Empirical test of Einstein's famous goof

[bcrowell]

I think it's fairly well known that Einstein made an incorrect prediction in his 1905 paper on special relativity. "Thence we conclude that a spring-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions." Once he developed GR, it became clear that there would be zero effect from latitude, because the Earth's surface is an equipotential.

What I hadn't realized was that there was a direct empirical test of this done in the 1970's by Alley et al. A description of the experiment is online here: http://www.pttimeeting.org/archivemeetings/index9.html Alley's group flew atomic clocks from Washington, DC to Thule, Greenland, left them there for four days, and brought them back. The difference between the clocks that went to Greenland and other clocks that stayed in Washington was 38+-5 ns, which was consistent with the 35+-2 ns effect predicted purely based on kinematic and gravitational time dilation while the planes were in the air. If Einstein's 1905 prediction had been correct, then there would have been an additional difference of 224 ns due to the difference in latitude.

 

[PAllen]

I think it's fairly well known that Einstein made an incorrect prediction in his 1905 paper on special relativity. "Thence we conclude that a spring-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions." Once he developed GR, it became clear that there would be zero effect from latitude, because the Earth's surface is an equipotential.

What I hadn't realized was that there was a direct empirical test of this done in the 1970's by Alley et al. A description of the experiment is online here: http://www.pttimeeting.org/archivemeetings/index9.html Alley's group flew atomic clocks from Washington, DC to Thule, Greenland, left them there for four days, and brought them back. The difference between the clocks that went to Greenland and other clocks that stayed in Washington was 38+-5 ns, which was consistent with the 35+-2 ns effect predicted purely based on kinematic and gravitational time dilation while the planes were in the air. If Einstein's 1905 prediction had been correct, then there would have been an additional difference of 224 ns due to the difference in latitude.

That's interesting but I guess I'm confused about one thing (never thought about this before). Independent of gravity, wouldn't an equatorial clock have a tiny bit different acceleration, plus a speed relative to a polar clock? Then if gravitational 'potential' is equal, shouldn't this make a small difference? Is there some simple explanation of why these facts should have no effect?

 

[Ich]

Then if gravitational 'potential' is equal, shouldn't this make a small difference? Is there some simple explanation of why these facts should have no effect?

Yes, not the gravitational potential is equal, but the http://en.wikipedia.org/wiki/Effective_potential" which includes rotational velocity.

 

[pervect]

If you look first at the classical sense (ignoring relativity for the moment) in which the Earth's surface is "equipotential", it's "equipotential" when you add the gravitational potential to the centrifugal potential.

As I recall (it's been a while) this is equivalent to writing the Hamiltonian in a frame that co-rotates with the Earth.

Now, the force of gravity should be normal to said equipotential surface. And if you go to a coordinate system that co-rotates with the Earth, gravity should point in the direction of $\partial/\partial g_{00}$. So you expect g_00 to be constant over the surface of the Earth.

It's really the idealized surface of the Earth that's being considered here, the geoid, rather than the physical surface. So you can think of all clocks "on the geoid" or roughly speaking "at sea level" as having the same metric coefficient g_00, which means they run at the same rate.

 

[bcrowell]

That's interesting but I guess I'm confused about one thing (never thought about this before). Independent of gravity, wouldn't an equatorial clock have a tiny bit different acceleration, plus a speed relative to a polar clock? Then if gravitational 'potential' is equal, shouldn't this make a small difference? Is there some simple explanation of why these facts should have no effect?

The basic idea is that you can adopt the frame that rotates along with the earth, so that all time dilations are gravitational, not kinematic. In that frame, the Earth's surface is an equipotential. Gravitational time dilation is $e^{-\Delta\Phi}$, based on the equivalence principle: http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.5 (subsection 1.5.5).

There is an issue because the spacetime is stationary but not static. Because it's stationary, you can rate-match clocks, and you can equate $e^{-\Delta\Phi}$ to the rate difference and use that as the definition of the gravitational potential. Because the spacetime isn't static, the metric isn't derivable from the potential (because the metric has information in it about stuff like the geodetic effect and frame-dragging, which are related to rotation and can't be encapsulated in a scalar potential). The non-staticity of the spacetime shows up because you get big Sagnac effects in these experiments. In the analysis of this type of experiment, they typically just take the Sagnac effect as a separate term in the theoretical equations.

[EDIT]Corrected a couple of mistakes.

 

[Passionflower]

For starters let's stop the handwaving:

We know the Earth's mass, radius, rotational speed and equatorial bulge so can we have the equation for the clock at the pole and the equator using the Kerr metric?

Note: if we can't do it please say so, but please no cop outs like "we don't need to", "we ignore rotation because the rotation is slow", we want to do GR here.

Any takers?

