  We work out the most general theory for the interaction of spacetime geometry
and matter fields -- commonly referred to as geometrodynamics -- for spin-$0$
and spin-$1$ particles. The minimum set of postulates to be introduced is that
(i) the action principle should apply and that(ii) the total action should by
form-invariant under the (local) diffeomorphism group. The second postulate
thus implements the Principle of General Relativity. According to Noether's
theorem, this physical symmetry gives rise to a conserved Noether current, from
which the complete set of theories compatible with both postulates can be
deduced. This finally results in a new generic Einstein-type equation, which
can be interpreted as an energy-momentum balance equation emerging from the
Lagrangian $L_{R}$ for the source-free dynamics of gravitation and the
energy-momentum tensor of the source system $L_{0}$. Provided that the system
has no other symmetries -- such as SU$(N)$ -- the canonical energy-momentum
tensor turns out to be the correct source term of gravitation. For the case of
massive spin particles, this entails an increased weighting of the kinetic
energy over the mass in their roles as the source of gravity as compared to the
metric energy momentum tensor, which constitutes the source of gravity in
Einstein's General Relativity. We furthermore confirm that a massive vector
field necessarily acts as a source for torsion of spacetime. Thus, from the
viewpoint of our generic Einstein-type equation, Einstein's General Relativity
constitutes the particular case for spin-$0$ and massless spin particle fields,
and the Hilbert Lagrangian $L_{R,H}$ as the model for the source-free dynamics
of gravitation.
