Coupled projection analysis for hybrid metric–Palatini gravity

Construct a metric-only Einstein-side reduction of hybrid metric–Palatini gravity by deriving and analyzing the coupled scalar–connection projection, its normal-operator matrix, kernel and cokernel, and the associated pushforward measure.

Background

The paper explains that hybrid metric–Palatini gravity contains both the metric Ricci scalar and an independent Palatini curvature, so its scalar–tensor representation includes a propagating scalar even though the pure Palatini scalar is algebraically constrained. Eliminating the connection and obtaining a metric-only Einstein-side description therefore requires treating scalar and connection variables together rather than applying the single-scalar normal-operator analysis developed for metric f(R) gravity.

The unresolved task is to construct this coupled reduction explicitly and determine how the scalar–connection normal-operator matrix, including its kernel and cokernel, controls locality, solvability, and possible obstructions. The analysis must also account for the pushforward functional measure, which is necessary for a consistent variational or quantum formulation.

References

A metric-only Einstein-side reduction would require the coupled scalar--connection projection, the corresponding normal-operator matrix, its kernel and cokernel, and the pushforward measure. That construction is model dependent and is left for future work.

— Differential Obstructions to Curvature-Dependent Conformal Transformations  (2608.13289 - Pereira et al., 13 Aug 2026) in Section “Beyond metric f(R): extensions and outlook,” subsection “Hybrid and higher-curvature constructions”