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The GPS satellite clock corrections (along with gravitational redshift) which are necessary for the proper operation of the GPS are fully described without invoking relativity theory as is the practice today.

In the Global Positioning System (GPS), there are atomic clocks on board the orbiting satellites which are necessary for the proper operation of the system. These clocks however experience clock retardation arising from their high-speed movement around the Earth (time dilation) and clock advancement resulting from reduced gravitational field intensity at the satellite altitudes (gravitational time dilation associated with gravitational redshift) [

Based on accurate experimental data, it has been rigorously confirmed that clocks that are stationary in the GPS satellites run slow relative to clocks that are stationary in the ECI frame. For clocks at the same gravitational potential, this time reduction satisfies the equation [

∫ p a t h d t ′ = ∫ p a t h ( 1 − v 2 2 c 2 ) d t (1)

where v is the velocity relative to the ECI frame, t is the time in the ECI frame and t ′ is the time in the frame moving relative to the ECI frame which here is onboard the orbiting satellites. When integrated along the satellite path, assuming an approximately constant value for v, this equation gives

t ′ = ( 1 − v 2 2 c 2 ) t (2)

This corresponds to the equation

t ′ = t γ (3)

given by Kelly [

t ′ = t / γ = t ( 1 − v 2 c 2 ) 1 / 2 ≃ t ( 1 − v 2 2 c 2 ) , v ≪ c (3a)

Equation (3) is the time transformation given by Selleri [

t ′ = γ ( t − v x c 2 ) = t γ − v x ′ c 2 (4)

where x is the space coordinate in the ECI frame and x ′ is the space coordinate in the moving frame. Therefore, in order that satellite clocks remain synchronized with clocks that are stationary in the ECI frame (corresponding to the Earth’s center), a continuous time adjustment has to be made to the satellite clocks. Thus from (3a), the difference between the time t ′ on the moving satellite clock and the time t on the clock that is stationary in the ECI frame is given by

t ′ − t = − t v 2 2 c 2 (5)

This represents a slowing of the clock rate as a result of the movement of the clock in the ECI frame. The velocity of the GPS satellites relative to the ECI frame is v = 3874 m / s . Therefore

t ′ − t = t v 2 2 c 2 = − t 3874 2 2 ( 2.998 × 10 8 ) 2 ≃ − 8.349 × 10 − 11 t (6)

Over a period of one day, the time change is given by

t ′ − t = − 8.349 × 10 − 11 × 60 × 60 × 24 ≃ − 7.214 × 10 − 6 s = − 7214 ns (7)

This means that the satellite clocks lose 7210 ns (rounded to 10 ns) each day compared with the clocks that are stationary in the ECI frame as a result of movement at velocity v = 3874 m / s relative to the ECI frame or the center of the Earth.

It has been experimentally observed that a light beam can change the momentum of an object upon which the light is incident. The corresponding pressure experienced by the object is referred to as radiation or light pressure. If the light energy E p is absorbed by the object, the momentum change experienced by the object is E p / c . From the principle of conservation of momentum, this means that photons of energy E p behave as if they have a mass m p given by [

m p = E p c 2 (8)

This derivation follows that given by Narlikar [

F m p = G M m p r 2 (9)

The work done in raising the photon through height d r is given by

F m p d r = G M m p r 2 d r (10)

Hence the work done in raising the photon from the surface of the massive object to an infinite height is given by

W = ∫ R ∞ F m p d r = ∫ R ∞ G M m p r 2 d r = G M m p R (11)

This work is done at the expense of the photon’s energy. This loss of energy by the photon results in a reduction of its frequency from f to f ′ given by

h f − h f ′ = G M m p R (12)

Using (8), this becomes

h f − h f ′ = G M h f R c 2 (13)

where E p = h f . Hence the fall in frequency Δ f = f − f ′ is given by

Δ f = G M R c 2 f (14)

This frequency change is referred to as gravitational redshift and has been confirmed to an accuracy of 1% or better [

The work done in raising the photon from an initial height r = R 1 to an increased height r = R 2 is given by

W = ∫ R 1 R 2 F m p d r = ∫ R 1 R 2 G M m p r 2 d r = − G M m p R 2 + G M m p R 1 (15)

Therefore, the fall in frequency is given by

Δ f = f − f ′ = G M f c 2 ( 1 R 1 − 1 R 2 ) (16)

