RELATIVITY AND GPS

Ronald R. Hatch

(Section I: Special Relativity)

Introduction
The satellites of the global positioning system (GPS) travel around the earth in 12-
hour periods in near-circular orbits. All of the satellites contain extremely precise atomic
clocks whose rates depend both upon satellite velocity and altitude. An observer bound to
the earth, in an airplane or in a satellite may determine his precise location by obtaining
signals from several satellites simultaneously. This paper discusses the implications of
GPS on Einstein's special theory of relativity. A subsequent paper will discuss the general
theory.

"Relativistic" effects within the Global Positioning System (GPS) are addressed.
Hayden [1] has already provided an introduction to GPS, so the characteristics of the
system are not reviewed.
There are three fundamental effects, generally described as relativistic phenomena,
which affect GPS. These are: (1) the effect of source velocity (GPS satellite) and receiver
velocity upon the satellite and receiver clocks; (2) the effect of the gravitational potential
upon satellite and receiver clocks; and (3) the effect of receiver motion upon the signal
reception time (Sagnac effect) . There are a number of papers which have been written to
explain these valid effects in the context of Einstein's relativity theories. However, quite
often the explanations of these effects are patently incorrect. As an example of incorrect
explanation, Ashby [2] in a GPS World article, "Relativity and GPS," gives an improper
explanation for each of the three phenomena listed above.
The three effects are discussed separately and contrasted with Ashby's explanations.
But the Sagnac effect is shown to be in conflict with the special theory. A proposed
resolution of the conflict is offered.
The Sagnac effect is also in conflict with the general theory, if the common
interpretation of the general theory is accepted. The launch of GPS Block II satellites
capable of intersatellite communication and tracking will provide a new means for a giant
Sagnac test of this general theory interpretation. Other general theory problems are
reviewed and a proposed alternative to the general theory is also offered.
Velocity Effects upon the Clock Rates.
The fundamental question of velocity is always: "Velocity with respect to what?"
Ashby, in the opening paragraph of his abstract, states:
Important relativistic effects arise from relative motions of GPS satellites and users, ...
And Ashby also states, at the start of a section on time dilation:
First, clocks in relative motion suffer (relativistic) time dilation,...
But these statements are patently untrue of GPS. It may appear to be a subtle
difference, 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

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chosen frame of reference—in this case the earth-centered, non- rotating, (quasi) inertial
frame.
The difference is easy to illustrate. GPS receivers are now routinely placed on
missiles and other spacecraft. Spacecraft receivers can be used to illustrate Ashby's error.
For illustrative simplicity, let us assume two "Star Wars" spacecraft are equipped with
GPS receivers. Let one of the spacecraft move in an orbit such that the spacecraft follows
a GPS satellite at a close and constant separation distance. Let the second "Star Wars"
spacecraft move in the same orbit but in the opposite direction. The nominal velocity of a
GPS satellite with respect to an earth-centered non-rotating frame is about 3.87
kilometers per second. Using this frame, the computed clock rates should slow by:
f = f 1 − (v / c) 2 (1)
0
For low velocity compared to the speed of light, the change in frequency is
approximated by 1/2 (v/c)2. Using this expression, one obtains a frequency decrease of
8.32 parts in 1011 for GPS satellites. Now, consider the first "Star Wars" receiver, which
is following the GPS satellite. Since it has the same velocity relative to the earth-centered
non-rotating frame, its frequency will be reduced by the same amount as the frequency of
the GPS satellite; and there will, therefore, be no apparent relativistic shift in frequency
of the received signal. This is, of course, also what one would get using the special theory
of relativity, since there is no relative velocity between the first "Star Wars" spacecraft
and the GPS satellite.
However, for the second "Star Wars" spacecraft moving in the opposite direction in
the orbit, the results are dramatically different. Relative to the earth-centered non-rotating
frame, this second spacecraft's speed is no different than the speed of the first spacecraft
or the speed of the GPS satellite. Thus, the expected frequency shift is the same 8.32
parts in 1011. This means that, in the earth-centered non-rotating frame, there is no
apparent relativistic shift in frequency between the second "Star Wars" spacecraft and the
GPS satellite, even when the relative velocity between the spacecraft receiver and the
GPS satellite is 7.74 kilometers per second (approaching each other at twice the orbital
speed). But, if Ashby were right, the relativistic induced difference in frequency between
this second "Star Wars" spacecraft and the GPS satellite would be 33.28 parts in 1011.
(Four times the amount a receiver stationary in the earth-centered non-rotating frame
would see.)
Is this large discrepancy in expected frequency difference detectable? Not really. The
special theory, in addition to claiming the frequency received is a function of the relative
velocity, also claims that the speed of light is isotropic relative to the (observer) receiver;
and the GPS system uses the earth-centered non-rotating frame and also assumes the
speed of light is isotropic in that frame. Jorgenson [3], ironically calling upon work by
Ashby, shows that, if one chooses a frame based upon the instantaneous velocity of the
second "Star-Wars" satellite receiver, one gets exactly the same received-frequency
difference when one combines the relativistic clock shift with the Doppler and aberration
effects. Jorgenson makes the following statement:
In considering alternative coordinate frames, the differences in special relativity exactly
counterbalance those in classical Doppler. Einstein's special relativity is the great equalizer of
coordinate systems. We are given the option of choosing the one most convenient to our needs,
and in the case of GPS, this is an earth-centered inertial frame.
But Jorgensen confuses the special theory claims with the claims of the Lorentz ether
theory. Indeed, many people claim that they are equivalent. However, as we shall see

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later, there is direct experimental evidence which supports the Lorentz ether theory over
the special theory. Whenever a frame is chosen which does not coincide with the receiver
or observer, experiment demands that the speed of light be treated as non-isotropic as far
as the receiver or observer is concerned. But this is anathema to the special theory, since
it is a direct contradiction of the special-theory teaching that the speed of light is always
isotropic relative to the observer (Einstein's "convention" that the round-trip time of a
light pulse is composed of two equal time intervals for the outgoing and incoming pulse).
And it is this aversion to non-isotropic light speed, as we will see later, which is
responsible for the myriad attempts to explain the Sagnac effect without admitting that it
simply arises from the choice of an isotropic frame in which the receiver is moving.
Ashby is guilty of claiming that clocks run at a rate determined by their relative
velocity. In fact, the rate at which clocks run must be computed using the clock velocity
with respect to the chosen isotropic light-speed frame. This is consistent with the Lorentz
ether theory but not with the special theory.
Gravitational Effects upon the Clock Rates
The experimental evidence shows that the gravitational potential affects: (1) the rate at
which clocks run; (2) the speed of light; and (3) the size of physical particles. In order to
demonstrate these effects without excessive use of mathematics, let us simply define a
scale factor, s, slightly less than one, which is used to multiply or scale the parameter of
interest. This scale factor is a direct function of, and can be computed from, the
gravitational potential. The lower the gravitational potential the smaller the scale factor
becomes. The scale factor is defined as:
2GM (2)
s = 1− 2
rc
where G is Newton's gravitational constant, M is the mass causing the gravitational
potential, r is the distance from the center of the gravitational potential.
Consider first those experiments which show that clocks run slower the lower they are
in the gravitational potential. The clocks run slow (measured time appears dilated) as
compared to the rate at which they would run if they were located external to the
gravitational field. The comparative clock rate is given in terms of the scale factor, s,
defined above as:
f = sf e (3)
where: f e is the rate the clock would run if it were external to the gravitational
potential
Several experiments show that clocks run slower the lower they are in the gravitational
field. There are three evidences for this within the GPS system itself. First, the GPS
monitor-station clocks demonstrate the effect. The monitor station at Colorado Springs
runs faster because of its near mile-high elevation than it would run if it were located at
sea level.
Second, the effect is also demonstrated by the reference clocks in the GPS tracking
stations. The tracking stations provide the data which are used to compute the predicted
GPS orbits for uploading and subsequent broadcast of the estimated GPS satellite
position. It is observed that all clocks at sea level in an earth-centered non-rotating frame
run at the same rate. A clock at sea level at the equator runs slower because of the earth's
spin, but that same spin via centrifugal force causes the earth to assume an oblate shape

