References
7 Global Positioning System and the twins’ paradox
Tom Van Flandern
Abstract
In the Global Positioning System (GPS), all atomic clocks in all reference frames (in orbit and on the ground) are set once and stay synchronized. We can use this same trick to place a GPS-type clock aboard the spacecraft of a traveling twin. That clock will stay synchronized with Earth clocks, allowing a clear resolution of the twins’ paradox in special relativity – why the traveler expects to come back younger, and why the stay-at-home twin is not entitled to the same expectation.
Background
About the Global Positioning System
The Global Positioning System (GPS) consists of a network of 24 satellites in roughly 12-hour orbits, each carrying atomic clocks on board. The orbital radius of the satellites is about four Earth-radii (26,600 km). The orbits are nearly circular, with a typical eccentricity of less than 1%. Orbital inclination to the Earth’s equator is typically 55 degrees. The satellites have orbital speeds of about 3.9 km/s in a frame centered on the Earth and not rotating with respect to the distant stars. Nominally, the satellites occupy one of six equally spaced orbital planes. Four of them occupy each plane, spread at roughly 90-degree intervals around the Earth in that plane. The precise orbital periods of the satellites are close to 11 hours and 58 minutes so that the ground tracks of the satellites repeat day after day, because the Earth makes one rotation with respect to the stars about every 23 hours and 56 minutes. (Four extra minutes are required for a point on the Earth to return to a position directly under the Sun because the Sun advances about one degree per day with respect to the stars.)
The on-board atomic clocks are good to about 1 nanosecond (ns) in epoch, and about 1 ns/day in rate. Since the speed of light is about one foot per nanosecond, the system is capable of amazing accuracy in locating anything on Earth or in the near-Earth environment. For example, if the
satellite clocks are fully synchronized with ground atomic clocks, and we know the time when a signal is sent from a satellite, then the time delay for that signal to reach a ground receiver immediately reveals the distance (to a potential accuracy of about one foot) between satellite and ground receiver. By using four satellites to triangulate and determine clock corrections, the position of a receiver at an unknown location can be determined with comparable precision.
Relativity in the GPS
General Relativity (GR) predicts that clocks in a stronger gravitational field will tick at a slower rate. Special Relativity (SR) predicts that moving clocks will appear to tick slower than non-moving ones. Remarkably, these two effects cancel each other for clocks located at sea level anywhere on Earth. So if a hypothetical clock at Earth’s north or south pole is used as a reference, a clock at Earth’s equator would tick slower because of its relative speed due to Earth’s spin, but faster because of its greater distance from Earth’s center of mass due to the flattening of the Earth. Because Earth’s spin rate determines its shape, these two effects are not independent, and it is therefore not entirely coincidental that the effects exactly cancel. The cancellation is not general, however. Clocks at any altitude above sea level do tick faster than clocks at sea level; and clocks on rocket sleds do tick slower than stationary clocks.
For GPS satellites, GR predicts that the atomic clocks at GPS orbital altitudes will tick faster by about 45,900 ns/day because they are in a weaker gravitational field than atomic clocks on Earth’s surface. SR predicts that atomic clocks moving at GPS orbital speeds will tick slower by about 7,200 ns/day than stationary ground clocks. Rather than have clocks with such large rate differences, the satellite clocks are reset in rate before launch to compensate for these predicted effects. In practice, simply changing the international definition of the number of atomic transitions that constitute a one-second interval accomplishes this goal. Therefore, we observe the clocks running at their offset rates before launch. Then we observe the clocks running after launch and compare their rates with the predictions of relativity, both GR and SR combined. If the predictions are right, we should see the clocks run again at nearly the same rates as ground clocks, despite using an offset definition for the length of one second.
We will refer to a clock whose natural ticking frequency has been corrected in this way as a ‘‘GPS clock’’. This will help in the discussion of SR effects such as the twins’ paradox. A GPS clock is pre-corrected for relativistic rate changes so that it continues to tick at the same rate as Earth clocks even when traveling at high relative speeds. So a GPS clock carried by the traveling twin can be used to determine local time in the Earth’s frame at any point along the journey – a great advantage for resolving paradoxes.
