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There have been numerous threads and lengthy discussions on many aspects of the speed of light on CR4. One aspect that has not really been addressed is the influence of gravity on the speed of light. This thread attempts to fill that void.
It is commonly known that gravity influences the relative rates of time passage. The ratio of the time passage of an observer at distance r from a body with mass M compared to the time passage of a distant observer (in free space) is given by:
[Eq. 1] 
where G is Newton's gravitational constant and c the speed of light in free space. This ratio is also denoted by √(gtt), where gtt is the 'time-time' coefficient of the Schwarzschild metric. The function is shown graphically in fig. 1 below.
Fig. 1: 
It is clear that gtt approaches unity when rc2 >> GM and becomes zero when rc2 = 2GM, at the event horizon of a Schwarzschild black hole. This ratio together with the curvature of space also regulates the speed of light in Schwarzschild coordinates, as depicted and quantified in fig. 2 below.
Fig. 2: 
The curve represents the curvature of space into a fictitious dimension (z). The spacing of the vertical dotted lines indicates the radial coordinate distance interval (Δr) that light travels in in every constant coordinate time interval (Δt). There are two equal effects at work: (i) the slowing of time lets light move a distance Δl = √(gtt)cΔt along the slope of space; (ii) this distance (Δl) is then projected onto the coordinate system as Δr = gttcΔt .
This is the case for light moving precisely radial relative to the mass. When the movement is precisely transverse (tangential) relative to the mass, there is no slope along the light's path and the projection is 1:1. It can hence be stated that the transverse coordinate velocity of light equals c√(gtt) while the radial coordinate velocity of light equals c gtt, which is smaller or equal to the transverse velocity, since gtt ≤ 1.
When combined in vector form, the general Schwarzschild coordinate velocity of light can be viewed as shown below, where the path of the light is making an angle θ with the radial to the mass.
Fig. 3: 
In the relatively low gravitational field strengths in our solar system, the effective coordinate speed of light remains very close to c, because gtt is very, very close to unity. Even at the surface of the Sun, it deviates from unity by only one part in 236 thousand. Shapiro et al determined experimentally that a photon passing very close to the Sun from afar is delayed by about 125 microseconds for every pass.[1][2]
If the Sun happened to be a neutron star with the same mass and a radius of about 10 km, a photon grazing the star's surface would have been delayed by around 10 seconds for the same pass. At the surface of the neutron star, the transverse coordinate velocity of light will be ~0.84c and the radial coordinate velocity ~0.71c. In the case of black holes, this situation obviously goes extreme at the event horizon, with both coordinate velocities zero.
Finally, just a reminder that a local observer measuring the photon's speed over a short distance will always get the value c and nothing else. Schwarzschild coordinate velocities are those measured by distant observers, located outside of the gravitational influence of the mass. A discussion of measurements made by 'distant observers' that are still inside the gravitational influence of the massive object will be the topic of the next post in this Blog.
The floor is open for discussion.
Jorrie
[1] "Was Einstein Right?" by Clifford M. Will.
[2] Tests of Relativity from Relativity 4 Engineers.
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