In Fig. we have shown a spherical massive body of mass `M` and radius `R`. Consider two planes one ar `r` and other at `(r + dr)` from the centre of massive body. Let the light rays be incident ai an angle `theta` on the plane at `r` and leave the plane at `(r + dr)` at an angle `(theta + d theta)`.
According to Snell's law,
`n(r ) sin theta = n(r + dr) sin (theta + d theta) = [n(r ) + (dn)/(dr) dr] (sin theta cos d theta + cos theta sin theta)`
As `d theta` is small, `cos d theta ~=` and `sin d theta ~= d theta`.
Neglecting products of differentials, we get
`n(r) sin theta = n(r ) sin theta cos theta d theta + (dn)/(dr) dr sin theta` or `-(dn)/(dr) dr sin theta = n (r ) cos theta d theta`
or `-(dn)/(dr) tan theta = n(r ) (d theta)/(dr)` ...(i)
As `n (r ) = 1 + (2GM)/(r c^(2)) :. (dn)/(dr) = (-2 GM)/(r^(2)c^(2))`
Put in (i) `(2GM)/(r^(2)c^(2)) tan theta = (1 + (2GM)/(rc^(2))) (d theta)/(dr) ~= (d theta)/(dr)` `( because (2 GM)/(rc^(2)) lt lt 1)`
`d theta = (2GM)/(c^(2)) (tan theta)/(r^(2))dr`
Integrating both sides, we get
`int_(0)^(theta_(0)) d theta = (2GM)/(c^(2)) int_(-oo)^(+oo) (tan theta dr)/(c^(2)) =(2 GM)/(c^(2)) int_(-oo)^(+oo) (tan theta r dr)/(r^(3))`
Now, from Fig.`r^(2) = x^(2) + R^(2) and tan theta = (R )/(x)`
`:. 2 r dr = 2 x dx :. int_(0)^(theta_(0)) d theta = (2 GM)/(c^(2)) int_(-oo)^(+oo) (R x dx)/(x(x^(2) + r^(2))^(3//2))`
If we put `x = R tan phi` so that `dx = R sce^(2) phi d phi`,
then `int_(0)^(theta_(0)) d theta = (2GM)/(c^(2)) int_(-pi//2)^(pi//2) (R^(2)sce^(2) phi d phi)/(R^(3) sec^(3) phi)`
`theta_(0) = (2 GM)/(R c^(2)) int_(-pi//2)^(pi//2) cos phi d phi = (4 GM)/(R c^(2))`
This is the deviation for the ray from the original path as it grazes the massive body.
