One-way Speed of Light Using the Global Positioning

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One-way Speed of Light Using the Global Positioning System Stephan J.G. Gift Department of Electrical & Computer Engineering Faculty of Engineering The University of the West Indies, St Augustine, Trinidad and Tobago, West Indies

Abstract The Principle of Light Speed Constancy is one of the tenets of modern physics and since 1983 has been the basis of the standard of length measurement. It was postulated in 1905 by Albert Einstein who employed it in the development of his special theory of relativity. Numerous tests have been performed to verify this postulate but after more than 100 years one-way light speed remains unverified. The advent of the global positioning system (GPS) has changed this unsatisfactory situation as it provides the means to accurately determine one-way light speed. It is a modern system that employs advanced synchronized time-measuring technology in its operation. It enables the accurate determination of elapsed time in a wide range of applications including time-stamping of financial transactions, network synchronization and the timing of events. According to the IS-GPS-200E Interface Specification GPS signals propagate in straight lines at the constant speed c (in vacuum) in an Earth-Centered Inertial (ECI) frame, a frame that moves with the Earth but does not share its rotation. This constancy of the speed of light in the ECI frame is utilized in the range equation of the GPS to accurately determine the instantaneous position of objects which are stationary or moving on the surface of the Earth. This author has used this very successful system in a variety of ways to determine the one-way speed of light and has found values that indicate variable light speed and therefore a violation of the principle of light speed constancy. In this chapter we review some of these contributions. Specifically we discuss the issue of the determination of the one-way speed of light on the rotating Earth using the synchronized clocks of the GPS. Here the algorithm for clock synchronization is derived using the Langevin metric and this is used in a simple kinematic calculation to demonstrate light speed variation. This analysis is supported by using the constancy of the speed of light in the ECI frame to derive the same result. Another powerful method discussed is the use of time transfer from a GPS receiver to determine one-way light speed. A fourth method to examine one-way light speed is the use of interplanetary tracking technology involving Coordinated Universal

2 Time in the solar barycentric frame. This approach also yields variable light speed and is consistent with light speed changes observed on the orbiting Earth for light from planetary satellites in the Roemer experiment and for light from stars on the ecliptic in the Doppler experiment. In the final section these results are reconciled with results presented by Selleri who introduced his set of “equivalent” transformations from which the correct space-time transformations can be derived. A full analysis is presented which shows that these correct transformations are the Selleri transformations and not the well-known Lorentz transformations.

1.

Introduction A central idea in modern physics is the principle of light speed constancy according to

which light travels at a constant speed c in all inertial frames [1-3]. It is one of the foundation postulates of special relativity introduced in 1905 by Albert Einstein and since 1983 has been used as the basis of the standard of length measurement. Numerous experiments have been conducted over the past century yielding light speed c (in vacuum) [4]. Those performed in recent years are generally modern versions of the famous Michelson-Morley experiment which compare the resonant frequencies of two orthogonal electromagnetic resonators [5-9]. Several experiments have attempted to test the one-way speed of light [10-12]. These consistent results have led to the almost universal belief that the principle has been experimentally confirmed. Zhang [4] has however shown that it is two-way light speed invariance that has been verified but not one-way light speed constancy which remains unconfirmed. While the principle as formulated applies strictly in inertial frames, the majority of light speed tests claiming its verification have been conducted in the non-inertial frame of the rotating Earth [4-12]. This is justified on the grounds that it is always possible to find in a sufficiently restricted area on the Earth’s surface a set of local inertial frames in which the principle can be successfully applied and therefore where the speed of light is locally isotropic [13]. This view is supported by many researchers including including Rizzi and Tartaglia [14], Rizzi and Tartaglia [15], Rizzi and Ruggiero [16], Pascual-Sánchez et al.[17] and Kassner [18]. This complete acceptance of the applicability of the principle of light speed constancy in the non-inertial frame of the rotating Earth is further demonstrated by its use in the SI definition of the metre which is used everywhere on the surface of the Earth.

