Generic Theory of Geometrodynamics from Noether’s Theorem for the $$\mathrm {Diff}(M)$$ Symmetry Group

Abstract

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.