The timekeeping of clocks

Like much else, the timekeeping of clocks is dealt with in modern physics with a disturbing lack of clarity and a number of falsehoods. The start point for any ‘explanation’ offered is in the theories of relativity of 1905 and 1915/16. By starting instead from what is currently known, we can be much clearer here.

## Gravitation

Clocks in planes and satellites tick ever so slightly faster than those on the ground. Since we wish to be entirely clear, we know this for atomic clocks, specifically. This difference in timekeeping is most noticeable in all communications with satellites and more distant craft. It is said to be vital that we know the details of this in calculating global positioning by satellite (but see later), where we use the time of travel of light (radio) signals and the precise time of arrival of those signals.

This effect is predicted by general relativity or, more strictly, the Schwarzschild metric. To be more precise still, the effect is predicted by a single expression in the metric. This gives us the adjustment to timekeeping compared to an imagined clock away from gravitational influence. The calculation is simple, provided we have a calculator, and is detailed below.

## Motion

Moving clocks run slow, according to Einstein in 1905, but not according to observation.

Hafele and Keating^{i} flew both ways around the world. The clocks travelling westward gained time (ran fast) and those travelling eastward ran slow overall. No one has done the experiment for clocks flying north or south. We need to break down the figures to see what is happening.

## Clocks in planes

Clocks in planes are affected in both ways, due to their height (the gravitational effect) and due to their motion (the Lorentz effect). In any experiment the actual amount due to gravitation will depend upon how long you are in the air (and this in turn will depend upon the route you take), and at what height. The Lorentz effect will depend upon how long you are in the air, and at what speed. The two effects are similar in amount, and both are measurable.

Put an atomic clock in a plane, as Hafele and Keating did in October 1971, and send it around the world, and you find that when it returns it has ticked faster than the one in the laboratory due to its additional height. This is combined with a Lorentz velocity effect, positive or negative, depending on direction. H&K flew both ways around the world, with four caesium clocks each time. Averaged over their four clocks, they found that clocks travelling westward had run fast by 273ns (275ns predicted) and those travelling eastward had run slow by 59ns (40ns predicted). A breakdown of the predicted (calculated) figures is available on the site of Epstein. This takes into account the longer flying time westwards, against the prevailing winds.

The researchers’ analysis of their results has been savaged by the late AG Kelly on his site. He makes a strong argument that the figures provided had been selectively distorted. Kelly’s work on a variety of anomalies in modern theory is generally excellent. My own re-analysis of the H&K calculations, which are repeated in more detail in Mike Berry’s standard text , ‘Principles of Cosmology and Gravitation’ (pages 76 to 79), is that, prior to applying the Lorentz transformation, they add and subtract velocities in a way that would be acceptable to Newton, but not to Lorentz or Einstein, and so their calculated figures related to motion are questionable.

## Other observations: gravity effect

The gravitational effect on the timekeeping of clocks is mirrored by the experiment of Pound and Rebka^{ii} in 1960. They found that light emitted by a particular atom at the bottom of the Jefferson Tower at Harvard University was not absorbed by an apparently identical atom at the top of the tower, 74 feet above. They went further. By moving the emitter as it emitted they added a Doppler shift to the signal (considered as a wave) until it was the right frequency to be absorbed. This enabled them to measure with good precision the difference in frequency between the light emitted at the bottom of the tower and that absorbed at the top.

This confirmed that light is emitted and absorbed at different frequencies in different gravitational fields, and also the formula used below. This ‘gravitational redshift’ is also seen in the spectrum of light emitted from stars, again confirming the formula.

The gravity effect is therefore viewed as a single effect on atoms and molecules, affecting three features of atoms in an identical way, namely rate of vibration, frequency of emission of light and frequency of absorption of light. This confirms what we already know, that these are different aspects of the same physical phenomenon, the vibration of atoms.

## Other observations: Lorentz effect

Definitive observations and calculations, clearly communicated, are harder to come by for the velocity effect. We have seen elsewhere on this site that we are often right to be suspicious when we cannot see the data and associated reasoning.

Certainly, the effect of motion on timekeeping has been tested for certain satellites, but these clocks have not returned for checking, an important part of the H&K protocol, and we do not know if the calculations harbour the same flaw.

We know also that particle accelerators worldwide find it harder to accelerate the particles as they speed up, and this appears to be in accord with the Lorentz transformation (considered to be affecting the mass).

## Bang! goes the reputation

Very few people know that clocks flying westward run fast instead of slow, and that includes physicists, though it is clear in the H&K papers and in some textbooks.
And physicists who do know this will almost never tell you about it!
In 2010, Dallas Campbell of ‘Bang!’, the snappy BBC TV science programme, flew an atomic clock around the world and concluded, apparently quite correctly, ‘I am categorically 230ns older by taking that flight than I would be if I had stayed on the ground.’ This appears to be the only time that the H&K experiment has been repeated.