Now if a radiation source emits photons at frequency f over time t registered on a clock at R 1 , then the number of cycles emitted is N = t f . If these photons travel to a greater height, then the frequency falls to f ′ . However, since the receiver must receive the same number of cycles, it follows that [

t f = N = t ′ f ′ (17)

where t ′ is the time registered on a clock at R 2 . Therefore, the decrease in light frequency is associated with an increase in clock rate at that position. This change in clock rate is referred to as gravitational time dilation. From (16) and (17),

t ′ = t f f ′ = t f f − G M f c 2 ( 1 R 1 − 1 R 2 ) ≃ t [ 1 + G M c 2 ( 1 R 1 − 1 R 2 ) ] , G M c 2 R ≪ 1 (18)

Equation (18) has been confirmed by accurate GPS data [

Therefore, in order that satellite clocks remain synchronized with clocks at rest on the surface of the Earth, a continuous time adjustment has to be made to these clocks. Thus from (18),

t ′ = t 1 1 − G M E a r t h c 2 ( 1 R E a r t h − 1 R S a t ) ≃ t [ 1 + G M E a r t h c 2 ( 1 R E a r t h − 1 R S a t ) ] , G M c 2 R ≪ 1 (19)

Hence the difference between the time t ′ on the moving satellite clock and the time t on the clock on the surface of the Earth is given by

t ′ − t = t G M E a r t h c 2 ( 1 R E a r t h − 1 R S a t ) (20)

Therefore, with mass of the Earth M E a r t h = 5.974 × 10 24 kg , polar radius of Earth R E a r t h = 6.357 × 10 6 m , radius of satellite orbit R S a t = 2.6541 × 10 7 m , G = 6.674 × 10 − 11 m 3 ⋅ kg − 1 ⋅ s − 2 and c = 2.998 × 10 8 m / s we get

t ′ − t = t G M E a r t h c 2 ( 1 R E a r t h − 1 R S a t ) = t 6.674 × 10 − 11 × 5.974 × 10 24 ( 2.998 × 10 8 ) 2 ( 1 6.357 × 10 6 − 1 2.6541 × 10 7 ) ≃ 5.307 × 10 − 10 t (21)

Over a period of one day, the time increase is given by

t ′ − t = 5.307 × 10 − 10 × 60 × 60 × 24 ≃ 45.850 × 10 − 6 s = 45850 ns (22)

This means that the satellite clocks gain 45,850 ns each day compared with the clocks that are stationary on the surface of the Earth as a result of the decreased gravitational field intensity above the Earth [

The net time gain each day from clock movement and reduced gravitational field intensity is then 45850 − 7210 = 38640 ns . The satellite clocks must be appropriately slowed in order to exactly compensate for this net time gain. From Equations (5) and (20), the total time change resulting from clock movement and gravitational time dilation is given by

t ′ − t = t G M E a r t h c 2 ( 1 R E a r t h − 1 R S a t ) − t v 2 2 c 2 (23)

Therefore, the fractional change in time is given by

t ′ − t t = G M E a r t h c 2 ( 1 R E a r t h − 1 R S a t ) − v 2 2 c 2 = 5.307 × 10 − 10 − 8.349 × 10 − 11 = 4.472 × 10 − 10 (24)

In order to correct for this, the clock frequency must be reduced by the same fraction i.e.

f − f ′ f = t ′ − t t = 4.472 × 10 − 10 (25)

where f ′ is the reduced frequency and f = 10.23 MHz is the clock frequency on Earth. From (25) therefore,

f ′ = f ( 1 − 4.472 × 10 − 10 ) = 10.23 MHz × ( 1 − 4.472 × 10 − 10 ) = 10.22999999543 MHz (26)

This is the adjusted frequency that is preset in the clocks at launch and can be found in IS-GPS-705F Interface Specification, May 2019 (page 9) [

In this paper, the clock corrections necessary in the GPS have been calculated without using relativity theory. As previously indicated, the speed of the satellite clocks is measured relative to the ECI frame corresponding to the center of the Earth or the poles. It turns out that (because of time change resulting from gravity and clock movement) clocks at the poles beat at the same rate as clocks along the Earth’s geoid across the surface of the Earth. This means that clocks that are stationary in the ECI frame are equivalent to clocks being stationary anywhere on the Earth’s geoid, regardless of latitude [

The author declares no conflicts of interest regarding the publication of this paper.

Gift, S.J.G. (2021) GPS Satellite Clock Corrections without Relativity Theory. Journal of Applied Mathematics and Physics, 9, 2476-2482. https://doi.org/10.4236/jamp.2021.910158