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Hafele - Keating Experiment Notes – 2.0
© GC - October 2018 / October 2024
Gianni Casonato – giacas11@gmail.com
Table of Contents
1 Background......................................................................................................................................2
1.1 Scope........................................................................................................................................2
1.2 References................................................................................................................................2
2 The Experiment...............................................................................................................................3
2.1 Description...............................................................................................................................3
2.2 Time Shift Model.....................................................................................................................4
2.2.1 Reference Frame Choice..................................................................................................4
2.2.2 Earth-Based Circular Path Case.......................................................................................5
2.2.3 Simplified Expression for Low Altitude Flight................................................................7
2.2.4 Model Error Sources........................................................................................................8
2.2.5 IAU Relativistic Model for Earth Flights........................................................................8
2.3 Predicted Values.......................................................................................................................9
2.3.1 HK Model Predictions.....................................................................................................9
2.3.2 IAU Model Predicted Values.........................................................................................10
2.4 Flight Data.............................................................................................................................10
2.4.1 Eastward Trip.................................................................................................................11
2.4.2 Westward Trip................................................................................................................13
2.5 Predicted Time Shift Re-calculation......................................................................................14
2.5.1 Eastward Trip.................................................................................................................15
2.5.2 Westward Trip................................................................................................................15
2.6 Observed Time Shift and Comparison with Predictions........................................................16
2.6.1 Experiment Measurement Results..................................................................................16
2.6.2 Measurement vs. Predictions Residuals.........................................................................19
3 Conclusions...................................................................................................................................21
© GC – HK Experiment Notes 2.0 1
1 Background
1.1 Scope
The note addresses the experiment conducted in the 70’s by two researchers, Hafele and Keating,
for measuring the effect on atomic clocks time of a relative motion of flying clocks with respect to a
ground based time reference station, and to compare the results against the predictions made by
special relativity theory.
This experiment is often referenced as demonstration of Special and General Relativity, and it is
interesting to clearly assess the limit of validity of its results for the objective of demonstrating
Einstein's relativity theory.
In the note the calculation of predicted time shift is re-calculated based on SR/GR model using
clocks flight data in [HK-3] and then compared with the measurements made by Hafele and Keating
as reported in [HK-1/HK-2].
1.2 References
The following documents are considered references for the content of this note:
[HK-1] – Around the World Clocks: Predicted Relativistic Time Gains – Hafele J. C., Keating R .E.
– Science Vol. 177 (1972)
[HK-2] – Around the World Clocks: Observed Relativistic Time Gains – Hafele J. C., Keating RE. –
Science Vol. 177 (1972)
[HK-3] – Performance and Results of Portable Clocks on Aircraft – Hafele J. C., – PTTI, 3rd
Annual Meeting, 1971
[HK-4] - Relativistic Time for Terrestrial Circumnavigations – Hafele J. C., American Journal of
Physics 40, 81 (1972)
[HK-5] - Relativistic Behaviour of Moving Terrestrial Clocks – Hafele J. C., Nature, Vol. 227, p.
270 (1970)
[SRT] – Special Relativity – French A., 1968
[GRVTD] – Gravitational Time Dilation – Wikipedia
[REL-ITU] – Relativistic Time Transfer – ITU 2018
© GC – HK Experiment Notes 2.0 2
2 The Experiment
2.1 Description
The Hafele-Keating (HK) experiment on time dilation was done in 1971 with the objectives of
measuring time shift on moving clocks with respect to an Earth based reference clock by flying
clocks in eastbound ans westbound trips using commercial flights (see [HK1] and [HK2]), and
proving the correctness of the prediction made using the theory of relativity.
The experiment consisted in using two high precision Caesium atomic clocks, synchronised with a
US reference ground station (located at USNO Washington DC laboratories) for flying them on
commercial airplanes in two separated closed flight trips, one eastward, and one westward. After
returning to USNO laboratories the clock times were compared with the reference time and the time
shifts recorded.
Initially all the clocks are co-located at USNO and synchronised between each other. The the
portable clocks are sent away for their eastward and westward flight paths, until they all get back to
USNO. Here the clock elapsed time measures are performed and compared with the predicted
values calculated applying the special and general relativity models to this specific experiment
setup. The expected accuracy for the portable Cesium clocks in the 70th used in the experiment was
about 3.6 ns/h i.e. 10-12 s/s [HK-1].
