Time Transfer and the Sagnac Correction in the GPS

Time transfer from a GPS satellite to a receiver fixed on the surface of the rotating Earth is investigated. Using experimentally confirmed GPS light signal transfer time and a simple kinematic calculation, light speed variation is demonstrated for light travelling between the satellite and the ground-based receiver. This variable speed is also determined using classical principles applied to light transmission relative to the terrestrial receiver moving as a result of the rotating Earth, all within the Earth-Centred Inertial (ECI) frame. These results produce the accurate time transfer algorithm without any correction and lead to the re-interpretation of the “Sagnac correction” considered by the ITU to be a necessary time correction in GPS time transfer. They also lead to a revision of the Lorentz Transformations that yields the Selleri Transformations which better accord with the observed light speed variation and confirmed relativistic phenomena in the physical world.


Introduction
Time Transfer is the process of communicating time information by way of electromagnetic signal transmission across space.It is necessary for example to maintain coordination of time and frequency in systems operating at or close to the Earth and beyond.The method by which this is accomplished is based on an algorithm published by the International Telecommunications Union (2012) that has been rigorously tested and verified.It involves the transmission of a signal from one station to another such that system synchronization can be effected.Today it is part of the standard procedure employed in time comparisons between separated laboratories on the rotating Earth and is widely accepted (Nelson, 2011;Ashby, 2010;Petit & Wolf, 2005;Blanchet, Salomon, Teyssandier, & Wolf, 2001).
The approach described in the ITU recommendation (2012) and supported by Nelson (2011), Ashby (2010), Petit and Wolf (2005) and Blanchet et al. (2001) is to use light (electromagnetic) signal transmission in the Earth-Centred Inertial (ECI) frame where it travels at speed c to determine the travel time from a satellite to a ground receiver which is moving at speed v .Because light speed in the ECI frame is known, this computation is very straightforward and yields a transfer time given by where R is the initial distance between the satellite and the receiver.This result has been experimentally confirmed.The publications by the ITU (2012) and several authors (Nelson, 2011;Ashby, 2010;Petit & Wolf, 2005;Blanchet et al., 2001) interpret the first term c R / in (1) as the time for the light to traverse the distance R at constant speed c while they interpret the second term 2 / .c R v as the "correction" necessary for accurate time arising because of the movement of the ground receiver as a result of the rotation of the Earth.These researchers refer to this latter time adjustment as the "Sagnac correction" and this interpretation is widely accepted as being correct.In the ITU recommendation it is also stated that "The speed of light is c in every inertial frame of reference".This claim of light speed constancy for one-way light speed in all inertial frames is a foundation principle in special relativity that has however not been confirmed (Zhang, 1997).
Light travel between a GPS satellite and a receiver on the surface of the Earth has been considered by Sato (2010).He concluded that for such light transmission there is light speed variation relative to an observer on the surface of the rotating Earth.Phipps (2006, p42) has also made reference to this phenomenon.These variable light speed results cast doubt on the interpretation by the ITU of constant light speed c with the "Sagnac correction".After careful examination of this situation, we have arrived at the conclusion that this interpretation of the time transfer equation (1) contained in the ITU recommendation and enjoying wide support is invalid and these results form the subject of this paper.We show using elementary analysis that the actual physical situation involves variable light speed for light travelling between the satellite and the moving ground receiver with no "Sagnac correction" being necessary for accurate results.These findings are reconciled with results previously presented by Selleri (2010) and the implications for space-time physics are thoroughly discussed.

Time Transfer in the GPS
In this section transfer time is derived following the approach by Ashby (2010).This is done by considering an electromagnetic pulse transmitted from a GPS satellite at position T . Ashby proceeds by squaring both sides of equation ( 2) and truncating the resulting (3) From this and again for leading order in v , The interpretation of the transfer time in equation ( 5) presented by the ITU and several researchers is that the first term c R / is the time for the light to travel the distance R at speed c as it travels toward the receiver fixed on the surface of the Earth and the second term is the "Sagnac correction" necessary because of the rotation of the Earth and consequent movement of the receiver.We now show using two approaches both involving elementary analysis that this interpretation is wrong.

