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Carlo Beenakker
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Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

Why was the minus sign dropped?
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.

Carlo Beenakker
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