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#### Welcome to Einstein Relatively Easy!

This web site is aimed at the general reader who is keen to discover Einstein's theories of special and general relativity, and who may also like to tackle the essential underlying mathematics.

Einstein's Relativity is too beautiful and too engaging to be restricted to the professionals!

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*"I have no special talents*

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Albert Einstein

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## Gravitational deflection of light

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- Category: General Relativity
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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**

## Geodesics in Schwarzschild spacetime

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- Category: General Relativity
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We saw in our previous articles Geodesic equation and Christoffel symbols and Geodesic that geodesic equations describe the paths of free falling particles in spacetime.

In order to understand how objects move in Schwarzschild spacetime, we therefore require the geodesic equations defined by the Schwarzschild metric.

We have shown that those equations are in the form of parameterised curves ^{[1]}

Replacing α by the four variables t, r, φ and θ gives us four complicated-looking differential Schwarzschild equations

For **α = t**, looking at the values of Γ^{t}_{μν} from our previous article, we see that only Γ^{t}_{tr} and Γ^{t}_{rt} are not null, leading to

Looking at α = r, we know that the four diagonal elements of the matrix Γ^{r}_{μν} are not null

so that we can write

If we replace now α by θ, and reminding us the Γ^{θ}_{μν} matrix, we can write

Finally, we retrieve the last geodesic equation by replacing α by Φ

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