© GC – HK Experiment Notes 2.0 3
lab+flying
clock sync
...
lab+flying
clock diffs
flying clock
path #1
flying clock
path #N
Δτg (full trip time)
Δτg=ΔτgΔτg≠Δτg
Δτg (actual flight time)
2.2 Time Shift Model
2.2.1 Reference Frame Choice
As explained in [HK-4] and [HK-5], the formula considered applicable for this experiment can be
derived from the GR base expression, by considering very special reference frame setup for getting
to a simplified model. More specifically the reference frame is a “fixed” frame centred in the Earth
and with the z-axis passing to the North pole, (ECI frame) and three moving reference frames
located respectively at each clock location, namely at USNO for the reference clock, and on the
various airplanes for the two flying clocks.
To be noted that the above reference frame, 3-dimensional Earth centred, is reduced in [HK-4] to a
2-dimensional one, without further explanation. A possible justification could the use of a 3D frame
with a z axis aligned with Earth’s rotation axis, and a frame origin very far from Earth’s centre, such
that the problem geometry can be seen as 2-dimensional within a certain accuracy level. This
simplification would also require to define a reference 2D plan height. Looking at [HK-4] it is that
assumed the reference plane is the equatorial plane, and that both ground clock and flying clocks
have a latitude and then 2D projected radius difference from the reference one.
Using that setup, the experiment can be modelled with some approximations as a relative motion of
USNO and flying clocks on circular trajectories projected in the equatorial plane with a fixed 2-
dimensional ECI-like frame. For the clocks, that would implies:
•
Rg=Recos λg
and
ve= ΩRecos λg
the ground clock 2D radial position and velocity,
given by Earth values scaled by the actual ground clock latitude cosine
•
Rf= (Re+h)cos λf
and
vf= ΩRecos λf+vgs
the flying clock 2D radial position and
velocity, scaled by actual flying clock latitude cosine
The error introduced by the clock’s position flattening into a 2-dimensional space to the equatorial
plane is linked to their relevant latitude, and it would cause a shortening of their actual position by a
factor function of the latitude cosine. Even though this error is expected to be small for low altitude
paths, this aspect is not elaborated at all in [HK-4] and [HK-5], and no error figure is provided.
The relative motion between the USNO located and the flying clocks would create a time shift
between them. This experiment is a demonstration of the so-called “clock measure paradox” [HK-1]
and [SRT], where two co-located and synchronised clocks are separated and then rejoined after one
© GC – HK Experiment Notes 2.0 4
y
x
z
Re
Pck
x
y
Re+h
G
Ω
ve
vf
has travelled far from the other, showing that the transported one has recorded a duration shorter
than the clock at rest.
To be noted that being the choice of the reference frame fully arbitrary in SR/GR, depending on the
choice it would happen that both clocks are either slower or faster than the other. Anyway the
experiment only assess the correctness of one (the more intuitive) of the two possible reference
frame choices (USNO clock at rest and airplane ones moving) and not the other one.
2.2.2 Earth-Based Circular Path Case
From a time drift perspective, the relative motion between the USNO located and the flying clocks
and their altitude difference would create a time shift. The starting point for an analytic derivation
of the model presented in [HK-4] is the general GR expression, where the differential expression for
the metric in the moving frame with respect to the one in the fixed frame is
ds2=(1+2χ
c2)c2dt2− [ dr 2
1+2χ
c2
+r2(dθ2+sin2θdϕ2)]
Introducing the experiment specific characteristic, namely:
•weak gravitational field (χ << c2) being
χ =− GM
R+h
•low velocity (v2 << c2)
and making some further simplifications in the expression terms (e.g. 2nd order effects negligible),
and assuming constant average velocity v, latitude and altitude h, the final formula for a generic
clock time shift becomes for both ground and flying clocks
Δ τ = ∫
t0
t1
(1−α(h ,v , λ)) dt = (1−α (h , v , λ)) Δ t
or
Δt= (1+ α(h , v , λ)) Δ τ
with
α(h , v , λ) = GM
c2R(1+h/R)+[RΩ(1+h/R)cos λ+v]2
2c2
where
•c is the speed of light
•h is the ground or flying clock altitude above Earth’s surface – assumed constant
•GM is the Earth’ gravitational parameter
•R is the Earth’s radius
•Ω is the Earth’s angular velocity
© GC – HK Experiment Notes 2.0 5
•λ is the ground or flying clock geographical latitude – assumed constant
•v is the ground or flying speed during the flight path (=0 for ground clock, >0 for flying
clocks) – assumed constant
•Δt is the coordinate elapsed flight time, i.e. elapsed flight time in the reference frame
•Δτ is the moving clock proper elapsed time
This expression contains two terms responsible for the offset between reference and moving frame
times:
•a gravitational term, depending on clock altitude
αgrav =GM
c2R(1+h/R)
•a kinematic term depending on clock velocity magnitude
αkin =[RΩ(1+h/R)cos λ+v]2
2c2
The above expression is general and valid for both ground and flying clocks, for paths where the
assumptions of constant altitude, velocity and latitude holds. In case of variability over time of such
parameters an integral expression has to be used, using the expression above as differential and
integrating over the whole time interval.
