In our previous article Gravitational redshift Part III - Experiments, we have mentionned the **Pound–Rebka experiment**, proposed in 1959 by Robert Pound and his graduate student Glen A. Rebka Jr, to test the gravitational redshift or Einstein effect, predicted as soon as 1907 in Einstein's paper On the relativity Principle and the conclusions drawn from it".

This experiment, as successfull as it was - the result confirmed that the predictions of general relativity were borne out at the 10% level, still had two limitations:

- it only tested **gravitational** time dilation

- it was **not** measured with **macroscopic clocks**

In October 1971, Hafele and Keating flew cesium beam atomic clocks^{[1]} around the world twice on regularly scheduled commercial airline flights, once to the East and once to the West.

In the opening statement of the first of two papers on the subject, the authors refer to the debate surrounding the "twins paradox" and how an experiment with macroscopic clocks might provide an empirical resolution.

In this experiment, both **gravitational time dilation** and **kinematic time dilation** are significant - and are in fact of comparable magnitude. Their predicted and measured time dilation effects were as follows:

Let us see how to calculate these relativistic predictions.

##### Kinematic effects (Special Relativity)

First let us consider as inertial referential the so called **ECI** (*Earth Centered Inertial*) with the center of Earth as origin but which does NOT rotate with the Earth^{[2]}. In this non-rotating referential, we will note +v the speed with respect to the earth of the plane flying eastwards and -v the speed with respect to the earth of the plane flying westwards.

Consequently, a plane will fly with the velocity **RΩ+v** (respectively **RΩ-v**) with respect to the** ECI referential**, R being the earth's radius at the equator and **Ω the rotational speed of earth. **We can use the Newton's addition of speed law there as we ignore the terms in higher order than v^{2}/c^{2}.

The ratio of the proper time interval (as mesured between two beats of the on-board clock) and the time interval as measured in ECI is then given by the usual Lorentz factor, as the consequence of the **Transverse Doppler Effect**

Therefore the relative shift of frequency between a flying clock and another one at rest in the ECI is given by:

In an analoguous way, we can deduce the relative shift frequency between a clock situated at the equator **but this time at rest with respect to the Earth** and another hypothetical clock at rest in the ECI referential as:

The shift of frequency between the flying clock and another one of reference at rest at the equator (with respect to the Earth) is then obtained by substraction of the two last terms:

Now it is unlikely that the flight happens exclusively along the equator, so the equatorial radius should be replaced by Rcosλ, λ being the latitude:

This kinematic relativistic effect is made of two terms:

- - the first one which is proportional to v
^{2}/c^{2}, often qualified as**second-order Doppler**effect - - and the second one, proportional to v/c
^{2}, and which**therefore depends on the direction of the travel**; this last term is due to the Sagnac effect, which will be covered in a further article.

##### Gravitational effects (General Relativity)

As we know, General relativity predicts an additional effect, in which **an increase in gravitational potential** due to altitude** speeds the clocks up**.

We have already derived the shift in frequency in this case in our article Gravitational redshift Part II - Derivation from the Equivalence Principle

which was giving a (negative) frequency shift when observed in a region of a weaker gravitational field; in our case, from Earth's perspective, the expression needs to be of opposite sign: on Earth's surface, the time slows down so the clock frequency speeds up and Δf/f>0.

So that the final expression of the shift between the Earth referential clock and the onbaord one becomes:

##### Results

Kinematic & Gravitational relativistic effects |
Eastbound |
Westbound |

Second-order Doppler Effect | -51 ns | -47 ns |

Sagnac Effect | -133 ns | +143 ns |

Gravitational Effect | +144 ns | +179 ns |

Predicted | -40 +/- 23 ns | +275 +/- 21 ns |

Measured | -59 +/- 10 ns | +273 +/- 7 ns |

**Remark 1**: These theoretical results could be approached by using the following values:

v=830km/h, RΩ=1700km/h, h=10km, c=300,000km/h, g=10m/s^{2}

**Remark 2: **In 1996 to commemorate th 25th anniversary of the Hafele and Keating experiment, NPL featured in a BBC horizon program that involved a flying single caesium flying clock from London to Washington and then back again. The predicted gain of 39.8ns for the flying clock compared very well with a measure of 39ns, providing once again a clear demonstration of relativistic effects.

[1] With an accuracy of from 2 to 3 parts in 10 to the 14th, i.e. 0.00000000000002 Hz, which corresponds to a time measurement accuracy of 2 nanoseconds per day or one second in 1,400,000 years, ** the second is the most accurate realization of a unit that mankind has yet achieved**.

[2] The inertial frame is fixed with its origin at the center of the earth. Its Cartesian axes remain fixed relative to the stars, and provide a reference frame for which the equations of motion are most simply expressed.

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