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 2h^{2} 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 c^{2} term in the final expression of light deflection.

##### Gravitational deflection of light

If we set **u _{0}(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

**u**

_{1}(Φ) (in the presence of gravitation).Replacing this expression in the second differential equation gives

In absence of gravitation, M = 0 so that

where u_{c} = 1/r_{c} is a constant.

which can be written as well as

or by using a famous trigonometric equality sin^{2}Φ = (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 10^{30} kg

R_{0} = 700 000 km

c = 3 x 10^{8} 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.