A symmetric twin paradox thought experiment is described and discussed. It raises some interesting questions about relativity. It is also open to experimental testing using atomic clocks orbiting the Earth. The data might be available already.
Introduction
The twin paradox is well known in Special Relativity and is usually explained by reference to the different levels of acceleration experienced by the twins.
To challenge that explanation a thought experiment is presented which is completely symmetric. It is argued that a distant observer will see each twin age slowly but at the same rate as the other, and that the twins must reach the same conclusion.
This conclusion lends support to the version of relativity that was being developed by Fitzgerald, Poincaré and Lorentz prior to publication of Special Relativity by Einstein in 1905.
This essay is part of a series investigating Mach’s Principle, the galactic rotation curve problem and associated issues related to dark matter, (Van de Vusse, 2024).
Basic Twin Paradox.
The Twin Paradox is thought experiment highlighting one of the main implications of Special Relativity. Identical twins meet, synchronise identical clocks and then one of them (B) sets off in a very fast spacecraft, eventually completing a return trip and coming back to meet their sibling (A). When they meet they compare clocks and they discover that B’s clocks give a lower elapsed time than A’s clock. Irrespective of the type of clock.
The Paradox is presented as follows: A has observed B travelling at very high speed, but B is entitled to think that it was A that was travelling at a very high speed relative to B. So should not A’s clock read lower than B’s? But the A clock cannot read less than the B clock if the B clock is reading less than the A clock. It’s a paradox!
The paradox is usually ‘explained’ by noting that Special Relativity is only simple if the observer is using an inertial reference frame. If Twin B experiences accelerations, rotations or decelerations then they have not been in an inertial reference frame. A’s clocks will show a bigger elapsed time than B’s clock and B will agree that this is so.
But what happens if both twins have exactly the same experience?
A Symmetrical Twin Paradox
Consider a dense massive star and imagine two identical spacecraft both in the same circular orbit around that star, but travelling in opposite directions. Each spacecraft is effectively in free fall. Each spacecraft can rotate once per orbit is a way that maintains their orientation relative to the distant stars and galaxies. Their spacecraft is as close to being an ‘inertial frame of reference’ as possible.
Now imagine that there is an astronaut aboard each craft and that the two astronauts are identical twins. Each twin has a telescope looking constantly at the other twin’s spacecraft. Each twin has a very precise and reliable atomic clock and the time it shows is displayed on a large screen which the other twin can see using their telescope.
Taking care not to crash into each other, the twins orbit at a very fast speed to counter the pull of the star. Each sees the other craft, twin and clock pass by at a very fast speed and recede into the distance and then come back again and whizz pass them once more. Over and over. They can compare clocks unambiguously each time they pass each other. Twice per orbit. Now whose clock runs the slower? The situation is entirely symmetric.
The answer has to be that each twin sees the other’s clock running exactly as fast as their own.
Now consider a third observer in a third space craft a billion or so kilometers away. ‘Floating gently’ in deep space at a constant distance from where the twins keep crossing paths. And imagine that such observer has an identical atomic clock and that each twin sends a reading from their clock every time they pass by each other. What will such observer notice about the time readings being received from the twins orbiting the star?
Using the symmetry argument again suggests such observer must receive identical readings from each twin.
If this was a real experiment the clock used by the distant observer would also contain a smaller gravitational time dilation factor, but careful calculations could correct for this.
Some may claim that Special Relativity is not designed to cope with systems featuring large amounts of gravity and this is true. However, the paradox can be modified as follows. Get rid of the dense star, add a second set of twins positioned just above the first pair, connect all the satellites to a central swivel using very strong fibres and make the motion of the second pair mirror that of the first pair so that they exactly counterbalance.
The thought experiment need not involve gravity. Figure 1 above shows fours clocks rotating in pairs in opposite directions. Each clock feels the same forces and acceleration. Each clock sees the clocks in the other pair moving very fast. But the symmetry argument suggest that they would not observe the other clocks to be running slow.
Ultra Precise Timekeeping and Global Positioning Satellites
Over the last forty years or so, impressive achievements have been made in developing global positioning systems (GPS) and ultra precise atomic clocks.
For GPS to work, atomic clocks on Earth have to be well synchronised with atomic clocks aboard specially designed satellites. There are a variety of relativistic effects to take into account. The main effect is explained by General Relativity. Gravity slows down time. Earthbound clocks are in stronger gravity than the orbiting satellites, as explained by Isaac Newton’s inverse square law. The effect of gravity is slightly reduced by centrifugal accelerations caused by the spin of the Earth. The overall gravitational effect is about 45 microseconds per day.
The gravitational time dilation effect is then adjusted for smaller relativistic effects, the main one being a Special Relativistic time dilation because the satellites are moving fast relative to the Earthbound clocks. This offsets the gravitational effect by about 7 microseconds per day, giving a net relative adjustment of 38 microseconds per day.