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so that the clock at the equator is located at a higher gravitational potential. At this higher
gravitational potential, the clock runs faster per equation (3). The net result is that the
velocity effect and the gravitational-potential effect exactly cancel, and the equatorial
sea-level clock runs at the same rate as the polar sea-level clock.
The third GPS demonstration that clocks run slower in a lower gravitational potential
is given by the GPS satellite clocks. The eccentricity of the GPS orbits causes the
satellites to move up and down in the gravitational field. When the satellite is at perigee,
it is closer to the earth, the gravitational potential is lower and the satellite speed is
higher. The lower gravitational potential causes the satellite clock to run slower per
equation (3), and the increased satellite speed also causes the satellite clock to run slower
per equation (1). The two effects are exactly equal and add together to give the net
change in the frequency of the satellite clock. At apogee the satellite is in a higher
gravitational potential and the velocity of the satellite is lower. Each of the two effects
causes the clock to run faster and again adds together to give the total change in the clock
frequency. The integral of the clock-rate effect gives the net correction to the clock time.
The cyclical-clock-time correction about the mean as a result of the orbital eccentricity is
given by:
∆t = −4.42807633eA1 / 2 cos E * 10 −10 (4)
where:: e is the eccentricity, A is the semi-major axis, and E is the eccentric anomaly.
The derivative of this equation gives the frequency as a function of the eccentricity,
semi-major axis and eccentric anomaly. The effect of both the velocity and the
gravitational potential is included.
There are a number of other experiments which have been performed which show the
gravitational effect upon clocks. One of the better known experiments was the flying of
atomic clocks around the world by Hafele and Keating [4]. In this experiment adjustment
had to be made for the faster rate at which the clocks ran at the altitude of the aircraft on
which they were ferried.
One other experiment is often directly cited as showing that clocks run faster at higher
altitudes. Specifically, Pound and Rebka [5] showed that the gamma rays emitted from a
radioactive source 22 meters above a tuned absorber of gamma waves was shifted to a
higher frequency so that the resonant absorption was reduced. The amount of shift in the
wavelength corresponded directly to that predicted by equation (3). However, this
experiment is often explained, not in terms of a changed clock rate (frequency of emitted
gamma rays), but instead as a change in the energy of the gamma waves as a result of
their falling in a gravitational field. Which explanation is correct? They appear to be
mutually exclusive. For, if the gamma waves are simply emitted with a higher frequency
and shorter wavelength, no extra energy and additional shortening of the wavelength
needs to be imparted as they fall in the gravitational field, else the effect would appear to
be double that actually observed. In addition, an increase in the frequency due to the
action of the gravitational field would violate the conservation of the number of cycles
transmitted. All experimental evidence is that cycles are always conserved. The number
received plus the number in transit must equal the number transmitted.
Ashby [2] calls upon the equivalence principle and uses an accelerating elevator to
show that one would expect the wavelengths and frequency of photons to increase as they
fall in a gravitational field. But this also violates the conservation of cycles and cannot be
a valid explanation for the observed change in frequency. As with the Pound and Rebka

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experiment, there is direct evidence that clocks at a higher gravitational potential run just
enough faster to explain the observed decrease in wavelength. No additional decrease in
wavelength appears to be needed from the gravitational fall of the photons.
Do electromagnetic waves pick up energy as they fall in a gravitational field? If it
does, why isn't the observed increase in frequency doubled and the conservation of cycles
violated? The experiments of I. I. Shapiro shed some light on this question. Beginning in
1966, Shapiro [6] showed that the gravitational potential of the sun causes radar signals
reflected back from Venus and Mercury to be delayed. The effect is strongest when
Venus and Mercury are almost directly opposite the earth in their orbits. The amount of
delay shows that the speed of light is decreased by two units of the scale factor. That is,
the computed gravitational-scale factor described above affects the speed of light by the
square of the scale factor—it is multiplied by the scale factor twice.
c = s 2 ce (5)
where ce is the speed of light external to the gravitational field.
Thus, we have direct and unambiguous evidence that the speed of light becomes
slower as the gravitational potential is decreased. But, if such a decrease in the speed of
light is locally undetectable and the clock rate used to measure that speed only
counteracts one-half of the decrease in speed, then lengths must also contract in a
decreased gravitational potential by the same scale factor as the clock rate. Thus, the two
effects combine together to make the change in the speed of light locally undetectable. In
fact, the bending of light near the sun also supports the decrease in length at lower
gravitational potentials. The bending effect is twice that computed classically by the
gravitational force upon the (mass equivalent) energy and is caused entirely by the speed-
of-light velocity gradient. The length in a gravitational potential is given by:
l = sle (6)
where le is the length scale external to the gravitational field.
Now we can see that photons falling in a gravitational field do not increase in energy.
Even though they do decrease in wavelength the frequency does not change. The
apparent change in frequency is caused by the change in frequency of the local unit of
comparison. Thus, claiming as Ashby did that the frequency of the GPS signals increase
as they fall is incorrect. It would violate the conservation of cycles. The apparent
gravitational increase in energy is not real. It appears to increase only because the
standard of comparison (the energy radiated by a similar atom at a decreased
gravitational potential) is decreased. The higher frequency of the GPS clock at its greater
gravitational potential is in fact the source of the increased frequency and decreased
wavelength of the received signal.
An expression has already been given in equation (4) for the clock-time variation due
to the eccentricity of the orbit. But there is a bias change in the clock frequency of the
GPS satellite clocks at the time of their launch. The change in the gravitational potential
at the surface of the earth to the gravitational potential at the satellite orbital height causes
an increase in the average rate at which the clock runs of 5.311 parts in 10 10. As stated in
the first section, the speed of the GPS satellites in orbit causes a clock frequency decrease
of 8.32 parts in 10 11. These two effects combine to give a net increase in frequency of
4.479 parts in 10 10. These two frequency-biasing effects and the additional small mean