We note that this post-launch rate comparison is independent of frame or observer considerations. Since the ground tracks repeat day after day, the distance from satellite to ground remains essentially unchanged. Yet, any rate difference between satellite and ground clocks continues to build a larger and larger time reading difference as the days go by. Therefore, no confusion can arise due to the satellite clock being located some distance away from the ground clock when we compare their time readings. One only needs to wait long enough and the time difference due to a rate discrepancy will eventually exceed any imaginable error source or ambiguity in such comparisons.
Theory and observations compared
The highest precision GPS receiver data is collected continuously in two frequencies at 1.5-second intervals from all GPS satellites at five Air Force monitor stations distributed around the Earth. An in-depth discussion of the data and its analysis is beyond the scope of this paper (Alley and Van Flandern 1998). These data show that the on-board atomic clock rates do indeed agree with ground clock rates to the predicted extent, which varies slightly from nominal because the orbit actually achieved is not always precisely as planned. The accuracy of this comparison is limited mainly because atomic clocks change frequencies by small, semi-random amounts (of order 1 ns/day) at unpredictable times for reasons that are not fully understood. As a consequence, the long-term accuracy of these clocks is poorer than their short-term accuracy.
Therefore, we can assert with confidence that the predictions of relativity are confirmed to high accuracy over time periods of many days. In ground solutions with the data, new corrections for epoch offset and rate for each clock are determined anew typically once each day. These corrections differ by a few ns and a few ns/day, respectively, from similar corrections for other days in the same week. At much later times, unpredictable pseudo-random errors due to imperfections in the clocks build up, so comparisons with predictions become increasingly uncertain unless these empirical corrections are used. But within each day, the clock corrections remain stable to within about 1 ns in epoch and 1 ns/day in rate.
The initial clock rate errors just after launch would give the best indication of the absolute accuracy of the predictions of relativity because they would be least affected by accumulated random errors in clock rates over time. Unfortunately, these have not yet been studied. But if the errors were significantly greater than the rate variance among the 24 GPS satellites caused by orbit differences, which is less than 200 ns/day under normal circumstances, it would have been noticed even without a study. So we can state that the clock rate effect predicted by GR is confirmed to within no worse than 200/45,900 or about 0.7%, and that predicted by SR is confirmed to within 200/7,200 or about 3%. This is a very conservative estimate. In an actual study, most of that maximum 200 ns/day variance would almost certainly be accounted for by differences between planned and achieved orbits, and the predictions of relativity would be confirmed with much better precision.
Twelve-hour variations (the orbital period) in clock rates due to small changes in the orbital altitude and speed of the satellites, caused by the small eccentricity of their orbits, are also detected. These are observed to be of the expected size for each GPS satellite’s own orbit. For example, for an orbital eccentricity of 0.01, the amplitude of this 12-hour term is 23 ns.
Contributions from both altitude and speed changes, while not separable, are clearly both present because the observed amplitude equals the sum of the two predicted amplitudes.
Speed of light constant?
Other studies using GPS data have placed far more stringent limits than we will here. But our goal here is not to set the most stringent limit on possible variations in the speed of light, but rather to determine what the maximum possible variation might be that can remain consistent with the data, allowing for systematic errors. The GPS operates by sending atomic clock signals from orbital altitudes to the ground. This takes a mere 0.08 seconds from our human perspective, but a very long (although equivalent) 80,000,000 ns from the perspective of an atomic clock. Because of this precision, the system has shown that the speed of radio signals (identical to the ‘‘speed of light’’) is the same from all satellites to all ground stations at all times of day and in all directions to within ±12 meters per second (m/s). The same numerical value for the speed of light works equally well at any season of the year.
Technical note: measuring the one-way speed of light requires two clocks, one on each end of the path. If the separation of the clocks is known, then the separation divided by the time interval between transmission and reception is the one-way speed of the signal. But measuring the time interval requires synchronizing the clocks first. If the Einstein prescription for synchronizing clocks is used, then the measured speed must be the speed of light by definition of the Einstein prescription (which assumes the speed of light is the same in all inertial frames). If some other non-equivalent synchronization method is used, then the measured speed of the signal will not be the speed of light. Clearly, the measured signal speed and the synchronization method are intimately connected. It is impossible in principle to measure the speed of light independent of theory using electromagnetic or slower phenomena. However, we can look for changes in the time interval required for a radio signal to travel a known distance, such as from a satellite in a circular orbit to the ground at different times of day or seasons of the year, to determine if the signal speed varies. And the preceding result shows that it does not vary by an amount comparable to Earth’s orbital
motions around the Sun or the Galaxy or relative to the microwave ‘‘background,’’ so ‘‘aether wind’’ theories are thereby falsified. Either SR is correct and the speed of light is invariant, or the density of any aether field that carries light must be entrained by the local gravity field.