When GPS satellites were first deployed the scientists in charge were not totally confident how much fine tuning would be required to get perfect synchronisation, so they allowed for a large degree of post launch adjustment. Now they make most of the adjustments before launch.
Of course gravity can have a direct physical effect on clocks. For example, a pendulum clock could not work without it. But that it not what we are talking about here. We are talking about an impact on time itself.
The GPS satellite clocks are proof that there are special and general relativity factors that affect time itself. (It is not easy to develop an intuitive appreciation of this. We are accustomed to assuming that time is a well behaved, well defined parameter. A universal parameter flowing gently into the future. It isn’t. In fact ‘time’ is probably the trickiest parameter in the whole of physics.
[It is difficult, and sometimes even impossible, for two separated observers to synchronise their clocks or to agree which events occurred first. The more fundamental and reliable concept is causality. If event A causes event B and event B causes event C then all and all observers should be able to agree that event A occurred ‘before’ event C.However, ‘time’ is more or less an illusion created by repetitive events.
Many cosmological models have a Universal time parameter as if there was some sort of Universal master clock. There isn’t. And talk of what happened in the first few seconds of the Big Bang should explain that the word time is being used very loosely.]
But let us return to the GPS clocks and orbiting twins.
An Experiment to Test the Symmetric Twins Paradox
GPS systems provide proof that there are General Relativistic and Special Relativistic factors that have to be taken into account to maintain synchronicity between atomic clocks at rest on the surface of the Earth and atomic clocks in orbiting satellites.
The smaller special relativity adjustment (also known as velocity time dilation) could be used as an experiment to explore the symmetrical twin paradox.
Imagine a satellite in a geostationary orbit above a time laboratory situated somewhere on the Earth’s equator and that the two locations have identical atomic clocks. The geostationary satellite clock has been adjusted for the gravity effect and any other possible distortions not related to Special Relativity. How much slower is the space based clock compared to the Earth based clock?
One argument goes as follows. An observer based at the Earth station sees the space based clock in its geostationary orbit as essentially standing still. They might reasonably expect there to be no Lorentzian time dilation effect at all.
Now imagine there is another satellite in the same orbit as the geostationary satellite bit travelling in the other direction i.e. retrograde. The ground based observer sees this travelling quite fast across the night sky. The observer is entitled to think that time aboard this retrograde satellite must be slowed by the Lorentzian gamma factor. The ‘transverse Doppler shift’ has even been demonstrated in laboratory experiments using high speed centrifuges, gamma rays and the Mossbauer effect. Ives and Stilwell first performed this experiment in 1938. Walter Kundig managed an even more accurate version 25 years later (Kundig, 1963).
However, observers aboard the two satellites do not need to know what the ground based observer thinks. They are both in inertial reference frames. Relative to a non-rotating reference frame that happens to be centered at the middle of the Earth they are both traveling equally fast. Looking through a telescope at the other satellite reveals it to be doing exactly the same thing that they are doing.
Maybe they both expect that the other satellite’s clock is running slower than theirs. But that would be a contradiction. The clocks can’t both be slower than each other when they are repeatedly observed in the same place and time.
Maybe they expect that the other satellite’s clock is running at the same rate as theirs. This is a not a paradox as far as they are concerned, but it is a paradox when they discuss this with the ground based observer. The ground based observer expects the retrograde satellite to be aging more slowly than the geostationary satellite.
What would the distant observers in the far off universe expect to see happen to the satellite based clocks? Depending on their angle of view they might see the two satellites going round and round, or back and forth and sideways. In addition to gravity effects and effects from the Sun orbiting around the centre of the Milky Way, they would expect a very small Special Relativity effect from the movement of the satellites in the Solar System, and they would expect this effect to be the same for both satellites.
Discussion and Resolution
The symmetric twin paradox challenges the usual explanations of the twin paradox.
The best placed observers in the above example are the ones located in deep space. The Earth based observers are not in an inertial reference frame and so what they expect to see is complicated. The satellite observers are in inertial reference frames but there is something wrong. Each satellite can consider itself to be stationary and the other satellite to be doing all the travelling. When looking at the other satellite they see it moving out and back in a loop, but without expending any rocket fuel.
What the thought experiment shows is that the definition and determination of an “inertial reference frame” as used by Einstein in Special Relativity needs careful thought. The safest option in most cases is to use a reference frame aligned to the cosmos.
What is the result as confirmed by experimental evidence from GPS and other satellites? These show a velocity dilation effect of about 7 seven microseconds per day between a typical GPS satellite and a typical Earth based observer. But all the GPS satellites seem to go around in the normal pro-grade direction. They are launched towards the east in order to benefit from the Earth’s own rotation. Launching into a retrograde orbit uses more fuel. It also increases the risk of collisions with other spacecraft – a bit like driving against the traffic.
Conjectures:
- The velocity time dilation in ultra precise satellite timekeeping is best modeled as a function of the rate of travel in a reference frame aligned partly to the Milky Way and partly to the background provided by distant galaxies.