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effects of the earth oblateness, sun and moon are compensated before launch by setting
the frequency low by 4.45 parts in 10 10.
The Sagnac Effect
There are probably more conflicting opinions expressed about the Sagnac effect than
any other "relativistic" effect. The review here of the Sagnac effect will be brief. The
reader is referred to Hayden and Whitney [7] for a more comprehensive discussion of the
effect. The most commonly held erroneous belief is that the effect is caused by rotation.
Ashby states:
In the rotating frame of reference, light will not appear to go in all directions in straight lines
with speed c. The frame is not an inertial frame, so the principle of the constancy of the speed of
light does not strictly apply. Instead, electromagnetic signals traversing a closed path will take a
different amount of time to complete the circuit.
In point of fact, rotation is only incidentally involved with the Sagnac effect. The
Sagnac effect is the result of a non-isotropic speed of light and arises any time an
observer or measuring instrument moves with respect to the frame chosen as the isotropic
light-speed frame. And it is here that the Sagnac effect runs into trouble with the special
theory. The special theory by postulate and definition of time synchronization requires
that the speed of light always be isotropic with respect to the observer. And this is where
the special theory is in error—the Sagnac effect illustrates that error.
Since relativists do not like to admit that non-isotropic light speed exists, they attempt
to explain the effect by other mechanisms. The most commonly referenced paper on the
Sagnac effect is by E. J. Post [8]. He claims:
Thus in order to account for the asymmetry [between the clockwise and counterclockwise
beams] one has to assume that either the Gaussian field identification does not hold in a rotating
frame or that the Maxwell equations are affected by rotation.
All existing evidence for the treatment of non-reciprocal phenomena in material media points
in the direction of modified constitutive relations, not in modified Maxwell equations.
Thus, Post claims the effect is caused by some underlying property of space which
arises during rotation. As we shall see, this is an inadequate explanation. To his credit,
Post also said:
The search for a physically meaningful transformation for rotation is not aided in any way
whatever by the principle of general space-time covariance, nor is it true that the space-time theory
of gravitation plays any direct role in establishing physically correct transformations.
In this quote, Post clearly excludes the general theory as a source of explanation for
the Sagnac effect.
But others have claimed the Sagnac effect is caused by acceleration and, thus, is
properly handled by the general theory of relativity. Ashtekar and Magnon [9] give an
analysis of the Sagnac effect within the general theory. Their development is very
abstruse, but it appears that they get the Sagnac effect from rotation precisely because
they do not get an isotropic velocity of light relative to the receiver at all times.
Another general relativity derivation of the Sagnac effect has been given by Deines in
a paper titled "Missing Relativity Terms in GPS" [10]. Deines ascribes his results to a
derivation by Nelson [11] for a rotating coordinate frame in a weak gravitational field.
Deines gives an equation (9) which he says contains three missing relativity terms which
arise due to the rotation of the earth. The last of the three terms is just the clock effect
due to the receiver velocity. While he is correct that this last term is real, its effect in
practice is insignificant. Since GPS receivers must solve for the clock time of the receiver

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in any case, they typically use low-quality internal clocks and any velocity correction due
to their motion is below the clock frequency noise level.
Deines claims that the other two terms, together with a changed coefficient of the third
term, give rise to the Sagnac correction. However, he uses two different integration
limits to get his desired result. The integration limits used to recover the Sagnac effect
are, in fact, magic—choose whatever limits are needed to recover the effect. Deines
argues that the two terms are the derivative of the Sagnac effect and that their integral is
therefore the Sagnac effect. However, the only way to get the effect by integration would
be to collocate the receiver and satellite at the time of launch and to integrate from the
time of launch of the GPS satellite to the present time instant.
To the extent the two additional clock-rate terms are real, they simple describe the
time derivative of the Sagnac effect and are exactly what one obtains using classical
means. But all high precision GPS applications correct for the Sagnac effect. Thus,
contrary to Deines' claims, these relativistic correction terms are not missing. Again, like
the Ashtekar and Magnon results, to the extent the results are valid, they simply indicate
that within the general theory the speed of light is not always isotropic with respect to the
moving observer; and, thus, they are in conflict with the special theory.
The presence of the Sagnac effect in the GPS system clearly shows that none of the
explanations listed above are adequate, for the path of the radiation from the GPS satellite
to the receiver clearly follows a straight line and the instantaneous velocity of the
receiver, while due to the earth's spin, is not affected significantly by the radial
acceleration during the instant of reception. This observation validates Ives' [12] claim
that the Sagnac effect is not caused by rotation. In 1938 Ives showed by analysis that the
measured Sagnac effect would be unchanged if the Sagnac phase detector were moved
along a cord of a hexagon-shaped light path rather than rotating the entire structure. Thus,
he showed the effect could be induced without rotation or acceleration. Let's assume we
fly a GPS receiver on an airplane in a slightly curved path with respect to the earth's
surface, such that its path with respect to the earth-centered non-rotating frame is a
straight line of constant velocity. I know of no one who would argue that such a receiver
would not be required to apply a correction for the Sagnac effect. The only way he could
avoid applying a Sagnac correction would be to change the chosen frame of reference to
his own inertial frame.
Furthermore, the general-theory results (assuming they have been properly derived)
are in conflict with the special theory to the extent that they do not give isotropic light
speed with respect to the moving observer. The conflict with the special theory is
illustrated by comparing the derivation of the Thomas-precession effect with what that
same derivation would give for a Sagnac effect.
Goldstein [13], in his development of the Thomas precession, states:
Consider a particle moving in the laboratory system with a velocity v that is not constant. Since
the system in which the particle is at rest is accelerated with respect to the laboratory, the two
systems should not be connected by a Lorentz transformation. We can circumvent this difficulty
by a frequently used stratagem (elevated by some to the status of an additional postulate of
relativity). We imagine an infinity of inertial systems moving uniformly relative to the laboratory
system, one of which instantaneously matches the velocity of the particle. The particle is thus
instantaneously at rest in an inertial system that can be connected to the laboratory system by a
Lorentz transformation. It is assumed that this Lorentz transformation will also describe the
properties of the particle and its true rest system as seen from the laboratory system.