 

[GoldPheonix]

I think it's fairly well known that Einstein made an incorrect prediction in his 1905 paper on special relativity. "Thence we conclude that a spring-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions." Once he developed GR, it became clear that there would be zero effect from latitude, because the Earth's surface is an equipotential.

What I hadn't realized was that there was a direct empirical test of this done in the 1970's by Alley et al. A description of the experiment is online here: http://www.pttimeeting.org/archivemeetings/index9.html Alley's group flew atomic clocks from Washington, DC to Thule, Greenland, left them there for four days, and brought them back. The difference between the clocks that went to Greenland and other clocks that stayed in Washington was 38+-5 ns, which was consistent with the 35+-2 ns effect predicted purely based on kinematic and gravitational time dilation while the planes were in the air. If Einstein's 1905 prediction had been correct, then there would have been an additional difference of 224 ns due to the difference in latitude.

I'm not familiar with the prediction Einstein was trying to make here, but I assume --given that this was a paper on special relativity-- that he simply used an incorrect example of the difference in time between two inertial frames of reference. The problem, of course, is that the two frames of reference are non-inertial, and Lorentz transforms are not valid in this regime. However, not all of the statements about the difference between non-inertial and inertial frames were well understood at the time Einstein wrote about them (and indeed, due to lack of interest, today they still aren't understood in general), and so this isn't exactly an interesting experiment because it just looks like Einstein misspoke. It makes no real statement about special relativity (other than that GR supersedes it, which we already know from tests of general relativity).(I mean, technically, one could state that since the Earth is approximately spherical, the points on its surface are closely approximated by Born coordinates --and they contain all the predictions that SR makes about rotating frames-- and use this to find the time difference between these non-inertial frames. From there you test the deviation that will occur due to the gravitational field, i.e. effects of the Kerr metric, as separated from the special relativistic effects coming from simple relativistic rotational motion. But again, I don't see why this is a very interesting experiment; Einstein simply misspoke.)

 

[PAllen]

It seems to me there is a significant physical prediction here, though one that does not really distinguish Newtonian vs. GR gravity. If I understand the various points made, it seems to me that a sufficiently rigid perfect sphere would not provide an equipotential surface under rotation. Instead, the equipotenial surface would extend above ground by miles (for the earth) at the equator. Thus, there is a prediction that a deformable body will shape itself to closely follow the equipotential surface. This seems very logical, because otherwise it would be under unbalanced stresses. So since super-sensitive clocks today probably have greater precision than force measurements, we have used clocks+GR to verify a classical equilibrium prediction that carries over to GR.

 

[bcrowell]

For starters let's stop the handwaving:

We know the Earth's mass, radius, rotational speed and equatorial bulge so can we have the equation for the clock at the pole and the equator using the Kerr metric?

Note: if we can't do it please say so, but please no cop outs like "we don't need to", "we ignore rotation because the rotation is slow", we want to do GR here.

Any takers?

The field of the Earth is not a Kerr metric. The equations needed in order to analyze the results are given on p. 23 of the reference I provided in #1. They are expressed in terms of the gravitational potential. If you consider that handwaving, then you should probably protest vigorously to Dr. Alley, assuming he's still alive.

 

[pervect]

I'm not familiar with the prediction Einstein was trying to make here, but I assume --given that this was a paper on special relativity-- that he simply used an incorrect example of the difference in time between two inertial frames of reference.

What happened was that Einstein hadn't worked out gravitational time dilation yet, so of course he didn't incorporate it in his SR paper.

Thus his analysis was based on treating the Earth as a rotating inertial frame. If you carried out the experiment of comparing clocks in a rotating inertial frame, there would be no gravitational effects, and you'd find that his prediction was correct. For instance, if you had a rotating wheel with a negligible gravitational field in flat space-time, you'd find that the clocks on the periphery of the wheel would run slower than clocks in the center because of their motion.

 

[Passionflower]

The field of the Earth is not a Kerr metric.

The field of the Earth? What field?

So are you saying that the Kerr metric cannot be used to handle spacetime outside of a rotating mass similar to the usage of the Schwarzschild metric outside a non rotating mass?

 

[PAllen]

The field of the Earth? What field?

So are you saying that the Kerr metric cannot be used to handle spacetime outside of a rotating mass similar to the usage of the Schwarzschild metric outside a non rotating mass?

See, for example:

http://arxiv.org/abs/gr-qc/0205127

 

[Passionflower]

See, for example:

http://arxiv.org/abs/gr-qc/0205127

Ah yes, that is correct.

So how do we approach it, I do not think there is an exact solution which comes close, so numerical relativity?

 

[pervect]

The moral of the story - a little "handwaving" (i.e. abstract theory) in advance can save one the time and wasted effort from doing totally pointless calculations. So, it's not a good idea to "skip over" all those "handwaving" sections in one's physics books - they're trying to tell the reader something important. If the reader is able to listen...