3 There is however some dissent regarding the physics of light transmission on a rotating platform. Specifically a few researchers argue for light speed anisotropy around such a platform and the consequent failure of the principle of light speed constancy [19, 20]. Selleri [19] for example advanced a proof of anisotropy in the speed of light in a reference frame that is commoving with the edge of a rotating platform. Even Rizzi and Tartaglia who initially argued for global light speed isotropy in a rotating frame [14], later modified their position to recognize only local light speed isotropy in a rotating frame [15]. In a paper presented at a Precise Time and Time Interval meeting more than thirty years ago describing the results of time transfer experiments using laser light, Alley and others [21] noted that general relativity predicts an asymmetry in the one-way speed of light travelling in the east-west and west-east directions and suggested then that this difference if real may eventually be measurable by experiment. In a second paper Alley and his colleagues [22] outlined the general relativity prediction of light speed asymmetry in the east-west direction and described an experiment that directly searched for this phenomenon. The availability of the Global Positioning System (GPS) where light transmission on the rotating Earth is a routine physical activity provides an excellent opportunity for direct experimental resolution of the controversy relating to the physics of light travelling on the surface of the Earth [23, 24]. GPS operation is based on time measurement using accurate synchronized atomic clocks of light travelling at constant speed c in the Earth-Centered Inertial (ECI) frame, a frame that moves with the Earth but does not share its rotation. Using the GPS Marmet [25] and Kelley [26] have argued that observed travel time differences for east-west light transmission is evidence of light speed anisotropy and this has been supported by Gift using the GPS clock synchronization algorithm [27] and the GPS range equation [28]. Wang [29] and Sato [30] have made similar observations and Hayden [31] previously arrived at this same conclusion after considering several experiments including Sagnac, Michelson-Gale and BrilletHall. This author has published a series of papers in which the GPS is used to determine oneway light speed for light travelling on or close to the surface of the Earth. Some of this material was collected in a recent book chapter [32]. In this chapter an update on the work done by this author on light speed measurement is presented. In section 2 we use the synchronized clocks of the GPS to determine one-way light speed on the surface of the Earth. This is done by deriving

4 the transmission time for light travelling on the surface of the Earth using the Langevin metric in general relativity. This yields the result accepted by the CCIR for clock synchronization which is then used to determine one-way light speed. In section 3 one-way light speed is determined by using the fact that light travels in the ECI frame at a constant speed c. The result corroborates the finding obtained using the synchronization algorithm. In section 4 one-way light speed is determined using time transfer methodology where time is transferred from a satellite to an Earth-fixed groung station. In section 5 one-way light speed in a space beyond the Earth’s surface is evaluated using interplanetary tracking technology. These results are followed in section 6 by a full discussion.

2.

One-Way Light Speed on the Rotating Earth Using GPS Time Measurement The one-way speed of light can be determined by using the synchronized clocks of the

GPS to measure the time it takes for light to travel between two points fixed on the surface of the Earth. The algorithm for that synchronization is based on the derived time of transmission for travelling light. In this section we derive the transmission time of a light signal travelling on the surface of the Earth and use this to determine light speed. We use a procedure from general relativity [Ashby] that has produced the correct result. Thus ignoring gravitational potentials, the metric in the ECI frame where space-time is Minkowskian is in cylindrical coordinates given by [33-35] − ds 2 = −(cdt ) 2 + dr 2 + r 2 dφ 2 + dz 2

(1)

The transformation from the coordinate system (t , r ,φ , z ) in the ECI frame where light speed is c to a coordinate system (t ′, r ′,φ ′, z′) in the Earth Centered Earth Fixed frame which is rotating at the uniform angular speed ω E is given by

t = t ′ , r = r ′ , φ = φ ′ + ω E t ′ , z = z′

(2)

The transformation t = t ′ in (2) means that time t ′ in the rotating frame is the same as the time t in the underlying ECI frame. This results in the so-called Langevin metric in the rotating frame given by − ds = −(1 − 2

where

ω E 2 r ′2 c

2

)(cdt ′) 2 + 2ωE r ′2 dφ ′dt ′ + dσ ′2

(3)

5 dσ ′2 = dr ′2 + r ′2 dφ ′2 + dz′2

(4)

is the square of the coordinate distance. Applying equation (3) to the propagation of light on the rotating Earth by setting ds 2 = 0 yields a quadratic equation in dt ′ given by (1 −

ω E 2 r ′2 c2

)(cdt ′) 2 −

2ω E r ′2 dφ ′ (cdt ′) − dσ ′2 = 0 c

(5)

Solving for (cdt ′) gives 2ω E r ′2 dφ ′ 2ω E r ′2 dφ ′ 2 ω E 2 r ′2 ± ( ) + 4(1 − ) dσ ′ 2 2 c c c cdt ′ = 2 2 ω r′ 2(1 − E 2 ) c

(6)

Taking the dominant term under the square root sign gives

ω E r ′ 2 dφ ′ cdt ′ =

c (1 −

Since

ω E 2 r ′2 c2

± dσ ′

ω E 2 r ′2 c2

(7) )

0 derived using the CCIR equation (10) formally establishes the fact that light takes longer to travel East than West between fixed points on the surface of the Earth. This experimental observation was first highlighted by Marmet [25] and Kelley [26]. Since the time of travel given by GPS equations (11) and (12) has been rigorously verified and is programmed into the operation of the GPS, this represents an actual time of travel measurement that can be used to determine the east-west speed of light. Thus using the GPS time ∆tWE in (11), the one-way speed of light cWE traveling eastward between two points fixed on the

surface of the Earth a distance l apart is given by cWE =

l l v = = c(1 + ) −1 = c − v, v