Δt=∑
i=1
N
(1+ αi(hi, vi,λi)) Δ τi
Above expression should be applied to both USNO ground and flying clocks, both bringing proper
time in their moving reference frames, for deriving the relevant coordinate times and to make a
comparison. In formulas
Δtg= (1+ αg(hg, v g,λg)) Δ τg
Δtf= (1+ α f(hf, v f,λf)) Δ τ f
Starting from above expressions, in order to be able to measure the clock difference using SR/GR, a
special setup must applied, to comply with the space-time events comparison restrictions imposed
by the theory. Specifically, SR/GR allows only to make time comparisons between clocks placed at
the same spatial location, so that the corresponding space-time events only differ by time. Vice
versa, clocks in different positions cannot be compared looking for pure time difference.
The experiment fulfil this constraint by performing a closed loop trajectory with the moving clock,
starting and ending at same space location. Under that conditions the delta time measure of co-
located clocks is an invariant observable between different inertial frames, and it can be
equivalently be performed in one of the clock frame [HK-1].
From another perspective, that can be seen as if the proper time differences between the two co-
located clocks at start and at end correspond to the same coordinate spacetime event in the reference
© GC – HK Experiment Notes 2.0 6
frame, i.e. Δtg = Δtf. Applying this result to the above expressions gives for the relative ground vs.
flying clocks proper time difference
(1+ αg) Δ τg= (1+ α f) Δ τ f
Δ τg , f = αgΔ τg− αfΔ τf
By noting that
Δ τf= Δ τg− Δ τg , f
it finally follows
Δ τg , f =αf− αg
1+ αf
Δ τg
The two Δτ values are available from measurement for ground and flying clocks at the end of the
experiment, and then above expression provides a theoretical model to compare measurements
against. To be noted that based on clocks’ respective position and velocity, it is expected to see the
flying clock eastward path time drift slower than ground clock (i.e. Δτg,f < 0) and for the westward
path faster (i.e. Δτg,f > 0).
In addition to the net time drift value, another useful parameter is the drift rate, which can be
compared with clocks “natural” drift specs (i.e. 10-13 s/s for Cesium based clocks used in the
experiment).
δg ,f =Δ τg , f
Δ τg
=αf− αg
1+ αf
2.2.3 Simplified Expression for Low Altitude Flight
As explained in [HK-4] and [HK-5], the general expression above can be further simplified
assuming h << R and neglecting the high order terms (but not justifying such simplification in terms
of order of magnitude of the errors introduced), giving
Δ τ = { 1 −GM
c2R−[RΩcos λ+v]2
2c2}Δt
= [1−gh
c2−(2RΩcosλv+v2)
2c2] Δ t
= [1− α*(h , v , λ)] Δ t
where, in addition to the parameters defined before for the general expression
•g is the “relativistic” gravitational acceleration on ground, namely
g=GM
R2−RΩ2cos2λ
To note that the low altitude simplification implies an error in the gravitational component which
was acceptable for the accuracy of this experiment.
For a generic path at latitude λ, the formula for the drift rate in this case can be expressed in a
simple way
© GC – HK Experiment Notes 2.0 7
δ = gh
c2− [(2RΩcos λ + v)v
2c2]
Finally, in [HK-1] is also provided an integral expression to be used in case of time variable
altitude, velocity or latitude. This expression is the following
Δ τ = ∫
ta
tb
[1−gh( τ)
c2−2RΩcos λ( τ) v(τ)+v( τ)2
2c2]d t
and has to be used for integrating the different delta time samples along the path for the time frame
measured at USNO. That implies in particular that the flight time to be used has to be the actual
flying time, i.e. during which velocity and altitude where not zero, and that “stop” periods between
one flight leg to another has to be excluded in the calculation.
