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This is a very good start, it gives you a synopsis of what is out there.

It has been done many times (correctly). I can provide you with links to the respective simulations, if you are interested.

on the rocket

Your intuition is correct. The effects cancel each other out completely because the geoid surface is equipotential. So, a clock at sea level at the equator and one at the pole are subjected to the same exact gravitational potential. Hence, same ticking rate. See here, for example. Vessot ran such an experiment . See here

Here is a decent reference that you can read on. The wave vector changes direction based on the frame of reference. The photons do NOT "take on the velocity of the source".

Actually, the direction of the light beam IS influenced by the relative motion between source and receiver. The phenomenon is called "relativistic aberration". It is the speed of light that is not influenced by relative motion, the direction IS influenced.

I already answered this question, it doesn't. The photon momentum is [math]\vec{p}=\hbar \vec{k}[/math] where [math]\vec{k}[/math] is the wave vector. So, no connection between photon momentum and proper time.

There is nothing to discuss, the result of the Opera experiment was corrupted by a bad cable connection. What I do not understand is why do you feel compelled to advertise your "book" in multiple internet physics forums/

There is no connection between photon frequency and proper time.

In flat spacetime, for example, [math]\tau=\int{\sqrt{1-(v/c)^2}dt}[/math] If [math]v=c[/math] then [math]\tau=0[/math].

The statement is not correct, QM and SR are fully compatible and they have been unified in a theory known as QFT. It is GR that hasn't been unified with QM, YET. Work is underway (see LQG, String Theory, etc).

In this case you would have indeed: [math]m=M \frac{r_s}{R}[/math] For example, for Earth, the ratio [math]\frac{r_s}{R}=\frac{10^{-2}}{6.4*10^6} =\frac{1}{6.4*10^8}[/math] Thus, [math]m=10^{16} kg[/math]

Yes, all observers agree.

The "book" is totally crackpot, please stop pushing it.

I looked at the "book", it is totally crackpot. Self-published by a crank called Syd Wilcox , quoting the "papers" of a (better known) crank, Paul Marmet. You wasted your money.

But physics is exactly about being able to do the calculations. You can't be doing physics by cherry-picking quotes from websites. As Markus points out, there is nothing physically particular about the Schwarzschild radius, there are other systems of coordinates that do not exhibit the features that the Schwarzschild system of coordinates exhibits. See for example, the Gullstrand-Painleve system of coordinates.

No, the correct formulas have been shown earlier here.

The simple answer is : coordinate length/time are variable in SR and in GR. Proper length/time is invariant in both SR and GR.

This is standard in cosmology: proper distance isn't available (for practical reasons). The only available (measurable) distance is radar distance.

One way light speed is not measurable, this is a well known fact in mainstream physics world. Only two-way light speed can be measured, the Shapiro delay type of experiments do this routinely at the cosmological levels (see any GR textbook).

The star is very far away, A and B are near a DIFFERENT gravitating body.

A and B are not at the same distance from the star, I thought that I made that quite clear. A is closer to the star and A clock ticks faster. B is farther from the star and B clock ticks slower. A radar signal sent from A to the star makes the roundtrip in [math]\Delta t_A=\frac{2d_A}{c}[/math] (google "Shapiro delay"). [math]d_A[/math] is the distance between A and the star. A radar signal sent from B to the star makes the roundtrip in [math]\Delta t_B=\frac{2d_B}{c}[/math]. In both cases, contrary to what Bjarne is trying to claim, the speed of the radar signal is the same , "c".

In order for the above to happen you need to have : [math]\sqrt{\frac{1-r_s/r_a}{1-r_s/r_B}}=2[/math] i.e. [math]r_B=\frac{4r_s}{3+r_s/r_A} \approx \frac{4r_s}{3}[/math] This is non-physical given that [math]r_s[/math] is the Schwarzschild radius. Even if it were possible, the disparity between [math]r_A[/math] and [math]r_B[/math] explains why the two observers have very differing distances to the star in question. The more normal case, as in the case of the Pound-Rebka experiment, the two observers (A and B) are separated by a few meters and theirs clocks are ticking at rates very close to each other. I am quite sure that the difference in clock rates is cancelled out by the differences in distances to the Schwarzschild observer (situated on the distant star).

What gives you this idea? (Hint: it is wrong).

This is an excellent question. The answer can only be given via experimentation, in the presence of strong gravitational fields, violations of Lorentz covariance start showing up over large "enough" extents. This is due to the effects of the gravitational fields, it is the presence of the gravitational field that makes SR no longer applicable. For example, if we were trying to run a Michelson-Morley experiment very close to the Sun, we might get a non-null result due to the fact that the Sun has a much stronger gravitational field than the Earth.