1.
What is the GPS?
The Global Positioning System (GPS)
consists of a network of 24 satellites in roughly 12hour orbits, each carrying
atomic clocks on board. The orbital radius of the satellites is about four
Earthradii (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 90degree 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 onboard 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 nearEarth 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.
2.
What relativistic effects on GPS atomic clocks might be seen?
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 nonmoving 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.
Special Relativity (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 onesecond 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 note that this postlaunch 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.
3.
Does the GPS confirm the clock rate changes predicted by GR and SR?
The highest precision GPS receiver
data is collected continuously in two frequencies at 1.5second intervals from
all GPS satellites at five Air Force monitor stations distributed around the
Earth. An indepth discussion of the data and its analysis is beyond the scope
of this paper. ^{[1]}
This data shows that the onboard 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, semirandom amounts (of order 1
ns/day) at unpredictable times for reasons that are not fully understood. As a
consequence, the longterm accuracy of these clocks is poorer than their
shortterm 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
errors in the clocks build up with time squared, 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, 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.
12hour 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 12hour 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.
4.
Is the 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. 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
oneway 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 oneway 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
nonequivalent 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 prescription are intimately connected.
Our result here merely points out that the measured
speed does not change as a function of time of day or direction of the
satellite in its orbit when the clock synchronization correction is kept
unchanged over one day. As for seasonal variations, all satellite clocks are
"steered" to keep close to the U.S. Naval Observatory Master Clock so
as to prevent excessive build up of errors from random rate changes over long
time periods. So we cannot make direct comparisons between different seasons,
but merely note that the same value of the speed of light works equally well in
any season.
5.
What is a "GPS clock"?
Cesium atomic clocks operate by
counting hyperfine transitions of cesium atoms that occur roughly 10 billion
times per second at a very stable frequency provided by nature. The precise
number of such transitions was originally calibrated by astronomers, and is now
adopted by international agreement as the definition of one atomic second.
GPS atomic clocks in orbit would run
at rates quite different from ground clocks if allowed to do so, and this would
complicate usage of the system. So the counter of hyperfine cesium transitions
(or the corresponding phenomenon in the case of rubidium atomic clocks) is
reset on the ground before launch so that, once in orbit, the clocks will tick
off whole seconds at the same average rate as ground clocks. GPS clocks are
therefore seen to run slow compared to ground clocks before launch, but run at
the same rate as ground clocks after launch when at the correct orbital
altitude.
We will refer to a clock whose
natural ticking frequency has been precorrected 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 precorrected 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.
6.
Is acceleration an essential part of resolving the "twins paradox"?
If the traveling twin carries both a
natural clock and a GPS clock on board his spacecraft, he can observe the
effects predicted by SR without need of any acceleration in the usual twins
paradox. That is as it should be because cyclotron experiments have shown that,
even at accelerations of 10^{19} g (g = acceleration of gravity at the
Earth's surface), clock rates are unaffected. Only speed affects clock rates,
but not acceleration per se.
Suppose that the traveling twin is
born as his spaceship passes by Earth and both of his onboard clocks are
synchronized with clocks on Earth. The natural onboard clock ticks more slowly
than the GPS onboard clock because the rates differ by the factor gamma that
SR predicts for the slowing of all clocks with relative speed v. [gamma =
1/sqrt(1v^{2}/c^{2})] But everywhere the traveling twin goes,
as long as his speed relative to the Earth frame does not change, his GPS clock
will give identical readings to any Earthsynchronized Earthframe clock he
passes along the way. And his natural clock will read less time elapsed since
passing Earth by the factor gamma. His biological processes (including aging),
which presumably operate at rates comparable to the ticking of the natural
clock, are also slowed by the factor gamma.
Since this rate difference is true
at every instant of the journey beginning with the first, there are no
surprises if the traveling twin executes a turnaround without change of speed
and returns to Earth. He will find on journey's completion what he has observed
at every step of the journey: His natural clock and his biological age are
slower and younger by the factor gamma than that of his Earthframe
counterparts everywhere along his journey, including at its completion. The
same would have been true if he had not turned around, but merely continued
ahead. He would be younger than his peers on any planet encountered who claim
to have been born at the same time that the traveler was born (i.e., when he
passed Earth) according to their Earthframe perspective.