2.2.4 Model Error Sources
The analysis of the uncertainties affecting he HK models presented in previous sections presented in
[HK-1] focuses on two main error sources:
1. errors or missing information in the flight data, assumed to affect the result with an order of
magnitude of ~60% of net value eastward and ~8% westward
2. theoretical approximations in the model expressions for neglecting Moon and Sun
gravitational field , assumed to be negligible for the accuracy of the experiment
2.2.5 IAU Relativistic Model for Earth Flights
Finally, for sake of completeness, a mention to the SR/GR IAU models for time transport provided
in [REL-ITU] is also given. Specifically considering a clock close to Earth’s surface, the
corresponding delta coordinate time is given by
ECI/TCG
Δt=∫
ta
tb
[1+U(r)
c2+v2
2c2]dτ
where:
•r and v are the clock frame position and velocity in ECI
•U(r) is the Earth gravitational potential at position r, excluding the centrifugal potential, and
expressed with different accuracy levels, e.g. as spheric harmonic expansion
U(r) = GM e
r+J2GMe(1−3 cos ² θ
2r3)
ECEF/TT
Δt=∫
ta
tb
[1−ΔU(r)
c2+v(τ)2
2c2]dτ + 1
c2∫
ta
tb
(
ωxr)⋅
v d τ
© GC – HK Experiment Notes 2.0 8
where:
•r and v are the clock frame position and velocity in ECEF
•ΔU(r) is the Earth gravitational positional difference between the clock at position r and
colatitude θ and one on the geoid, including the centrifugal potential
ΔU(r) = U(r) − Ug
with
U(r) = GM e
r+J2GMe(1−3 cos ² θ
2r3) + 1
2ω2r2sin2θ
and
Ug=U(Re)
Note: ΔU(r) is conventionally < 0 if calculated above the geoid.
•ω is the Earth angular velocity
2.3 Predicted Values
The pre-experiment time drift predictions were calculated using flight time average values of the
flight data recorded by flight crew. The other formula’s parameter values for velocity, altitude, and
latitude are instead not disclosed.
The values used in the note are the following reference values:
Parameter Eastward Trip Westward Trip
Trip time (flight time only) [hour] 41.2 48.6
Altitude [km] (*) 9.0 9.5
Ground speed [km/s] (*) 0.21 -0.21
Latitude [deg] (*) 34.0 31.0
(*) Missing values are extrapolated empirically for deriving the predicted values shown below.
2.3.1 HK Model Predictions
As reported in [HK-1], the predicted time shift calculated by the authors, including gravitational
and kinematic components too, is
Trip Direction Δτkinematic [ns] Δτgravitational [ns] Δτtotal [ns]
Eastward -184 ± 18 144 ± 14 -40 ± 23
Westward 96 ± 10 179 ± 18 275 ± 21
The deviation terms are calculated taking into account for the model error sources introduced
above.
To be noted that the error terms are in the same order of magnitude of the cesium clocks accuracy,
and then this has to be considered in using the clock measure values for the assessment of the
experiment objectives [HK-2].
© GC – HK Experiment Notes 2.0 9
The values obtained by recalculating the prediction using the HK models introduced in 2.2 are
provided in the following table.
Model / Trip Direction Δτkinematic [ns] Δτgravitational [ns] Δτtotal [ns]
HK Circular
Eastward -184.6 144.1 -40.4
Westward 96.8 179.5 276.3
HK Circular Low-Alt
Eastward -169.5 144.0 -25.5
Westward 114.1 179.4 293.5
Those values match the predicted ones from HK-1 within the provided error margins.
2.3.2 IAU Model Predicted Values
Applying the IAU models to the experiment parameters introduced above, the following values are
obtained.
Model / Trip Direction Δτkinematic [ns] Δτgravitational [ns] Δτtotal [ns]
SR/GR General
Eastward -197.7 144.1 -53.6
Westward 96.8 179.5 276.3
SR/GR Earth
Eastward -184.6 130.4 -54,2
Westward 96.8 163.3 260.1
The IAU predictions are pretty much in line with the values reported in previous section, and taken
from HK-1.
It is worth to mention that aside IAU model results reported above, similar results can be also
obtained with alternative relativity models, based or not on relative time (e.g. Selleri and others).
2.4 Flight Data
Detailed flight path data is provided in [HK-3], with departure and arrival airports and relevant
times, associated to overall trip flight duration and average ground speed, latitude and altitude
information.
The precise values for velocity and altitude, as well as for flight path latitude, needed for the
calculation of the experiment’s expected values, were not disclosed by the authors neither in [HK-3]
nor in any other reference document(!). That makes not possible an accurate recalculation by a third
party.