- The velocity time dilation will be a function of how fast the clocks are travelling in this frame.
GPS is very important in the modern world, including for space based research and defense, so presumably there is a community of scientists who are expert in ultra precise time keeping. They may already have the data and experience to answer the above conjectures.
Conventional Twin Paradox and Clock Postulate
The usual ‘explanation/resolution/conclusion’ of the standard (non-symmetrical) twin paradox is that one twin experiences accelerations and decelerations and this causes its clock to end up slower than the clock of the twin that stayed at rest.
Some presenters of this argument go further. They use the Principle of Equivalence to assert that accelerations are the same as gravity, and gravity is known to slow down the passage of time and so this explains why the accelerated clocks end up slower.
However, this all seems contrary to the Clock Postulate. ‘The ‘clock postulate’ is that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity. It was implicitly included in Einstein’s original 1905 formulation of Special Relativity, and it has been verified experimentally at very high accelerations within particle accelerators.
Since the travelling twin decelerates to zero and then has to accelerate to regain speed for the return journey, the Clock Postulate suggests a lessening in its overall time dilation. The decelerations/accelerations argument does not resolve the Twin Paradox – it makes it worse.
Conversely the conjectures offered above to resolve the Symmetrical Twin Paradox also resolve the normal Twin Paradox. Whichever twin travels the fastest for longest relative to a universal background reference frame is the twin that will end up the younger when they meet again. In a symmetrical situation they will age the same.
Conclusion
The symmetric twin paradox suggests that although both twins are inertial observers and each sees the other moving rapidly, there is no time dilation in the other’s clock relative to their own. This highlights the complexities presented by spinning and orbiting systems in Special Relativity. The complexity is reduced if is conceded that it is always possible to determine which system is rotating and which one is not. Physics involving spinning, rotating and orbiting is simplest in a reference frame that pays attention to what is going on in the local galaxy and in the Universe as a whole.
References
Van de Vusse, Sjoerd B.A., 2024, Some ideas and experiments for issues affecting modern physics, https://hereticalphysics.com.au/
Kündig, Walter, (1963). Measurement of the Transverse Doppler Effect in an Accelerated System. Physical Review. 129 (6): 2371–2375. Bibcode:1963PhRv..129.2371K. doi:10.1103/PhysRev.129.2371
Author contact: SBAvan@utas.edu.au Author’s location: Hobart, Australia
Addendum
The Wikipedia article on the Twin Paradox is well written and comprehensive (see https://en.wikipedia.org/wiki/Twin_paradox). The final section is relevant to the arguments advanced in the above essay. Look it up to get the links to the references. Here is a reformatted extract:
“No twin paradox in an absolute frame of reference
Einstein’s conclusion of an actual difference in registered clock times (or aging) between reunited parties caused Paul Langevin to posit an actual, albeit experimentally indiscernible, absolute frame of reference:
In 1911, Langevin wrote: “A uniform translation in the aether has no experimental sense. But because of this it should not be concluded, as has sometimes happened prematurely, that the concept of aether must be abandoned, that the aether is non-existent and inaccessible to experiment. Only a uniform velocity relative to it cannot be detected, but any change of velocity … has an absolute sense.”
In 1913, Henri Poincaré’s posthumous Last Essays were published and there he had restated his position: “Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one.”
In the relativity of Poincaré and Hendrik Lorentz, which assumes an absolute (though experimentally indiscernible) frame of reference, no paradox arises due to the fact that clock slowing (along with length contraction and velocity) is regarded as an actuality, hence the actual time differential between the reunited clocks.
In that interpretation, a party at rest with the totality of the cosmos (at rest with the barycenter of the universe, or at rest with a possible ether) would have the maximum rate of time-keeping and have non-contracted length. All the effects of Einstein’s special relativity (consistent light-speed measure, as well as symmetrically measured clock-slowing and length-contraction across inertial frames) fall into place.
That interpretation of relativity, which John A. Wheeler calls “ether theory B (length contraction plus time contraction)”, did not gain as much traction as Einstein’s, which simply disregarded any deeper reality behind the symmetrical measurements across inertial frames. There is no physical test which distinguishes one interpretation from the other.
In 2005, Robert B. Laughlin (Physics Nobel Laureate, Stanford University), wrote about the nature of space: “It is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed … The word ‘ether’ has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. … Relativity actually says nothing about the existence or nonexistence of matter pervading the universe, only that any such matter must have relativistic symmetry (i.e., as measured).”
In Special Relativity (1968), A. P. French wrote: “Note, though, that we are appealing to the reality of A’s acceleration, and to the observability of the inertial forces associated with it. Would such effects as the twin paradox (specifically — the time keeping differential between reunited clocks) exist if the framework of fixed stars and distant galaxies were not there? Most physicists would say no. Our ultimate definition of an inertial frame may indeed be that it is a frame having zero acceleration with respect to the matter of the universe at large.”
Author contact: SBAvan@utas.edu.au
Author’s location: Hobart, Australia