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Thus, with the help of this additional postulate, acceleration within the special theory
can be handled by successive infinitesimal Lorentz transformations (Lorentz boosts).
These Lorentz boosts give rise to the Thomas precession because successive Lorentz
transformations combine to form a single Lorentz transformation plus a coordinate
rotation. But, if we apply the same logic to the Sagnac experiment, no Sagnac effect can
be expected. Specifically, since the detector is always in an instantaneous inertial frame
(with isotropic light speed), the velocity of light arriving at the detector from both
directions ought to be the same at all times.
An Alternative to the Special Theory
We are left with a problem. The special theory, at least as amended for accelerations,
clearly disagrees with the Sagnac results. In addition, the velocity effects were also
inconsistent with the special theory in that they depended on the velocity relative to the
earth-centered frame rather than the velocity of the receiver relative to the source, as the
special theory predicted. Solutions have been offered which rely upon ether-drag
hypotheses, in which the speed of light is isotropic with respect to the earth's gravitational
field or the earth's gravitational potential or the earth's magnetic field. At one time I
thought that ether drag proportional to the earth's gravitational potential was a viable
solution. However, Charles M. Hill brought to my attention data from VLBI experiments
which could not be reconciled with the ether-drag hypothesis. More recently, Hill [14]
has shown, via an analysis of millisecond pulsar data, that clocks on the earth have cyclic
variations due to the eccentricity of the earth's orbit around the sun. The component of
this clock variation due to the earth's orbital velocity clearly indicates that the earth does
not drag the surrounding ether with it. Thus, while it is still true we cannot measure the
absolute ether drift caused by the earth's orbital motion, we can now measure the
variation in the ether-drift velocity.
There is, in my opinion, only one valid alternative to the special theory consistent with
the experimental evidence. Specifically, the Lorentz ether theory offers a valid
alternative. Many have claimed that the Lorentz ether theory is distinguished from the
special theory only by metaphysical considerations. However, as we shall see, such is not
the case. Figure 1 is a schematic illustrating the relationship between the Lorentz ether
theory and the special theory. On the right-hand side, the frame defined by the cosmic
background radiation (CBR) is designated by a circle. This is assumed to be the absolute
ether frame for the Lorentz ether theory. It is just another frame for the special theory. In
the middle of the figure, a circle designates the earth-centered frame with non-isotropic
light speed. The Mansouri and Sexl (MS) [15] transformation is used to map
experiments from the isotropic CBR frame to the earth-centered non-isotropic frame. ( I
will refer to the transformation as the MS transformation, however, it was earlier
described by Tangherlini and later its inverse and composite transformations by Selleri.)
The MS transformation is designated by the line connecting the two circles. The MS
transformation accounts for both clock slowing and length contraction as a function of
the speed relative to the CBR frame. Unlike the Lorentz transformation, the MS
transformation is reciprocal rather than symmetrical. Thus, an observer in the earth-
centered non-isotropic frame would see clocks run faster and lengths expanded in the
CBR frame. The MS transformation is nothing more than a Galilean transformation
adjusted for clock slowing and length contraction effects. The MS transformation
preserves an absolute simultaneity of time.

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On the left-hand side, a circle represents the earth-centered frame with isotropic light
speed. The special theory says that this frame can be directly related to the CBR frame
with isotropic light speed via a Lorentz transformation. This transformation is indicated
by the horizontal line connecting the two circles.
But there is another line which connects the earth-centered non-isotropic frame with
the earth-centered isotropic frame. This line represents Poincare's principle. Poincare's
principle states that, if lengths are contracted and time slows down as a function of the
velocity relative to the absolute frame, then there is no way via experiment to distinguish
between the non-isotropic ether frame and the isotropic frame. (Common means to
synchronize clocks within the frame lead to clock biases which make the speed of light
appear isotropic.) Thus, one is lead to the same Lorentz transformation via the two upper
paths of Figure 1 in the Lorentz ether theory. Specifically, the MS transformation and
Poincare's principle (clock biasing) together validate the Lorentz transformation.
Thus, we can arrive at the Lorentz transformation via two different paths; but the
interpretation of the transformation is profoundly different for the two paths. The special
theory says one must always transform to the observer's frame so that the speed of light is
always isotropic with respect to the observer. In fact, the special theory claims that light
in transit is automatically transformed to the new frame. By contrast, the Lorentz ether
theory says that any inertial frame we wish can be used as the isotropic light-speed
frame—we simply cannot tell which frame is the true frame. But, whichever frame is
chosen as the isotropic frame, that frame defines an absolute simultaneity and observers
moving with respect to that frame see non-isotropic speeds of light. Since the Lorentz
ether theory corresponds to an absolute ether theory (we simply do not know which
inertial frame is the absolute frame), we are not free to change frames in the middle of an
experiment. Thus, Lorentz boosts, which are valid in the special theory, are invalid in the
Lorentz ether theory.
The difference in the two theories can be clearly illustrated via their interpretation of
the famous twin paradox. Let Stella move away from Terrance at 0.6 times the speed of
light for two years per her own clock, turn around (instantly) and travel back at the same
speed. We will find that Terrance's clock will read 5 years when Stella returns and her
own clock will read only 4 years. If we put a video camera on Terrance's clock, transmit
it to Stella, have Stella show it on a video monitor next to her own clock and then video
record the two clocks, we will have a record of the combined effect of clock rate and
Doppler shift (transit time) between the two clocks.
This record will show that, for the first two years, the video of Terrance's clock as
shown on Stella's monitor will appear to be running one-half as fast as her own clock.
Thus, his clock will be reading one year on her monitor when she turns around. But, for
the next two years as she journeys back, Terrance's clock will appear to run twice as fast
on her monitor as her own clock. Thus, while her own clock reads two years plus two
years when she returns, Terrance's will read one year plus four years when she arrives
back.
As expected per Poincare's principle and the Lorentz ether theory, the video recording
of the clock rates is consistent with the choice of any absolute frame. Figures 2, 3 and 4
show the Minkowski diagram for the choice of (1) Stella's initial frame, (2) Terrance's
frame, and (3) Stella's final frame respectively. The magic of relativity is illustrated by
appending the first half of Figure 2 with the second half of Figure 4 and is shown in

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Figure 5. The special theory says that, as Stella accelerates, the signal in transit spread out
over 1.5 light years with a source Doppler which doubles the wavelength of the signal in
transit (Figure 2) has to be modified so that the signal in transit has a source Doppler
which cuts the wavelength in half (Figure 4). This magic is theoretically accomplished by
changing the time history of the signal in transit. The farther away the signal, the more its
history (time change) has to be modified. While the Lorentz ether theory says simply pick
a frame for the whole experiment and let the speed of light be non-isotropic for at least
one portion of the trip, the special theory resorts to magic so that the speed of light can
always be isotropic
The Sagnac effect is very simple to explain in the Lorentz ether theory. It is not a
direct result of rotation or acceleration. It simply arises any time the receiver is moving
with respect to the chosen isotropic light-speed frame. (The only unique feature of
rotation is that one cannot pick the observer as the isotropic frame because of the
observers acceleration.) If the receiver is moving in the chosen absolute frame, the speed
of light is not isotropic; and the Sagnac effect is simply the adjustment for the non-
isotropic light speed. It is easy to show that Sagnac's original rotating experiment will
give the same results (in agreement with his experiment) independent of the choice of
absolute isotropic frame. By contrast, the experimental evidence is loud and clear: It is
not valid to perform instantaneous Lorentz boosts per the special theory to keep the speed
of light isotropic with respect to the Sagnac phase detector. The Sagnac effect on GPS
signals in transit proves that the special theory magic does not keep the light speed
isotropic relative to the moving receiver..
Our objective has been met. There is more than a metaphysical difference between the
Lorentz ether theory and the special theory. The Sagnac effect clearly argues in favor of
the Lorentz ether theory. But, it must be admitted, a new problem has been created. If
Lorentz boosts are not valid, the standard explanation for the Thomas precession of the
electron has been invalidated. And, since the general theory did not support the special
theory with regard to the Sagnac effect, it cannot be expected to give an alternate
derivation of the Thomas precession. What is the solution? I have shown elsewhere [16],
that Thomas precession can be explained by unbalanced length contraction and mass
increase when one part of a spinning object adds to the velocity of translation while
another part subtracts. If the spinning object is then accelerated along its spin axis it will
experience a torque. Note that this explanation is valid only for spinning objects while the
special theory claims the effect occurs on all accelerated objects.
Finally, the claim that Lorentz boosts are invalid is also supported by the aberration of
the light from binary stars. Whitney [17] has developed this topic in some detail.
Conclusion
The three relativistic effects which must be considered in GPS have been addressed.
The gravitational effects are consistent with the general theory of relativity, even though
inadequate explanations are often provided. The effect of velocity on the rate at which
clocks run is not consistent with the special-theory predictions that it should be a function
of relative velocity between source and receiver. In addition, the presence of the Sagnac
effect is itself inconsistent with the special theory. It was shown that the Lorentz ether
theory provides a better explanation of the GPS relativistic effects than does the special
theory.

