# Are the Schwarzschild metric and the Geodesic Equation relevant in the context of the Earth? [closed]

The geodesic equation used in general relativity is the following: $${\mathrm d^2 x^\mu \over \mathrm ds^2} =- \Gamma^\mu {}_{\alpha \beta}{\mathrm d x^\alpha \over\mathrm ds}{\mathrm d x^\beta \over\mathrm ds}.$$ It states that the acceleration of the test particle is a function of the metric (Chistoffel symbol) and the derivative of coordinates with respect to "a scalar parameter of motion s ex.: proper time".

Also, the Wikipedia page on the Schwarzschild metric states the following: "[...] [Schwarzschild metric] is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero." and the metric is the following: $${c^2 \mathrm d\tau^2 =}\left({1-{r_s \over r}}\right)c^2\mathrm dt^2-\left(1-{r_s \over r}\right)^{-1}\mathrm dr^2-r^2\mathrm d\Omega^2$$

Assuming all these conditions are true, does the Schwarzschild metric apply to the context of a particle in the vicinity of Earth's gravitational field? If so, can you give an example?

If, for some reason, the metric in question does not apply to the context of Earth, why not?

## closed as unclear what you're asking by DilithiumMatrix, AccidentalFourierTransform, ACuriousMind♦, Asher, CuriousOneMay 25 '16 at 21:55

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• This is a good question, but should probably be restructured somewhat as it is borderline off-topic because of the homework-like question policy. – DilithiumMatrix May 24 '16 at 2:35
• I'm not sure what the question is - you present all the assumptions that are needed for Schwarzschild, and then ask "assuming all these conditions are true, does the Schwarzschild metric apply" - and of course it applies, because you just assumed it! Are you asking whether the assumptions are a good approximation for the conditions around Earth? – ACuriousMind May 24 '16 at 15:55