Lorentzian relativity
As history buffs may know, the Lorentz Ether Theory (LET) (Lorentz 1931: 208–11)
Both SR and LET explain all existing electromagnetic-based experiments and, in that sense, would remain viable theories of the relativity of motion. But the difference between them is much more than aesthetic. In addition to a great difference in practicality for use in systems such as the GPS (in favor of LET), the two theories differ about whether or not material bodies can exceed the speed of light in forward time. In SR, that is proved impossible because time ceases to advance for any entity traveling at the speed of light. By contrast, in LR, no speed limit for material bodies exists. It is true that speed relative to the preferred frame causes electromagnetic-type clocks (which include all ordinary mechanical, biological, and atomic clocks) to slow by the relativistic factor γ = 1/ 
just as in SR. But in LR, time itself is not affected. (Here, v is the relative speed of the body and c is the speed of light.) So the question of which theory better represents nature is of major importance to the future of physics, which is presently invested in the belief that speeds faster than light in forward time are not possible.
Today, our concepts of the ‘‘luminiferous ether’’ are considerably different than they were in Lorentz’s day. It is now widely recognized that the local gravity field serves as the ‘‘preferred frame’’ of LET. With this alteration from Lorentz’s original concept but without any change in the math or structure of the theory, LET has now become known simply as ‘‘Lorentzian relativity’’ (LR). Although LR has no intrinsic speed limit, it recognizes the innate difficulty of material bodies composed in part of electrons, while propagating in a luminiferous ether, being able to exceed the wave speed of that ether, the speed of light. LR treats this as analogous to a propeller-driven aircraft exceeding the speed of sound without any outside assists, such as from gravity. A force that cannot itself propagate faster than light cannot propel material bodies faster than light.
Of critical importance to choosing the model that best represents nature, none of the eleven independent experiments testing SR verify frame reciprocity or distinguish SR from LR. In fact, historically, de Sitter, Sagnac, Michelson, and Ives concluded from their respective experiments that SR was falsified in favor of the Lorentz theory. De Sitter argued that the forward displacement of starlight (aberration) depended on absolute, not relative, speeds because both components of a double star, each with some unique velocity, had the same aberration. Sagnac argued that the fringe shifts expected but not seen in the Michelson-Morley experiment are seen if the experiment is done on a rotating platform. Michelson argued in the 1925 Michelson-Gale experiment that the Earth was just such a rotating platform. Ives argued that ions radiated at frequencies determined by absolute, not relative, motion because they had to pick a specific frequency to radiate at. In each case, a complex-but-now-familiar SR explanation could account for the same observed results (Table 7.1).
Indeed, the GPS itself is a practical realization of Lorentz’s ‘‘universal time,’’ wherein all clocks (‘‘GPS clocks’’) remain synchronized despite being in many different frames with high relative speeds. However, subsequent reinterpretation of SR allowed that theory to survive these objections. This ‘‘magic’’ is envisioned to happen by virtue of each clock in the system being synchronized to an imaginary clock in the Earth-centered inertial (ECI) frame, instantaneously co-located with the moving clock, and assumed to be in a gravitational potential equal to that at sea level at Earth’s poles. (Note the ‘‘coincidence’’ that the magic makes use of the Lorentzian preferred frame, the local gravity field.) This trick makes the clock rates all the same as they would have been if they were at rest in the ECI frame and in a constant potential field. This is all very nice, but hardly what Einstein envisioned when speaking of two clocks in relative motion, one at a station and one on a passing train. How simple special relativity would have become all these years if physicists had realized that all they had to do was reset the clock rates so they all ticked at the same rate as the reference clock in the local gravity field!