© GC – HK Experiment Notes 2.0 10
To be noted that based on above data, for both trips a “flat” altitude profile is considered, i.e. the
given average flight altitude is used, neglecting altitude changes for take-off, landing and in flight
manoeuvres, assuming those interval to be “small” w.r.t. the whole flight duration.
2.4.1 Eastward Trip
The westward trip started on 04.10.1971 and finished on 07.10.1971. The detailed steps are in the
following table.
Date GMT Location (*)
Coordinates –
Lat [deg]
Coordinates – Lon
[deg] Distance [km]
4. Oct. 1971 07:30:00 PM USNO D (by car) 38.921674 -77.066884
5. Oct. 1971 12:12:00 AM Dulles D 38.9531162 -77.4565388 33.9
5. Oct. 1971 06:56:00 AM London A 51.4700223 -0.454295 5901.7
5. Oct. 1971 08:14:00 AM D
5. Oct. 1971 09:09:00 AM Frankfurt A 50.0379326 8.562152 653.5
5. Oct. 1971 10:36:00 AM D
5. Oct. 1971 12:48:00 PM Istanbul A 40.9829888 28.8104425 1862.4
5. Oct. 1971 01:57:00 PM D
5. Oct. 1971 03:13:00 PM Beirut A 33.819376 35.491204 990.6
5. Oct. 1971 04:19:00 PM D
5. Oct. 1971 06:13:00 PM Tehran A 35.6899882 51.311241 1458.5
5. Oct. 1971 07:40:00 PM D
5. Oct. 1971 10:41:00 PM New Delhi A 28.5561624 77.0999578 2546.0
6. Oct. 1971 12:00:00 AM D
6. Oct. 1971 03:33:00 AM Bangkok A 13.738007 100.645141 2935.6
6. Oct. 1971 05:13:00 AM D
6. Oct. 1971 07:45:00 AM Hong Kong A 22.308047 113.9184808 1694.8
6. Oct. 1971 08:55:00 AM D
© GC – HK Experiment Notes 2.0 11
havg
TTO1 TL,N
TL,1
...
TTO,N
6. Oct. 1971 12:16:00 PM Tokyo A 35.549393 139.779839 2902.1
6. Oct. 1971 02:32:00 PM D
6. Oct. 1971 09:10:00 PM Honolulu A 21.3245132 -157.9250736 6191.5
6. Oct. 1971 11:14:00 PM D
7. Oct. 1971 03:50:00 AM Los Angeles A 33.9415889 -118.40853 4108.0
7. Oct. 1971 04:47:00 AM D
7. Oct. 1971 07:13:00 AM Dallas A 32.848103 -96.851206 2001.3
7. Oct. 1971 07:53:00 AM D
7. Oct. 1971 09:59:00 AM Dulles A 38.9531162 -77.4565388 1869.5
7. Oct. 1971 12:55:00 PM USNO A (by car) 38.921674 -77.066884 33.9
(*) Entries with “D” means departure time from previous line’s location.
The additional average data provided in HK-3 this trip are:
•total flight time (including stops at airports): 65 h 25 m
•average altitude: 8.9 [km]
•average ground speed: 0.243 [km/s] (0.237 [km/s] if calculated from trip steps data above)
•average latitude: 34.0 [deg] (34.7 [deg] if calculated from trip steps data above)
From the above data the actual flight time relevant for time drift results 41 h 14 m.
Flight Path
© GC – HK Experiment Notes 2.0 12
2.4.2 Westward Trip
The westward trip started on 13.10.1971 and finished on 17.10.1971. The detailed steps are in the
following table.
