Hafele & Keating, Einstein, Dingle,Cocke, and Scott Murray

From: John Kennaugh (JKNG_at_kennaugh2435hex.freeserve.co.uk)
Date: 06/16/04 

Date: Wed, 16 Jun 2004 17:04:50 +0100

You may not have heard of Dr Scott Murray but he was a physicist with
the British Scientific Civil Service who had a distinguished career in
radio astronomy and satellite communications. This is mainly about an
article he wrote in 1986 "If you want to know the time...." Electronics
and Wireless World December 1986 P28 to 31 in which he looked at The
Hafele & Keating experiment and other related matters. As far as I can
see with my limited knowledge what he says is correct. You may judge
otherwise, but right or wrong I am certain he was thorough in his
researches. He ends his article:

"To sum up: In 1905 Albert Einstein predicted, as a consequence of
relativity theory, free of paradox, that an ideal clock (at sea level on
the equator would run more slowly than an identical clock (also at sea
level) at the earth's pole. An experiment performed in 1971, purported
to confirm the influence of both special and general relativistic
effects on the timekeeping of quasi-ideal clocks. The theory which was
said to underlie that experiment was also seen to support Einstein's
original prediction.
        However, it is observed that the differential slowing of clocks
due to latitude alone (independent of geographic motion and altitude),
which was predicted by Einstein, does not in fact take place. It seems
that the explanation put forward - and apparently generally accepted -
to account for this theoretical failure was physically unsound".

As far as I can see the explanation which Murray shows to be flawed
seems to be still generally accepted which is why I bring the article to
your attention.
           ---------------------------------------

Einstein said "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 under otherwise identical
conditions." - Einstein, A. On the Electrodynamics of Moving Bodies;
trans. in The Principle of Relativity. Methuen, 1923 and Dover, 1952.
pp.49-50.

Herbert Dingle - who I expect everyone other than Bjoern has heard of,
posed the question, - if speed is relative then why could not the clock
on the equator be considered stationary and the one at the pole be
considered as moving?
Dingle, H. Science at the Crossroads, Martin Brian and 0'Keeffe, 1972.
p.40ff.

Ian McCausland later published a brief summery of the varied
contemporary answers to Dingle's question of 1972 which makes it clear
that *at that time* there was no consensus answer from the physics
community. [Wireless World October 1983 P63-65]

Hafele & Keating were faced with a similar type of question and it was
necessary for them to choose an answer. Their report states "...the
relative timekeeping behaviour of terrestrial clocks can be evaluated by
reference to hypothetical co-ordinate clocks of an underlying
nonrotating (inertial) space." Hafele & Keating, Op.cit. (4): p.166
col.2 & p.167n(6).

They add the following footnote

"It is important to emphasise that special relativity purports to
describe certain physical phenomena only relative to (or from the point
of view of) inertial reference systems, and the speed of a clock
relative to one of these systems determines its timekeeping behaviour"
Builder, G. 'Ether and relativity', Aust..l. Plys. vol. 11, 1958 p.279.

Murray's comments - "Paraphrased, it says that an observer must be 'at
rest relative to an inertial frame of reference' if the explanations of
the physical observations are to be meaningful"

This interpretation answers Dingle's question by saying that the pole is
(or is a reasonable approximation to) an inertial frame and the equator
isn't. H&Ks assumption is therefore consistent with Einstein's
prediction and I understand (ref Bilge) that it is the way SR is
generally interpreted these days.

It would seem therefore that the H&K experiment not only showed that the
predictions of SR were correct and that GR is also correct (it used GR
to correct the data), it also showed by its success that the above
interpretation of how SR should be applied is the correct one, thus
potentially ending the confusion documented by McCausland.

According to Murray H&K used the equation (I've written dt for delta t)

        dt/t = gh/c^2 - (v^2/c^2)/2 -----------------. (1)

The term (v^2/c^2)/2 is Einstein's 1905 prediction due to motion at
velocity v (by special relativity theory), where

     v = Omega r cos (latitude) + u --------------------- (2)

Omega is the earths angular velocity, r is earths radius, and u is the
ground speed.

The term gh/c^2 is a GR correction to compensate for gravitational
potential where h is the height of the plane above mean sea level.
Not having read the H&K report I assume each flight was broken down into
segments and dt/t calculated for each and this compared with the
physical result giving good agreement. Murray assumes this was done with
the utmost of care but does note that acceleration isn't actually
mentioned.

However Murray points out that if two planes were left on the ground at
mean sea level at different latitudes i.e. put u = 0 and h = 0 into the
H&K equation then you get

     dt/t = 1/2 (Ohmega r Cos(lattitude))^2/c^2.

An experiment based on this would have the advantage that no correction
to the data would be needed, and it could be done over a long period to
provide any degree of accuracy required.

At this point I go back to "Thence we conclude that a balance-clock at
the equator must go more slowly ... than a precisely similar clock
situated at one of the poles under otherwise identical conditions."

Murray points out that while it is exceedingly difficult to synchronise
clocks it is not difficult to compare their rates. A comparison via a
two way radio link is self compensating if the comparison is mutual so
any two observatories in line of sight with a geo-stationary satellite
will do. While it would be nice to test Einstein's actual prediction no
geostationary satellites are visible above the horizon at the poles but
the same maths can be used to predict the difference expected with two
alternative latitudes.