10
 GPS and Relativity

In a second paper, we shall show that a modified Lorentz theory, with an elastic
compressible ether in which material particles are standing waves, is capable of replacing
the general theory and resolving some of the existing problems with the general theory.

EXPERIMENTAL FRAME EARTH

ISOTROPIC LIGHT SPEED CBR
ANISOTROPIC LIGHT SPEED EARTH
ct

sl rac ses olu

co ncre (a
ra

ow te

TA
nt a bs
bt

N
M ER
er d
su

G
AN LI
LE

H
IP

SO NI
as C

U TR
bi RIN

RI AN
cl en ss imu
oc gt
P

& S
tim RE'S

SE FO
k
l

XL R M
ra
te
A

m es

te
e
C

O AT
tim

h
a
IN

R
PO

LORENTZ
IO
N
fa and ses
ex cre

st e
lta

ETHER
de ity)
p a
er d
ne
d
ad

THEORY
LORENTZ TRANSFORMATION

slower clock rate slower

contracted length contracted
increases mass increases
EXPERIMENTAL FRAME: EARTH (relativity of time simultaneity) EXPERIMENTAL FRAME: CBR
ISOTROPIC LIGHT SPEED: EARTH ISOTROPIC LIGHT SPEED: CBR

Figure 1 The Relationship between Lorentz ether theory and the special theory

11
 GPS and Relativity

5.0 4.0
6.0

3.5
4.0 5.0

3.0
4.0

Time (years) in Stella's initial frame

3.0

2.5
3.0
At Stella turn-around
0.6 years of 2.3
Terrance's signal 2.25
2.0
is in transit over
2.125
1.2 light-years
distance
1.6 2.0
1.5

Source Doppler and 1.25

slower clock cut the
frequency in transit 1.0
in half 1.0

0.5

0.0
4.0 3.0 2.0 1.0
Distance (light years) from Stella's initial frame

Figure 2: Minkowski diagram for Stella's initial frame

12
 GPS and Relativity

5.0 4.0 At Stella turn-around

1.5 years of
3.75 Terrance's signal
4.5 is in transit over
3.5 1..5 light-years
distance

4.0 3.25

3.0
3.5 2.75

2.5
3.0
Time (years) in Terrance's frame

2.25

2.5 2.0

2.0

1.5
1.0
1.0

No source Doppler
0.5 or clock effects on
the signal in transit

0.0 0.5 1.0 1.5 2.0

Distance (light years) from Terrance's frame

Figure 3: Minkowski diagram for Terrance's frame

13
 GPS and Relativity

5.0 4.0
3.75
3.5
4.5
3.25
3.0

4.0 2.75
2.5
2.25
3.5
3.4 2.0

Time (years) in Stella's final frame

3.0

2.5

2.0
1.0
At Stella turn-around 0.0
2.4 years of
1.5 Terrance's signal
is in transit over
1.2 light-years
distance
1.0 -1.0

Source Doppler overcomes

the slower clock to double
0.5
the frequency in transit

-2.0

4.0 3.0 2.0 1.0

Distance (light years) from Stella's final frame

Figure 4: Minkowski diagram for Stella's final frame

14
 GPS and Relativity

At Stella turn-around
0.6 years of 5.0 4.0
Terrance's signal 3.75
in transit suddenly
changes to 2.4 years 3.5
in transit over 4.5
3.25

Time (years) in Stella's SRT frame

1.2 light-years
distance 3.0
4.0 2.75
2.5
2.25
3.5
1.6 & 3.4 2.0
2.5 2.0 1.5
1.5

1.0
The combined Doppler 1.0
and clock rate
frequency in transit
suddenly change from 0.5
one-half the transmit
frequency to twice the
transmit frequency
2.0 1.5 1.0 0.5 0.0
Distance (light years) from Stella's SRT frame

Figure 5: Minkowski diagram for Stella's SRT frame

15
 GPS and Relativity

RELATIVITY AND GPS — II

In the first paper we showed that the global positioning system (GPS) strongly
supports the Lorentz ether theory over that of Einstein's special theory. In this second
paper, we take a close look at several problems with the general theory. Particular
attention is focused on the claim of the general theory that an object in free-fall is not
acted upon by any forces and, hence, defines its own Lorentz frame. One aspect of this
claim can be refuted by the new GPS satellites which are capable of inter-satellite
tracking. A modification of the Lorentz ether theory is proposed which resolves the
general theory problems. In addition, the new theory predicts experimental results at
variance with the general theory for several experiments to be performed in the near
future.
Introduction
GPS clock and Sagnac effects were dealt with in the first paper. In that paper we also
dealt with the gravitational-potential effects upon the rate at which a clock runs. As we
saw, Ashby's explanation of the decreased wavelength of signals sent from the satellite
improperly ascribed the decrease to the falling of photons in a gravitational field.
However, even though his explanation was clearly incorrect, the results were still in
agreement with the general theory when a correct explanation was found. In this paper
we focus upon several real problems with the general theory—some acknowledged as
problems and some not. The claim that the earth in free-fall defines its own Lorentz
frame may be refuted by the GPS system in the near future. The launch of the new GPS
Block II satellites, capable of inter-satellite tracking, I believe, will provide evidence that
the general theory is incorrect—at least as it is generally interpreted.
To solve the problems with the general theory, a solid elastic ether is proposed and is
shown to be compatible with the Lorentz ether theory. This modified Lorentz ether theory
is a natural gauge theory and provides some simple explanations, as well as providing
some interesting insight into several puzzling aspects of modern physics.
Problems with the General Theory
There are at least four apparent problems with the general theory. Each will be
described briefly. First, Yilmaz [18] claims that the equations of the general theory
predict that no attractive force will be present between two parallel plates of infinite
extent. Most physicists believe that, if this is so, it represents a failure of the general-
theory equations.
Second, the general theory claims that space is curved due to the presence of energy.
But, as Schwarzschild [19] shows, the zero-point energy of vacuum fluctuations is so
large that, per the general-theory equations, space should have a curvature at least 120
powers of 10 greater than that actually observed.
Third, Shapiro and Teukolsky [20] show that a football-shaped distribution of mass
can collapse in such a way that, if the general theory is correct, a gravitational singularity
(infinite force) can be formed which is not inside of a black hole. This means that the
singularity should cause infinite forces on matter external to the singularity. Very few
physicists believe in infinite sources. Furthermore, prior to Shapiro and Teukolsky's