Date GMT Location (*)
Coordinates –
Lat [deg]
Coordinates –
Lon [deg] Distance [km]
13. Oct. 1971 07:40:00 PM USNO D (by car) 38.921674 -77.066884
13. Oct. 1971 11:22:00 PM Dulles D 38.9531162 -77.4565388 33.9
14. Oct. 1971 04:00:00 AM Los Angeles A 33.9415889 -118.40853 3674.0
14. Oct. 1971 05:03:00 AM D
14. Oct. 1971 10:14:00 AM Honolulu A 21.3245132 -157.9250736 4108.0
14. Oct. 1971 01:13:00 PM D
14. Oct. 1971 08:15:00 PM Guam A 13.497036 144.795309 6108.5
14. Oct. 1971 09:13:00 PM D
15. Oct. 1971 12:06:00 AM Okinawa A 26.20935 127.6503 2278.6
15. Oct. 1971 01:07:00 AM D
15. Oct. 1971 02:09:00 AM Taipei A 25.067566 121.552699 624.2
15. Oct. 1971 03:03:00 AM D
15. Oct. 1971 04:13:00 AM Hong Kong A 22.308047 113.9184808 835.5
15. Oct. 1971 12:48:00 PM D
15. Oct. 1971 03:14:00 PM Bangkok A 13.738007 100.645141 1694.8
15. Oct. 1971 04:32:00 PM D
15. Oct. 1971 08:06:00 PM Bombay A 19.09314 72.856753 3019.5
15. Oct. 1971 09:15:00 PM D
16. Oct. 1971 04:03:00 AM Tel Aviv A 32.005532 34.8854112 4045.5
16. Oct. 1971 05:09:00 AM D
16. Oct. 1971 06:45:00 AM Athens A 37.9356467 23.9484156 1193.6
16. Oct. 1971 07:33:00 AM D
16. Oct. 1971 09:03:00 AM Rome A 41.7998868 12.2462384 1086.0
16. Oct. 1971 10:01:00 AM D
16. Oct. 1971 11:38:00 AM Paris A 49.0096906 2.5479245 1101.0
16. Oct. 1971 02:25:00 PM D
16. Oct. 1971 03:57:00 PM Shannon A 52.6996573 -8.914691 901.8
16. Oct. 1971 05:06:00 PM D
© GC – HK Experiment Notes 2.0 13
16. Oct. 1971 11:38:00 PM Boston A 42.3656132 -71.0095602 4646.9
17. Oct. 1971 01:18:00 AM D
17. Oct. 1971 02:26:00 AM Dulles A 38.9531162 -77.4565388 662.8
17. Oct. 1971 04:00:00 AM USNO A (by car) 38.921674 -77.066884 33.9
(*) Entries with “D” means departure time from previous line’s location.
The additional average data provided in HK-3 in this case are:
•total flight time (including stops at airports): 80 h 20 m
•average altitude: 9.36 [km]
•average ground speed: 0.218 [km/s] (0.206 [km/s] if calculated from trip steps data above)
•average latitude: 31.0 [deg] (32.6 [deg] if calculated from trip steps data above)
In this case the actual flight time relevant for time drift results 48 h 39 m.
Flight Path
2.5 Predicted Time Shift Re-calculation
Based on the expression reported in par. 2.2, and using the flight trip data provided in par. 2.3, a re-
calculation of the predicted time shift has been performed using both HK and SR/GR models by
calculating the contributions from all the trip steps. Concerning USNO located clock, the following
values are used:
•altitude: 0.095 [km]
•ground speed: 0 [km/s]
© GC – HK Experiment Notes 2.0 14
•latitude: 38.921473 [deg]
2.5.1 Eastward Trip
For the eastward trip case time drift recalculation the following flying clocks model parameters
from table in 2.4.1 are used:
•altitude: 8.9 [km]
•ground speed: 0.243 [km/s]
•flight time: 41 h 14 m
The results for all models are shown in the tables below for time drift and drift rate.
SR Model Δτkinematic [ns] Δτgravitational [ns] Δτtotal [ns]
HK Circular -217.1 142.6 -74.5
HK Circular Low Altitude -201.9 142.5 -59.5
SR/GR -217.1 142.6 -74.5
SR/GR Earth -217.1 128.8 -88.3
SR Model ΔRtotal [ns/y]
HK Circular -1.0
HK Circular Low Altitude -0.6
SR/GR -1.
SR/GR Earth -1.3
2.5.2 Westward Trip
For the eastward trip case time drift recalculation the following flying clocks model parameters
from table in 2.4.1 are used:
•altitude: 9.6 [km]
•ground speed: 0.218 [km/s]
•flight time: 48 h 39 m
The results for all models are shown in the tables below for time drift and drift rate.
SR Model Δτkinematic [ns] Δτgravitational [ns] Δτtotal [ns]
HK Circular 93.4 177.0 270.4
HK Circular Low Altitude 122.8 176.8 299.7
SR/GR 93.4 177.0 270.4
SR/GR Earth 93.4 149.5 242.9
© GC – HK Experiment Notes 2.0 15
SR Model ΔRtotal [ns/y]
HK Circular 5.7
HK Circular Low Altitude 6.0
SR/GR 5.7
SR/GR Earth 5.4
2.6 Observed Time Shift and Comparison with Predictions
2.6.1 Experiment Measurement Results
The experiment’s results data were provided in two phases, with an initial report, and later on as
consolidated values, with some changes between the two sets.
