**6. Discussion**

We have applied the CDG decomposition to a Lorentz gauge theory and confirmed that it has a monopole condensate at one loop. Using the Clairaut formalism, we have found how

the monopole background modifies the canonical EOMs for the physical DOFs. Lorentz gauge theory has the problem of being non-positive semi-definite, which can be handled by adding a EH term. We did not add such a term but instead postponed the problem by Wick rotating the theory into Euclidean space, where the Lorentz gauge group becomes locally isomorphic to *SU*(2)*R* × *SU*(2)*<sup>L</sup>*. We found the spontaneous generation of a vacuum condensate which others have argued [10,16] leads to an effective Hilbert–Einstein term.

The CDG decomposition introduces an internal unit vector to indicate the local internal direction of the Abelian subgroup of the gauged symmetry group. However, the unit vector used to specify this subgroup does not form a canonical EOM and is degenerate. If we expand it in terms of its angular dependence, since its information content is purely directional, then those angles are also degenerate and we do not derive canonical EOMs for them. They do however add additional terms with important consequences for the theory's physics. They may not be ignored therefore, but require appropriate theoretical tools to analyse them. The authors addressed these issues in a previous analysis of QCD. The purpose of this paper was to do so for a theory relevant to gravity. The main advantages of working in a gauged Lorentz theory for us is that the gauge fields have quadratic kinetic terms well suited to our Clairaut-based approach in addition to the opportunity to apply analyses and even results from *SU*(2) Yang–Mills theories.

We have not considered the effects of matter fields in the fundamental representation. We do note in passing that differences in this part of the spectrum must lead to variations in the magnitude for the monopole condensate, so the differences in their matter spectra sugges<sup>t</sup> that this theory has significantly different infrared behaviour from that of *SU*(2) QCD.

We also observe that the net monopole condensate lies in a direction of a rotation generator. We have not been able to derive corresponding canonical DOFs to reflect this, so the physical significance of this observation, if any, remains obscure.

We have left the inclusion of translation symmetry to subsequent work. A full gravitational theory must of course include the full Poincaré symmetry group, but we submit that our Lorentz-only theory makes a sufficiently good approximation to indicate some relevant phenomenology.

**Author Contributions:** Conceptualization, M.L.W. and S.D.; methodology, S.D.; software, S.D.; validation, M.L.W. and S.D.; formal analysis, M.L.W.; resources, M.L.W.; writing—original draft preparation, M.L.W.; writing—review and editing, S.D.; visualization, S.D.; supervision, S.D.; project administration, S.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
