Pedagogically direct demonstrations of SR and GR

[bcrowell]

A few weeks back I posted about a test by C. Alley in the 1970's of Einstein's famous goof in predicting that a clock would run at a different rate at the poles than at the equator. Alley apparently was never big on publishing his results in journals, but I requested a conference proceedings via interlibrary loan and obtained a couple of his papers ( C. Alley, "Proper Time Experiments in Gravitational Fields with Atomic Clocks, Aircraft, and Laser Light Pulses," in Quantum Optics, Experimental Gravity, and Measurement Theory, eds. Pierre Meystre and Marlan O. Scully, Proceedings Conf. Bad Windsheim 1981, Plenum Press, New York, 1983, ISBN 0-306-41354-X, pp. 363–427. ) Some of his group's work was essentially an improved version of the famous Hafele-Keating experiment, and some was lunar laser ranging. The funding seems to have been because the military was pushing to create the GPS system.

In Alley's paper he mentions a couple of very pedagogically direct demonstrations of gravitational time dilation:

L. Briatore and S. Leschiutta, Evidence for the Earth gravitational shift by direct atomic-time-scale comparison, Il Nuovo Cimento B, 37B (2): 219 (1979)

S. Iijima and K. Fujiwara, An experiment for the potential blue shift at the Norikura Corona Station, Annals of the Tokyo Astronomical Observatory, Second Series, Vol. XVII, 2 (1978) 68.

In both of these experiments, atomic clocks were compared after one had been left in a valley and one at the top of a mountain. Briatore had 3250 m, 66 days, 15% precision, Iijima 2818 m, 1 week, and 5%.

It's hard to imagine a much more direct test of relativity than these. Unlike the Hafele-Keating and Alley experiments, these isolated one effect (gravitational time dilation, not kinematic). I have a hard time imagining what the anti-relativity kooks would make of something as conceptually simple as this.

Too bad there doesn't seem to be any atomic clock experiment that provides a similarly transparent test of kinematic time dilation. H-K and Alley both had a mixture of kinematic and gravitational. This was partly intentional. Alley's group had the navy planes fly as slowly as possible, because testing GR was more cutting edge than testing SR.

 

[atyy]

Did they control for temperature?

 

[Q-reeus]

It's hard to imagine a much more direct test of relativity than these. Unlike the Hafele-Keating and Alley experiments, these isolated one effect (gravitational time dilation, not kinematic)

Are you sure about that? If the experiments were at the north or south pole then OK. But they weren't, and we are dealing with a mix of 'pure' GR gravitational time dilation and a SR kinematic effect since the valley/mountain locations are rotating about the Earth's axis at different radii and hence different relative speeds. The latter is another variant of the Twin Paradox for rotating motion. Presumably the experimenters in both cases made appropriate allowance for this.

 

[matheinste]

Are you sure about that? If the experiments were at the north or south pole then OK. But they weren't, and we are dealing with a mix of 'pure' GR gravitational time dilation and a SR kinematic effect since the valley/mountain locations are rotating about the Earth's axis at different radii and hence different relative speeds. The latter is another variant of the Twin Paradox for rotating motion. Presumably the experimenters in both cases made appropriate allowance for this.

The clocks on Earth are not in relative motion with respect to each other.

Matheinste.

 

[Q-reeus]

The clocks on Earth are not in relative motion with respect to each other.

Matheinste.

I disagree. Static relative position here is an artifact of how one positions oneself in a rotating reference frame. While the clock on the mountain top appears static from the vantage point of the one in the valley and vice versa, notice by this same perspective the distant stars are whizzing around at ultra-hyper relativistic speeds, which is absurd. Instead, head out into deep space, positioning yourself in line with the Earth's spin axis. If your frame of reference is non-rotating, clearly the two clocks have different rotational speeds. To emphasize the differential, there's nothing stopping us in principle from having one clock being at the center of the earth. That is simply the 'extreme' situation discussed in another thread here:https://www.physicsforums.com/showthread.php?t=450236

 

[matheinste]

I disagree. Static relative position here is an artifact of how one positions oneself in a rotating reference frame. While the clock on the mountain top appears static from the vantage point of the one in the valley and vice versa, notice by this same perspective the distant stars are whizzing around at ultra-hyper relativistic speeds, which is absurd. Instead, head out into deep space, positioning yourself in line with the Earth's spin axis. If your frame of reference is non-rotating, clearly the two clocks have different rotational speeds. To emphasize the differential, there's nothing stopping us in principle from having one clock being at the center of the earth. That is simply the 'extreme' situation discussed in another thread here:https://www.physicsforums.com/showthread.php?t=450236

Yes, I see what you mean, the two clocks would need to be in orbit and complete one revolution counter to that of the earth, once per day, so in effect remaining "stationary" for want of a better word, at their different heights while the Earth effectively turned beneath them.