The converse is also true. Suppose we did not change the clock rates before launch, but instead let them tick at their design rates in accord with whatever speed and potential they experienced in orbit. Now, suppose we tried to Einstein-synchronize the system of clocks. Satellite and ground clocks would tick at different rates. And if we tried to work in any local, instantaneously co-moving inertial frame, the corrections needed to synchronize with each orbiting clock would be unique to that observer’s frame and different from moment to moment because both clocks are accelerating. The practical difficulties of operating the system would be virtually insurmountable. What we would gain by doing that is constancy of the measured speed of light in all inertial frames. But because all clocks are now re-synchronized to just the ECI frame in the GPS, the speed of light is constant in that one frame, and the invariance of the speed of light in other inertial frames is of no practical value.
Both Lorentz in 1904 and Einstein in 1905 chose to adopt the principle of relativity discussed by Poincare in 1899, which apparently originated some years earlier in the nineteenth century. Lorentz also popularized the famous transformations that bear his name, later used by Einstein. However, Lorentz’s relativity theory assumed an aether, a preferred frame, and a universal time. Einstein did away with the need for these. But it is important to realize that none of the eleven independent experiments said to confirm the validity of SR experimentally distinguish it from LR – at least not in Einstein’s favor.
Several of the experiments in 7.1 and 7.2 bearing on various aspects of SR gave results consistent with both SR and LR. But Sagnac in 1913, Michelson following the Michelson-Gale confirmation of the Sagnac effect for the rotating Earth in 1925 (not an independent experiment, so not listed in the tables), and Ives in 1941, all claimed at the time they published that their results were experimental contradictions of Einstein SR because they implied a preferred frame. In hindsight, it can be argued that most of the experiments contain some aspect that makes their interpretation simpler in a preferred frame, consistent with LR. In modern discussions of LR, the preferred frame is not universal, but rather coincides with the local gravity field. Yet, none of these experiments are impossible for SR to explain (Table 7.2)
For example, Fresnel showed that light is partially dragged by the local medium, which suggests a certain amount of frame-dependence. Airy found that aberration did not change for a water-filled telescope, and therefore did not arise in the telescope tube. That suggests it must arise elsewhere locally.
Michelson-Morley expected the Earth’s velocity to affect the speed of light because it affected aberration. But it didn’t. If these experimenters had realized that the aether was not a single entity but changed with the local gravity field, they would not have been surprised. It might have helped their understanding to realize that Earth’s own Moon does not experience aberration as the distant stars do, but only the much smaller amount appropriate to its small speed through the Earth’s gravity field.
Another clue came from De Sitter in 1913, elaborated by Phipps (Phipps 1989: 549–51), both of whom reminded us that double star components with high relative velocities nonetheless both have the same stellar aberration. This meant that the relative velocity between a light source and an observer was not relevant to stellar aberration. Rather, the relative velocity between local and distant gravity fields determined aberration. In the same year, Sagnac showed non-null results for a Michelson-Morley experiment done on a rotating platform. In the simplest interpretation, this demonstrated that speeds relative to the local gravity field do add to or subtract from the speed of light in the experiment, since the fringes do shift. The Michelson-Gale experiment in 1925 confirmed that the Sagnac result holds true when the rotating platform is the entire Earth’s surface.
When Ives and Stilwell showed in 1941 that the frequencies of radiating ions depended on their motion, Ives thought he had disposed once and for all of the notion that only relative velocity mattered. After all, the ions emitted at a particular frequency no matter what frame they were observed from. He was unmoved by arguments to show that SR could explain this too because it seemed clear that nature still needed a preferred frame, the motion relative to which would determine the ion frequencies. Otherwise, how would the ions know how often to radiate? Answers to Ive’s dilemma exist, but not with a comparable simplicity.
Richard Keating was surprised in 1972 that two atomic clocks traveling in opposite directions around the world, when compared with a third that stayed at home, showed slowing that depended on their absolute speed through space – the vector sum of the Earth’s rotation and airplane speeds – rather on the relative velocities of the clocks. But he quickly accepted that astronomers always use the Earth’s frame for local phenomena, and the solar system barycentric frame for other planetary system phenomena, to get results that agreed with the predictions of relativity. Being unaware of LR, he did not question the interpretation at any deeper level.
