Paradox
“How wonderful that we have met with a paradox.
Now we have some hope of making progress.”
Neils Bohr
Introduction
It is always impressive how skilfully supporters of SR demolish any perceived paradox raised by would-be critics. It’s therefore with some trepidation that we raise the following concerns based on the undoubted success of the GPS system.
At the heart of this system, in operation, is the accuracy of the clocks orbiting the Earth in their satellites and the degree with which they remain synchronised to their ground based counterparts. The paradox – if indeed it exists – concerns the engineering adjustments made to the tick rates (periods) of the orbiting clocks prior to their launch although the method of synchronisation used is also of interest with respect to the discussions on the Relativity pages of this site.
Pre-launch
Figures used in this section can be checked at the following link:
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
According to SR we should analyse the behaviour of the orbiting clocks from the rest frame of the ground based clocks. From this viewpoint a clock in orbit will be travelling at a speed of something like 14,000 km/hr and is designed with a clock period of 1 nanosecond. At this speed SR calculations show that the orbiting clock will fall behind the ground based clock by 7 microseconds per day because it will have a slightly slower tick rate (longer period). For the required accuracy of location, 5 to 10 metres, the clock periods from the orbiting clocks must be known to within 20-30 nanoseconds.
A further complication arises when we consider general relativity, GR. According to this theory the orbiting clock will tick at a faster rate (shorter period) than the ground based clock because it is further away from the centre of the Earth and so will be less affected by Earth’s gravity. Calculations show that the orbiting clock will gain 45 microseconds per day on the ground based clock.
The net result is that the orbiting clock will gain (45-7) 38 microseconds per day.
We are looking for an accuracy between 20-30 nanoseconds in comparing signals from orbiting clocks with those on earth. This problem is solved when, prior to launch, the satellite clocks have their rate of ticking decreased (periods increased) so that once in orbit they tick at the same rate as the ground based reference clocks.
So the GPS system, a significant engineering achievement, relies for its success on clock period adjustments made in accordance with both SR and GR.
So where is the paradox?
Post-launch
Suppose now we decide to return an orbiting clock to Earth. What would we expect its tick rate to be assuming no further tinkering occurred during its return journey?
Again from the rest frame of the ground based clocks we analyse the problem by applying SR and GR to the returning clock. Immediately prior to its descent the clock is ticking in synchrony with its ground based counterpart. Once at rest on the ground:
1. Its tick rate has increased (shorter period) because it is now ‘at rest’ so it will gain 7 microseconds per day (by SR).
2. Its tick rate has decreased (longer period) because of increased gravitational effects so it will lose 45 microseconds per day (by GR).
These are simply the reverse of the arguments used earlier.
The net result is that the clock tick rate is such that it will lose (45-7) 38 microseconds per day with reference to the ground based clock and so will be ticking exactly as it was immediately prior to the original launch.
In orbit
To keep the story going we now move our frame of reference to the orbiting GPS satellite just before returning its clock to Earth.
From this elevated position we note, following an exchange of signals, that our on-board clock is ticking at the same rate as its ground based counterpart as it should be.
By similar reasoning to that used above we find that:
3. Our now ‘stationary’ clock when returned to Earth will be moving at a speed, relative to us of 14000 km/hr. Our clock’s tick rate will have decreased (longer period) and so loses 7 microseconds per day compared to the stay-at-home clock (by SR); which does not agree with 1 above.
4. Our returned clock’s tick rate will have decreased (longer period) on the ground, because it is then closer to the centre of the Earth and the increased gravitational effect causes it to lose 45 microseconds per day (by GR); which agrees with 2 above.
The net result is that our unadjusted clock will lose (45+7) 52 microseconds per day on its return to Earth when, according to a ground based view it should only lose 38 microseconds per day.
The arguments from GR (2 and 4 above) are identical and, as expected, cause no problems.
The arguments from SR (1 and 3 above) suggest different outcomes and present us with the core of the paradox.
Resolution
If the discussion above seems rather familiar that’s because it is (almost) a re-presentation of the standard SR clock paradox and, as noted earlier, there are many ways supporters of SR explain it away. A quick web search will access many of these but for a criticism of them all see
Symmetrical Experiments to Test the Clock Hypothesis by Ling Jun Wang,
available at:
http://www.wseas.us/e-library/conferences/florida1999/Papers/p3.pdf
Of real interest is the question from which of the two frames of reference, Earth or orbit, do we get a practical solution to the problem of synchronizing the clocks?
The obvious answer is the Earth frame of reference because the GPS project works.
What then is wrong with the analysis of our orbiting scientists? If they constructed two clocks in orbit and sent one down to Earth to be synchronised with their satellite clock they would use arguments 3 and 4 above to tweak the Earth bound clock .
Their system wouldn’t work – but the GPS system does.
It looks as if we have to choose the Earth bound reference frame in preference to the orbiting one. But according to SR all inertial reference frames have equal status so how have we decided that, even so, the Earth frame is the correct one?
Stepping away form conventional SR solves the problem. The solution is available in Rel part II where the centre of mass (or more generally a centre of momentum) frame provides a locally preferred inertial reference frame. In this scenario clock rates do not vary as a result of the relative speed of one body with respect to another. They do vary in accordance with the speed of each body relative to the centre of mass of the system of masses of which they are a part.
Even if we consider the orbiting laboratory to be located on the moon we would still expect a similar result. The Earth is some 81 time more massive than the moon and orbits the Earth at a distance of roughly 240,000 miles and a speed of less than 2,400 mph. In reality both the Earth and the moon orbit around their common centre of mass which is something less than 3,000 miles from the centre of the Earth ie beneath the Earth’s surface. With these approximate figures the moon orbits this common centre of mass at something like 2370 mph and the Earth ‘orbits at 30 mph. In practical terms, even for scientists on the moon, the Earth is all but stationary relative to the Earth-moon centre of mass.
There is little wonder that the GPS community synchronise all their clocks to an imaginary, non-rotating reference frame through the centre of the Earth. This reference frame is (to better than the accuracy required of GPS clocks) the centre of mass frame of the Earth and its orbiting satellites, including the moon. The following apposite quotation says it all:
“…but it is very important to note that the GPS satellites’ clock rate and the receiver’s clock rate are not adjusted as a function of their velocity relative to one another. Instead, they are adjusted as a function of their velocity with respect to the chosen frame of reference – in this case the earth-centred, non-rotating, (quasi) inertial frame”
Ronald R Hatch, p2, Relativity and GPS (Section I; Special Relativity)
Available to download form:
http://www.worldsci.org/pdf/abstracts/abstracts_1783.pdf
Any experiment performed on Earth to demonstrate the validity of SR will always give the ‘correct’ answer. Movements on or within Earth, or orbitally or otherwise gravitationally linked to Earth, are usually measured with respect to some location on Earth having taken into account its rotation. Such movements expressed as speeds relative to that location will effectively be speeds measured in the centre of mass frame of the Earth and all its satellites because the Earth dominates our local system of masses just as the Sun dominates the solar system of masses.
The ‘paradox’ presented here offers a way, with current technology, of finding out which explanation passes the test of experiment.
We are placing our money on the off-piste, centre of mass version.