Matheinste.

 

[ghwellsjr]

A few weeks back I posted about a test by C. Alley in the 1970's of Einstein's famous goof in predicting that a clock would run at a different rate at the poles than at the equator.

Einstein said in his 1905 paper:

"Thence we conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles."

Doesn't the phrase "under otherwise identical conditions" get him off the hook here? He was talking about special relativity, not general relativity and he had been talking about a clock in motion in a circular pattern continually returning to its point of origin and how it would continually lose time with regard to a stationary clock relative to that point of origin. (This was the origin of the Twin Paradox.)

 

[Q-reeus]

Yes, I see what you mean, the two clocks would need to be in orbit and complete one revolution counter to that of the earth, once per day, so in effect remaining "stationary" for want of a better word, at their different heights while the Earth effectively turned beneath them.

Matheinste.

Agreed. You then have the correct arrangement for doing a pure GR test. A matter of taste and finances as to whether one mounts a polar expedition or jetsets around the world!

 

[PeterDonis]

Einstein said in his 1905 paper:

"Thence we conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles."

Doesn't the phrase "under otherwise identical conditions" get him off the hook here? He was talking about special relativity, not general relativity and he had been talking about a clock in motion in a circular pattern continually returning to its point of origin and how it would continually lose time with regard to a stationary clock relative to that point of origin. (This was the origin of the Twin Paradox.)

He also didn't mention the Earth's equatorial bulge, which is (on average) just the right height to make clocks at the equator, on the "surface" there, go at the same rate as clocks at the poles, on the "surface" there. ("Surface" meaning the reference height of the "geoid", the "average" surface of the Earth with local irregularities smoothed out.) The increase in height (which makes clocks go faster relative to a "reference clock" at the pole) is just enough to compensate for the velocity of rotation (which makes clocks go slower relative to a "reference clock" at the pole). But of course you need General Relativity to calculate this correctly.

 

[bcrowell]

I disagree. Static relative position here is an artifact of how one positions oneself in a rotating reference frame. While the clock on the mountain top appears static from the vantage point of the one in the valley and vice versa, notice by this same perspective the distant stars are whizzing around at ultra-hyper relativistic speeds, which is absurd. Instead, head out into deep space, positioning yourself in line with the Earth's spin axis. If your frame of reference is non-rotating, clearly the two clocks have different rotational speeds. To emphasize the differential, there's nothing stopping us in principle from having one clock being at the center of the earth. That is simply the 'extreme' situation discussed in another thread here:https://www.physicsforums.com/showthread.php?t=450236

You have the right qualitative description, but you're getting mixed up as far as which effects are big, which are small, and which are zero. When you analyze these atomic clock experiments, you can break down the predictions of GR into three terms: kinematic, gravitational, and Sagnac effect. In a mountain-valley experiment, the Sagnac effect is zero, because the path traveled by the clocks in the rotating frame (up and down the mountain) encloses zero area, and the Sagnac effect is proportional to area. The kinematic effect exists, in the non-rotating frame, but its size relative to the gravitational effect is $v^2/2\Delta\Phi=\omega^2 R \cos\lambda/g=2.8\times10^{-3}$, where $\lambda$ is the latitude of Tokyo and R is the radius of the earth. The fact that this number comes out so small tells you that matheinste was basically right; it's essentially a pure test of the gravitational effect, because although the clocks are not at rest with respect to one another, their relative velocity is extremely small. If you could do a mountain-valley experiment at the south pole, with both clocks within a kilometer of the pole, the ratio $v^2/\Delta\Phi$ would still be nonzero, but in both cases it's too small to have any measurable effect.

The other thing to realize is that the distinction between kinematic and gravitational time dilation is frame dependent, so, e.g., when you say we could do a pure test of the GR effect by going to the poles, that's not really true. I could take such an experiment and analyze it in a frame rotating about one of the two clocks, and in that frame I would have both Sagnac effects and kinematic time dilation in addition to gravitational time dilation. On the other hand, I can take any mountain-valley experiment, at any latitude, and analyze it in the earth-fixed rotating frame, in which case both clocks' velocities are exactly zero, and the effect is pure gravitational. (The centrifugal force is subsumed within the gravitational force.)

