  A new class of conserved currents, describing non-gravitational
energy-momentum density, is presented. The proposed currents do not require the
existence of a (timelike) Killing vector, and are not restricted to spherically
symmetric spacetimes (or similar ones, in which the Kodama vector can be
defined). They are based instead on almost-Killing vectors, which could in
principle be defined on generic spacetimes. We provide local arguments, based
on energy density profiles in highly simplified (stationary, rigidly-rotating)
star models, which confirm the physical interest of these 'almost-Killing
currents'. A mass function is defined in this way for the spherical case,
qualitatively different from the Hern\'andez-Misner mass function. An elliptic
equation determining the new mass function is derived for the Tolman-Bondi
spherically symmetric dust metrics, including a simple solution for the
Oppenheimer-Schneider collapse. The equations for the non-symmetric case are
shown to be of a mixed elliptic-hyperbolic nature.
