  In this work, we discuss the possibility of inertial sensing with positronium
in the $2^3 S$ metastable state for the measurement of optical dipole,
relativistic and gravitational forces on a purely leptonic matter-antimatter
system. Starting from the characteristics of an available $2^3 S$ beam, we
estimate the time necessary to measure accelerations ranging from $\sim10^5$
$m/s^2$ to 9.1 $m/s^2$ with two different inertial sensitive devices: a
classical moir\'e deflectometer and a Mach-Zehnder interferometer. The
sensitivity of the Mach-Zehnder interferometer has been estimated to be several
tens of times better than that of the moir\'e deflectometer, for the same
measurement time.\\ Different strategies to strengthen the $2^3 S$ beam flux
and to improve the sensitivity of the devices are proposed and analyzed. Among
them, the most promising are reducing the divergence of the positronium beam
through 2D laser Doppler cooling and coherent positronium Raman excitation from
the ground state to the $2^3 S$ level. If implemented, these improvements
promise to result in the time required to measure an acceleration of 9.1
$m/s^2$ of few weeks and 100 $m/s^2$ of a few hours. Different detection
schemes for resolving the fringe pattern shift generated on $2^3 S$ positronium
crossing the deflectometer/interferometer are also discussed.
