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I've recently read about the fact that the Christoffel Symbols in General Relativity can be interpreted as fictitious forces which one experiences as an artifact of being in the state of motion they're currently in; however, I have no idea how I could interpret them that way.

I've searched far and wide, and couldn't find any definitive answers for this question.

What I want to know is exactly how the Christoffel Symbols can be interpreted as such, and how I could distinguish the fictitious forces from components which arise as an artifact of the coordinate system.

Any help would be much appreciated. :)

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1 Answer 1

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It's actually very simple. Newton's second law in General Relativity may be written as $$\ddot{x}^{\mu}+\Gamma_{\nu \lambda}^{\mu}\dot{x}^{\nu}\dot{x}^{\lambda}=K^{\mu},$$ where $K^{\mu}$ is the four-force and the dots represent a derivative WRT an affine parameter. The second term in the LHS involves the Christoffel symbols, and its origin is purely geometrical, there's no actual physical force involved and this term includes the Coriolis, Euler and centrifugal forces. This equation may be interpreted as an standard $F=ma$ equation whith the force $-\Gamma_{\nu \lambda}^{\mu}\dot{x}^{\nu}\dot{x}^{\lambda}$ in addition to the actual physical force $K^{\mu}$, hence its interpetation as fictitious forces. Hope this helps.

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    $\begingroup$ Just to add to the good answer by @Don AI, for low velocity objects wrt to some reference frame, ${\Gamma^i}_{00}= -g^i$ where ${\bf g}$ is the gravitational acceleration as seen in that frame. $\endgroup$
    – mike stone
    Aug 8, 2023 at 11:29

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