\({v}^{2}_{s} ={c}^{2}[1-(1-\cfrac{1}{{n}^{2}_{e}}){ e }^{ -\cfrac { x }{ r } }]\)
\({v}_{s} ={c}\sqrt{1-(1-\cfrac{1}{{n}^{2}_{e}}){ e }^{ -\cfrac { x }{ r } }}\)
Which is an expression for the speed of a photon along the radial line joining the photon and the center of the planet. Given a path of light resolved into two components, one along the radial line and the other perpendicular, the radial acceleration is,
\(\cfrac{d{v}_{s}}{dx} =\cfrac{1}{2}{c}\cfrac{1}{\sqrt{1-(1-\cfrac{1}{{n}^{2}_{e}}){ e }^{ -\cfrac { 1 }{ r } x }}}(1-\cfrac{1}{{n}^{2}_{e}}).\cfrac{1}{r}{ e }^{ -\cfrac { x }{ r } }\)
This is the expression for gravity experienced by a beam of light around a planet. Notice that it is positive in the outward direction. Photons actually gain speed as it leave the gravitational field of a planet. As it approaches a planet, photon slows down, a gravitational acceleration develops that points away from the planet. An approaching beam of light is actually deflected AWAY from a planet gravitational field.
In the case of a black hole, \({n}_{e}\rightarrow\infty\)
\({g}_{B} =\cfrac{1}{2}\cfrac{c}{r}\cfrac{1}{\sqrt{1-{ e }^{ -\cfrac { x }{ r } }}}{ e }^{ -\cfrac { x }{ r } }\)
And the velocity field is,
\({v}_{r}(x)=c\sqrt{1-e^{-\cfrac{x}{r}} }- {v}_{or}\)
where \({v}_{or}\) is the radial component of the initial approach velocity. This is a field equation in \(x\), the radial distance from the planet surface outwards. Each value of \(x\) gives a corresponding value of velocity with an outward positive value. A sample plot of this curve (3000*(1-e^(-x/100))^(0.5)-1500) shows, that given an initial approach value, the radial velocity component of the beam eventually reverses and so is refracted away from the black hole.
The situation is just like total internal refraction where light bends away on its approach to a denser and denser medium. The same will happen around a massive body like the Sun. The conclusion here is however, opposite to popular believe that light bend towards the massive body. The conclusion here is that light is refracted away from the massive body.
Note:
1. If the beam of light is approaching at an angle \(\theta\) to the perpendicular of the radial line, then
\({v}_{or}=c.sin{\theta}\)
\({v}_{op}=c.cos{\theta}\)
where \(c\) is light speed.