Light Speed Determination Using GPS Transfer Time Measurement
The transfer time t Δ in (5) for light travelling from the orbiting satellite to a ground-based receiver has been fully tested and experimentally verified and is routinely used in time comparisons.It is therefore accurate and can be used to evaluate the light speed R c relative to the ground receiver.Using the distance R between the satellite and the receiver at the time of transmission of the signal, light speed R c is given by the receiver but is consistent with variable light speed results previously presented (Gift, 2013).
where vT is the distance moved by the pedestrian before interception.Equation ( 8) corresponds to Ashby's equation (2).Solving for time T gives Therefore velocity CR v of the automobile relative to the pedestrian is found by dividing the initial distance D between the automobile and the pedestrian by the time to interception T which gives This is the well-known result for relative velocity in classical mechanics and the elementary calculation in ( 10) is exactly that employed in ( 7 which is exactly the transfer time (5) given in the ITU recommendations that has been rigorously tested and confirmed.It follows therefore that the GPS transfer time is

Light Speed and Time Transfer Using Relative Velocity in the ECI Frame
The relative light speed R c can be determined in another way.The ECI frame is a frame that moves with the Earth in its orbit around the sun but does not share its rotation.In this frame GPS signals propagate in straight lines at constant speed c relative to the frame.We can use the ECI frame and classical velocity composition to determine the relative light speed R c for light propagating from the GPS satellite to the ground receiver.Thus consider again the light pulse from the GPS satellite travelling at speed c in the ECI frame and received by the ground receiver where the surface velocity is v at angleθ relative to the direction of light propagation as shown in Figure 3.In order to determine the velocity of the light relative to the moving receiver, a velocity v′ that is equal and opposite to the surface velocity v is applied in order to bring the receiver to rest relative to the ECI frame (Sadler, 1983).The light speed R c relative to the ground-based receiver is then the resultant of c and v′ given by and therefore Note that this light speed is the speed relative to the receiver fixed in the ECEF frame and not speed relative to the ECI frame which is c .It is exactly the speed obtained in the kinematic calculation in (7).In order to determine the angle α between the relative light speed R c and c , consider the right-angled triangle ABC in Figure 3 which is the transfer time t Δ presented in (5) and published by the ITU.
To further elucidate the arguments, consider the situation in the ECI frame where the surface velocity v is oriented in the direction of the light transmission corresponding to o 0 = θ in Figure 1.Then light transmission in the ECI frame yields This corresponds to equation (2) given by Ashby.Solving for t Therefore the light speed relative to the receiver is given by The expansion of ( 19) to first order in c v / gives  23) and ( 21) it can be seen that the two situations are completely analogous.In the case of the moving automobile however, equation ( 23) is not interpreted as the time  15) and ( 7).The possibility of demonstrating light speed variation v c ± relative to the rotating Earth by employing a GPS satellite operating in the ECI frame and transmitting to a receiver on the surface of the Earth was also pointed out by Phipps (2006, p42) and later demonstrated by Sato (2010).

