The dimensional analysis of the GR geodesic equation

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?

1 Answer

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.