  Spacetime geometry is supposed to be measured by identifying the trajectories
of free test particles with geodesics. In practice, this cannot be done
because, being described by Quantum Mechanics, particles do not follow
trajectories. As a first step to study how it is possible to read spacetime
geometry with quantum particles, we model these particles with classical
extended objects. We propose to represent such extended objects by its
covariant center of mass, which generically does not follow a geodesic of the
background metric. We present a scheme that allows to extract some of
components of an "effective" connection, namely, the connection that would be
obtained if the locus of the center of mass is regarded as a geodesic. We
discuss some issues that arise when trying to obtain all the components of the
effective connection and its possible implications.