Time Transfer and Space-Time Physics
The results of this research unmistakably demonstrate variable light speed for light transmission from a GPS satellite to a ground-based receiver, contrary to the principle of light speed constancy invoked in the ITU recommendation and by several authors.This light speed variation was determined firstly based on elementary kinematics utilizing confirmed GPS transfer time and a general distance measurement.This light speed value was then verified by using it to derive the experimentally confirmed value of time transfer.Light speed relative to the receiver was also determined using classical velocity composition for light transmission in the ECI frame.This then yielded the transfer time (5) by simple calculation.It means therefore that the GPS transfer time is Thus the detection of light speed variation is an objective fact validated by observational experience using accurate atomic clocks in the actual time measurement of travelling light.It follows therefore that the principle of light speed constancy is falsified on the surface of the Earth as well as in the immediate surrounding region within the ECI frame.Since the principle directly yields the Lorentz Transformations in space-time physics (Rindler, 2006), the invalidity of the principle implies that these transformations do not represent the physical world.Towards resolving this difficulty Selleri (1996Selleri ( , 2011)), using experimentally confirmed time dilation (Zhang, 1997) and two-way light speed constancy (Zhang, 1997) determined the set of "equivalent" transformations which contains all possible space-time transformations that connect two inertial frames (25) where . Interestingly Selleri (2011) has shown that despite the synchronization difference these transformations make the same predictions for a broad range of phenomena including Michelson-Morley, Romer, Doppler, Fizeau, and the International Atomic Time.However these transformations make different light speed predictions for light travel in the "moving" frame given by (Selleri, 1996(Selleri, , 2011) where c v / = β and θ is the angle between the direction of light propagation and v .Spavieri (2012) discussed the two significant members of this "equivalent" set namely the Lorentz transformations corresponding to and whose time transformation is and the Selleri transformations corresponding to The difference between the two transformations is clearly the single term Starting with the classical Galilean transformations given by measuring S t′ can be externally synchronized using synchronized clocks in where the light speed is c .They appropriately describe as "true speeds" the speeds calculated in S ′ using these synchronized clocks and ordinary rulers that measure S t ′ and x ′ respectively.This "true speed" of light as predicted by the Selleri transformations can be determined by setting in the "moving" frame S ′ (Selleri, 1996).This speed in (31) reduces to cos θ determined in this paper in the time transfer process using two methods in ( 7) and (15).Selleri (2010) used this variable light speed (31) to calculate the time for an electromagnetic signal to travel between two points on the surface of the earth via a geostationary satellite and thereby fully accounted for the "Sagnac correction" unnecessarily introduced by others in the time transfer process (ITU, 2012;Nelson, 2011;Ashby, 2010;Petit and Wolf, 2005;Blanchet et al., 2001).He argued therefore that "The procedure which we suggest to experimentalists is to avoid using a wrong velocity of light [ c c R = ] and correcting the result with an ad hoc term, but rather to use from the beginning the velocity of light ] of the [Selleri] transformations."This is fully consistent with the arguments presented in section 2 of this paper where we show that the light travels toward the receiver at relative speed θ cos v c − and notc .We therefore completely support Selleri's position.
Guerra and de Abreu describe the effect of the introduction of the term ".We entirely agree with their description of this process as "de-synchronizing" the synchronized measuring clocks which now measure time L t′ in S ′ .They unfortunately consider the resulting Lorentz transformations as mathematically equivalent to the original Selleri transformations which is incorrect since the two transformations predict different light speeds as given in (26).We nevertheless embrace their description "Einstein speeds" of the speeds calculated in S ′ using these "de-synchronized" clocks and ordinary rulers to measure L t′ and x ′ respectively.This speed of light corresponding to that predicted by the Lorentz transformations can be determined by setting It is clear therefore that the introduction of the "de-synchronizing" term results in the false prediction of light speed constancy in all moving frames.It constitutes an ad hoc mathematical convention which realizes non-existent light speed constancy that is unconnected to the real world.Almost any "speed" can be similarly obtained by suitably "de-synchronizing" the clocks through variation of 1 e in (26) and Will (1992) has observed that "a particularly perverse choice of [de-] synchronization can make the apparent speed…infinite"!This particular choice corresponds to in light speed equation ( 26) above which yields infinite light speed.These "apparent" speeds represented in general by (26) for  26).These results are entirely in accordance with the findings of Gift (2009a) and Selleri (2004Selleri ( , 2011) ) who identified the Selleri transformations given by z z y y vt x x

Conclusion
The main contribution of this paper is the demonstration of light speed variation in the time transfer process in the GPS which contradicts the principle of light speed constancy.This was effected using light travel times based on the GPS time transfer algorithm and simple kinematics as well as classical mechanics involving velocity composition in the ECI frame.Both approaches yielded the relative light speed on the surface of the Earth, thereby rendering any "correction" completely unnecessary (Selleri, 2010).
The failure of the principle of light speed constancy means that the Lorentz Transformations which are derived from this principle cannot represent the physical world and must be replaced.Selleri derived the "equivalent" set of transformations which provides the framework for the determination of the correct transformations from all those that are allowable.This turns out to be the Selleri transformations which differ from the Lorentz transformations by a single term 2 / c x v ′ in the time component of the Lorentz transformations.The elimination of this term effectively provides reconciliation with observed light speed variation while still satisfying all the experimentally confirmed predictions in special relativity as demonstrated by Selleri (2004Selleri ( , 2011)).The Lorentz transformations and special relativity have been at the center of ongoing controversy since the inception of the theory (Essen, 1971;Dingle, 1972;Bethell, 2009).In light of the elementary but unassailable results derived in this paper as well as the considerable contradicting evidence now available (Sato, 2010;Marmet, 2000;Kelly, 2005;Hayden, 1991;Gift, 2010aGift, , 2010bGift, , 2010cGift, , 2012aGift, , 2013;;Wallace, 1969;Navia et al., 2007;Nodland & Ralston, 1997), the continuing use of the Lorentz transformations and special relativity lacks intellectual integrity and is scientifically indefensible.
The rejection of the Lorentz transformations along with special relativity and the introduction of the Selleri transformations into mainstream physics are likely to open new and interesting areas of space-time research.For example the relaxation of the Lorentz covariance requirement of relativity theory has resulted in the development of the elements of a new quantum theory of magnetic interaction that offers plausible explanations for chemical reactivity, the covalent bond and the Pauli Exclusion Principle (Gift, 2009b).Another interesting development is the provision of a theoretical foundation for the existence of a preferred frame by the new transformations (Selleri, 2004(Selleri, , 2011)), the detection of which has been reported by several researchers (Galaev, 2002;Demjanov, 2010;Gift, 2006Gift, , 2012b)).Finally, these new transformations may assist in the development of an improved gravitational theory as attempted by Logunov (1998) and thereby cast new light on the major unsolved problem of the past century which is the unification of gravity and quantum theory (Smolin, 2006).The recent failure to verify supersymmetry (Wolchover, 2012;Lykken & Spiropulu, 2014) emphasizes the need for action. 2