From the figures reported above and calculated below, we can see that Campbell flew westwards, and this was confirmed by a graphic. The clock ran fast because of its additional height, and also ran fast because of its motion.

Yet the programme contained the following exchange:

National Physical Laboratory physicist expert: ‘Yes, relative to the clock left on the ground.’

Campbell: ‘OK, and general relativity, the further away I am from a massive object like a planet, the faster time is going to go.’

NPL: ‘That’s right.’

Campbell: ‘So which one of these is going to have the biggest effect on our time dilation experiment?’

NPL: ‘The bigger effect actually occurs due to the general relativistic effect.’

Note that Campbell’s advisor is correct on the gravitational effect but incorrect on the Lorentzian one.

Two factors make this more disturbing. One is that the programme was checked by the Open University, and the other is that it is hard to conceive how the NPL could arrive at its predicted time difference (245ns older) without considering both effects as speeding up, and adding them together.

## Clocks in planes: gravitational effect: the Schwarzschild metric

The adjustment to timekeeping due to height in the metric is: **km/r**

where m is the mass, r the radial distance, and k is a constant equal to 7.4 x 10^{-28} Ns^{2}kg^{-2}

The mass of the Earth is approximately 6 x 10^{24} kg, and so k times m = 0.00444

If we take the radius of the Earth, and therefore the r value for the laboratory, to be 6,380 km (6.38 x 10^{6} m) and the height of a flying plane to be 10 km (r = 6.39 x 10^{6} m), we can calculate the adjustment to timekeeping for each as:

Lab: 6.959 x 10^{-10} Plane: 6.948 x 10^{-10}

These adjustments tell us how much slower each clock ticks than the idealised one, as a proportion of the time taken. The plane adjustment is less than the laboratory one, and so the plane clock ticks faster, by a factor of approximately 1.1 x 10^{-12}. This is an exceptionally small difference, but we can measure it. A flight around the World, staying in the northern hemisphere, is in the air for about 160,000 seconds, and so the timekeeping discrepancy, measured when the flying clock returns, is around 175 nanoseconds, and with atomic clocks this is measurable.

This is in accordance with the figures in H&K and ‘Bang!’

(Note that k is calculated from k = G/c^{2}, where the values of c, the speed of light (2.998 x 10^{8} m/s) and G, the universal gravitational constant (6.672 x 10^{-11} Nm^{2}kg^{-2}) are well established.)

## Clocks in planes: speed effect: the Lorentz transformation

The Lorentz transformation is √(1 – v^{2}/c^{2}), and this gives a proportional adjustment to timekeeping of ½ x v^{2}/c^{2} for speeds much less than the speed of light. For a round the world flight of, say, 40,000 km at 240 m/s (around 540 mph), with the speed of light taken to be 3 x 10^{8} m/s, the proportional adjustment is 0.000 000 000 000 4 on a time of flight of, say, 160,000 seconds, for a time difference of 64 ns.

This is rather less than the adjustments for velocity in Hafele and Keating, but combined with the above gravitational figure is almost perfect for ‘Bang!’ This suggests that H&K were in the air for longer, as would be the case if their circumnavigation was south of that of Bang!

## Clocks in GPS satellites

Satellites orbit much higher than planes fly, and experience a gravitational field about one-sixteenth of that which we do. For timekeeping calculations, the important figure is that their height, from the centre of the Earth, is around four times ours. For a GPS satellite, at a height above sea level of 20,180 km, we get an r value of 2.66 x 10^{7} m. Compared to clocks in deep space, this gives a proportional adjustment of 1.67 x 10^{-10}, and a time discrepancy compared to the ground of around 45 microseconds each day. (The satellite clock ticks faster)

This is less important than we might think, as what we really require is that the satellites keep consistent time, rather than time correct to an earthly standard. This is because it is the difference in travel times between different satellites and the user that is important, rather than the absolute time at which those signals arrive.

## Consequences for theory

We have seen elsewhere on this site a cavalier attitude in physics towards care and precision in theoretical analysis, and this is true for the timekeeping of clocks. The implications of these results are therefore considered for special relativity here and for general relativity here.

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i. Hafele, J. C., Keating, R. E., Around-the-World Atomic Clocks: Predicted Relativistic Time Gains, Science, 177 (1972) page 166.

Hafele, J. C. and Keating, R. E., Around-the-World Atomic Clocks: Observed Relativistic Time Gains, Science, 177 (1972) page 168.

ii. Robert V. Pound and Glen A. Rebka, Jr., Resonant Absorption of the 14.4-keV Gamma Ray from 0.10-µsec Fe57, Phys. Rev. Letters (1959).

Robert V. Pound and Glen A. Rebka, Jr., Apparent Weight of Photons, Phys Rev Letters (1960).