The initial observed time drift values were presented in [HK-3] as raw measurements for all the
clocks involved in the experiment (four in total for the flying ones), calculated as both average
relative time drifts and average drift rates w.r.t. USNO clock time.
From the measures result table in [HK-3] it can be noted that:
•the four clocks have different behaviours with respect to each other, with some showing
positive time drift and rate and some with negative ones, regardless the trip direction, and
apparently showing an erratic kind of behaviour
•the time drift rate average and standard deviations are below the clock accuracy at rest (see
2.1) for both trip directions
Those differences are explained presenting the table above as “immediate” measurement after the
trips, not waiting for a stabilisation of the clock rates.
Because of the large clock drift rate at rest compared with experiment duration and expected time
drift values, the calculation logic used for clock time drift in [HK-3] took into account for such rate,
as shown in the figure below.
© GC – HK Experiment Notes 2.0 16
(a)
τiτf
Δτ = Δτf - Δτi
(b)
τiτf
Δτ = Δτf – (Δτi + R (τf - τi))
Ri
Rf
Rf
Ri
Δτ
τ
Δτi
ΔτfΔτi
Δτf
τ
R = (Rf + Ri)/2
The case (a) is when the clock rate at rest is negligible for the predicted relativity effect order of
magnitude (like for current atomic clocks), and then the time drift is simply the difference between
end and start times. Vice versa, case (b) considers a not negligible clock rate and the time drift is
also taking into account for such rate.
The actual values for both flight trips are shown in the table below as average values and standard
deviation.
Trip Direction Δτtotal [ns] ΔRtotal [ns/h]
Eastward
Ck 120 -52 -4.39
Ck 361 -110 1.72
Ck 408 35.00
Ck 447 -56 -1.25
Average -54 ± 46 (1σ) 0.27 ± 3.8 (1σ)
Westward
Ck 120 240 4.31
Ck 361 74 -2.93
Ck 408 209 -2.68
Ck 447 116 -2.25
Average 160 ± 78 (1σ) -0.89 ± 2.6 (1σ)
The clock samples and the above average values shown in the plot below for both trip directions.
© GC – HK Experiment Notes 2.0 17
About one year later, another paper was published, with the final experiment results [HK-2]. They
were obtained by recalculating the raw ones for compensating for clock drift rate stabilisation after
the trips end. The values are reported in the following table. To note that recalculated clock rates
were not provided.
Trip Direction Δτtotal [ns] ΔRtotal [ns/h]
Eastward
Ck 120 -57 n.a.
Ck 361 -74 n.a.
Ck 408 -55 n.a.
Ck 447 -51 n.a.
Average -59 ± 10 (1σ) n.a.
Westward
Ck 120 277 n.a.
Ck 361 284 n.a.
Ck 408 266 n.a.
Ck 447 266 n.a.
Average 273 ± 7 (1σ) n.a.
The recalculated samples are shown in the following plot.
© GC – HK Experiment Notes 2.0 18
Here the values looks more in line with the expectations from the theory, but they’re significantly
different from those in [HK-4], in both average values and standard deviations(!), without a specific
description of the type of data post-analysis and consolidation performed.
2.6.2 Measurement vs. Predictions Residuals
Here below a final summary table which provides an overall comparison between predicted values,
re-calculated values using “standard” SR/GR models, and raw and processed results from relevant
papers [HK-3] and [HK-2] respectively.
Raw measures
SR Model Eastward
Δtot-res [ns]
Westward
Δtot-res [ns]
HK Circular 20.5 110.4
HK Circular Low Altitude 5.5 139.7
SR/GR 20.5 110.4
SR/GR Earth 34.3 82.9
Processed measures
SR Model Eastward
Δtot-res [ns]
Westward
Δtot-res [ns]
HK Circular 15.5 2.6
HK Circular Low Altitude 0.5 26.7
SR/GR 15.5 2.6
SR/GR Earth 29.3 30.1
A bar-plot representation of above results is also very useful for highlighting model cross-
comparison and differences between raw and processed measure sets.
© GC – HK Experiment Notes 2.0 19
It can ben noted then that:
•model performance vs. observed values have significant differences between the two result
sets, with some models more accurate in one set and not in the other one, with no clear best
model highlighted
•HK simplified models are generally close to the processed measure set, but they are not so
much for the raw measures. A better HK models matching would be expected, being used as
experiment baseline. The reason for the difference is not explained apart from erratic drift
rate of used clocks(!)