Clearly, acceleration or the lack
thereof has no bearing on the observed results. If acceleration occurs, it is
merely to allow a more convenient comparison of clocks by returning to the
starting point. But since the traveler can never return to the same point in
spacetime merely by returning to the same point in space, the results of a
roundtrip comparison are no different in kind from those made anywhere along
the journey. The traveler always judges that his own aging is slower than that
in any other frame with a relative motion.
Then why isn’t the traveler entitled
to claim that he remained at rest and the Earth moved? The traveler is
unconditionally moving with respect to the Earth frame and therefore his clocks
unconditionally tick slower and he ages less as judged by anyone in the Earth
frame. However, if the traveler makes the same judgment, the result will depend
on whether he values his natural clock or his GPS clock as the better
timekeeper. If he takes readings on the GPS clock to represent Earth time, his
inferences will always agree with those of Earthframe observers. If he instead
uses the results of the exchange of light signals to make inferences of what
time it is at distant locations, he will conclude that the Earthbound twin is aging
less than himself because of their relative motion. But on the occasion of any
acceleration his spaceship undergoes, the traveler will infer a discontinuity
in the age of his Earthbound counterpart, which can be either forward or
backward in time depending on which direction the traveler accelerates. At the
end of any round trip after any number of such accelerations, the traveler and
Earthbound twins will always agree about who should have aged more.
7.
Does the behavior of GPS clocks confirm Einstein SR?
To answer this, we must make a
distinction between Einstein SR and Lorentzian Relativity (LR). 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 19^{th}
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 11 independent
experiments said to confirm the validity of SR experimentally distinguish it
from LR  at least not in Einstein's favor.
Experiment  Description  Year 
Bradley  Discovery of aberration of light  1728 
Fresnel  Light suffers drag from the local medium  1817 
Airy  Aberration independent of the local medium  1871 
MichelsonMorley  Speed of light independent of Earth's orbital motion  1881 
De Sitter  Speed of light independent of speed of source  1913 
Sagnac  Speed of light depends on rotational speed  1913 
KennedyThorndike  Measured time also affected by motion  1932 
IvesStilwell  Ions radiate at frequencies affected by their motion  1941 
FrischSmith  Radioactive decay of mesons is slowed by motion  1963 
HafeleKeating  Atomic clock changes depend on Earth's rotation  1972 
GPS  Clocks in all frames continuously synchronized  1997 
Table 1. Independent experiments bearing on Special Relativity
Several of the experiments bearing on various
aspects of SR (see Table 1) gave results consistent with both SR and LR. But
Sagnac in 1913, Michelson following the MichelsonGale confirmation of the
Sagnac effect for the rotating Earth in 1925 (not an independent experiment, so
not listed in Table 1), 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 is impossible for SR to explain.
For example, Fresnel showed that
light is partially dragged by the local medium, which suggests a certain amount
of framedependence. Airy found that aberration did not change for a
waterfilled telescope, and therefore did not arise in the telescope tube. That
suggests it must arise elsewhere locally. MichelsonMorley 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 for De Sitter in 1913, elaborated
by Phipps ^{[3]}, 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
nonnull results for a MichelsonMorley 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 MichelsonGale 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 Ives 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.
Experiment  Type  Notes on Reciprocity 
Bradley  Aberration  Moon exempt 
Fresnel  Fresnel drag  Existence of aether 
Airy  Existence of aether  Water in ‘scope ignored 
MichelsonMorley  No universal aether  Aether “entrained”? 
De Sitter  c independent of source  Double star aberration 
Sagnac  c depends on rotation  Local gravity field nonrotating 
KennedyThorndike  Clocks slow  Motion w.r.t. local gravity field 
IvesStilwell  Ions slow  " 
FrischSmith  Mesons live longer  " 
HafeleKeating  Clocks depend on rotation  Preferred frame indicated 
GPS  Universal synchronization  Preferred frame = local gravity 
Table 2. Independent experiments bearing on Special Relativity
Table 2 summarizes what the various
experiments have to say about a preferred frame. These experiments confirm the
original aetherformulated 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.
8.
How does the resolution of the "twins paradox" compare in LR and SR?
