Generalization of the f(R) Gravity Field Equations in Four-Index Form

While f(R) gravity, Gauss-Bonnet gravity, and other modified gravity theories have been extensively studied, a 4-index formulation of f(R) gravity remains unexplored. The primary objective of this work is to formulate and validate 4-index f(R) gravity field equations, establishing their consistency with existing physical results as a foundation for new physical solutions. A 2017 study on Generalization of Einstein's Gravitational Field Equations introduced a 4-index formulation of general relativity by generalizing the Einstein-Hilbert Lagrangian, introducing new scalars constructed from the Riemann and Ricci tensors, which ultimately reduce to the Ricci scalar. Building on this, we generalize the Lagrangian to an arbitrary function f(R), leading to the 4-index f(R) gravitational field equations, and employ this formulation to derive the effective energy-momentum tensor and new modified Weyl field equations. Imposing conservation of the total energy-momentum tensor on the 4-index field equations yields new compatibility equations. As a validation of the theory, maximally symmetric spacetime and power-law cosmology are investigated, where the curvature condition and the scale factor, respectively, successfully pass the consistency check against established results.