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.


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

$t = L/c = 12 sec$