Locally inertial coordinates with totally antisymmetric torsion

Understand

We show that the necessary and sufficient condition for erecting locally inertial coordinates at a point $p$ of a $U^4$-space, and therefore assuring the validity of the equivalence principle at that point, is the vanishing at $p$ of the symmetric part of the contortion tensor.

• This fact does not demand a vanishing torsion, but only a totally antisymmetric one.
• As an application, we derive the geodesic deviation equation; and prove the compatibility with the Newtonian limit.