Akiinbo,
"1. "The vc is how I represent the variable speed of light... It is the speed of light, ... that a remote observer in free space would calculate for anywhere else ..not in free space (like Earth)".
Since you know G, M and r for Earth, and the velocity of light in unfree space, try to do the math and get a value for light velocity in free space."
I know the magnitude of the velocity of light in free space. It is the same speed as measured here on Earth. The equation will not calculate it. The measured speed of light on the Earth is not the real speed of light on the Earth. Length contraction on the Earth affects the measurement causing the result to be the same as in free space.
Here on the Earth, assume there is in front of us a transparent tank containing water. The light that passes into the container will slow. We as outside observers use our outside measuring rod and clock to measure that slower speed of light. If the measurement occurred in the tank in the water, then the measured speed of light would be the same as in free space. This effect occurs because length contracts more in the water than exists outside the tank.
The measurement of the speed of light everywhere as measured locally will give the same free space magnitude because of locally occurring length contractions of various magnitudes.
"2. The statements, "Yes, the speed of light is faster in free space than on the Earth" and "Yes C has a greater magnitude than does vc everywhere that is not free space" violently contradict "I stated that the speed of light measures as the same constant everywhere. That means that the measurement of the speed of light on the Earth is the same as the measurement of the speed of light in free space. That is why I use the Earth's local measurement of the speed of light as the magnitude of C""
No they do not conflict. The locally measured speed of light everywhere is the same magnitude as exists in free space. The real local speed of light cannot be measured locally because of the existence of local length contraction to the measuring device. The real local speed of light is not the same as the speed of light in free space. A refined more exact form of my equation could be used to calculate the real speed of light here on Earth. It would not be the same as the locally measured speed of light here on the Earth.
"3. "The measurement cannot be carried out physically by placing the measuring device between the remote observer and Earth. The speed of light varies and length varies. The measurement process that I refer to is the same as used for expressing measurements from an observer's perspective in special relativity... ". Let us leave special relativity complications out of this for the moment since gravity is involved."
I include references to special relativity type effects because they are length contraction and time dilation. I am using the common name of 'time dilation' currently given to the slowing process that I say results from the slowing of the speed of light. The speed of light is slower here on the Earth than in free space. Events occur slower here on the Earth than they do in free space. My use of the name 'time dilation' does not refer to an effect upon time. It does refer to an effect upon clocks.
Both length contraction and the slowing effect called time dilation also occur due to gravity. My answer to you involved comparing lengths at two different locations. There is no length contraction in free space. There is length contraction on the Earth. Each observer relies on their local length of measuring device. They can't put that device between free space and Earth because length contracts in increasing amounts between free space and the Earth.
If I want to know what the remote observe in free space would measure for the speed of light on Earth, in the hypothetical case of using their own local measuring device, then I do it the same way as is done in special relativity. There are two different observers looking toward one another and neither moves nor reaches anythwhere out from their location. The variation of gravity inbetween the two is not a factor.
What matters is the length of their measuring devices locally. It is hypothetical, but it is the way to calculate the speed of light on Earth from the free space perspective. That equation of mine solves for Vc. Vc gives the speed of light everywhere other than in free space.
"4. "The measurement is hypothetical"
It is not. It has in fact been done. Several signals have been sent to freer space at time, t1 and received back on earth at time, t2. The analysis of the signals have been done and the facts are available with the Jet Propulsion Lab, JPL. The interpretation is what is contentious. Facts are sacred, comments are free.
FACT: The signals ARE catching up faster than expected with remotely located receiver.
COMMENT: Is this because the receiver is decelerating due to a solar attraction force, or is due to thermal problem in the receiver's engine slowing it down or is it as you say that light appears to be the one doing the acceleration and not the receiver? Give these a thought."
For the example of sending a light pulse out into the solar system away from the sun, the speed of light will be increasing with distance from the Earth and sun. My use of the word hypothetical applied to the example of the free space observer measuring the speed of light on the Earth solely from their vantage point in free space and using only their local measuring device. They do not reach out with anything across space. This hypothetical measurement process is analogous to the measurement process used in special relativity.
James Putnam