Did they control for temperature?

Alley certainly did, and I'm sure the Italian and Japanese groups weren't careless about anything that basic either. Alley describes various flights to Greenland, Hawaii, and New Zealand on military cargo planes, and having to use AC in Hawaii, then heat in the New Zealand winter. I haven't read the Iijima paper, but it sounds like they had the high-altitude clock in an observatory building. These clocks were big, delicate instruments that had to be carefully taken care of. The military guys referred to Alley's instrument as the "Two-Ton Timex."

[EDIT] Fixed factor of 2 mistake above.

 

[bcrowell]

There's another interesting section in the Alley paper, not directly related to the title of this thread, which is about the question of the effect of the sun's field.

For a clock at the Earth's surface, the sun's field creates a rate-change that oscillates with a period of one day. In the sun's frame it's 67 ns/day, but in the Earth's frame, the first-order effect vanishes, so it becomes 0.62 ps/day, which is too small to measure, even with today's technology.

There was also a debate about whether the sun would cause an effect that would oscillate with a period of one year. There was a prediction by R.U. Sexl, Seasonal differences between clocks, PL 61B (1976) 65, that there would be such an effect for clocks at two different latitudes, due to the tilt of the Earth's axis, and that the amplitude of the oscillation would be (14.8 ns/day)(sin lat2-sin lat1). Alley et al. flew a plane from Andrews AFB-Travis AFB-Hickam AFB, Hawaii-Christchurch-Hickam-Andrews, and found that Sexl's effect didn't exist. Sexl admitted he was wrong, and the shift would only be observable in the sun's frame.

 

[Q-reeus]

...The kinematic effect exists, in the non-rotating frame, but its size relative to the gravitational effect is $v^2/2\Delta\Phi=\omega^2 R \cos\lambda/g=2.8\times10^{-3}$, where $\lambda$ is the latitude of Tokyo and R is the radius of the earth...

Fair enough, I had not attempted calculations of relative effect, just an in principle thing. But the Tokyo figure of ~ 0.3% is not all that small, and I think we agree for a more sensitive experiment would need accounting for.

The other thing to realize is that the distinction between kinematic and gravitational time dilation is frame dependent, so, e.g., when you say we could do a pure test of the GR effect by going to the poles, that's not really true. I could take such an experiment and analyze it in a frame rotating about one of the two clocks, and in that frame I would have both Sagnac effects and kinematic time dilation in addition to gravitational time dilation.

Completely lost me here. I think we need a description of just how the experiment is physically setup and performed.
My arrangement: Let's go ideal and assume there really is a very tall 'north pole' sticking straight up at the geographical North Pole. One clock is mounted atop the pole and linked via optical fiber running straight down the pole to an identically made clock at the pole base, via a comparitor which records the frequency difference. For me this setup will be a 'pure' GR test. How does either Sagnac or SR kinematics enter?

On the other hand, I can take any mountain-valley experiment, at any latitude, and analyze it in the earth-fixed rotating frame, in which case both clocks' velocities are exactly zero, and the effect is pure gravitational. (The centrifugal force is subsumed within the gravitational force.)

Aren't we 'going around in circles' here (no pun intended)?

 

[bcrowell]

Completely lost me here. I think we need a description of just how the experiment is physically setup and performed.
My arrangement: Let's go ideal and assume there really is a very tall 'north pole' sticking straight up at the geographical North Pole. One clock is mounted atop the pole and linked via optical fiber running straight down the pole to an identically made clock at the pole base, via a comparitor which records the frequency difference. For me this setup will be a 'pure' GR test. How does either Sagnac or SR kinematics enter?

Go into a frame rotating at, say, 1 Hz, about an axis perpendicular to the tower and passing through its top. In this frame, the clock on the ground below is revolving in a vertical circle at some high speed. The clock on the ground suffers a kinematic time dilation because it's moving. It also picks up a Sagnac effect because its motion encloses a certain finite area with each revolution.

In general, a certain combination of kinematic time dilation, gravitational time dilation, and Sagnac effect will become a *different* combination of those three things under a change of coordinates. A less artificial-seeming physical example of this is cosmological red-shifts, which can be interpreted as kinematic, gravitational, or whatever you like.