Table 7.2 summarizes what the various experiments have so say about a preferred frame. These experiments confirm the original aether-formulated relativity principle to high precision. However, the issue of the need for a preferred frame in nature is, charitably, not yet settled. Certainly, experts do not yet agree on its resolution. But of those who have compared both LR and SR to the experiments, most seem convinced that LR more easily explains the behavior of nature.
Table 7.2 Independent experiments bearing on Special Relativity – type and notes on reciprocity
In LR, speed relative to the preferred frame (the local gravity field) causes clocks to slow. Electromagnetic-based forces become increasingly less efficient with increasing speed relative to the preferred frame, and approach zero efficiency as speed approaches c. There are natural, physical reasons why these things should be so (Van Flandern 1993). The frame of the local gravity field acts as a preferred frame. Universal time and remote simultaneity exist.
The single most important difference is that, in SR, nothing can propagate faster than c in forward time. In LR, electromagnetic-based forces and clocks would cease to operate at speeds of c or higher. But no problem in principle exists in attaining any speed in forward time using forces such as gravity that retain their efficiency at high speeds.
In a recent article, Ashby (Ashby 2002: 41–47) claimed that the clock-epoch correction term (also called ‘‘time slippage’’ term) in the Lorentz transformations, vX=c2 (see eq. (1) below), can be dropped even when its value is large, but he is very vague about why. However, this particular term is the only difference of consequence between Einstein synchronization of clocks in different inertial frames and Lorentz synchronization of clocks to an underlying ‘‘universal time’’. And the GPS system has been designed to use Lorentz synchronization, for which one frame, the local gravity field or ECI, is special; not Einstein synchronization, wherein clocks tick at their natural rates and all inertial frames are equivalent. By itself, this does not prove LR ‘‘right’’ or SR ‘‘wrong’’. But the practical difficulties for GPS of not changing the natural rates of clocks pre-launch, or with the use of SR for any frame but the Lorentzian preferred frame, are very great. If a ring of satellites (A, B, C, . . ., Y, Z) circled the Earth in a common orbit, and each satellite tried to Einstein-synchronize with the next in sequence, then when Z tried to complete the circuit by Einstein-synchronizing with A, the corrections required would lead to time readings for A different from the starting readings, making closure impossible.
Introducing the twins
The ‘‘twins’ paradox’’ is an illustration of the complexity of SR’s interpretations of nature. Suppose two identical twins start out at some common instant. One remains on Earth. The other (the ‘‘traveler’’) is on a spacecraft headed for Alpha Centauri (AC) four light-years away at 99% of the speed of light, for which speed, the time dilation factor is γ ≈ 7. (We choose a large value of γ so that the effects of the relativity of motion will be large and obvious, not subtle.) Upon arrival at AC, the traveler turns back to Earth at the same speed (or is replaced by a traveler already headed toward Earth of identical biological age at the moment they pass, to avoid need for an acceleration). The round trip requires slightly over 8 years Earth time; let’s say 98 months to be specific. This is path 1 in
That much is a clear prediction of SR. Note especially that no accelerations need actually occur at the beginning or end of the journey, nor even in the middle if we do the ‘‘twin replacement’’ trick, or by noting that half the age discrepancy for the full journey has already occurred by the mid-point before any turn-around event. That is consistent with cyclotron experiments showing that accelerations as such, even as great as 1019 g (where g is the acceleration of gravity at sea level), have no effect on clocks (Bailey et al. 1997: 301–5). At each stage of the journey where an event occurs, comparisons can be made without ambiguity between adjacent points, one in each of the inertial frames containing the clocks or twins to be compared. Despite the fact that many textbooks discussing the twins’ paradox treat accelerations as essential, that is illusory. Accelerations are unbounded in size, and in principle can be done in an instant, allowing no local time to elapse in any relevant frame. Accelerations do not change local clocks, clock rates, or biological aging.
Now we come to the paradox part: why isn’t the traveler entitled to claim that the spacecraft remained at rest and the Earth traveled away at 99% of the speed of light, then turned around and came back? From that perspective, the original traveler would argue that the Earth-bound twin should be the younger one. We will examine the rather different answers to that question offered by SR and by LR.
