Bruce,
Being able to do something and doing it are also two different things. I can tell you how it should be done. A second on the sun is shorter than a second on Mercury. What you do to use the Galileian transformation equations to figure relativity is convert Mercury time to sun time, use the Galileian transformation equations, and then convert anything that needs to be in Mercury time back from sun time.
The difference between Mercury time and sun time has to do with the velocity of Mercury in its orbit around the sun. Mercury is traveling at about 30 miles per second. So if the velocity is to be the same measured from Mercury as from the sun, there is a clock speed on Mercury at which the velocities will match. Note that in the classical interpretation of the Galileian transformation equations which you keep telling me I have to follow, a slower clock on Mercury would mean a faster velocity for the planet. I do not use that interpretation, but instead follow the equation t'=t, meaning that the slower clock on Mercury cannot be used as a time in the Galileian transformation equations until a conversion has been made to sun time, keeping the velocity the same from both frames of reference.
The same principle is true for your GPS satellites. I can't tell anything from the distances you give me. One appears to be the radius of the earth, the other the altitude of the satellite. You can tell the velocity of the satellite by the radius of its orbit. The principle of equivalence shows us that a baseball orbiting at the altitude of the moon would have the same velocity as the moon. So, depending on whether your satellite altitude is from the center of the earth or from the surface, you can figure its velocity. From velocity, you can get time according to an earth clock.
What you are claiming from your Schwarzchild equations is that what one clock reads relative to another depends on the distance between the clocks. I believe this is wrong. I think that if you have two GPS satellites in opposite orbits with identical clocks, the clocks read the same no matter how far from each other they are. In other words, the clocks read the same when they are on opposite sides of the earth from each other, and they read the same when they pass one another on the same side of the earth. The reason why your equations have a difference is because you have a distance contraction incorporated in your equations.
Robert B. Winn |