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Given the geodesic equation $$\ddot{x^a} + \Gamma^a_{\;bc}\,\dot{x}^b\dot{x}^c = 0$$ with initial conditions, say $x^a(\lambda=0)=0,\dot{x}^a(\lambda=0)=v^a$, and all the data of Christoffel symbols evaluated at each spacetime points, how to formally integrate the equation and solve for $x^a(\lambda)$?

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  • $\begingroup$ You should be able to write out eight first order ODEs and solve them directly using a nonlinear solver like RK4. $\endgroup$
    – m4r35n357
    Aug 22, 2019 at 12:30
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    $\begingroup$ What do you mean by "formally integrate?" If you mean finding a closed form solution, then in general that's impossible. $\endgroup$
    – user4552
    Aug 22, 2019 at 12:49
  • $\begingroup$ @BenCrowell Yes, that's what I mean. So it is impossible then? $\endgroup$
    – Shadumu
    Aug 22, 2019 at 13:16
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    $\begingroup$ Your geodesic equation is missing an index (also in initial conditions), and the indices you have on $x$ should be upper, not lower; $\dot{x}_b$ is not the same as $\dot{x}^b$. $\endgroup$
    – G. Smith
    Aug 22, 2019 at 16:25
  • $\begingroup$ @G.Smith Thanks. I am very sloppy. $\endgroup$
    – Shadumu
    Aug 23, 2019 at 18:49

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There is a concept of a "closed form" solution, which basically means an equation that you can write down in terms of some set of basic functions, without using infinite series or limits. Usually the set of basic functions is taken to include exponentials, logs, and other related functions such as trig functions and roots.

It's a general fact of life that if you write down a differential equation in closed form, there is typically no closed-form solution. In some cases this can be proved.

A good example from general relativity is the geodesics of the Schwarzschild spacetime. Their equations can't be written in closed form in terms of elementary functions, but can be written in terms of elliptic functions.

Keep in mind also that it's not even very common to be able to write down the metric in closed form. Very few interesting solutions of the Einstein field equations are known that can be written in closed form.

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