In 1911 the The Equivalence Principle led Einstein to believe that light would be deflected in the presence of a gravitational field.
It wasn't until 1915, however, after he had successfully incorporated curved spacetime into a gravitational theory of relativity, that he was able to make an accurate prediction as to the magnitude of such a deflection.
Returning to the mathematics, we have seen in our previous article Geodesics in Schwarzschild spacetime that the Schwarzschild geodesic equations can be used to derive null geodesic equations that describe the path of a light ray in spacetime.
Ultimately, for this purpose, we would like to express r as a function of Φ
Let's define u = 1/r
Replacing this expression in our original geodesic equation leads to
Deriving now this equation with respect to Φ
Which by dividing by 2h2 can be simplfied to
Remark: by commodity, we have replaced the Rs expression by 2GM supposing c=1. We should not forget to add the c2 term in the final expression of light deflection.
Gravitational deflection of light
If we set u0(Φ) the solution of the equation in the absence of gravitation, we can approximate a global solution by adding a perturbative correction of the form of u1(Φ) (in the presence of gravitation).
Replacing this expression in the second differential equation gives
In absence of gravitation, M = 0 so that
where uc = 1/rc is a constant.
which can be written as well as
or by using a famous trigonometric equality sin2Φ = (1-cos2Φ)/2
Let's try a function u(1) of the following form: u(1) = A + B cos 2Φ
We can now write down u(Φ):
We should now seek for the value of Φ when r is infinite, meaning u tends towards 0:
The deflection is symetric with respect to Φ = π/2, and replacing the light speed by its real value c (in the Schwarzschild expression above we assumed c=1), gives
Example of the light deflection near the Sun
Let's calculate the deflection near the Sun, taking the following values:
G = 6.67 x 10-11 SI
M = 2 x 1030 kg
R0 = 700 000 km
c = 3 x 108 ms-1
gives ΔΦ = 8.5 x 10-6 rad = 4.85 x 10-4 degrees = 0.029 ' = 1.74''
That's the value that Eddington confirmed in his famous Eddington experiment during total solar eclipse of 29 May 1919.