Quote by Chris Hillman
Time does not slow down and distances do not shorten! ...GTR says nothing like this.

I never claimed that what I said is a part of GTR. I simply looked at experimental observations and tried to understand them without prejudice in a simplest possible model.
I wouldn't say that "time slows down" in the gravitational potential (because I don't know what's the meaning of the expression "speed of time"). However, it is well established that identical clocks run slower in the gravitational potential than far away from it (e.g., the PoundRebka experiment, GPS,...). This difference in the clock rates is described well by formula
[tex] T' = T(1+\frac{GM}{c^2r}) [/tex]
where T' is the period of one clock tick in the potential.
It is also known that light takes longer time to propagate between two points (e.g., Earth and Venus) if there is a massive body (Sun) on its way. This Shapiro time delay can be explained within GTR, as you demonstrated. But one can offer a simpler explanation as well. One can simply assume that the speed of light depends on the gravitational potential as
[tex] c' = c(1\frac{2GM}{c^2r}) [/tex]
(independent on the light direction) It can be shown that the numerical value of the EarthVenusEarth time delay comes out exactly as in GTR.
Another fact is that the speed of light appears the same to observers in different gravitational potentials. All these facts can be reconciled by the assumption that the length of any rod decreases in the potential as
[tex] d' = d(1\frac{GM}{c^2r}) [/tex]
The slowingdown of clocks and the reduction of the speed of light can be explained within a simple Newtonlike theory of gravity, where gravitational interactions are distance and velocitydependent. Of course, this theory has nothing to do with GTR, and doesn't belong to the mainstream. However, in contrast to GTR, it is perfectly compatible with quantum mechanics, so, in my opinion, it can't be dismissed lightly.
I have no idea how to explain the shortening of lengths in the field. I think it would be great to design an experiment to show whether this length shortening exists or not. This question makes sense no matter which theory of gravity is correct. And it would be a great independent check on the general theory of relativity. Here I agree with kev.
Eugene.