  Energy conservation has the status of a fundamental physical principle.
However, measurements in quantum mechanics do not comply with energy
conservation. Therefore, it is expected that a more fundamental theory of
gravity -- one that is less incompatible with quantum mechanics -- should admit
energy nonconservations. This paper begins by identifying the conditions for a
theory to have an energy-momentum tensor that is not conserved. Then, the
trajectory equation for pointlike particles that lose energy is derived,
showing that energy nonconservation produces a particular acceleration. As an
example, the unimodular theory of gravity is studied. Interestingly, in
spherical symmetry, given that there is a generalized Birkhoff theorem and that
the energy-momentum tensor divergence is a closed form, the trajectories of
test particles that lose energy can be found using well known methods. Finally,
limits on the energy nonconservation parameters are set using Solar system
observations.
