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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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## 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

## Schwarzschild metric derivation

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- Category: General Relativity
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After a quick introduction to the Schwarzschild metric solution, it is now time to derive it.

According to his letter from 22 december 1915, Schwarzschild started out from the approximate solution in Einstein’s “perihelion paper”, published November 25th.

We will go through a more formal derivation, which could be broken down into the following steps:

- simplify the metric for a static and spherically symmetric solution with some coefficients as functions f(r), i.e depending on r only.

- deduce the corresponding Christoffel symbols

- calculate the Ricci tensor components

- deduce the exact form of the above coefficient functions f(r) by setting the above Ricci tensor components as null as they should be in vacuum.

##### Step 1 - Expression of the metric tensor for a static and spherically symmetric solution

We recall that in space-time the distance interval has the following form

In spherical coordinates t, r, θ, φ (which makes sense in case of a spherical solution..), the spacetime interval can be expanded as below:

First of all, because the solution is static, it should not depend on time. In particular, a change from t -> -t should not change anything **(time reversible**), So that we should not have any croseed term in the form of dtdr, dtdθ or dtdφ, but only of the form **dt ^{2}**.

Then, taking in account the spherical symmetry, we can start out by the Minkowski metric written in spherical coordinates:

We are free to multiply the terms by any arbitrary r-dependent coefficients, leading to:

where α(r), β(r) and γ(r) represent some unkwnown functions of r.

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