In LR, the answer is simple: The
Earth frame at the outset, and the dominant local gravity field in general,
constitutes a preferred frame. So the highspeed traveler always comes back
younger, and there is no true reciprocity of perspective for his or other
frames.
In SR, the answer is not so simple;
yet an explanation exists. The reciprocity of frames required by SR when
Einstein assumed that all inertial frames were equivalent introduces a second
effect on "time" in nature that is not reflected in clock rates
alone. We might call this effect "time slippage" so we can discuss
it. Time slippage represents the difference in time for any remote event as
judged by observers (even momentarily coincident ones) in different inertial
frames.
For example, we would argue that, if
it is 9/1998 here and now, it is also 9/1998 "now" at Alpha Centauri.
But an observer here and now with a sufficiently high relative motion (say, 99%
of c; gamma = 7) might judge that it is 9/1994 at Alpha Centauri
"now" (meaning that he just left there one month of Earth time ago,
and it was 8/1994 then). Or he might judge that it is 9/2002 at Alpha Centauri
"now" (meaning that he will arrive there in one month of
Earthelapsed time, and will find the time to be 10/2002). These differences of
opinion about what time it is at remote locations are illustrations of time slippage
effects that appear only in Einstein SR to preserve the frame independence of
its predictions.
So as a traveler passes Earth in
8/1994 at a speed of 0.99c , time slippage effects begin to build up. Seven
months later by his natural clock, the traveler arrives at Alpha Centauri. His
own GPS clock shows four years of elapsed time, and indeed Alpha Centauri
residents who think they are calendarsynchronized with Earth agree that the
twin arrives in 9/1998. But the traveler is convinced by Einstein SR that only one
month of Earth time has elapsed since he passed Earth and noted the time as
8/1994. The traveler, upon arriving at Alpha Centauri, claims that the time is
"now" 9/1994 on Earth. Alpha Centauri residents claim it is
"now" 9/1998 on Earth. The difference is the time slippage predicted
by SR.
If the traveler orbits Alpha
Centauri at a speed of 0.99 c, then whenever he is headed in the direction of
Earth his opinion changes to Earth time "now" is 9/2002. And whenever
he is again headed away from Earth, Earth time is once again 9/1994. Earth time
"now" changes continually, according to SR, because of these time
slippage effects needed to retain frame reciprocity. Earth residents  even
the ones who died in 1998  are oblivious to their repeated passages into the
future and past of the traveling twin, with concomitant deaths and
resurrections.
So when the traveler finally does
return, he will indeed find that time on Earth is 10/2002, just as his GPS
clock shows. He accounts for this as two months of elapsed time on Earth's
slowrunning clocks during his own 14month (by his natural clock) journey,
plus 8 years of "time slippage" when the traveler changed frames.
There is no logical or mathematical inconsistency in this resolution, which is
why SR remains a viable theory today.
We are, of course, free to question
whether or not this mathematical theory retains a valid basis under the
principles of causality. For those of us who answer "yes", LR is
unnecessary, and inelegant because it depends on a preferred frame. For those
of us who answer "no", LR is then the better descriptor of nature,
requiring the sacrifice of symmetry (“covariance”) to retain causality.
9.
What physical consequences arise from the differences between LR and SR?
In SR, speed causes changes in time
and space themselves, not just in clocks and rulers. Rest mass remains
unchanged, but resistance to acceleration increases toward infinity as speed
approaches c. There is no absolute time or space in the universe. The time at
remote locations depends on what frame one observes from. All frames are
equivalent.
In LR, speed relative to the
preferred frame (the local gravity field) causes clocks to slow and rulers to
contract. Electromagneticbased 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. ^{[2]}
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,
electromagneticbased forces and clocks would cease to operate at speeds of c
or higher. But no problem in principle exists in attaining any speed whatever
in forward time using forces such as gravity that retain their efficiency at
high speeds.
[1]
Alley, C.O. and Van Flandern, T. (1998). “Absolute GPS to Better Than One
Meter”, preprint not yet submitted for publication.
[2]
Van Flandern, T. (1993). Dark Matter,
Missing Planets and New Comets, North Atlantic Books, Berkeley, CA.
[3]
Phipps, T. (1989). "Relativity and Aberration", Amer.J.Phys. 57,
549551.
