If light passes near a massive object, the gravitational interaction causes a bending of the ray. This can be thought of as happening due to a change in the effective refractive index of the medium given by `n( r) = 1+ 2 GM//rc^2`
where `r` is the distance of the point consideration from the centre of the mass of the massive body, `G` is the universal gravitational constant, `M` the mass of the body and `c` the speed of light in vacuum. Considering a spherical object, find the deviation of the ray from the original path as it grazes the object.

Explanation

In Fig., we have shown a spherical massive body of mass `M` and radius `R`. Consider two planes one at `r` and other at `(r + dr)` from the centre of massive body. Let the light rays be incident at 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 d theta)`
As `d theta` is small, `cos d theta ~= 1 and sin d theta ~= d theta`
Neglecting products of differentials, we get
`n(r) sin theta = n (r) sin theta + n (r) 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)`
As `n(r) = 1 + (2 GM)/(r c^2) :. (d n)/(dr) = (-2 GM)/(r^2 c^2)`
Put in (i) `(2 GM)/(r^2 c^2) tan theta = (1 + (2 GM)/(rc^2)) (d theta)/(dr) ~= (d theta)/(dr)` `(because (2 GM)/(rc^2)lt lt 1)`
`d theta = (2 GM)/(c^2) (tan theta)/(r^2) dr`
Intergrating both sides, we get
`int_0^(theta 0) d theta = (2GM)/(c^2) int_(-oo)^(+oo) (tan theta dr)/(r^2) = (2 GM)/(c^2) int_(-oo)^(+oo) (tan thetar dr)/(r^3)`
Now, from (Fig. 11(EP).16), `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 d x)/(x(x^2 + R^2)^(3//2))`
If we put `x = R tan phi` so that `dx = R sec^2 phi d phi`,
then `int_0^(theta 0) d theta = (2 GM)/(c^2) int_(- pi//2)^(pi//2) (R^2 sec^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.
.