Photon’s delay in the gravitational field

In the post “The gem (1)”, the equation for photon’s velocity was derived, which tells us that the photon’s velocity is not constant in the gravitational field:


Before the equation for photon’s velocity, the equation for gravitational time was derived:


From these two equations, followed the equation for the photon’s path-element:


The Earth’s gravitational influence on the speed of light is practically neglectable. Namely, if we take that the speed of light is   , then, according to the equation


on the Earth’s surface it would be   

Even If we encounter the influence of Sun, the velocity of a photon would be


                         

But, very near the Sun’s surface, it would be  , which is   less than  .

Let’s assume that a photon travels from Earth to Mars and back, along some path which passes very near the Sun surface.

Pic1

If the speed of light would be constant, the photon would travel from Earth to Mars and back for some time . But, according to the equation


it should travel longer, for some time  


So, let’s calculate the  


The path-element of a photon is




Along the path    , we’d have
















 for Sun is approximately 3 km.

The minimal value for   is  , so we have that even the maximal value of exponent is very, very small


Hence, for our calculation, it is sufficient to take just the first two terms of the Taylor series for the function 



that is,







Let us divide the photon’s route in four sub-routes:

 ,   ,   ,  

Let us first consider the last sub-route 

If we take the point  to be the coordinate origin, and the direction of  -axis from   to   , we’d have








The additional short distance   which our photon has to travel is in the Earth’s vicinity. That is very far from Sun, so along the distance   we can consider that the velocity of our photon is practically   .

So, we get that the time dilation of our photon after it traveled from   to  is


The sum-time-delay along the last sub-route    and the along first sub-route   would be
The sum-time-delay along the second sub-route    and the third sub-route   would be


The total time delay would be



Pic2

If we use these data (the data from Shapiro experiment, shown in the Pic2), we would get the time delay which Shapiro got in his measurement: 


(Google: “Earth Mars Shapiro time delay”)