16
 GPS and Relativity

demonstration, almost all believed in "cosmic censorship", i.e. that all such singularities
were inside black holes where they could safely be ignored.
Fourth, it is the claim of most general-theory specialists that a freely falling object in a
gravitational field can always be described in its own Lorentz frame, i.e. that no force of
acceleration acts on the frame. This has several implications.
An Alternative to the General Theory
Each of the above problems will be addressed at least briefly. But, first, an alternate
theory based upon an elastic solid ether is proposed. This alternative is compatible with
the Lorentz ether theory and extends it to gravitational phenomena.
Our first assumption is that matter is composed of ether standing-wave structures.
There are a number of experiments which support such an assumption. But, for the
present development, it is simply a presupposition. See, for example, the model of the
electron proposed in Escape from Einstein [21]. Because of the spin and the nonlinearity
of the ether elasticity, such dynamic distortions cause a net decrease in the ether density
within the standing-wave structure and an associated increase in the ether density outside
the standing-wave structure. The decrease in the internal ether density is associated with
the particle's mass, and the increase in the external ether density is associated with the
particle's gravitational potential.
The effects of gravitational potential described above can be easily matched to the
effects of ether density. If we assume a classical relationship between density and
(longitudinal) waves of distortion in a solid, i.e. the speed of light is inversely
proportional to the square root of the density, it is clear that the effects of mass on the
external ether density must be given by:
ρ  4GM  (7)
ρ = 4e ≈ ρ e 1 + 2 
s  rc 
where ρ e is the ether density external to the gravitational potential, and s is the scale
factor defined in equation (2)
This equation shows that the expected change in ether density (gravity potential) is
proportional to the inverse of the radial distance. Thus, the excess ether density induced
by the mass of the particle is allocated approximately linearly between all the spherical
shells surrounding the particle, with the closest having the least excess ether. But,
because the nearby shells of the same thickness have smaller volume (by the square of
the radius), the increase in density in the nearby shells is larger. This is simply another
way of saying that the compressive pressure of the ether has reached a steady state value.
All physical units can be expressed in terms of a local scale (gauge) of length, time
and mass. Only the mass as a function of gravitational potential has not already been
determined directly from experiment. There are heuristic arguments which can be made;
but, again, for brevity, let us simply assume that the mass as a function of gravitational
potential is given by:
m (8)
m = 3e
s
where me is the mass external to the gravitational potential.
Equations (3), (6) and (8) describe the local gravitational gauge of the three
fundamental units. (The time scale is the inverse of the frequency scale given in equation
3.) All other units of measurement can be described locally in terms of these

17
 GPS and Relativity

fundamental units. This scale or gauge development of the effects of gravitational

potential (ether density) reveals several fascinating insights. For example, it becomes
immediately apparent that the source of gravitational potential energy is the decrease in
rest-mass energy with decreased gravitational potential. This decrease in rest-mass
energy results from the change in gauge of the speed of light and of mass. All presently
verified effects of gravity are consistent with this simple model of elastic-ether gauge
effects.
Before pursuing the differences between this gravitational gauge theory and the
general theory of relativity, it is instructive to relate this solid elastic-ether model to the
Lorentz ether theory. The velocity gauge effects are easiest to explain in two separate
steps: an energy-free step and an energy-dependent step. First, assuming particles are
(spinning) standing waves of disturbance in the ether, the structural integrity is
maintained by the propagation of disturbances at the speed of light from any one portion
of the structure to the other portions of the structure. An unchanged energy for the
structure can result only if the size of the structure is adjusted so that the average time for
a disturbance in one part of the structure to affect other portions of the structure remains
unchanged as the velocity is changed. This demands that the structural size be reduced in
each dimension by the mean two-way change in the velocity of light. Thus, if we define a
scale factor, γ , which is greater than one and given by
1 (9)
γ =
1 − v2 / c2
we find that the transverse dimensions of the standing-wave particles are reduced by
1/ γ and the longitudinal dimension is reduced by 1/ γ 2. The only change to the particle
caused by the apparent change in the speed of light relative to the moving structure is the
size of the particle. Its energy and mass remain unchanged.
The second step in relating the elastic-ether theory to the Lorentz ether theory is to
describe the energy-dependent changes which must occur. First, as stated above, mass is
related to a local reduction in ether density inside the particle structure. The gravitational
effect of a mass can be equated to a spherical volume of radius GM/c2 in which the ether
is completely excluded, i.e. a "vacuum ball." (The gravitational radius of the earth is 0.45
centimeters.) But, if we were to move such a mass, it is clear that the reaction rate of the
ether would cause the moving "vacuum ball" to become larger (by the factor γ ), because
the reaction rate of the ether around the ball can move only at the speed of light. But the
larger effective size due to movement, without a corresponding increase in the speed of
light, means that length, time and mass will become dilated as compared to the same
stationary particle.
When we combine the two steps described above, we match the two critical factors
associated with the Lorentz ether theory. Specifically, the dimension of a particle moving
through the ether shrinks by the factor 1/ γ in the longitudinal dimension but is
unchanged in the transverse dimensions. Time is dilated by the factor γ . While it is not
required to match the Lorentz ether theory, mass is increased by the factor γ .
Addressing the Problems
Finally, the structure is in place to address individually the problems with the general
theory mentioned above. First, we address the "Yilmaz" problem. Yilmaz has claimed
that the general theory predicts that no gravitational attraction would exist between two

