This is true that GPS do work, but under the assumption that 'speed of light' is constant for every observer. That is when a GPS receiver calculates its distance from the any GPS satellite it uses the formula d=c.t ....... (1), where 'c' is constant speed of light.
Now, before you say that it is experimentally proved that speed of light is same for every observer, let me stop you right there, and consider the same for a sound wave ....... does the speed of sound depend on the motion of the source ? Does the speed of sound is different for different frequencies ? the answer is NO. It is constant for any motion of source.
And for the second part of my argument, .......that the speed of sound depends on the motion of the observer and the same is shown for the speed of light by the Sagnac effect and Michelson-Gale type experiments, which measure the speed of light by the time of flight method,........ that is the speed of light does depend on the motion of the observer. But current understanding of these experiments from the the perspective of SR is that .... speed of light is still constant but time is different for different observers.
But it is perfectly consistent, mathematically also, that if instead of time we change the speed of light for different observers, then also the Lorentz Transformation can be formulated, for different observers. And the equation (1) will again give us the correct distance but assuming the time constant and speed of light different for different observers.
Note that above argument does not mean that a physical object can move FTL . It is just the transformation for EM waves for different observers. For experiments of EM waves only, as Sagnac effect, GPS etc..
Now you would think how a theory like that can explain the time dilation of decaying Muons or Relativistic Kinematics. Not only it has Time constant but also, it cannot be applied to the physical particles. That is Lorentz transformation is for EM waves only(since, it is derived from the Maxwell's equations) and cannot be applied to physical bodies (saving from every paradox that involve SR, including H Dingle).
This is the central part of my paper .... to explain Relativistic Kinematics and Time Dilation of radioactive particles. Remember, that radioactivity can be influenced by Sun.