•also general SR/GR models have a good accuracy w.r.t. processed measures, better for
westward trip case than eastward
•the absolute residuals are in the same order of magnitude of the experiment accuracy (tenth
of nanoseconds), and then in general they don’t look adequate to provide a conclusive
answer for providing the validity of the GR model
© GC – HK Experiment Notes 2.0 20
3 Conclusions
The HK experiment setup and supporting theoretical models have been analysed and its results
recalculated by means of HK baseline models in [HK-1] as well ad using standard SR/GR, using
flight data provided in [HK-3]. The recalculated results have been compared with the raw and
processed measured value sets provided [HK-3] and in [HK-2] respectively.
The experiments assumptions, constraints and limitations have been also addressed and highlighted,
specifically:
•Cesium atomic clock accuracy in the order of 10 ns/h (i.e. 10-12 s/s), also including their
unstable clock drift at rest
•prediction model based on a simplified 2D geometry and low-altitude / low-velocity
assumptions. Further simplifications are the exclusion of second order effects and the Moon
and Sun gravitational effects
Moreover, there are some key missing information in the supporting publications which leave gaps
for a full analysis and an accurate re-calculation, namely
•not all model inputs for prediction as well as for actual trip data use are available
•no justification or description of the experiment raw results post-processing applied for
getting to the final published results, noting those final value are significantly different from
the raw ones
Taking into account those points, the note provides a full recalculation of HK time drift values for
the eastward and westward trips using reported flight data, and it compares them with the two clock
measure sets published.
Last but not least, analysing the model vs. measures residuals, some interesting, not intuitive and
not straightforward aspects just looking at HK published data are reported. In particular model
performance are significant different between the two result sets, and in particular HK baseline
models are much closer to the processed measure set than to the raw one (without any explanation,
as mentioned above), but better for westward trip case than eastward one (no justification about this
either). SR/GR models and more in general any relativity model (even based on absolute time) have
a good accuracy w.r.t. processed measures. The absolute residuals are in the same order of
magnitude of the experiment accuracy (tenth of nanoseconds).
A final summary of the note key findings is the following:
1. the accuracy of the clocks used for the experiment, namely the standard deviation of their
raw measured times both on ground and in flight, is of the same order of magnitude of the
portable atomic clocks accuracy used for the flight paths. That raises doubts on the
possibility of using any type of result for the purpose of the experiment’s objectives
2. overall all analysed data, either predicted, recalculated and observed, are within the same
order of magnitude (tenths of nanoseconds for Eastward case and hundreds of nanoseconds
© GC – HK Experiment Notes 2.0 21
for Westward case), but the residual differences as significantly high (up to 40%), different
between trips, and non consistent at model level, meaning that some random effect should
have played a role on the clock drift during the trips, likely due to the the accuracy
limitations of the portable clocks used. That would imply that the experimental results
obtained are not good enough for providing a conclusive answer to the objective of
validating the SR/GR model as the only one valid for time shift
3. many important pieces of information for a more accurate reassessment of the experiment
and a recalculation of the prediction with other models are not provided in all the
publications produced by the authors, in particular with reference to the large post-
processing performed on the initial raw results
Based on that, the HK experiment is definitively a prove of a time drift occurring on atomic clocks
caused by their relative motion (i.e. different relative altitude and velocity) w.r.t. to ground located
ones. Anyway it is not possible consider the HK results are a conclusive evidence in favour of
SR/GR models, because of the experiment’s accuracy limitations as well as the possibility of
obtaining same time drift results with alternative relativity models based or not on relative time (e.g.
Selleri and others). This is a point open for future extensions of the present analysis.
© GC – HK Experiment Notes 2.0 22
Change Log
Date Change Description Version
Oct-2018 First issue 1.0
Jan-2019 Improved HK models description.
Expanded time drift recalculation, model vs. measures
comparison and conclusions sections.
1.1
Sep-2020 Described SR/GR models.
Added details for the flight data and for the recalculated values.
Updated model vs. measures comparison section.
1.2
Oct-2024 Simplified and clarified HK models description.
Described error sources.
Expanded values recalculation including also time drift rates.
Revised experiment measures analysis and described time drift
calculation logic.
Rewritten conclusion section.
Fixed typos.
2.0
© GC – HK Experiment Notes 2.0 23