The dimensional analysis of the GR geodesic equation

Question

The geodesic equation parametrized by the proper time contains two terms:

$$ {d^{2}x^{\mu } \over ds^{2}}=-\Gamma ^{\mu }{}_{{\alpha \beta }}{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\ $$ The dimensions of the different elements of the previous expresion are

$$ [x^{\mu }]=[s] $$ Both have dimnesion of length. The metric tensor being dimensionless implies the dimensionless of the Christoffel symbols $ \Gamma ^{\mu }{}_{{\alpha \beta }}$ and consequently the left and right sides of the equation of the geodesic have different dimensions. What is wrong?

Answer 1

The Christoffel symbols are obtained by differentiating with respect to $x^\alpha$, and since the metric is dimensionless if we write the dimensions we end up with:

$$ \left[{d^{2}x^{\mu } \over ds^{2}}\right] =\left[\frac{d}{dx^\alpha}\right]\left[{dx^{\alpha } \over ds}\right]\left[{dx^{\beta } \over ds}\right] $$

So both sides have dimensions of 1/length.