The traveler takes along a GPS clock
Let’s more closely follow what is happening to our two twins. Imagine that the inertial frame of reference containing Earth and AC is filled everywhere with synchronized clocks at rest, so that any traveler can always look out a window and read what time Earth-frame people think it is (see
Now let’s examine the journey details. When the traveler’s journey begins, the on-board native clock ticks slower than the GPS clock by a factor of 7. But isn’t that already an asymmetry present at a stage where there is simply a relative motion, and no way to decide which twin should be aging slower? LR answers simply ‘‘yes’’ because the frame of the local gravity field is the preferred frame in which clocks tick fastest, and time in all other relatively moving frames passes more slowly. But SR offers the opposite answer. And understanding that answer is the key to understanding SR.
SR is a mathematical theory built around the Lorentz transformations. Let the time T be the reading on a clock fixed in the Earth frame; and let X be the relative location in the Earth frame of a clock fixed in the spacecraft frame moving at speed v relative to the Earth frame. Let t be the time reading on the natural clock in the traveling spacecraft, and x be the relative location of Earth in the spacecraft’s frame. In that frame, Earth passes the spacecraft with a speed 
v. As before, the Lorentz time dilation/length contraction parameter is γ, having a value of 7 when v = 0:99c. So in general, the relation between the Earth-frame clock and the spacecraft-frame clock is:
(1)
Because the relationships are reciprocal (all inertial frames are equivalent in SR), the inverse relations must also hold:
(2)
Now let’s compare time in the two frames. First, let an observer at a clock fixed on Earth watch time on the spacecraft clock. Then the observer is
looking at a point X = vT. Substituting that into eq. (1), we get the unsurprising result that x = 0, meaning that the spacecraft clock remains fixed at the origin of its own coordinate system. And from the time transformation, using the definition of γ,we get t = T=γ,which restates the well-known prediction of SR we cited above that the spacecraft clock will appear to the Earth observer to be ticking γ = 7 times slower than the Earth-fixed clock.
That much is routine. But before we leave the Earth frame, let’s do one more calculation. Suppose the spacecraft frame is likewise filled with clocks everywhere, Einstein-synchronized with each other. Because all such clocks are at rest relative to one another, they can simply exchange light signals and assume that the light takes equal time for the uplink and downlink portions of its round trip. Then the average of the transmission and reception times on one clock must agree with the signal reflection time on the other clock by definition of ‘‘Einstein synchronization’’. Now, from the Earth frame, let’s peer into the spacecraft frame at the point X = 0, the fixed location of the Earth observer. From eq. (1), we now have t = γT. The meaning of this relation is that the Earth observer sees a succession of spacecraft-frame clocks parading by, and time on the succession of clocks goes by γ = 7 times faster than Earth time.
=You read that right. According to SR, at the same time that each and every clock in the spacecraft frame is seen by the Earth frame to tick γ = 7 times slower, time in the spacecraft frame on a succession of passing clocks is passing by γ = 7 times faster than Earth time. To repeat this essential point, in any inertial frame with a relative motion, all individual clocks tick slower but overall frame time moves forward at a faster rate. Such is the effect of the vX=c2 term in the transformation. The time difference between clocks in different frames is a function of the different rates they tick at and of the ‘‘time slippage’’ effect, whereby time is a function of location in any relatively moving frame. In SR, this would remain true even if we used the GPS trick and eliminated the rate differences between clocks. In general, the time slippage effect dominates the effect of a changed clock rate for any clock in a frame with relative motion. (N.B. time is everywhere the same when viewed from within an inertial frame. But it is everywhere different in that same frame from the perspective of any other frame with a relative motion because of time slippage.)
It is worth studying the points in the preceding two paragraphs because, while mathematically permissible, they defy our intuitions and are what makes relativity such an unintuitive theory.
The traveler looks back at the Earth twin
Now let’s switch to the spacecraft frame and look back at the Earth-frame clock at x = vt using eq. (2). Then we get X = 0 and T = t/γ. So the spacecraft clock sees the Earth clock ticking γ = 7 times slower than itself. Substituting x = 0, we get T = γt: the spacecraft twin sees Earth-frame clocks streaming by outside his window with time elapsing on a succession of them at γ = 7 time faster. So both effects, relative clock rates and frame time from time slippage, are reciprocal and symmetric between the two frames.
