Falling light. Gravitational lens

According to general relativity and the equivalence principle, light is bent by gravity. Consider two tall, perfectly reflecting mirrors exactly 4 m apart and facing each other. A beam of light is directed horizontally through a hole in one of the mirrors 10 m above the ground. (a) Determine the time it takes for the light to strike the ground
(b) The light will undergo N reflections (i.e., N/2 reflections from each mirror) before it strikes the ground. Find N.

Answers
The angle of deflection of the light by a gravitation is
$\theta = 4GM/rc^2$
(see https://en.wikipedia.org/wiki/Gravitational_lens#Explanation_in_terms_of_space.E2.80.93time_curvature)

where G is the gravitational constant , M the mass curving the path, r the distance from the light beam to the mass distribution and c the speed of light.
In the case of light $G = 6.67*10^{-11} m^3/kg/s^2$
$M =5.97*10^24$ kg
$r =6371 km$   is Earth radius (the height of the mirror is negligible)
$c= 3*10^8$m/s

$\theta = 2.778*10^{-9 }rad$
for such small angles  $\theta = tan(\theta) = d/D$
$d = D*\theta =4*2.778*10^{-9} =1.111*10^8 m =11.11 nm$ deflection/one reflection

total number of reflections is
$n =10 m/11.11 nm =9*10^8$ reflections from each mirror
total travelled distance is
$L = n*D = 4*9*10^8 =36*10^8 m$
time of travel is
$t = L/c = 12 sec$


valentin68

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