Permanent Sunset and Theory of Relativity
The research described in the reference work is based on the argumentative analysis from Galileo's relativity viewpoint of a realistic situation described as "permanent sunset", involving an airplane flying towards the West at a speed equivalent to that of the Earth's rotation. . This same situation can be useful for teaching Einstein's special relativity theory (SRT) at a later stage of teaching, for several reasons:
Students will be interested in deepening the analysis of this same situation with new conceptual tools.
The situation described as “permanent sunset” corresponds to the design of one of the first experiments carried out to empirically verify the reality of time dilation (which is one of the most counterintuitive consequences of special relativity): the experience of Hafele and Keating.
In light of the SRT and experimental evidence, the problem underlying permanent sunset has a defined answer that privileges the heliocentric model over the geocentric one.
Hafele and Keating experiment
In October 1971, Hafele and Keating synchronized three cesium atomic clocks (A, B, C) in order to check whether their measurements (as predicted by the theory of relativity) were affected by their state of motion (Schlegel, 1974).
Clock B was left in a base in Washington and the others were placed in commercial aircraft that flew around the world in 2 days. The geopolitical circumstances of the time (in the middle of the Cold War) prevented these aircraft from flying over most of the Asian continent, so half of this time was spent flying in the North-South direction (meridians). As a result, they only flew in the East-West direction (parallels) for one day, which is the time we will consider.
Description
Clock A flew in a West direction and clock C in an East direction, both at an altitude of 10,000 m.
Since the Earth rotates in one day, the speed of these planes relative to the Earth's surface was equal to the speed at which the Earth rotates around its axis.
The precision of the atomic clocks used in this experiment was 20 ns
Results
Experimental values measured (time difference compared to the base clock, B):
∆tAB = + 270 ns, ∆tCB = - 60 ns
Criticism and proposed analysis
According to the theory of relativity, time depends on both speed (kinematic effect predicted by special relativity) and gravitational acceleration (dynamic effect corresponding to general relativity).
The experimental design carried out is not suitable to separately validate both effects, since, although the clocks in A and B were subject to the same gravitational attraction because they flew at the same height and for the same time, this is not true for clock B which remained on the Earth's surface the whole time.
If a fourth clock G had been placed in a hot air balloon and kept stationary at a height of 10,000 m for the duration of the experiment, Hafele and Keating could have separated the effect of acceleration (by comparing the times of B and G) from that of velocity (by comparing the times of A, G, and C).
Therefore, to focus the analysis on the dependence on velocity we must be able to discount the influence of gravity.
One way to do this is to assume general relativity and apply it to the clock at B to calculate what the time would be marked if it had remained at the same height as the planes.
Cascading questions:
1- Is the result of the experiment compatible with the Galilean idea of absolute time (independent of the observer's state of motion)?
To do this, the three clocks should mark the same time upon returning to base.
(in case time depends on motion):
2- Is the result of the experiment compatible with the geocentric model of a stationary Earth at rest?
For this (due to the symmetry of the situation), the clocks in A and C should show the same time when they return.
(in case the Earth is not at rest but rotating towards the East):
3- Is the result of the experiment compatible with the relativistic concept of time dilation?
For this, the clocks in motion must be delayed with respect to the clocks at rest.
In this case, according to the model of Earth's rotation, the clock in A would be at rest while the Earth rotates below it, while the clock in B would be moving dragged by the Earth's rotation, and the clock in C would move at a faster speed than that in B.
(in case time dilation is confirmed qualitatively):
4- Is the quantitative result of the experiment compatible with the Lorentz formula for time dilation?
According to this formula, time for a moving observer dilates in proportion to the relativistic "gamma" factor. This effect is proportional to the square of the observer's speed when it is much lower than the speed of light (as is the case here).
Since the clock at C is moving eastwards on the Earth's surface at the same speed as the Earth's surface is rotating eastwards, both speeds will add up, so the speed of C will be twice that of B.
Since the effect depends on the square of the speed, this means that the time dilation of clock C should be 4 times greater than that of B.
Chained answers
1-The times measured by the three clocks do not coincide when returning to the base, which is not compatible with the Galilean idea of absolute time. That is, the time measured by a clock depends on its state of motion. This allows us to ask the second question.
2- The results obtained
∆tAB = + 270 ns, ∆tCB = - 60 ns
are not compatible with a static Earth, because if this were the case, nothing would distinguish the movement of an airplane flying towards the East from another flying towards the West.
Therefore, we must use the model of an Earth rotating towards the East to ask the third question.
3- If we accept the Earth's rotation, we must also assume that the clock in A is at rest, so it will be convenient to calculate the delays of the moving clocks (B and C) with respect to the clock at rest in plane A.
∆tBA= - 270 ns, ∆tCA = - 330 ns
In this way, we verify that time is delayed for an observer (clock) in motion, which qualitatively confirms the phenomenon of time dilation predicted by Einstein.
We can therefore also try to validate this phenomenon quantitatively (fourth question).
4- Since time dilation depends on both speed (kinematic effect) and acceleration (dynamic effect), we must separate both effects to focus on the kinematic aspect of our interest.
As explained in the critical analysis of the experimental design, one way to do this is to assume general relativity and apply it to the clock in B to calculate what the time would be if it had remained at the same height as the planes.
According to general relativity, a clock subjected to a gravitational action (which essentially consists of a curvature of spacetime) will suffer a temporal contraction, so its time will be ahead with respect to another clock that was free of this gravitational acceleration.
Under the conditions of the experiment carried out by Hafele and Keating, the calculations of general relativity indicate that the clock in B would be ahead (+ 170 ns) with respect to a hypothetical clock G that had remained in a hot air balloon at the same height as the planes.
Therefore, to make the comparison with these, it would be enough to delay the time of B by 170 ns to obtain what the clock in balloon G would mark compared to the clock at rest (A).
The result would be:
∆tGA = - 100 ns, ∆tCA = - 330 ns
We see that the time delay in C is much greater than twice that of G, although it is not exactly 4 times greater.
This discrepancy can be justified by taking into account that the flights carried out did not exactly compensate for the Earth's rotation due to the geopolitical circumstances of the moment.
Conclusion
From the analysis carried out, it can be seen that the results obtained by Hafele and Keating support the validity of the allocentric model of a rotating Earth and an airplane at rest as opposed to the egocentric description (airplane moving over an Earth at rest).
The application of these considerations to resolve the dilemma presented in the problem of permanent sunset, however, cannot be anticipated until later educational stages, once the students have learned, accepted and internalized the reality of the phenomenon of time dilation.
However, if we do not leave Galilean relativity, there is no reason to prefer a heliocentric system to a geocentric one (Lanciano, 1989)
References:
Lanciano, N. (1989). Ver y hablar como Tolomeo y pensar como Copérnico. Enseñanza de las Ciencias, 7(2), 173-182. https://doi.org/10.5565/rev/ensciencias.5018.
Schlegel, R. (1974). Comments on the Hafele-Keating Experiment. American Journal of Physics, 42(3), 183-187. https://doi.org/10.1119/1.1987645.