And that is SR’s answer to the symmetry challenge we posed at the outset of the spacecraft’s journey. Both LR and SR predict that the spacecraft’s clocks will appear to tick slower than all Earth-frame clocks as viewed by
Earth-frame observers. But SR (only) predicts that the situation looks reciprocal and symmetric for observers in the spacecraft frame looking at clocks in the Earth frame. The traveler is therefore not surprised by the behavior of the GPS clock on board, which correctly records a combination slower clock rate plus a fast time slippage for the Earth frame, and stays in agreement with the Earth-frame clocks passing by just outside the spacecraft window.
Now consider the traveler’s arrival at AC, when the spacecraft turns around or is replaced by a spacecraft heading Earthward at the same speed. Neither the natural clock nor the on-board GPS clock changes rate or reading significantly during the turn-around. However, before the turnaround, the Earth-frame clocks say 4 years (actually, 49 months) of Earth time have elapsed since the journey began, and the spacecraft natural clock says 7 months of spacecraft time have elapsed. But the spacecraft infers that only a single month has elapsed back on Earth because a single clock in another frame is affected only by clock slowing, not time slippage; whereas the spacecraft agrees that 49 total months have elapsed at AC because the journey began with a 48-month time slippage for AC and added one more month during the journey. As usual, time slippage corrections (where applicable) dominate clock readings in other frames.
We now come to the crux of the resolution of the paradox, which seems paradoxical only in SR. The traveler has inferred that only one month has elapsed back on Earth since his journey began, so by spacecraft-frame reckoning, Earth time is just one month later than the actual departure time. For example, if the journey commenced in 2000 January, when the traveler arrives at AC, the on-board GPS clock reads 2004 February; but the traveler infers that Earth clocks still read 2000 February. Because of the finite speed of light, the traveler can see Earth only as it was, not as it is now, and therefore cannot check this inference by direct observation. According to SR, all these clock-reading inferences are not just illusions, but reflect the real, physical time for each frame involved. So at the same time and place that an AC resident infers that Earth time is 2004 February, the spacecraft traveler infers it is 2000 February – a 4-year difference; and both are correct for their respective frames.
Then the spacecraft turns around. Nothing changes locally. But inferences about remote time change greatly because of time slippage, which now has the opposite sign. Now the traveler infers that Earth time is 2008 February – 4 years into the future instead of the past. As a consequence, the traveler will again infer that only one month of Earth time will elapse during the return journey, and all participants agree that Earth time upon the traveler’s return will be 2008 March. The traveler arrives back younger (14 months old) than the stay-at-home twin (98 months – over 8 years old). According to LR, this is because the traveler had a high speed relative to the preferred frame, the local gravity field; and there never was any symmetry between the two frames. But according to SR, the elapsed time is a
combination of slowed aging and time slippage effects, the latter changing discontinuously when the direction of the traveler changes. (We note in passing that the effect that SR expects accelerations or frame changes to have on remote clocks would constitute an instantaneous action at a distance, a violation of the causality principle.)
The Earth twin thinks the traveler should naturally age slower because his clocks run slower. And the traveler thinks the Earth twin should naturally age faster because the time slippage effect dominated the slowed aging effect when that twin turned around. Note that the traveler could have taken any path whatever as long as the spacecraft speed relative to the Earth frame remained 0.99 c. Then at every instant along the journey, the traveler’s biological age will be a factor γ = 7 less than the elapsed time on the on-board GPS clock, which will agree with the Earth clocks upon return.
Indeed, the spacecraft could have simply continued in a straight line past AC and on to Beta Centauri (BC), say (for purposes of this example only), 8 light-years from Earth. Then there was no turn-around event, but the traveler is still just 14 months old on arrival, and twins born on BC at the same time the Earth twin was born (according to Earth-frame clocks) will still be 98 months old. And the traveler will infer that the BC twins started out 96 months old at the journey’s beginning, and aged just 2 months during the journey. So clearly, neither the turn-around event nor any acceleration is essential to the result; and the SR resolution of the paradox retains its symmetry and the equivalence of all frames.