18
 GPS and Relativity

infinite (very large size compared to thickness) parallel plates of mass separated by a
small distance. Charles Misner and William Unruh [18] have reportedly taken up the
challenge to explain such "strange" behavior. But why is the non-attraction considered
strange? The elastic ether theory also predicts non-attraction from two infinite parallel
plates. Since all shells (plates) surrounding the two plates have the ether compressed by
equal amounts, there is no ether gradient and, hence, no force on either plate—once the
ether density is equalized between the outside and inside portions of the two plates. Since
finite plates should exhibit reduced gravitational attraction, the effect may be subject to
experimental confirmation.
The second problem with the general theory—the expected huge curvature of space
due to the huge energy of "zero point" oscillations in the vacuum (ether)—is easily
resolved by the ether-gauge theory and is quite similar to the prior solution. Yes, energy
causes a compaction of the surrounding ether. But the "curvature of space" arises from
the gradient of the ether density. Thus, a uniform energy density, such as is caused by the
"zero point" oscillations, does not result in any curvature, i.e. there is no ether-density
gradient.
Shapiro and Teukolsky demonstrate a real hole in the general theory (the third
problem mentioned above). But the same problem does not arise in the ether-gauge
theory. When the ether-density model is employed, it becomes evident that black holes
and infinite gravitational fields simply cannot exist. If a massive body were to shrink in
size due to gravitational attraction such that it approached its "gravitational radius," the
ether would be completely excluded from inside that radius; and, since particles are
standing waves in the ether, they could not exist inside the black-hole radius. And as
shown in a recent paper [24] the gravitational force is self-limiting so that black holes
cannot form. Presumably two very dense neutron stars colliding would largely
disintegrate into electromagnetic radiation. Even more intriguing, such disintegration
radiation may be the source of the strange gamma-ray bursts, which seem to be of
intergalactic origin. Tsvi Piran [25] presents a convincing argument that gamma-ray
bursts are the result of colliding neutron stars. Per the general theory, such collisions are
expected to be a significant source of gravitational waves; and the eventual detection of
such waves in coincidence with gamma-ray bursts, Piran says will confirm his diagnosis.
However, the modified Lorentz ether theory yields two significant differences from
Piran's predictions.
First, the modified Lorentz theory predicts that colliding neutron stars will not form a
black hole. Instead, the neutron stars would presumably explode in a gigantic burst of
radiation, in which a significant percentage of the mass would be converted into
radiation. This expectation actually is supported by Piran's data. He indicates that only a
small percentage of the neutron binaries' energy would be needed to generate the
observed gamma-ray bursts. However, it is clear from the mechanism which he proposes
(and the associated figure) that such radiation would be far from isotropic. A directional
beam of radiation, similar to the radiation from the neutron pulsar itself, could be
expected. But this, in turn, would imply that only a small percentage of the colliding
neutron binaries in the observable universe would emit gamma-ray bursts in the direction
of the earth. But, from Piran's calculations, the Compton Observatory apparently sees
every neutron binary collision in its field of view. This strongly suggests a near isotropic
generating mechanism.

19
 GPS and Relativity

The modified Lorentz ether theory also indicates that gravitational waves are nothing
more than electromagnetic waves [22]. The clear prediction of the new theory is that
gravitational radiation will never be detected. Thus, Piran's coincidence test will never be
executed.
The fourth and last problem with the general theory mentioned above is the claim that
no force of acceleration acts on a freely falling frame. From this claim many other claims
arise. As one example, Kip Thorne [26] claims, while discussing Stanford's Gravity
Probe B experiment, that a body orbiting the earth, since it is in free fall, will not
experience any Thomas precession. (Since our model ascribes the torque causing the
Thomas precession to a spin velocity induced mass imbalance, gravity which acts on
mass directly will not cause a precession—not because it is not a force but because it acts
on the center of mass.) Stanford's Gravity Probe B experiment involves the launching of
an extremely precise gyroscope into earth orbit. The launch is currently planned for about
the year 2000.
The Sagnac Effect Again
However, the claim, which we are most interested in pursuing here, is that the Sagnac
effect should not exist in a freely falling frame. In a rather thorough review of the Sagnac
effect, Anderson, Bilger and Stedman [27] make the following statement:
Incidentally, the final suggestion of Michelson [21], that the orbital motion of the Earth around
the sun be detectable in a sufficiently gargantuan ring interferometer, is not consistent with general
relativity: a freely falling point object (the whole earth in this context) defines a local Lorentz
frame.
This is a rather amazing statement. I know of no way to interpret it other than as a
claim that the Sagnac effect cannot be used to detect the approximately one degree of
earth rotation per day which is related to the earth's orbital motion. But, if this rotation is
undetectable, the measured rotation (Sagnac experiment fixed to the earth) with this
component removed must become the rotation per a standard 24-hour day rather than the
rotation per sidereal day. The implication, as Howard Hayden [28] points out, is that, if
true, a Sagnac experiment using the inter-satellite communication links of the newer GPS
satellites should yield a null result when computed relative to a frame rotating at a rate of
once per year. There is no reason that this experiment cannot be performed in the near
future, when a sufficient number of the new GPS satellites with inter-satellite
communication and ranging capability are in orbit. But there is already evidence that the
statement is untrue.
Let us look at the proposed experiment more closely. It turns out that the critical
concept is time versus clocks. If time is slowed by decreased gravitational potential,
Anderson et al. are interpreting the general theory correctly; and, in fact, no Sagnac effect
should be measured. On the other hand, if clocks simply slow down as a function of the
decrease in gravitational potential and a universal flow of time, independent of local
clock measurements, exists, then clearly the proposed Sagnac experiment can be used to
measure the angular rotation due to the orbiting earth. Note that the equations for an
elastic solid ether are virtually identical with the general theory equations. However, just
as with the Lorentz transformation, the ether interpretation is substantially different. The
general theory ascribes a change in the rate at which clocks run to a change in the flow of
time. By contrast, the ether theory ascribes the clock rate-change to an environmental
effect.

20
 GPS and Relativity

But it is not difficult to illustrate that treating the whole earth as a point object and,
hence, a single local Lorentz frame poses some problems. Several cases of four distinct
objects in free-fall orbit about the earth can be used to illustrate one of them. First, let's
impart a position and velocity to the four objects such that they all have the same orbital
period; but two are in a perfect circular orbit, one slightly behind the other. One of the
two remaining is placed slightly above the two that are in circular orbit, and one is placed
slightly below. Approximately 1/4 of an orbit later they will all have the same orbital
distance from the earth, and 1/2 an orbit later the above and below objects will have
interchanged their position. But the other two will remain in their same relative position.
There is no transformation, Lorentz or otherwise, which will properly map the changing
relationship of these freely falling objects.
To illustrate the problem further, if the objects above and below the two in circular
orbit are given the exact velocity such that they are also in precise circular orbits, their
orbital periods are changed such that the particles diverge from each other—not a very
useful Lorentz frame.
Finally, let's attach the four particles to each other. If they are all balanced in weight,
assuming no rotation, the four particles will orbit the earth; but the particular particle
farthest from the earth will change (i.e. no rotation in an inertial frame) as the combined
object passes through each quadrant of the orbit. This is clearly not an acceptable Lorentz
frame, since the direction of the velocity vector is constantly changing relative to the four
particles, which define the frame.
But what if the four particles are attached to each other and the upper and lower
particles are made heavier than the other two? Now there is a gravity gradient or tidal
force (sorry, Anderson et al. to use the word ‘force’—would you rather describe the
effect as a force-free divergence of the acceptable Lorentz frames?) which causes the four
objects to maintain their same relative positions with respect to the earth. The moon in
orbit around the earth illustrates this arrangement by keeping the same face directed
toward the earth. Now this gravity-gradient stabilized frame has some interesting
characteristics. The objects do maintain the same constant orientation with respect to the
common velocity vector. And it is not difficult to show that the gravity gradient and
differential velocity are precisely such as to cause all identical clocks located anywhere
upon such a composite object to run at the same rate. Saying this another way, the earth's
gravitational gradient and the orbital velocity gradient of the moon are such that their
effect on any clock located on the moon exactly cancels (in an earth-centered frame).
This is shown below.
The change of the clock rate with respect to a radial change in the distance from the
gravitational source is given by the derivative of equation (3). Specifically, the effect of
the gravitational gradient upon the clock rate is:
df GM (10)
≈ 2 2
dr r c
The orbital velocity at any point on a gravity-gradient stabilized object is given by, rθ! .
Substituting this expression into equation (1) and taking the derivative gives:
df rθ! 2 (11)
≈− 2
dr c

