "it's a fundamental property of the universe" when our best theories say that there isn't another answer.

I'm also objecting to another thing -- making factually false statements about GR. GR asserts that the gravitational constant is a fundamental property of the universe. If the gravitational constant is, in fact, not a fundamental property of the universe, that does not mean GR is incomplete; it simply means GR is wrong.

 

[yogi]

Hurkyl - I doubt that the "value" of G is a fundamental - We live in a dynamic cosmos - the value of the parameter that determines the attraction between masses may have been different when the universe was smaller. The tests performed to find some variation only measure the MG product in some fashion (e.g. radar ranging the moons of Mars). There will always be some number that defines the force between masses - in that sense the G factor is fundamental - but the value may change as the universe ages, but this does not affect the validity of GR

All the experiments show that the MG product appears to be invariant with time. From a historical perspective, as you probably already know, Newton originally lumped the Solar mass and G together when working his gravitational theory.

So I will take issue with your last statement - If the M and G are individually variable, but their product is constant, there is no reason to condemn GR.

Finally, I don't recall demanding anything - I do ask questions on these boards, and I do inject things I hope will provoke posters to be less certain in their assertions. This apparently bothers you and some others. There are many ways in which a theory can have great value even though new knowledge may require fine tuning. Einstein did not regard GR as the final expression of physics - he spent the last part of his life attempting to unify gravity with electrical phenomena - such a unification, we would assume, would be more fundamental than either of the theoies so unified. Albert finally concluding that he had given his best efforts and failed. If Albert didn't consider GR to be as fundmental as you suggest, why is so important that I do?

One more thing - we had a previous discussion reqarding the importance - or lack of importance, of interpreting units. Look at the units of G (volumetric acceleration per unit mass) That should make you wonder - does it make sense that the volumetric acceleration of the universe should be the same when it is 10^9 years of age as it was when it was one second?

 

[mendocino]

Different folks have weighed in on the question of whether two spaced apart clocks in an accelerating free space rocket will read different values when combined. While the rocket experiment is difficult to perform, I do not see why a spinning disk could not be used with 3 clocks- the first (R/0) at the center, the second at radial distance R/2 and the third at R. Run the disk for an extended period and compare the clocks at R/2 and R to the center clock from time to time while the disk is spinning. My guess is that the R/2 and R clock will exhibit only SR time losses during the experiment, i.e., there will be no additional time difference(s) due to the fact that the R/2 and R clocks experience different gravitational potentials.

Now stop the rotation and compare the accumulated readings on the three clocks to each other. What would you find?

Why isn't this a do-able experiment?

Since the distance between the clock at R/2 and clock at R do Not change with time, so the relative speed of clock at R/2 to clock at R is Zero.
So the time difference(s) is due to the fact that the R/2 and R clocks experience different gravitational potentials.

 

[yogi]

Neither clock is in an inertial frame - From the perspective of the center clock, R/2 has a different velocity that R. Centrifuge experiments do not show any time dilations other than those than can be attributed to Velocity - but you have raised a point - maybe the time dilation differences are in fact due to the potential and not velocity - most persons commenting on centrafugal clock measurments have immediately opted for the SR out - but that may not be correct.

I have frequently made the statement on these boards that experiments do not result in a permanent age differences unless acceleration takes place somewhere in the process. In the centrifuge, we can analyse the motion of R/2 and R from the perspective of R/0 using only SR - that is we can handle accelerated frames so long as the observer is inertial. but I think we cannot consider R/2 and R to be in the same frame.

 

[mendocino]

Neither clock is in an inertial frame - From the perspective of the center clock, R/2 has a different velocity that R. .

Why do you think R/2 has a different speed that R?
Since speed is the differential of distance with time and distance do Not change at all from the perspective of the center clock

 

[pervect]

Hi yogi,

I know only one experiment which (in a loose language) can be described as "measuring the influence of uniform acceleration on the rate of clocks"

C. E. Roos, J. Marraffino, S. Reucroft, J. Waters, M. S. Webster, E. G. H.
Williams, A. Manz, R. Settles, G. Wolf, "\Sigma^{\pm} lifetimes and longitudinal acceleration", Nature, 286, (1980), 244.

This experiment is somewhat similar to the lifetime dilation of muons experiencing "transverse" acceleration in a cyclotron, which I mentioned earlier. However, in this case, the acceleration is "longitudinal" (produced by breaking charged particles in hydrogen or photoemulsions). Of course, this is far from uniform acceleration of clocks, and not directly relevant to our discussion here, but it is the closest experimental thing that comes to mind. Just as in the muon experiment, these authors found that acceleration (up to 5 \cdot 10^{15}g) had no effect on the lifetime.

Eugene.

Just in case it isn't clear (in spite of my best efforts to clarify the situation), one does not expect any effect due to acceleration if one carries the calculations out in an inertial frame. So experiment matches what one would expect.

An inertial frame implies no "gravity", of any sort (so the Earth isn't an inertial frame, for instance). An inertial frame also has no "potential" of any sort (or rather, any such potential is uniform, and its gradient is zero everywhere) - there are no gravitational or psuedogravitational or inertial forces in an inertial frame).

An accelerating rocket in empty space can be analyzed in an inertial frame, though it's not necesary to do the analysis that way. In our rocket example, in an inertial frame, there is a force on the rocket, due to the rocket exhuast - but this force is not any sort of "inertial force".

Non-inertial frames (of any sort) can be characterized by metric coefficients that aren't Minkowskian, i.e. they are not diag(-1,1,1,1,). Note that the "potential" doesn't have any solid defintion in GR, in the Newtonian limit one can read the "potential" off of the time component of the metric coefficients, and that seems to be mostly how people have been using "potential" in this thread (but without definining what they've been talking about explicitly).

 

[pervect]

Why do you think R/2 has a different speed that R?
Since speed is the differential of distance with time and distance do Not change at all from the perspective of the center clock

Huh? In an inertial frame co-moving with the axis of rotation, the center clock is stationary, and if the disk is rotating, the speeds of the two clocks are

\omega R/2 for the clock at R/2 and \omega R for the clock at R, where \omega is the angular frequency at which the disk rotates.

 

[mendocino]

Huh? In an inertial frame co-moving with the axis of rotation, the center clock is stationary, and if the disk is rotating, the speeds of the two clocks are

\omega R/2 for the clock at R/2 and \omega R for the clock at R, where \omega is the angular frequency at which the disk rotates.

Let the center clock emit short pulse of light whenever it ticks,
Can you tell me if the clock at R see any Doppler shift of light pulse?
If yes, will it be red-shift or blue-shift?

 

[meopemuk]

Let the center clock emit short pulse of light whenever it ticks,
Can you tell me if the clock at R see any Doppler shift of light pulse?
If yes, will it be red-shift or blue-shift?

Experiments of this kind (but not exactly the same as you described, because the red shift was measured for energies of gamma quanta rather than for clock rates) were performed many times. This is probably the first such measurement

H. J. Hay, J. P. Schiffer, T. E. Cranshaw, P. A. Egelstaff, "Measurement of the red shift in an accelerated system using the Mossbauer effect in Fe^57", Phys. Rev. Lett. 4 (1960), 165.

But I've seen at least a dozen similar papers in later years.

Eugene.