Hence, from the traveler’s perspective, wherever the spacecraft goes in any direction without changing speed relative to the Earth frame, it will encounter Earth-frame clocks with more elapsed time than on the natural clock aboard the spacecraft (but with the same elapsed time as for the onboard GPS clock). But the traveler will infer that all clocks in the Earth frame are always ticking slower than the natural spacecraft clock, and beings in the Earth frame are always aging more slowly than the traveler. Everything encountered can be explained by a combination of clock rate changes and time slippage. The traveler cannot infer that the Earth-bound twin will be the younger one upon the spacecraft’s return because Earth experienced a time slippage event (whether sudden or gradual), rapidly aging everyone on Earth during the traveler’s journey. That time slippage event is no different in character than the one that a hypothetical twin on BC would experience if the traveler continued on past AC without a turnaround event.
The traveler takes different paths
What we have just described are careful and correct inferences of SR as applied to the twins’ paradox. This also shows the essentially mathematical nature of the theory, because it does violence to what we fondly call ‘‘common sense’’. The most important point to note carefully is that the theory is internally consistent, and no mathematical contradictions can be found no matter how the transformation equations are manipulated, or how many frames or twins are introduced. The next important point to note is that SR makes demands on our credulity that LR does not. Let’s examine why.
At the point of turn-around on the original journey from Earth to AC, the traveler’s inferences about time on Earth changed suddenly. Instead of the physically unrealistic instant turn-around, let’s assume the spacecraft ‘‘orbits’’ around AC to perform the turn-around. (To stay at a safe distance from the star at the same speed, this would require propulsion, not just gravity.) This can still take a time short enough to be neglected, especially at such a high relative speed. So the traveler’s spacecraft changes from headed away from Earth and inferring the Earth year is 2000, to traveling toward Earth and inferring the Earth year is 2008. Again, SR says this is real, physical time, and not an illusion.
So before commencing a journey back to Earth, let’s suppose the traveler orbits AC several times. Then each time the traveler heads away from Earth in that orbit, Earth time drops back to 2000; and each time the traveler heads toward Earth, inferred Earth time becomes 2008. The Earth-time is intermediate for intermediate orbital positions. Now the significance of repeating this situation several times is that, as Earth time goes to 2008, many people will have died and others will be born. And on each occasion that Earth time reverts to 2000, some of the dead will be resurrected and some living young children in 2008 will cease to exist in 2000. Note that while all this is happening according to SR, the on-board GPS clock representing LR’s ‘‘universal time’’ continues to insist that Earth time is the same as spacecraft time and AC time: 2004 everywhere. In SR, effects of this type are never observable because they ‘‘lie outside the observer’s light cone,’’ hidden from direct view by the finite speed of light. Nonetheless, SR insists that such changes affect real, physical time and are not mere illusions, because the viewpoint of each inertial frame is just as valid as that from any other frame.
In LR, one reference frame (the local gravity field) is preferred; and speed cannot affect time, but only the rate of ticking of mechanical, electromagnetic, or biological clocks. However, just as we do not assume that time has been affected when the temperature rises and causes a pendulum clock to slow down, LR says that changes in clock rates are changes in the rates of physical processes, and do not affect space or time. So by carrying an onboard GPS clock on the spacecraft, we are offered a clear choice between models: Earth time can be what SR infers it is, or it can be what the GPS clock says it is. In the former case, SR works, but leads to heavy-duty complexities and fantastic inferences about the nature of time at remote locations. Moreover, the proof that nothing can travel faster than light in forward time stands intact. In the latter case, LR works with great simplicity and in full accord with our intuitions about the universality of the instant ‘‘now’’. And the speed of light is no longer a universal speed limit because time itself is never affected either by motion or by gravity.
Aside from these practical difficulties with the use of SR in the GPS, Einstein’s special relativity is also under challenge in a more serious way from the ‘‘speed of gravity’’ issue, because the proven existence of anything propagating faster than light in forward time (as all experiments indicate is the case for gravity) would falsify SR outright (Van Flandern 1998: 1–11; Van Flandern and Vigier 2002: 1031–68). So it is entirely possible that reality is Lorentzian, not Einsteinian, with respect to the relativity of motion. In that case, physics may have no speed limit when the driving forces are gravitational or electrodynamic rather than electromagnetic in nature. And that may be the most important thing that the GPS has helped us to appreciate.
Notes
References
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