21
 GPS and Relativity

GM
But, for a circular orbit, the value of θ! 2 is given by 3 ; and, when this is substituted
r
into equation (11), it becomes clear that the two clock-rate effects cancel.
In conclusion, as far as clocks are concerned, it appears to be valid to claim that the
earth, or any object in free fall, can be treated as occupying its own local Lorentz frame—
at least if it is in a circular orbit. But, as already mentioned, Hill has shown, using
external pulsar timing sources, if the object is not in circular orbit, the local clock rate
will vary as a function of the changing gravitational potential and orbital velocity.
Clearly, this is behavior different from an unaccelerated frame. In addition, according to
the elastic-solid extension of Lorentz's ether theory, it is the clock behavior which is
changed, not time. Thus, we can still expect to detect the Sagnac effect caused by the
orbital rotation of the frame.
Now let's look directly at what our elastic-ether theory predicts for a GPS-based
Sagnac experiment around the earth. Can it detect the angular rotation due to the earth's
orbit of approximately 10 per day? It is easier to show the expected result if we modify
our Sagnac experiment a bit. Specifically, let us use a wedge-shaped light path and send
the two beams in different directions around only 1/2 of the total path. As shown in
Figure 1, the light source will be at about the mid-point of the left side of the wedge, and
the phase detector will be at about the same position on the opposite side. Because of the
small size of the light path as compared to the total orbit, it is valid to approximate the
outer leg of the light path as a segment whose length is roθ and the inner leg of the light
path as a segment of length riθ . The light source will be positioned such that the two
light beams arrive at the start of the outer and inner segments of the light path at the same
time. Similarly, position the detector such that, if the light beams arrive at the end of the
outer and inner segments at the same time, they will arrive at the detector at the same
time. Thus, our Sagnac detector simply measures the relative amount of light-travel time
across the outer and inner light paths. Clearly, the light-travel times must be equal, if this
detector is to rotate through an entire circle in one year without detecting any motion.
But it is not difficult to show that the outer light path is longer than the inner light
path, even when the paths are adjusted for the gravitational potential effect on the speed
of light. This can be shown easily by simply dividing the two distances by the respective
light speed given in equation (5). Thus, it is the clear prediction of the ether theory that a
Sagnac experiment using cross-linked GPS satellites should be capable of refuting the
Anderson et al. prediction of the general theory.

22
 GPS and Relativity

But there is already experimental evidence that the prediction is incorrect. Ironically,
in the very same paper, Anderson et al. point out how precise current Sagnac gyroscopes
have become. In fact, the third author of the paper, Stedman, works at a facility which
apparently has the world's most accurate ring-laser gyroscope. The precision is claimed to
be a 12 order of magnitude improvement over the Michelson-Gale experiment, while
using an encompassed area 276,000 times smaller. Because of temperature-induced
drifts, the authors indicate that the accuracy of the earth-rotation measurement is much
less but still good to about 0.1% of the rotation rate. But this accuracy is 2.7 times the
orbital-rotation rate and is thus easily measured by the existing ring-laser system. Why
did they not tell us the measured rotation rate? I am very sure that it includes the earth's
orbital-rotation rate.
The Stanford Gravity Probe B (GPB) experiment was mentioned above. It involves a
mechanical gyroscope, but I know of no physicist who would argue that a mechanical
and an optical gyroscope would give different results. It is the intent of GPB to measure
the Lense-Thirring frame dragging from earth rotation and the geodetic precession (spin-
orbit and space curvature effects). The former will amount to about 0.05 arc seconds per
year and the latter to about 6.9 arc seconds per year. By contrast, if the gyroscope were
affected by the orbital rotation, an additional anomalous precession of 1,296,000 arc
seconds per orbit results. This insensitivity of mechanical gyroscopes to orbital rotation is
clearly illustrated by the early TRANSIT (Navy navigation) satellites. During launch the
satellites acquired a large spin, and the satellites themselves acted like large mechanical

23
 GPS and Relativity

gyroscopes. In order to point the transmit antenna toward the earth, a boom with attached
mass had to be deployed to cause gravity-gradient stabilization. But the satellite spin had
to be removed before the gravity-gradient stabilization could occur—precisely because a
gyroscope can measure (i.e. is not itself affected by) the orbital rotation.
Another Prediction
Incidentally, I have already predicted [23] that Gravity Probe B will detect a different
amount of geodetic precession than that predicted by the general theory. I used a rather
long argument to conclude that the predicted spin-orbit component (2.3 arc seconds per
year) was only half the size it should be. The rest of the geodetic precession was due to
space curvature and contributed 4.6 arc seconds per year. A simple method of arriving at
my new prediction is to note that, if one measures time with a clock external to the
gravitational field (local clock rate is immaterial), the "space curvature" (gradient of ether
density) is twice what the general theory predicts. This leads directly to my prediction
that the total geodetic precession measured by GPB will be 9.2 arc seconds per year
rather than the general theory prediction of 6.9 arc seconds per year.
Before leaving the subject of freely falling frames and GPB, note the following quote
from Thorne [26]:
...In our gravitational problem the Thomas precession is absent because the gyroscope is
presumed to be in a free fall orbit i.e., it is not accelerated relative to local inertial frames; there are
no "boosts."
I argued earlier that Lorentz boosts are invalid. Thus, I have no problem with the
absence of Thomas precession. (Of course, as I have been arguing, I disagree with
Thorne's reason it is absent.) However, Thomas precession is real; and, if Lorentz boosts
are not the cause, another mechanism is needed. As mentioned in part I, Thomas
precession arises naturally as a result of the composite velocity effects (mass increase and
length contraction) on a moving spinning object which is accelerated orthogonal to its
translation velocity. But, it does not apply to gravitational acceleration because gravity
acts on the center of mass—not the center of spin.
Conclusions
Four problems with the general theory were presented. An alternative theory was
proposed of a solid elastic ether which constituted a particular representation of the
Lorentz ether theory. This new theory was shown to provide a simple resolution to the
general theory problems. The particular claim of the general theory that a freely falling
body is not acted upon by external forces was explored at length. It is clearly not valid. It
predicts gyroscopic behavior which is clearly not realized. In addition, it should be
capable of direct falsification with the launch of the new GPS satellites capable of inter-
satellite tracking.
Finally, several predictions have been made in the course of the development.
Specifically, it is predicted: (1) that gravitational radiation will never be detected; (2)
unambiguous evidence for a black hole will never be found; and, (3) the amount of
geodetic precession measured on the Gravity Probe B experiment will be one-third
greater than that predicted by the general theory.

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 GPS and Relativity

REFERENCES
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21. Ronald R. Hatch (1992) Escape from Einstein, pp 138-142.
22.ibid, pp 80-88.
23. ibid, pp 209-212.

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 GPS and Relativity

24. Ronald R. Hatch (1998) “Gravitation: Revising both Einstein and Newton,” Galilean
Electrodynamics, Vol.10, No.4, July/August, pp 69-5.
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28. Howard Hayden, (1994) personal letter dated 12 December.

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