The Observatory at Reykjavik 64deg N and one at Recife Brazil at 7deg S
should give a difference of 82.5ns per day. (one complete cycle of a
12MHz oscillator). They don't and this was know to be the case prior to
the H&KX. H&K in fact refer to it in a footnote in their report and
explain this apparently serious failure of relativity by a reference to
Cocke.

Cocke said:

"Clocks at rest on the earth's surface (at average sea level) keep the
same relativistic time independently of latitude differences. The effect
of the difference in surface speed at different latitudes is cancelled
to lowest order by a corresponding effect from the difference in surface
potential owing to the oblate figure of the earth."
Cocke. W.J. Relativistic corrections for terrestrial clock
synchronisation; Phas.Rw.Lett. vol.l6, 1966 p.662.

This it appears is the standard text book explanation of the failure of
Einstein's prediction. Murray followed up the reference to see how
Cocke came to his conclusion. The equation he uses is:

         dt/t = gH/c^2 - (v^2/c^2)/2 = 0

What it is saying is that the (first) GR term conveniently cancels the
(second) SR prediction because of the shape of the earth.

Note H is not as in the H&K equation the height above sea level but the
difference between local earths radius and polar radius. v is the speed
of the earths surface.

The following is my limited understanding of Murray's calculations. If I
have got it wrong don't blame Murray, look up what he actually wrote. I
do not feel confident to convert them into a suitable text format which
can be posted. While I think I understand the maths, even if I don't the
final conclusion he comes to appears to me to be blatantly obvious as
you will see.

He calculates the total gravitational potential at the surface of the
earth as (gH - (v^2)/2) - the first term coming from gravitational of
the earth and the second from centrifugal acceleration. This means that
Cocke's equation is wrong and should be:

        dt/t = (gH - 1/2 v^2)/c^2) - (v^2/c^2)/2
             = gH/c^2 - (v^2/c^2)/2 - (v^2/c^2)/2

The first term is the GR correction for potential due to the mass of the
earth, the second term is the GR correction due to centrifugal
acceleration and the third term is the SR term for time dilation. It
turns out that the last two terms are identical.

Cocke is correct when he says
               gH/c^2 - (v^2/c^2)/2 = 0

so the equation reduces to
               dt/t = 0 - (v^2/c^2)/2

In effect that is saying that there is no GR correction for clocks at
mean sea level. Mean sea level is a unipotential surface. Whether I have
understood the maths or not surely that is blatantly obvious. Water is
free to flow so some of it is not going to remain at a different
gravitational potential to the rest. The essential shape of the earth is
the shape the water takes up with a combination of mass and spin. The
land is simply bits of sea floor which poke through that surface. A
clock at mean sea level anywhere in the world is at the same
gravitational potential. Or as Murray puts it "The precise statement is
that the geoid is defined as a unipotential surface in Earth co-
ordinates - if one understands potential in its usual field theory
sense".

"...the difference in surface potential" caused by the oblate figure of
the earth which Cocke says cancels to lower order the effect of
difference speeds does not in fact exist.

This means that Einstein's prediction is proved wrong and there is no
comforting explanation to come to the rescue.

H&K's equations predict a change in rate of clocks at different
latitudes which does not occur and the interpretation of SR (Builder's)
which they used, supports Einstein's prediction which turns out to be
false.

I read the article in 1986 and was somewhat surprised when, having asked
in this NG as to the current explanation of the apparent failure of
Einstein's prediction Paul Anderson came up with Cocke's explanation
unmodified. While I am not qualified in these matters the argument does
not appear to depend on anything sophisticated. I therefore feel
competent to bring the matter to your attention and hope that I have
done justice to Murray's article.

If anyone would like a copy of the complete article - 89 kBytes - email
me by converting the number in my return address from Hex to decimal or
if you can't manage that by adding 6834 and removing 'hex' :o).

A list of the references used in the full article:

1. Einstein, A. On the Electrodynamics of Moving Bodies; trans. in The
Principle of Relativity. Methuen, 1923 and Dover, 1952. pp.49-50.
2. Dingle, H. Science at the Crossroads, Martin Brian and 0'Keeffe,
1972. p.40ff.
3. Essen, L. Einstein's Special Theory of Relativity, Proc. Royal
institution, vol 45, 1972, p141: p.l50ff.
4. Hafele, J.C. and Keating, R.E. Around-the world atomic clocks (two
papers), Science, vol. 177,1972.pp.166,168.
5. Hafele & Keating, Op.cit. (4): p.166 col.2 & p.167n(6).
6. Builder, G. Ether and relativity, Aust..l. Plys. vol. 11, 1958 p.279.
7. Farley, F.J.M. & Picasso. E. The muon (g-2) experiments,
Ann.Rev.Nucl.Part.Sci. vo1.29. 1979 p.243: pp.259,266.
8. Dingle, II. Op.cit.(2): pp.45-46.
9. Bondi, H. Assumption and Myth in Physical Theory. Cambridge, 1967.
pp.43-52.
10. Einstein, A. Dialogue concerning objections to Relativity Theory".
Naturvawiss. vol.48. 1918 p.697; German, untrans. (Part translated in
Dingle, Op.cit.(2): pp191-195).
11. Builder, G. Op.cit.(6): p.282.
12. Relativistic Effects in a Terrestrial Co-ordinate Time System CCIR
Report No. 439-3, 1982. (Defines TAI).
13. Cocke. W.J. Relativistic corrections for terrestrial clock
synchronisation; Phas.Rw.Lett. vol.l6, 1966 p.662.
14. Hafele & Keating, Op.cit.(4): p.168n (9). 

-- 
John Kennaugh
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