 

[Passionflower]

wasted effort from doing totally pointless calculations. So, it's not a good idea to "skip over" all those "handwaving" sections in one's physics books - they're trying to tell the reader something important. If the reader is able to listen...

I agree that it is not pointless to read handwaving. However I strongly disagree that calculations are pointless.

Look nobody is doing relativity calculations on this forum for any practical usage anyway. But in doing calculations one will learn. Too often students (and perhaps others) 'think' they know but only when they are able to practically apply they will know they know.

For instance on this forum I looked back in time and I did not go to the beginning but I never have seen anyone attempt to solve some hypothetical situation using the Kerr metric. You may say, "oh I know the metric, so in principle I can calculate, so it is pointless", but if you never have applied your presumed knowledge you might be in for a surprise when you actually try ("you" not being pervect but I mean any person here).

What is wrong in trying something new and calculate just for the love of math and physics? Perhaps you are professionally in this field but lost the love for it?

 

[John232]

If Einstein goofed makeing special relativity then I would like you to show us how you would be able to create a simple trajectory of a photon mathmatically that has two objects traveling at different speeds measureing a photon to have the same velocity.

The only way to do it is to give both observers there own time that is independent of each other. If they where both to use the same time the distance traveled of the photon would always come out to be zero using any type of geometry on it.

The triangle (ct)^2+(vt)^2=(ct)^2 would have zero distance of each side since two of the sides of the triangle would have the same distance. Then each side would have to lie on the same line. The only way to have to two objects to travel a real distance as they are seen to do is for them to have independent times.

 

[GoldPheonix]

What happened was that Einstein hadn't worked out gravitational time dilation yet, so of course he didn't incorporate it in his SR paper.

Obviously. What's your point?

Thus his analysis was based on treating the Earth as a rotating inertial frame. If you carried out the experiment of comparing clocks in a rotating inertial frame, there would be no gravitational effects, and you'd find that his prediction was correct. For instance, if you had a rotating wheel with a negligible gravitational field in flat space-time, you'd find that the clocks on the periphery of the wheel would run slower than clocks in the center because of their motion.

I think you misunderstand what the larger issue here is --there is no such thing as a "rotating inertial frame". It's a contradiction in terms.

Yes, clocks will run slower on the edge of the wheel, as observed by an inertial frame (i.e. moving constantly WRT the axis of rotation), but they will have different effects in a frame that is rotating along with those clocks (i.e. a rotating observer). What he said would be true (ignoring GR), if you were at rest somewhere outside of the Earth observing it. But since people on the Earth are comoving, in a sense, with the clocks that will also affect how we observe the flow of time. (Obviously, though, the existence of a gravitational field makes calculations like these quite invalid)

 

[bcrowell]

If Einstein goofed makeing special relativity then I would like you to show us how you would be able to create a simple trajectory of a photon mathmatically that has two objects traveling at different speeds measureing a photon to have the same velocity.

Post #1 doesn't say that SR is wrong, just that Einstein made a well known mistake in a particular prediction.

 

[pervect]

Obviously. What's your point?

I was trying to fill you on the history of Einstein's "error". You asked:

I'm not familiar with the prediction Einstein was trying to make here, but I assume --given that this was a paper on special relativity-- that he simply used an incorrect example of the difference in time between two inertial frames of reference.

Hopefully what I meant is now clear - that if you look at the history, Einstein's calculation was correct for what he knew at the time, and that it requires effects that he didn't discover until later to explain the actual, physical situation that we measure whereby all clocks on the geoid tick at the same rate.

I think you misunderstand what the larger issue here is --there is no such thing as a "rotating inertial frame". It's a contradiction in terms.

Sorry, I should have said "rotating frame" or possibly "rotating coordinate system".

Yes, clocks will run slower on the edge of the wheel, as observed by an inertial frame (i.e. moving constantly WRT the axis of rotation),

We don't really need to consider inertial frames at all - I think the notion of an inertial frame is unneeded and just confusing things here.

In order to compare the rate at which clocks tick, we don't even need to define the notion of simultaneity. All we have to insist is define how we compare clocks. The only important consideration is that the route over which we send the light or radio signals by which we compare the clocks be "stationary" and that it (the route) doesn't change with time. The notion of "stationary" being used here is imposed by the physical surface of the Earth - it's means the path which the signals move over by which we compare the clocks isn't moving with respect to the Earth. This would perhaps be a "frame", but not (as you point out) an inertial one.

In the hypothetical case of a rotating disk of negligible mass, or a rotating Earth without a gravitational field, we would find that the rate at which clocks ticked WOULD depend on their distance from the center (in the case of a disk) or how close they were to the North pole (in the case of a rotating Earth without a gravitational field).

We wouldn't need any inertial frames at all to make this observation. And I think it's clearer without them.