Discoveries at the Frontiers of Science pp 143-181 | Cite as

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

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## 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. Actually, we present a Hamilton–Lagrange–Noether formulation of the gauge theory of gravitation. It is based on the minimum set of postulates to be introduced, namely (i) the action principle and (ii) the form-invariance of the action under the (local) diffeomorphism group. The second postulate thus implements the *Principle of General Relativity*, also referred to as the *Principle of General Covariance*. 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 \(\mathscr {L}_{R}\) for the source-free dynamics of gravitation and the energy-momentum tensor of the source system \(\mathscr {L}_{0}\). 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-1 particles, this entails an increased weighting of the kinetic energy over the mass as the source of gravity, 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. From the viewpoint of our generic Einstein-type equation, Einstein’s General Relativity constitutes the particular case for scalar and massless vector particle fields, and the Hilbert Lagrangian \(\mathscr {L}_{R,\mathrm {H}}\) as the model for the source-free dynamics of gravitation.

## Notes

### Acknowledgements

First of all, we want to remember our revered academic teacher Walter Greiner, whose charisma and passion for physics inspired us to stay engaged in physics for all of our lives.

The authors thank Patrick Liebrich and Julia Lienert (Goethe University Frankfurt am Main and FIAS), and Horst Stoecker (FIAS, GSI, and Goethe University Frankfurt am Main) for valuable discussions. D.V. and J.K. thank the Fueck Foundation for its support.

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