Schouten–Codazzi gravity

We propose a new phenomenological second order gravity theory to be denoted as ‘Schouten–Codazzi’ Gravity’ (SCG), as it is based on Schouten and Codazzi tensors. The theory is related, but is clearly distinct from Cotton gravity. By assuming as source the energy momentum of General Relativity, we form a second order system with its geometric sector given by the sum of the Schouten tensor and a generic second order symmetric tensor complying with the following properties: (i) it must satisfy the Codazzi differential condition and (ii) it must be concomitant with the invariant characterization based on the algebraic structure of curvature tensors for specific spacetimes or classes of spacetimes. We derive and briefly discuss the properties of SCG solutions for static spherical symmetry (vacuum and perfect fluid), FLRW models and spherical dust fluids. While we do recognize that SCG is ‘work in progress’ in an incipient stage that still requires significant theoretical development, we believe that the theory provides valuable guidelines in the search for alternatives to General Relativity.