r
and GPS time T t travelling at speed c relative to the ECI frame to a ground receiver whose position at GPS time T t is R r and whose velocity because of the rotation of the Earth is v relative to the ECI frame.Let θ be the angle between the direction of propagation of the signal and v which, because c v << can be represented as shown in Figure 1.If the signal arrives at the receiver at time R t then for the signal transmission interval

Figure 1 .
Figure 1.Light Transmission in the ECI Frame from GPS Satellite to ECEF Receiver 2

Figure 2 .
Figure 2. Vehicle at A and Pedestrian at B on the surface S c travels at speed c relative to the receiver in the Earth-Centred Earth Fixed (ECEF) frame when in fact the light speed relative to the receiver is θ

Figure 3 .
Figure 3.Light Speed in the ECI Frame and Receiver Speed due to the Earth's Rotation

.
It follows therefore that the velocity of the light R c relative to the ground-based receiver on the surface of the rotating Earth is an angle θ relative to the receiver velocityv .The light transfer time to travel the distance R at speed θ 2 with the case of the automobile in Figure 2 at position A moving at velocity C v relative to the surface S (corresponding to light travel at velocity c relative to the ECI frame) and the pedestrian a distance D away at B walking at velocity v in the same direction as the automobile such that C v v << (corresponding to the receiver a distance R away from the satellite moving at speed c v << relative to the ECI frame).Then as shown previously, classical mechanics gives the speed of the automobile relative to the pedestrian as simply v v C − .Therefore as evaluated in equation (9) the time T for the automobile to intercept the pedestrian is given by

−
the automobile to travel at speed C v relative to the pedestrian with a "correction" the movement of the pedestrian.Instead the accepted and correct interpretation of (23) is this is the time) the automobile to travel to the pedestrian at speed v v C − relative to the moving pedestrian.In the same way in time transfer in the GPS, equation (21) should not be interpreted as the time c R / for the light to travel at speed c relative to the receiver in the ECEF frame with a "Sagnac correction" necessary because of the rotation of the Earth as advanced in the ITU recommendation and by several authors.Instead, as demonstrated for the automobile and the pedestrian, the correct interpretation of (21) is that it is the time ) for the light to travel to the receiver at speed v c − relative to the receiver.It should be noted that the relative speed v c − is exactly the relative speed in (15) and (7movement in a direction opposite to the direction of light transmission yields relative light speed v c .The interpretation in the ITU Recommendation that the time transfer is c R / with a "Sagnac correction" encourages the belief that light travels at speed c relative to the receiver when the speed is really θ cos v c − as has been conclusively demonstrated. Levy (2003) has shown that the incorporation of experimentally confirmed clock retardation and length contraction yields theSelleri transformations  Abreu (2006)  refer to these transformations as synchronized transformations since the clocks in ) the time component of the Selleri transformations (28) resulting in in the time component of the Lorentz transformations (27) as delaying the synchronized moving clocks "by a factor that is proportional to their distance x′ to the reference position 0 = ′ x moving" frame S ′ .This result (32) predicted by the Lorentz transformations constitutes light speed constancy that is a quite different speed from the variable light speed (31) derived from the Selleri transformations and is directly contradicted by the light speed c GPS signal transmission calculations in sections (2.1) and (2.2).
accord with the physical world and not the Lorentz transformations represented by z an angle θ with respect to the surface velocity at the receiver.This result is consistent with relative to the surface of the rotating Earth was used to derive the confirmed GPS time transfer algorithm that is published in the ITU recommendation.It was shown that accurate GPS time transfer uses )