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In the previous article, we have seen that the Schwarzschild coordinate speed of light (as measured by a distant observer) is dictated by the gravitational time dilation factor:
Eq. 1: 
If the distant observer is not at 'infinity', but at a radial distance ro from the central mass, the observer's coordinate time compared to the Schwarzschild time is given by:
Eq. 4-5: 
If we now express the time rate at radial distance r as a fraction of the observer's time rate (at ro), it gives:
Eq. 5: 
In analogy to the way the Schwarzschild speed of light has been calculated, one can use this ratio to calculate the effective radial and transverse components of the speed of light at radial distance r, as perceived in the coordinates of the observer located at ro, giving a radial velocity = c g'tt and a transverse velocity = c √(g'tt) respectively.
To get a graphical feeling for the values, it is best to place the observer not too far from a black hole, say at three times the event horizon radius and then plot the radial and transverse velocities of light at a range of radial distances, like shown below in the coordinate system of the local observer.
Fig. 4: 
At the observer's location (ro = 6GM/c2), both coordinate velocities of light must be equal to c, of course. At distances rc2 < 6GM, both coordinate velocities are smaller than c, with the transverse velocity of light larger than the radial velocity. At distances rc2 > 6GM, both coordinate velocities of light exceed c, with the radial velocity of light larger than the transverse velocity.
If the observer makes radar distance measurement to the red target, the return signal will be delayed, apparently giving a larger distance than what is expected. If the same radar distance measurement is made to the blue target, the return signal will arrive earlier than expected, giving the impression that the target is closer than it actually is.
I have calculated the delay and advance of the radar signal for the two targets respectively, assuming the following values: M~105 solar masses (~2x1035 kg), ro=109 m, yielding rc2~6GM as in the diagram. The red target is then at r=0.5x109 m and the blue target at r=1.5x109 m.[1]
The normal travel time of the radar echo from either target would be 2x0.5x109/(3x108) = 3.33 seconds in the observer's coordinates, if it was not for the delays and advances due to gravity. The actual radar signal time to the red target works out (by numerical integration) to be 4.26 seconds, meaning a delay of 0.93 seconds. Likewise, the return from the blue target arrives after 3.06 seconds, early by 0.27 seconds.
The average speed of the radar signal to the red target is 3.33c/4.26=0.78c and to the blue target it is 3.33c/3.06=1.09c, both correlating well with the dark blue curve in fig. 4 above.
Jorrie
[1] In black hole physics, radial distances are called 'circumferential radii' because one cannot measure distances from the center of a black hole. The observer can set up experiments in her own coordinate system to measure the circumference (C) of a concentric circle around the hole at her own or any other location. The Schwarzschild radial distance coordinate is then taken as r=C/2Π.
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