GPS clocks' apparent synchronization without navigation errors.

We consider the simplified situation where a GPS satellite would be in motion in the XY plane of the inertial Earth frame (360 km/s in space) along a circular and uniform trajectory in this moving frame (figure below).

If the radius of the trajectory is R, the motion equation in K' is:

[x'(t') , y'(t')] = R.[cos(w.t') , sin(w.t') ], with w.t' = a.

According to GTWMC: x = x'.sqrt(1-bb)+vt , y'=y , t =t'.sqrt(1-bb), the trajectory equation of the GPS satellite in the rest frame is:

Thus, the vectorial velocity is:

From them the radius around the Earth is :

and the square of the scalar velocity ( by neglecting weak terms with w =1.08 E(-3) rad/s and R= 6978 km) is:

Thus, the fractional frequency difference of the GPS clock relatively to a Earth ground clock chosen at the center of the Earth (origin of K' to simplify the calculations) is :

The phase change from time 0 to t is the integral of this fractional frequency difference, that is to say :.

This phase equation contains an isotropic term which varies of -27,44 us a day, but also an anisotropic term which varies non-cumalatively as much as 55.9 us each half-revolution of the satellite.

But fortunately for the GPS owner, this important anisotropic term is exactly cancelled by the anisotropy of the light speed.

To show that, I have to consider the anisotropic velocity of the light from the satellite toward the center of the Earth abtained with elementary plane geometry calculation:

(See figure above for the angle a).

Thus, the time delay variation for the light to reach the ground clock is from a=0 :

, the same as the anisotropic term of the phase variation of the clock.

That means that when the GPS clock is fast (for example) of delta-t, the time signal sent from it to the ground station will be late of delta-t and thus the clock will appear to be still synchronized for relativists who don't consider a phase shift of the clock , nor an anisotropy of the speed of light. But the result of my experiment has shown that such a result is an artifact.

That means also that to consider the phase of the clock as to be stable, and the fictitious relativistic isotropy of the speed of light don't lead to an error about the position of the satellite and thus permit a correct GPS navigation.

Nevertheless, the -27.44 us/day isotropic loss of time must be corrected several times a day , like the GPS owner do it.

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